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Texas Instruments Operating and Evaluating Quadrature Modulators for Personal Comm Systems Application notes
AN-899 Operating and Evaluating Quadrature Modulators for Personal
Communication Systems
Literature Number: SNAA009
National Semiconductor
Application Note 899
Ruth Umstattd
Wireless Communications
October 1993
ABSTRACT
The quadrature modulator is a fundamental radio component for worldwide digital wireless communication standards
such as Global System for Mobile Communications (GSM),
Digital Communications SystemsÐ1800 MHz (DCS1800),
Digital European Cordless Telecommunications (DECT),
American and Personal Digital Cellular (ADC/PDC), Japanese Personal Handy Phone (PHP RCR28). A typical system application and the theory of operation of the quadrature modulator are described along with two practical evaluation methods. Radio system considerations are discussed
for the quadrature modulator including phase and amplitude
imbalances, DC offsets, time dependent phase errors, modulated carrier harmonics, and nonlinear impairments.
In comparison, for a quadrature modulator information is
modulated onto a carrier with full freedom to manipulate the
amplitude, phase, or frequency of the carrier. A quadrature
modulator is implemented with a phase shifter, two mixers,
and a signal combining stage. Figure 2 shows a quadrature
modulator performing QPSK.
INTRODUCTION
Worldwide standards for the mobile communications environment are demanding spectrally efficient digital modulation techniques. Low power, low cost, integrated quadrature
modulators can be used in wireless personal communications for digital modulation (i.e., QPSK (Quadrature Phase
Shift Keying), q/4DQPSK (Pi over 4 Differential Quadrature
Phase Shift Keying), MSK (Minimum Shift Keying), GMSK
(Gaussian filtered Minimum Shift Keying)). The quadrature
modulator has the advantage that any parameter of a carrier
frequency (amplitude, phase, or frequency) can be simultaneously manipulated to represent information. Other modulators do not have this flexibility. For example, a single mixer
can be used to modulate a carrier. In this case, an information signal is limited to modulating a carrier by altering its
amplitude (or phase by 180§ ) or its frequency. Figure 1 illustrates a mixer performing a simple digital modulation technique, binary phase shift keying (BPSK), that consists of
shifting the phase of a carrier by 180§ (or changing the amplitude of a carrier between a positive and a negative state).
The result is two information states; where one could represent bit 0 and the other state could represent bit 1. The
constellation diagram of Figure 1 illustrates the two states
along one axis.
TL/W/11877 – 1
TL/W/11877 – 2
FIGURE 2. Quadrature Modulator Performing QPSK
Even bits of the information signal are directed to one mixer
and the odd bits are directed to the second mixer. The LO is
shifted by 90§ at one mixer. For QPSK, the result at the
output of the modulator is four states of information where
each state represents two bits. Compared to Figure 1, the
quadrature modulator transmits the same information in half
the time. QPSK is only one modulation scheme that can be
employed by the quadrature modulator. The information
bearing signal inputs can vary so that a vector from the
origin of the constellation diagram can have any amplitude
and phase value at any point in time. The quadrature modulator is a flexible modulator that is used with spectrally efficient modulation schemes.
Historically, a discrete version of the quadrature modulator
has been used in sophisticated communication systems.
Currently, the cellular and cordless telephony market demands an integrated quadrature modulator suitable for high
volume production. In this document, the description, operation, and evaluation of the quadrature modulator is given
from the current commercial applications perspective.
Operating and Evaluating Quadrature Modulators
for Personal Communication Systems
Operating and Evaluating
Quadrature Modulators for
Personal Communication
Systems
TYPICAL SYSTEM APPLICATION
The quadrature modulator provides the necessary interface
between digital data and the RF transmission channel. Figure 3 shows a typical transmit system block diagram using a
direct conversion quadrature modulator.
FIGURE 1. Mixer Performing BPSK
AN-899
C1995 National Semiconductor Corporation
TL/W/11877
RRD-B30M75/Printed in U. S. A.
TL/W/11877 – 3
FIGURE 3. Transmit System Block Diagram
This equation can be written in phasor format:
v(t) e k a(t) cos[0ct b x(t)]
A stream of information from the microprocessor is compressed and sent in bursts by the burst mode controller to
the baseband processor. The baseband processor mathematically manipulates the signal using a digital filter and a
digital to analog converter (DAC) to generate the in-phase
signal [i(t)] and quadrature signal [q(t)] for the modulating
inputs of the quadrature modulator. The local oscillator (LO)
input to the modulator is supplied by a frequency synthesizer that is composed of a phase detector, a low pass filter,
and a tunable VCO. The frequency of the LO or carrier
changes depending on the selected transmit channel. The
output of the modulator, the carrier frequency modulated by
the i(t) and q(t) signals, is amplified by a power amplifier,
filtered and then radiated through an external antenna.
(2)
where:
(3)
a(t) e 0 i2(t) a q2(t)
(4)
x(t) e tanb1 [q(t)/i(t)]
Figures 4, 5, and 6 are examples of three quadrature modulation schemes: QPSK, q/4DQPSK, and GMSK. The i(t),
q(t), the eye diagrams of i(t) and q(t), the frequency spectrum, and the constellation for each modulation scheme is
shown. All of these signals are plotted on the same time or
frequency scale for the sake of comparison. Sections a)
through d) of Figures 4 and 5 have a voltage by symbol time
reference. Sections a) through d) of Figure 6 have a voltage
by bit time reference. The dashed lines in the constellation
diagrams represent state transitions.
A raised cosine filter is used to shape the modulating signals before they enter the quadrature modulator. This reduces the bandwidth of the modulated signal out of the
quadrature modulator. A baseband processor is responsible
for this filtering not the quadrature modulator. For the sake
of understanding the input modulating signals, the equation
for this filter is shown below where T is the bit time normalized to 1 and a, the filter roll-off, is equal to 0.5 [2].
THEORY OF OPERATION
A typical monolithic implementation of a quadrature modulator consists of two multipliers and a signal combining amplifier stage. Each of the multipliers is provided an LO and a
modulating signal. One of the LO signals is shifted by 90§
with respect to the other LO signal at the inputs of the two
multipliers. The information signals i(t) and q(t) are each
mixed with one of the LO signals. The output signals of the
two multipliers are combined and amplified. Equation (1) describes the modulated output of an ideal quadrature modulator where i(t) is the in phase information signal, q(t) is the
quadrature information signal, cos(0ct) is the LO signal for
one mixer, sin(0ct) is the LO signal for the other mixer (Note
1), and k is a constant gain or loss.
v(t) e k [i(t) cos(0ct) a q(t) sin(0ct)]
(1)
h(t) e
Note 1: The LO signal is modeled as a sinusoidal signal for the sake of
simple mathematical demonstration The quadrature modulator limits the LO signal making a square wave representation of the LO
more precise.
2
sin (qt/T)
qt/T
cos(qta/T)
1 b 4(at/T)2
(5)
a) i(t)
b) q(t)
TL/W/11877 – 4
TL/W/11877 – 5
c) Eye Diagram of i(t)
d) Eye Diagram of q(t)
TL/W/11877 – 6
TL/W/11877 – 7
e) Frequency Spectrum
f) Constellation Diagram
TL/W/11877 – 9
TL/W/11877 – 8
FIGURE 4. QPSK Modulation
3
A raised cosine filter (Equation (5) is also used to shape the
modulating signals in Figure 5, a q/4DQPSK modulation
scheme. The modulating signals are differentially encoded
and prevent factors of a 90§ state transition. The spectral
efficiency of q/4DQPSK compared to QPSK is the same.
The advantage of q/4DQPSK over QPSK is that it reduces
the amplitude modulation of the modulated signal by preventing a zero crossing in the constellation.
a) i(t)
b) q(t)
TL/W/11877–10
TL/W/11877 – 11
c) Eye Diagram of i(t)
d) Eye Diagram of q(t)
TL/W/11877–12
TL/W/11877 – 13
e) Frequency Spectrum
f) Constellation Diagram
TL/W/11877 – 15
TL/W/11877–14
FIGURE 5. q/4DQPSK Modulation
4
The modulating signals shown in Figure 6 are Gaussian filtered. The equation used for the Gaussian filter, normalized
to a bit time (T) equal to 1, is given below [2]:
h(t) e B
0
2q
#e
ln(2)
GMSK is less spectrally efficient compared to QPSK or
q/4DQPSK. The advantage is seen in the constellation diagram. GMSK is strictly frequency modulation; there is no
amplitude modulation in a perfect GMSK signal.
b 2(Btq)2
ln(2)
(6)
a) i(t)
b) q(t)
TL/W/11877 – 16
TL/W/11877 – 17
c) Eye Diagram of i(t)
d) Eye Diagram of q(t)
TL/W/11877 – 18
TL/W/11877 – 19
e) Frequency Spectrum
f) Constellation Diagram
TL/W/11877 – 21
TL/W/11877 – 20
FIGURE 6. GMSK Modulation
5
One way to approach the choice of DC signal inputs is to
make instantaneous measurements that reflect the modulation scheme chosen for an application. For example, GMSK
is a constant envelope, frequency modulation scheme. This
translates by equation (2) to a vector with a(t) having a constant amplitude and x(t) rotating at the modulating frequency rate (or (/4 the bit rate). In this DC static test case, x(t) is
not a function of time. Instead this angle is determined by
tan-1 [q(i)/i(i)]. In order for a(i) to have a constant amplitude and x(i) to have a known angle, q(i) and i(i) can take
on the following instantaneous values:
i(i) e Vm cos i
(9)
EVALUATING QUADRATURE MODULATORS
For digital radios, an evaluation system for the quadrature
modulator would contain the following: a modulating signal
source that would generate the desired modulation scheme,
a local oscillator with, if needed, a highly accurate phase
shifter and power splitter, and an accurate demodulator that
would provide symbol error ratio (SER) results. Current test
and measurement equipment does not support a practical
solution for high volume production test of SER. Instead the
quadrature modulator is evaluated based on equations (1)
and (2). This provides the freedom of evaluating the amplitude and phase response of the quadrature modulator for a
variety of modulating input signals.
q(i) e Vm sin i
(10)
where Vm is the peak to peak modulating voltage and i is
the variable angle of the phasor both of which are determined by the user. Using equations (2), (3), and (4),
a(i) e Vm
(11)
STATIC MODULATING SIGNAL EVALUATION
One evaluation method for the quadrature modulator is to
use a vector network analyzer in conjunction with DC voltages as the modulating input to generate phasor information at the output of the modulator. Figure 7 describes the
necessary setup.
(12)
x(i) e i
v(t,i) e k Vm cos((0ct b i)
(13)
This allows for making discrete measurements of the points
identified by the phasor at a predetermined angle. If the
quadrature modulator is ideal and the input signals are accurate, the measurement points identity the trace of a perfect circle. Error terms like amplitude imbalance, phase imbalance, carrier leakage distort the circle in linear and nonlinear fashions. The data taken from a quadrature modulator
using a static modulating signal evaluation can be manipulated to predict frequency spectrum characteristics. An analysis of these error terms and the resulting effects are given
in the system considerations section of this document. The
single sideband up-conversion evaluation of the quadrature
modulator is necessary to circumvent the mathematics of
determining the frequency spectrum.
SSB UP-CONVERSION EVALUATION
The single sideband (SSB) up conversion test is currently
the most widely used evaluation method for the integrated
quadrature modulator. The implementation of this measurement can be done with a frequency synthesizer (for the LO
source), a spectrum analyzer, a dual arbitrary waveform
generator (for i(t) and q(t)), and an external coupler (if needed) for the LO phase shift and power split. Figure 5 illustrates the setup.
TL/W/11877–22
FIGURE 7. DC Static Test Setup
A vector network analyzer is set up in a continuous wave
mode to provide the LO source for the modulator. An external coupler is used to split and phase shift the LO signals for
the inputs of the modulator. (If the quadrature modulator
has an on board phase shifter, the external coupler is not
necessary). The output of the modulator is connected to the
second port of the network analyzer. The network analyzer
is set up for polar output of the forward transducer gain.
Manually changing the DC signals at the modulating input
ports rotates an information vector through the different
quadrants on the polar output of the network analyzer. The
DC voltage at the modulating input ports is the in-phase or
quadrature signal plus the necessary offset voltage VREF.
x(i) e i(i) a VREF
(7)
y(i) e q(i) a VREF
(8)
TL/W/11877 – 23
FIGURE 8. SSB Test Setup
6
In theory, this test is very much like the static modulating
signal test except that the modulating signals are sinusoids:
i(t) e Vm cos(0mt)
(14)
discrete frequency at f c b f m. The location of this impulse
can be changed to f c a f m by reversing the phase shift at
either the LO ports or the modulating ports.
(15)
q(t) e Vm sin (0mt)
Using equations (2), (3), and (4),
(16)
a(t) e Vm
x(t) e 0mt
(17)
v(t) e k Vm cos (0ct b 0mt)
(18)
Using equation (1),
v(t) e k Vm cos(0mt) cos(0ct) a k Vm sin(0mt) sin(0ct)
(19)
The modulating frequency for an SSB test is chosen to represent the peak frequency deviation of a given application.
For example, the bit rate of a GMSK signal for GSM is
270.8 kb/s. In this minimum shift keying (MSK) case, the
peak frequency deviation is one quarter the bit rate [5]. The
choice of modulating frequency for an SSB test for GSM is
67.7 kHz.
When making SSB measurements of an actual quadrature
modulator, the carrier, the undesired sideband, up-converted modulating signal harmonics, and harmonics of the modulated carrier are present in the frequency spectrum along
with the desired sideband. Figure 6 is an example of SSB
test data of a quadrature modulator. (Note 2)
Ð
1
0c b 0m)t a cos(0c a 0m)t
k Vm cos(
a cos(0c b 0m)t b cos(0c a 0m)t
2
which can be reduced to:
v(t) e k Vm cos((0c b 0m)t)
v(t) e
( (20)
(21)
For an ideal quadrature modulator, the output for this SSB
evaluation in the positive frequency spectrum would be a
TL/W/11877 – 24
FIGURE 9. SSB Test Data
Single sideband test data of a quadrature modulator includes traces of the undesired sideband, the carrier, up-converted modulating signal harmonics, and modulated carrier
harmonics. All of these terms are of critical interest to the
radio system designer. They can be correlated to symbol
error rate and adjacent channel interference. The sources
of performance degradation and features of the quadrature
modulator that must be addressed in a radio system design
are discussed in the following sections.
SYSTEM CONSIDERATIONS
The performance of the quadrature modulator is a function
of the accuracy of the quadrature modulator and the accuracy of the input signals. Imperfections in the input signals and
the quadrature modulator result in vector state amplitude
error resulting in unwanted envelope fluctuations, vector
state phase error that inhibit optimal demodulating sampling
time, and unfilterable out of band signals that may interfere
with other channels. The sources and affects of linear errors, slew rate limitations, modulated carrier harmonics, and
nonlinear impairments are shown for the quadrature modulator.
Note 2: The harmonics of the modulated carrier are outside the frequency
span of this plot.
7
This equation is expanded and then reduced to see the converted frequency terms.
Linear Errors
Sources of linear error consist of phase imbalance, amplitude imbalance, and DC offsets associated with the carrier
signal and the modulating signal. Each of these terms is
discussed and included in an equation to model the effects
of these error terms.
There are two different phase errors to model that are introduced either by the user’s signal input or by the quadrature
modulator. One phase error is introduced by imperfect
quadrature at the LO signal. This term is fixed at a given
carrier frequency and carrier power unless the quadrature
modulator has an LO phase adjust. The second phase error
is introduced at the modulating signals. During a SSB test,
the modulating signal phase error is controlled by the user.
For ideal operation, the quadrature modulating signals is
shifted by 90§ with respect to the in phase signal.
Amplitude imbalances are associated with the two LO signal
inputs and the two modulating signal inputs. Ideally, the two
LO signals are equal in amplitude and the two modulating
signals are equal in amplitude. The amplitude balance of the
two LO signals is far less critical than the balance of the two
modulating signals. (The LO signals operate transistors as
switches in the modulator while the modulating signals operate transistors in a linear fashion.)
DC offsets are introduced in both the LO signals and the
modulating signals. For integrated applications, the LO signal DC reference is set internally and the modulating signal
DC reference is set externally. DC offsets arise from improper supply of the DC reference and mismatch in transistors
internal to the quadrature modulator. The effect of a DC
offset is modeled as a ratio of the offset over the peak-topeak voltage swing. In an ideal case, this ratio is equal to
zero.
Equation (22) combines phase error, amplitude imbalance,
and DC offset errors into Equation (19) to predict vector
state amplitude error, vector state phase error, undesired
sideband suppression, and carrier suppression. The carrier
phase error is a. The modulating phase error is b. The carrier amplitude balance ratio is Kc. The modulating signal amplitude balance ratio is Km. The DC offset divided by the
peak-to-peak voltage swing at the carrier is Dc. The DC offset divided by the peak-to-peak voltage swing at the modulating signal is Dm. The error terms, Kc, Km, a, and b, are
lumped into half of the Equation (19) since they are relative
to the other half of the equation. The DC offset terms are
independent so Dc and Dm are divided into Dc1, Dc2 and
Dm1, Dm2.
v(t) e
Kc Km [sin(0ct a a) a Dc1] [sin(0mt a b) a Dm1]
k Vm
a [cos(0ct) a Dc2] [cos(0mt) a Dm2]
(22)
Ð
v(t) e
(/2 cos(0ct a 0mt) (1 b Km Kc cos(a a b))
a (/2 cos(0ct b 0mt) (1 a Km Kc cos(a b b))
a (/2 sin(0ct a 0mt) (Km Kc sin (a a b))
b (/2 sin(0ct b 0mt) (Km Kc sin (a b b))
a Km Kc Dc1 sin(0mt) cos b
a Km Kc Dc1 cos(0mt) sin b
k Vm
a Dc2 cos(0mt)
a Km Kc Dm1 sin(0ct) cos a
a Km Kc Dm1 cos(0ct) sin a
a Dm2 cos(0ct)
a Km Kc Dm1 Dc1
a Dm2 Dc2
(23)
The desired sideband term is a function of 0c b 0m. The
vector state amplitude error (in dB) and the vector state
phase error (in degrees) of the desired sideband are given
in Equations (24) and (25).
a(t)error e
10 log
# 2 #1
1
a 2 Km Kc cos(a b b) a Km2 Kc2
x(t)error e tanb1
Ð1
b Km Kc sin(a b b)
a Km Kc cos(a b b)
J J(24)
(
(25)
Equations (26) and (27) demonstrate the effects of these
error terms on sideband suppression and carrier suppression (both in dB) [3].
Sideband
Suppression e
10 log
1 b 2Km Kc cos(a a b) a Km2 Kc2
a 2Km Kc cos(a b b) a Km2 Kc2
Ð1
(
Carrier
Suppression e
Dm22 a 2Km Kc Dm1 sin a a Km2 Kc2 Dm12
10 log
(/2 (1 a 2Km Kc cos(a b b) a Km2 Kc2)
Ð
(26)
((27)
For example, if the LO coupler has a 3§ phase error and the
DC references at the modulating signals are accurate to
g 40 mV with a 1 VPP modulating signal, and the amplitude
ratio of the modulating signals is about 1.1, then the vector
state amplitude error is 0.24 dB; the vector state phase error
is 4.4§ ; the undesired sideband suppression is b25.3 dB;
the carrier suppression is b24.7 dB. To verify the effects of
various error terms on the undesired sideband suppression
and carrier suppression, an SSB evaluation can be used. To
verify the effects of these error terms on a(t) and x(t), the
DC static evaluation can be employed with Equations (28)
and (29).
(
8
Carrier
Vector
Amplitude e
Dm22 a 2 Km Kc Dm1 sin a a Km2 Kc2 Dm12
10 log
(/2 (1 a 2 Km Kc cos(a b b) a Km2 Kc2)
Ð
Time Dependent Phase Error
One other source of phase error that can be introduced to
the quadrature modulator is a time dependent phase error
associated with the slew rate limitations of the modulating
signals. As the peak-to-peak voltage of the modulating signal increases, the resistive and capacitive characteristics of
the device prevent full signal charge development. This results in a time dependent phase delay with a maximum
phase error occurring at the peak modulating voltage. The
effect of this error can be reduced by reducing the peak-topeak modulating voltage.
(
(28)
Carrier
Vector
Phase e
tanb1
ÐD
Km Kc Dm1 cos a
m2 a Km Kc Dm1 sin a
(
Modulated Carrier Harmonics
One source of modulated carrier harmonics is the internal
input transistors at the LO input of the quadrature modulator
that operate in a switching fashion. The quadrature modulator is effectively multiplying a square wave for the LO by the
modulating signals. Equation (30) represents a square wave
where A is the amplitude and 0o is the fundamental frequency.
(29)
Since the DC static measurements are made at the LO frequency, undesired carrier feed-thru will affect the measurement of a(i) and x(i). The undesired carrier feed-thru effectively adds as a constant vector, with an amplitude (in dB)
found in Equation (28) and a phase found in Equation (29),
to the vector defined by a(i) and x(i) (Figure 10).
4A
# sin 0 t
J
1
1
sin 30ot a
sin 50ot a . . .
3
5
(30)
Any signal that is modulated onto the carrier will also be
modulated onto the 3rd carrier harmonic and the 5th carrier
harmonic at reduced power levels. Most often, the band
limiting characteristics of the quadrature modulator prevent
the exit of these signals from the device. If this is not the
case, the signal about the 3rd and 5th harmonics must be
filtered out in the radio system.
f(t) e
q
o a
Nonlinear Impairments
Nonlinear impairment is associated with the modulating signals. Ideally, the modulating signals operate transistors in a
linear region. Deviations from this linear region arise during
single ended operation of the modulating signal inputs and
from overdriving the modulating signal voltages (Note 3).
For a memoryless two-port network with weak nonlinearity,
the output can be represented by a power series of the input
as
2
3
vo e k1 vi a k2 v i a k3 v i a . . .
(31)
TL/W/11877 – 25
FIGURE 10. Effect of Carrier Vector on a Measured
Vector for DC Static Evaluation
For the example given above, a carrier transmission vector
with an amplitude of 0.5 dB and a phase of 3.3§ are included
as an error source in a DC static measurement of a(t) and
x(t). The carrier transmission vector offset the center of the
circle traced in a DC static measurement. This constant vector is subtracted from the measurement to obtain a(i) and
x(i). The remaining errors in this vector are a(t)error and
x(t)error.
For a sinusoidal input,
vi e A cos 01t
the output is
1
3
k2 A2 a k1 A a
k3 A3 cos 01t
vo e
2
4
1
1
a
k2 A2 cos 201t a
k3 A3 cos 301t
2
4
#
Note 3: Most integrated quadrature modulators can be used differentially or
single-endedly. Single-ended operation eliminates the reflections of
the in-phase and quadrature input signals at a performance cost.
9
J
(32)
(33) [7]
Operating and Evaluating Quadrature Modulators
for Personal Communication Systems
Equation (33) represents the harmonics of a sinusoidal
modulating signal generated by nonlinearities. This means
that not only is the desired modulating signal up-converted
but the harmonics of this modulating signal are up-converted as well. These harmonics are kept to a minimum by operating the modulating signal ports differentially and within the
recommended peak-to-peak voltage.
CONCLUSION
The quadrature modulator is a universal modulator that is
used in contemporary digital wireless communications.
Evaluating and utilizing quadrature modulators requires an
understanding of the fundamental operation of the quadrature modulator along with the performance constraints. The
error terms that influence the quadrature modulator are interrelated and can be modeled by equations to predict performance. Other features of the quadrature modulator have
been discussed to guide a user in applying the quadrature
modulator in his or her radio.
ACKNOWLEDGMENTS
Special thanks goes to D. Fague, B. Madsen, and D. Bien
for their critique and contributions to this article.
REFERENCES
[1] P. Gray and R. Meyer, Analysis and Design of Analog
Integrated Circuits. John Wiley & Sons, 2nd ed., 1984.
[2] D. E. Fague, ‘‘An Integrated Baseband Processor for
Universal Personal Communications Applications’’,
Proceedings from the Third IEEE International Symposium on Personal, Indoor and Mobile Radio Communications , October, 1992, Boston, MA.
[3] J. W. Archer, J. Granlund, and R. E. Mauzy, ‘‘A BroadBand UHF Mixer Exhibiting High Image Rejection over
a Multidecade Baseband Frequency Range’’, IEEE J.
Solid-State Circuits, vol. SC-3, pp. 365 – 373, Dec.
1968.
[4] D. E. Norton, S. A. Massa, and P. O’Donovan, ‘‘I and Q
Modulators for Cellular Communications Systems’’, Microwave Journal, pp. 63 – 80, Oct. 1991.
[5] S Pasupathy, ‘‘Minimum Shift Keying: A Spectrally Efficient Modulation’’, IEEE Communications Magazine,
pp. 14 – 22, 1979.
[6] A. Papoulis, Circuits and Systems, A Modern Approach.
Holt, Rinehart and Winston, Inc., New York, 1980.
[7] A. Dao, ‘‘Integrated LNA/Mixer Basics’’, National
Semiconductor Application Note, AN-884, April, 1993.
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