Analogue Circuits For Low Power Communication - Spiral

Analogue Circuits For Low Power Communication - Spiral
Imperial College London
Department of Electrical and Electronic Engineering
Analogue Circuits For Low
Power Communication
Mark David Tuckwell
2010
Supervised by Dr. Christos Papavassiliou
A thesis submitted for the degree of
Doctor of Philosophy in Electrical and Electronic Engineering of
Imperial College London
and the Diploma of Imperial College London
1
2
Declaration
I herewith certify that all material in this dissertation which is not my
own work has been properly acknowledged.
Mark David Tuckwell
3
4
Abstract
Low power electronic circuits are required to extend the operational time of battery operated devices. They are also necessary
to reduce the power consumption of equipment in general, especially as the world tries to cut energy usage. The first section
of this thesis explores fundamental and implementation limits for
low power circuits. The energy requirements of amplification are
presented and a lower bound on the energy required to transmit
information over a point to point link is proposed.
It is evident from the low power limits survey that when a transistor is biased, significant thermodynamic energy is required to
reduce the resistance of the channel. A transmitter is presented
that turns on a transistor for 0.1 % of transmitted time. This
transmitter approximates a Gaussian pulse by allowing the impulse response of two 2nd order transmitting elements to sum in
free space. The transmitter is of low complexity and the receiver
architecture ensures that no on-line tuning is required. Measured
results indicate that by using coherent detection a 1 Mbps, 50
mm distance link with a bit error rate of 10−3 can be achieved.
The bandwidth of the transmitted pulse is 30-37.5 MHz and 30
dB of out of band attenuation is provided.
An analogue Gabor transform is described which splits a signal
into parallel paths of a lower bandwidth. This enables post processing at lower clock rates, which can reduce energy dissipation.
An implementation of the transform using sub-threshold CMOS
continuous time filters is presented. A novel method for designing
low power gmC filters using simple models of identical transconductors is used to specify transistor sizes. Measured results show
that the transform consumes 7 µW for an input signal bandwidth
of 4 kHz.
5
6
Acknowledgements
I would like to thank Dr. Christos Papavassiliou for his excellent
supervision and guidance during my PhD. I would also like to thank
the members of the CAS group for their useful advice and help during
my time at Imperial. I would like to thank my fianceé Katie for her patience and support over the last few years. Finally I would like to thank
my parents and my sister Hannah for many years of encouragement and
support.
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Contents
1 Introduction
25
1.1
Communication . . . . . . . . . . . . . . . . . . . . . .
25
1.2
Research Contributions . . . . . . . . . . . . . . . . . .
25
1.3
Thesis Structure . . . . . . . . . . . . . . . . . . . . . .
26
2 Low Power Limits
29
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
29
2.2
Information Transfer . . . . . . . . . . . . . . . . . . .
29
2.3
Classical Limits . . . . . . . . . . . . . . . . . . . . . .
30
2.3.1
The Shannon-Hartley theorem . . . . . . . . . .
30
2.3.2
Compression of a gas . . . . . . . . . . . . . . .
31
2.3.3
Thermal Equilibrium . . . . . . . . . . . . . . .
32
2.3.4
Achieving the kT ln 2 limit . . . . . . . . . . . .
33
2.3.5
Beating the kT ln2 limit . . . . . . . . . . . . .
34
2.3.6
Approaching the kT ln2 limit . . . . . . . . . .
35
Quantum limits . . . . . . . . . . . . . . . . . . . . . .
36
2.4.1
Uncertainty Principle . . . . . . . . . . . . . . .
37
2.4.2
Energy Per Operation . . . . . . . . . . . . . .
39
2.4.3
Blackbody Radiation . . . . . . . . . . . . . . .
39
2.4.4
Blackbody Communication Limit . . . . . . . .
40
2.4.5
Signal Representation Using Photons . . . . . .
41
Implementation Limits . . . . . . . . . . . . . . . . . .
43
2.5.1
Stein Limit . . . . . . . . . . . . . . . . . . . .
43
2.5.2
Driving An RC Interconnect . . . . . . . . . . .
44
2.5.3
Power Per Pole . . . . . . . . . . . . . . . . . .
46
2.4
2.5
9
2.6
2.7
Matched Transmission Line . . . . . . . . . . . . . . .
2.6.1
Energy Per Bit For Matched Information Transfer 47
2.6.2
Comparison of Analogue and Digital Signal Representation . . . . . . . . . . . . . . . . . . . . .
48
Exploration of Information Transfer Through an Amplifier 51
2.7.1
Information and Physical Energy . . . . . . . .
52
2.7.2
Information Flow Through a Practical LNA . .
55
2.7.3
Theoretical Vs Practical Output Bit Energy . .
58
2.7.4
Energy Required for Biasing . . . . . . . . . . .
58
2.7.5
Black Box Analogue Amplifier . . . . . . . . . .
62
2.7.6
Conclusion . . . . . . . . . . . . . . . . . . . . .
64
Electromagnetic Limits . . . . . . . . . . . . . . . . . .
65
2.8.1
Friis Limit . . . . . . . . . . . . . . . . . . . . .
65
2.8.2
Friis-Kraus Electromagnetic Limit . . . . . . . .
65
2.8.3
Near Field . . . . . . . . . . . . . . . . . . . . .
66
A New Electromagnetic Lower Bound . . . . . . . . . .
66
2.9.1
A numerical example . . . . . . . . . . . . . . .
67
2.9.2
Comparison With Current Standards . . . . . .
68
2.9.3
Limitations Of This Bound
. . . . . . . . . . .
69
2.9.4
Conclusion . . . . . . . . . . . . . . . . . . . . .
69
2.10 Biological Examples . . . . . . . . . . . . . . . . . . . .
70
2.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
71
2.8
2.9
3 Pulsed Communication
77
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
77
3.2
Comparison of FSK and PPM . . . . . . . . . . . . . .
78
3.2.1
Minimum PPM Power . . . . . . . . . . . . . .
81
3.2.2
Transmitter Complexity . . . . . . . . . . . . .
82
3.2.3
Channel Block Coding . . . . . . . . . . . . . .
86
PPM Simulation . . . . . . . . . . . . . . . . . . . . .
90
3.3.1
PPM Spectrum Shape . . . . . . . . . . . . . .
90
3.3.2
Orthogonal Signalling . . . . . . . . . . . . . . .
91
3.3.3
Pseudo Orthogonal BER . . . . . . . . . . . . .
92
3.3
10
47
3.3.4
PPM Detection . . . . . . . . . . . . . . . . . .
95
3.3.5
Correlation Detection . . . . . . . . . . . . . . .
96
3.3.6
Matched Filter Detection . . . . . . . . . . . . .
97
3.4
Truncated Gaussian . . . . . . . . . . . . . . . . . . . .
99
3.5
Gaussian Approximations . . . . . . . . . . . . . . . . 103
3.5.1
All Pole Approximation . . . . . . . . . . . . . 104
3.5.2
Cascade of Poles . . . . . . . . . . . . . . . . . 107
3.5.3
Padé Approximation . . . . . . . . . . . . . . . 110
3.5.4
Creating bandpass filters . . . . . . . . . . . . . 114
3.6
Gaussian Approximation BER . . . . . . . . . . . . . . 115
3.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 118
4 Communication using 2nd Order TX Elements
121
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2
Making Use of Antenna Topology . . . . . . . . . . . . 122
4.2.1
4.3
4.4
4.5
4.6
4.7
Efficiency of the 2nd Order Element . . . . . . . 122
Pulse Decomposition . . . . . . . . . . . . . . . . . . . 125
4.3.1
Minimal Decomposition . . . . . . . . . . . . . 126
4.3.2
Maximal Decomposition . . . . . . . . . . . . . 126
4.3.3
TX Element Coupling . . . . . . . . . . . . . . 130
2nd Order Receiving Element . . . . . . . . . . . . . . 130
4.4.1
Implicit Matched Filter . . . . . . . . . . . . . . 131
4.4.2
Coupling Matrix for N=2 . . . . . . . . . . . . 132
4.4.3
Transmission Distance . . . . . . . . . . . . . . 133
4.4.4
Attenuation Map . . . . . . . . . . . . . . . . . 134
Demodulation . . . . . . . . . . . . . . . . . . . . . . . 136
4.5.1
Coherent Detection . . . . . . . . . . . . . . . . 139
4.5.2
Sub Sampling . . . . . . . . . . . . . . . . . . . 139
4.5.3
Non Coherent Detection . . . . . . . . . . . . . 139
BER Performance . . . . . . . . . . . . . . . . . . . . . 139
4.6.1
Accuracy of Estimation . . . . . . . . . . . . . . 140
4.6.2
Preamble Length . . . . . . . . . . . . . . . . . 140
Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11
4.8
4.9
Circuit Design . . . . . . . . . . . . . . . . . . . . . . . 145
4.8.1
I-Q Delay Circuit . . . . . . . . . . . . . . . . . 146
4.8.2
Pulse Generation Circuit . . . . . . . . . . . . . 146
Measured Results . . . . . . . . . . . . . . . . . . . . . 147
4.9.1
Transmitted Pulse . . . . . . . . . . . . . . . . 148
4.9.2
Orthogonal Pulse . . . . . . . . . . . . . . . . . 154
4.9.3
Receiver Demodulation . . . . . . . . . . . . . . 154
4.9.4
Power Consumption . . . . . . . . . . . . . . . 157
4.9.5
Measured BER Performance . . . . . . . . . . . 157
4.10 Comparison with State of the Art . . . . . . . . . . . . 164
4.10.1 Integration
. . . . . . . . . . . . . . . . . . . . 166
4.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 167
5 An Analogue Gabor Transform
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2
The Gabor Transform . . . . . . . . . . . . . . . . . . 170
5.3
Implementation of Convolution . . . . . . . . . . . . . 172
5.3.1
Number of Filters and Coefficient Rate . . . . . 173
5.3.2
Noise Analysis . . . . . . . . . . . . . . . . . . . 175
5.4
Design of the State Space Filter from Impulse Response
Specifications . . . . . . . . . . . . . . . . . . . . . . . 177
5.5
Designing a Low Power gmC Filter . . . . . . . . . . . 182
5.6
12
169
5.5.1
Noise and Distortion . . . . . . . . . . . . . . . 182
5.5.2
Mismatch . . . . . . . . . . . . . . . . . . . . . 185
5.5.3
Bandwidth and Output Resistance . . . . . . . 187
5.5.4
A Low Power gmC Design Method . . . . . . . 189
Measured Results . . . . . . . . . . . . . . . . . . . . . 192
5.6.1
Impulse and Bode Response . . . . . . . . . . . 193
5.6.2
Centre Frequency Variation . . . . . . . . . . . 196
5.6.3
Power Consumption and SINAD . . . . . . . . . 196
5.6.4
Bit Error Test Performance . . . . . . . . . . . 198
5.6.5
Analysis Window Cross Correlation . . . . . . . 199
5.6.6
Transform Comparison . . . . . . . . . . . . . . 203
5.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 203
6 Conclusion
6.1
205
Future Work . . . . . . . . . . . . . . . . . . . . . . . . 209
7 Published Work
211
Bibliography
213
A Derivation of switching energy for a square wave
227
B A lower bound on the energy for a point to point communication link
231
B.1 System Temperature . . . . . . . . . . . . . . . . . . . 231
B.2 Antenna Noise Temperature . . . . . . . . . . . . . . . 233
B.3 Lower Bound on Transmission Energy . . . . . . . . . . 235
B.4 Energy per bit . . . . . . . . . . . . . . . . . . . . . . . 235
B.5 Isotropic and practical antennas . . . . . . . . . . . . . 236
B.5.1 Parabolic dish antenna . . . . . . . . . . . . . . 237
B.5.2 Dipole antenna . . . . . . . . . . . . . . . . . . 237
C The relationship between SNR and Eb /N0
239
C.1 Square wave SNR . . . . . . . . . . . . . . . . . . . . . 240
D Time-Frequency Uncertainty
241
D.1 Uncertainty Calculation . . . . . . . . . . . . . . . . . 241
D.2 Rectangular Pulse . . . . . . . . . . . . . . . . . . . . . 242
D.3 Sinusoidal Pulse . . . . . . . . . . . . . . . . . . . . . . 242
D.4 Gaussian Pulse . . . . . . . . . . . . . . . . . . . . . . 243
D.5 Gaussian Derivative . . . . . . . . . . . . . . . . . . . . 244
D.6 Truncated Gaussian Pulse . . . . . . . . . . . . . . . . 245
E BER Simulation
249
E.1 Correlation Detection . . . . . . . . . . . . . . . . . . . 249
E.2 Matched Filter Detection . . . . . . . . . . . . . . . . . 250
13
E.3 BER Results . . . . . . . . . . . . . . . . . . . . . . . . 251
F Impulse Approximation
255
G Inductive Coil Characterisation
259
G.1 Lumped Element Model . . . . . . . . . . . . . . . . . 259
G.1.1 Measured Coil Characteristics . . . . . . . . . . 260
G.2 Coupling Measurements . . . . . . . . . . . . . . . . . 262
H 2nd Order Approximation
265
I
267
Transmitter Coupling
J 2nd Order Element Temperature Change
271
K 2nd Order Element Tuning
273
L Pulse Generation and Receiver Schematics
275
14
List of Tables
2.1
2.2
2.3
2.4
2.5
Bit energies for several common communication standards.
Fundamental Classical Limits. . . . . . . . . . . . . . .
Fundamental Quantum Limits. . . . . . . . . . . . . .
Implementation Limits. . . . . . . . . . . . . . . . . . .
Biological vs Electronic Energy Examples. . . . . . . .
3.1
Examples of some pulse based transmitters for short
range communications. . . . . . . . . . . . . . . . . . . 84
Comparison of time-frequency uncertainty for a variety
of pulse shapes. . . . . . . . . . . . . . . . . . . . . . . 99
Comparison of bandwidth and attenuation for various
pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Approximated pulses with an attenuation of > 35 dB. . 115
3.2
3.3
3.4
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
5.3
69
73
73
74
75
Characteristics for the 3rd order baseband minimal pulse.126
Characteristics for the 3rd order baseband maximal pulse.129
Characteristics for the N=2 pulse. . . . . . . . . . . . . 130
Transmission distance using matched TX and RX 2nd
order elements. . . . . . . . . . . . . . . . . . . . . . . 134
Preamble frequency, phase and offset estimation algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Comparison of some transmitters suitable for short range
communication. . . . . . . . . . . . . . . . . . . . . . . 165
Tradeoffs in filter design. . . . . . . . . . . . . . . . . . 189
Comparison of model and BSIM3 simulation. . . . . . . 190
Variation in centre frequency for fixed bias currents. . . 196
15
5.4
5.5
5.6
16
Summary of measured results. . . . . . . . . . . . . . . 198
Cross Correlation of Cos Analysis Windows . . . . . . 200
Transform Comparison. . . . . . . . . . . . . . . . . . . 202
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
3.1
3.2
3.3
Compression of an ideal gas using a piston. . . . . . . .
Blackbody Radiation Spectrum also showing Wien’s displacement law. . . . . . . . . . . . . . . . . . . . . . .
Minimum energy required per bit when representing a
signal in terms of a Plank oscillator. . . . . . . . . . . .
The Ideal Integrator requires a power of 8kT SNRf . . .
A typical interconnect with matched source and load
impedances. . . . . . . . . . . . . . . . . . . . . . . . .
Information transfer using parallel digital, serial digital
and analogue representations. . . . . . . . . . . . . . .
Energy per bit required to represent a signal in analogue
and digital formats. . . . . . . . . . . . . . . . . . . . .
The mapping of free information to bound information.
Power and noise spectra of an idealised amplifier. . . .
The MOSFET acting as an information isolator. . . . .
Comparison of output bit energy for many LNAs across
a range of frequencies. . . . . . . . . . . . . . . . . . .
Low pass filter ideal spectrum. . . . . . . . . . . . . . .
Energy per bit lower bound for free space communications for a distance of 10 m. . . . . . . . . . . . . . . .
An example of FSK and PPM modulation for L=4. . .
Block Diagram of a transmitter capable of FSK or PPM
modulation. . . . . . . . . . . . . . . . . . . . . . . . .
Examples of some typical circuit topologies for wireless
transmitters. . . . . . . . . . . . . . . . . . . . . . . . .
32
40
43
46
47
49
51
53
56
57
59
63
68
79
80
82
17
3.4
Power consumption of PLL frequency generation circuits. 86
3.5
Comparison between 2PSK (antipodal) and M-ary FSK
(orthogonal) modulation schemes. . . . . . . . . . . . .
87
Comparison of bit error rates for convolutional block
coding. . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.7
Simulated power dissipation of a convolutional encoder.
89
3.8
Simulation of the power spectrum for the PPM information sequence. . . . . . . . . . . . . . . . . . . . . .
91
The effect of cross correlation on the energy requirements of ML detection. . . . . . . . . . . . . . . . . . .
94
3.10 Possible detection schemes when the impulse of the transmit filter is sent across a channel. . . . . . . . . . . . .
96
3.11 The effect of using the same approximated filter for the
transmit filter and matched filter. . . . . . . . . . . . .
98
3.6
3.9
3.12 Truncated Gaussian time domain pulse for T = 1. . . . 101
3.13 Truncated Gaussian pulse frequency domain response
for T = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.14 Attenuation achieved by the truncated Gaussian pulse.
102
3.15 Bandwidth-time product for the truncated Gaussian pulse.103
3.16 Out of band attenuation versus αT 2 . . . . . . . . . . . 105
3.17 All pole approximation of the Gaussian pulse function
for αT 2 = 6 and T = 1. . . . . . . . . . . . . . . . . . . 106
3.18 Frequency response of the all pole Gaussian pulse approximation for αT 2 = 6 and T = 1. . . . . . . . . . . . 106
3.19 Out of band attenuation vs αT 2 for the cascade of poles
approximation. . . . . . . . . . . . . . . . . . . . . . . 108
3.20 Cascade of poles approximation of the Gaussian pulse
function for N=2 and N=8. . . . . . . . . . . . . . . . 109
3.21 Frequency response of the cascade of poles approximation for N=2 and N=8. . . . . . . . . . . . . . . . . . . 109
3.22 Out of band attenuation versus αT 2 for the Padé Approximation. . . . . . . . . . . . . . . . . . . . . . . . . 111
18
3.23 Time domain response of the Padé approximation for
N=4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.24 Time domain response of the Padé approximation for
N=6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.25 Frequency response of the Padé approximation for N=4. 113
3.26 Frequency response of the Padé approximation for N=6. 113
3.27 Extra transmitter energy required over ideal orthogonal
pulses when using a correlation detector. . . . . . . . . 117
3.28 Extra transmitter energy required over ideal orthogonal pulses when using the approximated matched filter
detector. . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1
Circuit diagram showing a 2nd Order transmitting element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2
Using the inductor as a transmitter in a direct modulation approach and a pulse based approach. . . . . . . . 124
4.3
Sum of 2nd order impulse responses to form the desired
pulse using a minimal decomposition. . . . . . . . . . . 127
4.4
Spectral comparison of the minimally decomposed pulses.128
4.5
Sum of 2nd order impulse responses to form the desired
pulse using a maximal decomposition. . . . . . . . . . . 129
4.6
Diagram indicating the coupling between each transmitting element and each receiving element. . . . . . . . . 131
4.7
Coupling paths when using two transmit and receive
antennas. . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.8
Geometry of the transmit antennas. . . . . . . . . . . . 135
4.9
Out of band attenuation with the transmit antennas
placed 20mm apart. . . . . . . . . . . . . . . . . . . . . 137
4.10 Block diagram of the transmitter and receiver for the
pulse based communications link. . . . . . . . . . . . . 138
4.11 Effect of transmit distance on BER performance for an
N=2 pulse with 4 time slots. . . . . . . . . . . . . . . . 141
19
4.12 Effect of centre frequency shift on BER performance for
an N=2 pulse. . . . . . . . . . . . . . . . . . . . . . . . 142
4.13 Mean and variance of frequency, phase and symbol offset
estimation. . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.14 Top level schematic of the transmitter. . . . . . . . . . 146
4.15 Circuit diagram of the digital pulse delay circuit. . . . 147
4.16 Circuit diagram of the digital pulse generator. . . . . . 148
4.17 Photograph of the transmitter module. . . . . . . . . . 149
4.18 Photograph of the receiver module. . . . . . . . . . . . 149
4.19 Block diagram of measurement setup used to evaluate
BER performance. . . . . . . . . . . . . . . . . . . . . 150
4.20 Measured spectrum of the transmitter magnetic field on
the centreline of the inductors at a distance of 5 mm. . 152
4.21 Measured time domain pulses of the transmitter magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.22 Measured time domain orthogonal pulses at 33 MHz. . 155
4.23 Measured time domain orthogonal pulses after sub sampling operation. . . . . . . . . . . . . . . . . . . . . . . 155
4.24 A train of pulses measured at the receiver including the
preamble sequence. . . . . . . . . . . . . . . . . . . . . 156
4.25 Modelled and measured power consumption of the transmitter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.26 BER performance over distance for orthogonal coherent
detection. . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.27 BER performance versus Eb /N0 for orthogonal coherent
detection. . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.28 BER performance over distance for non coherent detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.29 BER performance versus Eb /N0 for non coherent detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.1
20
Information diagram illustrating the discrete time and
frequency energy. . . . . . . . . . . . . . . . . . . . . . 170
5.2
Truncation of the Gaussian pulse. . . . . . . . . . . . . 172
5.3
Generation of a single coefficient using the direct filter
approach. . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.4
Generation of a single coefficient using the time domain
approach. . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.5
Coefficient time line showing the generation of the coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.6
Block diagram of the gm-C filter. . . . . . . . . . . . . 180
5.7
Circuit diagram of the G matrix implementation using
identical transconductors and capacitors. . . . . . . . . 180
5.8
Circuit diagram of the C matrix implementation. . . . 181
5.9
Generation of the bias currents for a single complex filter
using current mirrors. . . . . . . . . . . . . . . . . . . . 181
5.10 Diagram of a simple transconductor. . . . . . . . . . . 182
5.11 Measured impulse response of each filter. . . . . . . . . 191
5.12 Die photograph showing a single complex filter. . . . . 192
5.13 Measurement Setup. . . . . . . . . . . . . . . . . . . . 193
5.14 Plot showing 200 overlaid cos 2500 Hz analysis windows
for a single chip . . . . . . . . . . . . . . . . . . . . . . 194
5.15 Measured Bode plot of the cos filters for each chip. . . 195
5.16 Measured Bode plot of the sin filters for each chip.
. . 195
5.17 Variation in the impulse response of the cos 2500 Hz
filter for each chip with a fixed bias current. . . . . . . 197
5.18 Variation in the frequency response of the cos 2500 Hz
filter for each chip with a fixed bias current. . . . . . . 197
5.19 Bit error comparison. . . . . . . . . . . . . . . . . . . . 199
5.20 Cross correlation error and bit error performance comparison between each chip. . . . . . . . . . . . . . . . . 201
A.1 A single pole RC filter. . . . . . . . . . . . . . . . . . . 227
D.1 Approximation of the time-frequency uncertainty for the
Gaussian, Gaussian derivative, sinusoidal and rectangular pulses. . . . . . . . . . . . . . . . . . . . . . . . . . 247
21
E.1 BER for L = 2 for correlation and approximated matched
receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . 252
E.2 BER for L = 4 for correlation and approximated matched
receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . 252
E.3 BER for L = 8 for correlation and approximated matched
receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . 253
E.4 BER for L = 16 for correlation and approximated matched
receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . 253
F.1 Approximation to the impulse function. . . . . . . . . . 256
F.2 Frequency Response of the approximated impulse response. . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
G.1 Measurement setup for characterisation of the inductive
transmitter element. . . . . . . . . . . . . . . . . . . . 260
G.2 Measured and modeled characteristics of a 10 mm long
coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
G.3 Measured and modeled characteristics of a 10 mm long
coil with a 100 pF ± 10% capacitor in parallel with the
coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
G.4 Ideal circuit for measuring the coupling constant. . . . 263
I.1
Circuit showing the coupling between two transmitter
elements. . . . . . . . . . . . . . . . . . . . . . . . . . . 268
I.2
Pole plot showing how the poles of two transmitting
elements varies as the coupling constant k is increased.
I.3
269
Frequency response of the Gaussian approximation with
various coupling constants. . . . . . . . . . . . . . . . . 270
L.1 Top level block diagram of the transmitter and receiver
circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
L.2 TX schematic . . . . . . . . . . . . . . . . . . . . . . . 277
L.3 TX2 schematic . . . . . . . . . . . . . . . . . . . . . . 278
L.4 TX3 schematic . . . . . . . . . . . . . . . . . . . . . . 279
22
L.5
L.6
L.7
L.8
L.9
TXpower schematic
RXamp schematic .
RXamp2 schematic
RX DAC schematic
RXpower schematic
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281
282
283
284
23
24
1 Introduction
1.1 Communication
Communication is a word that describes the transferal of information
from one point to another. The Oxford dictionary defines communication as [1]:
1 the action of communicating. 2 a letter or message. 3
(communications) means of sending or receiving information, such as telephone lines or computers. 4 (communications) means of travelling or of transporting goods, such as
roads or railways.
The subject of communication in engineering often deals with the
information theoretic concepts, for example Shannon’s paper [2]. A
transmitter and receiver are a special case where electronics is used to
transfer macroscopic information such as speech or video. In general all
circuits carry out communication between circuit elements. At a microscopic level electrons pass information around a circuit. Therefore,
communication is quite a general term that can be used to describe
both macroscopic and microscopic information transfer. In this thesis
an exploration of the fundamental communication limits is presented
and two low power circuits motivated by these limits are described.
1.2 Research Contributions
The following list describes the main research contributions of this
thesis:
25
• A comprehensive review of physical limitations of energy required
by electronic circuits has been undertaken
• A hypothesis for why energy is required by an amplifier is proposed
• A minimum energy bound for transmission between two antennas
is derived
• A low power analogue pulse based transmitter using 2nd order
transmitting elements has been described and implemented
• A low power analogue Gabor transform using 180 nm CMOS
technology has been fabricated
1.3 Thesis Structure
Chapter 2 contains a review and discussion of fundamental and implementation limits related to low power electronics circuits. A discussion
of the energy requirements of amplification is presented. A lower bound
on the transmit energy required to transmit information over a point
to point link is proposed.
In Chapter 3 the advantages of using a pulse based scheme such as
Pulse Position Modulation (PPM) over Frequency Shift Keying (FSK)
are discussed. Several continuous time approximations for generating
Gaussian pulse shapes are shown. The bit error rate performance of
using continuous time Gaussian approximations with PPM are presented.
A novel architecture for the transmission of information using tuned
2nd order transmitter elements is shown in Chapter 4. Here PPM is
used together with two approximately orthogonal pulses to reduce the
power consumption and complexity of the transmitter circuit. A possible application for this circuit is for low power medical transmitters.
Chapter 5 describes the design and implementation of an analogue
Gabor transform. The transform allows a signal to be split into several
26
parallel paths, each of a lower bandwidth than the original signal.
A possible application of this is in sensor networks, where an input
signal can be transformed and then sampled using several low rate
A/D converters instead of a single high rate converter. The transform
is implemented in 180 nm CMOS technology.
Finally conclusions and suggestions for future work are presented in
Chapter 6.
27
28
2 Low Power Limits
2.1 Introduction
The quest for low power electronics is of particular importance for
portable battery powered circuits where longer operational times are
required. Low power circuits are also required to reduce the power
consumption of fixed equipment, especially as the world tries to cut
energy usage. In this chapter a survey of physical and implementation
limits related to minimal energy electronics is provided. The aim of this
survey is to provide the reader with some understanding of the need
for energy when carrying out a task such as computation or transferal
of information. In addition to the survey a discussion of the energy
requirements of amplification is presented (Section 2.7) and a lower
bound on the energy required to transmit information over a point to
point link is proposed (Section 2.9).
2.2 Information Transfer
Information transfer can be defined as the movement of knowledge from
one place to another. There are many different ways in which to move
information. For example, information could be printed out on paper,
packed onto the back of a lorry, dropped off at a shop for someone to
buy, walk home with and read at their leisure. Alternatively, the same
information can be sent through the ‘air’ as electromagnetic waves
which arrive at a handheld terminal for the user to read. A widely used
technique of information transfer is via the internet, this may consist
29
of wired and wireless links which span many countries. Transferal of
information between people using speech happens on a day to day
basis.
Each form of information transmission and reception has its own
energy cost. For example, the energy required to transfer the newspaper contents in print form would include the energy to print the
pages, the energy to transport the newspaper and the energy used by
the customer when picking the newspaper up from a shop. This whole
process typically takes a few hours. The same information could be
broadcast using electromagnetic waves (i.e. television) and the energy
cost for this form of broadcast would include that for the transmitter
and the receiver.
Research into low power communication can be divided into two distinct subjects; fundamental physical limits and implementation specific
limits. The fundamental physical limits are those which describe limitations in terms of fundamental constants that do not change throughout the universe [3] 1 . The implementation limits are those where some
form of circuit structure, device or modulation scheme are assumed.
2.3 Classical Limits
2.3.1 The Shannon-Hartley theorem
The Shannon-Hartley theorem [2] describes the maximum rate at which
information can be transferred through a channel with additive white
Gaussian noise (AWGN):
C ≤ B log2
P
1+
N
(2.1)
where C is the channel capacity in bits/s, B is the channel bandwidth in
Hz, P is the signal power and N is the thermal noise power. By setting
1
The view that fundamental constants do not change throughout time is simplistic. According to many it is an open question whether fundamental constants
change throughout the universe or with time.
30
P
, the minimum
N = kT B and rearranging (2.1) to find the ratio C
energy per bit of information transfer can be found [4]–[8]:
−1
C/B
P
C
2
−1 .
= kT
C
B
(2.2)
The minimum energy is found in the limit of C/B → 0:
P
= Ebit ≥ kT ln2.
C
(2.3)
In order to derive the minimum energy it is necessary to set the
thermal noise power to N = kT B. This is the minimum noise floor
that is achieved when the link is impedance matched and the noise
factor is unity.
The limit shown in (2.3) is only valid when the bandwidth is infinite
or the rate of transmission is infinitely small. This is equivalent to
assuming that the information transmission is occurring sufficiently
slowly so as to maintain thermodynamic equilibrium [9].
In [8] the result of (2.3) is derived by considering an ideal CMOS
transistor operating as part of an inverter. This result is derived by
essentially ignoring the capacitance of the MOS transistor, which is in
effect implying an infinite bandwidth.
To add further evidence that this may be the lower bound on energy
per bit, Levitin [10] proved that (2.3) is the minimum energy per bit
by using quantum theory.
2.3.2 Compression of a gas
Using the laws of thermodynamics it is possible to prove that the
minimum energy required to compress a piston isothermally is kT ln2
[11]. Consider a piston immersed in a heat bath of temperature T which
remains constant throughout the process, figure 2.1. By undertaking
this compression the number of places where the gas molecules can
reside has been reduced by half, i.e. the loss of one bit of information.
The work done on the gas by compressing it from an initial volume,
31
Figure 2.1: Compression of an ideal gas using a piston. To reduce the
volume of the gas by half requires N kT ln2 J of energy.
V1 to a final volume
V1
2
can be written as:
W =
Z
V1
2
V1
N
kT
dV = −N kT ln2
V
(2.4)
where N is the number of gas molecules. In the limit of a single
molecule of gas kT ln2 J of energy is required for a loss of a single
bit of energy.
This is an interesting result because the same basic result (2.3) has
been found using a different set of physical assumptions, indicating
that the limit of kT ln2 joules per bit is fundamental.
2.3.3 Thermal Equilibrium
It is important to remember that circuits designed to date are not
operating in thermal equilibrium. For example, the temperature of a
resistor that is short circuited will tend towards the temperature of
the surroundings. However, as soon as a voltage is applied across the
resistor, or a current passed through it, then external energy is being
32
supplied to the resistor, which causes it to reach a temperature higher
than the surroundings. Therefore, the circuit is no longer in thermal
equilibrium as there is a flow of heat between the resistor and the
surroundings. The same is true for modern processors. A processor
containing millions of gates switching at a speed in the GHz region
becomes very hot. To stop the device from overheating a cooling fan is
added to remove heat from the device. This simple example shows that
practical electronic circuits are not operating in thermal equilibrium,
thus it should be expected that practical circuits will require more
power than the fundamental limit of kT ln2 joules per bit.
2.3.4 Achieving the kT ln 2 limit
Proakis [12] shows that the kT ln 2 limit can be met when representing
information using an infinite number of orthogonal signals. In this
region, C/B → 0, the system is power limited. This is in contrast to
bandwidth limited systems, such as amplitude and phase modulation
schemes which have C/B > 1. The upper limit on the probability of
error for using a set of M orthogonal signals is given by [12]:
Pe ≤

log M

exp− 22 −2 ln 2

2 exp− log2 M
„r
Eb
N0
Eb
−
N0
√
ln 2
«
> 4 ln 2,
ln 2 <
(2.5)
Eb
N0
≤ 4 ln 2.
Chapter 4 makes use of orthogonal signalling to reduce power consumption of a transmitter albeit with the number of orthogonal signals
much less than infinity.
Eq. (2.3) is of somewhat limited practical value due to the requirement of infinite bandwidth or infinitesimally small rate of transmission.
However, there are attempts in the literature to try and get closer to
the limit using adiabatic logic. There is also a school of thought which
suggests that a reversible computer which requires zero energy could be
constructed . These two cases are discussed in the following sections.
33
2.3.5 Beating the kT ln2 limit
In 1961 Landauer [13] (reprinted 2000 [14]) introduced the idea that
it may be possible to carry out computation without dissipation. The
reason why it may be possible to beat the limit lies in the fact that
physical reversibility exists under thermal equilibrium [9]. Under this
condition energy can theoretically be conserved when switching between states. Physical reversibility is a consequence of the Second Law
of Thermodynamics [9]. A reversible process is one where a change in a
system can be reversed in order to return the system and the surrounding environment to its original condition. Consider the example of the
piston (Section 2.3.2) where kT ln2 J of energy is required to compress
the gas to half its original volume under isothermal assumptions. If the
piston is now retracted so that the gas is allowed to expand to twice
its volume then the net energy loss between compression and expansion is zero. It should be noted that this example is a physical ideal
and neglects any friction due to the side walls. In fact reversibility
is a theoretical concept in any physical system as an infinite amount
of time would be required to change state isothermally. Any process
which involves friction, heat transfer across a temperature difference
or electrical resistance is irreversible.
In daily life, the concepts of Mr. Right and Ms. Right
are also idealisations, just like the concept of a reversible
(perfect) process. People who insist on finding Mr. or Ms.
Right to settle down with are bound to remain Mr. or Ms.
Single for the resist of their lives. The possibility of finding
a perfect prospective mate is no higher than the possibility
of finding a perfect (reversible) process. Likewise, a person
who insists on perfection in friends is bound to have no
friends. [15]
This quote gives some philosophical indication that finding a reversible
process in practice is highly unlikely. Even if physical reversibility
34
could be achieved then the very act of measurement requires energy [4],
[13], [16]–[18]. In essence the measurement apparatus is set to a standard state ready for the next measurement, i.e. information is erased
between measurements thus dissipating energy. Reversible computing
aims to prevent this information erasure thus reducing energy dissipation. This is achieved using logical reversibility, where a machine
can carry out a number of computations but is still able to return to
a previous state. Landuaer [13] makes the distinction that a machine
is logically reversible only if each of the individual steps are logically
reversible. Therefore, at each step the machine must remember some
information so that is can get back to the previous state.
Attempts at implementing circuits using reversible computation have
been made, but these typically use adiabatic circuits to try and achieve
thermal equilibrium and thus reduce power consumption.
2.3.6 Approaching the kT ln2 limit
There have been several attempts at producing adiabatic logic circuits
where gradual switching of a transistor is used to try and keep the
circuit close to thermal equilibrium throughout the switching process
[19]–[21]. The basic principle of adiabatic switching can be seen by
considering the energy required to make a logic transmission on an RC
interconnect. Typically a fast rising edge voltage is used in order to
make the transition and thus the energy dissipation is proportional to
CV 2 . Younis [20] has shown that by considering current steps instead
of voltage steps, the energy dissipation per cycle is:
2RC
(2.6)
T
where T is the period of the cycle. Hence by increasing the time of
the operation the energy per cycle can be reduced. For implementation
purposes an ideal current source can be replaced by a voltage ramp [20].
2
Ecycle = CVdd
Typically the clocking requirements of circuits using adiabatic logic
are more complex than ‘standard’ voltage stepped logic. Adiabatic
35
logic requires several voltage ramps at different phases, requiring a
clock generation circuit which consumes considerable energy. For an
8 bit microcontroller implemented using adiabatic logic, over 50 % of
energy is consumed by the clock generator [22], [23]. In this microprocessor the energy per operation is less than 10 pJ. Gong [24] has shown
that the implementation of an 8 bit shift register can reduce energy
consumption by 60-80 % compared with conventional logic. Fabricated
results at operating frequencies up to 15 MHz are presented in [24].
It is clear that adiabatic logic will play an important role in reducing
power, provided that the speed at which digital circuits operate can
be reduced. This can typically be achieved by using parallelism. To
this end Chapter 5 outlines a method to convert an analogue signal
into several parallel paths of lower bandwidth. This would then enable a digital processor to process the signal at lower switching speeds,
allowing exploitation of adiabatic logic circuits to reduce power in communication processing circuits.
2.4 Quantum limits
Quantum limits are those which look at the use of single particles to
store, compute or transfer information. A single particle in a potential
V is described deterministically by the Schroedinger Wave equation:
~2 ∂ 2 Ψ(x, t)
∂Ψ(x, t)
+ V (x, t)Ψ(x, t) = i~
(2.7)
2
2m ∂x
∂t
where ~ is Plank’s Reduced Constant, m is the mass of the particle, x is
the position of the particle, t is the time, V (x, t) is the potential energy
acting on the particle and Ψ(x, t) is the solution to the wave equation,
a wave function. The general form of the wave function (solutions) for
free particles (V = 0) is:
−
Ψ(x, t) = Ae−ikx e−iωt + Beikx eiωt .
(2.8)
Even though the wave functions deterministically describe the po-
36
sition of a single particle it is not possible to directly measure the
position, energy or momentum of the particle without introducing error because the act of measurement will affect the particles position,
energy or momentum. This is explained by the Heisenburg uncertainty
principle (see Section 2.4.1).
A useful property of the wave functions is that the probability density function of a particle can be found by taking the complex conjugate
of the wave function:
P (x, t) = Ψ∗ (x, t)Ψ(x, t)
(2.9)
The function P (x, t) can then be used to find expected values and
variances of the particles position. In effect the observation of the position of a single particle can only be made with a degree of uncertainty.
The quantum wave function, when used to describe a group of particles, leads to an equation for acceleration which matches that of Newtons law a = F/m in the limit of a large number of particles. Therefore,
quantum mechanics is an underlying principle of classical mechanics.
Formerly classical mechanics is quantum mechanics but in the limit
that ~ → 0.
2.4.1 Uncertainty Principle
The uncertainty principle is important as it embodies the impossibility of making a measurement without disrupting the particles being
measured.
The uncertainty relation that relates position and momentum is [25],
[26]:
∆x∆p ≥
h
.
4π
(2.10)
The corresponding energy-time uncertainty is given by:
∆E∆t ≥
h
.
4π
(2.11)
37
As pointed out by Bremermann [26], the interpretation of the energytime uncertainty is not trivial. Essentially the uncertainty bounds
explain that it is impossible to measure one aspect of a quantum system
without effecting the others. It is worth noting that only Gaussian type
functions satisfy this bound. The energy-time frequency relation could
be interpreted as noise [26]. For a given time of measurement there is an
uncertainty in the energy. This interpretation is used by Bremermann
to find the maximum capacity of a photon channel. Rearranging for
maximum energy gives [26]:
Emax =
˙
Ih
.
ln(1 + 4π)
(2.12)
where Emax is the maximum signal energy and I˙ is the information
rate. This result shows that increasing the rate at which information
is transferred requires the maximum signal energy to increase. The
energy per bit in this case can be expressed as:
Ebit ≥
Emax
h
=
τ ln(1 + 4π)
I˙
(2.13)
where τ is the length of the recorded signal. The shorter the received
signal, the more energy per bit is required; faster rates of information
transfer require more power. A similar result is found by Bekenstein
[27] where he derived his results by considering blackhole theory and
system entropy:
Ebit =
~ln2
.
πτ
(2.14)
A similar bound has also been derived by Pendry [28] which written
in terms of the energy per bit is:
Ė
3~ ln2 2I˙
= Ebit ≥
.
π
I˙
(2.15)
Both (2.15) and (2.14) show that the energy per bit increases as the
rate of transmission increases.
38
2.4.2 Energy Per Operation
In the quantum computation literature [29]–[31] there is evidence that
a single state change, i.e. a change of state which is distinguishable/orthogonal requires a minimum amount of energy:
π~
(2.16)
2∆t
where ∆t is the time required to undertake the operation. As in the
classical case it is evident that the longer the operation the less energy
is required. This result implies that quantum computing architectures
can potentially perform operations with much greater speed and less
energy than their classical counterparts. For example, with kT ln 2 J
a quantum evolution rate of 5.5 T operations per seconds could be
achieved. This result is not surprising as a classical system would
require many elementary particles to undertake a similar operation;
many quantum state changes are required for a single classical state
change.
E≥
2.4.3 Blackbody Radiation
Blackbody radiation theory explains the amount of energy radiated
by objects. The theory was originally devised by Plank with further
work carried out by Wein, Stephan and Boltzman [25]. Initially the
spectrum of radiated electromagnetic was found experimentally with
Plank formalising the spectrum by writing the intensity of radiation
as:
2hf 3 1
I(f, T ) = 2 hf
Js−1 m−2 Sr−1 Hz−1
(2.17)
c e kT −1
Wien’s Displacement law describes the frequency at which maximum
radiation occurs:
kT
(2.18)
h
where α = 2.821439 is a fitting constant. For example, the maximum
fmax = α
39
Figure 2.2: Blackbody Radiation Spectrum also showing Wien’s displacement law.
radiation from a blackbody at room temperature (300 K) occurs at
approximately 18 THz.
The Stephan-Boltzmann law gives the total power emitted from a
blackbody per unit area:
Pmax = ǫσT 4 Wm−2
(2.19)
where ǫ is the emissivity constant which lies between zero and unity. A
perfect blackbody has ǫ = 1. σ ≈ 5.66 × 10−8 Wm−2 K−4 is Stephan’s
constant. A plot of the blackbody spectrum is provided in figure 2.2.
2.4.4 Blackbody Communication Limit
In [32] a bound involving a point to point link is postulated:
40
I˙ ≤
512π 4 At Ar 3
P
1215h3 c2 d2
1/4
(2.20)
where At and Ar are the areas of the transmit and receive antennas and
d is the distance between the antennas. This limit assumes that the
receiver can measure the position and time of arrival of the photons.
In terms of energy per bit this limit can be written as:
Ebit
1215h3 c2 d2
≥
R
512π 4 At Ar
1/3
(2.21)
Notice that the bit energy in (2.21) increases as the rate of transmission
increases, however with a weaker relationship than is the case with the
bounds derived directly from the uncertainty bound. As an example
consider a transmitter and receiver which have an area of 1 m2 and are
1 m apart. For a bit rate of 1 Gbit/s the minimum bit energy would
be 86 × 10−27 Joules.
The above limits are fundamental limits and, as has been hinted
at, they require the measurement of momentum, position and time
of the photons. They provide no indication of how to achieve the
limits and thus their usefullness to practical engineering is somewhat
limited. In Section 2.9 a proposal for a lower bound on the energy per
bit for a point to point communications link is found via recourse to
blackbody radiation theory. This limit is based on a point to point
link with known antenna sizes and is more closely related to practical
engineering problems of this time.
2.4.5 Signal Representation Using Photons
In this section work by Gabor [33] is revisited in order to derive an
expression for the energy per bit required to represent a signal which
is valid for the quantum and classical regions. This comparison shows
that the energy required per bit increases as the operation frequency
increases for the quantum case, which suggests that less energy is required for signal representation when operating assuming the classical
41
case.
Gabor considers that a signal can be split into elementary information cells, each of which obeys the time frequency uncertainty. He
proposes that each of the elementary information cells can then be represented using a Plank oscillator resonating at a particular frequency.
One way of representing infomation is to encode the information in the
amplitude of each cell. Gabor [33] shows that the number of uniformally quantised levels (s) for a signal represented with N photons can
be given by:
s2 =
4N
(2.22)
1 + 2 ĒhfT
where ĒT is the energy associated with a Plank oscillator:
ĒT =
hf
e
hf
kT
−1
.
(2.23)
The total energy required by N photons is:
E = N hf
(2.24)
n = log2 (s).
(2.25)
and the number of bits is:
Thus the energy per bit of information represented by each information
cell can be written as:
Ebit = 22n−2 hf + 22n−1 kT.
(2.26)
Eq. (2.26) contains a low frequency and high frequency asymptote
which correspond to the so called quantum and classical regions of operation. Figure 2.3 shows the quantum and classical parts of (2.26).
This gives evidence to the previously seen fact that in the classical region energy per bit is independent of frequency whereas in the quantum
region energy is proportional to frequency.
42
Energy requirements to transmit a single bit of information
5
10
0
Energy/kT
10
−5
10
Quantum
Classical
Complete Picture
−10
10
6
10
8
10
10
10
12
10
Frequency (Hz)
14
10
16
10
Figure 2.3: Minimum energy required per bit when representing a signal in terms of a Plank oscillator.
2.5 Implementation Limits
In this section a number of results based on implied implementation
specifics are discussed. As expected, the minimum energy found from
these limitations is larger than the fundamental limits discussed above,
hinting that the optimal way of reaching the fundamental limits has
not yet been discovered.
2.5.1 Stein Limit
Stein [34] analysed the probability of distinguishing two voltage (current) levels when driving a capacitive (inductive) node in the presence
of thermal noise (akin to switching in CMOS digital circuits). With
a decision level half way between the supply voltage the energy per
switching operation (i.e. bit change) can be expressed as:
2
Eop = 4kT erfc−1 (2Pe )
(2.27)
43
where Pe is the probability of an error in the decision. As a numerical
example Stein argues for a bit error rate of 10−19 for a large digital
switching system. In this case 165 kT is needed per switching operation. For a communication link, where coding and error correction
are applied, a bit error rate of 10−3 may be suitable. In this case the
amount of energy required per switching operation is only 19 kT.
2.5.2 Driving An RC Interconnect
Krishnapura [35] has shown that the minimum energy required to drive
an RC circuit with a sinusoid is:
Pmin = 2πkT fs SNR
(2.28)
where fs is the signal frequency (which in this case is the same as the
cut off frequency) and SNR is the signal to noise ratio at the output of
the filter. By extending the analysis it is also possible to show that the
power required can be written in terms of the filter cut off frequency
as:
ω2
(2.29)
Pmin = kT SNR s
ωc
In [36] Andreou and Furth derive a similar result for driving an RC
interconnect:
f0
fp
P = 4kT SNRfp 1 − arctan
fp
f0
(2.30)
where fp is the message bandwidth and fo is the cut off frequency of
the filter. When the signal is at the 3 db cut off, the power reduces to:
P = 0.86kT SNRfp .
(2.31)
The difference between (2.28) and (2.31) occurs because (2.31) assumes
a uniformally distributed signal over a bandwidth fp . Eq. (2.28) assumes a sinusoidal input signal. This highlights the difficulty in producing universal lower bounds as much depends on the signal repre-
44
sentation.
In many practical cases an interconnect is not driven by a sinusoid.
An interesting case is that of a square wave, which is typically used by
digital computers as a clock signal. The square wave can be represented
as a Fourier Series where the number of terms taken is related to the
rise time of the pulse; see Appendix A for the derivation. The minimum
energy per switching operation in this case is:
Eop = 2kT SNRsq
(2.32)
where SNRsq is the required SNR of the square wave. The SNR can
be calculated by considering the probability that the signal is above a
certain threshold, as hinted at in Section 2.5.1. The SNR required for
binary detection is well known (see Appendix C):
2
SNRsq = erfc−1 (2Pe ) .
(2.33)
2
Eop = 2kT erfc−1 (2Pe )
(2.34)
Therefore, the energy per operation required for driving an RC load
with a square wave is given by:
Eq. (2.34) appears to be 3 dB better than the result derived by
Stein. However, Stein assumed unipolar switching, whereas bi-polar
switching was assumed here. If bi-polar switching is used with Stein’s
proof then the same result is obtained.
To put the result into some kind of perspective consider the number
of flip-flops in a modern digital-IC. There are many clocked devices
(flip-flops, registers and latches) present on each IC and each of these
devices requires a clock. For an IC with 1 × 109 transistors, where
there are on average 25 transistors per clocked device and 75% of the
chip contains clocked devices, 30 × 106 clock paths would be required.
Any misinterpretation of the clock would result in a system crash. For
an error rate of 10−19 the energy per clock path is 82.5 kT and for the
45
Figure 2.4: The Ideal Integrator requires a power of 8kT SNRf .
total chip is 2.5 × 109 kT J. If the chip were clocked at 3 GHz the power
dissipation solely from the clock paths would be around 4 mW at 100
C.
2.5.3 Power Per Pole
In [37] Vittoz derives a power limitation based on an ideal transconductor driving a capacitive load, figure 2.4. This limit is valid for
any technology which uses an active device to drive a capacitive load,
such as CMOS, bi-polar and other transistor technologies. The power
required per pole is:
Ppole = 8kT SNRf
Vdd
Vpp
(2.35)
where Vpp is the peak amplitude of the sinusoidal input and Vdd is the
voltage supply to the transconductor. The minimum power per pole is
then found by setting Vpp equal to the supply voltage:
Ppole (min) = 8kT SNRf.
(2.36)
The fact that the minimum power occurs when the signal amplitude is maximised suggests that the signal amplitude should be maximised at all times. To take advantage of this a technique known as
the analogue floating point technique [38] was invented. This can be
implemented by using an automatic gain control circuit.
46
Figure 2.5: A typical interconnect with matched source and load
impedances. The minimum energy per bit delivered by
the source is 2kT ln 2 J/bit.
2.6 Matched Transmission Line
In this section the energy per bit for transferring information using a
matched transmission line is derived. This analysis also enables a comparison between analogue and digital signal representations in terms
of energy required and number of bits resolution used.
Appendix C shows the relationship between SNR and Eb /N0 . Eb /N0
is the energy to noise ratio required at the load and this has a minimum
value of ln 2.
2.6.1 Energy Per Bit For Matched Information
Transfer
Consider the circuit shown in figure 2.5 where RS and RL are the source
and load resistances. The power delivered by the source is:
PS =
RS + RL
N0 BSNR
RL
(2.37)
with N0 given by:
N0 = 4kT
RS
RS + RL
2
.
(2.38)
47
The energy per bit for the circuit shown in figure 2.5 is thus:
Ebit
RS2
= 4kT
RL (RS + RL )
Eb
N0
.
(2.39)
For a matched link when RS = RL , the energy per bit in terms of
Eb /N0 is:
Ebit = 2kT
Eb
N0
.
(2.40)
This result indicates that 2kT ln 2 J/bit is required when transmitting information across a matched link, twice that of the minimum
limit.
2.6.2 Comparison of Analogue and Digital Signal
Representation
There have been several attempts to determine whether an analogue
or digital signal representation requires the lowest energy. Enz and
Vittoz [38], [39] compare the power per pole required by an analogue
filter to a digital filter which requires 50 operations per bit. For 8 kT of
energy per digital transition they find that an analogue representation
is only of benefit for SNR < 25 dB. Sarpeshkar [40] carried out the
analysis with MOSFET devices to conclude that an analogue representation has a benefit in terms of power consumption for SNR < 60 dB.
Hosticka [41] has provided a comparison by considering an analogue
signal driving a low pass filter and a digital signal represented using a
serial shift register; here analogue representation is shown to be lower
energy for SNR < 40 dB.
It is evident from these studies that the maximum SNR at which
analogue processing provides a benefit is very dependent on implementation specifics. In this work a comparison is made for the case
of information transmission using matched interconnects. The three
cases considered are shown in figure 2.6.
48
0
1
2
N-1
N-bit Parallel Matched Transmission
N-bit Serial Matched Transmission
Analogue Matched Transmission
Figure 2.6: Information transfer using parallel digital, serial digital and
analogue representations.
2 The two digital representations
−1
require 2kT erfc (2Pe ) J/bit and the analogue repre2n
sentation requires (2 n−1) kT J/bit
49
For bi-polar digital modulation over a matched line the energy per
bit for the parallel link is given by:
(digital)
Ebit
2
= 2kT erfc−1 (2Pe )
(2.41)
where Pe is the probability of a bit being in error. This result is
identical to (2.34) where a square wave was driving an RC interconnect.
The result is the same for parallel or serial representation because under
the classical region of operation the rate of information transfer does
not fundamentally effect the energy per bit.
For analogue transmission consider the case when the bandwidth
and the SNR of the channel are known. For a uniformally quantised
signal the number of bits can be related to the SNR by [40]:
n=
log2 (1 + SNR)
.
2
(2.42)
By substituting the SNR from (2.42) into (C.3) gives the minimum
analogue energy per bit for matched impedances as:
B
(2.43)
= 2 × (22n − 1) kT.
R
The rate of an analogue channel is provided by the Shannon-Hartley
law (2.1). Using (2.42) and (2.1) the rate can be found as:
(analogue)
Ebit
R = 2Bn.
(2.44)
Equivalently the rate can be found by considering the digitisation of
the analogue signal. In this case sampling must occur at the Nyquist
rate in order to preserve all the information. Therefore, the equivalent
digital data rate for the analogue signal is 2× B × n bits/s. The energy
per bit for the analogue representation can be written as:
(analogue)
Ebit
=
(22n − 1)
kT.
n
(2.45)
Figure 2.7 shows a plot of the energy per bit required for the ana-
50
3
Energy per bit [k T ln 2 Joules]
10
2
10
1
10
Digital
Analogue
0
10
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Number of bits
Figure 2.7: Energy per bit required to represent a signal in analogue
and digital formats. This shows that there is a benefit in
the analogue format for n < 3, which corresponds to a
SNR < 18 dB.
logue and digital representations of the signal; the probability of bit
error is taken as 10−5 . This shows that representing the signal using
an analogue channel is beneficial only up to a few bits of resolution;
SNR < 18 dB.
2.7 Exploration of Information Transfer
Through an Amplifier
In this section the energy required by Low Noise Amplifiers (LNA)
is discussed from a thermodynamic perspective. The link between
bound information and physical energy is shown and a proposal of
how information processing occurs in an LNA is made. It is shown
that current LNA implementations require seven orders of magnitude
greater energy than the thermodynamic limit of kT ln 2 J/bit. In order
51
to make this comparison a figure of merit based on information transfer
is presented.
2.7.1 Information and Physical Energy
It is possible to make a link between the disorder of a physical system
and the information present [16]. In order to do this the concepts of
entropy, elementary complexions and a definition of information are
required.
The first law of thermodynamics states that energy is conserved. For
an isolated system:
W −q =0
(2.46)
where W is the electrical work available and q is the heat transfer. The
second law of thermodynamics states that entropy (S) cannot decrease
in a closed system therefore:
∆S ≥ 0.
(2.47)
Entropy is related to the temperature (T ) and heat flow by:
∆q = T ∆S.
(2.48)
When considering physical states the definition of information should
be tightened to that of bound information. Bound information is the
amount of information transmitted along a physical channel and includes any necessary error coding or sequencing information. Figure
2.8 shows the mapping between ‘free’ and ‘bound’ information. This
work is not concerned with how this mapping takes place, but as an
example consider a microphone which captures an audio wave of somebody reciting a passage from a book. The text is the free information
and the mapping of the speech to an electrical wave is the encoding and
then the transmission of this wave along a wire to a tape recorder is the
bound information. Here the amount of energy required to represent
52
Figure 2.8: The mapping of free information to bound information.
the bound information is discussed.
The Boltzmann-Planck formula for the physical entropy is:
S = klnP
(2.49)
where P is the elementary number of complexions. An elementary
complexion is a discrete configuration of a quantised physical system.
It is a description of microscopic variables such as position, momentum
and velocity of individual atoms. A link between bound information
and entropy can be obtained if it is assumed that there is a one to
many mapping between the information and the number of elementary
complexions. For example, many electrons are required to represent
a single bit. To clarify the situation consider a system which starts
with zero bound information; this requires a number of elementary
complexions, P0 . As there are a positive number of elementary complexions the system has been overdetermined. There are more physical
states than are actually required to represent the information; such as
with a DC power line. A DC power line does not contain any bound
information but requires a number of elementary complexions in order
53
to define it. If the power line is modulated, by varying the voltage,
then bound information is present. In this case there will be number
P1 of elementary complexions.
Bound information may be measured by considering the number of
states:
I = Klog2 N
(2.50)
where K is a constant and N is the number of states.
The amount of information increase after modulation is:
1
∆I = log2
β
P0
P1
(2.51)
where β is a positive constant which makes the link between the possible cases in the bound information and the elementary complexions.
Effectively this means that the amount of bound information is β times
less than that available by considering the elementary complexions on
their own. When there is an increase in information more is known
about the system so the number of physical complexions P1 < P0 .
The entropy of the unmodulated line is:
S0 = klnP0 ,
(2.52)
and for the modulated line is:
S1 = klnP1 .
(2.53)
The change in entropy is then:
∆S = kln
P1
.
P0
(2.54)
The link between bound information and physical entropy is thus:
∆I = −
1
∆S.
kβln(2)
(2.55)
Eq. (2.55) shows that in order to increase the amount of information
54
the physical entropy must decrease. A decrease in entropy means that
an increase in the grade of energy has occurred, i.e. heat energy (low
grade) has been converted to electrical energy (high grade). Information loss on the other hand always requires an increase in entropy. An
increase in entropy is an irreversible process, i.e. a loss in grade of
energy so electrical energy is converted to heat energy.
Using the first and second laws of thermodynamics the link between
bound information and the energy supplied to the system can be written as:
E = +kT βln(2)∆I.
(2.56)
Note the positive sign on the right hand side of (2.56), this change
of sign occurs so that the energy supplied to the system is considered,
rather than the internal energy of the system (E = −W ). Eq. (2.56)
shows that if information is lost then dissipation occurs whereas if
information is created then energy is required. In the case that the
elementary complexions have a one-to-one mapping with the bound
information (β = 1) then (2.56) reduces to the thermal limit of kT ln 2.
If there is a decrease in information then E is negative, meaning
that energy is dissipated. This is equivalent to the case outlined in
the analysis of thermodynamic computing, where it is well known that
kT ln 2 J is dissipated per bit of information erased.
In the following section a value for β will be estimated by considering
information flow through a practical LNA.
2.7.2 Information Flow Through a Practical LNA
In this section it will be shown that information is always lost through
a simple electronic amplifier/attenuation system. Consider a single
input single output system immersed in a bath of temperature T, from
which energy can be provided to the system if required. The input
signal will contain a signal and noise component. Assume for simplicity
that the amplifier has a brick wall frequency response. In this case the
55
Figure 2.9: Power and noise spectra of an idealised amplifier.
input signal and noise will be amplified by the same amount over the
amplifiers bandwidth. The amplifying device will also add noise to the
output signal (amplification requires an active device, which will add
noise, even in a passive system; a resistor will add noise). The input
and output spectrum of the signal and noise for this idealised case are
shown in figure 2.9.
It is clear that the additive noise of the amplifier causes a bound
information loss between the input and the output of the amplifier.
By using the Shannon-Hartley law (2.1) the information gain of the
amplifier is:
∆I = −Blog2
1 + SNRIN
.
1 + SNROUT
(2.57)
This information loss should theoretically require no external energy
input, only dissipation of power from the input signal. However, this
is not seen in practical circuits where the bias current on a transistor
always requires significant external power.
A possible reason for the requirement of power is that an amplifying
device effectively isolates the input information from the output. For
example consider the MOSFET device shown in figure 2.10.
56
Figure 2.10: The MOSFET acting as an information isolator.
If the MOSFET is ideal then the information at the input is separated from the output, meaning that there is a loss of information
at the input. Consider the MOSFET as a device that is measuring
the input signal. In order to recreate the information at the output
external energy is required because there is an increase in information.
To make an estimate for β the output bit energy of the amplifier is
required. The maximum information rate at the output is given by:
I˙OUT = Blog2 (1 + SNRout ) .
(2.58)
The output SNR can be found by considering the noise factor (F )
of the amplifier [42]:
F =
SNRIN
.
SNROUT
(2.59)
The best input SNR occurs when the source is matched to the input and when the maximum power is being supplied from the source.
Typically the input 1 dB compression point provides a measure of the
maximum input power that can be applied to the amplifier. Thus the
maximum input SNR is:
SNRIN =
P1dB
.
kT B
(2.60)
57
A measure of the Output Bit Energy (OBE) is then:
OBE =
PDC
PDC
=
˙IOUT
Blog2 1 +
P1dB
F kT B
(2.61)
where PDC is the DC power supply to the amplifier. If the input 1 dB
compression point is not given then this may be estimated from the
output 1 dB compression point divided by the LNA gain.
Eq. (2.61) gives the energy per bit required to represent the information at the output of a low noise amplifier.
2.7.3 Theoretical Vs Practical Output Bit Energy
In this section a comparison of many CMOS LNA is made at a range
of different frequencies. The OBE for each amplifier is calculated using
(2.61). Figure 2.11 shows a comparison between the output bit energies
for many CMOS LNA circuits. It is clear that the bit energy is almost
independent of operating frequency as predicted by (2.56). From this
graph a value of β = 1×107 is observed, implying that many elementary
complexions are required to represent each bit of information. The
probable reason why some of the 800 MHz - 2.4 GHz circuits achieve
slightly better bit energies is due to the large amount of time and
money spent on these designs. It is clear that current circuits require
energy orders of magnitude greater than the theoretical minimum. In
the following section the energy required to bias a transistor due to
thermodynamic considerations is explored to try and explain this large
discrepancy.
2.7.4 Energy Required for Biasing
So far it has been assumed that the amplifier starts with P0 elementary
complexions. However, in reality to get to this state requires a change
in entropy, hence a change in energy. Consider a transistor at rest
with no connections; this physical system will contain P−1 elementary
complexions. The output of a transistor at a specified bias current
58
Output bit energy [kT ln 2 Joules]
1010
108
[48]
[45]
[46]
[47]
[49]
[43][44]
106
104
Biasing Limit
102
Thermal Limit
100
0
10
20
40
50
60
70
30
Centre Frequency [GHz]
80
90
Figure 2.11: Comparison of output bit energy for many LNAs across
a range of frequencies. Current LNA circuits are more
than 6 orders of magnitude greater than the fundamental
limit of kT ln 2 and 3 orders of magnitude greater than
the proposed biasing limitation.
59
with no input (i.e. the case with P0 complexions) can be modelled
by a resistor. A resistor produces thermal voltage fluctuations which
increase as the resistance increases [50]:
Vf2 = 4kT RB.
(2.62)
It is also well known that this small signal resistance is inversely proportional to the applied current [51]. Thus to reduce the fluctuations
a larger bias current is required. This can alternatively be viewed as
the lowering of the resistor’s temperature; refridgeration. Consider a
household fridge where an external energy source is used to lower the
temperature of the chamber. This results in a temperature rise outside
of the fridge. The same can be said about biasing a transistor.
To estimate the amount of energy required to get from P−1 complexions to P0 complexions, knowledge about the resistance and the
form of fluctuations is required. In order to reduce the fluctuations the
number of elementary complexions has to be compressed. Hence:
P0 << P−1 .
(2.63)
This implies that the information content has been increased and
more is now known about the system. Thus external energy is required
in order to make this happen, the same as when the line was modulated
with information.
The entropy rate of a signal with a Gaussian distribution probability
is given by Shannon [2]:
Ḣ = BN log2 2πeN bits/s
(2.64)
where N is the white noise average power and BN is the noise bandwidth. Here the difference between the entropy rate before and after
bias is required; this gives the change in entropy rate as:
∆Ḣ = Ḣ0 − Ḣ−1 = BN log2
60
R0
.
R−1
(2.65)
As R0 << R−1 the change in entropy rate is negative, implying an
increase in information akin to that shown in (2.55). Shannon entropy
is the negative of information so the information gain can be written
as:
∆I˙ = BN log2
R−1
bits/s.
R0
(2.66)
Thus the power required to reduce the thermal fluctuations is:
PN = +kT βBN ln
R−1
Joules/s.
R0
(2.67)
The energy per bit is then found by dividing the power by the output
bit rate, ∆I˙out , (2.58).
Numerical Example
In order to grasp the consequence of PN on the energy per bit required
by a amplifier a numerical example is presented. This will allow this
energy to be compared with that of current LNA designs. The noise
bandwidth BN is the bandwidth over which the thermal fluctuations
are reduced, R−1 is the semiconductor resistance at rest and R0 is the
small signal resistance after biasing.
The noise bandwidth will not be the same as the signal bandwidth
in this case. The small signal resistance of the transistor will be valid
at least until the transition frequency of the transistor; typically 10’s
of GHz for integrated transistors.
Assuming that the typical small signal resistance is 500 ohms and
that the resistance of a piece of semiconductor at rest is 1 T ohm. For
a data rate of 100 Mbps and a transition frequency of 10 GHz the
energy per bit is 2.14e3 βkT ln2 Joules. This is considerably more than
the amount of energy required to represent the information.
This example of bit energy is also plotted on figure 2.11 to show that
this explanation of energy usage due to biasing goes someway to explaining why current circuits are orders of magnitude greater than the
61
fundamental theoretical limit of kT ln 2 J/bit. In the following section
consideration of the energy purely due to amplification is discussed.
This is in contrast to the previous discussion where a transistor was
considered as an amplifying device.
2.7.5 Black Box Analogue Amplifier
In this section a lower bound on the energy required to process information through a single input single output filter is described. A filter
is a generalisation of an amplifier. As has already been mentioned,
the loss of information through a amplifier theoretically does not require any external energy because information is always lost, causing
dissipation of signal energy only.
From Shannon [2] the entropy at the output of a linear filter can be
given as:
1
HO = HI +
B
Z
ln|H(f )|2 df
(2.68)
B
where |H(f )| is the magnitude transfer response of the filter. The
entropy change through the filter is:
1
∆H =
B
Z
ln|H(f )|2 df.
(2.69)
Z
ln|H(f )|2 df.
(2.70)
B
The entropy rate is given by:
∆Ḣ = 2
B
It can immediately be deduced from this equation that amplification
of a signal results in positive entropy and therefore negative information. As maybe expected, an amplified signal requires more degrees of
freedom to represent it. Therefore, the energy in the output signal will
be less than the input.
Figure 2.12 shows the spectrum of an ideal brick wall input and
output response. The voltage gain of this system is:
62
Figure 2.12: Low pass filter ideal spectrum.
Av =
AOUT
= |H(f )|.
AIN
(2.71)
By treating the signal and noise spectra independently and considering that the noise figure is given by [42]:
F =
NOUT
SNRIN
=
SNROUT
NIN A2v
(2.72)
the respective entropy rates are:
∆Ḣ(signal) = 4SBW ln Av
(2.73)
∆Ḣ(noise) = 4NBW ln F A2v .
(2.74)
When considering amplification, F > 1 and Av > 1. Therefore, the
entropy gain of the amplifier is always positive. Thus an amplifier in
thermal equilibrium does not theoretically require external energy in
order to operate. This analysis backs up the intuition that the information gain through an amplifier is negative, c.f (2.57). However, there
is a serious flaw with this argument if power gain (Gp ) is considered.
For a positive Gp the power of the output signal is greater than the
power of input, therefore power is required. In fact the entropy rate
shown here is simply the entropy required to carry out the amplification operation with no consideration of input and output impedances.
The act of voltage amplification, with constant power, is equivalent
to increasing the output impedance of the amplifier while keeping the
63
power of the signal the same:
Av =
r
RL
.
RS
(2.75)
In contrast power amplification, with constant voltage, requires a decrease in resistance for the same signal voltage throughout amplification:
RS
Gp =
.
(2.76)
RL
Eq. (2.67) shows that a decrease in effective resistance requires external energy (cooling effect), whereas an increase in resistance does
not (heating effect). It is well known that voltage amplification does
not require an external energy source; for example a transformer or
a cascade of RC T-networks [52]. For all known power amplification
tasks external energy is required. Perhaps the most useful observation
from the entropy rate equations for an amplifier is that it is physically
impossible to obtain a 100 % efficient amplifier. There must always be
power dissipation from the amplifier signal energy in order to satisfy
the entropy increase.
2.7.6 Conclusion
It has been shown that the energy required by an LNA is many orders
of magnitude larger than that required by the fundamental limit of
kT ln 2 J/bit.
The derivation of the energy per bit in order to achieve a given bias
is by no means an exact derivation and many assumptions have been
made. However, this work gives insight into the amount of energy required to carry out information transfer operations and should provoke
the reader to consider circuits from a thermodynamic perspective.
One key point from this analysis is that the effective cooling in order
to reduce the fluctuations almost certainly increases the amount of
energy required per a bit of information. In effect this means that
keeping the output resistance of the transistor high will reduce the
64
energy requirement. Keeping the output resistance high is equivalent
to keeping the transistor in the off position for as long as possible. This
low duty transistor usage is typical of digital circuits and is exploited
in Chapter 4, where a low power transmitter circuit is described.
2.8 Electromagnetic Limits
2.8.1 Friis Limit
In 1946, Friis proposed a key formula that is now used in communications engineering to estimate the amount of transmission power
required for a point to point communications system [53]:
Pr = Pt
At Ar
λ2 d2
(2.77)
where Pr is the signal power at the receiver, Pt is transmitter power,
At and Ar are the effective areas of the transmit and receive antennas
respectively, λ is the wavelength and d is the distance between the
transmitter and the receiver. Appendix B.5 shows how the effective
area of an antenna is related to physical size.
2.8.2 Friis-Kraus Electromagnetic Limit
An extension to the Friis formula (2.77) has been derived by Kraus [54].
In this extension the transmit power required to achieve a given SNR
at the receiver is found to be:
Pt =
SNRd2 c2 BkTsys
At Ar f 2
(2.78)
where k is the Boltzman constant (1.38 × 10−23 ), B is the bandwidth,
f is the centre frequency of transmission and Tsys is the system temperature. Eq. (2.78) can be derived by considering, SNR = Ps /Pn and
Pn = kTsys B.
65
2.8.3 Near Field
Near field communication is becoming increasingly important for short
range links, particularly in the medical arena and for short range smart
payment devices. Yates [55] has analysed a tuned narrow band inductive link using loop antennas to show that the minimum power required
for transmission is:
PT X =
d6
PRX
ω 2 GA
(2.79)
where GA is a constant that depends on the antenna geometry and
directivity.
One difference between near field and far field is that the field rolls
off much more quickly with distance with near field in comparison to
far field radiation. Capps [56] attempts to summarise the distance
at which near field communication is advantageous. More details on
near field communication are provided in Chapter 4 where the design
and measured results of a near field link operating at 33.75 MHz are
presented.
2.9 A New Electromagnetic Lower Bound
Eq. (2.77) and (2.78) imply that an ever increasing centre frequency
of transmission will enable lower and lower energies to be used for
communications. In Appendix B a derivation of an electromagnetic
limit based on the fact that the transmitting antenna is a blackbody is
shown. By taking this into account it shows that the energy of a link
is bounded by:
SNR
2εĀt f 2 (F − 1)d2 c2
Ebit ≥
.
kT
+
log2 (1 + SNR)
c2
At Ar f 2
Eq. (2.80) has a minimum of:
66
(2.80)
s
Ebit (MIN) = 2d
2εĀt (F − 1)
kT ln 2,
At Ar
(2.81)
at a centre frequency of:
f0 =
(F − 1)d2 c4
2εĀt At Ar
41
.
(2.82)
The frequency at which minimum energy of transmission occurs is
based on the dimensions of the antenna, the transmission distance and
the receiver noise figure, so (2.82) could be used to optimise the areas of
the transmit and receive antennas. The energy per bit given by (2.80)
is dependent on the signal to noise ratio at the receiver. The ability
to minimise the signal to noise ratio will ensure that lower energy per
bit transmissions are made. Lowering the SNR in (2.80) will result
in lower energy per bit at the expense of increased bandwidth for a
given transmission rate. Schemes like Ultrawide Broadband (UWB)
are trying to achieve this by trading wider bandwidth utilisation with
lower transmission power [57].
2.9.1 A numerical example
In this example, a typical size of parabolic antenna is considered together with a half wavelength dipole. The physical area of the parabolic
antenna is taken to be 0.01 m2 . The same transmit and receive antennas are used; their efficiency is assumed to be 50 % and the transmission distance is 10 m. It is assumed that the antennas are made from
a material with a blackbody emissivity of 0.1, and that the receiver
noise factor is 1.1.
The plot in figure 2.13 for these conditions shows that there is a
minimum in the theoretical transmit energy required to transmit a bit
of information using parabolic antennas. In this case the lowest energy
transmission can be achieved around 35 GHz, with a bit energy of 39
kT J/bit.
67
Energy per bit requirement for transmission
Zigbee
Bluetooth
10
Energy [kT J/bit]
10
DS−UWB
5
10
Parabolic Antenna
Dipole Antenna
0
10
9
10
10
10
11
10
Frequency, Hz
12
10
13
10
Figure 2.13: Energy per bit lower bound for free space communications
for a distance of 10 m. Parabolic antenna areas are 0.1
m3 . The emissivity of the transmit antenna is 0.1. The
receiver has a noise figure of 1.1.
The frequency at which the minimum energy occurs is proportional
to the square root of the transmitted distance. So for a 100 m transmission the minimum power point would occur at about 111 GHz for
the sizes and conditions as above and the energy per bit would increase
to 390 kT J/bit.
The transmit energy required for the dipole at fo is approximately
five orders of magnitude greater than for the parabolic antennas. This
implies that lower energy systems for point to point links can be optimised by choosing an antenna whose effective area is independent of
frequency. This is known as an electrically large antenna as its dimensions are greater than λ/2π [58].
2.9.2 Comparison With Current Standards
There are several standards that define the amount of transmit power
required for a given data rate. The Bluetooth, Zigbee and DS-UWB
transmit energies are plotted together with the numerical example in
figure 2.13. Table 2.1 shows the protocol specifications used in deriving
68
Standard
Tx distance (m) Tx Power (mW) Data Rate (Mbps) Ebit (pJ/bit)
Bluetooth (Class 2)
10
2.5
3
833.33
DS-UWB
10
0.23
110
2
Zigbee
70
1
0.25
4000
Ref
[59]
[57]
[60]
Table 2.1: Bit energies for several common communication standards.
the bit energies. Note that any extra transmission due to error control
has not been included, however, this does not detract from the fact
that these standards require more than four orders of magnitude more
power than the limit suggests.
The DS-UWB transmission power has been calculated by assuming
that the FCC limit of -41.25 dBm/MHz power has been adhered to
over a 3 GHz bandwidth.
2.9.3 Limitations Of This Bound
The Friis-Kraus limit, (B.6) assumes plane wave propagation and is
only valid for distances of [53]:
2a2 f
(2.83)
c
where a is the largest linear dimension of the transmit or receive
antenna. For the numerical example above when d = 10 m the maximum frequency for which the result is valid is 150 GHz. The limit
(2.81) assumes that the transmitter is the only blackbody radiator in
the system and that the temperature of the transmitter and receiver
are the same.
Despite these limitations the lower bound derived in this paper shows
that there will be a frequency at which minimum transmit power is
required, which to the author’s knowledge, has not been shown before.
d>
2.9.4 Conclusion
The work in this section shows that there is a fundamental lower limit
to the amount of energy required to transmit a bit of information be-
69
tween two points in free space. In order to calculate the new limit
the blackbody radiation of the transmit antenna has been taken into
account. For practical antenna sizes this limit will present itself somewhere in the 10’s of GHz region. This result shows that as the frequency
of communication systems are increased beyond this limit more transmit power will be required. The type of antenna used is important
in minimising the transmission energy. The dipole antenna shows an
increasing bit energy with frequency, whereas the use of a parabolic or
horn antenna allows for a minimum energy point to be found, which
can be four orders of magnitude lower in power than the dipole.
2.10 Biological Examples
It is interesting to be aware that some biological processes also carry
out information processing. There is a trend towards trying to mimic
biological processes by using electronic circuits [40], [61] or by directly
manipulating cells in order to create biological microprocessors [62].
Bennett has described DNA replication via RNA polymerase in nature [17]. A nucleotide represents a single code of a DNA sequence. He
states that approximately 20 kT J is dissipated per nucleotide insertion,
which occurs at a rate of 30 nucleotides per second with a probability
of error 1 × 10−4 . He also notes that if the reaction is slowed down
by reducing enzyme concentration then the energy required decreases;
that is the reaction occurs closer to thermal equilibrium. These values are much closer to the classical lower bounds than any electronic
implementation has achieved so far.
Abshire [61] considers the energy required by a biological and silicon
photo receptor. In this paper it is found that the bit energy of a silicon
blowfly receptor varies between 20 pJ/bit and 2 pJ/bit, depending on
light intensity. The measured silicon model typically requires 5-6 orders
of magnitude more energy per bit than the biological receptor, similar
to that found in Section 2.7.3 for the low noise amplifiers.
Another example is that of the human brain. Sarpeshkar [40] has
70
estimated that the human brain carries out 3.6 × 1015 synaptic operations per second and consumes 12 Watts of power. This equates to
3.33 fJ/bit or 805, 000 kT joules per operation. In comparison, consider the latest Intel Itanium microprocessor [63], [64]. The Dual Core
9120N is able to perform up to 6 operations per clock cycle, equating
to 109 operations per second with a power consumption of 104 Watts;
10 nJ/bit or 2.4 × 1012 kT joules per operation. Therefore, the human
brain uses 6 orders of magnitude less energy per operation than one of
the fastest processors built to date.
2.11 Conclusion
This chapter has presented many limitations to energy requirements in
electronic circuits. These limits are summarised in tables 2.2, 2.3 and
2.4. The first table contains limits due to classical physics, the second
limits due to quantum physics and the third shows implementation
limits. From this summary it is evident that obtaining a true measure
of the energy required to undertake an information processing task
is extremely difficult as heavy reliance on implementation specifics is
required. Table 2.5 shows that biological systems operate closer to the
fundamental limits.
The rate of information transfer depends on the amount of energy
available. This is explicitly shown in the quantum limits and hinted at
in the classical limits as the minimum energy relies on the SNR tending
towards zero, i.e. zero rate of transmission. For classical information
processing the energy required per bit is independent of the operating
frequency.
The exploration of the power required by an amplifier has shown that
current designs require 6 orders of magnitude greater power than the
fundamental limit. It has been shown that the biasing of a transistor
requires considerable energy from a thermodynamic point of view.
A lower bound for the energy per bit for a free space point to point
link has been proposed. This is based on the fact that the transmitter
71
antenna is a blackbody radiator which contributes to receiver noise as
the frequency of operation increases.
An important observation is that the digital representation of a signal could fundamentally reach the lower limit whereas an analogue
representation requires considerably more energy per bit. The input
and output stimuli of the real world are not digital in nature. Therefore
analogue signals and circuits are required for information processing.
In Chapter 4 a scheme of transmitting information using short pulses
is shown which requires the main transmission transistor to be on for
only 0.1 % of the transmitted time. The output of the transmitter is
a well defined analogue pulse. This circuit takes advantage of the fact
that there are no biased transistors. The transmitter is able to operate
at bit energies of < 10 nJ/bit over short range links.
Chapter 5 shows a scheme where an analogue signal can be split
into several parallel channels each of lower bandwidth. This allows
information to be channelled into a digital processor at lower rates
suiting adiabatic logic which consumes less power at lower rates.
In the next Chapter the advantages of using a Gaussian pulse generator for communication over short distances is discussed. This is
followed by some theory on the Gaussian pulse and how it can be
approximated. The approximations are used for the transmitter described in Chapter 4 and as a filter prototype for the Gabor transform
described in Chapter 5.
72
Limit
Ebit > kT ln 2
Brief Description
Classical limit which can
be derived by considering Shannon’s Law,
Compression of a gas
and from quantum theory
Reversible computation
could theoretically require zero energy
Adiabatic
computing
aims to get closer to the
limit
Ebit < kT ln 2
Ebit → kT ln 2
Section Ref.
2.3.1
[4]–[8], [10], [11]
2.3.5
[13]
2.3.6
[19]–[21]
Table 2.2: Fundamental Classical Limits.
Limit
Ebit >
h
τ ln(1+4π)
Ebit >
~ ln 2
τπ
Ebit >
3~ ln2 2I˙
π
Eop >
π~
2∆t
Ebit ≥
h
Brief Description
Bremermann’s
bound
based on the uncertainty
principle
Bekenstein’s
bound
based on black hole
theory and entropy
Pendry’s bound based
on quantum information
flow
Limit on the energy required to distinguish between orthogonal quantum states
1215h3 c2 d2
R
512π 4 At Ar
i1/3
Ebit = 22n−2 hf + 22n−1 kT
Section Ref.
2.4.1
[26]
2.4.1
[27]
2.4.1
[28]
2.4.2
[29]–[31]
Limit on the energy for 2.4.4
black body communications. The entire black
body spectrum is used
for transmission.
n-bit signal representa- 2.4.5
tion in terms of photons.
Valid for quantum and
classical regions
[32]
[33], This Work
Table 2.3: Fundamental Quantum Limits.
73
Limit
Eop = 4kT erfc−1 (2Pe )
Brief Description
Stein’s limit on uni-polar
digital switching
Minimum energy to
drive an RC interconnect with a sinusoid
Energy required per
clock cycle for driving
an RC interconnect with
a 50-50 duty cycle clock
Pmin = 2πkT fp SNR
2
Eop = 2kT erfc−1 (2Pe )
h
P = 4kT SNRfp 1 −
f0
arctan
fp
Ppole = 8kT SNRf
Ebit = 2kT
(analogue)
Ebit
=
Eb
N0
22n −1
kT
2n
E = +kT β ln 2∆I
PN = +kT βBN ln RR−1
0
Pt =
SNRd2 c2 BkTsys
At Ar f 2
q
Ebit = 2d 2ǫĀAtt(FAr−1) kT ln 2
PT X =
d6
P
ω 2 GA RX
i
fp
f0
2.5.2
[35]
2.5.2
This
Work
Minimum power re- 2.5.2
quired to drive an RC
interconnect with a
uniformally distributed
signal
Minimum power re- 2.5.3
quired per pole of a
filter
Minimum energy per bit
to drive a matched transmission line
Minimum amount of energy required to represent an n-bit resolution
signal on an analogue
line
Energy required to represent an information
change. β links the physical elementary complexions to macroscopic information
Power required to make
a change in resistance
Transmission power required for electromagnetic communication
[36]
[37]
2.6
This
Work
2.6
This
Work
2.7
This
Work
2.7.4
This
Work
[54]
2.8.2
Minimum transmission 2.9
energy required when
the transmitter antenna
is considered to be a
black body
Transmission power re- 2.8.3
quired for a tuned near
field link using loop antennas
Table 2.4: Implementation Limits.
74
Section Ref.
2.5.1
[34]
This
Work
[55]
Biological
20kT J/code
2 − 20 pJ/bit
805, 000kT J/op
Electronic
-
200 nJ/bit
2.4 × 1012 kT J/op
Brief Description
DNA replication in
nature via use of the
enzyme RNA polymerase at a rate of 30
codes a second
Energy cost of information
transfer
through a blowfly
photoreceptor and a
silicon replication
Estimation of the energy per synaptic operation for the human brain compared
with that of an Itanium processor
Section Ref.
2.10
[17]
2.10
[61]
2.10
[40], [63], [64]
Table 2.5: Biological vs Electronic Energy Examples.
75
76
3 Pulsed Communication
3.1 Introduction
The amount of transmission power required by orthogonal modulation
schemes decreases as the the number of symbols is increased. Two
commonly used orthogonal modulation schemes are Frequency Shift
Keying (FSK) and Pulse Position Modulation (PPM). The first part of
this Chapter considers why a pulse based transmitter using PPM could
lead to lower power and lower complexity transmitters. An expression
for the power consumption of a generic PPM transmitter is presented,
which shows that the transmitter power can be optimised by choosing
the number of time slots. The PPM transmitter is argued to be lower
complexity than an FSK transmitter because power hungry elements,
such as a PLL, are not necessarily required. It is also suggested that
block coding should be used to improve the bit error rate of data
transmission as the power overhead of an encoder is relatively small.
In the second part of this Chapter, the characteristics of PPM are
explored. It is shown that as the number of symbols increases the output spectrum tends towards that of the pulse shape used. There are
many examples of pulse based transmitters in the literature that use
approximations to different pulse shapes; see section 3.2.2. However,
little attention is paid to the effect of the approximation on the bit
error rate (BER). Due to pulse approximation the symbols will not be
strictly orthogonal, giving overlap between symbols. The effect of this
overlap is to introduce Inter Symbol Interference (ISI), which degrades
the ability to detect the symbols at the receiver. Simulation schemes
for estimating the BER for matched filter and correlation detectors
77
with arbitrary pulse shapes are described. The channel model used is
AWGN and the detector used is the Maximum Likelihood (ML) estimator. These allow quick evaluation of the BER for an approximated
pulse. The channel and estimator choice are only valid for systems
where the effects of fading are small. This is not the case in UWB as
the ISI is dominated by multipath components, so different receiver
architectures are required [12]. However, the simple AWGN channel is
valid for wireline communications and for directional wireless systems
such as the inductive transmitter shown in Chapter 4.
The final part of the Chapter explores the approximation of the optimal time-frequency localised Gaussian pulse. Firstly the properties of
the ideal truncated Gaussian pulse are explored. This is followed by a
comparison of the truncated pulse with three approximation methods;
All Pole, Cascade of Poles and Padé. Simulation results of the BER for
the approximated pulses using coherent correlation and approximated
matched filter detectors are presented. These show the performance
penalty of using the approximated matched filter. For filter orders of
N > 3 using the correlation detector the ideal BER can be achieved.
Finally, methods for converting the approximated baseband pulses to
orthogonal bandpass filters are presented.
The All Pole Approximation is used in the low power PPM transmitter described in Chapter 4. The superior Padé approximation is
used in Chapter 5 as the window function for the Gabor transform.
3.2 Comparison of FSK and PPM
For FSK there are M possible pulses of length T; the frequency spacing between the pulses is 1/T. FSK modulation leads to a continuous
emission of power from the transmitter. In PPM modulation there are
L time slots, each with a slot of length T. The transmitter remains in
an idle state for L − 1 time slots. A combination of these two schemes
results in time-frequency modulation. An example of FSK and PPM
modulation is shown in figure 3.1.
78
1
0.5
FSK 0
−0.5
−1
0
0.5
1
1.5
2
Time [s]
2.5
3
3.5
4
4
6
8
Time [s]
10
12
14
16
T
sym
1
0.5
PPM 0
−0.5
−1
0
2
Figure 3.1: An example of FSK and PPM modulation for L=4. For
the same number of symbols PPM modulation requires L
times longer than FSK.
Tang [65] carries out an analysis between FSK and PPM for wireless
sensor networks which includes non-linear battery effects. The conclusion of this study is that in dense networks PPM can outperform FSK,
but in sparse networks FSK is generally a better choice. However, the
model used for the transmission power does not take into account circuit complexity. It assumes that the transmitter contains only a power
amplifier. An alternative power model is shown in figure 3.2. This
generic circuit is capable of implementing FSK or PPM modulation.
For FSK a continuous stream of pulses is generated, each with a length
of T . For PPM modulation a pulse is generated on average once every
LT . In this case the rates for FSK and PPM are:
log2 (L)
,
T
log2 (L)
=
.
LT
RF SK =
(3.1)
RP P M
(3.2)
The DC energy per bit for the transmitter operating in FSK and
PPM are given by:
79
PDC
PPULSE
Data
R bits/s
Pulse
Generator
PTX
RF
Converter
PRF
PQUIES
Figure 3.2: Block Diagram of a transmitter capable of FSK or PPM
modulation. PP U LSE is the power required by the circuit
to generate a continuous stream of pulses. PQU IES is the
quiescent power of pulse generator and RF conversion when
no pulses are being generated. PT X is the power required
by the RF converter, PRF = ηPT X is the transmitted power
where η is the efficiency of the conversion.
DC
Ebit
(F SK)
DC
Ebit
(P P M ) =
RF
T
Ebit
=
[PP U LSE + PQU IES ] +
,
log2 (L)
η
(3.3)
T
E RF
[PP U LSE + LPQU IES ] + bit .
log2 (L)
η
(3.4)
Eq. (3.4) shows that the PPM scheme requires more energy than the
FSK scheme due to the quiescent power of the circuit. The value of
PP U LSE will generally not be the same for the FSK and PPM schemes.
FSK schemes typically require a PLL in order to accurately switch the
frequencies.
RF
RF
In (3.3) and (3.4) the value of Ebit
is also dependent on L. If Ebit
is kept the same for a given distance of transmission, a better error
rate is achievable when L is increased. It is assumed that PRF can
be increased with the same efficiency η in order to keep the symbol
energy the same when the number of time slots in the PPM scheme is
increased.
80
3.2.1 Minimum PPM Power
The transmitter energy for PPM (3.4) can be minimised by increasing
the efficiency of the RF output stage and by optimising the number of
symbols, L:
d
dL
PP U LSE LPQU IES
+
log2 (L)
log2 (L)
= 0.
(3.5)
Eq. (3.5) has a closed form expression which gives the optimal value
of L as:
„
W e−1
Lopt = e
PP U LSE
PQU IES
«
+1
where W (z) is the Lambert W function. For
approximation to this equation is:
Lopt ≈
PP U LSE
+ 5.
3PQU IES
,
PP U LSE
PQU IES
(3.6)
> 2 a good linear
(3.7)
In order to get closer to the fundamental energy per bit for transmission the number of time slots needs to be large. Hence the quiescent
power must be reduced as much as possible in order to minimise the
transmitter energy. This result is used in Chapter 4 to optimise the
number of time slots for low power transmitter implementation.
In many PPM based transmitters a power amplifier is used to couple
the pulse generator to the antenna. This will require large amounts of
quiescent power. Power amplifiers have a filtering effect so when they
are turned on or off the step response of the filter’s transfer function
would be sent to the antenna. Thus, powering off a PA in between
pulses in PPM is difficult to achieve. In Chapter 4 a novel architecture
that does not use a power amplifier or oscillator is shown, which is
designed to dissipate small amounts of quiescent power.
81
Baseband Pulse Shaping LO
PA
Oscillator OOK
Vdd
VDATA
G(s)/G(z)
fsamp(digital)
DSP based Systems
LO
PA
DSP
VDATA
VDATA
e.g. OFDM
Pulse Based System
Impulse
or
G(s)/G(z)
Step
PA
fsamp(digital)
Analog/Digital Filter
Figure 3.3: Examples of some typical circuit topologies for wireless
transmitters.
3.2.2 Transmitter Complexity
For short range links the RF transmission power is typically very low;
< 1 mW. One of the latest Texas Instruments Low Power DSPs (the
TMS320C5402-100) consumes 180 mW of power with a 100 MHz clock
[66]. Even an ultra low power MSP430CG microcontroller typically
consumes 6.6 mW at a 10 MHz clock rate [66]. These figures suggest
that the use of a digital processor for implementing the modulation
scheme directly would be highly inefficient at low transmit powers. In
addition to the digital processor, a multiplier, phase locked loop (PLL)
and power amplifier (PA) are typically required.
Some typical block diagrams of transmitters are shown in figure 3.3.
The OOK oscillator is an interesting example and could potentially
consume very low power for short range links [55]. The advantage of
this scheme is that the inductor forms the antenna. However, the main
drawbacks of such a scheme are that it is difficult to predict when the
oscillator will turn on and also what the startup phase will be. This
restricts use of this transmitter to non-coherent detection. The out-
82
put spectrum is limited to either a exponentially shaped pulse or an
approximation to a rectangular pulse. Examples of DSP intensive systems are narrow band and wideband OFDM [67], [68]. These schemes
employ Fast Fourier Transforms in order to generate the transmission
signal over many frequency bands. Basedband pulse shaping schemes
are used extensively in commercial systems. For example, the Bluetooth and GSM standards use Gaussian filters to shape the data [69],
[70]. Most developments in UWB pulse systems rely on either mixing
an envelope of the baseband pulse or by directly generating a pulse
using a filter. See table 3.1 for some examples of pulse based transmitters. The overall efficiency of the transmitter is defined as:
ηT X =
PRF (average)
× 100%
PDC
(3.8)
83
84
b
Triangular
YES
NO
Rectangular
YES
YES
YES
YES
NO
YES
NO
NO
pulse
impulse
YES
Exponential
shape
Gaussian
Biphase
train
YES
NO
YES
YES
NO
Digitally designed
from the available
spectral mask
Exponential pulse
shape
Raised Cosine
YES
NO
Rectangular
c
a
PLL/VCO
NO
PA
NO
Pulse Shape
Digitally designed
from the available
spectral mask
6
5
4
4.3
8
4
4
6
0.4
f0 [GHz]
6
0.5
0.5
0.5
2
4
2
0.528
9
0.003
B [GHz]
6
0.04
0.015
0.1
0.002
0.75
0.036
0.1
1.8
0.00012
R [Gbps]
2.5
2
2
31.3
0.236
9
29.7
1.82
227
0.35
PDC [mW]
1.5
b
Table 3.1: Examples of some pulse based transmitters for short range communications.
only operational during pulse
suppressed operation by ramping up and down to prevent spurious output. On for 1/3 of continuous operation
c
only operational during pulse
d
external frequency synthesiser power not included
a
Mixing of a baseband triangular pulse
with the output from a ring oscillator
Pulse Generation Technique
Distributed pulse generation with combination using transmission lines. Contains several impulse generators to generate the impulse response of an FIR
filter.
Direct modulation of the antenna using
a digitally controlled oscillator.
FIR pulse generation using delay lines.
Delays are derived from a high frequency clock
Switching of an oscillator to produce a
pulse. No pulse shaping filters
Summation of offset triangluar pulses
to create an approximated raised cosine
Formed by the combination of 4 digitally produced monocycles to produce
a monocycle train.
Switching of an LC oscillator to create
a pulse.
Uses a tanh pulse shaping mixer to produce a Gaussian pulse
Gating of a local oscillator.
0.2
7.5
d
0.01
8.5
0.01
0.44
0.78
0.03
-
ηT X [%]
0.7
[80]
[79]
[78]
[77]
[76]
[75]
[74]
[73]
[72]
[71]
Ref.
A majority of the circuits in table 3.1 are for GHz frequencies. Although [72] operates at around 400 MHz and switches an oscillator on
and off to produce a rectangular pulse shape. The efficiency of [72]
could not be determined due to insufficient data. The architecture
which has the best overall efficiency is [77], which turns on and off an
LC oscillator to create an exponentially shaped pulse. The high efficiency results from designing the LC oscillator so that it has a large
voltage swing. The circuit in [75] uses a power amplifier but this is
only switched on for one third of the time. To prevent spurious transmissions, the gate voltage of the PA is ramped up slowly to turn the
device on.
Many of the circuits in table 3.1 require a frequency control device to
act as the carrier or for synchronisation of a set of sub pulses which are
summed to form a single pulse. Figure 3.4 shows the power dissipation
for a number of PLL designs which could be used as a frequency control
device. All of these results have been extracted from journal papers
containing fabricated PLL integrated circuits. The figure clearly shows
that the power required increases as the operating frequency increases.
A power improvement of approximately 2 orders of magnitude is seen
over the last 10 years. The use of a phase locked loop typically requires
several mW at operating frequencies in the 100’s MHz region, thus
making it a power hungry component for a low power transmitter.
As seen in Chapter 2, every interconnect and clock signal fundamentally dissipate power. Many of the direct digital pulse based circuits [71] contain many hundreds of interconnects and typically require
clocks at least twice the carrier frequency. These clocks and the information transferral between interconnects can be reduced by considering simple analogue pulse based architectures. To this end the
transmitter presented in Chapter 4 has a simple topology with few
interconnects and requires no clocks at the carrier frequency.
85
4
10
2000 and before
2001 to 2005
2006 to 2009
3
Power [mW]
10
2
10
1
10
0
10
−1
10
−2
10
−1
10
0
10
1
10
2
10
Frequency [GHz]
Figure 3.4: Power consumption of PLL frequency generation circuits.
3.2.3 Channel Block Coding
Channel block coding is applied to the information sequence before it
is modulated. Essentially k input bits are mapped to n output bits
in order to provide an improvement in the error rate. When n > k
a decrease in the rate of transmission occurs, i.e. some redundancy
is added to the information sequence. The underlying modulation
scheme (e.g PSK, FSK or PPM) will influence the performance of the
block coding. Figure 3.5 shows the bit error performance between PSK
(which is an example of an antipodal modulation scheme) and FSK
using AWGN channel, coherent detection and no block coding. It is
clear that at an error rate of 10−4 , orthogonal signalling outperforms
PSK for M ≥ 8.
Figure 3.6 shows the bit error rates for convolutional block coded
modulation schemes. The trellis structure used causes a rate reduction
of half but it shows significant improvement in the bit error rate for
86
0
10
2PSK
2FSK
−1
10
4FSK
8FSK
−2
10
32FSK
−3
BER
10
−4
10
−5
10
−6
10
−7
10
−8
10
0
2
4
6
8
Eb/N0 (dB)
10
12
14
16
18
Figure 3.5: Comparison between 2PSK (antipodal) and M-ary FSK
(orthogonal) modulation schemes. M-ary orthogonal modulation out performs antipodal signalling for M ≥ 8. Channel model is AWGN.
a given Eb /N0 . When orthogonal signalling with 32 symbols is used
a reduction of 20 % in energy per bit can be seen for a Pe of 10−4 .
Alternatively for the same Eb/N0 required for uncoded 32-FSK, an
increase of over 2 orders of magnitude in the probability of error can
be seen. Convolutional coding is a powerful tool for reducing energy
requirements in communication of information. The performance of
block coding depends on the modulation scheme used and it is clear
that although orthogonal signalling requires much larger bandwidths
(or lower rates) than schemes such as 2PSK, it can fundamentally
achieve lower energy per bit.
The convolutional encoder is straightforward to implement. An estimate of the dynamic and static power for implementation of the encoder using an Xilinx Spartan3E device is shown in figure 3.7. For
data rates of 1 Mbps, the power consumption is 170 µW. It is certainly worth considering adding this circuit block to any low power
implementation of a transmitter as its cost in terms of power is small.
87
0
10
2PSK−Convolutional
−1
10
2PSK
8FSK−Convolutional
−2
10
8FSK
32FSK−Convolutional
32FSK
−3
BER
10
−4
10
−5
10
−6
10
−7
10
−8
10
0
2
4
6
8
10
Eb/N0 (dB)
12
14
16
18
Figure 3.6: Comparison of bit error rates for convolutional block coding
using a constraint length of 7 and a generator code of [171
133]. The convolutional code error rates are the upper
bounds. It is evident that the performance still relies on
the underlying modulation scheme. The channel model is
awgn.
88
Static and Dynamic Power [dBm]
20
15
10
5
0
−5
−10 6
10
7
8
10
10
9
10
Data Rate [bps]
Figure 3.7: Simulated power dissipation of a convolutional encoder
with a constraint length of 7. The target FPGA was a Spartan3E. The static power for the entire chip was 34 mW but
the encoder uses only 0.21 % of the available logic. This
gives an estimated static power consumption for an ASIC
implementation of 71.4 µW. At bit rates of 1 Mbps, the total power consumption required by the encoder is 170 µW,
making this a suitable circuit block for improving circuit
performance of low power transmitters.
89
3.3 PPM Simulation
In this section the simulation of a PPM modulation scheme for an arbitrary pulse shape is presented. It is shown that the output spectrum
of PPM tends towards that of the pulse shape chosen. An overview of
detecting orthogonal and pseudo-orthogonal symbols is shown. Simulation methods for a correlation and a matched filter coherent detector
are then presented. The use of these simple models enables the performance of approximated pulses to be compared quickly without carrying
out time intensive continuous time simulations.
3.3.1 PPM Spectrum Shape
The use of PPM has an effect on the output spectra of the transmitter.
The output power spectrum is given by [12]:
1
|G(f )|2 SI (f )
(3.9)
T
where SI (f ) is the power spectrum of the discrete information sequence, G(f ) is the Fourier transform of the pulse function and T is the
length of the pulse. The power spectrum of the discrete information
sequence is:
S(f ) =
SI (f ) =
k=+∞
X
RI (k)e−j2πkf T .
(3.10)
k=−∞
RI (k) is the autocorrelation of the discrete information sequence, In .
For the case that In contains either +1 or -1 (with equal probability)
the power spectrum is equal to unity so the pulse shape G(f ) entirely
determines the spectral usage. For PPM this is not the case. A single
pulse is transmitted every L time slots and the position of the pulse is
in one of the L slots. Figure 3.8 shows a simulation of the PPM information sequence power spectrum, SI (f ). As the number of time slots
increases then the spectrum tends to unity. Therefore, the spectrum of
a PPM modulation with a large number of time slots is approximately
90
5
0
Power Spectrum
−5
L=2
−10
L=4
−15
L=8
L=16
−20
−25
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Normalised Frequency
Figure 3.8: Simulation of the power spectrum for the PPM information
sequence, SI (f ). As the number of time slots increases the
power spectrum tends to unity.
that of the pulse function used.
3.3.2 Orthogonal Signalling
The crux of orthogonal signalling is that the set of symbols used to
represent the information must be orthogonal. Each symbol represents
a number of bits:
n = log2 L
(3.11)
where L is the number of symbols. For the set of continuous time real
symbols:
h
iT
s = s1 (t) s2 (t) s3 (t) · · · sL (t) .
(3.12)
In the notation used in (3.12), each row of s contains discrete sam-
91
ples of si (t). The sampling frequency must be much larger than the
Nyquist rate to ensure each row of s approximates its continuous time
representation. s is then a L×Nsamp matrix where Nsamp is the number
of samples.
Orthogonality exists if the cross correlation of the symbols is zero
and the autocorrelation is constant:
Z
0
T

1, if i = j
si (t)sj (t)dt =
0, if i =
6 j
(3.13)
Alternatively the normalised cross correlation matrix (inner product)
has only 1’s on the diagonal:

1

0


′
Rss = s s = 0
.
 ..

0

··· 0

· · · 0

· · · 0

. . . .. 
.

0 0 ··· 1
0
1
0
..
.
0
0
1
..
.
(3.14)
Any symbol set which obeys 3.14 is orthogonal and therefore is a
power constrained modulation scheme.
3.3.3 Pseudo Orthogonal BER
In this section a general method for determining the bit error for a
signalling scheme that is not completely orthogonal is described. The
ML (Maximum Likelihood) detector for the case when each symbol has
equal energy, is equiprobable and the noise is additive white Gaussian
is given by [12]:
m̂ = arg max r · sm
1≤m≤M
where r · sm is the inner product and can be written as:
92
(3.15)
r · sm =
Z
+∞
r(t)sm (t)dt
(3.16)
−∞
where r(t) is the received signal corrupted by noise and sm (t) is the
mth symbol.
The measure of signal orthogonality is the cross correlation matrix,
which for orthogonal signals is shown in (3.14). For the case of M = 2
the closed form expression for the energy per bit required is:
2
Eb
[erfc-1 (2Pe )]
.
=
N0
1 − Rss [1, 2]
(3.17)
Eq. (3.17) is not dependent on the pulse shapes, only the cross correlation between the two pulses.
Figure 3.9 shows a plot of how the SNR varies for different values
of cross correlation at a fixed probability of error. It is evident that
antipodal signal schemes (i.e. {g(t), −g(t)}) require the least amount
of energy. On-Off keying (i.e. {g(t), 0}) requires 3 dB more energy
than antipodal signals. Strongly correlated signals require a further
increase in energy; an extra 3 dB for Rss = 0.5.
For M > 2 there are closed form expressions for the upper and lower
probability bounds [12]. In this work a simple numerical simulation
method will be presented which is valid for any value of M and which
can be used to also model the ISI.
Cross correlation between symbols results in the spread of transmitted energy amongst all the symbols. The energy components of the
symbol set can be found by decomposing Rss to find s, where here s
is a M × M matrix of energy coefficients.
With s symmetrical the symbol set that corresponds to the cross
correlation matrix Rss is:
1
1
2
s = Rss
= VD 2 V−1
(3.18)
where V are the eigenvectors of Rss and D is a diagonal matrix containing the corresponding eigenvalues. Each row of s represents the
93
20
18
−3
P = 10
e
−4
Pe = 10
16
−5
P = 10
e
12
b
0
E /N [dB]
14
10
8
6
4
2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Cross Correlation, R [1,2]
ss
Figure 3.9: The effect of cross correlation on the energy requirements
of ML detection for a symbol set of size M = 2 for various
error rates.
94
energy of each symbol.
To simulate the probability of detecting the wrong symbol, a row
of s is chosen at random and white noise is added to this vector to
produce the received vector:
r = sm + n
(3.19)
where sm is the mth row of s. When the autocorrelation of each of the
symbols is unity the noise vector n contains white noise samples with
a variance given by:
σn2 =
1
2 log2 (M )
Eb
N0
.
(3.20)
Eq. (3.15) is then used to make an ML estimate of the received symbol.
As only the cross correlation of the symbols is required, the algorithm
can be implemented efficiently in a few lines of Matlab code, thus
allowing fast evaluation of the error rate for a symbol set. The following
sections extend the method to take into account ISI and use of an
approximate matched filter at the receiver.
3.3.4 PPM Detection
The received pulse will be the transmitted pulse, plus the overlap from
the previously transmitted pulses. This implies that this transmission
scheme has memory. It is also the same mechanism by which intersymbol interference occurs (ISI). The optimum receiver for this type of
interference is a correlator followed by a maximum likelihood sequence
estimator (MLSE) [12].
A sub optimum estimate, yet much easier to implement, is to use
the ML detector for the case of no ISI, as described in section 3.3.3.
In this case the estimator is no longer a maximum likelihood estimator
because the symbol overlap has not been taken into account when
finding the maximum likelihood function. However, using this detector
in the presence of ISI still provides a good estimate of the transmitted
95
{In}
Impulse
Generator
g(t)
g*(-t)
z(t)
ML
{In}
MLSE
{In}
nT
s
ML
{In}
Figure 3.10: Possible detection schemes when the impulse of the transmit filter is sent across a channel. Using the ML estimator
for the case of no ISI is suboptimal but straightforward to
implement. The MLSE uses a trellis to search to find the
most-likely sequence in order to mitigate ISI effects. The
correlation method acts on a symbol by symbol basis to
find the most likely transmitted symbol
data. This detection scheme can also be formed by using a matched
filter followed by a sampler followed by the ML estimator for the case
of no ISI. A block diagram of three estimation methods are shown in
figure 3.10.
3.3.5 Correlation Detection
As ISI depends on the previously transmitted symbols, the current
transmitted signal can be written as:
s̃(t) =
P
X
i=0
g(t − Ii T + iLT ) for 0 < t < LT
(3.21)
where P are the number of previous symbols to take into account. Ik
is the symbol information sequence that takes on values between 0 and
L − 1. Thus, the current symbol can be written as:
s̃(t) = g(t − I0 T ) +
96
P
X
i=1
g(t − Ii T + iLT ).
(3.22)
Eq. (3.22) shows that each transmitted symbol is the sum of the the
current symbol and a contribution from the past symbols. Appendix
E shows the derivation of the ML detector for correlation detection.
Using correlation detection is not the optimum way to detect a signal
corrupted with ISI but, as shown in section 3.6, the approximated
pulses performance is close to theoretical limit for orthogonal detection.
3.3.6 Matched Filter Detection
The matched filter impulse response is the time-reversed complex conjugate of the transmit filter. For the ideal Gaussian pulse the transmit
and matched filter response are the same:
2
g ∗ (−t) = g(t) = e−παt .
(3.23)
This allows the same filter to be used at the transmitter and the receiver. However, only an approximation to the Gaussian pulse can be
made so the implemented matched filter will not be perfect. Figure
3.11 shows the effect of using the same approximated Gaussian filter
for the transmit and matched filters. This is for the all pole approximation, see section 3.5.1. It clearly shows that as the order is increased,
a better approximation to the ideal matched filter is produced; as expected from (3.23).
To find an equivalent cross correlation matrix which can be used for
ML detection the approximated filter characteristics need to be taken
into account. Appendix E shows the derivation of the ML estimator
for approximate matched filter detection.
A comparison of the BER performance using approximated pulses is
shown in section 3.6. In the following section the choice and approximation of the pulse g(t) is discussed.
97
Matched Filter Response
Transmitted Pulse
2
N=2
1
g(t)
0.5
1
0
0
2
3
4
5
6
3
4
5
6
10
11
5
6
7
8
9
10
11
6
7
8
9
10
11
6
7
8
9
10
11
0
3
4
5
6
5
1
2
0.5
1
0
0
2
9
0.5
1
0
N=8
8
1
2
2
7
0
0
N=6
6
0.5
1
2
5
1
2
N=4
g(t) ∗ g(−t)
g(t) ∗ g(t)
g(−t)
3
4
5
6
5
Figure 3.11: The effect of using the same approximated filter for the
transmit filter and matched filter. g(t)∗g(t) is the convolution when using the same approximated filters. g(t) g(−t)
is the convolution when using an ideal matched filter. As
the order increases, the difference between g(t) ∗ g(t) and
g(t) ∗ g(−t) is reduced.
98
3.4 Truncated Gaussian
Having seen that the pulse shape directly affects the spectrum of the
modulated signal, it is important to choose a pulse that provides optimum data rate and spectral use. The choice of the Gaussian function
is optimal in the sense that it provides the best spectrum usage for a
given pulse width. It also meets the time-frequency uncertainty constant. The calculation of the time-frequency uncertainty for a number
of pulse shapes is shown in Appendix D. A summary of the timefrequency uncertainty is shown in table 3.2. ǫ is the normalised timefrequency uncertainty bound that varies between zero and unity (D.1).
Pulse Shape
Time-Frequency Uncertainty, ǫ
Rectangular
0
Sinusoidal
0.3
1
Gaussian Derivative
3
Truncated Gaussian
ǫ → 1, for αT 2 >> 1
Gaussian
1
Table 3.2: Comparison of time-frequency uncertainty for a variety of
pulse shapes. ǫ is the normalised time-frequency uncertainty
(D.1).
The Gaussian pulse is an analytic function and has a Taylor series
expansion. However, there are several properties that make a practical
implementation difficult. For implementation of the pulse using linear
systems a causal transfer function is required. The Gaussian function
is not causal so it must be approximated by shifting the signal so that
it is zero for negative time. This results in truncation of the pulse. For
a digital representation, for example when using a look up table, the
pulse would have to be truncated.
When comparing linear approximated Gaussian pulses it is more
relevant to compare them to the ideal truncated Gaussian pulse because this can be implemented by a digital process. In this section the
bandwidth-time product and the achievable attenuation of the truncated pulse are compared with the ideal Gaussian.
99
The ideal Gaussian pulse is given by:
g(t) = e−παt
2
(3.24)
where α describes the width of the pulse. The corresponding Fourier
transform [81], ignoring the scaling constant is:
π
2
G(f ) = e− α f .
(3.25)
The bandwidth of the Gaussian pulse is dependent on the attenuation required and the value of α:
r
A ln 10
α
(3.26)
5π
where A is the attenuation in dB. This equation shows that the bandwidth and the attenuation are linked.
B=
The Fourier transform of the truncated pulse can be found by evaluating the integral:
Gt (f ) =
Z
T /2
2
e−αt e−j2πf t dt
(3.27)
−T /2
where T is the length of the pulse in the time domain. Evaluation of
(3.27) ignoring any scaling factors results in:
π
Gt (f ) = R(M )e− α f
2
(3.28)
with M given by:
M = erf
√
παT
+ jf
2
r π
.
α
(3.29)
The term αT 2 appears in these equations (and also for subsequent
equations involving the truncated Gaussian pulse). Therefore αT 2 , is
the parameter that will be used when describing the truncation effects
on the Gaussian pulse. Figures 3.12 and 3.13 show the time and frequency domain truncated Gaussian pulse for different values of αT 2 .
100
3
α T2 = 1
α T2 = 5
2.5
α T2 = 10
Magnitude
2
1.5
1
0.5
0
−0.5
0
Time [s]
0.5
Figure 3.12: Truncated Gaussian time domain pulse for T = 1. As αT 2
goes towards 0 a square wave is obtained.
In these examples the window width is set to T = 1 and the energy of
the pulses has been set to be equal.
The truncated pulse attenuation can be found numerically from
|Gt (f )|. This is achieved by comparing the height of the peak and
the 1st side lobe. Figure 3.14 shows a plot of the attenuation versus
αT 2 , which is valid for any value of T . The numerical result can be
approximated using a linear equation:
A[dB] = 7.5αT 2 + 12.6.
(3.30)
The bandwidth-time product is also approximated using a linear
equation derived from numerical calculations. Figure 3.15 shows a
plot of the numerical and approximated bandwidth-time for different
value of αT 2 . The side lobe merges with the main lobe periodically,
causing the discontinuities at αT 2 ≈ 3.5 and αT 2 ≈ 7.75. The linear
approximation is:
BT ≈ αT 2 + 1.4.
(3.31)
101
50
α T2 = 1
α T2 = 5
α T2 = 10
Magnitude [dB]
0
−50
−100
−150
−200
−10
−5
0
Frequency [Hz]
5
10
Figure 3.13: Truncated Gaussian pulse frequency domain response for
T = 1. The bandwidth of the pulse is dependent on the
value of αT 2 .
120
Numerical
Linear Approximation
Attenuation [dB]
100
80
60
40
20
0
0
2
4
6
α T2
8
10
12
Figure 3.14: Attenuation achieved by the truncated Gaussian pulse for
various value of αT 2 .
102
12
10
BT
8
6
4
Numerical
Linear Approximation
2
0
0
2
4
6
8
10
α T2
Figure 3.15: Bandwidth-time product for the truncated Gaussian pulse
for various values of αT 2 .
The use of the truncated Gaussian pulse allows a trade-off between
attenuation and time-bandwidth product that is not available when
considering square or sinusoidal pulses. Table 3.3 shows a comparison
of the attenuation and bandwidth-time product for the rectangular,
sinusoidal and truncated Gaussian pulses.
Rectangular Sinusoid
BWml T
A [dB]
2
13.3
3
23
Gaussian
αT 2 = 1 αT 2 = 5 αT 2 = 10
2.4
5.4
11.4
20
50
90
Table 3.3: Comparison of bandwidth and attenuation for various
pulses. BWml is the main lobe bandwidth.
3.5 Gaussian Approximations
A pulse can be approximated using a variety of methods, some of which
were discussed in section 3.2.2. To reduce complexity of the transmitter, the approach used here is to apply an impulse to an continuous
103
time filter with the resulting response being an approximation to the
Gaussian pulse:
g(t) = L−1 [G(s)GI (s)]
(3.32)
where g(t) is the Gaussian approximation, G(s) is the frequency domain response of the linear filter and GI (s) is the impulse function.
For an ideal impulse (delta-Dirac) GI (s) = 1. In reality an ideal impulse function is not realisable as this would require infinite amplitude
and infinitesimally width in time. Therefore, an approximation to the
impulse function has to be made. Details of this approximation and
its effects on obtaining the impulse response are shown in Appendix
F. In the following sections an overview and comparison of signalling
performance for three approximation methods is shown.
3.5.1 All Pole Approximation
This is the simplest of approximations that can be made from the
frequency description of the Gaussian pulse. The time shifted Laplace
representation of (3.25) is:
1
2
T
G(s) = e 4πα s e− 2 s
(3.33)
where the time shift of T /2 has been added to ensure that the approximation is causal. This function has a Taylor series expansion, but
finding this would result in a transfer function with no poles, which is
not realisable. Therefore, the following expansion is used in order to
ensure an all pole approximation:
G(s) =
1
1
h
i= N
.
N
1
s + aN −1 s −1 + · · · + a0
Taylor G(s)
(3.34)
Figure 3.16 shows how the out of band attenuation varies according to the value of αT 2 and the order of the filter. The out of bound
104
120
Truncated
Attenuation [dB20]
100
N=6
N=5
80
N=4
N=3
60
N=2
40
N=1
20
0
0
2
4
6
2
8
10
12
14
αT
Figure 3.16: Out of band attenuation versus αT 2 . The solid line shows
the attenuation for the ideal truncated pulse.
attenuation is calculated at the bandwidth described by (3.31). The
solid line shows the attenuation vs αT 2 for the truncated pulse, (3.30).
This figure shows that the order of the filter needs to be increased to
provide a larger attenuation. The stability of the filter is also dependent on αT 2 . In particular, higher order all pole filters do not exist for
low values of αT 2 .
Figure 3.17 shows the all pole approximation of a pulse where αT 2 =
6 and T = 1. This figure shows the spreading of energy into adjacent
time slots, which will cause non-zero cross correlation between the
PPM symbol set. The corresponding frequency response is shown in
figure 3.18. As the order of the filter is increased then the approximated
pulse is closer to the ideal Gaussian.
105
1.2
Ideal
N=1
N=2
N=3
N=4
N=5
T
1
Amplitude
0.8
0.6
Energy spread into
adjacent time slots
0.4
0.2
0
−0.2
0
0.5
1
1.5
Time [s]
2
2.5
3
Figure 3.17: All pole approximation of the Gaussian pulse function for
αT 2 = 6 and T = 1. As the order is increased the approximation is improved and there is less energy in adjacent
time slots.
Magnitude Response [dB]
0
−20
−40
Truncated
N=1
−60
N=2
N=3
−80
−100
−5
N=4
N=5
0
Frequency [Hz]
5
Figure 3.18: Frequency response of the all pole Gaussian pulse approximation for αT 2 = 6 and T = 1. As the order is increased
the approximated pulse is closer to the truncated pulse.
106
3.5.2 Cascade of Poles
The crux of the approximation is to use a cascade of 1st order systems.
Consider the following time domain function:
h(t) =
tN −1 −at
e .
(N − 1)!
(3.35)
Eq. (3.35) has a well known Laplace transform, which is the cascade
of identical poles [82]:
H(s) =
1
.
(s + a)N
(3.36)
To approximate the Gaussian function firstly, centre this function so
its maximum is at t = 0. This can be achieved by substituting t in
(3.35) with [83]:
t=
p
2(N − 1)t̃ + N − 1
a
(3.37)
to form a new function h̃(t̃). As the order, N , is increased then h̃(t̃)
approximates a Gaussian pulse:
2
lim h̃(t̃) = e−t̃ .
n→∞
(3.38)
Expressions for α and T can be obtained by rearranging (3.37):
a2
2π(N − 1)
2(N − 1)
T =
a
2
αT 2 = (N − 1)
π
α=
(3.39)
(3.40)
(3.41)
In this case the value of αT 2 is fixed by the order of the filter, unlike
in the all pole approximation. Also the approximation is only defined
for N > 2.
Figure 3.19 shows a plot of the attenuation for different values of
107
120
Truncated
Attenuation [dB20]
100
80
N=20
60
N=18
N=16
N=14
N=12
40
N=10
N=8
N=6
20
N=2
0
0
N=4
2
4
6
8
10
12
14
α T2
Figure 3.19: Out of band attenuation vs αT 2 for the cascade of poles
approximation. The solid line shows the attenuation for
the ideal truncated pulse.
αT 2 . The solid line shows the ideal truncated Gaussian pulse attenuation. With reference to figure 3.16 it is evident that the order of the
cascade of poles filter must be much greater than the all pole approximation for a given attenuation. For example, 40 dB of attenuation
required a N = 10 cascade filter or a N = 3 all pole filter.
Figures 3.20 and 3.21 show the time and frequency response of the
approximated pulses for N=2 and N=8. The time domain pulses show
much greater overlap between symbols in comparison to the all pole
approximation. For large values of N a very good approximation of
the Gaussian pulse can be made.
An extension of this approximation is used by several authors [83],
[84] to produce a Gaussian pulse using the sum of complex first order
systems. This extension makes the approximation bandpass with real
and imaginary outputs.
108
1.2
2
N=2 α T = 0.64
T
N=8 α T2 = 4.5
α T2 = 0.64 Ideal
1
α T2 = 4.5 Ideal
0.8
Amplitude
Energy spread into
adjacent time slots
0.6
0.4
0.2
0
−0.2
0
0.5
1
1.5
Time [s]
2
2.5
3
Figure 3.20: Cascade of poles approximation of the Gaussian pulse
function for N=2 and N=8. For low order filters this approximation shows considerable spread in energy in adjacent time slots.
0
Magnitude Response [dB]
−10
−20
−30
−40
−50
N=2 α T2 = 0.64
N=8 α T2 = 4.5
−60
−70
−5
Truncated α T2 = 0.64
Truncated α T2 = 4.5
0
Frequency [Hz]
5
Figure 3.21: Frequency response of the cascade of poles approximation
for N=2 and N=8.
109
3.5.3 Padé Approximation
The Padé approximation includes zeros as well as poles in order to
provide a better estimate of the pulse function. [85] provides a derivation of the Padé approximation for an arbitrary function. The idea of
the Padé approximation is to rationalise the Taylor series expansion of
G(s):
G(s) = ck+1 sk+1 ck sk + · · · + c1 s + c0
(3.42)
where k + 1 is the order of the approximation and G(s) is shown in
(3.33). The Padé approximation is then:
Ĝ(s) =
P (s)
pM sM + · · · + p1 s + p 0
=
Q(s)
qN s N + · · · + q1 s + q0
(3.43)
where k = M + N and N > M to ensure a realisable transfer function.
See [85] for a succinct derivation for finding the values of the coefficients
pi and qi . From this derivation the matrix of numerator coefficients are
given by:

cM +1

 cM +2
q = Nullspace 
 ..
 .
· · · c0
c1
cM +N · · ·
h
iT

0

c0 
.. 

. 
(3.44)
cM
where q = q0 q1 · · · qN . For normalisation scale q such that
qN = 1. As N > M the denominator coefficients are then given by:

h
where p = p0 p1
c0

 c1
p=
 ..
 .
0
c0
···
···
...

0

0
 · qM


c0
cM cM −1 · · ·
iT
· · · pM and qM is the 1st M rows of q.
(3.45)
Figure 3.22 shows a plot of the achievable attenuation when using
110
110
110
M=0, N=4
100
M=1, N=6
M=2, N=4
90
M=0, N=6
100
M=1, N=4
M=2, N=6
90
M=3, N=6
Truncated
80
Attenuation [dB20]
Attenuation [dB20]
M=3, N=4
70
60
50
80
Truncated
60
50
40
40
30
30
20
20
10
0
2
4
6
8
2
αT
10
12
M=4, N=6
70
10
0
2
4
6
8
10
12
2
αT
Figure 3.22: Out of band attenuation versus αT 2 for the Padé Approximation.
the Padé approximation for an N = 4 and N = 6 filter. As M is
increased the out of band attenuation of the pulse decreases. This occurs because the Padé approximation introduces zeroes in the transfer
function, which tend to add 20 dB/decade of gain.
Figure 3.23 shows the time domain approximation for the N = 4
filter and figure 3.24 shows the time domain approximation for N = 6.
The corresponding frequency domain pulses are shown in figures 3.25
and 3.26.
It is evident from figures 3.25 and 3.26 that a better approximation
of the frequency response can be made when using the Padé approximation, compared to the other approximation methods. This is useful
for signal processing applications where a bank of filters is required.
For example, the Padé approximation is used for the Gabor transform
shown in Chapter 5. However, for a PPM modulation scheme the Padé
approximation results in worse out of band attenuation, compared with
the other approximation methods. The all pole approximation is superior in terms of attenuation. An advantage of the Padé approximation
is that the intersymbol interference is reduced when more zeros are
used.
111
T
M=1,N=4
M=2,N=4
M=3,N=4
Ideal
0.8
Amplitude
0.6
Energy spread into
adjacent time slots
0.4
0.2
0
−0.2
0
0.5
1
1.5
Time [s]
2
2.5
3
Figure 3.23: Time domain response of the Padé approximation for
N=4.
T
M=1,N=4
M=2,N=4
M=3,N=4
Ideal
0.8
Amplitude
0.6
Energy spread into
adjacent time slots
0.4
0.2
0
−0.2
0
0.5
1
1.5
Time [s]
2
2.5
3
Figure 3.24: Time domain response of the Padé approximation for
N=6.
112
0
−10
Magnitude Response [dB]
−20
−30
−40
−50
M=1,N=4
M=2,N=4
−60
M=3,N=4
Truncated
−70
−80
−90
−5
−4
−3
−2
−1
0
1
Frequency [Hz]
2
3
4
5
Figure 3.25: Frequency response of the Padé approximation for N=4.
0
Magnitude Response [dB]
−10
−20
−30
−40
−50
−60
M=1,N=4
M=2,N=4
M=3,N=4
Truncated
−70
−80
−5
0
Frequency [Hz]
5
Figure 3.26: Frequency response of the Padé approximation for N=6.
113
3.5.4 Creating bandpass filters
So far the low pass (base band) filter approximation has been shown.
In this section several methods of creating a bandpass filter are shown.
When there are many oscillations in the transfer function the Taylor
series has difficulty in converging, making direct approximation of a
bandpass filter difficult. To create a bandpass filter create a prototype
low pass filter with T = 1 and make the following substitution for s:
s→
ω02 T
s2
ω02
+1
.
(3.46)
2s
Making this substitution causes the poles to be reflected about +/−ω0
on the imaginary axis. Therefore, the order of the filter is doubled.
Using (3.46) will also introduce transmission zeros which are undesired
if an all pole filter is required.
To make the all pole approximation bandpass firstly scale the bandpass representation such that the width of the pulse is correct. Then
reflect the modified poles about the required frequency on the imaginary axis. For each pole (pi ) of the base band filter apply the following
equation to find the new pole (pn ) positions:
pn = Re[pi ] ± j(Im[pi ] + w0 ).
(3.47)
An orthogonal filter can be created by multiplying the bandpass
filter by s, i.e. taking the differential of the transfer function. This
introduces a zero at the origin.
In the cascade of pole approximation a common way to produce the
bandpass filter is to make a in (3.35) complex. This technique is known
as the cascade of first order complex systems (CFOS) [84].
For the case of the Padé approximation a set of orthogonal filters
can be created by finding the inverse Laplace transform of G(s) and
multiplying this by a sinusoid or cosinusoid waveform of the desired
frequency. This technique is used to produce the orthogonal bandpass
filters in Chapter 5.
114
3.6 Gaussian Approximation BER
A comparison of the bit error performance for the all pole, cascade of
poles and Padé approximations for the correlation and approximate
matched receiver are made using the simulation framework described
in section 3.3.
To prevent interference between adjacent channels the out of bound
attenuation must be limited. For many channels the out of bound
attenuation is defined by a spectral mask which is made available via
communications regulators, for example Ofcom [86]. In Chapter 4 a
modulator operating in the 30 - 37.5 MHz band is presented which
has a measured out of band attenuation of 30 dB. At the time of
writing no spectral mask is available for this band. In order to make
a performance comparison between the approximate Gaussian pulses
several pulse prototypes with an attenuation of greater than 35 dB are
compared. Table 3.4 shows the approximations to the truncated pulse
that meet this criteria.
Approximation Method Filter Order αT 2 BT
All Pole
N=2
8
9.4
All Pole
N=3
5
6.4
All Pole
N=4
5
6.4
Cascade
N=10
6
7.4
Padé
M=1, N=4 4.75 6.15
Padé
M=2, N=6
7
8.4
Table 3.4: Approximated pulses with an attenuation of > 35 dB and a
bandwidth-time product given by αT 2 + 1.4.
For the pulses shown in table 3.4, the BER performance for L=2, 4,
8 and 16 are shown in Appendix E.3. Figures 3.27 and 3.28 show the
extra energy required over ideal orthogonal signalling for correlation
detection and matched filter detection, respectively.
Figure 3.27 shows that the bit error performance tends to that of
ideal orthogonal signalling as the number of symbols is increased. The
N = 3 all pole approximation performs better than the orthogonal
115
signalling limit because the normalised distance between adjacent time
slots is greater than unity. This can be seen by examining the cross
correlation matrix for L=4:

1.0021 −0.0540 0.0073 −0.0003


−0.0540 1.0021 −0.0539 0.0073 
.

R̃ss = 

0.0073
−0.0539
1.0020
−0.0539


−0.0003 0.0073 −0.0539 0.9938

(3.48)
The Padé approximation is able to meet the ideal orthogonal performance for both correlation and matched filter receivers. The N = 2
All Pole filter using the matched filter requires significantly more energy, however, as the number of time slots increases the performance
also increases. The reason for the poorer performance can be seen in
figure 3.11 which shows that the approximation of the matched filter
for N = 2 is poor, resulting in a smaller distance between the symbols.
The same is true of the N=10 Cascade of Poles filter, which shows a
roughly constant performance penalty as the number of time slots is
increased.
To summarise, it is possible to create a pseduo-orthogonal approximate Gaussian pulse PPM symbol set using continuous time filters
whose BER performance approaches that of an ideal orthogonal symbol set as the number of symbols is increased.
116
20
N=2 All Pole
N=3 All Pole
N=4 All Pole
N=10 Cascade
M=1, N=4 Pade
M=2, N=6 Pade
Increase in Transmitter Energy, %
15
10
5
0
−5
−10
0
2
4
6
8
10
12
14
16
18
Number Of Symbols, L
Figure 3.27: Extra transmitter energy required over ideal orthogonal
pulses when using a correlation detector with the approximated pulses.
30
25
N=2 All Pole
Increase in Transmitter Energy, %
N=3 All Pole
20
N=4 All Pole
N=10 Cascade
M=1, N=4 Pade
15
M=2, N=6 Pade
10
5
0
−5
−10
0
2
4
6
8
10
12
14
16
18
Number Of Symbols, L
Figure 3.28: Extra transmitter energy required over ideal orthogonal
pulses when using the approximated matched filter detector with the approximated pulses.
117
3.7 Conclusion
In this chapter it is suggested that a low complexity transmitter, such
as PPM, is beneficial for low power transmitters. It has been shown
that the quiescent power of a PPM transmitter determines the optimum number of symbols, which minimises the overall power consumption of the transmitter.
Typically block coding circuits require a few hundred µW of power
so would be useful addition to dramatically improve performance of
low power transmitter architectures. This allows the specification of
a raw BER of 10−4 ; using block coding with L = 32 will improve the
error rate to 10−7 .
The spectrum of PPM for a large number of symbols tends towards
the spectrum of the pulse function used. A simulation framework for
finding the BER for an arbitrary pulse using a correlation and approximated matched filter have been described. These approaches are for
AWGN channel, which is only valid for cases where the multipath contributions are small. However, it allows a methodical comparison of
arbitrary pulse shapes.
The ideal truncated Gaussian pulse has been analysed in depth.
The advantage of using a Gaussian pulse over a rectangular or sinusoid pulse is that the bandwidth-time product can be chosen so as to
provide better out of bound attenuation. Approximate expressions for
the attenuation and bandwidth for a truncated Gaussian pulse have
been derived. A comparison between three continuous time approximation methods and the truncated Gaussian pulse have been made.
This showed that the all pole approximation has better out of bound
attenuation than the other pulses. The Padé approximation is superior
in approximating the shape of the Gaussian in both frequency and time
but it leads to higher out of bound attenuation due to the addition of
zeroes in the transfer function.
The simulated BER for the approximated pulse shapes show many
of the pulses can reach or exceed the performance of ideal orthogonal
118
signalling. As the number of time slots is increased, the performance of
the correlation receiver converges to the ideal orthogonal performance.
The effect of implementation errors in the filter, for example due to
mismatch of components, can be analysed by simulating the BER with
the erroneous g(t).
The result of the analysis in this Chapter gives confidence that high
performance PPM transmitters and receivers can be constructed using
approximated Gaussian continuous-time filters. The all pole Gaussian
pulse approximation is used as the pulse prototype for the PPM scheme
described in the following chapter, and the Padé approximation is used
for the analogue Gabor transform described in Chapter 5.
119
120
4 Communication using 2nd
Order TX Elements
4.1 Introduction
The UK Office of Communications has made the 30-37.5 MHz band
available for short range medical implant devices [87]. In this chapter
a transmitter and receiver operating at a centre frequency of 33 MHz
with an out of band attenuation of 30 dB is analysed and implemented.
The transmitter is based upon using 2nd order transmitting elements,
which are naturally available when using inductive coils. It is shown
that using the resonant properties of an inductive coil provides much
better coupling of DC power to the magnetic field than direct modulation of the inductor with a current source. The All Pole Gaussian
pulse approximation shown in the previous chapter is used as the prototype pulse. This pulse can be decomposed into the sum of shifted
and scaled 2nd order systems. Each of these systems is then implemented using a separate transmitting element. If the same number of
receiving elements as transmitting elements are used at the receiver
then an implicit matched filter is implemented.
The N=2 All Pole Gaussian baseband pulse is used as the prototype function to implement a coherent modulation scheme. The out
of phase pulse is generated by delaying the impulse response to the
transmitting elements. BER results are presented which show that the
variation in centre frequency of the approximated pulse has the most
significant effect on performance. To mitigate the error due to centre
frequency drift, an estimation technique for use at the receiver is pro-
121
posed. Simulated results show that using a preamble overhead of 1 %
allows sufficiently accurate estimation of the centre frequency. Thus
the transmitter requires no on-line tuning, which makes the transmitter topology simple.
A discrete circuit implementation of the transmitter and front end
of the receiver are described. Measured results are presented.
4.2 Making Use of Antenna Topology
Many antennas can be modelled as lumped element circuits. In particular, a solenoid coil can be modelled as an inductor in parallel with a
capacitor. This immediately gives rise to a 2nd order circuit element,
which can be used as part of the pulse generating filter. Figure 4.1
shows an example of a 2nd order transmitting element. Assuming that
RX > RS >> RL the 2nd order response of the resonant circuit is
approximately given by:
IT X
ω02
= 2
IIN
s + s ωQ0 + ω02
Q ≈ ω0 CRX
r
1
ω0 ≈
.
LC
(4.1)
(4.2)
(4.3)
These equations show that Q and f0 can be tuned independently to
obtain the required 2nd order response. The amplitude of the pulse
can be controlled by adjusting VIN or RS and negative pulses can be
generated by reversing the orientation of the coil. Experimental results
of coil characterisation are described in Appendix G.
4.2.1 Efficiency of the 2nd Order Element
Figure 4.2 shows two circuits that can be used as inductive transmitters. The current sources can be implemented using a transistor. In
122
RS IIN
ITX
L
RX
C
VIN
RL
Impulse
Figure 4.1: Circuit diagram showing a 2nd Order transmitting element.
The magnetic field in the inductor follows a damped sinusoidal response.
the direct modulation approach all of the current flowing through the
inductor must also flow through the current source. In contrast the
pulse based modulation scheme applies an impulse, hI (t), to a resonant tank, which produces an exponentially decaying magnetic field,
only the impulse current is provided by the source.
To make a comparison between the two cases it is ensured that the
same rms current flows through the inductors in both schemes. In the
pulse based scheme the current is an exponentially decaying sinusoid.
In the direct modulation scheme the current could be a sinusoid, i.e.
an OOK modulator.
The RMS current flow through the inductor is defined as:
IRMS =
s
1
T
Z
T
iL (t)2 dt
(4.4)
0
where iL (t) is the current in the inductor and T is the length of the
transmitted pulse.
The 2nd order decaying current to be generated in the pulse based
modulation circuit due to an impulse is:
123
Direct Modulation
Pulse Based Modulation
Vdd
Vdd
iL(t)
R
L
L
C
hI(t)
iL(t)
Figure 4.2: Using the inductor as a transmitter in a direct modulation
approach and a pulse based approach. The direct modulation is typically found in low power OOK implementations
where the current through the inductor is determined by a
oscillator.
ω0
iL (t) = e− 2Q t sin ω0 t.
(4.5)
Take the transmitted pulse length to be the time that it takes the
envelope of (4.5) to decay to 95 % of its starting value:
T ≈
6Q
.
ω0
(4.6)
The rms current flowing in the inductor is then:
iRMS
≈
L
1
3.5
(4.7)
The rms power consumption of the direct modulation approach is
thus:
Vdd
(DM)
PRMS ≈
.
(4.8)
3.5
For an impulse whose rise times are given by τ = 0.1TI the length
of the impulse can be approximated by, see Appendix F:
TI ≈
2π
.
3ω0
(4.9)
The amplitude of the impulse function can be found by considering
124
that the Laplace transform of (4.5) must have a DC value of 1/ω0 , see
Appendix H. The amplitude of the impulse function required is:
a≈
1
≈ 0.531.
ω0 0.9TI
(4.10)
Therefore the rms power consumption of the pulse based modulation
scheme is given by:
(PM)
PRMS
≈ Vdd a
r
TI
1
√ .
≈
T
3.2 Q
(4.11)
The ratio of power dissipation of the pulse based modulation to the
direct modulation is:
(P M )
PRMS
(DM )
PRMS
1.1
≈√ .
Q
(4.12)
Therefore pulse based modulation requires less power to generate the
same rms current in the inductor when the Q of the resonant tank is
greater than 1.2. This result is important as it shows the advantage in
terms of power consumption when using high Q 2nd order transmitting
elements. In the following sections the use of 2nd Order transmitting
elements to produce shaped pulses is considered.
4.3 Pulse Decomposition
The pulse shape of a 2nd order transmitter element excited by an impulse is a damped exponential. A single element is capable of implementing the All Pole Gaussian approximation with an order of N = 1.
The 1st order approximation to a Gaussian is far from ideal, as shown
in Chapter 3. To implement higher order Gaussian pulses the transfer function can be decomposed into the sum of 2nd order transfer
functions. These can then be implemented using separate 2nd order
transmitting elements. The magnetic fields produced in the inductors
then sum in free space to give the required pulse approximation. In this
125
Section f0 [MHz] Q Delay [nS] Amplitude
1
33.63
41
0
-0.52
2
33.00
28
1.5
1
3
32.40
40
2.9
-0.54
Table 4.1: Characteristics for the 3rd order baseband minimal pulse.
αT 2 = 4.6, T = 1 µs and f0 = 33 MHz.
section a minimal and maximal decomposition of the transfer function
will be considered.
4.3.1 Minimal Decomposition
For hardware minimisation it is important to consider the minimal
representation where the least number of components is required. In
this case each bandpass transfer function can be decomposed into a
sum of N 2nd order transfer functions each with a transmission zero:
G(s) =
aN s + b2N
a1 s + b21
+
·
·
·
+
.
s2 + c1 s + d1
s2 + CN s + dN
(4.13)
In Appendix H it is shown how an approximation to each of these
sections can be constructed by using the delayed and scaled impulse
response of a standard 2nd order low pass filter. The characteristics
of each 2nd order element for a 3rd order All Pole Approximation to
the Gaussian pulse with αT 2 = 4.6, T = 1 µs and f0 = 33 MHz is
shown in table 4.1. Figure 4.3 shows the individual impulse responses
that sum to form the desired pulse. Due to the use of the delays in
approximating the transfer function zeros there is an impulse at t = 0.
This has the effect of raising the out of bound attenuation, see figure
4.4.
4.3.2 Maximal Decomposition
For the maximal representation split (4.13) into its constituent low pass
and bandpass filters, thus creating 2N 2nd order filters. The bandpass
filter can be approximated by applying an impulse delayed by 4f10 , i.e.
126
0.5
0.4
0
0.3
−0.5
0
0.2
0.5
1
1.5
2
1
0.1
0.5
0
0
−0.5
−1
0
−0.1
0.5
1
1.5
2
0.5
−0.2
0
−0.3
−0.5
0
−0.4
0.5
1
Time [µS]
1.5
2
0
0.5
1
1.5
2
Time [µS]
Figure 4.3: Sum of 2nd order impulse responses to form the desired
pulse using a minimal decomposition. αT 2 = 4.6, T = 1 µs
and f0 = 33 Mhz. The impulse at t = 0 is due to the delay
used to approximate the zeroes of the 2nd order transfer
functions. The dashed line is the envelope of the All Pole
Gaussian approximation.
127
0
−10
N=1
−20
Attenuation [dB]
−30
N=2
N=3 min
−40
−50
−60
Ideal Truncated
−70
−80
−90
25
N=3 max
30
35
40
45
Frequency [MHz]
Figure 4.4: Spectral comparison of the minimally decomposed pulses.
Due to the approximation of the transfer function zeroes
the 3rd order minimal decomposition has worse out of band
attenuation than the maximum decomposition.
approximating orthogonality. With this scheme there is no undesired
impulse at the beginning of the pulse. Also only a single delay is
required. However, generally twice as many 2nd order transmitting
elements are required.
For the 3rd order representation one of the amplitudes required is
an order of magnitude lower than the others so can be neglected, thus
only five 2nd order sections are required. The characteristics for the 3rd
order maximal pulse with αT 2 = 4.6, T = 1 µs and f0 = 33 MHz are
shown in table 4.2. The resulting sum of the 2nd order filter impulse
responses is shown in figure 4.5. The frequency response is shown in
figure 4.4.
The 2nd order baseband approximation is a special case where the
minimal and maximal representations are the same. The 2nd order
function also requires no delays and the amplitudes are of the same
magnitude. This will make implementation of the 2nd order approximation much easier, and less suspectable to component tolerances.
128
Section f0 [MHz]
1
33.63
2
33.63
3
33.00
4
32.38
5
32.38
Q Delay [nS] Amplitude
41
0
0.15
41
7.58
0.5
28
7.58
-1
40
0
-0.15
40
7.58
0.5
Table 4.2: Characteristics for the 3rd order baseband maximal pulse.
αT 2 = 4.6, T = 1 µs and f0 = 33 MHz.
0.4
0.3
Amplitude
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
0.5
1
1.5
2
Time [µS]
Figure 4.5: Sum of 2nd order impulse responses to form the desired
pulse using a maximal decomposition. αT 2 = 4.6, T = 1 µs
and f0 = 33 Mhz. There is no undesired impulse at t = 0
for the maximal representation. The dashed line is the
envelope of the All Pole Gaussian approximation.
129
Section f0 [MHz] Q Delay [nS] Amplitude
1
33.32
45
0
-1
2
32.69
44
0
1
Table 4.3: Characteristics for the N=2 pulse with αT 2 = 4.6, T = 1 µs
and f0 = 33 Mhz
The characteristics of each transmitter element for the 2nd order baseband approximation are shown in table 4.3. The frequency response is
shown in figure 4.4.
4.3.3 TX Element Coupling
As several transmitting elements are required there will be mutual
coupling between all the transmitting inductors. This will affect the
transmitted pulse shape. Appendix I describes the effect of coupling
on the pulse shape when using two coupled transmitting elements.
This analysis shows that as the coupling is increased it becomes difficult to compensate for the change in pole positions. As coupling is
increased the bandwidth of the pulse increases and the Q decreases.
The coupling effect can be mitigated by ensuring that the transmitting
antennas are far enough apart so that the mutual coupling is small.
From experimental coupling results, see Appendix G, a distance of 20
mm between the centres of the TX coils gives a coupling constant of
1.8 × 10−3 . This value of coupling is small enough so the transmitting
elements can be assumed to be independent.
4.4 2nd Order Receiving Element
The use of a parallel tuned tank at the receiver is advantageous as it
allows a matched filter to be implemented and the higher the Q the
better the receiver sensitivity. In order to use a parallel tank as a
receiving element a high impedance amplifier is required to couple the
coil to the measuring circuit. The high impedance is required so that
130
k11
GTX1
Y
k22
GTX2
GRX2
k2N
kN2
k1N
kN1
GTXN
GRX1
k21
k12
kNN
GRXN
Figure 4.6: Diagram indicating the coupling between each transmitting
element and each receiving element.
the Q of the tank is not affected.
4.4.1 Implicit Matched Filter
When there are N transmitting and N receiving elements then an approximated matched filter receiver can be implemented. Figure 4.6
shows a transmitter and receiver consisting of N transmitting elements
and N receiving elements.
In the frequency domain the sum of the individually transmitted
pulses is defined as:
h
GTX = GT X1 GT X2
 
1

i
1
(N ×1)

.
· · · GT XN 
 ..  = ĜTX 1
.
(4.14)
1
In a similar way the transfer function of the receiver can be defined as:
GRX = ĜRX 1(N ×1) .
(4.15)
Each receiving element will see a contribution from each of the transmitting elements, i.e. N separate paths. The coupling constant of the
ij th path is given by kij . Defining the received signal as the sum of the
outputs of the individual receiver elements gives:
131

k11

 k21
Y = ĜRX 
 ..
 .
k12 · · ·
k22 · · ·
...
kN 1 · · ·

k1N

iT
k2N  h

Ĝ
.
TX
.. 
. 
(4.16)
kN N
For the approximated matched filter the received signal will be:
h
iT
Y = GRX [GTX ]T = ĜRX 1(N ×N ) ĜTX .
(4.17)
Therefore, if all the coupling constants between the transmitter and
receiver elements are equal then the matched filter is explicitly implemented at the receiver. For non equal coupling constants the matched
filter approximation does not hold and this will affect the bit error
performance.
4.4.2 Coupling Matrix for N=2
Figure 4.7 shows the coupling paths when using two solenoid transmit
and receive antennas and when the receiver is aligned with the the
transmitter. In practice the user may orientate the receiver at any
angle. For the purposes of simplicity it is assumed that the transmit
antenna is aligned. This will provide a reasonable estimate for the
coupling matrix which can be used to evaluate BER performance.
For identical transmit and receive solenoids the coupling constant is
given by:
lR2 sin θtx sin θrx
,
(4.18)
2d3
where l is the length of the solenoid, R is the radius of the solenoid,
θtx and θrx describe the coil orientation and d is the distance between
the two solenoids. The coupling matrix is then:
k=
132
atx
TX
θtx1
θtx2
d
d11
d22
d21
d12
RX
Figure 4.7: Coupling paths when using two transmit and receive antennas. The coupling matrix for an N=2 transmitter can
be derived from this simple model.
2
k=

lR 
2d3
1+
1
a2tx
d2
−1
a2tx
d2
1+
1
−1 
.
(4.19)
It is clear that in the limit of a large distance between the transmitter
and receiver the matched filter approximation achieved.
4.4.3 Transmission Distance
Due to the impulse applied to the 2nd order transmitting element the
peak inductor current in the transmit inductor is:
(Peak)
IT X
=
VDD
ω0 (T − τ ) .
RS
(4.20)
The peak induced voltage in the receiver element will be:
(Peak)
Vind
(Peak)
= ω0 M IT X
(4.21)
where M is the mutual inductance given by:
133
Distance
[mm]
10
15
20
50
100
200
500
1000
G = 0 dB
VRX [mV ]
564
166
70
4.6
0.56
0.08
0.004
0.0006
G = 10 dB
ADC [bits]
8
7
3
-
VRX [mV ]
524
222
14.2
1.78
0.22
0.01
0.002
G = 60 dB
ADC [bits]
8
4
1
-
VRX [mV ]
560
80
4
0.6
ADC [bits]
6
2
-
Table 4.4: Transmission distance using matched TX and RX 2nd order
elements.
M =k
p
LT X LRX .
(4.22)
The peak voltage across the receiver coil is thus:
(Peak)
VRX
= ω0 QRX k
p
LT X LRX
VDD 3π
.
RS 5
(4.23)
Table 4.4 shows the received voltage versus transmission distance
16×10−9
when VRDD
, LT X = LRX = 200 nH and QRX =
=
10
mA,
k
=
d3
S
45. The resolution in bits at a 10 bit ADC with a 1 Vpp input range
is also shown with and without amplification. It is evident from table
4.4 that a gain of at least 10 dB is required in order to transmit data
up to 50 mm. For longer distances a gain of 60 dB could be realised,
which would take the detection range up to 500 mm.
4.4.4 Attenuation Map
The transmit antennas have to be placed a certain distance apart to
reduce the coupling between them. The consequence of this is that
the sum of the magnetic fields will be different at each point on the
transmitting plane. Therefore, the pulse shape seen at each point will
vary. Figure 4.8 shows the geometry of the problem with two transmit
antennas. Here only a 2-D plane is shown, however, the analysis could
easily be extended to three dimensions if required.
134
atx
H1
y
H2
θtx1
dtx1
θ
θtx2
d
dtx2
H
x
Figure 4.8: Geometry of the transmit antennas. The transmit antennas
are placed far enough apart such that the effect of coupling
between the antennas can be ignored. The magnetic field
at an arbitrary point can be found if θ, d, atx and the
magnetic fields found in the centre of the two solenoids are
known.
135
The magnetic field at an arbitrary point can be written as:
H = H1
sin θtx2
sin θtx1
+ H2 3 .
3
dtx1
dtx2
(4.24)
The path lengths are given by:
dtx1
dtx2
r
atx 2
=
+ d2 sin θ2
d cos θ −
2
r
atx 2
+ d2 sin θ2
d cos θ +
=
2
(4.25)
(4.26)
and the angles of the paths are given by:
d sin θ
dtx1
d sin θ
.
=
dtx2
sin θtx1 =
(4.27)
sin θtx2
(4.28)
Figure 4.9 shows the out of band attenuation when the transmit
antennas are 20 mm apart. The magnitude of the magnetic field has
been set constant on the θ = 0 axis so that the comparative attenuation
at any transmit power can be observed. Figure 4.9 shows that the
attenuation of 31 dB is close to the expected attenuation at distances
far from the transmit antennas. To either side of the centre line the
attenuation is smaller than the ideal because the contribution from
each antenna is no longer equal.
4.5 Demodulation
A block diagram of the transmitter and receiver are shown in figure
4.10. This receiver samples the output of both receive elements and
sums them to form the approximated matched filter output.
136
0
27
40
32 33
31
35
27
28
31
30
30
2278 29
26
0.05
26
24
28
24
26
29 27
30
31
29
33
32
24
28 29
3130
35
24
40
0.1
30
31
29
29
30
28
28
31
0.2
29
27
27
28
26
0.15
28
y [m]
29
26
0.25
29
0.3
30
30
29
31
31
0.35
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x [m]
Figure 4.9: Out of band attenuation when the transmit antennas are
20mm apart. The approximate pulse is centred at 33 MHz
with T = 1 µS and αT 2 = 4.6. The position of the two
transmit antennas are shown by the two solid points at the
top of the graph.
137
138
Vdd
Vdd
Gain
Gain
A/D
A/D
lpf
lpf
Carrier f0 Estimation
Symbol Synchronisation
Q
I
Digital Processing
nT
nT
Q
I
ML
Estimate
^I
n
Figure 4.10: Block diagram of the receiver for the pulse based communications link. The digital processing estimates
the carrier frequency of the transmitted pulse from a preamble sequence.
Generator
Impulse
Symbol Symbol
Clock Sequencer
µP
In
4.5.1 Coherent Detection
For coherent detection it is important that the receiver knows the frequency and phase of the transmitted carrier. The carrier frequency
with a frequency offset maybe written as:
gc (t) = sin (2π(f0 + δf0 )t) = sin (2πf0 t + 2πδf0 t)
(4.29)
which shows that the error in frequency can be considered as a phase
error term. Considering a centre frequency of 33 MHz and a variation
in frequency of 1 % the phase error at the centre of a 1 µS pulse will
be almost 60o . Unless an estimate of the frequency at the receiver is
made then using orthogonal signals without tuning the transmitter will
require considerably more energy than the ideal receiver.
4.5.2 Sub Sampling
The receiver ADCs operate using subsampling. The sampling frequency is 30 MHz so the centre frequency of the resulting pulses is
around 3 MHz.
4.5.3 Non Coherent Detection
The receiver is capable of implementing non coherent detection where
the frequency and phase of the carrier do not need to be estimated. The
1
rate of non coherent detection is reduced by LT
because the orthogonal
symbol cannot be used.
4.6 BER Performance
In this section the BER performance of the order N=2 transmitter is
analysed using the narrow band receiver with coherent demodulation.
The simulation framework for the matched filter, see Chapter 3, is used
to obtain an idea of BER performance.
139
Figure 4.11 shows the effect of moving the receiver closer to the
transmitter. At a transmit distance of 100 mm the difference in performance compared to the best achievable with k = 1(2×2) is small.
The noncoherent BER for L=4 is plotted to show that using coherent
detection has a definite advantage in terms of Eb /N0 when the transmit distance is greater than 25 mm. The sensitivity of the BER to
variations in Q is very small. Variations in Q of over 10 % do not
cause any significant variation in the BER performance.
The result of a BER test when changing only the centre frequency
of the pulse is shown in figure 4.12. The sensitivity of the BER due
to the difference in the centre frequency is large. Therefore, a good
frequency estimation scheme at the receiver is required. It is clear that
the centre must be estimated to within 0.1 % in order to provide a
performance advantage over using ’non-coherent’ detection. The main
reason for the large error is due to phase accumulation as indicated by
(4.29).
4.6.1 Accuracy of Estimation
The procedure for estimating frequency, phase and symbol offset from
the preamble data is outlined in table 4.5. Standard techniques are
used which can be implemented in a digital processor at the receiver.
Figure 4.13 shows the bias and variance for frequency, phase and
offset estimates of the subsampled preamble packet with random sampling phase. This figure shows that the estimation method is suitable
for obtaining a frequency estimate well within 1 % and a phase estimate withing a few degrees. The variance of the estimates could be
reduced further by increasing the length of the preamble.
4.6.2 Preamble Length
As shown in figure 4.12, the frequency estimate for a 33 MHz carrier
needs to be less than 0.1 %. This equates to a phase difference of 12o
per time slot. For an equivalent phase shift on the sub sampled pulse
140
0
10
L=4 (8) exact
L=4 ’noncoherent’
L=4 (8) d = 25 mm
L=4 (8) d = 50 mm
L=4 (8) d = 100 mm
L=4 (8) d = ∞
−1
10
−2
10
−3
BER
10
−4
10
−5
10
−6
10
−7
10
−8
10
2
3
4
5
6
7
E /N (dB)
b
8
9
10
11
12
0
Figure 4.11: Effect of transmit distance on BER performance for an
N=2 pulse with 4 time slots and coherent demodulation;
total of 8 symbols. The distance between the transmit
antennas is 20 mm. The performance decreases as the
transmit distance is decreased because the accuracy of
matched filter approximation decreases due to k 6= 1(2×2) .
Step
1 - Offset estimation
Description
Find the baseband preamble sequence by finding the
square of yp (t) and low pass filtering the result. Cross
correlate the baseband preamble sequence with the
known sequence to find the offset.
2 - Carrier Frequency Obtain a frequency estimate of each preamble pulse by
estimation
computing the periodogram of each pulse. Estimate
the centre frequency from the peak of the periodogram.
Average the resulting estimates to find the centre frequency.
3 - Carrier Phase esti- Using the frequency estimates for each pulse from step
mation
2, find the most likely phase of the carrier for each pulse.
4 - Phase Accumula- Compute the average difference in phase between the
tion
estimated phases of the pulses in step 3. Use this to
estimate the symbol synchronisation
Table 4.5: Preamble frequency, phase and offset estimation algorithm.
The received preamble sequence is yp (t)
141
0
10
−1
10
−2
10
−3
BER
10
−4
10
L=4 (8) df = 0 %
L=4 (8) exact
−5
10
L=4 (8) df = 0.1%
L=4 (8) df = 0.2%
L=4 (8) df = 0.5%
−6
10
L=4 ’noncoherent’
−7
10
−8
10
0
2
4
6
8
10
12
E /N (dB)
b
0
Figure 4.12: Effect of centre frequency shift on BER performance for
an N=2 pulse with 4 time slots and orthogonal signalling;
total of 8 symbols. The plot clearly shows the performance
penalty when the centre frequency of the pulse is not estimated closely enough. A percentage change in f0 of 0.2 %
makes the performance worse than the non-coherent L=4
scheme.
142
0.8
0.05
0.7
Frequency Std [%]
Frequency [%]
0.1
0
−0.05
−0.1
−0.15
6
8
10
12
0.6
0.5
0.4
6
8
EbNo [dB]
10
12
10
12
10
12
EbNo [dB]
2
10
Phase Std [deg]
Phase [deg]
9
1
0
−1
8
7
6
5
−2
6
8
10
4
6
12
8
EbNo [dB]
4
35
2
30
Offset Std [ns]
Offset [ns]
EbNo [dB]
0
−2
−4
6
8
10
EbNo [dB]
12
25
20
15
6
8
EbNo [dB]
Figure 4.13: Mean and variance of frequency, phase and symbol offset
estimation. 100 runs at each Eb /N0 value. The parameters that are varied are f0 (T X) with standard deviation
(sd) 1 %; f0 (RX) with sd 1 %; sampling phase between 0
and 2π; transmit delay between 0 and 4 µs. The preamble
length is 32 time slots.
143
the frequency needs to be estimated to within 1 %. The preamble
sequence, when the transmitter and receiver clocks are synchronised,
has been chosen to be 4h empty time slots followed
by 32 time slots
i
with the pulse sequence gI (t) 0 · · · gI (t) 0 , where gI (t) is the in
phase pulse. The preamble ends with 4 empty slots. Using a sequence
of the same pulses allows the pulses to be averaged, thus reducing the
noise when carrying out the estimation. 2048 time slots containing
data follow the preamble, a 2 % overhead.
When the clocks are not synchronised, as will typically be the case
a preamble length of 64 was chosen in order to improve the estimates.
For both cases a moving average of the last twenty preamble estimates
is used to smooth the estimates which prevents catastrophic packet
failures. For the non synchronised clocks 1024 time slots containing
data follow the preamble, a 7 % overhead.
4.7 Tuning
The analysis in the previous sections shows that the accuracy of the
centre frequency of each 2nd order element plays a significant role in
BER performance. Using a preamble approach with an estimation
algorithm relaxes the requirement of ensuring the coils are perfectly
tuned. The main reason for centre frequency drift will be due to environmental factors such as temperature. Appendix J shows the variation of centre frequency and Q over temperature for each 2nd order
element. The percentage change over a temperature of 55o for the centre frequency is ±0.4% and for Q is ±0.85%. These values indicate
that the estimation algorithm would be capable of estimating the centre frequency in the presence of temperature change. Therefore, the
2nd order coils could be tuned once at start up in order to take care of
drift due to ageing. 2nd order circuit elements are popular sub circuits
for building higher order filters, so there is a large amount of literature on tuning circuits for 2nd order filter elements. A survey of these
techniques can be found in chapter 7 of [88]. This survey outlines a
144
digital frequency tuning method which requires four clock signals. This
tuning method is able to tune the centre frequency to within 0.3 %.
The problem with the standard tuning techniques described in [88]
is that they rely on being able to use the filter on an input signal.
This is fine for tuning the receive coils but in the case of the 2nd order
transmitting element, figure 4.1, the only input available is the impulse
response. The output of the filter to the impulse can be measured by
using a sense coil located close to the transmitting inductor. Using a
sense coil with lower inductance and lower Q than the transmitting coil
reduces the effect of pole shifting caused by excess coupling. Appendix
K outlines a method for tuning the 2nd order Transmitting element
using the impulse response and a single sinusoid. This shows that a
theoretical error of 0.02% in the tuning of the centre frequency of a
coil with a Q of 40 is obtainable.
4.8 Circuit Design
Having determined some key performance measures in the previous
sections an implementation of a transmitting circuit is presented. This
circuit was designed using readily available discrete components. The
top level design of the transmitter circuit is shown in figure 4.14. Detailed schematics for both the transmitter and receiver circuits can be
found in Appendix L.
The PULSE and IQ SELECT signals are generated as part of the
measurement setup, see Section 4.9. For every transmitted symbol
there are L time slots and for each time slot either an I or Q pulse can
be transmitted. Only a single pulse is transmitted per symbol. The
most significant bit of the incoming symbol data determines whether
an I or Q pulse is transmitted. The rest of the bits determine which
time slot the pulse is to be transmitted in.
In the following sections a description of the main circuit blocks is
provided. The supply voltage to the digital gates was chosen to be 2.8
V and the supply voltage to the TX elements was chosen to be 5 V.
145
I-Q SELECT
I-Q DELAY
BUFFER
PULSE
GENERATOR
PULSE
TX
ELEMENT
TX
ELEMENT
IQ SELECT
Figure 4.14: Top level schematic of the transmitter.
4.8.1 I-Q Delay Circuit
The I-Q Delay circuit is required in order to select whether the transmitted pulse is in phase or quadrature phase. Figure 4.15 shows the
circuit diagram for the I-Q delay circuit. The pulse delay is the difference between the time constants of each path. Coarse control of
the width was made using the variable capacitor. Fine control can
be made by adjusting the voltage supply to an inverter. This also allows automatic tuning of the pulse width. The inverters used are the
NC7SV04, which have a quiescent current of less than 1 µA. The I-Q
Select switch is a SN74LVC2G66, which has a quiescent current of less
than 10 µA.
4.8.2 Pulse Generation Circuit
Figure 4.16 shows an implementation of a digital pulse generator. This
uses the same operating principles as the I-Q delay circuit. The pulse
width is the difference in the time constant between the two paths. The
NAND gate used is the 74LVC1G00, which has a quiescent current of
less than 10 µA.
The buffers between the pulse generation circuit and the TX elements are in the same package to minimise the mismatch in the propagation delay, thus ensuring that the impulse arrives at each TX element
at the same time. The 74LVC2G04 dual gate inverter was used for this
purpose. This has a quiescent supply of less than 10 µA.
The TX element is shown in figure 4.1. RX is the series combination
146
500
I-Delay
10pF
Vdd (Fine Control)
500
Q-Delay
6-30pF
Figure 4.15: Circuit diagram of the digital pulse delay circuit. Coarse
pulse width control is obtained using a variable capacitor. Fine pulse width control is obtained by adjusting the
supply voltage to an inverter.
of a fixed and variable resistor, used to adjust the Q of the resonant
tank. C is the capacitance of the coil, a 68 pF fixed capacitor and a
6-30 pF variable capacitor in parallel. A varactor diode can be added
to enable voltage controlled tuning. RS is 500 ohms to set the impulse height to 10 mA. The transistor switch used is a BFG410W wide
bandwidth NPN transistor.
4.9 Measured Results
Figures 4.17 and 4.18 shows photographs of the transmitter and receiver modules. For this prototype tuning circuits were not implemented. The centre frequencies and Q were set once using a spectrum
analyser to determine the values. Despite the lack of tuning circuits
the resulting measured pulse is close to the desired response. In order
to measure the BER performance an FPGA development board with
two ADCs was used. The measurement setup for BER performance is
shown in figure 4.19. Separate clock domains for the transmitter and
receiver circuit were used so that the transmitter and receiver clocks
147
Width Control
Vdd
500
6-30p
500
4.7p
Figure 4.16: Circuit diagram of the digital pulse generator. Coarse
pulse width control is obtained using a variable capacitor. Fine pulse width control is obtained by adjusting the
supply voltage to an inverter.
could be chosen to be synchronised (derived from the same clock) or
unsynchronised (derived from independent clocks). The receiver front
end consists of two 2nd order elements followed by a 10 dB FET amplifier.
4.9.1 Transmitted Pulse
The transmitted pulse in the frequency domain was measured using
an HP 11940A spectrum analyser and an HP E4403B magnetic field
probe. The measured spectrum of the resulting sum of the 2nd order
responses is shown in figure 4.20. The spectral contribution from each
transmitting element, measured on the centreline at a distance of 5
mm, are also shown in figure 4.20. The noise floor of the instrument
is at -69.4 dBm, which prevents accurate measurement of the out of
bound attenuation. Extrapolation of the spectrum indicates that the
attenuation at 30 MHz is around 30 dB, as expected.
The time domain measurements of the individual transmitting elements are shown in figure 4.21. The 33.32 MHz pulse has a higher Q
148
Figure 4.17: Photograph of the transmitter module. The centres of the
two transmit elements are separated by 20 mm.
Figure 4.18: Photograph of the receiver module. A single FET stage
has been used to buffer the tuned coil from the ADC input.
The FET stage has a gain of 10 dB.
149
Host PC
USB
MATLAB
Control
40 MHz
RAM
Symbol Data
TX
2 MHz
RX
DDR SDRAM
128 MBytes
30 MHz
ADC0
ADC1
ch1
ch2
FPGA
(3 clock domains)
Pulse
Orthog
Transmitter
Receiver
Figure 4.19: Block diagram of measurement setup used to evaluate
BER performance.
150
than desired but the 32.69 MHz pulse shows a good match to the ideal
envelope. The resulting sum of these pulses shows a very good match
to the ideal envelope. The measured pulse shows slightly more energy
spread into the adjacent time slot. Therefore, inter-symbol interference
will be increased resulting in a BER performance worse than expected.
151
152
(b) Contribution of the 33.32 MHz transmitting element.
(c) Contribution of the 32.69 MHz transmitting element.
Figure 4.20: Measured spectrum of the transmitter magnetic field on the centreline of the inductors at a distance of
5 mm. The pulse repetition rate used was 200 kHz.
(a) Magnetic field due to both transmitting
elements.
153
−0.03
0
−0.02
−0.01
0
0.01
0.02
0.5
1
1.5
Time [µS]
2
2.5
Measured
Ideal Envelope
3
−0.04
0
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.5
1
1.5
Time [µS]
2
2.5
Measured
Ideal Envelope
3
(b) Contribution of the 33.32 MHz transmitting element.
Amplitude [V]
0.5
1
1.5
Time [µS]
2
2.5
Measured
Ideal Envelope
3
(c) Contribution of the 32.69 MHz transmitting element.
−0.04
0
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
Figure 4.21: Measured time domain pulses of the transmitter magnetic field, averaged over 128 pulses.
(a) Resulting sum of the transmitting elements.
Amplitude [V]
0.03
Amplitude [V]
4.9.2 Orthogonal Pulse
The I-Q delay used to produce the orthogonal pulse, was tuned by
hand. A high speed oscilloscope was used to measure the pulse delay.
The measured orthogonal pulses at 33 MHz are shown in figure 4.22.
The cross correlation of these two pulses was measured at the receiver
following the sub sampling operation. The sub-sampled measured orthogonal pulses are shown in figure 4.23. The cross correlation of these
two pulses is:
CXX
#
0.997 −0.0359
.
=
−0.0359 1.0030
"
(4.30)
Equation 4.30 shows that the orthogonality of the pulses is close to the
ideal identity matrix.
4.9.3 Receiver Demodulation
Figure 4.24 shows a train of pulses measured at the receiver and also
shows the result of the IQ demodulation scheme. The preamble can
be seen at the beginning of the pulse train, which is used to estimate
the centre frequency and phase of the carrier.
154
0.03
I measured
Q measured
Amplitude [V]
0.02
0.01
0
−0.01
−0.02
−0.03
0
0.1
0.2
0.3
0.4
0.5
Time [µS]
Figure 4.22: Measured time domain orthogonal pulses at 33 MHz, averaged over 128 pulses to increase SNR.
20
Q pulse
15
I pulse
Voltage [mV]
10
5
0
−5
−10
−15
−20
0
0.5
1
1.5
2
2.5
3
Time [µS]
Figure 4.23: Measured time domain orthogonal pulses after sub sampling operation.
155
Voltage [mV]
0
20
40
50
50
Time [µS]
Demodulated Data
Time [µS]
Received Data
100
100
Q Channel
I Channel
150
150
Figure 4.24: A train of pulses measured at the receiver including the preamble sequence. The result of the IQ
demodulation is also shown.
−5
0
0
5
10
15
20
−40
0
−20
Amplitude
156
4.9.4 Power Consumption
Using the same notation for PPM power consumption as described in
Chapter 3, the overall energy per bit of the transmitter for coherent
detection using two orthogonal pulse shapes is given by:
DC
Ebit
=
T
[PP U LSE + LPQU IES + PT X] .
log2 (2L)
(4.31)
The quiescent supply to the circuit was measured as 3.6µW . PP U LSE
and PT X are measured when L = 1, which corresponds to a pulse
repetition rate of 1 MHz. The dynamic power measurements were:
PP U LSE = 2.8 mW and PT X = 3.6 mW. The optimal number of time
slots that result in minimum transmitter energy is given by (see Chapter 3):
Lopt =
PP U LSE + PT X
+ 5.
3PQU IES
(4.32)
Thus the optimum number of time slots is Lopt = 512. The measured
and modelled (4.31) power consumption for various values of L are
shown in figure 4.25. As the minimum lies on a shallow curve the
reduction in the rate at values of L > 16 shows only a small improvement in the energy per bit. If non coherent detection were used the
energy per bit would be the same but the rate would be reduced. For
example, for two time slots the non coherent rate would be 500 kbps
compared with 1 Mbps for the coherent case. In the next section the
BER performance of the system for L < 16 is presented.
4.9.5 Measured BER Performance
To measure the BER performance 100,000 bits were transmitted and
then demodulated for each test. Figures 4.26 and 4.27 show the BER
performance for coherent detection. For a target BER of 10−3 the
maximum transmission distance when the TX and RX clocks are synchronised is 60 mm. For non synchronised clocks the maximum transmission distance is 50 mm. Figure 4.27 clearly shows the advantage
157
3.5
Modelled Energy
Measured Energy
1 Mbps
3
0.75 Mbps
2
E
bit
[nJ/bit]
2.5
0.5 Mbps
1.5
0.3125 Mbps
0.1875 Mbps
1
0.5
1
0.10938 Mbps
2
3
4
5
6
7
8
9
10
log (L)
2
Figure 4.25: Modelled and measured power consumption of the transmitter. The model allows extrapolation of the measured
results to find the minimum transmitter energy.
158
of increasing the number of symbols in order to reduce the required
transmission energy. In the high SNR regime the BER performance
levels out. This effect can be attributed to the fact that the matched
filter approximation is worse the closer the receiver is to the transmitter, (4.19). The ideal BER performance for two time slots (four
symbols) shows that an extra 10 dB of transmit power is required by
the fabricated transmitter when using synchronised clocks. This is
larger than predicted by the previous theoretical BER analysis. There
is a performance penalty because the circuit does not use continuous
tuning methods and the accuracy of the initial tuning is only within
1 %. When using non synchronised clocks an extra 15 dB of transmitter power is required compared to ideal modulation. The extra
5 dB increase in power over using synchronised clocks is due to the
variance in estimating the frequency and phase parameters. By using
longer preamble sequences the variance of the estimates will decrease
and the performance when using non synchronised clocks will improve.
However, this is at the expense of increased overhead.
Figures 4.28 and 4.29 show the BER rate performance when using
non coherent detection. In this case the range can be extended to
70 mm when using synchronised clocks and to 65 mm when using
non synchronous clocks. These results show that using non coherent
detection can achieve a BER with lower energy than the coherent case,
albeit at lower transmission rates. The reason why the non coherent
modulation achieves better results than coherent detection is due to
the performance of the frequency and phase estimator.
In summary, the BER performance of coherent detection is poorer
than non coherent detection, primarily due to the frequency and phase
estimator. The advantage of using coherent detection is that higher
bit error rates can be achieved using the same bandwidth. Using non
synchronised clocks a bit error rate of 10−3 at a rate of 1 Mbps and a
distance of 50 mm is achievable using coherent detection. The maximum rate of non coherent detection is 500 kbps over a distance of 65
mm for a BER of 10−3 .
159
160
BER
40
60
Distance [mm]
50
70
80
L=16 (32)
L=8 (16)
L=4 (8)
L=2 (4)
40
50
60
Distance [mm]
70
80
L=16 (32)
L=8 (16)
L=4 (8)
L=2 (4)
(b) RX and TX clock independent. Preamble overhead is 7 %.
−5
30
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
Figure 4.26: BER performance over distance for orthogonal coherent detection.
(a) RX and TX clocks synchronised. Preamble overhead is 1 %.
−5
30
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
BER
161
0
25
30
35
b
10
15
b
0
E /N
20
25
L=2 (4)
L=4 (8)
L=8 (16)
L=16 (32)
L=2 (4) Ideal
30
Figure 4.27: BER performance versus Eb /N0 for orthogonal coherent detection.
(b) RX and TX clock independent. Preamble overhead is 7 %.
−6
5
20
E /N
−6
5
15
−5
−5
10
−4
−3
−2
−1
0
−4
−3
−2
−1
L=2 (4)
L=4 (8)
L=8 (16)
L=16 (32)
L=2 (4) Ideal
(a) RX and TX clocks synchronised. Preamble overhead is 1 %.
BER
0
BER
162
BER
40
60
Distance [mm]
50
70
L=16
L=8
L=4
L=2
80
40
50
60
Distance [mm]
70
L=16
L=8
L=4
L=2
80
(b) RX and TX clock independent. Preamble overhead is 7 %.
−5
30
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
Figure 4.28: BER performance over distance for non coherent detection.
(a) RX and TX clocks synchronised. Preamble overhead is 1 %.
−5
30
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
BER
163
15
20
E /N
0
25
30
35
b
10
15
20
b
0
E /N
25
30
L=2
L=4
L=8
L=16
L=4 Ideal
35
Figure 4.29: BER performance versus Eb /N0 for non coherent detection.
(b) RX and TX clock independent. Preamble overhead is 7 %.
−5
5
−4.5
−4.5
10
−4
−4
−5
5
−3.5
−3
−2.5
−2
−1.5
−1
−3.5
−3
−2.5
−2
−1.5
L=2
L=4
L=8
L=16
L=4 Ideal
(a) RX and TX clocks synchronised. Preamble overhead is 1 %.
BER
−1
BER
4.10 Comparison with State of the Art
Table 4.6 shows comparisons between the transmitter presented in this
Chapter and some of the latest published inductive transmitters. Inductive circuits for inter-chip communication have not been included
as these are designed for transmission distances in the order of µm.
164
165
b
a
Power
[µW]
6
2300
6750
10,500
3,200
3,200
400
f0
[MHz]
≈ 200
≈ 400
433
400
33
33
33
b
0.3125
0.5
Rate
[Mbps]
0.25
200
0.330
0.8
1
1,300
6,400
Energy
[pJ/bit]
24
12
20,500
13,000
3,200
Distance
[mm]
59
20
50
65
50
10−2.5
10−3
10−3
10−3
10−3
BER
2009 This Work
2009 This Work
2007 [89]
2009 [90]
2007 [91]
2007 [92]
2009 This Work
Year Ref.
Table 4.6: Comparison of some transmitters suitable for short range communication.
ZL70101 commercial device. Limited information available.
Includes receiver power
Antipodal
Antipodal
FSK
Unknown a
PPM coherent
L=2
PPM non coherent L = 2
PPM coherent
L = 16
Modulation
[90] describes a high data rate inductive link suitable for multimedia
applications. [91] uses FSK modulation to transmit neural data from
a body sensor over a 433 MHz inductive link. Direct modulation of
the inductor by a VCO is used generate the transmitted data. [89],
[90] both contain a pulse based transmitter where a bi-polar voltage
impulse is applied to the transmitting inductor. The current through
the inductor is then approximately a triangular pulse. [89], [90] and
this work all apply an impulse to an inductor. However, in this work
the resonant aspects of the antenna are exploited in order to shape the
pulse. [89], [90] apply a voltage across the inductor for the whole duration of the transmitted pulse, which is akin to the direct modulation
approach shown in figure 4.2. Applying a voltage across the inductor
for the length of the pulse helps to reduce the intersymbol interference.
However, in this work the intersymbol interference is reduced by using
the resonant properties to approximate a Guassian pulse shape.
4.10.1 Integration
The power consumption of the discrete circuit presented is significantly
more than the state of the art transmitters shown in table 4.6. The
power consumption of the pulse generator could be reduced significantly if implemented as an integrated circuit. For a digital circuit the
power consumption of each gate is given by [93]:
Pgate = CV 2 f.
(4.33)
Equation (4.33) shows that the supply voltage and load capacitance
play a significant role in the dynamic power consumption. For the
discrete implementation the supply voltage was 2.8 V and the capacitance per gate (load and power dissipation capacitor to model the
internal gate) is approximemetely 35 pF. With integration the capacitance could be easily reduced to 1 pF and the supply voltage reduced
to 1 V. With these conservative figures a reduction in power of approximately 270 times can be achieved per switching element. This
166
would lead to a power consumption of the pulse generation circuitry,
approximately PP U LSE = 10 µW. The quiescent power due to leakage
will be much less than 0.5 µW.
The use of a larger inductance would also decrease the RF power
required, as described by the efficiency of generating the magnetic field,
section 4.2.1. In [90] a planar inductor with a diameter of 10 mm was
used which has an inductance of 2µH, ten times the inductance of
the coil used in this Chapter. The use of the planar inductor would
increase the efficiency of creating the magnetic field by a factor of
ten. Therefore, the required transmitter power to achieve the same
performance as the discrete implementation shown in this Chapter
would be reduced to PT X = 50 µW. For L = 8 the estimated energy
per bit for a fabricated IC would be 15 pJ/bit.
4.11 Conclusion
A novel circuit that makes use of the resonant properties of 2nd order
transmitting elements to form an approximated pulse has been shown.
It has been shown that applying a current impulse to a resonant tank
rather than directly modulating an inductor reduces the rms power by
√
a factor of Q.
Theoretical decomposition of an all pole Gaussian pulse approximation using the sum of 2nd order responses is described. The N = 2
approximation does not require the creation of delays, and the amplitudes of the pulses from the two 2nd order elements is the same, thus
simplifying implementation.
It is shown that by using the same number of transmitting and
receiving elements an implicit approximated matched filter can be implemented at the receiver. The coupling matrix and radiated magnetic
field profile of the transmitter with two elements spaced at 20 mm are
shown.
As the carrier frequency of the transmitting elements can only be
tuned to within 1 %, an estimation of the carrier frequency is required
167
in order to carry out coherent demodulation. An algorithm that estimates the parameters from a known preamble sequence is proposed.
The BER performance of coherent detection is poorer than non coherent detection, which is primarily due to the frequency and phase
estimator. The advantage of using coherent detection is that higher
bit error rates can be achieved using the same bandwidth. Using non
synchronised clocks, a bit error rate of 10−3 at a rate of 1 Mbps and a
distance of 50 mm is achievable using coherent detection. The maximum rate of coherent detection is 500 kbps, which can be achieved at
a distance of 65 mm for a BER of 10−3 .
The power consumption of the discrete circuit implementation is
much higher than state of the art. However, it is shown that the
projected power consumption would be 15 pJ/bit if the circuits were
fabricated on an integrated circuit.
In summary a transmitter circuit and receiver topology suitable for
a medical implant transmitter operating in the 30-37.5 MHz band have
been presented. The transmitter achieves 30 dB of out of band attenuation and can transmit at rates of up to 1 Mbps over a 50 mm distance
by using coherent detection.
168
5 An Analogue Gabor
Transform
5.1 Introduction
In the Gabor transform (also known as the Short Time Fourier Transform) [94], a set of basis functions are used to represent the input
signal. The input signal is projected onto these basis functions to form
a set of discrete time continuous amplitude coefficients. An approximate reconstruction can then be made by multiplying these coefficients
by the same basis functions.
The algorithm transforms a signal into multiple parallel channels,
each of which occupies a lower bandwidth than the original signal. A
possible application of this is for low power sensor networks; apply the
Gabor transform and then use a set of analogue to digital converters
with lower sampling rates. The downstream digital processor can then
process the received data using parallel channels, each at the lower
rate. A lower clock rate processor would be required, which would
reduce dynamic power dissipation [39].
In Section 5.2 an introduction to the Gabor transform is provided,
followed by a comparison between a time domain and a direct filter
approach in Section 5.3. This analysis shows that using the direct filter
approach requires significantly less hardware than the time domain
approach. Section 5.4 discusses how the filters were designed based
on the framework devised by Haddard [85]. Section 5.5 shows a novel
method to quickly design state space continuous time filters. Finally,
Section 5.6 presents measured results.
169
Energy
Frequency [k]
2
B
1
∆f
0
∆t
0
1
2
Time [p]
Figure 5.1: Information diagram illustrating the discrete time and frequency energy resulting from the projection of a signal onto
the pseudo basis functions.
5.2 The Gabor Transform
The Gabor transform [94] was proposed in 1946. A useful way of
describing this process is via the information diagram, figure 5.1. In
this diagram the time and the frequency are split up into a set of units.
Each unit is of size ∆t∆f and the time uncertainty as defined by Gabor
is:
1
∆t∆f ≥ .
(5.1)
2
The arrow protruding from each unit is the complex coefficient obtained by projecting the signal onto an analysis window. For the case
of Gaussian windows, the bound in (5.1) is met so ∆f ∆t = 21 . The
continuous time and frequency Gabor transform is given by:
Cc (τc , fc ) =
Z∞
s(t − τc )hc (fc , t) dt
(5.2)
−∞
where Cc are the coefficients, τc and fc are both continuous variables
describing the time and frequency localisation and hc is the window.
170
The Gabor transform is a special case of a wavelet transform that
has a uniform time and frequency grid. It is shown in [95] that the
continuous wavelet transform can be made discrete in frequency and
time without loss of information, provided that the sampling in both
frequency and time is sufficiently dense.
C(k, p) =
Z∞
s (t − p∆t) h(k, t) dt
(5.3)
−∞
where C(k, p) are the coefficients, s(t) is the input signal, p is the time
index, k is the frequency index and h(k, t) is the analysis window for
each frequency.
The window functions are complex sinusoidal Gaussian functions
which are given by:
2 2
“
”
j2π f0 + LB k t
h(k, t) = e−α t e
k
(5.4)
where B = ∆f Lk is the bandwidth and f0 is the frequency of the first
analysis window. α is the spread of the pulse envelope given by:
α=
√
2π∆f.
(5.5)
In order to reconstruct the original signal from the coefficients an
approximate inverse can be used:
s(t) ≈
Lp −1 Lk −1
X X
p=0 k=0
C(k, p)h(k, t − p∆t)
(5.6)
where Lp is the total number of time samples and Lk is the total number
of frequency divisions. Eq. (5.6) uses the same windows as used in the
forward transform. If different window functions were chosen for the
inverse then an exact reconstruction could be arrived at. However, the
inverse (5.6) still allows reconstruction of the original information in
the signal albeit with some error. Refer to [95] for more details on this
171
2 2
g(t)=eα
t
∆t
t
T
w
Figure 5.2: Truncation of the Gaussian pulse.
issue. A bit error test is used in the measurement section to observe
the performance of the transform and its inverse.
For a practical implementation the analysis window must be truncated to a length Tw as shown in figure 5.2. In this case the integration
limits of (5.3) are from −Tw /2 to Tw /2. To implement the window in
the analogue domain the complex sinusoid is split into its real and
imaginary parts and these are processed separately; figure 5.11 shows
a set of analysis windows. The analysis windows must also be causal.
This is achieved by shifting the window, figure 5.2, by Tw /2, thus ensuring the function is zero for t < 0.
5.3 Implementation of Convolution
There are several ways in which the convolution operator of (5.3) can
be implemented using analogue circuits. In [96], [97] complex demodulation is used to convert the signal to the baseband. The complex
baseband signal is then passed through two low pass filters to pro-
172
duce the real and imaginary coefficients, which can then be sampled
to obtain the discrete time transform. Both these schemes require two
multipliers and two low pass filters per complex coefficient.
The scheme described in [98] produces a time domain representation
of the analysis window using a baseband filter and a local oscillator.
The input signal is then mixed with the analysis window and integrated
to find the coefficients. This scheme requires four multipliers and a
single low pass filter to obtain one complex coefficient.
The advantage of the circuits described above is that the complexity
of the filters is reduced so half the number of poles are required compared to using bandpass filters. However, these circuits require clock
generation circuits and/or demodulation circuitry. By using advances
in higher order complex filter design [85], methods to directly implement the convolution using bandpass filters are achievable. This means
that there is no requirement for clock generation circuits, potentially
reducing the amount of hardware and power required.
The generation of a complex coefficient using the direct filter approach requires two of the circuits shown in figure 5.3 per complex
coefficient. The two filters have the same centre frequency but are
orthogonal to each other.
By using the direct time domain approach the impulse response of
the bandpass filter is multiplied with the input signal then integrated
using a resettable integrator, figure 5.4. Again two of these circuits
are required per complex coefficient. The analysis in the following
section compares the direct filter approach and the direct time domain
approach in terms of hardware and noise.
5.3.1 Number of Filters and Coefficient Rate
In the direct filter approach two filters are required per analysis window, one for each of the real and imaginary parts. In the time domain
approach, integration is carried out over the length of each analysis
window and the windows overlap in time. Figure 5.5 shows the over-
173
n(t)
s(t) + p(t)
y[n] + q[n]
H(s)
Figure 5.3: Generation of a single coefficient using the direct filter approach. n̄(t) is the output noise of the filter.
s(t) + p(t)
m(t)
Tw
Impulse Train
h(t) + n(t)
y[n]+q[n]
H(s)
TW
0
Figure 5.4: Generation of a single coefficient using the time domain
approach. n̄(t) is the filter noise, p̄(t) is the signal noise
and m̄(t) is the multiplier noise.
lap in the calculation of coefficients for a truncated window of length
2∆t. Here four filters are required per analysis window; two per complex coefficient.
In the general case the length of the window is Tw = m∆t, where m
describes the amount of overlap. Thus for the time domain approach
there will be 2mN channels with a coefficient rate of 1/m∆t. In constrast the direct filter approach requires 2N channels with a rate of
TW
C(k,1)
C(k,3)
C(k,2)
0
∆t
2∆t
C(k,4)
3∆t
4∆t
5∆t
Figure 5.5: Coefficient time line showing the generation of the coefficients for TW = 2∆t.
174
1/∆t. Each channel requires a bandpass filter, so the time domain approach requires m times as many filters, but the rate of the coefficients
is m times less. Therefore, in terms of hardware requirement, the direct
filter approach is superior to the direct time domain approach.
5.3.2 Noise Analysis
Generation of a single coefficient using the time domain approach is
shown in figure 5.4. n̄, p̄ and m̄ are independent noise sources with zero
mean. n̄ is the noise added by the filter used to generate the impulse
response, p̄ is the signal noise, m̄ is the multiplier noise, h(t) is the
analysis window, s(t) is the input signal, Tw is the length of the analysis
window and y[n] is the discrete time continuous amplitude coefficient.
The integrator is reset to zero every Tw . At the output there is a signal
RT
component, y[n] = 0 w s(t)h(t)dt and a noise component:
q̄[n] =
ZTw
[p̄(t)h(t) + n̄(t)s(t) + p̄(t)n̄(t) + m̄(t)] dt.
(5.7)
0
As the random variables are independent and have zero mean it is
straightforward to see that the expectation of q̄[n] will be zero. The
variance of q̄[n] in terms of the autocorrelation function of the noise
source, R(τ ), is [99]:
2
E[q̄[n] ] =
Z
0
Tw
Z
0
Tw
[Rp (t1 − t2 )h(t1 )h(t2 )
+ Rn (t1 − t2 )s(t1 )s(t2 ) + Rp (t1 − t2 )Rn (t1 − t2 )
+ Rm (t1 − t2 )]dt1 dt2 . (5.8)
For the case of wide sense stationary white noise the autocorrelation
function is:
R(τ ) = σ 2 δ(τ )
(5.9)
where σ 2 is the variance of the white noise source and δ is the Dirac
175
Delta function. Thus (5.8) can be simplified to:
2
E[q̄[n] ] =
σp2
ZTw
2
h(t) dt +
σn2
ZTw
2
s(t)2 dt + σp2 σn2 Tw + σm
Tw
(5.10)
0
0
where σi2 is the variance of the ith white noise source. The peak coefficient will occur when s(t) = h(t) thus the peak signal to noise ratio
(SNR) is:
T
Rw
h(t − T2w )2 dt
SNRTD = 0 p
.
(5.11)
E[q̄[n]2 ]
For the Gaussian analysis window (5.4), the maximum value of the
coefficient output is:
ZTw
h(t −
Tw 2
) dt
2
≈
A2 Tw
2m
(5.12)
0
where A is the peak amplitude of the window function and m is the
length of the window as defined in Section 5.3.1. The peak SNR of the
signal, filter output and multiplier output computed over the signal
bandwidth, ∆f are:
A
√
σp ∆f
A
SNRn = √
σn ∆f
A2
√
SNRm =
.
σm 2∆f
SNRp =
(5.13)
(5.14)
(5.15)
Eq. (5.11) can then be approximated by:
SNRTD
v
u
1
u
≈t 1
1
4 SNR
2 + SNR2 +
p
n
m
SNR2m
.
(5.16)
In the direct filter approach of figure 5.3 the peak SNR can be written
176
as:
SNRDF
t
R
Tw
2
− τ ) dτ
s(τ )h(t −
max
0
= v
u f0 +∆f /2
u R
t
σn2 + σp2 |H(f )|2 df
(5.17)
f0 −∆f /2
where H(f ) is the filter transfer function. When the input signal has
a maximum amplitude of A and the filter has unity gain then (5.17)
reduces to:
v
u
1
u
.
SNRDF ≈ t (5.18)
1
π 1
4 SNR
+
2
4 SNR2
n
p
In the direct filter the noise in the signal path is reduced due to the
action of the filter. This is not the case when using the time domain
approach. In the trivial case where the SNR of the filter, multiplier
and signal are the same then:
SNRDF ≈ 1.5 × SNRTD .
(5.19)
This result implies that the SNR of the coefficients computed using the direct filter method would give almost 2 dB improvement over
the time domain case using windows of length of 2∆t. The result of
the analysis shows that from a noise point of view there is no benefit in using the time domain approach for implementing the Gabor
transform.
5.4 Design of the State Space Filter from
Impulse Response Specifications
From the time domain function a Padé approximation, as shown by
Haddad [85] and described in Chapter 3, has been used in order to
create causal and rational 8th order bandpass transfer functions in the
Laplace domain. For the implementation of this chip the bandwidth
has been chosen to be 4 kHz and f0 = 500 Hz, so the spacing in the
177
frequency domain for N = 4 is 1 kHz, (5.4). An example of a transfer
(s)
function, H(s) = N
, for the imaginary (sin) and real (cos) windows
D(s)
with Tw = 2 s and a centre frequency of 0.75 Hz is:
N (s)(sin) = 56.14s5 + 284.6s4 + 5332s3 + 34.33e3 s2
+ 28.64e3 s − 20.16e3
N (s)(cos) = 5.956s6 + 63.75s5 + 583.3s4 + 5376s3
− 48.73e3 s2 − 48.73e3 s − 10.45e3
(5.20)
D(s) = s8 + 21.97s7 + 313.5s6 + 2850s5
+ 18.93e3 s4 + 87.66e3 s3 + 286.9e3 s2
+ 581.2e3 s + 626.0e3 .
In order to convert this Laplace transfer equation into a state space
realisation, a method documented in [100] is used. In this method an
LC ladder network for the characteristic polynomial is created, from
which the entries of the state matrix can be found. The entire transfer
function is then created by taking a weighted sum of the nodes of the
LC filter in order to realise the output matrix. For the filter in (5.20),
scaled so that it is centred at f0 = 1500 Hz, the state space model
using CN = 20 pF of capacitance at each node, ignoring any parasitics
and using the notation in [101] is:


0
112.1
0
0
0
0
0
0


−112.1

0
127.6
0
0
0
0
0


 0
−127.6
0
157.0
0
0
0
0 




 0

0
−157.0
0
177.5
0
0
0

G =
 0
0
0
−177.5
0
212.3
0
0 




 0

0
0
0
−212.3
0
278.4
0


 0
0
0
0
0
−278.4
0
542.2 


0
0
0
0
0
0
−542.2 −878.8
(5.21)
178
h
i
C(sin) = 605.4 −275.1 −538.2 200.1 −107.5 112.5 0 0
C(cos)
(5.22)
i
= −152.3 −781.5 291.4 169.4 −37.5 127.8 −83.1 0
h
(5.23)
h
i
BT = 0 0 0 0 0 0 0 200
TC = diag CN I(1×8)
H (sin) (s) = C(sin) (sTC − G)−1 B
H (cos) (s) = C(cos) (sTC − G)−1 B.
(5.24)
(5.25)
(5.26)
(5.27)
where all the values in G, C and B above are written in nano Siemens.
The characteristic polynomial determines the centre frequency of the
filter, and multiplication of the G matrix by a constant enables control
over the centre frequency of the filter. This feature is exploited when
tuning the filter, as only a single bias current is required to set the
centre frequency. Another feature of the state space implementation
is that the same G matrix can be used together with two different
C matrices in order to implement the sin and cos components of the
analysis windows, figure 5.6.
The circuit design of the filter is based on a gmC ladder approach.
The topology of the ladder implementation is similar to that used
in [85]. Figure 5.7 shows the implementation of the G matrix using
identically sized transconductors. The bias current of each transconductor is set such that the value of transconductance defined by the G
matrix is realised. The implementation of the sin and cos C matrices is
shown in figure 5.8. The individual bias currents are generated on-chip
using a simple current mirror, figure 5.9.
In order to design the filter for low power there are several tradeoffs
that need to be addressed. In the following section a methodology for
the design of a low power gmC filter is shown.
179
VIN
V-I
(sin)
C
IOUT(sin)
C(cos)
IOUT(cos)
G
b81
Figure 5.6: Block diagram of the gm-C filter.
IG12
0
+
-
0
IB81
VIN
IG23
0
+
-
IG34
+
IG21
0
0
+
-
IG45
+
IG32
0
+
-
0
+
-
IG56
+
IG43
0
0
+
-
IG67
+
IG54
0
0
+
-
IG78
+
IG65
0
0
+
-
0
+
+
IG76
0
+
-
IG87
C1
V1
C2
V2
C3
V3
C4
V4
C5
V5
C6
V6
IG88
C7
V7
C8
V8
Figure 5.7: Circuit diagram of the G matrix implementation using
identical transconductors and capacitors. The gm of each
transconductor is set via the bias currents IB81 and IGij .
180
V1
0
0
+
-
+
-
V1
IC11(sin)
0
V2
IC11(cos)
0
+
-
+
-
V2
IC12(cos)
IC12(sin)
0
V3
+
-
0
+
-
V3
IOUT(sin)
IC13(sin)
V4
0
IC13(cos)
V4
+
-
+
-
0
IC14(cos)
IC14(sin)
0
V5
0
+
-
+
-
V5
IC15(cos)
IC15(sin)
V6
0
IOUT(cos)
V6
+
-
+
-
0
IC16(sin)
IC16(cos)
0
+
-
V7
IC17(cos)
Figure 5.8: Circuit diagram of the C matrix implementation. A
weighted sum of the state voltages shown in figure 5.7 is
created using transconductors. ICij are the bias currents of
the transconductors.
IBIAS(OFF CHIP)
IB81
W=20µ
L=20µ
IG12
IG88
IC11(sin)
(sin)
(W/L)B81
(W/L)G12
(W/L)G88
(W/L)C11
IC16(sin)
(sin)
(W/L)C16
IC11(cos)
IC17(cos)
(cos)
(cos)
(W/L)C11
(W/L)C17
Figure 5.9: Generation of the bias currents for a single complex filter
using current mirrors. (W/L) of each transistor is scaled
to provide the required bias current.
181
Vdd
M3
M4
i0
v+
M1
M2
v-
I0
Figure 5.10: Diagram of a simple transconductor. The differential input voltage is defined as: vin = v+ − v− .
5.5 Designing a Low Power gmC Filter
The widths and lengths of the transistors used in the transconductor
need to be chosen in order to meet a certain specification. In the following analysis an insight into how to size the transistors for optimum
performance is shown. The design is based on the use of identical simple differential pair transconductors with no linearisation, figure 5.10.
The following sections show some simplified analyses which enable
the filters performance to be established before using a full blown analogue simulator, leading to faster design times.
5.5.1 Noise and Distortion
For a given filter topology it is important to maximise the SNR and
minimise the distortion. This can be achieved by maximising the
182
SINAD (signal to noise+distortion ratio) of the filter which is defined
as:
Pf
(5.28)
SINAD =
Pn + Pd
where Pf is the power in the fundamental at the output, Pn is the
integrated output noise and Pd = Pf (THD)2 is the distortion power.
THD is the total harmonic distortion.
Estimating THD
The transfer function of the transconductor in figure 5.10 using the
EKV model [102] is:
io = I0 tanh
vin
2nUt
(5.29)
where n is the subthreshold slope factor and Ut = kT /q is the thermal
voltage. The gm can be found by taking the first term of the Taylor
series expansion:
gm =
I0
.
2nUt
(5.30)
To compute the THD for the single transconductor a series expansion of (5.29) can be used to find the first few terms of the Fourier
series [103]. There are also several methods reported for estimating
the distortion of state space systems [104]–[107]. Any of these methods could be used to estimate the distortion of the filter, however,
they rely on extracting a weakly non-linear model of the transconductor. For the subthreshold case it is known that the V-I characteristic
is a tanh function (5.29). Therefore, the distortion estimate can be
simplified by forming a non-linear state space model:
1
VIN
v
v̇ =
+ IB tanh
IG tanh
CN
2nUt
2nUt
v
IOUT = IC tanh
2nUt
(5.31)
(5.32)
183
where IG , IB and IC are the bias current matrices for the transconductors, for example:
IG = 2nUt |G|.
(5.33)
VIN is the input voltage to the filter shown in figure 5.7 and the state
variable vector is defined as:
iT
v = V 1 V 2 ··· V 8 .
h
(5.34)
The non-linear state space system can then be simulated using Euler
numerical integration:
vd+1
vd
1
VIN (td )
IG tanh
= vd + ts
+ IB tanh
CN
2nUt
2nUt
(5.35)
where td is the time index and the time step is given by:
ts = td+1 − td .
(5.36)
Higher order integration methods could also be used for improved speed
and accuracy [108], however, the simplicity of the Euler method is suitable for this case. The cumulative error over a time period is proportional to ts .
To find the THD a simulation over time T is carried out, where T is
large enough to reach the steady state response. The last period of the
simulation is then used to calculate the distortion by using a Fourier
transform. This method of simulating the behaviour of the filter is
much quicker than using the full model in an analogue simulator. For
example, calculation of the THD for the 1500 Hz filter block took 0.5
s in MATLAB compared to 25 s using the Spectre simulator with the
analogue models.
184
Noise
The input referred spectral noise densities for subthreshold MOS transistors defined in [109] are:
Sfgate =
Stgate
KF 1
W LCOX f
2kT n
=
gm
(5.37)
(5.38)
where COX is the gate oxide capacitance and KF is the SPICE flicker
noise coefficient. For a transconductor the input referred noise spectrum may be written as [101]:
Sin (f ) =
St
Sf
+ .
gm
f
(5.39)
By referring the input noise voltage of each transistor to the input
for the simple transconductor the following expressions can be derived
for St and Sf (see Chapter 5 of [51]):
Sf =
2KF
COX
"
St = 8kT n
#
2
1
1
n
+
(W L)(1,2)
np
(W L)(3,4)
(5.40)
(5.41)
where np is the slope factor for the pmos transistor. This shows that
the thermal noise is dictated by the process and not by the sizes of
the transistors. In contrast, the flicker noise decreases as the area of
the nmos and pmos transistors are increased. Using these equations
together with Kozeil’s method [101], the total output referred noise for
the filter can be found and Pn in (5.28) can be calculated.
5.5.2 Mismatch
Mismatch in the subthreshold region predominantly causes variation
in threshold voltages, which in turn causes variation in the offsets and
the gm of the transconductor [103]. Offsets sum together to produce
185
an offset at the output and do not affect the AC response. There
will also be mismatch from the capacitors, but as long the ratios of
capacitors are well matched then this error will be small compared
to that produced by the transconductor. Assuming that the areas of
the transistors supplying the bias currents are large compared to the
transconductor transistors, the main error is due to the mismatch in
the active load transistors. Making the bias transistors large does not
affect the output resistance, bandwidth or power consumption. The
variation in gm due to M3 and M4 can be written as [103]:
′
gm
= gmo
X
1+ √
2np Ut
where X is a random variable with a normal distribution:

!2 
ξ

X ∼ N 0, p
(W L)(3,4)
(5.42)
(5.43)
where ξ is the threshold voltage mismatch constant, which is provided
by the foundry. The minimum value of (W L)(3,4) can be found independently of the bias currents and capacitance value by considering
the relative change in gm value:
∆gm
X
.
=√
gmo
2np Ut
(5.44)
A Monte Carlo analysis using the standard state space model with
the normalised matrices Gn , Bn and Cn can be used to determine
m
the maximum value of ∆g
. The normalised matrices are those where
gmo
the node capacitance (CN ) is equal to unity. The state space model
including mismatch is:
186
v̇ = [Gn + δGn ] v + [Bn + δBn ] Vin
(5.45)
IOU T = [Cn + δCn ] v
(5.46)
XG
δGn = √
◦ Gn
2nUt
XB
δBn = √
◦ Bn
2nUt
XC
◦ Cn
δCn = √
2nUt
(5.47)
(5.48)
(5.49)
where ◦ is the Hadamard product. XG , XB and XC are matrices containing random variables sampled from the normal distribution (5.43).
The error due to mismatch can be calculated by computing the magnitude of the transfer function over the frequencies of interest and comparing this to the desired transfer function. By running a Monte Carlo
simulation the standard deviation of this error can be computed which
is then the mismatch error, emm :
emm
v q

u
P
2
u
′
u
ω (|H(jω)| − |H (jω)|)

P
= tE 2 
ω |H(jω)|
(5.50)
where H(jω) is the desired frequency response and H ′ (jω) is the simulated response. Summations are used here as ω is considered to be a
discrete variable that ranges over the bandwidth of the filter passband.
5.5.3 Bandwidth and Output Resistance
The bandwidth and output resistances are very dependent on the transfer function being implemented. For a simple transconductor the bandwidth can be approximated by a single pole:
ωp ≈
√ gm3
2
Cx
(5.51)
I0
where gm3 = 2nU
and Cx = Cgs3 + Cgb3 + Cgs4 + Cgb4 . M3 and M4
t
are in the weak inversion saturated region so Cx can be approximated
by [102]:
187
vgs − Vth
np − 1
Cx = 2(W L)(3,4) COX exp
+
np Ut
np
(5.52)
where Vth is the device threshold voltage. The gate source voltage, vgs
is given by:
vgs = Vth + np Ut ln
I0
4np µCOX (W/L)(3,4) Ut2
(5.53)
where µ is the channel mobility.
To take account of the bandwidth a function F is defined, which is
used to modify the gm of each transconductor:
Fij (s) =
s
ωp
1
+1
(5.54)
where ij is the index of the matrix entry affected.
The output resistance of a transconductor can be modeled as:
Rout ≈
1
,
λI0
(5.55)
where λ is the channel length modulation.
The effect of finite output resistance is modeled by subtracting the
total node conductance from the diagonal of G. A diagonal conductance matrix is defined as:
Dg = 2λnUt diag G1(mx1)
(5.56)
where 1(mx1) is a one’s row vector equal in length to the order of the
transfer function, m. The total effect of bandwidth and output resistance can then be modeled as:
H ′ (s) = (FC ◦ βC C)(sTC − (FG ◦ (G − Dg )))−1 (FB ◦ B)
(5.57)
where βC is a constant that scales the bias currents required for the C
matrix. FG , FB and FC are matrices whose elements are formed using
188
Analysis
Distortion
Noise
Mismatch
Bandwidth & Rout
Outputs
THD
Pn
emm
ebwro
Inputs
CN , Vin
(W L)(1,2) , (W L)(3,4)
(W L)(3,4)
CN ,W(3,4) ,L(3,4) ,βC
Table 5.1: Tradeoffs in filter design.
(5.54). The error between the desired and modeled H(s) can then be
found over the bandwidth of the filter. The error is defined in a similar
way to the mismatch error (5.50):
ebwro =
qP
ω
(|H(jω)| − |H ′ (jω)|)2
P
.
ω |H(jω)|
(5.58)
5.5.4 A Low Power gmC Design Method
The overall DC current consumption of the complex gmC filter is:
IDC =
X
IG +
X
IB +
X
IC (sin) +
X
IC (cos) .
(5.59)
The method presented here is one way in which to methodically
obtain a low power realisation given the set of tradeoffs. The analyses
used to investigate the tradeoffs are shown in table 5.1. All fixed
process parameters and the normalised state space matrices are also
passed to this analysis.
To obtain a low power implementation, first find the SINAD for
differing values of capacitance, then select a value of capacitance so
that the required SINAD is achieved. Increase the size of (W L)(1,2) to
reduce flicker noise if greater SINAD is required.
A suitable value for (W L)(3,4) can be found from the mismatch analysis and then a value for W(3,4) can be found by carrying out the Rout
and bandwidth analysis with βC set to a large value. When W(3,4) has
been found, βC should be reduced until the maximum error is reached.
As there are four G matrices and eight C matrices and the design
189
Filter
cos 500 Hz
sin 500 Hz
cos 1500 Hz
sin 1500 Hz
cos 2500 Hz
sin 2500 Hz
cos 3500 Hz
sin 3500 Hz
SINAD [dB]
model
simulation
41
40
36
38
40
39
40
39
39
38
39
38
38
37
38
37
emm [%]
model
simulation
2.1
4.5
2.0
2.2
2.3
3.2
2.2
2.9
2.8
2.9
2.7
2.8
3.4
3.8
3.5
3.7
ebwro [%]
model
simulation
1.3
8.6
0.5
3.2
1.9
1.1
1.6
1.0
2.4
3.5
2.6
2.7
3.2
3.9
3.5
3.7
IDC [µA]
model
simulation
0.9
1.1
1.2
1.2
1.2
1.3
1.3
1.4
Table 5.2: Comparison of model and BSIM3 simulation.
uses identical transconductors for each filter, each stage of the analysis
needs to be carried out on each filter in order to determine the worst
case.
For a SINAD of greater than 35 dB with an input voltage of 10 mV,
a capacitance of 20 pF is suitable. For emm < 5% and ebwro < 5%,
W(1,2) = L(1,2) = 10 µm and W(3,4) = L(3,4) = 5 µm were calculated.
The bias transistors were chosen to have an area at least ten times
larger than W L(3,4) so that the mismatch due to these could be ignored.
Table 5.2 lists the analysis results and the corresponding simulated
results for each filter.
The predication of SINAD using the model shows a good match with
the simulated results. In terms of estimating the bandwidth and mismatch effects the models generally underestimate the error. However,
using the model has enabled a specification for the device sizes to be
sought before resorting to using the simulator. Balanced transconductors were used for the implementation. These have a similar model to
the simple transconductor, except that the bandwidth is slightly less
due to extra internal nodes and the power supply is twice that of the
simple transconductor.
190
191
1
2
Time [ms]
sin 500 Hz
1
2
Time [ms]
3
3
−20
0
−10
0
10
20
−20
0
−10
0
10
20
1
2
Time [ms]
sin 1500 Hz
1
2
Time [ms]
cos 1500 Hz
3
3
−20
0
−10
0
10
20
−20
0
−10
0
10
20
1
2
Time [ms]
sin 2500 Hz
1
2
Time [ms]
cos 2500 Hz
3
3
−20
0
−10
0
10
20
−20
0
−10
0
10
20
1
2
Time [ms]
sin 3500 Hz
1
2
Time [ms]
cos 3500 Hz
3
3
Figure 5.11: Measured impulse response of each filter. The solid line shows the ideal state space impulse response
as found in Section 5.4
−20
0
−10
0
10
20
−20
0
−10
0
10
cos 500 Hz
Amplitude [mV]
Amplitude [mV]
20
Amplitude [mV]
Amplitude [mV]
Amplitude [mV]
Amplitude [mV]
Amplitude [mV]
Amplitude [mV]
Figure 5.12: Die photograph showing a single complex filter. The entire chip contains four of these complex filters
5.6 Measured Results
A chip was fabricated using UMC 180 nm technology. Each fabricated
chip contains four complex filters centred at 500 Hz, 1500 Hz, 2500 Hz
and 3500 Hz. The chip area for each complex filter is 0.11 mm2 . Figure
5.12 shows the die photograph for a single complex filter. Nine dies
were packaged for testing to obtain an idea of the statistical spread in
the response of the filters. The measurement set-up is shown in figure
5.13. The impulse function and the A/D trigger were generated using
an FPGA running at 50 MHz; this meant that the jitter on the impulse
response would be less than 10 ns. The length of the impulse signal
was chosen to be 10 µs in order to give a bandwidth of the impulse
response up to 33 kHz (Appendix F). The analogue output of the NI
DAQ was used to provide sinusoidal stimulus in order to measure the
Bode response of each filter.
192
Figure 5.13: Measurement Setup. A National Instruments aquisition
board was used capture the output from the four complex
filters (8 channels)
5.6.1 Impulse and Bode Response
Figure 5.11 shows the measured impulse responses, averaged over 200
frames, for fabricated ICs. The filters were tuned by adjusting the
bias currents to alter the centre frequency of the filters. The output
amplitude of the filters was also adjusted by tuning the bias current
of the output I-V converter. All offsets have been removed from the
measured results. The output offsets for the chips have a mean of
25.4 mV and a standard deviation of 6.61 mV. These offsets are large
(due to subthreshold CMOS mismatch) but can be removed by AC
coupling the output of the filter. figure 5.14 shows the overlaid plot
of 200 measured frames for the cos 2500 Hz window. The rms jitter
around the zero crossing point (0.5 ms) for all chips is less than 21 µs,
the simulated result was 33 µs.
The Bode plot of the measured results for the tuned cos filters are
shown in figure 5.15 and for the tuned sin filters in figure 5.16. These
figures clearly show the effect of mismatch on the frequency response.
It is also clear that the low frequency response of the filter deviates
significantly from the ideal state space response. The reason for this is
193
15
Amplitude [mV]
10
5
0
−5
−10
−15
0
0.5
1
Time [ms]
1.5
2
Figure 5.14: Plot showing 200 overlaid cos 2500 Hz analysis windows
for a single chip.
due to the variation in the finite output resistance of the transconductors. The bias currents required to tune the centre frequencies for each
chip are different due to mismatch. This directly translates into different conductance matrices (5.56) for each chip, resulting in a different
low frequency gain for each chip (5.57).
194
0
Amplitude [dB20]
−10
−20
−30
−40
−50
−60
−70
3
10
Frequency [Hz]
4
10
Figure 5.15: Measured Bode plot of the cos filters for each chip. The
thick line shows the ideal state space frequency response.
0
Amplitude [dB20]
−10
−20
−30
−40
−50
−60
3
10
Frequency [Hz]
4
10
Figure 5.16: Measured Bode plot of the sin filters for each chip. The
thick line shows the ideal state space frequency response.
195
Filter
500 Hz 1500 Hz 2500 Hz 3500Hz
Ibias [nA]
61.0
59.1
58.5
57.9
Average f0 [Hz]
554
1559
2542
3547
Standard Deviation [Hz]
49.8
50.8
68.5
76.12
Quality Factor
0.5
1.5
2.5
3.5
Table 5.3: Variation in centre frequency for fixed bias currents.
5.6.2 Centre Frequency Variation
Table 5.3 shows the variation in the centre frequency between the chips
for fixed bias currents. It is evident from these results that the variation
in the centre frequency is larger for the higher centre frequency filters.
The reason for this is that the Quality factor of the filters increases
with increasing centre frequency. The Quality factor is defined as:
Q = f0 /B, where f0 is the centre frequency of the filter and B is the
bandwidth. For these filters the bandwidth is fixed at 1000 Hz. As
Q increases then the position of the poles becomes more sensitive to
errors in the G matrix. This effect can also be seen in the modeled
mismatch results in table 5.2.
For the cos 2500 Hz window, figure 5.17 shows the impulse response
for each chip at a fixed bias current of 58.5 nA and figure 5.18 shows
the frequency variation.
5.6.3 Power Consumption and SINAD
Table 5.4 shows the measured power consumption, the range of bias
currents required and the range of measured SINAD. The average
power consumption of all four complex filters is 7.06 µW when operating from a 1.2 V supply. The power shown for each complex filter
includes the power consumed by all of the transconductors that make
up the filter, figure 5.7 and figure 5.8. It also includes the generation
of all the bias currents, figure 5.9, but excludes the output buffers that
are used to take the signal off-chip for measurement. A single bias current per complex filter is generated off-chip using a variable resistor so
196
20
15
Amplitude [mV]
10
5
0
−5
−10
−15
−20
0
0.2
0.4
0.6
Time [ms]
0.8
1
Figure 5.17: Variation in the impulse response of the cos 2500 Hz filter
for each chip with a fixed bias current of 58.5 nA.
0
Amplitude [dB20]
−10
−20
−30
−40
−50
−60
3
10
Frequency [Hz]
Figure 5.18: Variation in the frequency response of the cos 2500 Hz
filter for each chip with a fixed bias current of 58.5 nA.
197
SINAD (avg) [dB]
SINAD (min) [dB]
SINAD (max) [dB]
Ibias (avg) [nA]
Ibias (min) [nA]
Ibias (max) [nA]
Power (avg) [µA]
Power (min) [µA]
Power (max) [µA]
500 Hz
cos sin
37 41
12 29
41 47
58.3
56.2
61.4
1.42
1.35
1.48
1500Hz
cos sin
34 35
30 26
42 37
56.9
54.1
61.0
1.62
1.58
1.73
2500Hz
cos sin
39 38
38 36
40 39
57.7
55.2
60.8
1.95
1.87
2.00
3500Hz
cos sin
37 35
36 31
40 38
57.3
55.6
58.6
2.07
2.01
2.15
Table 5.4: Summary of measured results.
that the centre frequencies can be tuned. As an alternative to using a
variable resistor, the voltage supply to a fixed resistor could be varied.
To produce a filter where each value of the G and C matrix can be
individually controlled then a scheme such as the “Stochastic I-Pot”
presented in [110] could be used. The measured average SINAD, table
5.4, corresponds well with the modeled and simulated results shown in
table 5.2.
5.6.4 Bit Error Test Performance
The direct filter technique of figure 5.3 was used to test the performance
of the transform. A pseudo random binary signal is applied to the input
of the filters and the outputs of the four complex filters were sampled
to obtain the complex coefficients. The rate at which the coefficients
were obtained is equal to 1/∆t, which for this implementation is 2 kHz.
The effective resolution of the A/D converter is 10 bits. Therefore, the
noise floor of the measuring instrument is much lower than the test
circuit. The measured coefficients were then used to reconstruct the
binary input using the ideal windows, (5.6). The results of a bit error
test of 10000 bits for the ideal and measured transform are shown in
figure 5.19. This figure shows that with the measured windows, bit
rates up to 7 kbit/s can pass through the transform at error rates of
198
−0.5
Bit error rate [dB10]
−1
−1.5
−2
−2.5
−3
Chip Average
Chip 2
Chip 1
Ideal
−3.5
0
2000
4000
6000
8000
Bit Rate [bps]
10000
12000
14000
Figure 5.19: Bit error comparison. The chip average bit error is the
average measured result from 9 chips. Chip 1 showed the
best bit error performance and chip 2 the worst.
about 10−2 . The worst and best performing chips are also shown to
provide an idea of the spread in the bit error rate. The departure from
the ideal curve is due to the error in the cross correlation of the analysis
windows, which is a result of mismatch error.
5.6.5 Analysis Window Cross Correlation
The cross correlation is computed using the average of 200 measured
windows. The set of basis functions is not orthogonal, due to the
truncation of the Gabor pulses and also due to errors in the implented
response caused by mismatch. The cross correlation of the real (cos)
windows for ideal (Cii ) and measured (Cmm ) windows is shown in
table 5.5. The cross correlation has been computed over 5 ms to allow
the measured impulse response to decay to zero. The measured cross
199
Table 5.5: Cross Correlation of Cos Analysis Windows
Cii


0.36 −0.12 0.01 −0.00
−0.12 0.25 −0.11 0.01 


 0.01 −0.11 0.25 −0.12
−0.00 0.01 −0.12 0.24
Caa


0.37 −0.14 0.02 −0.00
−0.14 0.25 −0.12 0.02 


 0.02 −0.12 0.25 −0.12
−0.00 0.02 −0.12 0.25
Cmm


0.36 −0.15 0.03
0.01
−0.15 0.25 −0.12 0.02 


 0.03 −0.12 0.25 −0.12
0.01
0.02 −0.12 0.24
correlation of the analysis windows is close to the ideal windows of
(5.4). However, a consequence of the error in the cross correlation
is that the transform will not perform as expected at high bit rates.
This is clearly shown in figure 5.20 where the cross correlation error
for each chip is plotted together with the bit error at 8 kHz. The
cross correlation error is defined as the sum of the mean squared error
between the ideal and the measured cross correlation coefficients.
200
3.5
−1
Cross Correlation Error
−1.5
2.5
−2
2
Bit Error
−2.5
1.5
1
1
2
3
4
5
6
Chip Number
7
8
−3
9
Figure 5.20: Cross correlation error and bit error performance comparison between each chip. There is a strong link between
the amount of cross correlation error and the bit error
performance of the chip.
201
Bit Error @ 8kbps [dB10]
Cross Correlation Error
3
202
Table 5.6: Transform Comparison.
This Work
Graham [111]
Haddad [85]
Haddad [112] Moreira-Tamayo [98] Edwards [96]
Justh [97]
Year
2009
2007
2005
2005
1995
1993
1999
No. of complex windows
4
8
1/2
1/2
1
6
16
Overall Transform Bandwidth
4 kHz
10 kHz
10 kHz
45 MHz
Maximum filter centre frequency
3.5 kHz
1 MHz
5.8 kHz 58 MHz
25 kHz
2 kHz
9 kHz
50 MHz
Process
180 nm CMOS
0.5 µm
180 nm BiCMOS
BiCMOS
Discrete
2 µm CMOS 2 µm CMOS
Area
0.44 mm2
2.25 mm2
0.28 mm2
5.43 mm2
Transform Power Consumption
7 µW
320 µ
1.5 µW 24.3 mW
6.75 µW
1.6W
Supply Voltage
1.2
3.3
1.2
1.5
Transform Methodology
Direct Filter
Time domain
Complex Demodulation
Bit Rate [10−2 error]
7kbps
Measured
Yes
Yes
No
Yes
Yes
Yes
5.6.6 Transform Comparison
A comparison between this work and other attempts at implementing
similar transforms are shown in table 5.6. The work in this Chapter
shows a complete working transform, many of the other designs in
the literature are for the individual filters. In [111] sixteen 10th order
filters were fabricated. These would be suitable for implementing eight
complex analysis windows. The implementation shown in this Chapter
requires 0.44 mm2 for four complex analysis windows. This is a much
smaller area than required by the most recent work in [112], which
requires 0.28 mm2 to implement one half of a complex window, and
in [111], which requires 2.25 mm2 to implement eight complex analysis
windows. The most likely reason for the reduced area requirement is
that in this work the size of the transconductors has been optimised
using a given set of performance criteria (see Section 5.5). The power
consumption per filter is similar to the simulated work shown in [85].
5.7 Conclusion
In this Chapter the design and implementation of an analogue Gabor
transform have been discussed. An approach using bandpass filters
has been compared to a time domain filtering approach. The results of
the comparison show that using the bandpass filter approach requires
significantly less hardware and has slightly better noise performance
than the time domain approach. The use of a state space filter allowed
a complex filter to be implemented using the same characteristic equation but using different output summing networks for the sin and cos
windows. A technique for optimising the transistor sizes to produce a
low power implementation has been presented. Simplified models to
analyse the effect of distortion, noise, mismatch, output resistance and
bandwidth on the filter transfer function are derived. The simplified
models allow rapid evaluation of the tradeoffs in the filter design.
An approximation to the Gabor transform has been successfully im-
203
plemented in 180 nm CMOS technology using devices operating in the
sub-threshold region. Measured bit error rate tests confirm that the
transform is operating as expected up to 7 kbit/s. Performance is limited by the approximation of the filter and the variation in the gm of
the transconductors. A set of tradeoffs have been described together
with mathematical descriptions so that device sizes can be found from
a set of given constraints. This has led to the fabrication of a low
power, low area circuit that carries out the analogue Gabor transform.
This circuit can be used to split a 4 kHz signal into eight separate 2
kHz bandwidth channels for further processing.
204
6 Conclusion
In todays society the reduction of energy consumption in electronic
circuits is important for increasing longevity of battery operated equipment, reducing heat dissipation to enable tighter integration of circuits
and to reduce our burden on the planets energy resources. In Chapter 2
a survey of energy consumption limits has been presented which shows
fundamentally how and why energy is dissipated by electronic circuits.
The survey looks at fundamental limits as well as implementation specific limitations. It shows that modern circuits typically require many
orders of magnitude more energy than the fundamental limits. For
the case of a transistor amplifier a possible reason for the large energy consumption is that significant energy is required to reduce the
fluctuations in the transistor semiconductor. This is akin to cooling
(refridgerating) the transistor, which requires energy. This observation
implies that circuits that bias a transistor using a low duty cycle will
benefit from reduced power consumption. The survey also highlights
the fact that the rate of information transfer depends on the amount
of energy available. This is explicitly shown in the quantum limits and
is hinted at by the classical limits as the minimum energy relies on the
SNR tending towards zero, i.e. zero rate of transmission.
As part of the survey a lower bound for the energy per bit for a free
space point to point link has been proposed. This is based on the fact
that the transmitter antenna is a blackbody radiator that contributes
to receiver noise. This bound is interesting because it shows that there
is an operating frequency at which a minimum energy per bit occurs.
This frequency and the minimum energy are dependent on the antenna
geometry and distance between the transmitter and receiver. A pos-
205
sible application of this lower bound would be to create a transmitter
whose centre frequency is adjusted depending on transmission distance
in order to minimise the energy required.
Inspired by the survey, two low power circuits designed to reduce
energy consumption have been described and measured results have
been presented. The PPM circuit in Chapter 4 shows a low complexity circuit which uses a transistor with a very low duty cycle. The
analog Gabor transform described in Chapter 5 decomposes a signal
into smaller bandwidths to allow post processing to be carried out at
a reduced rate, thus enabling the possibility of reducing energy consumption in the post processing circuits.
Chapter 3 provides an overview of PPM and presents the Gaussian
pulse as a suitable pulse prototype. It is suggested that low complexity PPM schemes are well suited to low power transmitter circuits.
In particular, analogue pulse based systems are advantageous as they
eliminate clocks at the carrier frequency and typically contain fewer
interconnects than their digital counterparts.
For PPM modulation, theory shows that by increasing the number of
orthogonal symbols, time slots in this case, the transmission energy can
be reduced towards the fundamental limit. However, in practice the
quiescent energy consumption of the transmitter circuit means that
there is an optimum number of time slots which provides minimum
energy.
The Gaussian pulse is superior to a rectangular or sinusoid pulse
because the bandwidth-time product of the Gaussian pulse can be
chosen arbitrarily to provide a desired out of band attenuation. This is
particularly important as improving the out of band attenuation gives
better spectral efficiency and reduces electromagnetic interference. A
performance comparison between three continuous time approximation
methods shows that the All Pole approximation has better out of bound
attenuation for a given filter order than the Cascade of Poles or Padé
approximations. This is important as the number of poles is directly
related to analogue circuit complexity and thus should be minimised
206
in order to reduce power consumption. As the approximate pulses are
not completely orthogonal, extra transmission energy is required to
achieve the same BER as the ideal Gaussian pulse. However, results
show that increasing the number of time slots reduces this extra energy
requirement. This means that the performance of the approximate
pulses is very close to the ideal Gaussian pulse when the number of
time slots is large. In the case of using a correlation detector with 16
time slots, the extra transmitter energy due to the approximation of
the pulse is less than 1 %.
Chapter 4 presents the analysis and implementation of a PPM architecture suitable for use within the UK 30-37.5 MHz medical implant
band. The transmitter makes use of two 2nd order RLC resonant tanks
whose impulse responses sum in free space to approximate a Gaussian
pulse. This significantly reduces circuit complexity as no clocks are
required at the carrier frequency.
Unlike simple OOK transmitters, the PPM transmitter shown in this
thesis is able to use coherent transmission by utilising two orthogonal
pulses. These pulses are produced by delaying the impulse response
to the resonant circuits to approximate orthogonality. The advantage
of this is that the rate of transmission can be increased for a given
bandwidth of operation.
The implemented transmitter provides 30 dB of out of band attenuation and has a maximum rate of 1 Mbps. The power consumption
of the discrete circuit at 1 Mbps is 3.2 nJ/bit and at 312.5 kbps is 1.3
nJ/bit. The high power consumption of the discrete implementation
is due to the large node capacitance of discrete logic gates. The estimated power consumption of a CMOS implementation is 15 pJ/bit for
a rate of 500 kbps.
A major advantage of the transmitter circuit presented in Chapter
4 is that no dynamic tuning circuitry is required by the transmitter.
This reduces complexity and size of the transmitter. To overcome the
drift in frequency of the transmitting and receiving elements a preamble
sequence sent by the transmitter is used by the receiver to estimate the
207
centre frequency and provide symbol synchronisation. Measurements
using the proposed estimation scheme indicate that a bit error rate of
10−3 can be achieved using coherent transmission at a rate of 1 Mbps
and a distance of 50 mm.
In Chapter 5 the design and implementation of an analog Gabor
transform has been presented. The implementation of the Gabor transform in the analog domain provides a low power circuit which can separate an input signal into several parallel paths, each of lower bandwidth
than the original signal. As pointed out in Chapter 2, processing signals at a lower rate may potentially allow circuits with lower energy
dissipation to be designed.
The transform topology in this chapter employs a bandpass filter
approach as this requires significantly less hardware and has slightly
better noise performance than a time domain approach. Using a state
space filter allows a complex filter to be implemented using the same
characteristic equation but using different output summing networks
for the sin and cos windows, hence reducing the amount of hardware
required. A technique for optimising the transistor sizes to produce a
low power implementation has been presented. This technique involves
using simplified models to analyse the effect of distortion, noise, mismatch, output resistance and bandwidth of the filter transfer function.
The simplified models allow rapid evaluation of the tradeoffs in the
filter design, which lead to a low power, low area circuit.
The Gabor transform was successfully implemented in 180 nm CMOS
technology using devices operating in the sub-threshold region. Measured bit error rate tests confirm that the transform operates as expected up to 7 kbit/s. Performance is limited by the approximation of
the filter and the variation in the gm of the transconductors. This particular implementation can be used to split a 4 kHz signal into eight
separate 2 kHz bandwidth channels for further processing. Using a
modified optimisation technique with bi-polar technology would allow
the bandwidth of the transform to be increased, potentially making
the circuit suitable as the front end of a receiver circuit, such as the
208
one presented in Chapter 4.
From the work carried out in this thesis it is surmised that in order
to achieve lower energy dissipation in the future the complexity of the
circuits needs to be kept to a minimum, the duty cycle of transistor
operation needs to be reduced and the rate of information transfer
within a circuit needs to lowered.
6.1 Future Work
The fundamental limitations described in Chapter 2 assume that the
system is operating under thermal equilibrium. Generally thermal
equilibrium is not met in practical circuits and so the fundamental energy is many orders of magnitude lower than seen in practical circuits.
Analysing a system that is not in thermal equilibrium is difficult, however, future research from the physics community into non-equilibrium
systems may provide further insight into the minimum energy requirements of electronic circuits. [113] provides some details of analysing
systems that are not in thermal equilibrium.
Chapter 4 shows the feasibility of a low power, low complexity coherent transmitter suitable for the 30-37.5 MHz band. To reduce power
further an integrated circuit of the transmitter electronics is required.
To improve the BER, a convolutional block encoder should also be
added. The use of an integrated circuit will significantly reduce the
node capacitance of the pulse generation gates, thus reducing power
consumption. The convolutional encoder is a digital circuit which
would benefit, in terms of lower power consumption, by using adiabatic logic, see Chapter 2.
The algorithm used to estimate the carrier frequency at the receiver
is not optimal and could be improved, thus reducing the amount of
overhead, currently 7 %, when using non synchronised transmit and
receive clocks.
Using the same techniques as shown in Chapter 4 the transmitter
analysis could be extended to use 2nd order transmitting elements at
209
higher frequencies. For example, the capacitive property of a short
electrical dipole could be used. Using an electric field instead of a
magnetic field would enable longer distance links to be achieved.
The Gabor transform in Chapter 5 shows a general circuit that is
suitable for splitting an analogue signal into several parallel paths, each
of which has a lower bandwidth than the original signal. This would
be particularly beneficial if the output of the transform is processed
using adiabatic logic, as the lower coefficient rate would reduce the
required clock rate. The Gabor transform could be implemented at a
higher bandwidth and used as the front end to the receiver described in
Chapter 4. The digital side of the receiver could then operate on the
frequency domain coefficients using adiabatic logic. Doing this may
offer considerable savings in energy over the standard logic approach.
However, the adiabatic logic circuit would not be trivial as the operations on the received coefficients need to occur in parallel in order to
take advantage of the lower coefficient rate.
210
7 Published Work
• An Analogue Gabor Transform Using Sub-Threshold 180
nm CMOS Devices. Mark Tuckwell, Christos Papavassiliou.
IEEE Transactions on Circuits and Systems I: Regular Papers,
December 2009. Volume 56, Issue 12, pp. 2597 - 2608.
• Exploration of energy requirements at the output of an
LNA from a thermodynamic perspective. Mark Tuckwell,
Christos Papavassiliou. IEEE International Symposium on Circuits and Systems, 2007. Page(s):2810 - 2813
211
212
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A Derivation of switching
energy for a square wave
The proof in [35] for a sinusoid driving an RC interconnect, figure
A.1, assumes that the driving signal is at the cut off frequency of the
filter. For a square wave there are many components which are well
below the cut off frequency. The proof in this section expands on that
in [35], by considering a square wave driving an RC interconnect.
The input signal is a square wave which is the sum sinusoids given
by [114]:
Vin (t) = VP
4
π
N
X
1
sin(nω0 t)
n
n=1,3,5,···
(A.1)
where N is the maximum harmonic index, ω0 is the fundamental
frequency of the clock and VP is the peak amplitude of the clock.
Here the clock is assumed to swing between +Vp and −Vp .
By using superposition the total input power required can be found
by considering the input power for each component of Vin . The cut
off frequency of the filter is related to the number of harmonics which
are included in the representation of the square wave, ωc = N ω0 . The
Figure A.1: A single pole RC filter.
227
transfer function for the nth harmonic is thus given by:
H(n) =
s
1
ejθ(n)
1 + (n/N )2
(A.2)
where θ(n) = −tan−1 (n/N ). The output voltage of the nth harmonic
is given by:
s
4
1
VP sin(nω0 t + θ(n))
.
Vout (n, t) =
πn
1 + (n/N )2
(A.3)
The mean squared output voltage of the nth harmonic can then be
written as:
2
Vout
(n)
1
=
2
4
VP
πn
2
1
.
1 + (n/N )2
(A.4)
The output noise of an RC filter is V 2 = kT /C, so the output SNR
for the nth harmonic is:
4 2
C
VP πn
SN R(n) =
2kT (1 + (n/N )2 )
(A.5)
By considering the voltage drop across the resistor the input power of
the nth harmonic can be written as:
4
VP πn
Pn =
2R
2 1
1−
1 + (n/N )2
(A.6)
which reduces to:
n2
Pin (n) = kT SN R(n)ω0 .
N
(A.7)
The SNR of an ideal square wave where N is taken to infinity is:
SN Rsq =
VP2 C
kT
(A.8)
The total input power can then be expressed in terms of the number
of harmonics and the SNR of the ideal square wave as:
228
16
1
PT = kT SN Rsq f0
π
N
N
X
1
1 + (n/N )2
n=1,3,5,···
(A.9)
In the limit of N going to infinity the series sum can be replaced by:
1
N
N
X
1
π
=
2
1 + (n/N )
8
n=1,3,5,···
(A.10)
The energy per switching operation for a square wave driving an RC
interconnect is PT /f0 :
Eop = 2kT SN Rsq
(A.11)
229
230
B A lower bound on the
energy for a point to
point communication link
In this section a lower bound on the energy required for transmitting
a bit using electromagnetic radiation is sought in terms of distance,
antenna sizes and noise factor of the receiver.
The Friis equation for free space path loss is widely used in radio
communication calculations [53]. This equation, together with
Plank’s blackbody radiation formula [25], is used to show that there
is a lower bound on the energy required to transmit information
between two antennas in free space. This lower bound takes into
account the fact that the transmit antenna is a blackbody and
radiation from it increases with the square of signal frequency. This
blackbody radiation can be treated as noise generated by the
transmitter.
In this work, the noise generated at the transmitter due to blackbody
radiation will be considered. This will show that there is a frequency
at which an energy minima occurs.
B.1 System Temperature
The system temperature (Tsys ) is the overall noise temperature of the
system, taking into account the noise temperature that the receive
antenna sees, together with the noise introduced in the receiver
electronics. The definition of Tsys in [115] is:
231
Tsys = TA + TAP
1
1
1
− 1 + TLP
− 1 + TR
ǫ1
ǫ2
ǫ3
(B.1)
TA is the noise temperature seen by the antenna when no
transmission is taking place, but includes the noise temperature due
to the blackbody radiation emitted from the transmitting antenna.
TAP is the physical temperature of the receive antenna. If the
thermal efficiency of the antenna is 100% then:
TAP
1
−1 =0
ǫ1
(B.2)
TLP is the physical temperature of the transmission line and again, if
there are no thermal losses due to this, then:
TLP
1
−1 =0
ǫ2
(B.3)
TR is the noise temperature of the receiver electronics, which is
dictated by the gain and number of stages as:
TR = T1 +
T3
Tn
T2
+
+ ... + Qn−1
G1 G1 G2
i=1 Gi
(B.4)
where n is the number of stages, Ti is the noise temperature of the
ith stage and Gi is the gain of the ith stage. For a multistage
receiver the overall noise temperature, TR , can be related to the
physical temperature of the receiver, To , by the noise figure:
TR = (F − 1)To
(B.5)
By setting Tsys = TA + (F − 1)To , the following relationship for the
minimum transmit power can be written:
Pt ≥
SNRd2 c2 kB
[TA + (F − 1)To ]
At Ar f 2
(B.6)
Eq. (B.6) shows that the transmit power has an inverse relationship
to signal frequency. This implies that as the frequency of
232
transmission increases, then the power required to transmit
information becomes smaller. In the following sections a lower bound
will be derived which shows that this inverse relationship to frequency
does not hold because TA is also frequency dependent (∝ f 4 ) due to
the blackbody radiation from the transmit antenna. Note also that
increasing the quality factor (Q = Bf ) will reduce the power, implying
that narrowband systems require less transmit power.
B.2 Antenna Noise Temperature
The antenna noise temperature, TA , seen by the receiving antenna is
a function of the coupling of the antenna to all radiating sources. In
general, there are many radiating objects that couple with the
antenna. The radiation of the sky is approximately 3K, so some of
this may couple into the receive antenna. Also the ground is at
around 290K, so some of the antenna field will couple with this.
However, in order to derive a lower bound on transmit power these
effects are ignored and only the radiation due to the transmit
antenna is taken into account.
Every object can be considered as a blackbody that radiates energy
according to Plank’s radiation formula [25]:
I(f, T ) = ε
1
2hf 3
hf
2
c e kT − 1
(B.7)
2f 2 kT
c2
(B.8)
where I is the intensity of radiation at frequency f and temperature
T, h is Plank’s constant (6.626068 × 10−34 ) and ε is the emissivity of
the blackbody. Dealing only with frequencies below approximately
100T Hz, (B.7) can be simplified by keeping up to the linear term of
the Taylor expansion, ex = 1 + x:
I(f, T ) = ε
For an antenna with a physical area of Āt , the noise power generated
233
by the transmitter antenna due to blackbody radiation can be given
by:
2kT
Pn(Tx) = εĀt 2
c
Z
f 2 df
(B.9)
B
For narrowband systems with Q > 1 it is possible to approximate the
integral in (B.9) by the area of the rectangle:
Pn(Tx) ≈
εĀt 2kT f 2 B
c2
(B.10)
This blackbody noise source originates at the transmit antenna and is
attenuated by the path loss (2.77), therefore, the noise power at the
receive antenna is given by:
2εĀt At Ar f 2 kT B
Pn(Rx) ≈
λ2 d2
c2
(B.11)
The equivalent noise temperature of this received power is given by:
TA =
Pn(Rx)
kB
(B.12)
which results in the noise temperature at the receiver due to
transmitter blackbody radiation as:
TA =
2εĀt At Ar f 4 T
d2
c4
(B.13)
Eq. (B.13) shows that the temperature due to the blackbody
radiation of the transmitter antenna is proportional to the 4th power
of frequency, which means that the antenna will appear increasingly
hotter as the centre frequency of the transmitted information is
increased.
234
B.3 Lower Bound on Transmission
Energy
By substituting the antenna temperature due to blackbody radiation
(B.13) into (B.6) and rearranging the resulting equation, a new limit
on the amount of power required for communication between two
points in free space can be shown to be:
2εĀt f 2 (F − 1)d2 c2
Pt ≥ SNRkT B
+
c2
At Ar f 2
(B.14)
Here the temperature of the transmit antenna and the receive
antenna have been set equal to T, which is not unreasonable for
systems which transmit over limited distances. Doing this simplifies
the analysis in order to clearly show the lower bound on the energy
required for transmission.
Notice that the increase in power due to the blackbody radiation
increases as the cube of frequency, whereas the increase in power due
to the thermal noise of the receiver electronics decreases as 1/f . This
is interesting as it will provide a minimum amount of power required
for transmission.
B.4 Energy per bit
In order to make comparisons between systems, it often makes more
sense to compare the energy per bit as this takes into account the
information processing ability of the circuit. The energy per bit can
be written as:
E=
P
I˙
(B.15)
where I˙ is the channel capacity in bits/s. The upper bound on
channel capacity is given by Shannon [2] as:
235
I˙ ≤ Blog2 (1 + SNR)
(B.16)
By using this equation the energy per bit can be found from (B.14):
SNR
2εĀt f 2 (F − 1)d2 c2
Ebit ≥
kT
+
log2 (1 + SNR)
c2
At Ar f 2
(B.17)
This equation has its minimum at f0 given by:
f0 =
(F − 1)d2 c4
2εĀt At Ar
14
(B.18)
and at low signal to noise ratios (SNR << 1):
SNR
= ln(2)
log2 (1 + SNR)
(B.19)
which gives the minimum energy per bit as:
s
Ebit (MIN) = 2d
2εĀt (F − 1)
kT ln 2
At Ar
(B.20)
B.5 Isotropic and practical antennas
In order to analyse this lower bound further and make comparisons
with practical scenarios, some knowledge about how the physical area
relates to the effective area for an antenna is required. An isotropic
antenna is a hypothetical antenna which radiates equally in all
directions and one in which all power at the input to the antenna is
transmitted. The effective area of an antenna is defined, via the
antenna gain G, as:
λ2
Aeffective =
G
4π
(B.21)
For the isotropic case the gain is equal to unity and it is found that
the antenna noise temperature, TA , becomes a constant, which means
that the minimum energy occurs at DC. However, for other types of
236
antenna that show a direct relation between effective area and
physical area, the minimum energy can be selected to be at much
higher frequencies.
B.5.1 Parabolic dish antenna
For the parabolic reflective dish the antenna gain is:
π 2 D2
(B.22)
λ2
where D is the reflector diameter. Essentially the effective antenna
area is proportional to the area of the transmitter with the constant
of proportionality being the efficiency [54]:
Gparabolic =
Aparabolic = η Ā
(B.23)
where Ā is the physical antenna area and η < 1 is the efficiency.
B.5.2 Dipole antenna
For a half wavelength dipole antenna the gain of the antenna is [53]:
π
2
which leads to a frequency dependent effective area:
Gdipole ≈
Adipole ≈
0.125c2
f2
(B.24)
(B.25)
In the case of antennas whose physical area is proportional to the
effective area there will be a minimum, at a frequency other than
zero, of (B.20). Whereas for those cases where a dipole antenna is
used the contribution due to the blackbody relationship becomes a
constant for all frequencies, which means that:
Ebit (MIN) ∝ f
(B.26)
237
238
C The relationship between
SNR and Eb/N0
In digital communications a widely accepted measure of system
performance is Eb /N0 [116]. This measure gives the amount of energy
over the noise floor required in order to transmit a single bit of
information. Typically graphs showing Eb /N0 against bit error are
used to compare communications protocols. Signal to noise ratio is
defined as:
SN R =
PS
PN
(C.1)
where PS is the average signal to noise ratio and PN is the average
noise power. For white noise it is more convenient to write:
PN = N0 B
(C.2)
where B is the bandwidth and N0 is the spectral height of white
noise. Eb /N0 is thus related to the SNR:
Eb
SNRB
PS
=
=
N0
N0 R
R
(C.3)
Using (C.3) and the channel capacity formula (2.1) the minimum
value of Eb /N0 can be obtained:
SNR
Eb
(min) = lim
= ln 2
SN R→0 log2 (1 + SNR)
N0
(C.4)
For matched source and load impedances N0 = kT , thus in a
straightforward manner the minimum energy per bit required at the
239
load is kT ln 2 joules. As this is only true in the limit of SNR → 0
then the rate of information also tends towards zero. This is the same
result as shown several times in Section 2.3.
C.1 Square wave SNR
For detection of a bi-polar binary signal the following relationship
can be written [116]:
2
Eb
= erfc−2 (2Pe ) .
N0
(C.5)
Thus the SNR required in order to represent this signal is:
2
R
erfc−2 (2Pe ) .
(C.6)
B
For a rectangular pulse the ratio R/B is approximately equal to
unity, thus the SNR for a square wave is:
SNR =
2
SNRsq = erfc−2 (2Pe ) .
240
(C.7)
D Time-Frequency
Uncertainty
In this appendix the time-frequency uncertainty bound is calculated
for the Square, Gaussian, Gaussian derivative and Sinusoidal pulse
functions.
D.1 Uncertainty Calculation
For a pulse shape, g(t) the degree of time-frequency uncertainty can
be written as [117]:
1
4π∆t∆f
ǫ=
(D.1)
where ∆t and ∆f can be computed from the second moments of the
time and frequency representations of the function:
R 2
t |g(t)|2 dt
(∆t) = R
|g(t)|2 dt
R 2
f |G(f )|2 df
.
(∆f )2 = R
|G(f )|2 df
2
(D.2)
(D.3)
The Fourier transform of g(t) is:
G(f ) =
Z
∞
g(t)e−j2πf t dt.
(D.4)
−∞
241
D.2 Rectangular Pulse
The baseband rectangular pulse can be written as:
g(t) =
q
 2
T
0
0 < t ≤ T,
(D.5)
elsewhere.
The Fourier transform of the rectangular pulse is:
G(f ) =
√
2T sinc(f T ).
(D.6)
The frequency uncertainty is given by:
2
df
f 2 sinc 2πf2 T
= ∞.
(∆f ) = R 2
2πf T
df
sinc 2
2
R
(D.7)
Therefore, the time-frequency uncertainy for the rectangular pulse is
ǫ = 0.
D.3 Sinusoidal Pulse
The sinusoidal pulse function is:
q
 4 sin( π t) 0 < t ≤ T ,
T
T
g(t) =
0
elsewhere.
(D.8)
The Fourier transform of the sinusoidal pulse is:
G(f ) =
√
4T
1 + e−j2πf T
.
π (1 − (2f T )2 )
(D.9)
The time uncertainty can be found as follows:
Z
0
242
T
Z
T
|g(t)|2 dt = 2
0
1
2
2
2
2
−
t |g(t)| dt = T
3 π2
(D.10)
(D.11)
∆t = T
s
2
3
− π12
.
2
(D.12)
The frequency uncertainty is:
∞
Z
|G(f )|2 df = 2
−∞
∞
2
Z
−∞
f |G(f )|2 df =
∆f =
(D.13)
1
2T 2
1
.
2T
(D.14)
(D.15)
Therefore, the time-frequency uncertainty is ǫ = 0.3. Although much
better than the rectangular pulse, this is still much less than unity
which is achieved by the Gaussian pulse.
D.4 Gaussian Pulse
The Gaussian pulse is given by:
g(t) =
√
T 2
2(2α)1/4 e−πα(t− 2 )
(D.16)
The Fourier transform of this pulse is:
G(f ) =
√
14
π 2
2
e− α f .
2
α
(D.17)
The time uncertainty is:
Z
Z
∞
−∞
∞
2
−∞
|g(t)|2 dt = 2
t |g(t)|2 dt =
1
∆t = √
2 πα
1
2πα
(D.18)
(D.19)
(D.20)
243
The frequency uncertainty is:
Z
∞
|G(f )|2 df = 2
(D.21)
−∞
∞
2
α
f |G(f )|2 df =
2π
−∞
r
1 α
∆f =
.
2 π
Z
(D.22)
(D.23)
The time-frequency uncertainty of the Gaussian pulse is ǫ = 1. This
is the optimum time-frequency which is only reached by the Gaussian
pulse.
D.5 Gaussian Derivative
The 1st Gaussian derivative is given by:
g(t) = 4
α3
2π
14
2
te−αt .
(D.24)
The Fourier transform of the pulse is:
G(f ) = 2j
8π 5
α3
14
f e−
π2 f 2
α
.
(D.25)
The time uncertainty is:
Z
∞
−∞
∞
2
|g(t)|2 dt = 2
3
t |g(t)|2 dt =
2α
−∞
r
3
∆t =
4α
Z
The frequency uncertainty is:
244
(D.26)
(D.27)
(D.28)
Z
∞
−∞
∞
2
|G(f )|2 df = 2
(D.29)
3α
f |G(f )|2 df = 2
2π
−∞
r
3α
.
∆f =
4π 2
Z
(D.30)
(D.31)
For the derivative of a Guassian pulse the time-frequency uncertainty
is ǫ = 31 which is slightly better than for the sinusoidal pulse.
D.6 Truncated Gaussian Pulse
The truncated Gaussian pulse is:
g(t) =
√
 2(2α)1/4 e−πα(t− T2 )2
0
0<t≤T
(D.32)
elsewhere.
The Fourier transform of this pulse is:
G(f ) =
1
π
1
4
(M + M ∗ ) e− α f
2
(2α)
r √
παT
π
.
M = erf
− jf
2
α
(D.33)
(D.34)
The magnitude squared function is then:
|G(f )|2 =
4
(2α)
π
1
4
2
(Re(M ))2 e−2 α f .
(D.35)
The frequency uncertainty does not have a closed form expression so
a numerical approximation is required. The complex error function
may be approximated using:
245
∞
2 X (−1)n z 2n+1
2
z7
z3 z5
erf(z) = √
=√
+
−
+ ··· .
z−
3
10 42
π n=0 n!(2n + 1)
π
(D.36)
Figure D.1 shows a plot of the approximated time-frequency
uncertainty evaluated over a bandwidth of 20/T . The plot also shows
the approximation of the Gaussian derivative, sinusoid and
rectangular pulses over the same bandwidth. The figure clearly shows
the superiority of the truncated Gaussian pulse over the other pulses
in terms of the time-frequency uncertainty. The reason why the
rectangular uncertainty value is greater than 0 is because the
frequency uncertainty is only evaluated over 20/T which results in a
value of ǫ = 0.2725. As expected as the value of αT 2 decreases
towards zero the time uncertainty becomes smaller, thus
approximating a square wave.
246
1
Time−Frequency Uncertainty − ε
0.9
0.8
0.7
Truncated Gaussian
Gaussian Derivative
0.6
Sinusoid
Rectangular
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
2
αT
Figure D.1: Approximation of the time-frequency uncertainty for the
Gaussian, Gaussian derivative, sinusoidal and rectangular
pulses. The approximation is taken over a bandwidth of
20/T
247
248
E BER Simulation
E.1 Correlation Detection
The entire set of symbols for a given past history can be written in
matrix form as:
s̃(ψ)

g(t)
g(t − T )
..
.





 + 1(L×1)

=



g(t − (L − 1)T )
p
X
i=1
g(t −
(ψ)
Ii T
!
+ iLT )
(E.1)
where 1(L×1) is a vector containing L one’s. ψ is an integer between 0
and LP − 1. For every transmitted symbol the previous information
sequence will be different, so there are LP possible values for s̃. To
ensure that the transmitted power is equal to unity a scaling factor
can be found such that:
P
L −1
1 X h (ψ) (ψ) T i
tr s̃
s̃
= L.
LP ψ=0
(E.2)
A cross correlation matrix can be formed for each of the previous
information sequences. Assuming that each symbol is equally likely
and that ML detection on a symbol by symbol basis is used then an
average cross correlation matrix can be formed:
P
L −1
1 X (ψ)
R̃ss = P
s̃ [s]T
L ψ=0
(E.3)
249
iT
where s = g(t) g(t − T ) · · · g(t − (L − 1)T ) .
The L × L matrix created by (E.3) together with the L × L matrix
Rss = s sT can then be used to find the bit error rate performance.
Given these two matrices, the transmitted symbol set and the
original symbol set can be found as:
h
s̃ = s−1 R̃ss
1
s = [Rss ] 2 .
(E.4)
(E.5)
The ML estimate (3.15) then becomes:
m̂ = arg max [s̃i + n] · sm
(E.6)
1≤m≤L
where n has a variance shown in (3.20) and s̃i is the ith row of s̃
selected at random.
E.2 Matched Filter Detection
In the same way as in (E.1) the complete time domain signals for the
current symbol and the past P symbols needs to be constructed. The
difference here is that s(t) must be defined over the interval
−P LT < T < LT . For unity transmit energy ensure that s̃(ψ) is
scaled so that (E.2) is satisfied.
Each row of s̃(ψ) can be convolved with g(t), the approximated
matched filter:
ỹm (t)(ψ) = s̃(ψ)
m ∗ g(t).
(E.7)
The result of the convolution is sampled at the times of:
t = {T, 2T, · · · , LT }
(E.8)
in order to construct an L × L matrix of samples for each value of ψ:
250


ỹ1 (T )(ψ) ỹ1 (2T )(ψ) · · · ỹ1 (LT )(ψ)


..
..
..
..
.
ỹψ = 
.
.
.
.


(ψ)
(ψ)
(ψ)
ỹL (T )
ỹL (2T )
· · · ỹL (LT )
(E.9)
The variance of the noise at the output of the matched filter will be:
σy2
=
σz2
Z
+∞
−∞
|G(f )|2 df = σz2 Eg
(E.10)
where Eg is the energy of g(t).
Create an averaged matrix ỹ:
P
L −1
1 X ψ
ỹ = P
ỹ .
L ψ=0
(E.11)
The ML estimate (3.15) then becomes:
m̂ = max
"
#
ỹi
p +n
Eg
(E.12)
where n has a variance shown in (3.20) and ỹi is the ith row of ỹ
chosen at random.
E.3 BER Results
Figures E.1, E.2, E.3 and E.4 show the simulated BER performance
for a variety of different pulse approximations.
251
Correlator Receiver
Approx. Matched Filter Receiver
0
0
−1
−1
L=2
−2
−2
N=2 All Pole
−3
N=3 All Pole
N=10 Cascade
M=1, N=4 Pade
M=2, N=6 Pade
−3
log10 Pe
log
10
P
e
N=4 All Pole
−4
−4
Ideal
−5
−5
−6
−6
−7
−8
2
4
6
8
10
E /N
b
12
−7
2
14
4
6
8
10
E /N
0
b
12
14
0
Figure E.1: BER for L = 2 for correlation and approximated matched
receivers.
Correlator Receiver
Approx. Matched Filter Receiver
−1
−1
−1.5
−1.5
L=4
−2
−2
−2.5
−2.5
N=2 All Pole
N=10 Cascade
log
M=1, N=4 Pade
−3.5
M=2, N=6 Pade
e
P
−3
10
N=4 All Pole
log
−3
10
P
e
N=3 All Pole
−3.5
Ideal
−4
−4
−4.5
−4.5
−5
−5
−5.5
2
3
4
5
6
Eb/N0
7
8
9
10
−5.5
2
3
4
5
6
Eb/N0
7
8
9
10
Figure E.2: BER for L = 4 for correlation and approximated matched
receivers.
252
Correlator Receiver
Approx. Matched Filter Receiver
−1
−1
−1.5
−1.5
L=8
−2
−2.5
−2
−2.5
N=2 All Pole
log
10
P
e
N=4 All Pole
N=10 Cascade
−3
M=1, N=4 Pade
log10 Pe
N=3 All Pole
−3
M=2, N=6 Pade
Ideal
−3.5
−3.5
−4
−4
−4.5
−4.5
−5
2
3
4
5
6
Eb/N0
7
−5
2
8
3
4
5
E /N
b
6
7
8
0
Figure E.3: BER for L = 8 for correlation and approximated matched
receivers.
Approx. Matched Filter Receiver
Correlator Receiver
−1
−1.5
−1.5
−2
L=16
−2
−2.5
−2.5
N=2 All Pole
log
M=1, N=4 Pade
P
10
log
e
10
N=10 Cascade
P
N=4 All Pole
e
N=3 All Pole
−3
−3
M=2, N=6 Pade
−3.5
Ideal
−3.5
−4
−4
−4.5
−5
2
−4.5
3
4
E /N
b
5
0
6
7
−5
2
3
4
E /N
b
5
6
7
0
Figure E.4: BER for L = 16 for correlation and approximated matched
receivers.
253
254
F Impulse Approximation
It is impossible to implement a delta-dirac impulse because it is
defined as having infinite height and infinitesimally small width with
unity energy. Any implementation is an approximation of the ideal
impulse. Figure F.1 shows an approximation to an impulse. The
length of the pulse is TI , the rise and fall times are τ and it has an
amplitude of a.
The time domain representation of this pulse is:

a


t

τ
h(t) = a



− a t +
τ
0<t≤τ
aTI
τ
τ < t ≤ TI − τ
(F.1)
TI − τ < t ≤ TI
Using the following two relationships the Laplace domain impulse
response can be formulated.
Z
t2
te−st dt =
t1
Z
e−st1 + t1 se−st1 − e−st2 − t2 se−st2
s2
t2
t1
e−st dt =
e−st1 − e−st2
s
(F.2)
(F.3)
Due to much symmetry in the shape of the pulse the following
simplified form is obtained:
a −sτ
−s(TI −τ )
−sTI
1
−
e
−
e
+
e
(F.4)
τ s2
A plot of this function over frequency for several values of T, with
τ = 10TI , is shown in figure F.2. On this figure the ideal impulse
response is also plotted. It is evident that the approximate pulse is
H(s) =
255
a
TI
τ
t
Figure F.1: Approximation to the impulse function. The length is TI
and the rise and fall times are τ
flat across frequency, up to a certain point. Therefore it can be
expected that the the correct shape of the impulse response can be
obtained albeit with a loss of amplitude.
To find the low frequency constant value of the approximated impulse
response the limit of H(s) as s → 0 is required. One way of doing
this is by using the Taylor expansion of e−x . The first 3 terms of the
2
expansion are required in order to find the limit; e−x = 1 − x + x2 .
Using this expansion on (F.4) results in:
H(s → 0) = a(TI − τ )
(F.5)
(F.5) shows that the amplitude of the impulse response at the output
of the filter is directly related to the area underneath the
approximate pulse in the time domain. This can be verified by
referring to figure F.2. Here it is assumed that a is equal to unity.
From the graph the frequency up to which the approximation is valid
can be found. This is:
fmax ≈ 1/(3TI )
(F.6)
This analysis shows that the approximation of the impulse provides a
constant value in the Laplace domain up to a certain frequency,
256
Attenuation [dB]
0
−50
Ideal
T = 10 µ s
T=1µs
T = 0.1 µ s
T = 0.01 µ s
−100
−150 2
10
3
10
4
5
10
6
10
10
Frequency [Hz]
7
10
8
10
9
10
Figure F.2: Frequency Response of the approximated impulse response
for τ = 10Ti for several values of Ti and a = 1.
which is dependent on the length of the pulse. Provided the pulse
width is small enough then the roll off in frequency can be neglected.
For a triangular pulse where TI = 0 (F.4) simplifies to:
2a
[1 − cos(ωτ )] .
τ s2
In this case the cutoff frequency is:
HT (s) =
fTriangle =
1
2πτ
(F.7)
(F.8)
and the DC gain is:
HT (0) = τ.
(F.9)
257
258
G Inductive Coil
Characterisation
This appendix shows the characterisation of an inductive coil for use
at 33 MHz. The measured results of the characterisation have been
used in order to simulate the behaviour of the inductive
communications link.
G.1 Lumped Element Model
The measurement setup is shown in figure G.1. The vector network
analyser (VNA) is setup to measure the reflection s-parameter, S11 .
These can then be converted into Z parameters.
The impedance function of the transmitter element is:
Z=
s
C
+
R
LC
s2 +
sR
L
+
1
LC
.
(G.1)
The magnitude squared of the impedance at resonance is:
2
2
|Zres | = Q
R2
1
+
ω02 C 2
ω08
.
(G.2)
Assuming that R << ω0 then (G.2) can be simplified and the
magnitude impedance at resonance is then:
|Zres | ≈
Q
.
ω0 C
(G.3)
The resonance frequency and Q can be found from the measured Z11
parameters, thus a value for C can be found. From (G.1) the centre
259
Network Analyser
Lumped element model
of inductive transmitter
RS
L
C
R
Figure G.1: Measurement setup for characterisation of the inductive
transmitter element. The vector network analyser measures the the reflection S parameters, S11 .
frequency and Q are given by:
1
LC
Lω0
,
Q=
R
ω0 = √
(G.4)
(G.5)
which enables L and then R to be found.
G.1.1 Measured Coil Characteristics
For a 10 mm long coil with a radius of 2.5 mm using 1 mm diameter
enamelled copper wire, the measured and approximated impedance of
the coil are shown in figure G.2. The modeled parameters were:
260
L = 190 nH
(G.6)
C = 715 fF
(G.7)
R = 31.15 Ω
(G.8)
Q = 16.5
(G.9)
80
Measured
Approximation
78
76
Impedance [dBohm]
74
72
70
68
66
64
62
60
350
400
450
500
Frequency [MHz]
Figure G.2: Measured and modeled characteristics of a 10 mm long
coil with radius 2.5 mm using 1 mm diameter enamelled
copper wire.
f0 = 432 MHz
(G.10)
With a 100 pF ± 10% placed in parallel with the coil, figure G.3, the
following modeled parameters where obtained:
L = 212 nH
(G.11)
C = 98.5 pF
(G.12)
R = 0.36 Ω
(G.13)
Q = 128.8
(G.14)
f0 = 34.85 MHz
(G.15)
The results of this characterisation show that this coil is suitable to
be used in a transmitting element with a Q of less than 128. The
reason for the large discrepancy between the resistance of the coil at
432 MHz and 35 MHz is probably due to the proximity effect. Wires
261
80
Measured
Approximation
Impedance [dBohm]
70
60
50
40
30
20
10
0
20
40
60
80
100
Frequency [MHz]
Figure G.3: Measured and modeled characteristics of a 10 mm long
coil with radius 2.5 mm using 1 mm diameter enamelled
copper wire. A 100 pF±10% capacitor is placed in parallel
with the coil.
placed in close proximity causes current crowding, which results in
much higher losses. If the coil windings were spaced out then the
proximity effect would be smaller. However, this is not a problem
with this coil at frequencies around 35 MHz.
G.2 Coupling Measurements
To estimate the coupling between the inductors two series R-L
circuits are used. In this case the coil resistance and capacitance can
be neglected, see figure G.4. The coupling constant of this ideal
circuit is given by:
Vout
M sR
=
Vin
L2 s2 + s 2R
+
L
where M is the mutual coupling given by:
262
R 2
L
.
(G.16)
R
Vin
L
L
R
Vout
Figure G.4: Ideal circuit for measuring the coupling constant.
M = kL.
(G.17)
The value of R can be chosen to place the centre frequency of the
filter to 30 MHz. The gain at the centre frequency is given by:
k
kL
= .
(G.18)
2L
2
By measuring the gain at the centre frequency an estimate for the
coupling constant can be found. A resistance value of 50 Ω was used
in the experiment. The Q of this circuit is low (< 0.5) so any
deviation in the centre frequency between the TX and RX coils will
not produce a large error.
G=
The coupling when the antennas are aligned along the axis of the
magnetic field (i.e the coupling between transmit and receive
antennas) was found to be:
k=
16 × 10−9
.
d3
(G.19)
For the perpendicular coils (i.e. between a pair of transmit antennas)
the coupling was found to be:
kp =
6.3 × 10−9
.
d3
(G.20)
The theoretical model predicts that the coupling constant between
axis aligned coils is:
ki =
lR2
2d3
(G.21)
263
were l is the length of the coil and R is the radius of the coil. This
equates to a coupling constant of 31.25 × 10−9 . This is double the
value that was measured. The reason for this error is likely to be due
to the orientation of the two coils during the measurement.
264
H 2nd Order Approximation
For the low pass and bandpass filter the Laplace transfer function
and time domain impulse response for Q >> 1 are:
HLP F (s) =
b2i
s2 + ωQ0 s + ω02
ω0 t
hlpf (t) = bi e− 2Q sin ω0 t
ai s
HBP F (s) = 2 ω0
s + Q s + ω02
ω0 t
hbpf (t) = −ai e− 2Q cos ω0 t
(H.1)
(H.2)
(H.3)
(H.4)
(H.5)
where ω0 is the centre frequency of the filter and Q is the Quality
factor. ai and bi are constants. A second order filter with a
transmission zero may be represented as the sum of low pass and
bandpass filters:
ai s + b 2
HZERO (s) = 2 ω0 i 2
(H.6)
s + Q s + ω0
ω0 t
hzero (t) = Ge− 2Q sin ω0 [t − dt].
(H.7)
where G and dt can be found by considering the vector sum of the
low pass sinusoid and bandpass cosinusoid. When dt << ω2π0 then the
impulse response of the filter with the transmission zero is well
approximated by a scaled and delayed low pass filter:
ω0
ĥzero = Ge− 2Q [t−dt] sin ω0 [t − dt].
(H.8)
265
266
I Transmitter Coupling
In this appendix the statespace model for the coupling between two
adjacent transmitter elements is derived. The impulse response of the
sum of the inductor currents is then found. From this expression the
resulting transfer function in the limit of small and large coupling is
shown.
The coupling of two transmitter elements is shown in figure I.1. For
simplicity assume that the same impulse function is applied to each
of the elements; this is true for the 2nd order Gaussian
approximation. The inductive elements are also assumed to have the
same inductance.
As the inductances are equal the induced voltages can be written as:
Vind1 = kLsi1
(I.1)
Vind2 = kLsi2
(I.2)
Thus the state space model is:
   
˙ 
1
1
− C11R1
0
− C1
0
v1
v1
C1





  
v 2  
0
− C12  v2   C12 
0
− C21R2
   +   IP
 =
k
1
   
i  
0
0 
  i1   0 
 1   L(1−k2 ) − L(1−k2 )
k
1
− L(1−k
0
0
i2
0
i2
2)
L(1−k2 )
(I.3)
IP is the input pulse which is assumed to be the ideal delta-Dirac
function, i.e. IP (s) = 1. This formulation could easily be extended to
a greater number of coupled elements if required. The characteristic
267
Vdd
Vdd
i1
v1
C1
R1
i2
v2
k
L
C2
R2
L
Vind1
Vind2
IP
IP
Figure I.1: Circuit showing the coupling between two transmitter elements.
polynomial of the sum of the inductor currents is given by:
1
1
+
+
Isum (s) = s + s
R1 C1 R2 C2
1
1
1
2
s
+
+
+
LC1 (1 − k 2 ) LC2 (1 − k 2 ) R1 C1 R2 C2
1
1
1
s
+ 2
+
(I.4)
2
2
LC1 C2 R1 (1 − k ) LC1 C2 R2 (1 − k )
L C1 C2 (1 − k 2 )
4
3
When there is no coupling this factors into the product of two
separate state space systems:
Isum (s)|k=0 =
1
1
+
s +s
R1 C1 LC1
2
1
1
s +s
+
R2 C2 LC2
2
(I.5)
This is not the case when k 6= 0. To see the effect of the coupling
constant consider the pole plot shown in figure I.2. This figure shows
that for small coupling constants there is little effect on the pole
positions. When the coupling is greater than 0.5 then there is a rapid
change in the frequency difference between the poles, which causes
the bandwidth of the pulse to increase. It is possible to adjust the
268
8
8
x 10
k=0.9
6
4
k=0
k=0.26
k=0.0166
Imaginary
2
k=0.068
0
−2
−4
−6
−8
−2.23
−2.225
−2.22
−2.215
−2.21
Real
−2.205
−2.2
−2.195
−2.19
6
x 10
Figure I.2: Pole Plot showing how the poles of two transmitting elements vary as the coupling constant k is increased. The
pulse is centred at 33 MHz with T = 1 µS and αT 2 = 4.6.
values of R and L to compensate for the coupling when k is small.
However, when k is less than 10 × 10−3 there is only a small
difference in pole position. Figure I.3 shows the frequency domain
plot for a pulse centred at 33 MHz with T = 1 µS and αT 2 = 4.6 for
a variety of coupling constants. As the coupling constant is increased
the frequency difference between the poles increases and the quality
factor decreases. For k = 1 × 10−3 there is very little difference
between the ideal and coupled responses; the difference cannot be
seen in figure I.3. Therefore, provided that the coupling is of the
order of 1e − 3 then it can be assumed that the transmitting elements
are not coupled, thus simplifying the analysis and implementation of
the transmitter coils. In practice this means that a certain separation
in space between the coils is required.
269
35
k=0
−3
k=1 × 10
30
k=1 × 10−4
−5
k=1 × 10
Magnitude [dB20]
25
20
15
10
5
0
30
31
32
33
34
35
36
Frequency [MHz]
Figure I.3: Frequency response of the a Gaussian approximation with
the pulse centred at 33 MHz with T = 1 µS and αT 2 = 6.
As the coupling constant is increased the frequency difference between the poles increases and the quality factor
decreases. For k = 1 × 10−3 there is very little difference
between the ideal and coupled responses.
270
J 2nd Order Element
Temperature Change
The centre frequency and Q of the transmitting element will be
affected by the temperature coefficients of the capacitor, inductor and
resistor. In this section the change in centre frequency and Q over a
standard temperature range is determined. This shows that the
percentage change in centre frequency and Q due to temperature
changes is small compared to that achievable using tuning techniques.
Therefore, the 2nd order elements may be tuned once at circuit build
and left to run without tuning. The effects of age on the components
has not been taken into account, however, it is typically smaller than
the change due to temperature. The inductive coils used for this
transmitter are custom built. A change in temperature has the effect
of altering the dimensions of the inductor.
The linear dimensions of a metal wire follow the thermodynamic
expansion:
∆x = αx0 ∆T
(J.1)
where ∆x is the change in linear dimension due to thermal expansion
and x0 is the linear dimension at 20 C. α is the expansion coefficient,
which is 17 ppm for copper.
The inductance of a solenoid coil depends on the area of the core and
the length of coil. Thus the inductance after thermal expansion can
be written as:
271
µ0 N 2 πr2 (1 + α∆T )2
= L0 (1 + α∆T ) .
L̂ =
l (1 + α∆T )
(J.2)
The number of turns N is not affected by thermal expansion.
Therefore, the temperature coefficient for the inductance is the same
as for copper, i.e. 17 ppm.
The change in centre frequency of a 2nd order element when using a
combination of fixed and variable capacitors is given by:
fˆ0
=
f0
s
CV + CF
[CV (1 + AV ∆T ) + CF (1 + AF ∆T )] (1 + AL ∆T )
(J.3)
where CF is the fixed capacitance with temperature coefficient AF ,
CV is the variable capacitance with temperature coefficient AV and
AL is the inductor temperature coefficient. For a 2nd order element
where a maximum of 25 % of the capacitance contribution is from a
variable capacitor, the change in centre frequency can be written as:
fˆ0
=
f0
s
1+AV ∆T
4
1.25
.
+ (1 + AF ∆T ) (1 + AL ∆T )
(J.4)
For an NPO ceramic capacitor the temperature coefficient is typically
30 ppm, however a trimmer capacitor (Murata TZC03 series) has a
much larger temperature coefficient of 500 ppm. Over a 55o C
temperature range (J.4) shows a ±0.4 % variation in the centre
frequency. In a similar way the variation in Q can be written as:
Q̂
=
Q
s
(1 + AF ∆T ) + (1 + AV ∆T )
(1 + AR ∆T )
1.25 (1 + AL ∆T )
(J.5)
where AR is the temperature coefficient of the resistance. A typical
trimmer resistor has a temperature coefficient of 100 ppm. Over a 55o
C temperature range (J.5) shows a variation in Q of approximately
±0.85 %.
272
K 2nd Order Element
Tuning
This appendix describes a tuning method to set the centre frequency
of each 2nd order transmitting element. A sense coil is used to
measure the response of the transmitting coil to an impulse response.
The voltage across the sense coil due to current in the transmitter
coil, ignoring gain factors, will be:
v(t) = g(t) sin ωc t
(K.1)
ct
is the exponentially decaying envelope, ωc is
where g(t) = exp − ω2Q
the centre frequency. Multiply (K.1) by I and Q sinusoids, with the
required centre frequency ω0 , and low pass filter the product:
yI = g(t) sin(ωe t + θ)
(K.2)
π
)
(K.3)
2
where ωe = ωc − ω0 is the error in the frequency. θ is the phase of the
reference clock, which is unknown. Taking the integral of yI and yQ
over a time period, T, which is long enough such that it can be
assumed the pulse has decayed to zero results in:
yQ = g(t) sin(ωe t + θ +
Z
0
Z
0
T
yI(t) dt =
2Q [2ωe Q cos(θ) + ωc sin(θ)]
ωc2 + 4ωe2 Q2
T
yQ(t) dt = −
2Q [2ωe Q sin(θ) − ωc cos(θ)]
ωc2 + 4ωe2 Q2
(K.4)
(K.5)
The dependence on the unknown phase of the reference clock can be
273
removed by finding the sum of the squares of (K.4) and (K.5):
16ωe2 Q4 + 4Q2 (ωe + ω0 )2
R=
((ωe + ω0 )2 + 4ωe2 Q2 )2
(K.6)
which when simplified and assuming Q >> 1 results in:
R≈
4Q2
.
4ωe2 Q2 + 2ωe ω0 + ω02
(K.7)
Equation (K.7) has a maximum which occurs when:
ωe =
−ω0
.
4Q2
(K.8)
Therefore, the percentage error in tuning the 2nd Order transmitting
coils using this method can be at best:
100
fe
=
[%].
f0
4Q2
For a Q of 40 the percentage error due to tuning is 0.02 %.
274
(K.9)
L Pulse Generation and
Receiver Schematics
The following pages show the schematics of the circuits used for the
pulse transmitter and receiver. These circuits include the provision of
digital to analogue converters to enable calibration of the centre
frequency and Q of the second order elements. However, this
functionality was not required so the DAC ICs were not fitted.
Both schematics and PCBs were created using Kicad (an open source
programme for circuit development).
275
276
Figure L.1: Top level block diagram of the transmitter and receiver circuits
277
Figure L.2: TX schematic
278
Figure L.3: TX2 schematic
279
Figure L.4: TX3 schematic
280
Figure L.5: TXpower schematic
281
Figure L.6: RXamp schematic
282
Figure L.7: RXamp2 schematic
283
Figure L.8: RX DAC schematic
284
Figure L.9: RXpower schematic
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