Master Thesis, Tomas Elgeryd, IR-SB-EX-0224

KUNGL TEKNISKA HÖGSKOLAN
Institutionen för
Signaler, Sensorer & System
Signalbehandling
100 44 STOCKHOLM
ROYAL INSTITUTE
OF TECHNOLOGY
Department of
Signals, Sensors & Systems
Signal Processing
S-100 44 STOCKHOLM
Iterative algorithms for linearising
non-linear systems by digital predistortion
Tomas Elgeryd
November 2002
IR–SB–EX–0224

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REPORT IR-SB-EX-0224 1 (68)
Prepared (also subject responsible if other) No.
RSA/RRU/R Tomas Elgeryd RRU/R-02:043
Approved Checked Date Rev Reference
RSA/RRU/R Leonard Rexberg 2002-11-12 A Master Thesis
Iterative algorithms for linearising non-linear systems by
digital predistortion
Abstract
This report considers iterative algorithms for linearising a non-linear system
by digital predistortion. In this case, a power amplifier for the 3G, third
generation, base station constitutes the non-linear system. Algorithms based
on the recursive estimation methods Recursive Least Squares and Kalman
are tested. The performances of these algorithms are benchmarked to
predistorters based on the Least Squares and Least Mean Square algorithms.
The study focuses on rate of convergence, complexness, accuracy and how
well the algorithms apply to the special case of non-linear predistortion.
Various base functions are utilized to form the linear combination in the
estimator that approximates the non-linearity. The bases considered here are:
polynomial bases, triangular bases, constant bases and a combination of
polynomial and triangular bases.
The results of the simulations indicate that the performances of the Least
Squares, the Recursive Least Squares and the Kalman algorithms are
equivalent. The performance of the Least Mean Square algorithm is lower
than the other algorithms. It is concluded that the best performance of the
digital predistorter is achieved when using the polynomial bases, closely
followed by the combination of polynomial and triangular bases and then
finally the triangular bases. The constant bases do not perform well, unless a
considerable number of base functions in the estimator are used.

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REPORT 2 (68)
Preface
This master thesis report is the result of the last individual project of my
education in the M.Sc. Electrical Engineering program at the Royal Institute of
Technology, Stockholm.
I would like to take the opportunity to thank especially two persons who have
been involved in my work. The first person to thank is my supervisor and
examiner Peter Händel at the Department of Signals, Sensors and System
(S3) at the Royal Institute of Technology. The second person to thank is my
supervisor Dr Leonard Rexberg, Senior Specialist of Radio Modelling,
RSA/RRU/R at Ericsson AB.
Tomas Elgeryd
Stockholm

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REPORT 3 (68)
Contents
1 Introduction ............................................................................................. 5
1.1 Objectives ................................................................................... 5
1.2 Report outline.............................................................................. 5
2 Background ............................................................................................. 7
2.1 Problem description.................................................................... 7
2.2 The system environment ............................................................ 8
2.3 Non-linear systems..................................................................... 8
2.4 General on PA linearisation...................................................... 10
2.5 Signal estimation by minimization of a related cost function ... 12
3 Theory of linear estimation algorithms .............................................. 14
3.1 Least Squares........................................................................... 14
3.2 Least Mean Square................................................................... 15
3.3 Recursive Least Squares.......................................................... 16
3.4 Kalman...................................................................................... 17
4 Initial case studies................................................................................ 20
4.1 Structure of initial cases............................................................ 20
4.2 Results ...................................................................................... 22
4.3 Conclusions............................................................................... 33
4.4 Discussion................................................................................. 34
4.5 Summary................................................................................... 34
5 The digital predistorter......................................................................... 35
5.1 Predistorter or postdistorter...................................................... 36
5.2 The structure of the estimator................................................... 37
5.3 Various base functions.............................................................. 39
5.4 Implementations in context of linearisation .............................. 41
5.5 Parameter settings.................................................................... 44
6 Linearisation of a memory-less PA..................................................... 45
6.1 The WCDMA signal .................................................................. 45
6.2 The number of base functions in the PD estimator.................. 47
6.3 The choice of parameter settings in the algorithms................. 48
6.4 Performance measurements of the Matlab implementation.... 51
6.5 Results ...................................................................................... 52
6.6 Conclusions............................................................................... 63
6.7 Discussion................................................................................. 64
6.8 Summary................................................................................... 64
7 Future work............................................................................................ 65
8 Appendix................................................................................................ 67
8.1 Appendix 1................................................................................ 67

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Abbreviations
ACLR Adjacent Channel Power
BB Base Band
CW Continuous Wave
dB decibel
dBc decibel below carrier
IF Intermediate Frequency
IM Inter Modulation
IMD Inter Modulation Distortion
LS Least Squares
LMS Least Mean Square
LUT Look Up Table
M Misadjustment of the MSE
MCPA Multi Carrier Power Amplifier
MSE Mean Square Error
MMSE Minimum Mean Square Error
PA Power Amplifier
PD Predistorter
PDF Probability Density Function
PSD Power Spectral Density
RF Radio Frequency
RLS Recursive Least Squares
ROC Region of Convergence
SNR Signal to Noise Ratio
UMTS Universal Mobile Telecommunication System
WCDMA Wideband Code Division Multiple Access
Symbols
.T
Transpose
.H
Hermitean transpose
.*
Complex conjugate
( )1ˆ −nnx The estimate of the state vector x(n) given the observations y(k)
for k ≤ n-1.
∆ACLR The reduction in Adjacent Channel Power

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1 Introduction
All types of power amplifiers (PA) that are used in the RF-stage of a radio
base station suffer from the effects of non-linearity. The non-linear behaviour
appears, among other things, due to the characteristics of the transistors in
the electrical circuits.
The non-linearity causes intermodulation distortion (IMD), which degrades the
overall performance of the radio base station. The IMD is distinguished by
frequency leakage into adjacent frequency intervals. The effect of IMD
appears for example in radio systems with broadband signals, e.g. the
Universal Mobile Telecommunication System (UMTS) that has a carrier
bandwidth of approximately 5 MHz. Therefore, this has become an important
aspect to consider in the design of the Multi Carrier Power Amplifiers (MCPA).
In order to counteract the IMD, caused by the non-linear PA, digital
predistortion could be applied to the input signal. In this case, the digital
predistorter (PD) is chosen to operate on the base band signal in front of the
PA. Then, the base band signal could be adaptively filtered in such a way that
the gain of the PA and PD, connected to each other, will be linear. Hence, the
digital PD could be considered to work as the inverse of the nonlinearities in
the PA.
1.1 Objectives
The purpose of this master thesis project is to analyse various recursive
algorithms, which will be used in the implementation of the digital PD. There is
a special interest in recursive algorithms that are computationally efficient and
have fast tracking ability, e.g. Recursive Least Squares (RLS) and Kalman.
Digital PDs will be designed, based on the RLS and Kalman algorithms
mentioned above. Simulations will be accomplished to measure the
performance of the digital PDs. A suitable benchmark is an implementation of
a PD based on the Least Squares (LS) or Least Mean Square (LMS)
algorithms.
The preceding work in this topic has been focused on the implementation of
the PD by the LS algorithm. Also Neural Networks have been considered [1].
Therefore, it would be interesting to implement the PD with recursive
algorithms for comparison of the performance.
1.2 Report outline
The outline of this Master Thesis is as follows:
Chapter 1 contains the introduction of the predistortion topic. Chapter 2
contains the background and an overview of the general on PA linearisation.
Chapter 3 contains the necessary theory on the numerical algorithms for
linear estimation.

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REPORT 6 (68)
Chapter 4 contains some initial case studies to get acquainted with the
numerical algorithms. Chapter 5 contains a thorough description on digital
predistorters. In Chapter 6, the linearisation is performed. Chapter 7 contains
some aspects about the future development on this subject.

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REPORT 7 (68)
2 Background
2.1 Problem description
This Master Thesis project comprises theoretical studies and practical
implementations of algorithms to compute parameters for digital PDs for
linearisation of non-linear memory-less systems. The systems considered
here are PAs that suffer from IMD, due to the nonlinearities. The objective is
to test several algorithms for solving digital PDs based on recursive
algorithms, e.g. RLS and Kalman, which will suppress the IMD in an iterative
way rather than by large matrix products and matrix inversions.
The primary task is to analyse the RLS and Kalman algorithms in the pre-
distorter case. The performance of these algorithms will be measured and
benchmarked with algorithms based on non-recursive solutions, i.e. LS or
LMS. The comparison of the performance of the different algorithms will
mainly be visualised through plots of the Power Spectral Density (PSD) of the
output signal of the PA, but also by plots showing the parameter convergence
in terms of number of iterations. The performance of the algorithms can also
be measured by using ordinary tools, such as: misadjustment (M) of the mean
square error (MSE), tracking ability and the reduction in adjacent channel
power (∆ACLR).
To keep the connection to the real case, measured input and output signals of
a physical PA will be used in the simulations, which will be performed in
MATLAB 6.0.
Interesting areas to become engrossed in are among others:
• The influence of noise to the different estimation algorithms.
• How the different base functions used in the estimator affect the
performance concerning the suppression of the IMD.
• How the rate of convergence depends on different estimation
algorithms and the choice of base functions.
• How the type of the input signal to the PD affects the performance.
Conclusions to be drawn are:
• Performances of the different algorithms based on different base
functions; the suppression of IMD, rate of convergence, computational
cost and noise influence.
• Suggestions for the best choice of base functions in the estimator.
• Best choice of recursive scheme according to the suppression of the
IMD, rate of convergence, computational cost and noise influence.

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2.2 The system environment
In this section, the system environment that surrounds the predistorter is
presented. The predistorter works in the discrete time domain, where the
sample frequency of the complex discrete time signal is approximately Fs =
66.44 MHz. The WCDMA signal that enters the predistorter is of base band
(BB) type. A simple block diagram of the communication system is described
in Figure 2.1. Furthermore, it is assumed that the nonlinearity in the PA is
dominating compared to the quantization effects due to the A/D and the D/A
conversions in the system. Thus, these effects are considered infinitesimal.
IF/RFPDTX
DSP
PAD/A
A/D
RF/IF
IF/RFPDTX
DSP
PAD/A
A/D
RF/IF
IF/RFPDTX
DSP
PAD/A
A/D
RF/IF
Figure 2.1 The block diagram of the system environment. The different blocks are:
TX = transmitter, PD = predistorter, D/A = digital to analog converter, IF/RF = up
converter, PA = power amplifier, RF/IF = down converter, A/D = analog to digital
converter, DSP = digital signal processor. Following components operate on the base
band: TX, PD, D/A , A/D and the DSP.
The common time delay between the measured signals, which is due to the
propagation along different signal paths in the system, has already been
compensated for. Hence, this effect is not needed to be considered in the
signal processing.
Also, the measured signals are normalized according to the largest sample
magnitude of each respective signal. Therefore, the largest signal magnitude
that appears in the measurement data is 1.
2.3 Non-linear systems
The definition of a non-linear system is a system that violates the conditions
of a linear system, i.e. the superposition and scaling properties. A linear
system is thus characterised by Eqn. (2.1), where a and b are arbitrary
constants and x1 and x2 are arbitrary input signals [2].
)()()( 2121 xbLxaLbxaxL +=+ (2.1)

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In our case, the primary task for the PA is to multiply the input signal with a
constant gain factor. In theory, this seams to be an easy operation. But in
practice, there are problems concerning the linearity of the input to output
signal relation. The non-linear behaviour of the PA originates from the
electrical circuits which deviate from their ideal linear operation, e.g. due to
saturation. The non-linear behaviour of the PA causes IMD that degrades the
performance of the amplifier.
2.3.1 Memory effects
A memory-less system is time-invariant, which implies that the output is only
dependent of the input at the same time instant [2]. The time-invariance, in
the continuous time domain, can be expressed according to Eqn. (2.2),
)()()()( TtyTtxtytx −→−⇒→ (2.2)
Only memory-less non-linear systems are considered in this master thesis.
This is due to simplify the characterization and linearisation of the non-linear
systems. A common phenomenon of memory effects is that the PSD of the
output signal often is unsymmetrical with respect to the carrier.
2.3.2 Intermodulation distortion
The underlying reason for the IMD is that the transfer gain of the PA is not
constant as the amplitude of the input signal varies. As the instantaneous
amplitude of the input signal varies at every time sample, spurious harmonics
will turn up on the output port of the PA.
The IMD can be illustrated by supposing that two CW signals, in this case two
sinusoids, are fed to the input of the PA. For a memory-less nonlinearity, the
output signal can be described by the polynomial (2.3), where x is the input
signal.
...)( 432
+++++= exdxcxbxaxf (2.3)
If the input signal x is composed of two sinusoids, with angular frequencies ω1
and ω2, the Eqn. (2.3) will result in a number of products of the form:
etc.,...)23cos(),3cos(
),3cos(),2cos(),2cos(),cos(
2121
21212121
ωωωω
ωωωωωωωω
±±
±±±±
(2.4)
The products in the expression (2.4) are called IM products or IMD. Most of
the products will fall in the interval around the harmonics of the sinusoids at
the input. These products can easily be suppressed by conventional band
pass filtering. The problem appears when the products are situated close to
the fundamentals, see Figure 2.2. For example, the products cos(2ω1 - ω2)
and cos(3ω1 - 2ω2) will cause greater problems, assumed that the angular
frequencies ω1 and ω2 are fairly close to each other. Because, these products
cannot be suppressed by band pass filtering.

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For a multi-carrier signal, the products falling in the band of the input signal
itself are referred to as cross modulation, which will interfere with the original
signal. The parts of the output signal falling just outside the band of the multi-
carrier signal will interfere with adjacent channels.
It can be shown from Eqn. (2.3) that the IM products close to the
fundamentals, i.e. the two sinusoids, originate from the odd order terms in the
polynomial (2.3). The even order terms in the polynomial (2.3) will fall in the
frequency interval of the harmonics.
1st harmonic zone 2nd harmonic zone 3rd harmonic zone
Fundamentals with the angular frequencies ? 1 and ? 2
IM products
?
1st harmonic zone 2nd harmonic zone 3rd harmonic zone
Fundamentals with the angular frequencies ? 1 and ? 2
IM products
?
Figure 2.2 The IMD of two sinusoids with angular frequencies ω1 and ω2. The IM
products appear to the left and to the right of the two fundamentals.
The IM products near the fundamentals can effectively be suppressed by
using the technique of predistortion.
2.4 General on PA linearisation
In this section, the general concepts of PA linearisation are described. There
are several techniques to linearise a PA, for example by using techniques like
a Cartesian Loop, feedforward or predistortion. Though, in this report only the
predistortion technique is of interest. Therefore, no focus has been spent on
the other methods. In addition to this brief introduction a thorough description
of the PD is given in Chapter 5.
The principle of a predistorter is that the input signal to the PA is distorted in
such a way that the gain of the total system will be linear. Hence, the
nonlinearities in the PA will distort the already predistorted input signal back
into its original shape. Thus, ideally the two signals should only differ by a
constant gain factor. In Figure 2.3, a block diagram shows the principle of the
predistortion concept.
A simple mathematical interpretation of the PA and PD constellation in Figure
2.3 c) can be written as Eqn. (2.5).

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( )
( )
( )( )
( ) ( )xPAxPD
xxPDPAxx
xPDz
zPAx
1
ˆIf
ˆ
−
≡⇒
≡⇒≡
=
=
(2.5)
This equation indicates that the PD can be considered to work as the inverse
of the non-linear PA. The difficult task here is that the inverse PA-1
(x) is
nontrivial and hard to determine analytically. Therefore, numerical solution
methods will be applied instead to solve this problem.
G
x y = Gx
PA(·)
x y = PA(x)
PA(·)PD( ·)
x z = PD(x)
xˆ
a)
b)
c)
GG
x y = Gx
PA(·)
x y = PA(x)
PA(·)PD( ·)
x z = PD(x)
xˆ
a)
b)
c)
Figure 2.3 a) This figure shows the expected linear behaviour of the PA.
b) This block diagram shows the real behaviour of the PA, which is non-linear.
c) This block diagram shows the PD and PA coupled in cascade to linearise the PA.
A common approach to design the PD is to assume that the transfer function
is composed of several base functions, organized in a linear combination.
Usually, determination of the coefficients of that linear combination could be
done by minimization of a related cost function, see Section 2.5.
There are various numerical methods to solve these, usually overdetermined,
equation systems that emanate from the minimization of the cost function.
The numerical algorithms considered here are: LS, LMS, RLS and the
Kalman algorithm.
If the linearisation has been performed properly, the performance of the PA
will improve by the suppression of the IM products. The typical performance
measure is the plot of the PSD of the input and output signals of the PD and
PA constellation, see Figure 2.4.

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Figure 2.4 The typical PSD plot of the output signals of the PA, with and without
linearisation.
2.5 Signal estimation by minimization of a related cost function
It is very common in applications of signal processing that estimators and
quadratic cost functions are used. These tools are also frequently used here
when linearising the non-linear PA by applying a PD to distort the input signal.
The estimator is a model that is designed to create an estimate of some
desired signal. The structure of the estimator is typically a linear combination
of base functions, designed from measurements of a signal that is correlated
with the desired signal, [3] and [4].
The closeness of the estimate to the desired signal could then be computed
by applying the quadratic cost function, which indicates the deviation between
the estimate and the desired signal. Hence, the optimal estimate is found
when the quadratic cost function assumes its minimum value.
An example of a simple quadratic cost function, J(α), is given in Eqn. (2.6). In
order to simplify, the linear combination that forms the estimate will in this
case only consist of one parameter. This linear combination will of course be
extended with more terms in the real case to get the optimal performance
from the estimator.
Hence, the signals that are used to form this simple quadratic cost function
are: xn that is the desired signal, yn that is the base function and finally α that
is the unknown parameter in the estimator.
( ) LnyxJ nn ,...,1,0for,min
2
=−= αα
α
(2.6)
By plotting the quadratic cost function, Figure 2.5, with respect to the
parameter α, one can see the minimum value.

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A various number of iterative numerical methods could be used to find the
minimum of the quadratic cost function J(α). The common approach is to
make use of the gradient. As known, the gradient of J(α) points towards the
direction where the quadratic cost function increases the most.
To find the minimum we only have to design an iterative algorithm that follow
the quadratic cost function, J(α), in the direction of the steepest descent, see
Figure 2.5. In this way, the estimate of α will converge to its optimum value as
approaching the minimum of J(α).
Figure 2.5 The quadratic cost function J(α). Three iterations have been plotted to
show the principle of the iterative algorithms to find the minimum value of J(α). Of
course the one dimentional quadratic cost function will be extended to a multi variable
quadratic cost function, when the real PD is designed.
All the estimation algorithms that will be treated in this report utilize some kind
of cost function with its related gradient to find the optimal estimate.

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3 Theory of linear estimation algorithms
This chapter contains the essential theory on linear estimation algorithms that
will be treated in this report. In this case the focus has been on the four
algorithms: LS, LMS, RLS and Kalman. The LS and LMS algorithms are non-
recursive, whereas the RLS and Kalman algorithms work in a recursive
manner.
For simplicity and to be in accordance with the literature on this subject only
the general expressions of the algorithms are presented here. In further
chapters these expressions will be modified to suit the estimator of the PD in
a better way.
3.1 Least Squares
The LS algorithm in Eqn. (3.2) is a deterministic, non-recursive, algorithm for
linear estimation. This algorithm gives the optimum estimate by minimizing
the general quadratic cost function J(x), stated in Eqn. (3.1). The weight
matrixes Π0 and W are positive-definite and the vector x0 contains the initial
guess of the estimate x. By choosing appropriate values of the design
parameters x0, Π0 and W the performance of this algorithm can be controlled.
( ) ( ) ( ) 2
0
1
00 W
H
HxyxxxxxJ −+−Π−= −
(3.1)
The reason for using the general quadratic cost function in Eqn. (3.1) is that
the least squares solution xˆ becomes unique, even when the matrix H is not
full rank, [4].
The matrix H contains the base functions of which the linear combination of
the estimator will be formed, while the matrix Π0 and the vector x0 give
additional a priori knowledge to the LS problem. The matrix Π0 indicates the
confidence of the closeness of the initial guess x0 to the true value x. Large
values of the matrix Π0 indicate low confidence of the initial guess, whereas
small values indicate high confidence.
The minimum of Eqn. (3.1) is obtained when the estimate xˆ fulfils the Eqn.
(3.2).
[ ] [ ]0
11
00
ˆ HxyWHWHHxx HH
−+Π+=
−−
(3.2)
As can be seen from the solution in Eqn. (3.2), the computational complexity
grows with the increasing dimension of matrix H. The numerical problems
occur when performing the matrix inversion in Eqn. (3.2). The matrix that is
inverted has the dimension N × N, where N equals the number of elements in
the vector x.
Although this algorithm will be used as a reference to which the other
algorithms will be benchmarked. The computational complexity of inverting a
matrix is in the order of N3
/3 for real matrices, and 2N3
for complex matrices.

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REPORT 15 (68)
3.2 Least Mean Square
Another iterative algorithm is the LMS algorithm as described by Eqn. (3.3).
Due to its simple structure, the LMS algorithm does not need to compute
direct correlation of long signal vectors or to perform time-consuming complex
matrix inversions. Hence, the computational complexity of each iteration is of
order N.
( ) ( ) ( ) ( ) ( ) ( )( )1ˆ1ˆˆ *
−−+−= nnYnxnYnn T
θµθθ (3.3)
According to Eqn. (3.3), the estimate of the desired vector θ is updated with a
small step in the negative gradient direction of the quadratic cost function.
The quadratic cost function in the LMS algorithm is stated in Eqn. (3.4) and is
a measure of the distance between the desired signal x(n) and our estimate
YT
(n)θ.
( ) ( ) ( )( )2
, θθ nYnxnMSE T
LMS −= (3.4)
The vector Y(n), given by Eqn. (3.5), contains the N base functions y(n-N+1)
up to y(n) and is used to form the linear combination YT
(n)θ in Eqn. (3.4).
( ) ( ) ( ) ( )[ ]T
NnynynynY 11 +−−= L (3.5)
By using the framework of averaged error system for the LMS algorithm, the
stability criterion of the LMS algorithm can be derived, which indicates that the
stability is foremost dependent of the step size µ, see Appendix 1.
A large step size µ results in fast convergence but also risk of instability. A
small step size gives the opposite property, slow convergence and stability.
Hence, there is a trade-off between stability and tracking ability.
The non-zero step size will result in an inevitable misadjustment, M, of the
optimal MSE(θ), even for large values of the sample time n [5]. The
misadjustment M could be approximated with Eqn. (3.6).
2
2
yN
M
σµ
≈ (3.6)
As observed, the misadjustment is proportional to the step size µ, the number
of parameters N, and the signal variance σy
2
.
This means that the LMS algorithm does not converge to the optimal θ,
irrespective of the assigned value of µ. The approximation of the MSE(n), for
large values of n, is then given by Eqn. (3.7).
( ) ( )( )MMSEnMSE optLMS +≈ 1θ (3.7)

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3.3 Recursive Least Squares
A natural extension of the LS algorithm in Eqn. (3.2) is the RLS algorithm as
described in Eqn. (3.9), where the estimates are computed in a recursive
manner [4]. Hence, the RLS algorithm is more computationally efficient
compared to the LS algorithm. The computational complexity of the RLS
algorithm is of order N2
, which is though more expensive than the LMS
algorithm.
Compared with the LMS algorithm, the rate of convergence is faster, which is
due to that the RLS algorithm utilizes information in the input data extending
back to the instant of time when the algorithm was initiated.
Unlike the LMS algorithm, the RLS algorithm also uses the information of the
second order derivative of the MSE(θ) to form the estimate of θ. In addition,
also a less noise sensitive approximation of the MSE(θ) is used than stated in
Eqn. (3.4). The reduction in noise sensitivity is due to the introduction of the
forgetting factor λ [5].
The forgetting factor λ is a design parameter that indicates the influence of
past measurements on the current estimate. It assumes values according to
the inequality (3.8).
10 ≤< λ (3.8)
If the value of λ is chosen to be unity, all the past measurements contribute
equally to the estimate. On the other hand, if the value is chosen to be less
than one, past measurements have less influence. This implies that the
algorithm can adapt to certain non-stationary signal properties.
The forgetting factor is also closely connected to the rate of convergence of
the RLS algorithm. Smaller values of λ imply faster convergence and
adaptation to certain non-stationary signals. On the contrary, a large value
close to unity implies slower convergence.
One common form to state the RLS recursion is according to Eqn. (3.9), see
[4].
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )( )11
1
1
ˆ1ˆˆ
1
1
1ˆˆ
*
*
−−−=+
−+−=
−+
−
=
−=
nPnYnKnPnP
nxnxnKnn
nYnPnY
nYnP
nK
nnYnx
T
RLS
T
T
RLS
λ
θθ
λ
θ
(3.9)
The vector Y(n) is the observation vector, of dimension N, given by the Eqn.
(3.10). This vector contains the base functions that form the linear
combination in the estimate.

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REPORT 17 (68)
( ) ( ) ( ) ( )[ ]T
NnynynynY 11 +−−= L (3.10)
The matrix ( )nP of Eqn. (3.9) is the inverse of the covariance matrix formed of
the vector Y(n) and it has the dimension N × N. Finally, the vector K(n), with
dimension N, is a type of gain that corrects the estimate according to the
estimation error that occurs in the Eqn. (3.9). Note the similarity between the
gain vector K(n) of Eqn. (3.9) and the corresponding one, µY*
(n), of Eqn.
(3.3).
The RLS equations in Eqn. (3.9) have to be initialised with
( ) estimate.parameterinitialThe0ˆ −θ
( ) ( ) 0.someforIastaken,0ˆofyuncertaintinitialThe1 NN >− × δδθP
There are two different approaches to compute the misadjustment, M, for the
MSE of the RLS algorithm. Here, the value of the forgetting factor λ has a
significant importance, see for example [5]. The MSE of the RLS algorithm
can be approximated as Eqn. (3.11) assumed that λ = 1, where the variable N
is the number of parameters in the estimator and n is the sample time.
( ) ( ) 





+=
n
N
MSEnMSE optRLS 1θ (3.11)
The conclusion from Eqn. (3.11) is that the RLS estimate converges to the
optimal parameter θopt when the time n approaches infinity and λ equals 1.
If the forgetting factor is close to unity, i.e. 0<<λ < 1, and the sample time n is
large the MSE is approximated as in Eqn. (3.12).
( ) ( ) ( )





 −
+=
2
1
1
N
MSEnMSE optRLS
λ
θ (3.12)
The conclusion from Eqn. (3.12) is that the RLS estimate does not converge
to the optimal parameter θopt when λ ≠ 1.
3.4 Kalman
The Kalman algorithm is based on the state space (SS) model, written in Eqn.
(3.14). The SS model describes the future evolution of the physical system to
be modelled. And the state vector, x(n), contains the information required to
determine the future evolution of the system when the input u(n) is given.
The state vector x(n) is written as Eqn. (3.13), where xk(n) is the kth state at
time n.

Open
REPORT 18 (68)
( )
( )
( )
( )











=
− nx
nx
nx
nx
N 1
1
0
M
(3.13)
In our case, the state vector will contain the PD parameters that we want to
determine. However, this will be scrutinized in Chapter 5. In this section we
only present the general expressions of the algorithm, as stated in literature.
According to theory, the Kalman filter is the linear MMSE estimator of the
state vector x(n), see [5]. The solution is computed recursively, where the
updated estimate is based on the previous estimate and the new input data.
Hence, only storage for the previous estimate and a correlation matrix are
needed for each update.
The state space model is represented by:
( ) ( ) ( ) ( )
( ) ( ) ( )nvnHxny
nGwnuGnFxnx u
+=
++=+1
(3.14)
where
tmeasuremenvector tostaten vector,transitio
statetheandinputebetween thmatrixntransitioG
statesdifferentobetween twmatrixntransitioF
signalmeasuredy(n)
noisetmeasuremenv(n)
noiseprocessw(n)
signalinputu(n)
vectorstatex(n)
u
=
=
=
=
=
=
=
=
H
The process noise {w(n)} and the measurement noise {v(n)} are assumed to
be zero mean white noise processes, related to each other as shown in Eqn.
(3.15), where R1 and R2 are the variances of {w(n)} and {v(n)}, respectively.
( )
( )
( ) ( )( ) ( )mn
R
R
mvmw
nv
nw
E −





=











δ
2
1HH
0
0
(3.15)
To estimate the state vector x(n+1) from the observations y(k), for nk ≤ , one
can apply an observer of the form
( ) ( ) ( ) ( )( )
( ) ( )nnxHnny
nnynyKnnxFnnx
1ˆ1ˆ
1ˆ1ˆ1ˆ
+=+
−−+−=+
(3.16)

Open
REPORT 19 (68)
The Kalman gain K=K(n) in Eqn. (3.16) corrects the state estimate according
to the size of the prediction error. The optimal trade-off between the noise
sensitivity and the measurement information is achieved by applying the
Kalman equations, stated in Eqns. (3.17) and (3.18).
The measurement update:
( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )[ ] ( )nHPRHnHPHnPnPnQ
nnxHnynnxnnx
RHnHPHnP
HH
HH
1
2
1
2
1ˆnL1ˆˆ
nL
−
−
+−=
−−+−=
+=
(3.17)
The time update:
( ) ( )
( ) ( )
( ) ( ) HH
GGRFnFQnP
nnxHnny
nnxFnnx
11
1ˆ1ˆ
ˆ1ˆ
+=+
+=+
=+
(3.18)
The Kalman recursion has to be initiated by assigning initial values of the
estimates ( )10ˆ −x and P(0). Where, the matrix P(0) is given by the Eqn. (3.19)
and δ is a positive constant.
( )
NN
P
×












=
δ
δ
δ
L
OMM
L
L
00
00
00
0 (3.19)
The design parameters R1, R2, P(0) and ( )10ˆ −x will considerably affect the
performance of the Kalman algorithm. This will be treated in Section 6.3.
There is a relationship between the rate of convergence and the noise
sensitivity of the algorithm. Here, the ratio between R1 and R2 will influence of
the estimate.
• sensitivenoiseisestimatebut theing,fast tracksmall
1
2
⇒
R
R
• sensitivenoiselessisestimatebut thetracking,slowlarge
1
2
⇒
R
R

Open
REPORT 20 (68)
4 Initial case studies
The purpose with the initial case studies is to get a better overview and
comprehension of the algorithms for linear estimation, that will be used for
predistortion. The thought of the initial case studies is to separate the
algorithm itself from the context of the predistortion problem. These studies
should be regarded as a complement and visualisation of the theoretical
background that was presented in Chapter 3.
By using deterministic signals in the linear estimation, one can monitor the
behaviour of each algorithm in a strict way. Therefore, it is suitable to design
some basic estimation problems, where the number of unforeseen effects is
minimized. Complex estimation problems could then be partitioned into
smaller parts to be further analysed by using the background of these studies.
4.1 Structure of initial cases
To maintain the simplicity of the estimation problem only one parameter, α, is
considered in the initial cases. The block diagram of the linear estimation is
illustrated in Figure 4.1.
1 Estimator
f(n)
+ -
a(n)v(n)
1 Estimator
f(n)
+ -
a(n)v(n)
Figure 4.1 The block diagram of the simple estimation problem in the initial case
studies. The optimal estimate of the parameter α is 1.
The algorithms that will be considered here are those mentioned in Chapter 3.
Each algorithm is studied for three different types of input signals:
• Constant signal
• Sinusoid with constant amplitude and frequency
• WCDMA signal
By choosing these signals, the performance of the algorithms can be
examined in a clear way. The WCDMA signal is of the same type that will be
present when linearising the PA.

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REPORT 21 (68)
Also, estimates will be computed when noise perturbs the observed signal.
This will of course bring uncertainty to the estimated parameter. The
interesting thing here is how the noise affects the estimate. The additive noise
will be uniformly distributed with zero mean and variance σ2
. In all cases of
the initial studies, the noise variance assume following values, unless other
values are stated:
• 4.02
=σ for the constant signal
• 2.02
=σ for the sinusoid
• 016.02
=σ for the WCDMA signal
The reason for choosing these values of the noise variance is due to the
characteristic of the different signals. The ambition is to keep the signal to
noise ratio equal to SNR = 10 dB.
The focuses of the initial case studies are especially on:
• The rate of convergence for each algorithm, with respect of signal type
and the parameter settings in the algorithm.
• The influence of noise on the estimates for each algorithm.
• The misadjustment, M, of the optimal MSE.
The structure of the simplest noiseless estimation problem to solve is shown
in Eqn. (4.1), where the signal f(n) is a deterministic signal and α is the
parameter to be estimated. In Figure 4.1 there is a set-up for this estimation
problem, presumed that the noise processv(n) = 0.
( )
( )
( )
( )
( )
( ) 











−
−
=












−
−
0
2
1
0
2
1
f
nf
nf
f
nf
nf
MM
α (4.1)
The structure of this problem extended with noise is shown in Eqn. (4.2),
where the signal f(n) is disturbed by zero mean uniformly distributed noise,
v(n), with variance σ2
. In Figure 4.1, there is a set-up for this estimation
problem.
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) 











−
−
=












+
−+−
−+−
0
2
1
00
22
11
f
nf
nf
vf
nvnf
nvnf
MM
α (4.2)

Open
REPORT 22 (68)
As seen in Eqn. (4.1) and Eqn. (4.2) the optimal solution of α is unity, i.e. α =
αopt = 1.
The three different deterministic signals that are used in the estimation
problems are the following:
2)( =nf (4.3)
( ) 





= nnf
10
sin2
π
(4.4)
( ) ( )nWCDMAnf = (4.5)
The WCDMA signal in Eqn. (4.5) has the PSD that is plotted in Figure 4.2 and
the bandwidth of the signal is approximately 5 MHz.
Figure 4.2 The power spectral density of the WCDMA signal used in the initial case
studies.
To make a fair comparison between the different algorithms, the same noise
contribution has been used in each test case.
4.2 Results
The results of the initial cases studies are presented separately for each
algorithm. The noiseless and noisy cases are treated in parallel for an easier
comparison. But, the focus has mostly been spent on estimates based on
signals with the noise contribution, which highly reflects the real environment
where these algorithms are supposed to operate. To make the result simpler
to summarize, some results will just be mentioned here without an explicit
illustration.

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REPORT 23 (68)
4.2.1 The LS algorithm
The plots in this section give a little feeling of the optimum estimate in
comparison with the estimates produced by the recursive algorithms, treated
in subsequent sections. It is also seen that the convergence of the estimated
parameter α is very fast for the LS algorithm, irrespective of the type of signal.
Constant signal
In this case, the constant signal has been applied. The convergence of the α
estimate is illustrated in Figure 4.3. In the left plot the estimate converges to
its optimal value, because the observed signal is free from noise according to
Eqn. (4.1). In the right figure the estimate does not converge to its optimal
value, due to the additive noise v(n) according to Eqn. (4.2).
Figure 4.3 The convergence of the estimate of α with respect to the sample time n,
when the constant signal was applied.
a) This plot illustrates the convergence of the estimate when no noise is present.
Hence, the estimate of α converges to its optimal value.
b) In this figure the estimate differes from the optimal value, due to the additive noise.
The sinusoid
In this case, the sinusoid has been applied. The convergence of the α
estimate is illustrated in Figure 4.4. In the left plot, the estimate converges to
its optimal value, due to there is no noise in the signals according to Eqn.
(4.1). In the right figure, the estimate does not converge to its optimal value,
due to the additive noise v(n) in Eqn. (4.2).

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REPORT 24 (68)
Figure 4.4 This figure illustrates the convergence of the estimate of α with respect to
the sample time n, when the sinusoid was applied.
a) This plot illustrates the optimal estimate of α, when f(n) is free from noise.
b) This plot shows the estimate of α, when noise is present.
WCDMA signal
In this case, the WCDMA signal has been applied. The convergence of the α
estimate is illustrated in Figure 4.5. In the left plot, the estimate converges to
its optimal value, because no noise is present according to Eqn. (4.1). In the
right figure the estimate converges, but not to its optimal value due to the
additive noise v(n) in Eqn. (4.2).
Figure 4.5 This figure illustrates the optimal estimate of α with respect to the sample
time n, when the WCDMA signal is applied.
a) There is no bias of the estimate of α, when the signals are free from noise.
b) The estimate deviates from its optimum value, due to the additive noise.
4.2.2 The LMS algorithm
As expected the step size µ of the LMS algorithm in Eqn. (3.3) is closely
connected to the rate of convergence and the stability of the algorithm. If the
step size µ increases the rate of convergence increases also, but there is a
limit where a too large value of µ results in divergence of the estimated
parameter α. Hence, there is a trade-off between rate of convergence and
divergence.

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REPORT 25 (68)
In an equation system like Eqn. (4.1) the magnitude of the signal f(n) is
connected with the range of appropriate values of the step size µ. Signals
with larger magnitudes claim smaller step sizes to preserve the same shape
of the tracking curve. The decrease in step size is also needed to maintain the
convergence of the algorithm.
In this section, the misadjustment M of the MSE stated in Eqn. (3.7) is
illustrated by various plots of the estimated parameter α for the systems in
Eqn. (4.1) and Eqn. (4.2). The plots will show that the estimate of α does not
converge to the optimal solution when noise v(n) perturbs the signal f(n). This
property is valid irrespective of the applied signal.
In fact the misadjustment M is proportional to the step size µ, the signal
variance σv
2
and the number of parameters N, as shown in Eqn. (3.6).
Constant signal
In this case, the constant signal in Eqn. (4.3) is used. It can be seen in Figure
4.6 that the estimated parameter α converges asymptotically to its optimal
value when the signal f(n) is free from noise (Figure 4.6 a). When noise is
present, the estimate of α deviates from its optimal value and does not
converge asymptotically to the optimum value (Figure 4.6 b).
Figure 4.6 a) This figure shows the noiseless case where α converges asymptotically
to its optimum value, which is equal to the misadjustment M = 0. The rate of
convergence increases when the step size µ becomes larger.
b) This figure illustrates the misadjustment of the estimate α when noise perturbs f(n).
Due to the noise contribution the estimate α does not converge to its optimal value,
which is a result of the misadjustment M ≠ 0.
The misadjustment increases when the step size, µ, or the noise variance, σ2
,
increases. These properties are illustrated in Figure 4.7 and Figure 4.8. The
same behaviour is discovered for each type of input signal f(n).

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REPORT 26 (68)
Figure 4.7 a) This plot shows the estimate for the step size µ =0.001, before the step
size was increased by a factor two.
b) This plot shows the estimate when the step size is µ =0.002.
Figure 4.8 These two plots illustrate the increase of misadjustment when the noise
variance increases. In these two plots the step size is kept constant to µ = 0.001.
a) The convergence of the estimate for noise variance σ
2
= 0.4. The estimate is
biased due to the misadjustment.
b) The convergence of the estimate for noise variance σ
2
= 0.8. The misadjustment of
the MSE is larger.
The sinusoid
When the sinusoid is applied to the system of equations in Eqn. (4.1) the
estimate will look like Figure 4.9. Sometimes, for certain settings, the estimate
will get an oscillative convergence. Probably, this characteristic is due to
some relation between the period time and the sampling frequency of the
sinusoid. Though, this is not visible in these figures.

Open
REPORT 27 (68)
Figure 4.9 a) The estimated parameter α for the noisless case. The estimate
converges to its optimum value.
b) This plot illustrates that the estimate of α does not converge in an asymptotically
sense when noise is present. The estimate is biased due to the misadjustment.
WCDMA signal
One difference to the other signals is that the WCDMA signal is complex
valued. Besides that, the LMS algorithm proves to have the same properties
as when applying the other signal types, which is shown in Figure 4.10. This
is an interesting point to have in mind, because the WCDMA signal is going to
be used when linearising the real PA.
Figure 4.10 a) The estimate converges to its optimum value in the noisless case.
b) The estimate does not converge to its optimal value when noise is present.

Open
REPORT 28 (68)
4.2.3 The RLS algorithm
According to theory the misadjustment of the MSE can assume two different
forms, as described in Eqns. (3.11) and (3.12), depending on the value of the
forgetting factor λ in the algorithm. Therefore, it is of interest to examine the
estimate for various settings of the design parameters. The two parameters in
the RLS Eqns. (3.9) that have been focused on are λ and δ.
The forgetting factor λ indicates the influence of past measurements on the
current estimate. The parameter δ , on the other hand, is connected to the
confidence of the initial guess of the estimate. These parameters affect the
estimate through the rate of convergence and the influence of noise.
Estimates resulting from four different constellations of the parameters λ and
δ were examined for each type of signal. Only one single parameter value has
been changed from one plot to another.
However, only the results from the noisy estimation will be examined.
Because, in the noiseless case all the estimates converge asymptotically to
the optimal value as expected. Hence, there is no misadjustment of the MSE
in that case. The interesting part here appears in the noisy environment
where the misadjustment of the MSE can be studied.
The changes in the rate of convergence are the same irrespective of the
noise contribution. There are no constraints on the design parameters to
preserve the stability, as opposed to the step size µ in the LMS algorithm.
Constant signal
In Figure 4.11 the estimates from four different constellations of the
parameters λ and δ are plotted. From these plots, it is concluded that the
forgetting factor λ is closely connected to the rate of convergence. The rate of
convergence increases when the forgetting factor decreases. But instead, the
influence of noise appears more prominently for smaller values of λ. Hence,
there is a trade-off between the noise contribution and the rate of
convergence when assigning this value.
The parameter δ is a measure of the confidence in the choice of the initial
settings of the RLS Eqns. (3.9). From Figure 4.11, it can be concluded that
the parameter δ also affects the rate of convergence, but does not affect the
misadjustment of the estimate. The influence is rather moderate compared
with the influence of λ.
In Figure 4.11, it can be seen from the smooth estimate that the
misadjustment decreases when λ = 1 as the sample time increases. When λ
< 1, the misadjustment becomes larger than the previous case and the
estimate does not converge to its optimal value.

Open
REPORT 29 (68)
In general, the rate of convergence is faster than that of the LMS algorithm.
But, the circumstances could be reversed if improper parameter settings in
the RLS algorithm have been performed.
The noise sensitivity has also been improved in the RLS algorithm. There is a
great freedom of choice, without risk of instability, by choosing appropriate
parameter settings of λ and δ. The choice of these design parameters will
greatly affect the total behaviour of the algorithm.
Figure 4.11 These four plots show the dependences of the two design parameters λ
and δ. The estimate converges asymptotically to the optimal estimate as λ = 1, i.e. the
misadjustment becomes approximately zero for large n. The misadjustment appears
when λ < 1. Hence, the estimate does not converge to its optimal value.
The sinusoid
When applying the RLS algorithm to the sinusoid, the estimates will look like
those in Figure 4.12. The same relationship between the tracking curves of
the estimate and the changes in the parameter values λ and δ are still valid as
in the case of the constant signal.

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REPORT 30 (68)
Figure 4.12 These four plots show the dependences of the two design parameters λ
and δ. The estimate converges asymptotically to the optimal estimate as λ = 1, i.e. the
misadjustment, M, becomes approximately zero for large n. The misadjustment
appears when λ < 1. Hence, the estimate does not converge to its optimal value.
WCDMA signal
The WCDMA signal is complex, consisting of a real part I(t) and an imaginary
part jQ(t), i.e. Inphase and Quadrature components. As can be viewed in
Figure 4.13, the same convergence characteristic appears as the other
signals types.

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Figure 4.13 These four plots are similar to the plots of the estimate when the constant
singal was applied. In this figure the same characteristic is shown as before. The
estimate converges asymptotically to the optimal estimate as λ = 1, i.e. the
misadjustment, M, becomes approximately zero for large n. The misadjustment
appears when λ < 1. Hence, the estimate does not converge to its optimal value.
4.2.4 The Kalman algorithm
There are a few design parameters that affect the performance of the Kalman
algorithm. Two of those are the process covariance R1 and the measurement
covariance R2. Hence, to get an optimal trade-off, these parameters must
assume appropriate values.
In each of the following three signal cases, plots illustrate the dependence
between the ratio R2/R1 and the shape of the tracking curve. From these
plots it is concluded that the ratio of R2 and R1 has a great significance on the
rate of convergence and the influence of noise on the estimate, which also is
according to theory in Chapter 3.4.
The difficult thing here is to determine the values on R1 and R2 to get the
optimal trade-off between the rate of convergence and the noise sensitivity.

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REPORT 32 (68)
Constant signal
Figure 4.14 From this figure it is verified that the rate of convergence decrease for
larger ratios of R2/R1. In this figure we also see that a faster convergence also is
accompanied by a noisier iterative solution, as illustrated in the left plot.
The sinusoid
Figure 4.15 From these plots it is verified that the rate of convergence decrease for
larger ratios or R2/R1. The influence of noise is bigger in the left figure, which is
according to theory.

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REPORT 33 (68)
WCDMA signal
Figure 4.16 In this figure the same characteristic is shown as before. From these plots
it is verified that the rate of convergence decrease for larger ratios of R2/R1. There is
also seen that a faster convergence also is accompanied by a noisier iterative
solution, as illustrated in the left plot.
4.3 Conclusions
The LS algorithm, with the matrix inversion, computes the optimal estimate of
the parameter α. The algorithm works in a non-recursive manner where the
dimension of the observation vector Y(n) equals the number of base functions
in the estimate. This algorithm becomes very expensive with respect to the
memory usage as the number of observations increases. Also, the matrix
inversion that is a part of the solution is very computationally demanding.
The RLS algorithm is a recursive extension of the LS algorithm. Here, the
preceding estimate of the parameter α is updated when the next observed
sample becomes present. In this way, only a fixed size of memory is used in
each iteration. The forgetting factor λ determines the relative importance
between past and new measurements. As λ equals unity, all measurements
become equally weighted. Instead, if λ is less than one, past measurements
have less influence on the estimate. The rate of convergence increases with
decreasing values of the forgetting factor λ. As a consequence, the influence
of noise instead becomes bigger. The misadjustment of the MSE converges
to zero when λ = 1 and conditioned that the number of observations
approaches infinity.
The LMS algorithm is more noise sensitive compared with the RLS algorithm.
There is a problem with the stability if a too large step size µ is applied. Also,
the rate of convergence is slower than that of the RLS algorithm. The
misadjustment M is proportional to the number of parameters N, the step size
µ and the signal variance σy
2
.

Open
REPORT 34 (68)
The Kalman algorithm is the linear MMSE estimator, presupposed that the
design parameters R1 and R2 assume appropriate values. Both the LMS and
RLS algorithms are special cases of the more general Kalman algorithm.
Hence, these algorithms are considered suboptimal in general. The ratio
R2/R1 determines the rate of convergence and noise contribution to the
estimate. The rate of convergence is fast and the noise contribution is big
when the ratio is small. The rate of convergence is slow and the noise
contribution is small when the ratio R2/R1 is large.
4.4 Discussion
As can be concluded from the initial case studies the choice of the design
parameters is very important and will affect the performance. Therefore, it is a
little bit complicated to compare the algorithms to each other, because a bad
parameter setting will degrade the performance of that algorithm.
4.5 Summary
The performance is totally an open question. One cannot rank these four
algorithms without performing further tests in a more realistic environment.
Hence, further conclusions will be drawn later on.

Open
REPORT 35 (68)
5 The digital predistorter
The aim for the PD is to compensate for the nonlinearities in the PA by
predistorting the input signal such that the gain of the PA will be linear. The
mathematical description of predistortion is presented as shown in Eqn. (2.5),
which will be solved numerically. As mentioned earlier there are several
numerical algorithms at hand to solve this estimation problem.
In Figure 5.1, a block diagram of the system for linearisation is shown. In this
figure, the predistorter is situated in front of the non-linear power amplifier.
This is the physical emplacement, which corresponds to the reality.
xˆ
PDpre(·) PA(·)
x y = PDpre(x) xˆ
PDpre(·) PA(·)
x y = PDpre(x) xˆ
PDpre(·) PA(·)
x y = PDpre(x)
Figure 5.1 The PD connected to the input of the PA. This is the predistorter approach
which agrees with the physical arrangement.
If the absolute value of the gain transfer function of the PA is plotted, we will
get the characteristic plot as illustrated in Figure 5.2. Here we have assumed
that the input and output signals of the PA are normalized, i.e. the ideal gain
transfer function ought to be unity. But, the gain of the non-linear PA deviates
from this ideal gain for large signal amplitudes, perhaps due to saturation.
Figure 5.2 The gain transfer function of a non-linear amplifier. Ideally, the gain should
be constant along the whole dynamic range of the amplifier. Due to nonlinearities the
true PA differs from the ideal characteristic.

Open
REPORT 36 (68)
The task for the PD is to compensate for the deviation from the ideal gain by,
in simple words, multiplying the input signal to the PA with some variable gain
factor so the total gain will be unity at the output.
5.1 Predistorter or postdistorter
To simplify the estimation of the unknown parameters in the estimator, the
predistorter approach will be rearranged as a postdistorter estimation problem
instead, where the PD is placed after the PA, see Figure 5.3. This will
considerably simplify the calculations of the PD parameters. Hence, only the
input and output signals of the PA are needed. Otherwise, the unknown
transfer function of the PA has to be known, which is basically the problem
that should be solved.
The main restriction when performing simulations in Matlab is that the test
bench with the real transfer function of the PA is not available. This implies
that the outputs from predistorted inputs cannot be produced. Therefore, the
postdistorter approach will be used when calculating the PD parameters.
The difference of the physical emplacement of the PD is that the transfer
function of the PA will be available implicitly in the system. Hence, the
predistorter approach will be used instead.
Now, it should be appropriate to conduct a discussion about the relation
between the two approaches of the predistorter and postdistorter. Assume, for
example, that the characteristic of the PA is known. Thus, there exist an
operator that can model the relation between the input and output of the PA. It
is also assumed that there exist an inverse operator that can invert the output
signal of the PA back to the input signal.
The basic conditions are now fulfilled to prove the equivalence between the
postdistorter and the predistorter approaches. By comparing the conditions of
the Eqns. (5.1) and (5.2), one can see that the two approaches are equal.
xˆ
PA(·) PDpost(·)
x y = PA(x)
xˆ
PA(·) PDpost(·)
x y = PA(x)
xˆ
PA(·) PDpost(·)
x y = PA(x)
Figure 5.3 The postdistorter approach, where the PD is placed after the PA.
( )
( ) ( )( ) xxPAPDyPDx
xPAy
postpost ≡==
=
ˆ
iff. ( ) ( )yPAyPDpost
1−
= (5.1)

Open
REPORT 37 (68)
xˆ
PDpre(·) PA(·)
x y = PDpre(x) xˆ
PDpre(·) PA(·)
x y = PDpre(x) xˆ
PDpre(·) PA(·)
x y = PDpre(x)
Figure 5.4 The predistorter approach, where the PD is placed before the PA.
( )
( ) ( )( ) xxPDPAyAPx
xPDy
pre
pre
≡==
=
ˆ
iff. ( ) ( )xPDxPA pre=−1
(5.2)
But, in practice it is very difficult to determine the transfer function of the PD
exactly. Therefore, we have to be satisfied with an approximate solution
instead. However, this gives a satisfying performance relative to the great
simplifications we have accomplished.
5.2 The structure of the estimator
A suitable estimator for the PD output is a linear combination of several
weighted base functions, see Eqn. (5.3).
( ) ( ) ( ) ( )( )nNNnnnn xBxBxBxxPD 111100 −−+++= ααα L (5.3)
The base functions, Bk(xn), are real and can be of various types, e.g.
polynomials, triangular, constants or combinations of triangular and
polynomial bases. The parameters αk, for k= 0,..,(N-1), often assume complex
values.
Worth to notify here is that the estimate PD(xn) is only based on one observed
sample, xn, in time and the base functions, Bk(xn), are constructed from that
single sample. The reason for that is the assumption that the PA is memory-
less, which means that the output is independent of past inputs, such as xn-1,
xn-2, xn-3 and so on. Hence, the output from the memory-less PA is only
dependent on the input signal at the present sample time n, which means that
the PA is a static system.
5.2.1 Complex gain transfer function
From the estimator given in Eqn. (5.3) the complex gain transfer function in
Eqn. (5.4) can be extracted. The characteristics of the complex gain transfer
functions of an arbitrary PD and a PA are plotted in Figure 5.5.
( ) ( ) ( ) ( )nNNnnn xBxBxBxg 111100 −−+++= ααα L (5.4)

Open
REPORT 38 (68)
From Figure 5.5 we can see that the gain transfer function of the PD,
|gPD(xn)|, compensates for the non-linear behaviour of the PA, |gPA(xn)|, by
deviation in the opposite direction.
Figure 5.5 The characteristic of the complex gain transfer functions of the PD and the
PA. It can be seen that the PD compensates for the non-linear behaviour of the PA.
Ideally, the product of these two curves will be equal to unity, for all signal
amplitudes.
5.2.2 Number of parameters
The number of parameters, αk, in the estimator PD(xn) is also an important
design issue. The experience from simulations indicates that there is a trade-
off between the number of parameters and the performance. If the number of
parameters is too low there will be problems to represent the desired gain
characteristic. On the other hand, if the number of parameters is too high,
numerical uncertainties will influence the estimate that will degrade the
performance.
However, since the assumption was that the non-linearities in the PA are
slowly time varying, we can consider the number of parameters in the PD
model to be constant. Hence, only one determination of optimum number of
parameters has to be performed at the beginning, to be used thereafter.
In Section 6.2 there is an explanation of how to determine the number of
parameters N of the estimator.

Open
REPORT 39 (68)
5.3 Various base functions
The estimator of the PD, Eqn. (5.3), can be formed in several ways, just by
altering the type of base function. Natural choices of base functions are the
polynomial bases, which suit well to represent the non-linearities in the PA.
Other base functions, which are interesting to analyse are: overlapping
triangular pulses, constant base functions and combinations of polynomial
and triangular base functions. Each type of base function has its own
advantages and disadvantages.
5.3.1 Polynomial bases
The polynomial base functions are represented by Eqn. (5.5) and a few of
them are plotted in Figure 5.6. These bases are suitable to model the
nonlinearities in the PA, because they give smooth estimates.
( ) k
nnk xxB = (5.5)
Figure 5.6 The polynomial bases.
5.3.2 Triangular bases
The overlapping triangular bases could be written as in Eqn. (5.6), where N is
the number of parameters used in the linear combination in Eqn. (5.3).
Obviously, these base functions result in linear interpolation when they are
applied in the estimator.
( )
( )
( )





++−−
−+−
=
0
11
11
kxN
kxN
xB n
n
nk for
( ) ( )
( )
otherwise
kxNk
kxNk
n
n
11
11
+≤−≤
≤−≤−
(5.6)
Only two bases will contribute to the linear combination for every incoming
sample xn, which is illustrated in Figure 5.7. Though, one drawback by using
triangular base functions is that the smoothness of the estimate is lost.

Open
REPORT 40 (68)
Figure 5.7 The triangular bases.
A smother estimate could be achieved if the number of bases is increased. In
this way the distances between the interpolation points will get closer, which
results in a better accuracy. Another way to achieve smoother estimates is by
combining the triangular bases with polynomial bases.
5.3.3 Constant bases
It should be interesting to view the performance of estimates based on
constant functions, Eqn. (5.7). The shape of one constant base function is
visualised in Figure 5.8. These bases are often used in practice, due to their
simplicity, e.g. in a Look Up Table (LUT) of the physical PD. But, one
drawback from the rectangular pulses is that there is needed a considerable
number of bases, even to get an acceptable performance.
( )



=
0
1
nk xB for
otherwise
1+<≤ kxNk n
(5.7)
Hence, there is always a trade-off between the simplicity of implementation
and an acceptable performance. This will be discussed in further chapters.
Figure 5.8 The constant non overlapping base function. The shape of the pulse is
rectangular.

Open
REPORT 41 (68)
5.3.4 Combination of triangular and polynomial bases
The purpose with the combination of the triangular and the polynomial bases,
Eqn. (5.8), is to combine the advantages from the two different base
functions. The combination of these base functions is plotted in Figure 5.9.
( )
( )
( )






++−−
−+−+
=
k
n
n
k
n
n
k
n
nk
x
kxNx
kxNx
xB 11
11
for
( ) ( )
( )
otherwise
kxNk
kxNk
n
n
11
11
+≤−≤
≤−≤−
(5.8)
The ambition is to bring together the smoothness of the estimates based on
polynomial base functions, but still keep some of the properties of the linear
interpolation from the triangular base functions.
Figure 5.9 Combination of a triangular base function and a polynomial base function.
5.4 Implementations in context of linearisation
In this section, the aspects on the specific implementation of each algorithm
are discussed in context of the estimation of the PD parameters. In Figure
5.10, a block diagram shows the estimation arrangement according to the
post distorter concept, discussed in Section 5.1. This figure also shows the
pertinent signals that will be used for estimation of the PD parameters.

Open
REPORT 42 (68)
PA PD
xn yn
+ -
nxˆ
PA PD
xn yn
+ -
nxˆ
Figure 5.10 Block diagram that shows the estimation problem with the relevant
signals included.
When performing estimation with the post distorter solution, only two
measured signals will be needed. Those signals are:
• The desired signal, xn, which is the input signal to the PA.
• The output signal of the PA, yn, which constitutes the observation that
enters into the estimate.
Generally, in the following sections of this chapter, the base function Bk(yn) is
an arbitrary base function, chosen among the four different types that were
scrutinized in Section 5.3.
Beside these general statements, only short presentations of the components
of each algorithm will be presented in the four following sections.
5.4.1 LS
To keep the notations of the algorithm and the signals apart, the LS Eqns.
(3.1) and (3.2) are restated in the following equations:
( ) ( ) ( ) 2
0
1
00 Wnnn
H
n xHyxxxxxJ −+−Π−= −
(5.9)
[ ] [ ]0
11
00
ˆ xHyWHWHHxx nn
H
nn
H
n −+Π+=
−−
(5.10)
The assignments of the components of the LS algorithm, Eqn. (5.10), are as
follows:
( )
( )
( )











=
− 0ˆ
0ˆ
0ˆ
1
1
0
0
N
x
α
α
α
M
,
1)1(0
1
×+
−












=
n
n
n
n
x
x
x
y
M
,
Nn
n
n
n
h
h
h
H
×+
−












=
)1(0
1
M
and
( ) ( ) ( )[ ] NmNmmmm yByByByh ×−= 1110 L

Open
REPORT 43 (68)
5.4.2 LMS
The assignments of the components of the LMS algorithm, Eqn. (3.3), are as
follows:
( )
( )
( )
( )











=
− n
n
n
n
N 1
1
0
ˆ
ˆ
ˆ
ˆ
α
α
α
θ
M
, ( )
( )
( )
( )











=
− nN
n
n
n
yB
yB
yB
ynY
1
1
0
M
5.4.3 RLS
The assignments of the components of the RLS algorithm, Eqns. (3.9), are as
follows:
( ) nxnx =
( )
( )
( )
( )











=
− n
n
n
n
N 1
1
0
ˆ
ˆ
ˆ
ˆ
α
α
α
θ
M
, ( )
( )
( )
( )











=
− nN
n
n
n
yB
yB
yB
ynY
1
1
0
M
, ( )
NN
P
×












=
δ
δ
δ
L
OMM
L
L
00
0
00
00
0
5.4.4 Kalman
The SS model is formed like the Eqn. (5.11), where the state vector contains
the PD parameters to be estimated. These parameters are assumed to be
independent of each other. Therefore, the transition matrix F assumes the
form of an identity matrix, see Eqn. (5.13). The process noise, w(n), is
assumed to affect each state separately. Hence, the matrix G also assumes
the form of an identity matrix. Both the measurement noise, v(n), and the
process noise, w(n), are assumed to be zero mean white noise processes.
Those processes are related as stated in Eqn. (3.15), with their respective
covariance R1 and R2.
There is a difference to the general SS model expressed in Eqn. (3.14). In the
current SS model, Eqn. (5.11), the states are assumed not to be affected by
any input signal, u(n). Hence, our system is unforced!
The state space model is written as
( ) ( ) ( )
( ) ( ) ( )nvnHxny
nGwnFxnx
+=
+=+1
(5.11)
where the state vector x(n) is expressed as

Open
REPORT 44 (68)
( )
( )
( )
( )
( )
( )
( )











=












=
−− n
n
n
nx
nx
nx
nx
NN 1
1
0
1
1
0
α
α
α
MM
(5.12)
and the matrixes F and G assume the form
NN
IGF
×












===
100
010
001
L
MOMM
L
K
(5.13)
The elements of the time-variant matrix H=H(n), Eqn. (5.14), contains the
base functions that form the PD estimator in Eqn. (5.3) .
( ) ( ) ( ) ( )[ ]nNnnn yByByBynHH 110 −== L (5.14)
The last thing to perform before the Kalman equations can be applied is to
initialise the algorithm. Since we chose to start with the measurement update,
the initial values of P(0) and ( )10ˆ −x are needed. Suitable choices of the initial
values are expressed in Eqn. (5.15).
( ) ( )
NN
x
×












=












=−
δ
δ
δ
000
0
00
00
0Pand
0
0
0
10ˆ
OMM
L
L
M
(5.15)
The variable δ in the posterior error covariance function, P(0), is assumed to
be large, i.e. δ → ∞.
5.5 Parameter settings
To perform a fair comparison between the different algorithms it is suitable to
make the right or optimal choice of the parameter settings for each algorithm.
This turned out to be a complex task, because there are many dependencies.
However, the principle to determine these design parameters is to reduce the
degrees of freedom, i.e. to lock as many parameters as possible to constant
values, while investigating the behaviour when changing one single
parameter. This method was used when the different algorithms were tuned,
during the linearisation. This topic will be treated further in Section 6.3.

Open
REPORT 45 (68)
6 Linearisation of a memory-less PA
In this case, true WCDMA signals measured at the input and output of a PA
are used. Now, the effects of measurement noise and other kinds of
disturbances become present, which will degrade the performance of the PD.
In this section, a predistorter will be designed to linearise the memory-less
PA. A suitable number of parameters in the linear combination in Eqn. (5.3)
will be determined. Also estimates based on the various types of base
functions, mentioned in Section 5.3, will be examined.
6.1 The WCDMA signal
The complex valued WCDMA signal is composed of two components. These
are the Inphase and Quadrature components, I(t) and Q(t), according to Eqn.
(6.1). Both I(t) and Q(t) are real valued signals. The continuous WCDMA
signal is sampled with a sample frequency of Fs = 66.44MHz, where each
batch of data contains 16294 samples.
( ) ( ) ( )tjQtItfBB += (6.1)
The sampled input and output signals of the PA are normalized with respect
to the largest sample magnitude that appears in the set of data. Hence, during
the simulations the ideal gain of the PA is forced to be unity. However, this will
not affect the estimate of the PD parameters.
The PSD of the WCDMA signal, at the input and output of the PA, is given in
Figure 6.1. From that plot one can see that the bandwidth of the signal is
approximately 2.5 MHz.
The input signal of the PA is formed by a signal generator. Hence, the input
signal is considered to be almost noiseless, with a noise level of about 100
dBc. On the other hand, the noise of the output signal stays at a level of about
55 dBc, due to noise from various parts of the measurement set-up.

Open
REPORT 46 (68)
Figure 6.1The PSD of the WCDMA signal at the input, x(n), and the output, y =
PA(x(n)), of the PA. One can see that the difference in the noise level between the
input and the output of the PA is about 40 dB.
The probability density functions of the magnitudes of the WCDMA signals,
i.e. the input and output signals of the PA, are plotted in Figure 6.2 and Figure
6.3. Here, one can observe the clear tendency of the WCDMA signal that
there is a loss of samples at larger magnitudes.
Figure 6.2 The PDF of the input signal to the PA, i.e. xn.

Open
REPORT 47 (68)
Figure 6.3 The PDF of the output signal from the PA, i.e. yn.
This means that the estimator will have a problem with the adaptation to the
characteristic for larger magnitudes of the signals. Therefore, the gain
characteristic of the PD will be best in the region of the middle, where the
probability density is high.
6.2 The number of base functions in the PD estimator
The intention with this section is to determine a suitable number of base
functions, or equivalently the number of parameters N, in the estimator stated
in Eqn. (5.3). This task can be carried out in several ways. Here, the MSE of
various numbers of base functions are calculated to find the dependence
between the MSE and N.
A suitable number of parameters in the linear combination can be determined
by performing linearisation using the LS algorithm. By computing the MSE for
various number of parameters, one can plot the dependence between those
two quantities, see Figure 6.4. This procedure was performed for each type of
base function that is intended for the performance measurements.
Figure 6.4 The MSE for various number of parameters. The estimates were computed
by the LS algorithm.

Open
REPORT 48 (68)
From Figure 6.4 one can see that the MSE decreases rather quickly in the
beginning for increasing number of parameters. But, the MSE curves flatten
and change insignificantly for larger number of parameters.
According to this behaviour the number of parameters is chosen to N = 9. For
this number of parameters, the MSE of the estimates based on polynomial
bases and triangular bases are very close to each other, whereas the MSE of
the estimate based on constant bases still is poor. The MSE of the estimate
based on the combined bases is squeezed between the MSE of the
polynomial and triangular estimates. However, in general it can be concluded
that the MSE in each case converges to the same limit as the number of
parameters increases.
Already at this point the estimates based on constant bases can be excluded
from the performance measurements. This conclusion is made on basis of
Figure 6.4. Here, I have reasoned that there is needed about ten times more
parameters in the case of constant bases to match the performance of the
three other base functions.
In practice this implies more than 40 parameters to get the performance
comparable with the results of the polynomial base functions. A large number
of parameters will increase the computational burden and will not be a
relevant choice because there are other bases that perform equally, but to a
lower cost.
Further the measurement results of the combined bases will also be
excluded. This is due to that the performance of these bases will be
somewhere in between the performance of the polynomial and triangular base
functions. Hence, I have reasoned that it is more appropriate to present the
performance of the outer bounds instead, to make the results foreseeable.
6.3 The choice of parameter settings in the algorithms
It was found out that it is cumbersome to choose the parameters settings in
the different algorithms, to get a fair comparison. Bad parameter settings will
not generate a true picture of the relation of the different algorithms.
One way to proceed is by reducing the degrees of freedom in the parameter
settings of each algorithm. The performance measurements then become
easier to carry out. The principle is to lock as many parameters as possible to
constant values, while only one or a few parameters are variable. In this way
it is simpler to monitor the effect when changing one single parameter.
One parameter setting that is general, for all algorithms, is the initial guess of
the PD parameters in the estimator given by Eqn. (5.3). The initial values of
the PD parameters were chosen to be all zero, because then it becomes
clearer to see the rate of convergence of each PD parameter. Naturally one
can expect that the first parameter α0 of the estimator given by Eqn. (5.3)
should be 1, because that is the ideal gain of the PA.

Open
REPORT 49 (68)
The conclusion from this is, in a real practical implementation one should use
as much a priori information as possible to improve the performance. But
here, we disregard from this fact and feel satisfied with the choice of the initial
values. The primary thing here is to be consequent during measurements, to
retain the algorithms comparable.
The parameter settings in the LS algorithm
In the LS algorithm, given by Eqn. (3.2), there are three initial parameters to
set. The first is the confidence matrix Π0, the second parameter is the
weighting matrix W and the third is the initial estimate x0. According to the
discussion above, x0 is set to a zero vector. The confidence matrix Π0
contains large elements to indicate low confidence of the initial guess x0.
Finally the weighting matrix W is set to the identity matrix. These parameter
settings are gathered in the Eqns. (6.2), where the dimension N is the number
of parameters in the PD estimator.
NNNN
N
IWIx ××
×
=∞≈Π












= ,,
0
0
0
0
1
0
M
(6.2)
In Eqn. (6.2) it can be observed that δ approaches infinity, i.e. δ → ∞.
In this way the LS algorithm will compute a satisfying estimate of the PD
parameters, which will be used as a reference towards the estimates
computed by the other algorithms.
The parameter settings in the LMS algorithm
In the LMS algorithm, given in Eqn. (3.3), there are only two parameters to
assign. The first one is the initial value of the PD parameters estimate, ( )1ˆ −θ ,
which is a zero vector that is shown in Eqn. (6.3).
( )
1
0
0
0
1ˆ
×












=−
N
M
θ (6.3)
The second parameter to assign is the step size µ of the LMS algorithm,
which affects the stability of the LMS algorithm, the accuracy of the estimate
and the rate of convergence. Hence, this value should be chosen with care.
To determine a suitable step size, the MSE of the estimated input signal to
the PA was plotted for various µ. The minimum of the MSE corresponds to
the optimal choice, which implies that this step size should be used in the
proceeding measurements. Hence, the optimal step size that corresponds to
the minimum of the MSE is presented in Eqn. (6.4).
1≈optµ (6.4)

Open
REPORT 50 (68)
This value was found valid irrespective of the type of base functions that was
applied.
The parameter settings in the RLS algorithm
The parameters in the RLS algorithm, Eqns. (3.9), are assigned in the Eqn.
(6.5). To get the RLS algorithm comparable with the LS algorithm we have to
set the forgetting factor equal to one, λ = 1. By doing this all the samples are
considered equally in the estimate, exactly like the LS algorithm.
( ) ( ) NN
N
IP ×
×
=












== δθλ 1,
0
0
0
0ˆ,1
1
M
(6.5)
The initial estimate of the PD parameters, ( )0ˆθ , is a zeros vector and the
initial uncertainty of that vector, ( )1P , equals a scaled identity matrix. The
scaling factor δ is determined in the same manner as the step size, µ, in the
LMS algorithm. That is, by searching the MMSE of the estimated input signal
to the PA, the value of δ can be determined. In this way δ was determined to δ
= 109
.
The parameter settings in the Kalman algorithm
The Kalman algorithm in Eqns. (3.17) and (3.18) contains four design
parameters, which are R1, R2, P(0) and ( )10ˆ −x . These values are assigned as
in Eqns. (6.6). It turns out that only δ and R1 have to be determined, because
the initial estimate of the state vector is assumed to be a zeros vector and the
measurement variance is approximately R2=-60 dB, according to the noise
floor in Figure 6.1.
( )
( )
{ }
1
6
6
2
1
2
1
10
10
0
0
0
0
10ˆ
R
R
R
R
IP
x
NN
N
−
−
×
×
=≈=
=












=−
δ
M
(6.6)
Also in this case, the MSE is computed for various constellations of δ and R1.
The MMSE corresponds to the optimal choice of these design parameters.
Hence, δ and R1 were chosen to δ ≈ 370000 and R1 ≈ 10-15
.

Open
REPORT 51 (68)
6.4 Performance measurements of the Matlab implementation
The number of parameters, N = 9, is kept constant during the performance
measurements. The results from the constant base functions and the
combination of triangular and polynomial base functions have been excluded.
Hence, only results from the linearisation of the PA with the polynomial base
functions and triangular base functions will be presented.
6.4.1 Tools to measure the performance
There are different ways to measure the performance of an algorithm. In the
performance measurements four tools are used to compare the algorithms.
The visual tools that will be used here are plots of the PSD, MSE and the
parameter tracking.
Mean Square Error
By computation of the mean value of the squared estimation error one can get
a hint of the accuracy of the estimate and the performance of the algorithm.
This measure is very useful to utilize when we are adjusting the parameter
settings of each algorithm. Though, the suppression of IM cannot be
visualized by using the MSE.
Power Spectral Density
The PSD is a useful tool when we want to visualize the IM suppression in the
output signal of the PA. In this way we can get a picture of the reduction of the
adjacent channel power.
The PSD is computed in a regular way by applying a Welch-Periodogram of
the signal. A further explanation is given in [6].
Adjacent Channel Power
This measure is equal to the mean value of the PSD in a 5 MHz wide band
just beside the carrier. By using the data from the computation of the PSD,
one can get a numerical value of the ACLR to the left and to the right of the
carrier. These values are presented relatively the PSD of the carrier, i.e. dBc.
Parameter tracking
In this method the estimated parameters are stored for each iteration. From
this information the rate of convergence can be estimated. We can also see if
the parameters have settled to a constant level or if they are still changing.

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6.5 Results
The estimations based on the constant base functions are not comparable
with the estimations based the polynomial or triangular base functions. It was
found out that there is needed about ten times more parameters for these
bases even to get near the performance of the polynomial or triangular base
functions. Therefore, the results from the measurements of the constant base
functions are omitted. Also, the results from the combined bases are
excluded. This is because that the performance of those bases is in between
the performance of the polynomial and triangular bases. Hence, we have
already got the outer bounds.
6.5.1 Linearisation by using the LS algorithm
It was observed that the performance of the triangular base functions is very
close to that of the polynomial base functions. The difference can hardly be
distinguished in the PSD plots of the output signals, Figure 6.5, when the
number of parameters is larger or equal to N = 9.
Figure 6.5 The triangular bases give almost as good result as the polynomial bases.
The green curve represents the PSD of the output signal when the triangular base
functions have been used. The blue curve represents the PSD of the output signal
when the polynomial base functions have been used.
The result from the PSD plot can be summarised by computing the adjacent
channel power of the output signals.
The ACLR performance:
ACLR lower side band ACLR upper side band
Polynomial bases 53.6 dBc 51.8 dBc
Triangular bases 53.2 dBc 51.7 dBc

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The reduction in ACLR is:
∆ACLR lower side band ∆ACLR upper side band
Polynomial bases 17.3 dB 15.2 dB
Triangular bases 16.8 dB 15.0 dB
The plots in Figure 6.6 show the magnitudes of the parameters, both for the
polynomial bases and the triangular bases. The parameter magnitudes in the
left plot increases rapidly, while the magnitudes in the right plot are
approximately constant up to k = 5.
Figure 6.6 The magnitudes of the estimated PD parameters.
The plots in Figure 6.7 show the gain transfer functions of the PD, both for the
polynomial bases and the triangular bases.
Note the similarities between the left plots in Figure 6.6 and Figure 6.7. The
shape of the respective plot is due to the linear interpolation when using
triangular base functions.
Figure 6.7 Gain transfer functions of the PD in the case of polynomial bases and
triangular bases.

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The performance of the LS algorithm will be used like a reference to which the
performance of the other algorithms can be compared.
6.5.2 Linearisation by using the LMS algorithm
From the results of the measurements in this section, one can say that the
iterative LMS algorithm does not produce as good estimates like the LS
algorithm. If the suppression by the LMS algorithm is compared with the
corresponding measures of the LS algorithm, one can see that the IM
suppression differ more than 4 dB.
This was maybe predictable, but the surprising thing is that the two estimates,
produced by the polynomial and the triangular base functions, differ quite a lot
mutually. This can be seen in Figure 6.8, which shows the PSD plot of the two
different output signals. In general, one can say that the PD formed by the
polynomial bases suppresses the IM products better than the case of
triangular bases.
Figure 6.8 The PSD plots of the output signals, with and without linearisation. There is
a clear difference between the polynomial bases and the triangular bases.
The ACLR performance confirms the differences that were discovered in the
PSD plots. There is a difference of approximately 4 dB of the ACLR.
The reduction in ACLR:

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The plots in Figure 6.9 show the magnitudes of the parameters, both for the
polynomial bases and the triangular bases. The parameter magnitudes in
these plots are different compared with the plots from the LS algorithm. The
LMS algorithm is the only algorithm, among the three iterative algorithms, that
differs from the characteristic of the LS estimates.
Figure 6.9 This figure shows the magnitudes of the estimated PD parameters. Note,
the difference between these plots and the plots given in the preceeding section of
the LS algorithm.
As a consequence of the parameter estimates illustrated in Figure 6.9, the
corresponding gain transfer functions get the shapes like Figure 6.10. The
deviation from the LS case is obvious; at least, in the right plot that shows the
gain transfer function formed by the triangular bases.
Figure 6.10 Gain transfer functions of the PD in the case of polynomial bases and
triangular bases. These gain transfer functions differe compared with those of the
other algorithms.

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The convergences of the parameters are illustrated in Figure 6.11. From
these plots there is seen that almost every parameter has problem with the
convergence. In principle, only the first three estimates converge to constant
levels during the time interval of survey. The rest of the parameters do not
settle to fix values. In addition to the poor convergences, the estimates also
suffer from noise. Probably, this is an effect of the large misadjustment, stated
in Eqn. (3.6), of the LMS algorithm.
Figure 6.11 This figure illustrates the convergence of the parameters estimated by the
LMS algorithm. The parameters are plotted in ascending order from the left top to the
right bottom. One can see that the estimates suffer from noise and the convergence
is poor. This was also reflected in the bad IM suppression in the PSD measurements.

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6.5.3 Linearisation by using the RLS algorithm
The performance of the RLS algorithm is much better than the previous LMS
algorithm. The results are definitely comparable with those of the LS
algorithm. In fact, there is almost impossible to separate the PSD plots in
Figure 6.12 from the corresponding plots in Figure 6.5. The performance of
the RLS algorithm is almost as good as optimal. The difference is hardly
noticeable, even when comparing the ACLR below. Mutually, the PSD plots
from the polynomial bases and the triangular bases are hardly separable, as
shown in Figure 6.12.
Figure 6.12 The PSD plot of the output signal, with and without linearisation. The
performance is similar to the PSD plot in the LS case.
The reduction in ACLR:

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The parameter magnitudes in Figure 6.13 and the gain transfer functions in
Figure 6.14 are very similar to those given by the LS algorithm. Hence, the
performance is much better than the performance of the LMS algorithm.
Figure 6.13 This figure shows the magnitudes of the estimated PD parameters.
Figure 6.14 Gain transfer functions of the PD in the case of polynomial and triangular
bases. Note the steep flank on the right gain transfer function. This characteristic is
due to the loss of a sufficient number of samples of those magnitudes, in combination
with the triangular bases. Hence, the last parameter will seldomly be updated during
the iteration, which implies that the gain transfer function gets this shape.

Master Thesis, Tomas Elgeryd, IR-SB-EX-0224

Master Thesis, Tomas Elgeryd, IR-SB-EX-0224

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