Dynamic nonlinear pre-distortion of signal generators for improved dynamic range ITB/Electronics

Dynamic nonlinear pre-distortion of signal generators for improved dynamic range  ITB/Electronics
ITB/Electronics
Dynamic nonlinear pre-distortion of signal generators
for improved dynamic range
Suzan Jawdat
June 2009
Master’s Thesis in Electronics/Telecommunication
Supervisor : Dr. Niclas Björsell
Examiner : Dr. Magnus Isaksson
Abstract
In this thesis, a parsimoniously parameterized digital predistorter is derived for
linearization of the IQ modulation mismatch and the amplifier imperfection in the signal
generator [1]. It is shown that the resulting predistorter is linear in its parameters, and
thus they may be estimated by the method of least-squares. Spectrally pure signals are an
indispensable requirement when the signal generator is to be used as part of a test bed.
Due to the non-linear characteristic of the IQ modulator and power amplifier, distortion
will be present at the output of the signal generator. The device under test was the IQ
modulation mismatch and power amplifier deficiencies in the signal generator.
In [2], the dynamic range of low-cost signal generators are improved by employing
model based digital pre-distortion and the designed predistorter seems to give some
improvement of the dynamic range of the signal generator.
The goal of this project is to implement and verify the theory parts [1] using data program
(Matlab) to improve the dynamic range of the signal generator. The design digital predistortion that is implemented in software so that the dynamic range of the signal
generator output after predistortion is superior to that of the output prior to it. In this
project, we have observed numerical problems in the proposed theory and we have found
other methods to solve the problem.
The polynomial model is commonly used in power amplifier modeling and predistorter
design. However, the conventional polynomial model exhibits numerical instabilities
when higher order terms are included, we have used the conventional and orthogonal
polynomial models. The result shows that the orthogonal polynomial model generally
yield better power amplifier modeling accuracy as well as predistortion linearization
performance then the conventional polynomial model.
i
Acknowledgements
Firstly, I would like to give my deep and sincere gratitude to my supervisor, Dr. Niclas
Björsell, for giving me the opportunity to perform this thesis and helping me during my
work.
I would like to thanks Prof. Peter Händel for his wonderful project description.
I also would like to thanks, Charles Nader, Per Landin and Carl Karlsson for helping me
in my project.
Finally, I would like to gratitude my family they had always helped and supported me
during my live and the academy.
ii
Contents
1 Introduction...........................................................................................1
1.1 Background.......................................................................................................1
1.2 Objective ...........................................................................................................3
1.3 Thesis outline ....................................................................................................3
2 Theory .................................................................................................... 4
2.1 Introduction ......................................................................................................4
2.2 Linear systems ..................................................................................................6
2.2.1 Predistortion ..........................................................................................6
2.2.2 Analog predistortion .............................................................................7
2.2.3 Digital predistortion..............................................................................8
2.3 The baseband equivalent ................................................................................8
2.4 Predistorter design based on physical modelling .........................................10
2.4.1 IQ imbalance.........................................................................................12
2.4.2 Predistortion mirror distortion...........................................................14
2.4.3 Power amplifier deficiencies................................................................16
2.5 Parsimonious predistorter structures.............................................................18
2.5.1 Linear part of the predistorter...........................................................19
2.5.2 Non linear part of the predistorter ....................................................21
2.6 Predistorter properties ....................................................................................22
2.7 Training of the predistorter based on N-sequences of data .........................24
2.7.1 Model output........................................................................................24
2.7.2 Least-squares problem....................................................................... 27
2.7.3 Determination of the function β (⋅) ....................................................28
3 Method ................................................................................................... 29
3.1 Introduction ......................................................................................................29
3.2 Test setup ..........................................................................................................29
3.2.1 The implemented system......................................................................29
3.2.2 Vector Signal Generator ......................................................................31
iii
3.2.3 Vector Signal Analyzer .........................................................................32
3.3 Simulation model..............................................................................................33
3.4 Determinate parameters ..................................................................................34
3.4.1 Gamma (γ ) ...........................................................................................34
3.4.2 Model order (M & L ) ............................................................................34
3.4.3 Basis function ψ m (⋅) or φ m (⋅) ..............................................................35
3.4.4 Realization.............................................................................................35
4 Results .................................................................................................... 36
4.1 Introduction ......................................................................................................36
4.2 Simulation results.............................................................................................36
4.3 Measurement results ........................................................................................43
5 Discussion............................................................................................... 48
6 Conclusions ............................................................................................ 49
7 References .............................................................................................. 50
iv
List of Figures
Figure 1.1. The implemented system............................................................................. 2
Figure 2.1. IQ signal transmission ................................................................................ 5
Figure 2.2. Predistortion main ideas ............................................................................. 7
Figure 2.3. Baseband equivalent................................................................................... 9
Figure 2.4. Digital predistortion design ........................................................................ 11
Figure 2.5. IQ imperfection in the signal generator...................................................... 12
Figure 2.6. Digital predistortion design for IQ imbalance............................................ 13
Figure 2.7. Training of the digital predistorter F ( s n ;θ ) . .............................................. 15
Figure 2.8. Digital predistortion design for power amplifier........................................ 16
Figure 2.9. Digital predistorter where input signal x n produces output signal z n .......... 21
Figure 3.1. Calibration set-up ....................................................................................... 30
Figure 3.2. Hardware from IF to the process ................................................................ 32
Figure 3.3. Gamma (γ ) ................................................................................................. 34
Figure 4.1. Power spectrum of two-tone signal without predistortion ......................... 37
Figure 4.2. Power spectrum of two-tone signal with predistortion............................... 38
Figure 4.3. Power spectrum of two-tone signal φ m (⋅) function without predistortion .. 39
Figure 4.4. Power spectrum of two-tone signal φ m (⋅) function with predistortion ....... 40
Figure 4.5. Power spectrum of two-tone signal ψ m (⋅) function without predistortion.. 41
Figure 4.6. Power spectrum of two-tone signal ψ m (⋅) function with predistortion ....... 42
Figure 4.7. Power spectrum of two-tone signal φ m (⋅) function without predistortion .. 44
Figure 4.8. Power spectrum of two-tone signal φ m (⋅) function with predistortion ....... 45
Figure 4.9. Power spectrum of two-tone signal ψ m (⋅) function without predistortion... 46
Figure 4.10. Power spectrum of two-tone signal ψ m (⋅) function with predistortion ....... 47
v
Abbreviations
ADC
Analog to digital converter
AWG
Arbitrary waveform generator
ATT
Attenuation
DAC
Digital analog conversion
DUT
Device under test
FT
Fourier transfer
IF
Intermediate frequency
IQ
In-phase Quadrature phase
LAN
Local area network
LO
Local oscillator
LUT
Look up table
NCO
Numerical controlled oscillator
PA
Power amplifier
PC
Personal computer
PD
Pre-distortion
RBW
Resolution bandwidth
REF
Reference level
RF
Radio frequency
SA
Signal analyzer
SG
Signal generator
VBW
Video bandwidth
vi
1 Introduction
1.1 Background
When measuring the performance of high-quality components, such as analog to digital
converters (ADC) and power amplifiers (PA), one must ensure that the test setup has
superior performance compared to the device under test (DUT). In some test setups the
signal generator (SG) is the weak link. Even state-of-the-art signal generators can have
problem to generate spectrally pure enough signals for some applications. Nonlinearities
and other imperfection in the generator results in problems with harmonic distortion and
intermodulation products in the generated signal. Today, most of the electrical signals are
processed in the digital system performance, and as a result of the ADC present on the
border to the digital domain [3].
The evolution of digital signal processing enables cost-efficient trade-offs between
performance boosting by digital operations and cost-reduction by reduction the
requirements on the analog hardware. Even though the SG hardware is insufficient,
spectrally pure signals can be generated by using software. A modern SG is equipped
with an arbitrary waveform generator (AWG), where the waveform is a time series
created in a computer program for example (Matlab). A method is to generate the wanted
signal and measure the actually generated signal. Thereafter calculate a pre-distorted
(PD) signal that is adjusted to compensate for the distortion. The device under test was
the IQ modulation mismatch and the amplifier imperfection in the signal generator. We
consider digital pre-distortion of radio frequency signal generators by means of
parametric dynamic modelling of the inverse of the nonlinear artifacts, in order to
improve the dynamic range of the analyzer.
1
The aim of this project is to implement and verify the theory parts using data program
(Matlab) in software to give some improvement of the dynamic range of the signal
generator. The signal is measured by a signal analyzer (SA) that is connected to a person
Figure 1.1. The implemented system
computer (PC) via a local area network (LAN) as shown in Figure 1.1. The same PC is
used to generate time series to the SG (that also is connected to the LAN).
A parsimoniously parameterized digital pre-distorter is derived starting with ’grey box’
models of the IQ modulation mismatch and the amplifier imperfections. The pre-distorter
may be estimated by the method of least-squares. The resulting pre-distortion algorithm –
containing of linear time invariant filters, summations and multiplications, only – is
expected to handle frequency varying IQ modulation mismatch and amplifier deficiencies
with memory [1].
2
1.2 Objective
In this project, we will implement an automatic method for generating spectrally pure
signals using digital pre-distortion design. This works includes theory parts from [1], my
task is to implement and verify the theory parts using data program (Matlab) in software
to give some improvement of the dynamic range of the signal generator, under an
assumption that the distortion produced by the signal generator dominates the distortion
by the instruments in the set-up.
1.3 Thesis Outline
The first Chapter of this thesis is introduction that provides general information about the
content of the thesis. It intends to set the introduction, background and the objective of
this project. The second Chapter includes all the theoretical and mathematical
information required to understand the work. The third Chapter contains the technical and
theoretical information that was used to have the appropriate test setup, simulation and
measurement procedure along the whole thesis. Chapter fourth includes the simulation
and measurement results. Chapter fifth presents the analyze and discussion of the results.
3
2 Theory
2.1 Introduction
In-phase quadrature (IQ) signal processing is a widely used tool in modulated systems
and radio communications in order to take full advantage of the available resources (such
as the transmission bandwidth). In communications signal processing, it is common to
use the notation of complex-valued signal. As an illustration, two oscillator signals with a
90º phase difference, cos(2πf 0 t ) and sin( 2πf 0 t ) , can be conveniently modelled as a
complex oscillator cos( 2π f 0 t ) + j sin( 2π f 0 t ) = e j 2 πf 0 t . In practice, a complex-valued
signal is simply a pair of two real-valued signals carrying the real and imaginary parts.
The benefit of employing and processing complex-valued signals is most conveniently
described in frequency domain. For a real-valued signal, say x(t ) , the Fourier transform
(FT) X ( f ) obeys the Hermitian symmetry, i.e., X (− f ) = X ∗ ( f ) where the superscript
(⋅)∗
denotes complex conjugation [4]. In radio communications, the concept of complex-
valued or IQ signals was initially enabled and justified by the virtue of bandpass signal
transmission. In general, using the lowpass–to-bandpass transformation, a complexvalued baseband signal z (t ) = z I (t ) + z Q (t ) can be transmitted in a real-valued channel.
[
]
U (t ) = Re z (t )e jwt = z I (t ) cos (wt ) − z Q (t )sin (wt )
4
Z I (t )
cos(wt )
U (t )
Z Q (t )
sin(wt )
Figure 2.1. IQ signal transmission
where w denotes the carrier angular frequency in rad /s and Re[x] refers to the real part of
a complex-valued quantity x, according to Figure 2.1 two real-valued messages z I (t ) and
z Q (t ) can be transmitted over the same bandwidth resulting in increased spectral
efficiency. The IQ signal processing has the ability to process the negative and positive
frequencies separately is strictly valid only if the I and Q branches are perfectly matched
but some unintentional variations between the amplitudes and phases of such a twobranch structure will always take place. The I and Q branch mixers are two separate
physical components, their characteristics always differ from one another to some extent.
The two local oscillator (LO) signals should ideally have equal amplitudes and an exact
phase difference of 90º [4]. In practice, there exists a small non-90° phase error between
the I and Q channels of an IQ modulator. Also, the I and Q channels may not always have
the same amplitude. The phase error and amplitude imbalance cause unwanted image
signals that are generated in-band and consequently, degrades the system performance
[5]. Within this project digital pre-distortion will be designed to increase the system
performance and minimize the phase error and amplitude imbalance in the IQ channel.
5
The radio frequency (RF) power amplifier (PA) is a key component in modern
telecommunication systems since its power consumption dominates the other parts in the
system. The purpose of the RF PA is to amplify the radio signal to a necessary power
level for transmission to the receiver. RF PAs are divided into different classes, i.e. A,
AB, B etc. with respect to their power efficiency. In RF PAs there is a trade-off between
efficiency and linearity. High efficiency and high linearity cannot be achieved at the same
time [6]. In the case of amplifier for the transmitter so the efficiency is an important
parameter of an amplifier; however, to obtain the maximum efficiency the amplifier is
usually pushed into non-linear region. This in turn induces intermodulation products. The
non linear region is the region where the gain of the amplifier does not increase linearly
with the increment of the input. One of the most promising applications of dynamic
behavioural modelling of RF PAs is digital predistortion. The digital predistortion is
achieved by designing the predistorter to be as close as possible to the inverse of the
power amplifier function to get a desired output signal from the signal generator.
2.2 Linear systems
Linear system is a mathematical model of a system based on the use of a linear operator.
As a mathematical abstraction or idealization, linear systems find important applications
in automatic control theory, signal processing, and telecommunications. Linearization is
used to make (nonlinear) systems behave more linear. This means less spectral distortion.
Predistortion is one of the most promising linearization techniques
2.2.1 Predistortion
Predistortion is a technique consisting of introducing the inverse of the unwanted
characteristic of the DUT of the power amplifier, in series with the DUT, to eliminate the
distortion introduced by the unwanted DUT’s characteristic [7].
6
X(f )
Nonlinear
Device
Pre-distortion
Y( f )
Figure 2.2. Predistortion main ideas
In this work, the predistortion design used to improve the dynamic range of the IQ
modulation and power amplifier of the signal generators. It is a cost-saving technology
and it can be done in an analog as well as digital manner. Pre-distortion is an application
of behavioural modelling in which the input signal is predistorted -or precorrected- in
order to achieve a desired output signal. An ideal predistorter is the inverse of the
system’s response function. The predistorted input signal is calculated from the desired
output signal by the predistorter [8]. The goal of the predistortion system is to make the
cascade of the predistorter for IQ modulation and power amplifier. This is achieved by
designing the PD to be as close as possible to the inverse of the IQ modulation and power
amplifier function.
2.2.2 Analog predistortion
The advantages of analog predistortion are relatively simple low-cost, low energy
consumption, wideband signal handling capability and integrity. If they are implemented
adaptively, then the system complexity may increase significantly. There are different
ways to implement the analog predistortion. It can be a simple circuit composed of diodes
or transistors as in RF predistorters, or it can be composed of multipliers to realize
polynomial nonlinearities [9]. The system can be adaptive or fixed depending on
environmental conditions and system specifications. However, a reliable system must
have a kind of adaptation adjusting the predistorter according to the environmental
conditions especially in today’s mobile communication systems, which may operate
7
under extreme conditions and still must fulfil the specifications. High linearity system
based on RF predistortion is extremely difficult to achieve and are not widely available.
There are three analogue linearization techniques; power back-off, feed forward
linearization techniques, and Cartesian-loop linearization, details are given in [10].
2.2.3 Digital predistortion
Digital predistortion is usually implemented in digital baseband but it is also possible to
do it at intermediate frequency (IF). The theory behind is the same as in analog
predistortion. This method is in general used for base stations in mobile communication
systems in order to improve linearity, which is very important in systems with wide
bandwidths. A significant improvement can be achieved for class B and AB in
applications requiring high linearity. Digital predistortion is simple compared to feed
forward linearization widely used in base stations. It is unconditionally stable and a
precise linearization is possible [9].
There are two digital linearization techniques; Look up table (LUT) and polynomial predistorter. Look up table based predistorter stores the pre-distortion coefficients for all
input values in the LUT and the incoming signal is multiplied sample with this
coefficient. In the polynomial pre-distorter case, the characteristics of the signal generator
and the pre-distorter are described by polynomial functions. The polynomial coefficients
of the pre-distorter are adjusted to compensate the DUT nonlinearity, resulting in a linear
system [7].
2.3 The baseband equivalent
In signal processing, baseband is an adjective that describes signals and systems whose
range of frequencies is measured from zero to a maximum bandwidth or highest signal
{
frequency. The physical signal corresponds to z I (t ) cos(wt ) − z Q (t )sin (wt ) = Re z (t )e jwt
where ω is the carrier angular frequency in rad /s.
8
}
Power amplifier is a device that changes, usually increases, the amplitude of a signal. The
signal is usually voltage or current. The relationship of the input to the output of an
amplifier usually expressed as a function of the input frequency is called the transfer
function of the amplifier, and the magnitude of the transfer function is termed the gain.
In this work, the baseband equivalent is given in Figure 2.3, the signal generator
equipment chain is modelled by the nonlinear dynamic function G(⋅) and H (⋅) ,
respectively. G(⋅) and H (⋅) describing the IQ and power amplifier deficiencies are
unknown and have to be estimated. The signal u n is a baseband representation of the IQ
modulator output prior to the power amplification. The predistorter transforming the
reference stimuli rn into the signal generator input z n is denoted by F (⋅) . The aim of the
predistorter F (⋅) is to minimize a figure of merit based on the error ε n typically the sumsquared error [1]. The function β (⋅) is a know function to match the properties of the
reference stimuli rn to the level of the signal analyzer output s n . With reference to Figure
2.3, the resulting baseband signal s n should typically have the same spectral support as
the reference signal rn .
Figure 2.3. Baseband equivalent
9
In order to handle the inherent delays in the measurement, ideally the signal generator
output equals
s n = α e iΦ rn − k
(2.1)
that is, the analyzer baseband output is an attenuated or amplified (by the real valued
factor α >0) delayed (by k samples) and phase-shifted ( φ radians) replica of the baseband
reference signal rn . In order to simplify the notation, introduce the normalized reference
signal ~
rn as
~
rn = β (rn )
(2.2)
where, for example, β (rn ) = α e iφ rn − k .
2.4 Pre-distorter design based on physical modelling
There is a basic observation in the identification system to look for the physical process
of improving the quality of models derived. Predistortion in almost a linear or vaguely
non-linear, dynamic systems can be performed by a variety of model structures. Here, we
rely on the physical behavioural and study a clear impact on the IQ imbalance and
deficiencies of the amplifier between the nonlinearities of the main sources for this type
of equipment. In such a way, physically motivated structures for predistorter design are
achieved. The aim of this section is to derive the structure of the Parsimonious
parameterized digital predistorter [1]. Then, in Section, 2.7 we focus on training of the
derived predistorter.
10
Figure 2.4. Digital predistortion design
Figure 2.4 shows digital predistorter design where input signal x n produces the output
signal z n . In operational mode x n = rn (see Figure 2.3) producing output z n and in
calibration mode x n = s n (see Figure 2.7) producing ẑ n , respectively.
The function f n and f n are pulse responses, f (⋅) denotes the nonlinear branch of the
parallel Hammerstein structure, and (⋅) denotes conjugate operation. Conceptually, we
*
employ a cascade structure of the digital pre-distorter, see Figure 2.4 the first stage of the
predistorter is designed to deal with imperfections in the power amplifier and the second
stage the imperfections in the IQ modulator, so that ideally for β (r n
)=
r n in (2.2) the
distorter output in operational mode is given by
(
)
z n = F (rn ) = G −1 H −1 (rn )
(2.4)
yielding
[ (
s n = H [G ( z n )] = H { G G − 1 H
under an assumption that the involved inverses exist.
11
−1
(rn ))] } =
rn
(2.5)
Figure 2.5. IQ imperfection in the signal generator
2.4.1 IQ imbalance
IQ imbalance mainly attribute to the mismatched components in the I and the Q branches.
Examples include but not limited to an imperfectly balance local oscillator and/or
baseband low pass filters with mismatch frequency responses [11]. The digital predistortion design in Figure 2.4 will give improvement to the IQ imbalance. In Figure 2.3
the baseband equivalent of the IQ modulator output by u n . Due to the IQ imbalance we
have that u n of the IQ modulator output is given by [12]:
u n = g n ∗ z n + g n ∗ z n*
(2.6)
g n and g n are some unknown linear time invariant pulse responses. The physical
interpretation of the pulse responses is referred to [12]. A detailed discussion on the topic
is available in [4]. The model in (2.6) captures the IQ mixer amplitude and phase
imbalance, impulse response due to the digital-analog conversion (DAC) and analog lowpass filters. The model is flexible enough to capture the behaviour of frequency
dependent IQ imbalance, as well. The second term in (2.6) is the mirror distortion. With
z n being a complex-valued cisoid z n = e jw0 n , the IQ modulator outputs both an amplitude
and phase shifted cisoid with angular frequency ω 0 , but also a component at angular
frequency − ω 0 .
12
Figure 2.6. Digital predistortion design for IQ imbalance
The IQ imbalance is to be compensated by the second stage of the predistorter. With
reference to Figure 2.6, we consider the second stage of the predistorter defined by its
input vn and output z n given by:
z n = v n + f n ∗ v n*
(2.7)
where f n is the pulse response of a linear time invariant filter. We have chosen to
exclude a linear filter operating on the first term in (2.7), a filter that without loss of
generality can be include in the first stage of the predistorter. The second stage output
(2.7) results in an IQ modulator output u n in (2.6) given by:
(
)
(
u n = g n ∗ v n + f n ∗ v n* + g n ∗ v n + f n ∗ v n*
(
)
)
*
= g n + g n ∗ f n* ∗ v n + (g n ∗ f n + g n ) ∗ v n*
(2.8)
The mirror distortion is cancelled by forcing the second term in (2.8) or g n ∗ f n + g n to
zero for all time instants n , by a proper selection of the predistorter coefficients gathered
in f n [12]. For the subsequent discussion, it is assumed that the second stage of the
predistorter is perfectly tuned. That is, subject to cancelled mirror distortion one has that
the input to the power amplifier is a filtered replica of vn :
u n = an ∗ vn
13
(2.9)
For a pulse response a n implicitly given by a n = g n + g n ∗ f n* subject to the condition
g n ∗ f n + g n = 0 , that is
(
)
a n = g n ∗ 1 − f n ∗ f n* . Under the null-mirror-distortion
assumption the structure for the first stage of the predistorter for power amplifier
deficiencies is discussed in Section. 2.4.3.
2.4.2 Predistortion mirror distortion
In order to determine the parameter values of the digital predistorter, the training set-up
in Figure 2.7 is considered. With reference to Figure 2.7, the model output ẑ n describing
the inverse of the IQ imbalance reads
zˆ n = s n + f n ∗ s n*
(2.10)
The objective in this section is to formulate the least-squares problem to obtain the
parameters of the unknown pulse response f n . Thus,
Δ
z n = zˆ n − s n = f n ∗ s n*
(2.11)
Consider N samples of the user generated stimuli z n resulting in the corresponding
signal analyzer output s n , that is data {z 0 ,L , z N −1 } and {s 0 , L, s N −1 }, respectively. Denote
the column vectors with the sought for predistorter pulse response by f . Then, the
predictor output z = zˆ − s with z =
(z 0 , K
, z N − 1 ) (where T denotes transpose)
T
can be written
z = s 0* f
where s0 in (2.12) is given by
14
(2.12)
Figure 2.7. Training of the digital predistorter F ( s n ;θ )
⎛ s0
⎜
⎜ s1
⎜ M
⎜
⎜ sL
s 0 = ⎜⎜ M
⎜ s N −1
⎜
⎜ 0
⎜ M
⎜⎜
⎝ 0
0
s0
⎞
⎟
⎟
0 ⎟
⎟
s0 ⎟
M ⎟⎟
s N − L −1 ⎟
⎟
M ⎟
s N −2 ⎟
⎟
s N −1 ⎟⎠
0
M
L
O
L
L
O
s N −1
L
0
(2.13)
z −z
With ~z = β ( z ) defined by (2.2), the least-squares solution fˆ = arg min ~
reads
(
fˆ = s 0H s 0*
)
−1
s 0H z
(2.14)
Leading to the predistorter in operation
z n = r n + fˆ ∗ r n*
15
(2.15)
2
Figure 2.8. Digital predistortion design for power amplifier
2.4.3 Power amplifier deficiencies
In order to minimize nonlinear distortion that produced from the power amplifier, digital
predistortion was applied to the signal generator to correct the nonlinearities and cancel
distortion introduced by power amplifier. In digital predistortion the signal is distorted in
the digital domain to compensate for the power amplifier’s signal distortion. In model
based digital predistortion an approximate inverse of the PA’s transfer function is used
[13]. The first stage of the predistorter handles the nonlinear artifacts introduced by the
power amplifier. A digital predistortion was used to generate an input signal to the power
amplifier to reduce the non-linear region in the amplifier. A parallel Hammerstein or
memory polynomial models are used for inverse as well as direct modelling of power
amplifiers subject to memory effects [14]. The employed predistorter has the form of a
parallel Hammerstein model that is consists of the nonlinearity followed by a linear filter
is often used to present certain higher-order nonlinear systems, which input-output
relation formally can be written as reference to Figure 2.8.
vn = f n ∗ xn + f (xn )
(2.16)
where f n is the pulse response of the linear time invariant filter of the linear path in the
16
parallel Hammerstein structure. The function f ( x n ) is a short notation for the sum of the
nonlinear branches. In this project, both the traditional parallel Hammerstein structure
with odd order polynomial coefficients has been considered, that is
f ( xn ) =
(
M
∑
f m ,n ∗ x n
m =1
2m
xn
)
(2.17)
From (2.17), we note that each branch (that is, for each m= 1,…, M) is described by linear
2m
filtering (determined by the filter coefficients { f m , 1 , K , f m , M }) of the input x n x n
for all n. Introduce φ m ( x n ) = x n
2m
x n for m=1,…, M, then (2.17) reduces to
f ( xn ) =
M
∑
m =1
f m ,n ∗ φ m ( x n )
(2.18)
The basis functions φ m (⋅) are summarized in Table I.
Table I
Basis function for a conventional parallel Hammerstein model (memory polynomial)
φ m ( x ) respectively the complex-valued counterpart to the shifted Legendre
polynomialsψ
m
m
φm (x)
( x ) [15].
ψ
m
(x)
2
4 x x − 3x
4
15 x x − 20 x x + 6 x
6
56 x x − 105 x x + 60 x x − 10 x
8
210 x x − 504 x x + 420 x x − 140 x x + 15 x
1
x x
2
x x
3
x x
4
x x
2
3
2
4
3
2
17
For a robust implementation, numerical aspects have to be taken into account. For that
purpose, the basis functions in [15] are employed as well. Thus the basis function φ m (⋅) in
(2.18) is replaced by ψ m (⋅) given by the complex-valued counterpart to the shifted
Legendre polynomials [15]:
(−1) l + m +1 (m + 1 + l)!
l −1
ψ m ( x) = ∑
x x
l =1 (l − 1)!(l + 1)!(m + 1 − l )!
m +1
(2.19)
Table I shows the four orthogonal polynomials for ψ m (⋅) . If we replace the complex
valued
basis
functions x
m −1
x, m = 1,2,....., m
with
real-valued
basis
function
x , m = 1,2,...., m , the real valued orthogonal polynomials defined in the region [0, 1],
m
0
which are known as the shifted Legendre polynomial, except that the x polynomial is
not included. The construction of an orthogonal basis is often an iterative procedure. One
may note that φ m (⋅) only contains odd powers of x n , whereas ψ m (⋅) also includes even
powers.
2.5 Parsimonious predistorter structures
Parsimonious predistorter structure used in this project, the predistorter F (⋅) in Figure
2.3 consists of three filtering blocks, namely the linear time invariant filtering by f n of
the input x n , the linear time invariant filtering by f n of the conjugate of the first-stage
output vn , and the M branches of time invariant linear filtering by the { f m,n }m =1 of the
M
static nonlinear mapping of the input x n by the appropriate basis function φ m (⋅) and
ψ m (⋅) , respectively. Accordingly, the design objective of the predistorter is to find the
suitable parameters of the pulse responses f n , f n and f 1, n ,L, f M , n where the integer M and
the lengths of the individual pulse responses are still to be determined (in a structural
way). Combination of (2.7) and (2.19), rearranging terms yields
18
z n = f n ∗ xn + f n ∗ f n∗ ∗ xn∗ + f ( xn ) + f n ∗ f (xn )
123
*
(2.20)
f n'
where the pulse response f n′ is introduced to simplify the notation. Divide z n in (2.20)
into its linear and nonlinear parts, viz as [1].
z n = z nL + z nN
(2.21)
z nL = f n ∗ x n + f n′ ∗ x n*
(2.22)
where by definition
and
z nN = f ( xn ) + f n ∗ f ( xn )
*
(2.23)
The linear (2.22) and nonlinear (2.23) parts are analyzed in the following sections.
2.5.1 Linear part of the predistorter
In the linear part of the predistoerter, there is no overlap in the spectral support of the
source and its mirror distortion. Accordingly, there is an ambiguity in the linear part in
(2.22). This fact is clearly seen by transforming the model to the frequency domain. The
frequency domain representation of (2.22) is given by F (ω )X (ω ) + F ′(ω )X * (− ω ) , where
F (ω ), F ′(ω ) and x(ω ) are the Fourier transform of the corresponding time domain
quantities. For a cisoid input x n = exp(iω 0 n) with frequency ω 0 , the output signal is
given by F (ω 0 ) exp(iω 0 n ) + F ′(− ω 0 ) exp(− iω 0 n ) , where the gains and phase-shifts are
determined by F (ω 0 ) and F ′(−ω 0 ) , respectively.
19
In a similar vein, the output from another filter, say H ' (ω ) , driven by the sum
exp(iω 0 n) + exp(−iω 0 n)
follows
by
the
superposition’s
principle
as
H ' (ω 0 ) exp(iω 0 n ) + H ′(− ω 0 ) exp(− iω 0 n ) . Accordingly, we cannot distinguish the two
separate branches determined by F (ω ) and F ′(ω ) and the single branch determined
by H ′(ω ) , with
H ′(ω 0 ) = F (ω 0 )
and
H ′(−ω 0 ) = F ′(−ω 0 ) . Accordingly, for a
parsimonious model structure we replace the two pulse responses f n and f n′ in (2.22)
with hn′ [1], viz.
(
z nL = h n′ ∗ x n + γ x n*
)
(2.24)
The scalar γ in (2.24) is a complex –valued scaling and rotation. The wisdom of γ is to
reduce the dynamic range of the transfer function corresponding to the pulse response hn′ .
Intuitively, the magnitude of γ should be chosen proportional to the actual mirror
distortion of the signal generator, that is the injected mirror component γ x n* produced
by the predistorter should be of the same magnitude as the actual mirror distortion of the
signal generator. Furthermore, it is necessary that the coefficient of the pulse response
corresponding to the direct term, which is the term at time instant n equals unity since the
scaling is taken care of by β (⋅) in (2.2). In order to manage imbalance in the tuning of
β (⋅) the pre-distorter consist of a direct term, that is (2.24) is replaced by
z nL = x n + h n ∗ (x n + γ x n* )
(2.25)
where the requirement on a unity direct term in the pulse response hn is relaxed.
20
Figure 2.9. Digital predistorter where input signal x n produces output signal z n .
2.5.2 Non linear part of the predistorter
The nonlinear part z nN in (2.23) is analyzed to find a parsimonious representation. In fact,
it is sufficient to analyze the second term in (2.23), that is
f n ∗ f (xn )
*
⎛M
⎞
= f n ∗ ⎜ ∑ f m , n ∗ Φ m ( x n )⎟
⎝ m =1
⎠
*
(2.26)
where the equality follows from (2.18). A rearrangement of terms yields
M
f n ∗ f ( xn ) = ∑ f n ∗ f m*,n ∗ Φ m ( xn )
*
*
(2.27)
m =1
*
where the pulse responses f m , n = f n ∗ f m , n for m=1,…, M are introduced to simplify
the notation. Further, we note that our basis function Φ m ( x ) (andψ m ( x ) ) obey
( )
Φ m ( x ) = Φ m ( x * ) (andψ m ( x ) = ψ m x * ). Thus, (2.27) is reduced to
*
*
21
f n ∗ f (x n ) =
*
M
∑
m =1
( )
f m , n ∗ Φ m x n*
(2.28)
One may note that the right side of (2.28) exactly has the same structure as (2.18), but for
some other parameter values f m,n . Therefore we can write the right hand side of (2.28)
as f (x n* ) , where f (x n* ) satisfy (2.18). In summary, gathering the derived results (2.21),
(2.24), (2.28) the predistorter is parsimoniously described by [1]
(
)
(
z n = x n + h n ∗ x n + γ x n* + f ( x n ) + f γ x n*
)
(2.29)
where f (⋅) and f (⋅) are nonlinearities according to (2.18) (eventually with φ m (⋅) in
(2.18) replaced byψ m (⋅) ) with two different set of parameter values. We notes that the
scaling γ has been introduced in f (⋅) , again motivated by the difference in power
between the direct path and the conjugate path. The resulting predistorter structure is
given by Figure 2.9.
2.6 Predistorter properties
The predistorter input-output relation that introduced in (2.4) is written as
z n = F ( x n ;θ )
(2.30)
where x n is the input signal, and z n the output, θ is introduced as a generic parameter
vector of proper length. The entries of θ consist of the parameter of the gain γ and
sought for pulse responses hn , f m ,n and f m,n for m = 1,…, M, where f m ,n and f m,n are
defined through (2.18). The following properties of F (x n ;θ ) is:
Property 1: F ( x n ;0 ) = x n
22
(2.31)
where 0 is the null-vector of appropriate size. The implication of (2.31) is that generatoranalyzer set-up that only introduces a gain and phase shift. Let
xn
be a
cisoid x n = exp(iw0 n ) , then it holds that:
(
)
Property 2: F e iω0 n ;θ = c1e iw0 n + γ c 2 e − iw0 n
(2.32)
c1 and c2 is the complex – valued constants. The implication of (2.32) is that a
predistorter linear in x n + γ x n* is sufficient for signal tone stimuli, that is M = 0. Another
property includes the least-squares training properties given in Section 2.7
Property 3a: θˆ = arg min V (θ )
(2.33)
θ
where
N −1
Property 3b: V (θ ) = ∑ (β ( z n ) − F (s n , θ ))
2
(2.34)
n =0
where N is the number of collected samples in the training phase, z n is the input stimuli
to the signal generator, and s n is the baseband output from the signal analyzer; see
Figure 2.7.
( ) ( )
Property 4: V θˆl ≤ V θˆl −1
(2.35)
where l denotes the number of parameters in θ . Due to the nonlinearities involved
carefulness is required in the process of model order selection and number of parameters
in the different filter branches.
23
2.7 Training of the predistorter based on N-sequences of data
To determine the values of the parameters of the digital predistorter, the training set-up in
Figure 2.7 is considered. In Figure 2.7, β (⋅) is assumed known.
In this Section the least- squares problem is formulated and studied subject to a given
structure that is subject to a given set of parameters that are included in the generic
parameter vector θ .
2.7.1 Model output
The model output ẑ n with reference to Figure 2.7 describing the inverse of the chain of
equipment reads
(
)
( )
zˆn = sn + hn ∗ sn + γ sn* + f (sn ) + f γ sn*
(2.36)
The aim of this section is to formulate a least squares problem to obtain the unknown
pulse response hn , f m ,n and f m,n for m = 1,…,M, where f m ,n and f m,n implicitly are
defined through (2.18). We introduce the simplified notation for the different transformed
versions of the output of the signal analyzer s n , that is the sum of the signal and it’s
conjugate
s n = s n + γ s n*
(2.37)
and the nonlinear static mappings of s n , that is
s m , n = φ m (s n )
(2.38)
s m , n = φ m (γ s n* )
(2.39)
and
24
with the notation (2.37)-(2.40), the model output (2.36) is rewritten as
M
M
m =1
m =1
zˆ n − s n = h n ∗ s n + ∑ f m , n ∗ s m , n + ∑ f m , n ∗ s m , n
(2.40)
Let the order of the linear filters be L , that each filter comprises L + 1 filter coefficients.
This is not an absolute requirement, but makes the notation significantly less complex.
The number of nonlinear branches M in f (⋅ ) and f (⋅) is, at the moment equal.
Then, one may then note that the total number of coefficients to be estimates is (2M + 1)
(L + 1). For example, with tenth order filters and 2+2 nonlinear branches, there are some
55 complex-valued parameters to estimate. Then, write out the explicit form of the
convolutions, the predistortor output reads
L
M
L
z n = ∑ hl s n − l + ∑ ∑ f m , l s m , n − l + f m , l s m , n − l
l =0
m =1 l = 0
(2.41)
where we also introduced z n = zˆ n − s n . Consider N samples of the user generated
stimuli z n resulting in the corresponding signal analyzer output s n , that is data
{z 0 , K , z N −1 }
and {s 0 , K , s N −1 }, respectively. Denote the column vectors (still, all of
length L + 1 ) with the sought for predistorter pulse responses by h , f 1 , K , f M , f 1 , K , f M ,
and left f and f be the augmented column vectors of length M ( L + 1) , given by:
⎛ f1 ⎞
⎜
⎟
f =⎜ M ⎟
⎜f ⎟
⎝ M⎠
and
25
(2.42)
⎛ f1 ⎞
⎜
⎟
f =⎜ M ⎟
⎜f ⎟
⎝ M ⎠
(2.43)
Then, the predictor output z = zˆ − s = ( z 0 , K , z N −1 )T (where T denotes transpose) can
be written
z = (s
S
⎛h⎞
⎜ ⎟
⎜f⎟
⎜f⎟
⎝ ⎠
S)
(2.44)
where s in (2.44) is given by s = s 0 + γ s 0* where
⎛ s0
⎜
⎜ s1
⎜ M
⎜
⎜ sL
s 0 = ⎜⎜ M
⎜ s N −1
⎜
⎜ 0
⎜ M
⎜⎜
⎝ 0
0
L
s0
O
L
L
O
s N −1
L 0
⎞
⎟
M ⎟
0 ⎟
⎟
s0 ⎟
M ⎟⎟
s N − L −1 ⎟
⎟
M ⎟
s N −2 ⎟
⎟
s N −1 ⎟⎠
0
(2.45)
Further, S in (2.44) (of size (N + L ) × M (L + 1) ) and its companion s * are data matrices
formed from the signal generator output. Explicitly, S is composed out of M + 1 submatrix as:
S = (s1 L s M )
(2.46)
where s m is the of size ( N + L ) × (L + 1) Toeplitz data matrix (for m = 1,…,M):
26
⎛ s m,0
⎜
⎜ s m ,1
⎜ M
⎜
⎜ s m, L
⎜
sm = ⎜ M
⎜ s m , N −1
⎜
⎜ 0
⎜ M
⎜⎜
⎝ 0
0
s m,0
O
L
L
O
s m, N −1
0
L
⎞
⎟
⎟
0 ⎟
⎟
s m,0 ⎟
M ⎟⎟
s m , N − L −1 ⎟
⎟
M ⎟
s m, N −2 ⎟
⎟
s m, N −1 ⎟⎠
0
M
L
(2.47)
where s m , n was introduced in (2.38). The matrix s is constructed in a similar vein
replacing s m , n in (2.47) with s m , n as defined in (2.40) [1].
2.7.2 Least-squares problem
Method of least squares or ordinary least squares (OLS) is used to solve the over
determined system that is a system of approximately linear equations. Least square is
often applied in statistical context, particularly regression analysis. The method was first
described by Johann Carl Friedrich Gauss around 1794.
Least squares can be interpreted as a method of fitting data the best fit in the Leastsquares sense is that instance of the model for which the sum of squared residuals has its
least value, a residual being the difference between an observed value and the value given
by the model. In this work, the least-squares solution of the predistorter coefficients is
2
z = β (z ) . In short,
given by minimizing ~z − z . From the vector ~z given by (2.2), i.e. ~
the linear set of equations reads
⎛h⎞
⎜ ⎟ ~
⎜ f ⎟= z
⎜ f ⎟
⎝{⎠
(1s 42
S S)
43
R (γ )
θ
Here, the dependency of R on the scaling γ is explicitly indicated.
27
(2.48)
2.7.3 Determination of the function β (⋅ )
The source signal rn in Figure 2.3 is a complex - valued wide sense stationary process
[
]
with auto-correlation function R p = E rn+ p rn* , R0 is the signal power σ r2 . Then, the crosscorrelation between the signal generator output s n and the reference signal rn , that is
[
]
[
]
S p = E s n + p rn* directly follows from (2.1) as S p = α e iφ E rn −k + p rn* . The inherent delay K
in (2.1) is found as the bin location of the maxima of S p , that is
K = arg max S P
(2.49)
P
For the sought for K given by (2.49), it holds that the cross-correlation yields
S p = k = α e iϕ σ
2
r
(2.50)
Accordingly, the real-valued and positive quantity α can be determined by peak picking
the maximum magnitude value of the cross-correlation normalized with the power of the
source signal, that is α = S p = k σ r2 . The phase follows by ϕ = ∠ [S p = k ] . In practice, the
cross-correlation S p and power σ r2 are replaced by the corresponding estimated
quantities based on measured data [1].
28
3 Method
3.1 Introduction
This Chapter presented the technical and theoretical information that was used to have
the appropriate test setup, simulation and measurement procedure. It includes an
overview of the test setup, implemented system, measurement system, internal structure
and configuration of the different instruments used in this work for generating spectrally
pure signal. It consists of the simulation model and parameter determination that is
consists of the gamma (γ ) , model order M and L , basis function ψ m (⋅) or φ m (⋅) and
realization. The device under test was the IQ modulation mismatch and amplifier
deficiencies in the signal generator Rhode & Schwarz SMU200A. The design digital predistortion that is implemented in software so that the dynamic range of the signal
generator output after pre-distortion is superior to that of the output prior to it.
3.2 Test Setup
We consider measurement set-ups where the signal generator is synchronized with the
signal analyzer. The test setup consists of a Rhode & Schwarz SMU200A Generator, a
Rhode & Schwarz FSQ 26 Signal Analyzer, and a personal computer interconnected to
arrange an appropriate measurement environment for the DUT characterization, these
instruments are essential for the generating a spectrally pure signals.
3.2.1 The implemented system
We consider the set-up in Figure 3.1 where the digital arbitrary wave-form baseband test
stimuli z n (a complex - valued quantity) is generated by software and transmitted in
digital format to the signal generator for DAC of the in-phase and quadrature branches,
29
followed by a modulation wheel to produce the analog radio frequency signal rt . In the
calibration set-up, the signal generator radio frequency output rt is connected directly to
the input of the signal analyzer, producing the down- converted baseband signal s n of the
radio signal.
Figure 3.1. Calibration set-up.
By the design of a predistorter a high quality radio frequency output rt will produce to
calculate the proper stimuli z n by nonlinear dynamic filtering of a baseband reference rn .
In other words, rn is the desired baseband representation of the radio frequency signal rt .
⋅ and ℑ[]
⋅ denote the real and imaginary part of the complex-valued
The notation ℜ[]
quantity within the brackets. The quality measure is typically directly related to the
squared error, where the error is defined as the difference between a reference, or target,
signal and the feedback baseband signal s n . It is assumed that the performance of the
analyzer is superior to that of the generator, and that the instruments are synchronized,
both during the phase of calibration and the subsequent measurement campaign. The
device under test is the IQ imbalance and power amplifier deficiencies in the signal
generator. Due to the derived digital predistortion function the signal generator now
behaves as a better performance to excite the device under test.
30
3.2.2 Vector Signal Generator (VSG)
The R&S SMU 200A vector signal generator used in this measurements, that had two
DAC each with clock frequencies up to 100 MHz and it was equipped with an AWG,
with IQ modulation. In this work, the two-tone baseband signal where generated in
Matlab program, all the different parameters were specified, such as amplitude, frequency
and phase difference between I and Q. Coherent sampling was used to ensure higher
spectral resolution. The signal was generated in Matlab as I and Q data files and down
loaded to the AWG. The AWG generates the analogue I and Q signals at baseband as
N
I (t ) =
∑
A k (t ) cos (w k t + φ k (t ) + ϕ k
)
N
A k (t )sin (w k t + φ k (t ) + ϕ k
)
K =1
Q (t ) =
∑
K =1
(3.1)
(3.2)
Where Ak , φ k , wk and ϕ k are the amplitude, phase, frequency and initial phase of the
k:th sub-carrier [16]. The I and Q signals are then fed to the IQ-modulator in the signal
generator. The modulation is described in (3.3), the output of the modulator is represent
by s (t )
s (t ) = r (t ) cos( wct + ϕ (t ))
(3.3)
where wc is the carrier frequency and:
r (t ) = (I 2 (t ) + Q 2 (t ))
1/ 2
⎛ I (t ) ⎞
⎟⎟
⎝ Q(t ) ⎠
ϕ (t ) = arctan⎜⎜
The signal r (t ) is the envelope of s (t ) and ϕ (t ) is the phase of s (t ) [17].
31
(3.4)
(3.5)
3.2.3 Vector Signal Analyzer (VSA)
The R&S FSQ26 vector signal analyzer used in this project that has an IQ demodulator
that returns the values of the signals in a complex format, Figure 3.2 shows the analyzer
hardware.
Figure 3.2. Hardware from IF to the process [18].
The radio frequency input signal is down-converted to an intermediate frequency (IF) of
20.4 MHz. The IF signal is digitized using ADC with 81.6 MHz sampling rate. An analog
bandpass filter in front of the ADCs limits the spectrum. The digitized (IF) is downconverted to IQ baseband using a digital mixer fed by a numerical controlled oscillator
(NCO). The digital equalizer filter corrects the amplitude and phase distortion. Then the
signal resample’s which reduces the sampling rate from 81.6 MHz to 40.8 MHz to adapt
the actual signal bandwidth. Finally the output sampling rate can be adjusted from 81.6
MHz to 10 KHz. Then the IQ signal is filtered by low pass filters and written
continuously into the IQ memory in parallel. Most of the functions in the VSA can be
control from a program Matlab, for example resolution bandwidth (RBW), video
bandwidth (VBW), Reference level (Ref), attenuation (Att) and span.
32
3.3 Simulation model
The simulation model comprises of an IQ imbalance given by first order linear filtering,
and a power amplifier model described as a parallel Hammerstein system. White
Gaussien noise is added to the IQ modulator output, and is thus propagated through the
parallel Hammerstein model. Each filter in the parallel Hammerstein model consist of
three filter taps employing a two tone stimuli with positive frequencies.
A parsimonious predistorter structure has been used in this project that is consisting of
three filtering blocks, with reference to Figure 2.9. Where
f (⋅) and
f (⋅) are
nonlinearities according to (2.18) and hn is the pulse response of the linear time invariant
filter. We consider the relation introduced in Figure 2.9. the signal x n is the complex
input signal, and z n the output signal. A generic parameter vector θ is introduced as a
proper length. The entries of θ consist of the parameters of the gain γ and sought for
pulse response hn , f m ,n and f m,n for m = 1,..., M , where f m ,n and f m,n implicitly are
defined through (2.18). The least squares problem has used to obtain the parameters of
the unknown pulse response hn , f m ,n and f m,n for m = 1,..., M . The least squares
problem have calculated from equation (2.48), the equation (2.37) has been calculated
and then S and S matriser have been determined. We have used basis function φ m (⋅) to
generate the simulation signal, then we have used basis function ψ m (⋅) to generate it.
33
3.4 Determinate parameters
3.4.1 Gamma (γ )
Figure 3.3. Gamma (γ )
The scalar γ is a complex-valued scaling and rotation. The rational for using γ is to
reduce the dynamic range of the transfer function. The magnitude of γ has been chosen
proportional to the actual mirror distortion of the signal generator. The γ has been
chosen from the measuring data between a two-tone signal and the mirror distortion that
we had, with reference to Figure 3.3.
3.4.2 Model order
The model order L is the linear filter that each filter comprises L + 1 filter coefficients
and the model order M is the number of nonlinear branches in f (⋅) and f (⋅) at the
moment equal. In the theory parts says that we should calculate V (θ ) from (2.34) for
different values of M and L . Then we should make a three dimensional plots. The
parameter values of M and L , which gave the lowest value of V (θ ) , we will chose as
the best parameter choice. Because of problems with numerical solutions so unfortunately
this has not been implemented in practice
34
3.4.3 Basis function ψ m (⋅) or φ m (⋅)
The polynomial model is communly used in power amplifier modeling and predistorter
design. Higher order polynomials present a challenge for both power amplifier modelling
and predistortion design. However, the basis function φ m (⋅) (see Section 2.4.3) exhibits
numerical instabilities when higher order terms are included. Theoritically, the basis
function φ m (⋅) or ψ m (⋅) are equivalent and thus should behave similarly. In practise, the
two approaches can perform quite differently in the presence of finite precision
processing[15]. One may note that basis function φ m (⋅) only contains odd powers of the
input signal x n , whereas basis function ψ m (⋅) also includes even power. we have tested
both models to evaluate which is most appropriate. So, the basis function ψ m (⋅) had a
better performance than the basis function φ m (⋅) .
3.4.4 Realization
In this project, it was a shortage of mathematics, because we calculated S matrix with γ .
Since γ is a small speech, then all columns in S will be small compared to the columns
of S matrix. This gives the ill-conditioned matrix R. This problem will effect the
calculation of θ in (2.48). For f , f and h filters in (2.48), we have used the same
model order but in real there are not. These shortcomings in the method have also been
highlighted by others during the project.
35
4 Result
4.1 Introduction
In this Chapter, different simulation and measurement results have been presented. The
results of two-tone complex signal, with equal amplitude and equal phase shift used to
obtain the spectrally pure signal. Both the conventional polynomial φ m (⋅) and orthogonal
polynomial ψ m (⋅) have been used to generate the signals. The device under test was IQ
modulation mismatch and amplifier deficiencies in the signal generator. The least squares
method was used for system identification and design of the digital predistorter.
For the measurements performed in this work, the signals have been generated using the
vector signal generator and the measurements are performed using the spectrum analyzer.
Synchronization has been used to find the delay between two signals. Synchronization is
a definition of cross-correlation between input and output signal and phase shift, with
reference to Section 2.7.3.
4.2 Simulation results
In this Section, a two-tone complex signal has been generated. A two-tone signal was at
frequencies 34.53 MHz and 45.16 MHz with equal amplitude at 6.9 dBm and equal phase
shift. A tow-tone signal without predistortion has the mirror distortion at frequencies 34.53 MHz and -45.16 MHz, with equal amplitude at -73.0 dBm. The tow-tone signal
with predistortion have the mirror distortion at frequencies -34.53 MHz and -45.16 MHz,
with amplitude -86.3 dBm and -85.6 dBm, respectively. With reference to Figure 4.14.2 the mirror distortion have been reduced after predistortion design. The mirror
distortion have been reduced 13.3 dBm at frequency -34.53 MHz and 12.6 dBm at
frequency -45.16 MHz. we could reduce mirror distortion by using γ , but we have not
used it. Because γ is a small speech, then all columns in S will be small compared to
the columns of S matrix. This gives the ill-conditioned matrix R. This problem will effect
36
the calculation of θ in (2.48). So we haven’t used γ in the S
matrix. The
intermodulation products for two-tone signal for φ m (⋅) function without predistortion
were at frequency 23.91 MHz with amplitude -59.0 dBm and at frequency 55.78 MHz
with amplitude -59.0 dBm. The intermodulation products for two-tone signal for φ m (⋅)
function with predistortion were at frequency 23.91 MHz with amplitude -97.1 dBm and
at frequency 55.78 MHz with amplitude -96.4 dBm (see Figures 4.3-4.4). The
intermodulation products for two-tone signal for ψ m (⋅) function without predistortion
were at frequency 23.91 MHz with amplitude -59.0 dBm and at frequency 55.78 MHz
with amplitude -59.0 dBm. The intermodulation products for two-tone signal for ψ m (⋅)
function with predistortion were at frequency 23.91 MHz with amplitude -93.6 dBm and
at frequency 55.78 MHz with amplitude -94.9 dBm (see Figures 4.5-4.6).
Two tones signal
50
X: 34.53
Y: 6.992
X: 45.16
Y: 6.992
Power level [dBm]
0
-50
X: -45.16
Y: -73.07
X: -34.53
Y: -73
-100
-150
-200
-400
-300
-200
-100
0
100
Frequency [MHz]
200
300
400
Figure 4.1. Power spectrum of two-tone signal without predistortion
37
Two tones signal
50
X: 34.53
Y: 6.979
X: 45.16
Y: 6.979
Power level [dBm]
0
-50
X: -45.16
Y: -85.69
X: -34.53
Y: -86.3
-100
-150
-200
-400
-300
-200
-100
0
100
Frequency [MHz]
200
300
Figure 4.2. Power spectrum of two-tone signal with predistortion
38
400
Two tones signal
50
X: 34.53
Y: 6.992
X: 45.16
Y: 6.992
Power level [dBm]
0
X: 23.91
Y: -59.03
-50
X: 55.78
Y: -59.06
-100
-150
-200
-400
-300
-200
-100
0
100
Frequency [MHz]
200
300
400
Figure 4.3. Power spectrum of two-tone signal for φ m (⋅) function without
predistortion
39
Two tones signal
50
X: 34.53
Y: 6.979
X: 45.16
Y: 6.979
Power level [dBm]
0
-50
X: 23.91
Y: -97.11
X: 55.78
Y: -96.43
-100
-150
-200
-400
-300
-200
-100
0
100
Frequency [MHz]
200
300
400
Figure 4.4. Power spectrum of two-tone signal for φ m (⋅) function with
predistortion
40
Two tones signal
50
X: 34.53
Y: 6.992
X: 45.16
Y: 6.992
Power level [dBm]
0
X: 23.91
Y: -59.02
-50
X: 55.78
Y: -59.04
-100
-150
-200
-400
-300
-200
-100
0
100
Frequency [MHz]
200
300
400
Figure 4.5. Power spectrum of two-tone signal for ψ m (⋅) function without
predistortion
41
Two tones signal
50
X: 34.53
Y: 6.979
X: 45.16
Y: 6.979
Power level [dBm]
0
-50
X: 23.91
Y: -93.65
X: 55.78
Y: -94.97
-100
-150
-200
-400
-300
-200
-100
0
100
Frequency [MHz]
200
300
400
Figure 4.6. Power spectrum of two-tone signal for ψ m (⋅) function with
predistortion
42
4.3 Measurement results
The signal used for this measurement was a two-tone complex signal that is generated by
personal computer software and sent to the vector signal generator. The measurement
was performed using the spectrum analyzer. In the SA the center frequency of the signal
was 2.14 GHz and a bandwidth of 20 MHz. The resolution bandwidth is an important
parameter in the configuration of the spectrum analyzer was set to 10 KHz, in order to
achieve high enough signal to noise ratio without falling in to long sweep time. The
measurement was done at a sampling frequency of 80 MHz.
A two-tone complex signal has been generated. A two-tone signal was at frequencies
3.012 MHz and 8.088 MHz with equal amplitude at 6.2 dBm and equal phase shift. The
intermodulation products for two-tone signal for φ m (⋅) function without predistortion
were at frequency -2.065 MHz with amplitude -51.0 dBm and at frequency 13.16 MHz
with amplitude -55.2 dBm. The intermodulation products for two-tone signal for φ m (⋅)
function with predistortion were at frequency -2.065 MHz with amplitude -58.0 dBm and
at frequency 13.16 MHz with amplitude -71.3 dBm (see Figures 4.7-4.8). The
intermodulation products for two-tone signal for ψ m (⋅) function without predistortion
were at frequency -2.065 MHz with amplitude -50.8 dBm and at frequency 13.16 MHz
with amplitude -55.1 dBm. The intermodulation products for two-tone signal for ψ m (⋅)
function with predistortion were at frequency -2.065 MHz with amplitude -59.8 dBm and
at frequency 13.16 MHz with amplitude -88.6 dBm (see Figures 4.9-4.10).
43
Two tones signal
40
20
X: 3.012
Y: 6.159
X: 8.088
Y: 6.251
0
Power level [dBm]
-20
X: -2.065
Y: -51.03
-40
X: 13.16
Y: -55.26
-60
-80
-100
-120
-140
-15
-10
-5
0
Frequency [MHz]
5
10
15
Figure 4.7. Power spectrum of two-tone signal for φ m (⋅) function without
predistortion
44
Two tones signal
40
20
X: 3.012
Y: 6.232
X: 8.088
Y: 6.239
0
Power level [dBm]
-20
-40
X: -2.065
Y: -58.02
X: 13.16
Y: -71.39
-60
-80
-100
-120
-140
-15
-10
-5
0
Frequency [MHz]
5
10
15
Figure 4.8. Power spectrum of two-tone signal for φ m (⋅) function with
predistortion
45
Two tones signal
40
20
X: 3.012
Y: 6.185
X: 8.088
Y: 6.277
0
Power level [dBm]
-20
X: -2.065
Y: -50.86
-40
X: 13.16
Y: -55.13
-60
-80
-100
-120
-140
-15
-10
-5
0
Frequency [MHz]
5
10
Figure 4.9. Power spectrum of two-tone signal for ψ m (⋅) function without
predistortion
46
15
Two tones signal
40
20
X: 3.012
Y: 6.212
X: 8.088
Y: 6.213
0
Power level [dBm]
-20
-40
X: -2.065
Y: -59.83
-60
X: 13.16
Y: -88.66
-80
-100
-120
-140
-15
-10
-5
0
Frequency [MHz]
5
10
15
Figure 4.10. Power spectrum of two-tone signal for ψ m (⋅) function with
predistortion
47
5 Discussion
In this project, the problems I have encountered were that the matrix R is ill-conditioned
and the reason of that was the column of the matrix S has low values because we
calculate S matrix with γ . Since γ is a small speech, then all columns in S will be small
compared to the columns of S matrix. This gives the ill-conditioned matrix R. For f ,
f and h filters we have used the same model order but in real there are not. These
shortcomings in the method have also been highlighted by others during the project.
We have studied parallel project, [19] there they have achieved a good result by using a
novel predistorter structure for the joint mitigation of power amplifier and IQ modulator
impairments in wideband direct-conversion radio transmitters. The predistorter was based
on the parallel Hammerstein or memory polynomial predistorter, yielding a predistorter
which is completely linear in the parameters. In the estimation stage the indirect learning
architecture is utilized. The proposed technique is the first technique to consider the joint
estimation and mitigation of frequency-dependent PA and modulator impairments. It is a
similar solution, but do not have the same disadvantages.
In [19], the PA predistorter is a parallel Hammerstein or memory polynomial predistorter
with the static nonlinearities given by the orthogonal polynomials, the IQ predistorter is
of the general two-filter type where one filter G1 (z ) , is filtering the original or nonconjugate signal, and the other, G2 (z ) , is filtering the conjugated signal and LO leakage
compensator. The filters of the PA PD and IQ PD are in cascade, making their separate
estimation difficult. A joint power amplifier and IQ modulator predistorter was suitable
for mitigating frequency dependent impairments. The PD is completly linear in the
parameters thus allowing easy estimation of PD parameters with linear least-squares. The
simulation and measurement analysis show good performance.
48
6 Conclusion
A simulation system and a measurement system have been designed in a data program
(Matlab) and a parsimoniously parameterized digital predistorter design used to generate
a spectrally pure signal. The objective of this project was to implement and evaluate the
theory parts using data program (Matlab). The parallel Hammerstein structure has been
used that is consist of the nonlinearity followed by a linear filter and it is useful for digital
predistortion of power amplifier. In this work, two polynomial model have been used in
power amplifier modelling and predistorter design. the conventional polynomial φ m (⋅)
and orthogonal polynomial ψ m (⋅) have been used to generate signals.
The simulation results for two-tone signal presented in Section 4.2, the plots showed that
the intermodulation products have been reduced after predistortion design. The
intermodulation products for φ m (⋅) function at frequency 23.91 MHz have been reduced
38.0 dBm and at frequency 55.78 MHz have decreased 37.3 dBm. The intermodulation
products for ψ m (⋅) function at frequency 23.91 MHz have decreased 34.6 dBm and at
frequency 55.78 MHz have decrased 35.9 dBm. The measurement results for two-tone
signal presented in Section 4.3, the plots showed that the intermodulation products have
been reduced after predistortion design. The intermodulation products for φ m (⋅) function
at frequency -2.065 MHz have decreased 6.9 dBm and at frequency 13.16 MHz have
decreased 16.1 dBm. The intermodulation products for ψ m (⋅) function at frequency 2.065 MHz have decreased 8.9 dBm and at frequency 13.16 MHz have decreased 33.5
dBm. The results showed that the ψ m (⋅) function generally yield better power amplifier
modeling accuracy as well as predistortion linearization performance then the φ m (⋅)
function.
We did not achieve the performance increase that we wish or hope we have identify
possible causes of it and also studied alternative solutions, unfortunately, we have not had
the opportunity in this project to implement and verify these theories.
49
7 References
[1]
P. Händel, “Dynamic nonlinear pre-distortion of signal generators for improved
dynamic range”, Draft for a journal paper, Oct. 2008.
[2]
C. Luque and N. Björsell, “Improved dynamic range for multi-tone signal using
model-based pre-distortion,” 13 Th Workshop on ADC Modeling and Testing,
florence, Italy, 2008, September 22-24.
[3]
N. Björsell,” Modeling Analog to Digital converters at Radio Frequency”. Doctoral
thesis in Telecommunications, KTH School of Electrical Engineering, Stockholm,
2007.
[4]
M. Valkama, “Advanced I/Q signal processing for wideband receivers: models and
algorithms, PhD thesis, Tampere University of Technology, November, 2001.
[5]
G. Yang, G. Vos and H. Cho. “I/Q modulator image rejection through modulation
pre- distortion”, IEEE Vehicular Technology Conference, 1996.
[6]
M. Isaksson,” Behavioral Modeling of Radio Frequency Power Amplifiers “.
doctoral Thesis in Telecommunications, KTH School of Electrical Engineering,
Stockholm, 2007.
[7]
C. Luque,” Model-based pre-distortion for signal generators”, Department of
technology, vol. Master thesis. Gävle: University of Gävle, 2007.
[8]
M. Isaksson, D. Wisell and D. Rönnow, “Wide-Band Dynamic Modeling of Power
amplifiers Using Radial-Basis Function Neural Neworks”, Microwave Theory and
Techniques, Volume 53, Issue 11, Nov. 2005.
[9]
N. Ceylan, “Linearization of power amplifiers by means of digital predistortion,” in
Technische Fakultät, vol. Doctor-Ingenier. Erlangen: Universität ErlangenNurnberg, 2005.
[10] E. Aschbacher, “Digital Pre-distortion of Microwave Power Amplifiers”, doctoral
thesis, university of technology, wien, 2005.
[11] G. Xing, M. Shen, and H. Liu, “Frequency Offset and I/Q Imbalance Compensation
for Direct-Conversion Receivers”, IEEE transaction on Wireless Communication,
Mars 2005.
50
[12] L. Anttila, M. Valkama and M. Renfors, ”Frequency selective I/Q mismatch
calibration of wideband direct-conversion transmitters”, IEEE Transactions on
Circuits and Systems – II: Express briefs, Vol. 55, No. 4 April 2008, pp. 359-363.
[13] D. Rönnow and M. Isaksson, ”Digital predistortion of radio frequency power
amplifiers using Kautz-Volterra model”, Vol. 42, issue 13, pp. 780-782, University
of Gävle, June 2006. .
[14] P.L. Landin, M. Isaksson and P .Händel, “On accurate power amplifier behavioural
models,” IEEE Transactions on Microwave theory and techniques, submitted, April
2008.
[15] R. Raich, H. Qian and T. Zhou, “Orthogonal polynomials for power amplifier
modelling and predistorter design”, IEEE Transactions on Vehicular Technology,
Vol. 53, No. 5, Septembe2004,pp.1468-1479.
[16] D. Wisell, “A Baseband Time Domain Measurement System for Dynamic
characterization of Power Amplifiers with High Dynamic Range over Large
bandwidth”. Instrumentation and Measurement Technology Conference, 2003.
[17] J. Proakis, Digital Communications: McGraw-Hill, 1995.
[18] Rhode & Schwarz FSQ 26 Operating Manual.
[19] L. Anttila, M. Valkama, P. Händel, “Joint Mitigation of Power Amplifier and
Modulator Impairments in Wideband Direct-Conversion Transmitters”, Draft for a
journal paper, April. 2009.
51
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertising