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

KUNGL TEKNISKA HÖGSKOLANInstitutionen förSignaler, Sensorer & SystemSignalbehandling100 44 STOCKHOLM

ROYAL INSTITUTEOF TECHNOLOGY

Department ofSignals, Sensors & Systems

Signal ProcessingS-100 44 STOCKHOLM

Iterative algorithms for linearisingnon-linear systems by digital predistortion

Tomas Elgeryd

November 2002

IR–SB–EX–0224

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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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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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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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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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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

xxPDPAxxxPDzzPAx

1

ˆ If

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.

Gx y = Gx

PA(·)x y = PA(x)

PA(·)PD( ·)x z = PD(x)

x̂

a)

b)

c)

GGx y = Gx

PA(·)x y = PA(x)

PA(·)PD( ·)x z = PD(x)

x̂

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 consis t 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.

( ) ( ) ( ) 20

100 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 x̂ 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 x̂ fulfils the Eqn. (3.2).

[ ] [ ]011

00ˆ 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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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 TLMS −= (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).

( ) ( ) ( ) ( )[ ]TNnynynynY 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

2yN

Mσµ

≈ (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].

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )1111

ˆ1ˆˆ1

1

1ˆˆ

*

*

−−−=+

−+−=

−+−=

−=

nPnYnKnPnP

nxnxnKnn

nYnPnYnYnPnK

nnYnx

T

RLS

T

TRLS

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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( ) ( ) ( ) ( )[ ]TNnynynynY 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.parameter initial The 0ˆ −θ

( ) ( ) 0. somefor I as taken ,0ˆ ofy uncertaint initial The 1 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.

( ) ( )

+=nN

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).

( ) ( ) ( )

−+=

21

1N

MSEnMSE optRLSλ

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.

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( )

( )( )

( )

=

− nx

nxnx

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

tmeasuremen vector tostate n vector, transitiostate theandinput ebetween thmatrix n transitioG

statesdifferent obetween twmatrix n transitioFsignal measured y(n)

noiset measuremen v(n)noise processw(n)

signalinput u(n) vectorstate x(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.

( )( )

( ) ( )( ) ( )mnR

Rmvmw

nvnw

E −

=

δ

2

1HH

00

(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

1ˆ1ˆ1ˆ1ˆ1ˆ

+=+

−−+−=+ (3.16)

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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

nnxHnynnxnnxRHnHPHnP

HH

HH

12

12

1ˆnL1ˆˆnL

−

−

+−=

−−+−=

+=

(3.17)

The time update:

( ) ( )( ) ( )( ) ( ) HH GGRFnFQnP

nnxHnny

nnxFnnx

11

1ˆ1ˆˆ1ˆ

+=+

+=+

=+

(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

0000

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.

• sensitive noise is estimate but the ing,fast track small 1

2 ⇒RR

• sensitive noise less is estimate but the tracking,slow large 1

2 ⇒RR

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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 Estimatorf(n)

+ -

a(n)v(n)

1 Estimatorf(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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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 process v(n) = 0.

( )( )

( )

( )( )

( )

−

−

=

−

−

0

21

0

21

f

nfnf

f

nfnf

MMα (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

21

00

2211

f

nfnf

vf

nvnfnvnf

MMα (4.2)

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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

10sin2 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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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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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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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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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 σ2 = 0.4. The estimate is biased due to the misadjustment. b) The convergence of the estimate for noise variance σ2 = 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.

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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.

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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.

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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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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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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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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.

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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.

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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.

x̂PDpre(·) PA(·)

x y = PDpre(x) x̂PDpre(·) 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.

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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.

x̂

PA(·) PDpost(·)x y = PA(x)

x̂


x̂


Figure 5.3 The postdistorter approach, where the PD is placed after the PA.

( )( ) ( )( ) xxPAPDyPDx

xPAy

postpost ≡==

=

ˆ iff. ( ) ( )yPAyPDpost1−= (5.1)

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x̂PDpre(·) 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)

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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.

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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.

( ) knnk 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.

( )( )( )

++−−

−+−

=

011

11kxNkxN

xB n

n

nk for ( ) ( )

( )otherwise

kxNkkxNk

n

n

1111

+≤−≤

≤−≤−

(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.

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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.

( )

=01

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.

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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

nk

n

nk

n

nk

xkxNxkxNx

xB 1111

for ( ) ( )

( )otherwise

kxNkkxNk

n

n

1111

+≤−≤

≤−≤−

(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.

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PA PD

xn yn

+ -

nx̂PA PD

xn yn

+ -

nx̂

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:

( ) ( ) ( ) 20

100 Wnnn

Hn xHyxxxxxJ −+−Π−= − (5.9)

[ ] [ ]011

00ˆ xHyWHWHHxx nn

Hnn

Hn −+Π+=

−− (5.10)

The assignments of the components of the LS algorithm, Eqn. (5.10), are as follows:

( )( )

( )

=

− 0ˆ

0ˆ0ˆ

1

1

0

0

N

x

α

α

α

M,

1)1(0

1

×+

−

=

n

n

n

n

x

xx

yM

,

Nn

n

n

n

h

hh

H

×+

−

=

)1(0

1

M and

( ) ( ) ( )[ ] NmNmmmm yByByByh ×−= 1110 L

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5.4.2 LMS

The assignments of the components of the LMS algorithm, Eqn. (3.3), are as follows:

( )

( )( )

( )

=

− n

nn

n

N 1

1

0

ˆ

ˆˆ

ˆ

α

α

α

θM

, ( )

( )( )

( )

=

− nN

n

n

n

yB

yByB

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

nn

n

N 1

1

0

ˆ

ˆˆ

ˆ

α

α

α

θM

, ( )

( )( )

( )

=

− nN

n

n

n

yB

yByB

ynY

1

1

0

M, ( )

NN

P

×

=

δ

δ

δ

L

OMM

L

L

0000000

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

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( )

( )( )

( )

( )( )

( )

=

=

−− n

nn

nx

nxnx

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

010001

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

00000000

0P and

0

00

10ˆOMM

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.

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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.

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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.

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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.

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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.

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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

00

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).

( )

10

00

1ˆ

×

=−

N

Mθ (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)

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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

00

0ˆ ,1

1

M (6.5)

The initial estimate of the PD parameters, ( )0θ̂ , 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

66

21

2

1

1010

0

0

00

10ˆ

RR

RR

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.

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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 ACLR performance: ACLR lower side band ACLR upper side band Polynomial bases 49.5 dBc 47.7 dBc Triangular bases 45.7 dBc 44.9 dBc

The reduction in ACLR: ∆ACLR lower side band ∆ACLR upper side band Polynomial bases 13.1 dB 11.0 dB Triangular bases 9.4 dB 8.3 dB

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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 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 The reduction in ACLR: ∆ACLR lower side band ∆ACLR upper side band Polynomial bases 17.3 dB 15.2 dB Triangular bases 16.8 dB 15.2 dB

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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.

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The convergences of the parameters estimates are illustrated in Figure 6.15. From this figure it is discovered that almost every parameter has settled, during the time of survey. The noise contribution is relatively small compared with the noise contribution in the LMS case.

Figure 6.15 The parameter convergence for the RLS algorithm. The parameters are plotted in ascending order from the left top to the right bottom.The large scaling on the y-axis is due to big impulse changes close to the origin of time. However, the purpose is to get a general view of the evolution of the different parameters.

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6.5.4 Linearisation by using the Kalman algorithm

The performance of the Kalman algorithm is completely in the same class as the RLS or the LS algorithms. The PSD plots in Figure 6.16 are almost identical to the plots from the two other algorithms. Hence, the performance of this algorithm is very close to the optimal LS algorithm. Neither the ACLR performances differ from those of the LS or the RLS algorithms.

Figure 6.16 The PSD plot of the output signal, with and without linearisation. 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 The reduction in ACLR: ∆ACLR lower side band ∆ACLR upper side band Polynomial bases 17.3 dB 15.2 dB Triangular bases 16.8 dB 15.0 dB

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The parameter magnitudes in Figure 6.17 and the gain transfer functions in Figure 6.18 are similar to those given by the LS algorithm. Hence, the performance is better than that of the LMS algorithm.

Figure 6.17 This figure shows the magnitudes of the estimated PD parameters.

Figure 6.18 The gain transfer functions of the PD in the case of polynomial and triangular bases. Note also the steep flank on the right gain transfer function. This characteristic was 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.

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The convergences of the estimates are very distinct and one can clearly see that every parameter converges to a stable value. The estimates do not seem to be significantly influenced by noise either, which is a good property.

Figure 6.19 The parameter convergence for the Kalman algorithm. The convergence is distinct and all the parameters settle. The parameters are plotted in ascending order from the left top to the right bottom.

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6.6 Conclusions

From these measurements one can conclude that the LMS algorithm is the only one that deviates from the performance of the LS algorithm. The rest of the algorithms perform close or equal to the optimal. Still, the LS algorithm is considered to be the optimal choice with respect to the suppression of IMD.

However, there are some drawbacks with the method to compute the estimates by the LS algorithm. Foremost, the matrix inversion of the normal equations is very computational demanding; therefore, it is suitable to use the recursive alternatives instead.

From the different PSD plots from the four algorithms we saw that the performances of the LS, RLS and the Kalman algorithms are well comparable with each other. Thus, it is very hard to rank one of those in front of the other. Other aspects have to be considered in the process of choosing the most appropriate recursive scheme.

For example, the recursive algorithms, Kalman and RLS, are quite similar to each other. Through further investigations one can conclude that the Kalman algorithm is the general one, from which the RLS algorithm could be derived. Hence, the RLS algorithm is considered to be suboptimal. In fact, the same relationship is valid when studying the Kalman algorithm versus the LMS algorithm. Thus, the LMS algorithms is also said to be suboptimal compared with Kalman. These properties are summarized in literature on recursive algorithms, e.g. [3] or [5].

One can also speculate in the poor performance regarding the LMS algorithm. Either, there were problem with the tuning of the parameter that belongs to the algorithm, or it could possibly be a consequence of the static assignment of the step size µ and the number of parameters N in a combination with noise. The second explanation is probably the most reasonable.

Another drawback that is present in the LMS algorithm is that there is only a finite window of measurement data present in the algorithm during the iterations, as opposed to the recursive algorithms. In the recursive algorithms the measurements are generated continuously from the instant of time, when the recursion started. This lack of past measurements probably effects the suppression of IMD in a negative way.

The different types of base functions affect the estimate considerably. This fact becomes obvious when the number of bases, used in the estimate, decreases to only a few. This property is seen in Figure 6.4, where the MSE for each base function is plotted against the number of bases N. But, when the number of bases is sufficient large, this effect cannot be discovered that easily.

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However, the polynomial base function performs the best, closely followed by the combination of base functions and the triangular base functions. The constant base functions do not perform that well. One could at least see from Figure 6.4 that the four MSE curves converge to the same limit as the number of bases increase.

6.7 Discussion

It is interesting to discuss if the recursive algorithms could be applied in the real case. It is clear that the performance of those algorithms will not be better than the batch version LS algorithm, which is used in the physical implementations at this moment. The basic problem or inconvenience is that the sample frequency of Fs = 66.44 MHz, in the communication system, is too high for real-time computations of the PD parameters. Therefore, we are still locked to the technique of collecting samples into a buffer to perform the computation off-line anyway. Due to these circumstances the whole principle of collecting consecutive samples to perform real-time updates of the parameters falls.

The recursive algorithms should become appropriate when the dedicated hardware has improved to such an extent that the computations can be performed in between consecutive samples. But, as long as we wait for the hardware to get fast enough, these recursive algorithms will maybe be applicable in a buffer solution instead. One can think that the recursive algorithm could be applied on the samples in a filled buffer and then to proceed with the next buffer, when the last sample of the previous buffer has been reached.

6.8 Summary

The three algorithms: LS, RLS, and Kalman have equal performances, assumed that the recursive algorithms have been properly tuned. The LMS algorithm did not perform that well, probably due to the noise sensitivity.

The polynomial bases turn out to be the best choice of base functions. But, the performance of the triangular bases is almost the same. The difference in performance between those becomes clearer if the number of bases in the estimator decreases.

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7 Future work An interesting continuation from the simulations is to implement the iterative algorithms in a physical measurement set-up. In this way we can verify the results from the simulations. This physical test part was unfortunately outside the scope of this master thesis.

It would also be of great value to carefully examine the relation between the predistorter and the postdistorter. In our case, it was assumed that the two concepts were identical, presumed that the PD equals the inverse of the PA. Though, in the real case the PD is always an approximation, which probably influence a bit on the performance. Anyway, the postdistorter concept proved to work satisfying according to the results from the simulations.

Another area related to the iterative algorithms is to design a framework for the tuning of the parameters. In this way it would be much easier to assign the parameters to get the optimal performance out of each algorithm.

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References

[1] D. Spångberg, Master Thesis: Digital Pre-Distortion of Power Amplifiers, Uppsala University, Uppsala, Sweden, 2002

[2] A.V Oppenheim, A.S Willsky and S.H Nawab, Signals and systems, 2:d edition, Prentice-Hall, Upper Saddle River, New Jersey, U.S.A, 1997

[3] S. Haykin, Adaptive Filter Theory, 3:d edition, Prentice-Hall, Upper Saddle River, New Jersey, U.S.A, 1996

[4] T. Kailath, A.H Sayed, B. Hassibi, Linear Estimation, Prentice-Hall, Upper Saddle River, New Jersey 07458, U.S.A, 2000

[5] H. Hjalmarsson and B. Ottersten, Lecture Notes In Adaptive Signal Processing 2E1350, KTH – Royal Institute of Technology, Stockholm, Sweden, 2000

[6] J.G Proakis and D.G Manolakis, Digital Signal Processing, 3:d edition, Prentice-Hall, Upper Saddle River, New Jersey, U.S.A, 1996

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8 Appendix 8.1 Appendix 1.

Least Mean Square

Another iterative algorithm is the LMS algorithm as described by Eqn. (A.1.5). This algorithm emanates from the steepest-descent algorithm in Eqn. (A.1.1), where the expectation value of the squared error has been replaced by an estimate, see e.g. [5]. Due to its simple structure, the LMS algorithm does not need a priori measurement of pertinent correlation functions or computation of any matrix inversion. Hence, the computational complexity of each iteration is of order N. However, there are some constraints on the choice of parameters that will affect the stability and the rate of convergence of this algorithm.

The basic structure of the steepest-descent algorithm could be expressed by Eqn. (A.1.1). The estimate of the desired variable θ is updated with a small step in the negative gradient direction of the quadratic cost function.

( ) ( ) ( ) ( )1ˆ,2

1ˆˆ−=∂

∂−−= nnMSEnn

θθθ

θµ

θθ (A.1.1)

For the steepest-descent iteration scheme, the expectation of the squared error (MSE) is

( ) ( ) ( )( )[ ]2, θθ nYnxEnMSE T−= (A.1.2)

The vector Y(n) contains the observations y(n) to y(n-N+1) and takes the form of

( ) ( ) ( ) ( )[ ]TNnynynynY 11 +−−= L (A.1.3)

Y(n) will be used to form the linear combination YT(n)θ in Eqn. (A.1.2).

The LMS algorithm follows if the expectation value in Eqn. (A.1.2) is replaced by the instantaneous error, Eqn. (A.1.4), instead.

( ) ( ) ( )( )2, θθ nYnxnMSE TLMS −= (A.1.4)

The LMS algorithm could then finally be expressed as

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )1ˆ1ˆ

,2

1ˆˆ

*

1ˆ

−−+−=

=∂∂

−−=−=

nnYnxnYn

nMSEnn

T

nLMS

θµθ

θθ

µθθ

θθ (A.1.5)

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By using the framework of averaged error system for the LMS algorithm the stability criterion of the LMS algorithm can be derived. The stability of the LMS algorithm is foremost dependent of the step size µ. The algorithm is stable if the step-size satisfies the inequality

1

20λ

µ << (A.1.6)

where λ1 is the maximum Eigenvalue of the transition matrix of the averaged system describing the error propagation of the LMS algorithm, i.e. the maximum eigenvalue of Eqn. (A.1.7).

( ) ( )[ ]nYnYE HYY =Σ (A.1.7)

The upper bound of the inequality (A.1.6) can be approximated with the bound of the inequality (A.1.8), to simplify the computation.

( ) ( )[ ]nynyNE *

20 << µ (A.1.8)

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. This means that the LMS algorithm does not converge to the optimal θ. The approximation of the MSE(n), for large values of n, is given by Eqn. (A.1.9).

( ) ( )( )MMSEnMSE optLMS +≈ 1θ (A.1.9)

The misadjustment M could then be approximated with Eqn. (A.1.10), according to [5].

2

2yN

Mσµ

≈ (A.1.10)

The misadjustment is therefore proportional to the step size µ, the number of parameters N, and the signal variance σy

2.

In general, there are some constraints with the LMS algorithm:

• The convergence is poor when the matrix in Eqn. (A.1.7) is ill conditioned.

• The algorithm is noise sensitive, as seen by studying the misadjust-ment in Eqn. (A.1.10).

Documents

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