Analysis and Simulation of an Adaptive Predistorter

Gurmail S. Kandola

B.A.Sc., Simon Fraser University, 1989

THESIS SUBMWED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in the

School of Engineering Science L

O Gurmail Kandola 199 1

SIMON FRASER UNIVERSITY

June 1991

All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other

means, without permission of the author.

APPROVAL PAGE

Name: Gurmail Singh Kandola

Degree: Master of Applied Science

Title of Thesis: Analysis and Simulation of an Adaptive Predistorter

Examining Committee:

Chair: Vladimir Cuperman Graduate Chairman and Professor S c w l of En eering Science pt"

Senior Supervisor:

Supervisor:

7-

James Cavers Director and Professor School of Engineering Science

Supervisor:

Jamal Deen Associate Professor School of Engineering Science

Supervisor:

Examiner:

- Steve Hardy Professor School of Engineering Science

Mkei'i)leh Associate Professor School of Engineering Science

Date Approved: & 1~ 2 , / @/

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T i t l e o f Thesis/Project/Extended Essay

"Analysis and Simulation of an Adaptive re distorter"

Author:

(s ignature)

Gurmail S. Kandola

(name)

r j ~ j w 21 \ I 4 ( (date)

ABSTRACT

An adaptive predistorter for linearizing the power amplifier in a radio transmitter is studied, and

both analytical and simulation results are presented. Unlike most other predistorters found in the

literature, this predistorter does not have the problem of loop delay or phase shift in its feedback

path. The feedback is used only periodically to update the predistorter parameters so that it adapts

to changes in the amplifier characteristics due to aging and changes in temperature or biasing. The

predistorter is a 5" order nonlinear system represented by a complex power series in the analysis.

The power contained in the 3* and 5"order intermodulation distortion products, denoted by IM, is

computed by a new technique whereby a sample of the input signal is mixed with a sample of the

amplifier output. The process of bandpass filtering the resulting signal and computing its average

power yields the value of IM. It is established analytically and by simulation results that IM has

parabolic shapes in the planes of the 3rd and 5" order complex coefficients of the predistorter. Thus

a simple adaptation algorithm finds the globally minimum IM and adjusts the predistorter coeffi-

cients accordingly. With the optimum predistorter coefficients found by the algorithm, a minimum

of 11 dB improvement in the intermodulation distortion levels is achieved.

For Mom and Dad

with love.

ACKNOWLEDGEMENTS

I wish to express deep appreciation to my senior supervisor Dr. Shawn Stapleton for providing

continuous support and encouragement throughout my entire degree program. I am very grateful

to Dr. Jim Cavers for several discussions we had on the subject and for providing to me his literature

that was very useful in directing and guiding the analytical investigations presented in this document.

Thanks are also due to both Dr. Stapleton and Dr. Cavers for making valuable suggestions and

comments on the earlier versions of the thesis report. I wish to thank the other committee members,

Dr. Vladimir Cuperman, Dr. Jamal Deen, Dr. Steve Hardy and Dr. Marek Synycki, for taking the

time out of their busy schedules to be on the thesis committee. The help of Cameron Alakija and

Flaviu Costescu in creating a pleasant working environment in the RF/Microwave Laboratory is

much appreciated. I acknowledge the assistance of Ms. Brigitt Rabold (the graduate secretary) for

coordinating the thesis defence. I extend my thanks also to my friend Kuldip Rai for lending me

his microcomputer for typing in this report. Another pair of friends, Shinder Bra. and Shinder

Purewal, deserves appreciation as well for making several comments and suggestions on the format

and flow of the information presented in the document.

Table of Contents

APPROVAL PAGE ......................................................................................................... ...................................................................................................................... ABSTRACT

.............................................................................................. ACKNOWLEDGEMENTS

.......................................................................................................... 1 INTRODUCTION

2 BACKGROUND ............................................................................................................ 2.1 Linearization Techniques ...................................................................................... 2.2 Predistorter Updating Techniques .........................................................................

.................................................................................... 2.3 Amplifier Characterization 2.4 Model of a Nonlinear System ................................................................................

............................................................................. 2.4.1 The Quadrature S tnrc ture 2.4.1.1 Real Bandpass Form (RBF) and Complex Envelope Form (CEF) ......

3 ANALYSIS .................................................................................................................... ................................................................................................... 3.1 System Modeling

3.1.1 Composite Model (in CEF) .......................................................................... 3.1.2 Composite Model (in RBF) .......................................................................... 3.1.3 Real Bandpass Model (RBM) ...................................................................... 3.1.4 Relationship Between RBM and Complex Envelope Model (CEM) ...........

3.2 Autocorrelation Function (ACF) and Power Spectral Density ............................. .......................................................................................... 3.2.1 Amplifier Output

3.2.1.1 ACF By Direct Method ....................................................................... ..................................................................... 3.2.1.2 ACF By Indirect Method

3.2.1.3 Power Spectral Density ........................................................................ 3.2.1.4 Two Tone Input ...................................................................................

.......................................................................................... 3.2.2 The Output Stage 3.2.2.1 Autocorrelation Function ..................................................................... 3.2.2.2 Power Spectral Density ........................................................................ 3.2.2.3 Two Tone Input ...................... .. .........................................................

.............................................................................................................. 4 SIMULATION 4.1 Simulation Model ..................................................................................................

4.1.1 Input and Output Expressions ...................................................................... 4.1.2 PredistorterIAmplifier Complex Gains .........................................................

4.2 Running Simulation ............................................................................................... 4.3 IM Dependence on Predistorter Coefficients ........................................................ 4.4 Adaptive Algorithm ............................................................................................... 4.5 Results ...................................................................................................................

5 CONCLUSIONS ............................................................................................................ 6 REFERENCES ...............................................................................................................

Table of Tables

Table 3.1 Relating real BP model with complex envelope ......................................... Table 4.1 Initial predistorter coefficients .......................................................................... Table 4.2 Statistics on IMa values .................................................................................... Table 4.3 Optimum predistorter coefficients ..................................................................

Table of Figures

Figure 1.1 Predistortion ..................................................................................................... Figure 1.2 System block diagram .................................................................... Figure 1.3 Quadrature model of bandpass nonlinear system ............................................ Figure 1.4 Composite system (PDIPA) and the output stage ............................................ Figure 2.1 Feedforward linearization technique ............................................................... Figure 2.2 LINC transmitter .............................................................................................. Figure 2.4 Quadrature model for nonlinear system ........................................................... Figure 2.5 Two equivalent forms of quadrature structure ................................................. Figure 3.1 System block diagram ...................................................................................... Figure 3.2 Simplified system block diagram .................................................................... Figure 3.3 System model (real bandpass form) ................................................................. Figure 3.4 System model (complex envelope form) ......................................................... Figure 3.5 System composite model (complex baseband form) ...................................... Figure 3.6 System composite model (real bandpass form) ............................................... Figure 3.7 System composite model with real bandpass nonlinearities ........................... Figure 3.8 Contours of constant P3 in a, plane ................................................................ Figure 3.9 Contours of constant P3 in a, plane ................................................................ Figure 3.10 System composite model with the output stage ............................................. Figure 4.1 Simulation model (complex envelope form) ................................................... Figure 4.2 Amplifier and predistorter complex gains .......................................................

..................................................................... Figure 4.3 Defining IM and distortion level Figure 4.4 IMD power versus amplifier output back-off .................................................. Figure 4.5 Contours of constant IM, on a, plane .............................................................. Figure 4.6 Contours of constant IM, on a, plane .............................................................. Figure 4.7 Convergence of PD coefficients with adaptation algorithm ............................ Figure 4.8 Convergence of IM. with adaptation algorithm ............................................... Figure 4.9 Output spectra as PD coefficients converge .................................................... Figure 4.10 Values a, converges to in 5 trials .................................................................. Figure 4.11 Values a, converges to in 5 trials .................................................................. Figure 4.12 Input (16-QAM) and amplifier output spectra ............................................. Figure 4.13a Output stage spectra S, (n with 16-QAM input .......................................... Figure 4.13b Output stage spectra Sq (n with 16-QAM input .......................................... Figure 4.14 Amplifier output spectrum without PD (two tones) ...................................... Figure 4.15 Amplifier output spectrum with PD (two tones) ...........................................

List of Symbols and Abbreviations

ACF AWAM AMPM BPF CEF CEM

HPF IM IM, 1% 1% IM, IMD PA PD PSD RBF RBM

autocorrelation function amplitude to amplitude conversion (of PA) amplitude to phase conversion (of PA) bandpass filter complex envelope form (of quadrature structure) complex envelope model (nonlinearities functions of complex envelope of the input; used extensively in the document) high pass filter (average) power of intermodulation distortion products IM obtained from V,(t) IM obtained from VJt) IM obtained form V&) IM,=IM,+UI, intermodulation distortion power amplifier predis torter power spectral density real bandpass form (of quadrature structure) real bandpass model (nonlinearities functions of real bandpass signal; used just to make reference to results in the literature) complex envelope of the input signal complex envelope of PD output complex envelope of PA output

I Vm(t> l2 I Vd(f) l2 (inphase) output of the output stage (quadrature) output of the output stage real bandpass input signal real bandpass predistorter output real bandpass amp iifier output predistorter complex coefficient amplifier complex coefficient complex coefficient of composite (PDPA) system coefficient of RBM coefficient of RBM

Re {qi I Im {rli I predistorter complex gain

amplifier complex gain

complex composite (PDPA) gain (of the CEM)

complex composite (PDPA) gain of the RBM

autocorrelation function of V,(t) au tocorrelation function of Va ( t ) autocorrelation function of V,,(t) PSD of V,,,(t) PSD of Va( t ) PSD of V,,(t) PSD of V , ( t )

1 INTRODUCTION

Increasing demand for spectral efficiency in radio communications makes multilevel linear mod-

ulation schemes such as Quadrature Amplitude Modulation (QAM) more and more attractive. Since

their signal envelopes fluctuate, these schemes are more sensitive to the power amplifier

non-linearities which is the major contributor of nonlinear distortion in a microwave transmitter.

An obvious solution is to operate the power amplifier in the linear region where the average output

power is much smaller than the amplifier's saturation power (i.e. larger output back-off). But this

increases both cost and inefficiency as more stages are required in the amplifier to maintain a given

level of power transmitted and hence greater DC power is consumed. Power efficiency is certainly

a critical consideration in portable systems where batteries are often used. Another approach to

reducing nonlinear distortion is the linearization of the power amplifier.

Several techniques of linearizing amplifiers have been studied in the literature (Section 2.1)

such as 1) Cartesian feedback, 2) feedforward, 3) LINC transmitter and 4) predistortion. The basic

idea behind predistortion is to insert another nonlinear system, the predistorter, with characteristics

inverse to those of the power amplifier so that the overall output of the entire system is linear with

respect to the input (Figure 1.1).

i I

I ' I I t

I Predistorter Power Amplifier ! 'W j

* I output

I ! I

Figure 1.1 Predistortion.

Since the arnplifier characteristics change due to timing and changes in temperature or biasing, a

practical predistorter must adapt to those changes to maintain acceptable performance quality.

A block diagram of a radio transmitter with the adaptive predistorter implemented in the

course of this thesis work is shown in Figure 1.2 below.

Predistorter

Modem

I output Stage I+-

Figure 1.2 System block diagram.

A brief functional description of the system follows. The modem generates the baseband quadrature

components of the information signal which is up-converted to an intermediate frequency (IF) o,.

The resulting real bandpass signal v , ( t ) is fed into the predistorter block. The advantage of having

an IF input to the predistorter is that it's function is independent of the modulation format of the

input. The predistorter circuitry is controlled by two polynomials F, and F, generated by the block

labeled "polynomial generator." These polynomials are functions of the complex envelope of v , ( t )

coming into this block through the envelope detector. The predistorted output v d ( t ) is then up-

converted to the carrier frequency and passed through the power amplifier. A sample of the amplifier

output V,( t ) is down-converted and fed into the "output stage" along with a sample of V,,,(t). Both

of these samples are "mixed" and low pass filtered (with cut-off at q) to generate signals named

V,(t) and V,(t) as shown. These signals an then passed through HPFs (with cut-off equal to twice

the input signal bandwidth) so that only the components of va(t) that are the intermodulation dis-

tortion (IMD) products are considered. The power detectors measure the average powers (IMP and

%) in the IMD products, which are then summed to get IM,,. Then D&,, also referred to as IM, is

fed into a microprocessor (through an A D converter, not shown) which uses an adaptation algorithm

to determine how the predistorter polynomial coefficients are to be changed in order to minimize

IM. Note that it is advantageous to use the feedback path just for adaptation as explained in Section

3.1.

The main objective of this thesis work is (1) to establish the relationship between IM and the

predistorter coefficients, (2) to find a practical method of computing IM, and (3) to implement a

mechanism for adaptively changing the predistorter coefficients from IM computations. Section 2

presents some background material useful for understanding the analysis. It begins with an intro-

duction to some linearization techniques found in the literature. Since amplifier characteristics do

change, the linearizer should adapt to those changes. Some current techniques of adaptation,

including a new one introduced in this report, are presented in Section 2.2. A high power amplifier

used in a radio transmitter may by characterized by its A N A M (amplitude to amplitude) and

AM/PM (amplitude to phase) conversion measwements as discussed in Section 2.3. Section 2.4

describes a quadrature structure as a model of a narrow band bandpass (or simply bandpass) nonlinear

system. The quadrature structure consists of two independent amplitude-dependent non-linearities:

one in phase and the other in phase quadrature with the input as shown below in Figure 1.3.

nonlineari

(t) - nonlineari

Figure 1.3 The Quadrature structure model of a bandpass nonlinear system.

The inphase and quadrature non-linearities may in turn be represented by real polynomials which

are functions of the magnitude squared of the complex envelope (a baseband signal) of the bandpass

input.f(t). For this reason such a model of a bandpass nonlinear system with baseband nonlinearities

is referred to here in this document as the complex envelope model. Another configuration for

representing the above system is where the inphase and the quadrature nonlinearities are functions

of the real bandpass input f ( t ) itself (instead of its complex envelope). This configuration with real

bandpass non-linearities is termed as the real bandpass model. Some results available in the literature

for the real bandpass model derived by Heiter [7] are appropriately modified and used for the

complex envelope model which makes the analysis somewhat easier.

In Section 3.1, the system is first described and then reformed repeatedly to yield a simple

composite system better suited for analytical investigations. The composite system formed from

cascading the predistorter and the power amplifier depicts the relationship between the input VJt)

and the amplifier output Va(t) as shown below in Figure 1.4.

qt) composite system i(1)

1 (PD and PA)

Figure 1.4 Composite system (PDIPA) and the output stage.

The autocorrelation functions (hence power spectral densities (PSDs)) of the outputs of the amplifier

va(t) and the output stage Vop(t) and V,(t) are derived in Section 3.2. From the PSDs, the amount

of intermodulation distortion power (IM) is found in terms of the predistorter and the amplifier

coefficients (represented by complex power series). It is shown analytically that IM is a quadratic

function of both the 3rd and the 5& order predistorter coefficients.

The results arrived at analytically in Section 3 are verified with the simulation in Section 4.

Section 4.1 begins with a detailed description of the simulation model itself including individual

representations of the predistorter and the amplifier with complex gains. The power spectral densities

saw of the amplifier output, and S o P o andS,(n of the output stage are obtained from the simulation

with and without the predistorter present. To illustrate the parabolic shapes, plots of IM computed

from the PSDs are shown in the plane of each predistorter coefficient (Section 4.3). An algorithm

is given in Section 4.4 to change the predistorter coefficients adaptively in order to keep IM as close

as possible to its globally minimum value. Section 4.5 shows same simulation results with and

without the predistorter. The conclusions drawn form this thesis work and some recommended

future enhancements are presented in Section 5.

2 BACKGROUND

This section begins with an introduction to some linearization techniques found in the literature.

Since amplifier characteristics change with time, the linearizer should adapt to those changes. Some

current techniques of adaptation including a new one introduced in this report are presented in

Section 2.2. A high power amplifier used in a radio transmitter may be characterized by its AWAM

and AM/PM conversion measurements as discussed in Section 2.3. Section 2.4 describes a quad-

rature structure as a model of a bandpass nonlinear system.

2.1 Linearization Techniques

( 1 ) Cartesian feedback. As described by Bateman [I], Cartesian feedback uses a feedback loop in

which a sample of the amplifier output (down-converted) is subtracted from the input to generate

an error signal. A serious problem with this mechanism is the loop delay that limits the bandwidth

and linearity, and causes instability in the feedback path. This is a common problem with feedback

control loops when delay is present which may cause 180' phase shift of the feedback signal; hence

oscillations can occur.

(2) Feedforward. As shown in Figure 2.1, a delayed sample of the input is subtracted from the

power amplifier output, having gain Go and delay t, to generate a distortion signal [2]. The distortion

signal is further amplified separately by another amplifier, with the same gain and delay t', and

subtracted from the fmt amplifier output (also delayed by t') to generate the output signal. A major

drawback of this method is that it uses two amplifiers instead of one and extra delay circuits. It is

very difficult to implement the two circuits with proper delays and to find the two amplifiers with

the same gain (and phase shift) as required.

input

A wwer

t' delay

t delay

Figure 2.1 Feedforward linearization technique.

(3) LJNC transmitter. The LINC (Llnear amplification using Nonlinear Components) transmitter

represents any arbitrary bandpass signal by means of two signals S,(t) and S,(t), as shown in Figure

2.2, which have phase variations but constant amplitudes [3]. These two angle modulated signals

are amplified separately using high nonlinear devices and combined to produce an amplitude

modulated signal. This technique requires rather complicated structures to produce the two angle

modulated signals. Again, two amplifiers with the same gain and phase shift are required.

Separator

Figure 2.2 LINC transmitter.

(4) Predistortion. As illustrated earlier in Figure 1.1, the predistorter, having inverse characteristics

to those of the amplifier, predistorts the input signal before amplification so that the overall output

is linear with respect to the input. A predistorter may be digital or analogue by the nature of its

implementation. Recently an adaptive digital predistorter has been reported by Nagata [4]. The

prrdistorter is implemented with a huge (2 mega words) random access memory and with a loop

delay and a phase shifter in its feedback path which is not a very practical arrangement. Moreover,

the adaptation process is very slow: takes 10 seconds at 10 ksym/sec. These shortcomings with the

digital predistorter are overcome by another digital predistorter implemented by Dr. J. Cavers [ 5 ] .

To be useful, a predistorter or any other linearizer must be adaptive to changes in the amplifier

characteristics. Thus some technique should also be incorporated in the design to update the pre-

distorter parameters.

2.2 Predistorter Updating Techniques

AS amplifier characteristics often change with time and changes in temperature or biasing, the

predistorter is required to adapt to those changes. A number of techniques are found in the literature

for updating the linearizer parameters to achieve adaptation.

(1) Tone technique. The input to the predistorter is supplied by a function generator that outputs a

composite signal of two or three sinusoid tones of certain frequencies. The output of the amplifier

is connected to a spectrum analyzer to measure IMDs falling inside the band. Amplitude and phase

adjustments (usually with potentiometers) are done until the IMD levels are minimized. For example,

the AT&T DR6111-135 Digital Microwave Radio requires this three-tone nonlinear cancellation

technique in both factory and field. This seems to be the current field practice. The major setbacks

of this technique are that the amplifier is taken out of service for adjustments and that these

adjustments are time-consuming and expensive.

(2) Noise loading. The amount of nonlinearities at the amplifier output can not be measured directly.

This technique uses a notch filter to eliminate a small band of a Gaussian input signal before it is

applied to the predistorter. Then the level of nonlinearities in the band are measured at the amplifier

output and the adjustments are done as in (1) above. This also requires the system be out of service

while the same expensive adjustments are carried out.

(3) New technique. As shown in Figure 1.2 and briefly described in the Introduction, IM (the average

Power in IMDs) is computed using a new method while the system remains in service. The pre-

distorter described in this document is represented by a complex power series. The amplifier output

(down-converted to IF frequency) is coupled and "mixed" with the input bandpass signal to

give two lowpass signals. These lowpass signals are further passed through high pass filters with

c~t-off frequency equal to twice the input bandwidth. Then the average powers in the resulting

signal are measured and their sum IM is fed into a microprocessor. The microprocessor then adjusts

the predistorter coefficients adaptively according to an updating algorithm in order to minimize IM.

The IM is found to have parabolic shape with respect to both the third and the fifth order predistorter

coefficients (Section 3.2 and Section 4.3). The updating algorithm therefore easily finds the optimum

~oefficients. Major advantages of this approach are that the system always remains in service and

that the adjustments are done by the system itself on a regular basis.

2.3 Amplifier Characterization

An amplifier may be characterized roughly with the tone test or the noise loading techniques

mentioned above. Most commonly, AMIAM and AM/PM conversion measurements are made to

characterize the power amplifier by applying an amplitude modulated sinusoid input. As input signal

amplitude is varied, the amplifier power gain and transmission phase shift are recorded often as a

function of the input power. From the data, one can easily obtain the complex gain.

2.4 Model of a Nonlinear System

The design of a predistorter requires a suitable analytical model for the amplifier with the following

properties:

(1) the model should require only those amplifier parameters which can be measured in a

straight-forward manner such as gain and phase;

(2) the model should be invertible in order to suggest predistorter structure; and

(3) it must be general enough to incorporate all relevant amplifier characteristics (i.e. amplifier

W A M and AM/PM conversion effects). However, it should be as simple as possible to facilitate

analytical investigations.

It is rather cumbersome to model nonlinear systems with memory. With respect to the instantaneous

input and output voltages or currents, all microwave amplifiers have memory. In fact amplifier

AMEM conversion could not occur without memory. Volterra series is a general way to model a

nonlinear system with memory. This can be viewed as a Taylor series with memory or as a nonlinear

generalization of the linear convolution integral. Although a major advantage of Volterra series is

that it allows one to obtain inverse characteristics, its practical application is hindered by its

requirement of model parameters that can not be obtained from straight forward amplifier mea-

surements.

2.4.1 The Quadrature Structure

The quadrature structure, depicted in Figure 2.3 and briefly introduced in Section 1, seems to be

an appropriate model for an amplifier exhibiting both amplitude and phase distortions to a nar-

rowband bandpass input signal [6].

Figure 2.3 Quadrature structure model for nonlinear system.

The quadrature structure consists of two amplitude-dependent and memoryless non-line

depicted in the figure. Because the input process is assumed to be narrowband, the 90' phase shifter

has no effect on the statistics of the signal passing through. Thus the inputs to both the "inphase

nonlinearity" and the "quadrature nonlinearity" have the same statistics. It is desirable to represent

the inphase and quadrature non-linearities by functions that (1) can be easily computed from the

measured data of the device characteristics, and (2) have the form convenient for power spectra

computations.

2.4.1.1 Real Bandpass Form (RBF) and Complex Envelope Form (CEF)

Powers series are used here to represent each nonlinearity in the quadrature structure. Let vi(t) be

a real bandpass input signal with center frequency o, that may be written as

vi(t) = ~ e { Vi(t) eJmor } where Vi(t ) = i (t) + jq (t) is its complex envelope.

Rewriting

vi(t) = i(t) cos(w0t) - q (t) sin(oot) (2.4.2)

The real bandpass form (called so as the input andoutput are real bandpass signals) of the quadrature

structure with inphase and quadrature nonlinearities represented by power series S, and S, is

presentedin Figure 2.4a. The following is a brief mathematical derivation to show that the quadrature

structure in its real bandpass form is equivalent to its complex envelope form presented in Figure

2.4b.

(a) real bandpass form

(b) complex envelope form

Figure 2.4 Two equivalent forms of the quadrature structure.

In Figure 2.4a, the output of the phase shifter using (2.4.2) becomes

A

vi(t) = -i(t) s in(~ , t ) - q(t)cos(o,t) The real bandpass output is

where

Vo(t) = vi(t) [ sl(l vi(t) 12) +jS2(I Vi(t) 12) I This is exactly the same expression for the output complex envelope Vo(t) as indicated by the block

diagram in Figure 2.4b. Thus the real bandpass form of Figure 2.4a and the complex envelope form

of Figure 2.4b are two equivalent representations of the quadrature structure. Both forms are used

interchangeably in Sections 3 and 4 to represent the predistorter and the amplifier.

3 ANALYSIS

In Section 3.1, the system is first described and then reformed repeatedly to yield a simple composite

system better suited for analytical investigations. The composite system formed from cascading the

predistorter and the power amplifier depicts the relationship between the input v,(t) and the

amplifier output va(r). The autocorrelation functions (hence power spectral densities PSDs) of the

outputs of the amplifier va(t) and the output stage V,(t) and V,(t) are derived in Section 3.2. From

the PSDs, the amount of average power in IMDs (denoted IM) is found in terms of the predistorter

(represented by complex power series) parameters.

3.1 System Modeling

A block diagram of a radio transmitter with the adaptive predistorter implemented in the

course of this thesis work is shown in Figure 3.1 below.

Modem

micro- , processor

- I output Stage

Figure 3.1 System block diagram.

A brief functional description of the system follows. The modem generates the baseband quadrature

components of the information signal which is up-converted to an intermediate frequency (IF) o,

using an analogue quadrature modulator. The resulting real bandpass signal V,(t) is fed into the

predistorter block. The advantage of having an IF input to the predistorter is that it's function is

independent of the modulation format of the input. The predistorter circuitry is controlled by two

polynomials F , and F2 generated by the block labeled "polynomial generator." These polynomials

are functions of the complex envelope of v,(t) coming into this block through the envelope detector.

The predistorted output vd(t) is then up-converted to the carrier frequency and passed through the

power amplifier. A sample of the amplifier output va(t) is down-converted and fed into the "output

stage" along with a sample of v,(t). Both of these samples are "mixed" and low pass filtered (with

cut-off at Q) to generate signals named VJt) and V,(t) as shown. These signals are then passed

through HPFs (with cut-off equal to twice the input signal bandwidth, with stop-band rejection of

-55 dB, and bandwidth of about four times the input bandwidth) so that only the components of

va(t) that are the intermodulation distortion (IMD) products are considered. The power detectors

measure the average powers (q and IMJ in the IMD products, which are then summed to get

I&. Then m, also referred to as IM, is fed into a microprocessor (through an AD converter, not

shown) which uses an adaptation algorithm to determine how the predistorter polynomial coeffi-

cients are to be changed in order to minimize IM. The details of the hardware implementation are

provided by Costescu [13].

It should be noted that the feedback path here is not used to compute the predistorter output

vd(t) in real time but rather for adaptation of the predistorter coefficients. Unlike some other pre-

distorters found in the literature [4], this eliminates the need of a delay circuitry for simultaneous

comparison of the input VJt) and the output va(t), since no such comparison is made. As just

described, the sole purpose of the output stage is to compute the IMD power by mixing the input

and the amplifier output. This mixing process shifts down the IMD products to the baseband where

they are filtered rather easily. The output stage requires only a filter, a phase shifter and two mixers

for implementation. Another distinctive advantage of this predistorter is that the IMD power is used

-14-

to control the predistorter parameters. Because the IMD power is found to have a parabolic shape

in the planes of the 3d and 5' order predistorter coefficients, a simple adaptation algorithm can

determine the globally minimum IMD power value and the optimum predistorter coefficients

associated with it. Moreover, the predistorter structure allows pulse shaping to be done before

predistorting, and hence the system is independent of the modulation format. That is, the system

does not need to know what modulation scheme the input signal is generated with, whether it is

FM, QAM or whatever.

Figure 3.2 below shows a simplified system block diagram.

Predistorter ,-------...----------

Modem

1 output stage 1

I I

Figure 3.2 Simplified system block diagram.

The modem generates a complex baseband signal denoted as

V,(t> = i(t> + J&>

where i (t) is the inphase component and q (t) is the quadrature component. When up-converted

(using a quadrature modulator) to Q, the real narrowband bandpass signal becomes

vm(f) = ~ e { vm(t)eJ*}

= i (t) COS(O,~) - q (t) sin(o,t) (3.1.2a)

The predistorter is implemented with a quadrature structure (described in Section 2.4.1) consisting

of two independent nonlinearities that are inphase and in quadrature with the input signal and are

represented, respectively, by

F, = F,(x,(t)) = c, + cgm(t) + c&t) + . . . and F2 = F2(x,,,(t)) = dl + @,,,(t) + d+.i(t) + . . . where x,(t) = I V,,,(t) l2

and ci's and d,'s are real predistorter coefficients. Note that the even order terms are ignored as they

do not produce distortion products within the first frequency zone (which is of interest only).

Alternatively, in terms of complex gain

F(xm(t)) = Fl(xm(t)) +jF2(xm(t))

= a, + agm(t) + cc&)x:(t) + . . . (3.1.4a)

where q = ci + jdi are the complex predistorter coefficients. (3.1.4b)

The coefficients ci's and di's are updated periodically to make the predistorter adapt to any changes

in the amplifier characteristics. The predistorter output can be written as

V d ) =~e{v~(t)e'']

For analytical purposes, one can model the power amplifier by another quadrature structure with

two amplitude-dependent nonlinearities in exactly the same way as done for the predistorter.

Modem

Figure 3.3 System model (real bandpass form).

Figure 3.3 above shows a system model in which each of the predistorter and the amplifier is

represented with a quadrature structure in the real bandpass form (Section 2.4.1.1). The amplifier's

inphase nonlinearity is represented by

G, = ~,(x,(t)) = r1 + rgd(t) + r&(t) + . . . and the quadrature by

and ri's and si's are the amplifier real coefficients. Alternatively, in terms of complex gain

where Pi = ri + js, are the amplifier complex coefficients.

The amplifier output becomes

v,(t) = Re[ va(t)eJw}

= Re{ v ~ ( ~ ) G (xd(f ))e '.I' } =Re{ v.(~)F(~,"(~N ~(x , ( t ) ) e '~}

(3.1.8)

It is clear from the above equation that the complex envelope Va(t) of the amplifier output is simply

the product of VJt) (input signal's complex envelope), F (xm(t)) (predistorter's complex gain), and

G(xd(t)) (amplifier's complex gain). This leads to a more compact analytical model in the form of

complex envelopes (making use of the equivalent forms in Figure 2.5) as shown in Figure 3.4 below.

l Predistorter I

Power Amp I n -1

Modem I

I y,(t) I I

I I I

I output stage I I I 1 I I I I A

Figure 3.4 System model (complex envelope form).

All double lined signal paths depict complex signals, and the output V,(t) is the complex envelope

of the amplifier output.

3.1.1 Composite Model (in CEF)

For further simplification, a composite system is formed by cascading the predistorter and the

amplifier as shown in Figure 3.5.

Figure 3.5 System composite model (complex envelope form).

Composite System

From (3.1.4a) and (3.1.5), the predistorted complex envelope is

V&) = Vm(t) F(x.0)) = Vm(t) [a, + w m ( t ) + a$;(t) + . . .]

-18-

Modem

(predlstortor 6 amplifier)

YXt)

I I

5 + iK2

Substituting for Vd(t) and forxd(t) =I Vd(t) from (3.4.9a) into (3.1.9b)

predistorter and the amplifier with qi's being its complex coefficients.

Let pi + jqi = qi. (3.1.100

The diagram in Figure 3.5 relates the complex envelope Vm(t) of the input signal vm(t) with the

complex envelope Va(t) of the amplifier output v,(t) in terms of the composite complex gain

K(xm(t)).

3.1.2 Composite Model (in RBF)

Equivalently, one can represent the same system in the real bandpass form (instead of the complex

envelopes form) as shown in Figure 3.6. It should be noted that the composite nonlinearities dis-

cussed thus far (in Figures 3.5 and 3.6) are "baseband nonlinearities," that is, they are functions of

the input complex envelope which is a baseband signal. In consistence with the terminology

introduced in Section 1, they are the complex envelope models of the composite system.

Composite System (predbtonor 6 mpllner)

r 1

Figure 3.6 System composite model (real bandpass form).

3.1.3 Real Bandpass Model (RBM)

In the literature, a similar system has been analyzed [7] which is shown in the dotted box in Figure

3.7 below. This system from now on is referred to as the real bandpass model reflecting the fact

that composite nonlinearities are functions of the input bandpass signal v,(t) itself instead of its

complex envelope V,(t). This is the only, but considerable, difference between the real bandpass

model and the complex envelope model of Figure 3.6. Specifically, the polynomials K, and & in

the real bandpass model are functions of the input signalx(t) = v,(t) itself, whereas the polynomials

K, and K, in the complex envelope model are functions of the complex envelope squared 1 V,(t) 12.

Composite System (real bandpass model)

i ( t ) BPF

Figwe 3.7 System composite model with real bandpass nonlinearities.

Here in the real bandpass model, the inphase and quadrature nonlinearities are represented by

- K, = gl(x(t)) = a, + ug2(t) + aG4(t) + . . . (3.1.11a)

and - K, = g2(x(t)) = b1 + bG2(t) + bG4(t) + . . . (3.1.11b)

where ai and bi are real coefficients. For this analysis it is assumed that x(t) = vm(t) is a zero-mean

wide-sense stationary (WSS) Gaussian stochastic process (defined in Proakis [8]). Heiter [7] has

in this case given an expression for the autocomlation function of the real bandpass output y (t) in

terms of that of x(t) as shown below:

where J is the order of nonlinearities and

with exactly the same expression for B, as (3.1.12b) except ai is replaced by bi; moreover

I J - n for J-n even

for J-n odd.

Only the odd order terms are retained here; hence n is odd in the above equations. (Even order terms

produce IMD products outside the band of interest.) To be able to apply these results, which are

applicable only to the real bandpass model (in Figure 3.7), to the complex envelope model (Figure

3.6) under investigation, one must find the relationship between the two models.

3.1.4 Relationship Between RBM and Complex Envelope Model (CEM)

Now it is necessary to establish the relationships between Kl(I V,(t) 12) and zl(v,(t)) and between

K2(I Vm(t) 12) and z2(vm(t)) SO that the complex envelope model in Figure 3.6 is equivalent to the

real bandpass model with the bandpass filter inFigure 3.7. Then Heiter's results for the real bandpass

model can be applied to the complex envelope model. Towards this end, one may compute the

outputs from both models with the same input. A two tone input seems to be an appropriate and

convenient choice.

Let the input complex envelope signal be

Vm(t) = A cos(at)

then x,(t) = I V,(t) l2 = ~ ~ c o s ~ ( a t )

The complex envelope of the amplifier output of the complex envelope model in Figure 3.6 with

the two tone input becomes

Now assume that the signal is up-converted to o, so that the real bandpass output becomes

where p , + j q , = q ,

Now the output of the real bandpass model in Figure 3.7 with the same input is to be determined.

With input complex envelope V,(t) as given by (3.1.13), the real bandpass input becomes

3.2 Autocorrelation Function (ACF) and Power Spectral Density

In this section the autocorrelation functions of the outputs from the amplifier v,(t) and from the

output stage VJt) and V,(t) are derived for two different inputs: a zero-mean WSS Gaussian

random process (defined by Proakis [8]) and a deterministic signal with two tones. The power

spectral densities are obtained by finding Fourier transforms of the autocorrelation functions.

3.2.1 Amplifier Output

Assume that the input v,(t) is a sample function of a narrowband bandpass WSS stochastic process

with zero mean. The input may be represented in terms of its complex envelope (equivalent low-pass

process) V,(t ) or in terms of quadrature components i (t) and q (t ) as follows

vm(t) = ~ e { ~,(~)e"} (3.2.la)

= i(t) cos(a0t) - q(t) sin(aot) (3.2.lb)

The complex envelope here in terms of the quadrature components is

v,(t) = i(t) +jq(t) (3.2.2)

Since v,(t) has zero mean, both i(t) and q(t) must have zero mean as well. The autocorrelation

function of the real process i(t) and the cross-correlation function of both i(t) and q(t) are,

respectively, given by

and

The stationarity of v,(t) implies that the autocorrelation and cross-correlation functions of i(t) and

q(t) satisfy the following properties [8]:

R"(T) = R,(T) (3.2.3a)

R, (T) = -Rqi (T) (3.2.3b)

From the definition of cross-correlation function, it readily follows that

-25-

Rqi (7) = Riq (-5)

From (3.2.3b) and (3.2.3~)

That is, the cross-correlation R,(z) is an odd function of z and hence

R,(O) = 0 (3.2.3e)

Using the properties (3.2.3a) and (3.2.3b), and the form of v,(t) given in (3.2.lb), it follows that

the autocorrelation R,(T) of the real bandpass input v,(t) is

The autocorrelation function of the complex envelope Vm(t) is by definition

Substituting (3.2.2) directly into (3.2.5) and using linearity of expected values

This equation in turn, with use of the symmetry properties in (3.2.3a) and (3.2.3b), becomes

R,(z) = R,(z) - jR, (z) (3.2.6)

Equation (3.2.6) expresses the autocorrelation function of the complex envelope (the equivalent

low-pass process) in terms of the autocorrelation and cross-correlation of the quadrature compo-

nents. From (3.2.4) and (3.2.6), the relationship between the autocorrelation function (R,(T)) of the

real bandpass process v,(t) and that of its complex envelope V,(t) follows

Gaussian Input

In addition to the input being zero-mean WSS stochastic process, now it is further assumed that the

I input process is also Gaussian. The autocorrelation function of the amplifier output VJt) is found

in two ways: 1) directly using the formula for mixed moments of zero-mean jointly Gaussian random

variables, and 2) indirectly by making use of Heiter's results for the real bandpass model of Figure

3.6. Knowing the relationship between the real bandpass model and the complex envelope model

which is illustrated in Table 3.1, one can easily extend these results to the complex envelope model

under consideration here.

3.2.1.1 ACF By Direct Method

The autocorrelation function of the complex envelope Va(t) of the amplifier output is by definition

1 Ra(T) = ?E{Va(t) v:(t '7))

The amplifier complex envelope (3.1.10) is

Va(f) = Vm0) [%+?I3 I V,"(f) f+qJ Vm(f) f + ( t + 7) = v:(t + 7) [q; + q; I Vm(t + T) P +q;

For convenience, denote Vm(t) by

V, =i t + jq,

and vm(t + 2) by

V2=i2+ jq2

where i, = i(t), i2 = i(t + T), q, = q(t), and 4, = q(t + 7). Expanding the argument to the

expected value function in (3.2.8) using (3.2.9) in terms of V, and V2, the autocorrelation function

becomes

2R&) = q,q; E {v,v:}

+ q IT: E{ v1v2vf} + ?l n; E{ v,v:v:} + r1;713 E{V:V;V,'I

+ fl3q: E[ v:v;v2v;? + q3q: E{ v:v;v;v;? + q h E[ v;v:v:} + q:qs E{ v:v:v2v:I

3 . 2 2 +tlAE{V1Vl v2$} (3.2.10) Each of the expected values in the above equation is further expanded in terms of the zero-mean

jointly Gaussian real random variables i's and q's. For instance, the second term yields

All the expected values appearing in the above equation can be determined by using the following

formula for the mixed moments of L zero-mean jointly Gaussian random variables x,,x2, . . .,xt [9].

where & = E {xsk)

L

C [A&,,,,. . .h,] if L is even 11 &tinct

For example

{ x ~ x z % ~ = %2%4 + h3h2L) +

If some of the variables appear in the moment expression with a power of 2 or higher, each repeated

subscript is treated as if it were distinct when applying the formula. In the case when L is even, the

number of terms entering into the formula is

Now, applying the formula (3.2.12) to Equation (3.2.1 1) and collecting terms

,5[ VlV2v~} = gRij(0) -I'Rk(z)I

= 8Rm(0)R,(z) (3.2.14)

Continuing in this manner with the other terms in Equation (3.2. lo), that is, expanding each expected

value expression in (3.2.10) in terms of its real random variables i's and q's and then applying the

formula (3.2.12), one obtains the autocorrelation function of the amplifier complex envelope Va(t)

3.2.1.2 ACF By Indirect Method

First of all, the autocorrelation function R,(z) of the real bandpass model output y (t) in Figure 3.6

is given by (3.1.12) with the coefficients An's being as follows:

For the coefficients Bn's the same equations as above hold except that ai's are replaced by hi's.

Then the expression for R,(z) is expanded and all the second and higher order harmonics are

-29-

eliminated (effect of the bandpass filter in Figure 3.6). The resulting expression is equal to the

auto-correlation function R.(T) of the amplifier output. Furthermore, all the composite bandpass

coefficients ai's and hi's may be replaced by the composite baseband coefficients pi and qi

respectively from Table 3.1. To facilitate comparison with the result obtained by the direct method

(3.2.15), the complex envelope Ra(z) of this R,(T) is determined using the relation (3.2.7)

R .(TI = Re{ Ra (z)eJ"'}

The result is

2 2 ' a ( ' ) = [ 1 ~ 1 IZ + ~ R ~ { % ~ S I R , ( O ) + 4 8 ~ e { q l ~ ; l ~ ; ( o ) + 16 1 7 3 I Rm(0)

2 4 + 1 9 z e {%~~;IR;(O) + 576 1 rl, I R,(O) lR,(z)

2 2 + [ 8 1 T3 l 2 +19~e{r l3r l ; l~ , (o) + 1152 1 rl, I R,(O) I R : ( ~ : ( T )

+ [ 192 1 % lZ I ~ h R : ( z )

which of course is identical to previous results (3.2.15).

3.2.1.3 Power Spectral Density

The spectral characteristic of a stationary stochastic process is obtained by finding the Fourier

transform of its autocorrelation function. Since i ( t ) is real, it follows directly from the definition of

autocorrelation that R&) is real and even, that is

R,(z) = R,(-t) (3.2.18)

From (3.2.6)

RJz) = RJz) - jR,(z)

Using (3.2.3d) and (3.2.18), the above equation becomes

R~(-T) = R s O - iRi(z)

R&) = R m ( ~ ) (3.2.19)

The power spectral density function S,(f) of the complex envelope V,(t) is obtained by taking

Fourier transform of its autocorrelation function R,(z)

Replacing f by -fin (3.2.20) and then taking the complex conjugate to get the spectral characteristic

The Fourier pairs are

Using (3.2.22), the Fourier transform of R,(T) in (3.2.17) gives the spectral characteristic of the

complex envelope of amplifier output as

2 4 + 1 9 2 e {r13r1;I~d(o) + 576 I q, l R,@) I S ,m

2 2 + [ 8 I 113 IZ + 1 9 ~ e { W l ; l ~ , ( o ) + 1152 1 q5 1 R,(O) I s : ( - . * s , ( ~ * s , ( ~

+ [ 192 1 175 l' 1 s:(-f)*~:(-fl*~,(n*~,(n*~,m (3.2.23)

Let P1= I V I + 4q@,(O) + 24r1&(0) 12 (3.2.24a)

p3 = 8 1 r13 + 12r1Pm(0) 12 (3.2.24b)

P, = 192 1 q, 1' (3 .2 .24~)

s,(n = ~ : ( - f l * ~ , ~ n * ~ , ~ n (3.2.24d) and sd(i? = ~ : ( - n * ~ : ( - f l * ~ , m * ~ , m * ~ , m (3.2.24e)

Now Equation (3.2.23) can be written as

s o w = P 1 s m w + P 3 S d w + P 5 s d ~ (3.2.25)

Here PISm(f) is the desired spectrum density, whereas P$,Cf) and P,S,Cf) are the spectra of the

3rdand 5"order (respectively) IMD products which from (3.1.10) and (3.2.24) are quadratic functions

of the 3d and 5m order predistorter coefficients (q and q). This relationship is further illustrated

graphically in Figures 3.8 and 3.9. For these graphs, the values of W's used were obtained by

curve-fitting the amplifier complex gain, and those of 4 ' s were computed by curve-fitting the

predistorter complex gain as given in Table 4.1 (Section 4.1.2). The contours of constant P3 in the

oc, plane (when ol, is kept constant as given by Table 4.1) and a, plane (when a, is kept constant

as given by Table 4.1) help one to visualize the quadratic dependence. We do not see smooth circular

contours because of the limited number of data points imported into the plotting software.

-8.882 -8.001 8 8.001 8.002 8.883 8.064 8.885 8.886

Re Alpha 3

Figure 3.8 Contours of constant P, in a, plane.

-14 -12 -10 -8 -6 -4 -2 0 2

Re Alpha 5 ( x 0.00001)

Figure 3.9 Contours of constant P, in g plane.

This quadratic dependence is of course desirable, because it follows that the total IMD power has

a parabolic shape in the planes of a, and g. Hence a simple adaptation algorithm can find the

globally minimum value of the IMD power which in turn gives the optimum values of g and g.

3.2.1.4 Two Tone Input

For the two-tone case, the complex envelope of the input becomes

VJt) = A cos(at)

From (3.1.15), the amplifier output is

where C, = - p , A 1 + 8 p 4 3 + 1 6 ~ ~ 3 5 5

2

The time autocorrelation function %(z) of the real Va( t ) is by definition

Replacing 7 by -7

Hence the time autocorrelation is an even function of Z, that is

a* =

The time autocorrelation function of a real signal

N g (t) = C A, cos(a,t + 0,)

r = l

is given by [lo]

g ( 2 ) that is th ~e sum

(3.2.37)

of N sinusoids

(3.2.38a)

Applying (3.2.38) to va(t) in (3.2.35a) yields the desired autocorrelation

where

C

& (r) = B, [cos((oo - a )r) + cos((q, + a ) ~ ) ]

+ B3 [cos((a, - 3a)r) + cos((o, + 3a)r)l

+ B, [cos((o, - 5a)r) + cos((o, + 5a)z)l (3.2.39~)

1 3 9 C:+D: =-A21ql ~ 2 + - ~ 4 ~ c { q l q ~ + 1 2 8 ~ 6 1 q 3 ~ Bl= 2 8 16

5 15 25 (3.2.39b) +-A 6 ~ e {q,$ +-A 'Re {q3q5 +-A1' I 11, l2

32 128 5 12

1 5 25 (3.2.39~) C:+D: = - A ~ ~ ~ ~ ~ + - A ' R ~ { ~ ~ ~ ~ } + ~ A ~ ~ I ~ , I ~ B3= 2 128 256

1 G+D: - (3.2.39d) Bs= 2 -2048A101 %12

The PSD Sa(a) of va(t) is simply the Fourier transformation of its time autocorrelation &(r)

$,(a) =dl {6[a+(oo-a)]+6[a-(a,-a)]

+6[a+(oO +a)]+6[a- (a, +a)] }

+ ~ 4 {6[a+(ao-3a)]+6[a-(a,-3a)I

+6[a+(ao+3a)]+6[a-(c~.1,+3a)]}

+d5 {6[a+(a0 -5a)l +6[a-(a, -5a)]

The PSD of the amplifier complex envelope form (3.2.40a) becomes

S,(o) = 2 d , {6[a-a)]+S[a+a)])

+ 2 d , {6[a-3a)]+6[a+3a)] } 3d IMD density

+ 2 d , {6[a-5a)]+6[a+5a)]} 5* IMD density (3.2.40b)

Again form (3.1.10), (3.2.39) and (3.2.40b), one observes that the PSD of the Ydand 5" order IMD

products are quadratic functions of the 3dand 5"' order predistorter coefficients (a, and Q.

3.2.2 The Output Stage

The main objective, which has been achieved in Section 3.2.1, of the analysis is to show the rela-

tionship between the IMD power and the predistorter coefficients. The function of the output stage

is just to make the measurements of the IMD power practically easier. Thus the outputs VJt) and

V,(t) of the output stage express the IMD products alternatively. Finding the autocorrelation

functions and the PSDs of these outputs in this section, one arrives at the same result of Section

3.2.1 which shows the quadratic dependence of the IMD power on the predistorter coefficients.

Figure 3.10 below shows the analytical model with baseband composite nonlinearities and

the output stage. The purpose of the output stage is to provide signals labeled V,,(t) and V,(t) which

measure discrepancies in the input VJt) and the amplifier output ba(t). These signals are then

passed through HPFs (with cut-off equal to twice the input signal bandwidth, with stop-band

rejection of -55 dB, and bandwidth of about four times the input bandwidth) so that only the

components of ba(t) that are the intermodulation distortion (IMD) products are considered. The

power detectors measure the average powers (IM, and IM,) in the IMD products, which are then

summed to get IM,,. Then IM,,, also referred to as IM, is fed into a microprocessor which adjusts

the predistorter coefficients (coefficients of polynomials F , and F,) in order to minimize IM.

Figure 3.10 System composite model with the output stage.

In this Section, the autocorrelation functions of the outputs VJt) and V,(t) are derived with

zero-mean Gaussian stochastic input and with two tone input. First of all, expressions for VJt)

and V,(t) are found as follows:

vm(t> = i(t> + jq (0 (3.2.50)

vm(t) = ~ e { Vm(t) e'"1 = i(t) cos(oot) - q(t) sin(oot) (3.2.5 1)

The amplifier output is

v,(t) =Re{ Vm(t) [ K,(x,(t)) + jK2(xm(t)) ] eJmO'] (3.2.52)

For convenience, let i = i ( t ) , q = q ( t ) , K , = K,(xm(t)), and K2 = K2(x,(t)). Expanding v , ( t )

p,(t) = [K,i - K2q] cos o,t - [ K,i + K,q ] sin w,t (3.2.53)

The output of the upper mixer in the output stage (Figure 3.10), from (3.2.51) and (3.2.53) becomes

t a t = [ ~ ~ ( i ' + ~ ~ ) ]

+ [K1i2 - 2K2iq - K1q2 ] cos 2 q t

- [ K2i2 + 2 ~ , i ~ - KZq2 ] sin 2 q t (3.2.54a)

After filtering to remove all components with frequencies above o,, the result is the desired signal

Let input the V,(t) after being passed through the 90' phase shifter be

A

v m ( t ) = -i sin mot - q cos oot The output of the lower mixer in the output stage becomes

A

v m ( t ) 'a( t) = [ ~ ~ ( i ' + ~ ' ) ]

- [ K2i2 + X 1 i q - KZq2 ] cos 2o.r

- [ K,i2 - 2K2iq - K1q2 ] sin2o0t

After filtering to remove all components with frequencies above o, as before, the result is the signal

Expanding the polynomials K1 and K2 with qi = p i + jqi as in (3.1.10), Equations (3.2.54b) and

3.2.2.1 Autocorrelation Function

The autocorrelation function of Vop(t) by defmition is

R o p ( ~ ) = E{Vop(t) Vop(t + t)I For convenience, denote I V,(t ) I by

where i1 = i (t), i2 = i (t + z), q, = q (t ), and 4, = q (t + z). Expanding the argument to the

expected value function in (3.2.59) using (3.2.57) in terms of Vl and V2, the autocorrelation function

becomes

Each of the expected values in the above equation is further expanded in terms of the zero-mean

jointly Gaussian random variables i's and q's. For instance, the first term yields

All the expected values appearing in the above equation can be determined by using the formula

for the mixed moments (3.2.12). This gives

Repeating this procedure for all the other terms in (3.2.60), that is, expanding each expected value

expression in (3.2.60) in terms of its real random variables i ' s and q's and then applying the formula

(3.2.12), one finally obtains the auto-correlation of the output in terms of that of the input process

3.2.2.2 Power Spectral Density

The power spectral density Sop(f) of Vop(t) is the Fourier transform of its autocorrelation function

After passing Vop(t) through the HPF (with cut-off at twice the input bandwidth, stop-band rejection

of about -55 dB and bandwidth of about four times the input bandwidth), only the last two terms

in (3.2.64) remain. The constants of these terms are quadratic functions of a, and a, as can be seen

from (3.1.10). The PSD of the other output V,(t) of the output stage is identical to that given in

(3.2.64) except that pi's are replaced by qi's.

3.2.2.3 Two Tone Input

Substituting the two tone input

V,,,(t) = A cos at

into (3.2.57)' one gets the output of the output stage

1 +- [C, + C3] A cos k t 2

1 +- [C3 + C,] A cos 4at 2

1 +- [C,] A cos 6at 2 (3.2.65)

Same expressions for V,(t) hold as above except that Ci's are replaced by Di's, where Ci's and Di's

are as given by (3.2.35). Applying the formula (3.2.38) to (3.2.65), one obtains the time

autocorrelation function 1&,(z) of the output V J t )

where 1 El = 4 [ ~ 1 ] 2 ~ 2

The PSD Sop(o) of Vop(r) is simply the Fourier transformation of its time autocorrelation % p ( ~ ) .

S o p ( o ) = rn,W) +ICE,, { 6 ( o + h ) + 6 ( o - h ) }

+nE3, { 6(0+4a)+6(0 -4a ) } (3.2.68)

+nE, { 6 ( 0 + 6 a ) + 6 ( 0 - 6 ~ ) }

4 SIMULATION

The results arrived at analytically in Section 3 are verified with the simulation in Section 4. Section

4.1 begins with a detailed description of the simulation model itself including individual repre-

sentations of the predistorter and the amplifier with complex gains. The power spectral densities

S a m of the amplifier output and S,,m and S& of the output stage are obtained from the simulation

with and without the predistorter present. To illustrate the parabolic shapes, plots of IM computed

from the PSDs are shown in the plane of each predistorter coefficient (Section 4.3). An algorithm

is given in Section 4.4 to change the predistorter coefficients adaptively in order to keep IM as close

as possible to its globally minimum value. Section 4.5 shows some simulation results with and

without the predistorter.

4.1 Simulation Model

4.1.1 Input and Output Expressions

The simulation model is the system model in the complex envelope form as shown in Figure 4.1

below.

Figure 4.1 Simulation model (complex envelope form).

As before the input complex envelope generated by the modem is

V&> = i(t> + im

The predistorter is implemented by having its complex gain given by the following complex power

series

F W ) ) = a1 + a;c,(t) + a&)

where xm(t) = I Vm(f) f and ai 's are the predistorter complex coefficients.

Letting ci = R e { a , } and di = Im{a, } .

one may write

F(xm(t)) = Fl(xm(t)) + jF2(xm(t)) (4.1.3b)

where Fl(xm(t))=cl+c+m(t)+c.&(t) (4.1.3~)

and F2(xm(r))=dl+@m(t)+d+:(t) (4.1.3d)

The real polynomials F, and F, represent the two amplitude-dependent inphase and quadrature

nonlinearities respectively. The complex envelope of the predistorter output is

VdO) = V&) F(xm(t)) (4.1.4)

The power amplifier is represented by a look-up-table (LUT) containing the complex gain

(determined from amplifier measurements as explained in Section 4.1.2) as a function of input

power

G (xdW = Gl(xd(t)) +jG2(xd(t)) (4.1.5)

where xd(t) = I vd(f) l2 It should be noted that for the analysis in Section 3 both the predistorter and the amplifier are

represented by power series and then they are cascaded to form the composite system (which in

turn is represented by another series), whereas here in the simulation, the amplifier is represented

separately by a LUT instead and no composite system is formed.

Amplifier Output

The complex envelope of the amplifier output is simply given by

V,(t> = Vd(f G (~d(t))

Substituting Vd(t) from (4.1.4) into (4.1.6)

va(t) = vm(t> F(xm(t)) G(xd(t)) (4.1.7)

The Output Stage

From the diagram in Figure 3.1, one can easily right expressions for the outputs of the output stage

in the following form

vv(t) =R~{v~v:(~)I (4.1.8)

and VqO) =~m{vav:(t)~ (4.1.9)

Thus in the simulation, the amplifier complex envelope is computed by (4.1.6) and the outputs of

the output stage by (4.1.8) and (4.1.9). These outputs are eventually passed through HPFs (with

cut-off at twice the input bandwidth, stop-band rejection of about -55 dB, and bandwidth equal to

four times the input bandwidth) as explained in Section 3.1 to obtain the IMD products.

4.1.2 Predistorter/Amplifier Complex Gains

The amplifier AMIAM conversions are measured in terms of power output (in dB) as a function of

power input (in dB). After converting the power into watts or milliwatts, the magnitude of the

amplifier gain is obtained as a function of input power (in milliwatts). The phase is obtained likewise

from AM/PM measurements. For the particular class AB amplifier used here, the magnitude and

the phase of the amplifier complex gain G are given by the upper left and right (respectively) plots

in Figure 4.2.

input magnitude squared 0 20 40 60 80

input magnitude squared

.% 0.8 0 b 5 0 75

7

0.65

input magnitude squared input magnitude squared

Figure 4.2 The amplifier and predistorter complex gains.

The predistorter gain is determined by solving the following equation (obtained form 4.1.5 and

4.1.7) for a finite number of x, values and then fitting a c w e through the data.

~(x,")G[x", lF(x,) 123 = KO

where KO is the desired magnitude of the composite gain. The gain and phase of the predistorter

complex gain, along with second order polynomial approximations, are graphed in the lower two

plots in Figure 4.2. The two second order polynomials have the coefficients given in Table 4.1.

-..-:me*ured ......................... ............*.- ... interpolated j 1 ; ;.:

: c: ............... + ................................. i ....... t: ......-

; .3 . . ' I : :' , . . . . I ,. 8 .. , ... ............... j ................................... .............-

. . . . .::. : : * / - : -. ' -+-" 3,-zz .+=. I I I

Table 4.1 Initial computed predistorter coefficients.

0 20 40 60 80

4.2 Running Simulation

Two different inputs are used - a 16-QAM signal and a two-tone signal. The complex envelope of

the 16-QAM bandpass signal can be represented as

where an's are the complex symbols in the 16-QAM constellation, T is the symbol rate (the bit rate

is 4T), and g(t) is the transmit filter or pulse shape. The raised-cosine pulse with 50% access

bandwidth is used in the simulation which can be written as

where the roll-off factor a is set to 0.5. To obtain the power (IM) contained in the IMD products

with given predistorter coefficients and given input, the simulation is run for about 20,000 symbols.

The complex envelope Va(t) of the amplifier output and the output stage's outputs VJt) and V,(t)

are recorded and their PSDs S a m and S,m and S,m (respectively) are estimated using Welch's

method [I 11. Figure 4.3 below shows a plot of S a m with the 16 QAM input and without the

predi s torter (PD) .

PSD (dB)

Figure 4.3 Defining IM and distortion level.

As illustrated in the figure, intermodulation distortion power at the amplifier output, denoted IM,,

is defined as the average value (in dB) of all those points in the Sam graph that are above the

reference line and that have frequencies (normalized with the sampling rate) between I fH -f, I. The

reference line is somewhat arbitrarily set at "reasonably" low level for computation convenience.

The low frequency cut-offf, is chosen to be equal to the bandwidth of the input signal and the high

frequency cut-off f, is set at high "enough" value to include all IMD power. The definition of the

distortion level (DL) is also illustrated in the figure; it is the highest value of the IMD product

density on the graph. The IMD power from the outputs of the output stage, denoted as I M , = q + w ,

is similarly defmed from a plot of S,(n and S,lf) (instead of Sam). In this case however, the low

frequency cut-offf, is set equal to twice the bandwidth. It must be mentioned here that the IM value

obtained by running the simulation for several thousand symbols has some uncertainty associated

with it. Generally, the ter the number of symbols the simulation is run for, the better the IM

approximation is.

Uncertainty in IM

To quantify the uncertainty in IM, two tests were performed. The simulation was run 100 times on

a certain set of predistorter coefficients, each time with a different sequence of 10,000 symbols.

Thus 100 different values for IM, were obtained having minimum, maximum, mean and variation

as shown in Table 4.2 below. In the second test, 20,000 symbols were used instead of 10,000. From

the statistics in Table 4.2 of both tests, one observes that 20,000 symbols give "acceptable"

approximations. Thus it should be noted that every value of IM given in this Section is an outcome

of a single trial and it is uncertain to the degree reflected by the statistics in the table.

Table 4.2 Statistics on IM, values.

I 10.000 syms. 20.000 syms. I No. of trials 100 100

Minimum (dB) 6.52 6.87

Mean.(dB) 6.95 7.10

Maximum (dB) 7.1 8 7.28

Variance (dB) 0.020 0.0046

Output Back-Off (BO)

The output back-off (BO) power is computed using the following equation

where P, is the saturation power of the amplifier

and P, is the average amplifier output power for the entire simulation

session.

The value of IM (the IMD power) obtained by running the simulation once (for a few thousand

symbols) also depends heavily on the amplifier output back-off as illustrated in Figure 4.4 below.

This dependence is not so strong with back-off greater than about 5 dB. This follows from the fact

that the amplifier gain and phase nonlinearities are more severe as the amplifier is operated closer

to saturation.

Output Back-Off (dB)

Figure 4.4 IMD power (IM) versus amplifier output back-off.

4.3 IM Dependence on Predistorter Coefficients

Recall from Section 3.2 that IM is a quadratic function of the 3rd and 5th order predistorter coef-

ficients. This time some simulation results are shown to confirm the same relationship. The initial

values of a,'s are set equal to those found by having second degree polynomials F , and F, curve-fit

the inverse characteristics of the power amplifier as presented in Table 4.1.

Then the simulation is started and is varied while a, and a, are kept constant. For each

value of g , the simulation is run for about 20,000 symbols and the PSDs, Sam are estimated from

which the IMD power (IM,) is computed. Figure 4.5 shows contours of constant IM, on the complex

g plane.

This time a, is varied while keeping a, and a, (reset to its initial value in Table 4.1) constant.

The contour plots, from this simulation, of constant 1% are presented in Figure 4.6.

-0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005 0.006

Re Alpha 3

Figure 4.5 Contours of constant IM, on a, plane.

Re Aloha 5

Figure 4.6 Contours of constant IM, on a~ plane.

4.4 Adaptive Algorithm

Figures 4.5 and 4.6 show that IM, have parabolic shapes on a, and g planes. The globally minimum

value of IM, (6.7dB for this configuration) or IM, is associated with the optimum values of g and

a,. The IM, without the predistorter is about 11 dB. This information is used to implement a very

simple algorithm for updating predistorter coefficients. The algorithm starts with arbitrary initial

coefficients and yields the optimum set of coefficients that has lowest IM.

Algorithm

- denote the coefficients as

G = R e { a, 1

G = I m I a , 1

G = Re ( a, 1 C, = Im ( a5 ).

- let Ci denote the optimum value of Ci ( i=1,2,3,4).

- let "compute IM (IM, or IM,)" refer to the process of running the simulation for about 20,000

symbols with given set of coefficients, and obtaining PSD and hence the IM value.

- initially set all C, to arbitrary values, and set ci = C,.

- compute IM. Set 1% = IM.

(1) change Ci: C, = Ci + A,

(2) compute IM

(3) if IM > 1% then

unchange C,: Ci = Ci - A , and

set A, = - A,

otherwise - C, = Cj (forj=1,2,3 and 4), and

1% = IM.

(4) i = next i (i varies cyclically). Goto (1)

Notes:

(1) The optimum coefficients associated with IM- are used as a reference to measure further

improvements. As any change in PA characteristics is suspected, IM- itself could be

updated periodically (that is, compute IM with coefficients set to ti's.

(2) As 1% becomes "low", the magnitudes of 4 ' s could be decreased.

(3) All the coefficients are updated cyclically.

The adaptation algorithm begins with an arbitrary initial predistorter coefficients (a, and g). The

first order coefficient a, is always kept constant to maintain constant power in the desired signal

at the amplifier output. During each step or cycle through the algorithm (1) the real or imaginary

part of oc, or a, is modified, (2) the simulation is run for about 20,000 symbols, and (3) PSD and

hence IM is estimated. The increase or decrease in IM from the previous cycle indicates the direction

in which the coefficient should be changed in the next cycle. The goal is to get the lowest possible

IM value which corresponds to the optimum values of a, and &. Figure 4.7 shows four plots of the

values of a, and a5 after each cycle of the algorithm starting with some arbitrary initial values. For

instance, the upper left graph in the figure shows that Re{a3}=-0.001 at the beginning and it

converges to its optimum value of 4.0019 after about 30 cycles. Figure 4.8 shows that as the

coefficients converge to their optimum values, IM converges from 16.5 dB (with arbitrary coeffi-

cients) to its optimum value of 6.7 dB. A value of 11 dB corresponds to having no predistorter at

all; therefore, its initial value of 16.5 dB indicates that convergence is achieved in a wide domain

of the predistorter coefficients.

0 20 40 60

Cycle Number 0 20 40 60

Cycle Number

0 20 40 60 0 20 40 60

Cycle Number Cycle Number

Figure 4.7 Convergence of predistorter coefficients with the adaptation algorithm.

-0 10 20 30 40 50 70

Cycle Number

Figure 4.8 Convergence of UI, with the adaptation algorithm.

Figure 4.9 shows that as the predistorter coefficients converge to their optimum values, the amplifier

output spectrum gets closer and closer to the "clean" input spectrum. The graph marked (1) in the

figure sketches the output spectrum with the initial arbitrary predistorter coefficients. The graph

(2) is the spectrum as the coefficients get "closer" to their optimum values and the optimum values

produce the spectrum given by graph (3). The graph (4) marks the spectrum of the 16-QAM input.

One observes from the graphs that the out-of-band power, or the IMD power, reduces drastically

as the convergence takes place as also illustrated in Figure 4.8.

PSD (dB) POWER SPECTRUM

Figure 4.9 Output spectra as PD coefficients converge.

The domains of a, and a, in which they are allowed to vary are shown in Figures 4.10 and

4.11 respectively. Also shown in these domains or planes are the values to which they converge in

five separate trials starting with arbitrary initial values. Of course their closeness to each other is

controlled by the amounts (deltas) by which they are allowed to vary from one cycle to another.

" -2 -1 0 1 2 3 4 5 6

Re { Alpha 3) (x 0.001)

Figure 4.10 Values a, converges to in 5 trials.

-14 -12 -10 4 -6 4 -2 0 2

Re { Alpha 5) (x 0.00001)

Figure 4.1 1 Values g converges to in 5 trials.

The set of optimum coefficients found by the algorithm is shown below in Table 4.3.

Table 4.3 Optimum predistorter coefficients.

4.5 Results

In this Section, graphs of power spectrum densities Sam, S,@ and S,(n with and without the

predistorter are shown for different inputs (16 QAM and two-tone). The optimum set of predistorter

coefficients computed by the algorithm is used to represent the predistorter, and the power amplifier

is implemented with the look-up-table. Figure 4.12 shows three plots of (1) amplifier output spectrum

S a m with the predistorter (PD), (2) without PD, and (3) the 16 QAM input spectrum smm. Improvement in the distortion level of 11 dB is achieved as shown with the predistorter at the output

back-off power of 5.88 dB.

PSD (dB) POWER SPECI'RUM

FREQUENCY f / f~ (fi=rampling f q . )

Figure 4.12 Input (16 QAM) and amplifier output (with and without PD) spectra.

Similarly, spectra at the output stage of SOpm and S,m are shown in Figure 4.13a and in Figure

4.13b, respectively. In each figure, the third graph, marked with (3), displays the spectrum of the

output when there is no predistorter (PD) and no power amplifier (PA) present (there complex gains

set to unity). That is, the output stage in this case mixes the 16-QAM modem output with itself and

produces the output Vop(r) (or V,(t)) from which this spectrum S o P o (or S,o) is computed. The

other two graphs, one with the PD (2) and the other without it (I), show significant improvement

in the IMD level. The two tones that we see at frequency of f0.0667 occur at the symbol rate of 16

k symlsec (with f,=240 k samplesls). The occurrence of the tones is explained by considering the

mean E {I V,,, 1') (ignoring nonlinearities for simplicity) over the data ensemble and realizing that

this quantity peaks at multiples of the symbol rate [12]. They are not predicted by the output PSDs

derived analytically, because the input here in the simulation is a 16-QAM signal, not a zero-mean

Gaussian random process as was the assumption in the derivations.

PSD (dB) POWER SPECTRUM

FREQUENCY f/fa (fs=tampling fnq.)

Figure 4.13a Output stage spectra S,(n with 16 QAM input.

PSD (dB) POWER SPECTRUM 0 1 I

1 I 1

FREQUENCY f/fa (fs=sarnpling fnq.)

Figure 4.13b Output stage spectra S,Cf) with 16 QAM input.

Now consider a two-tone input. Figures 4.14 and4.15 show the spectra of the amplifier output

without the predistorter and with the predistorter, respectively. From these plots, one observes that

the 5" order IMD product is suppressed by the predistorter from -37 dB to -5 1 dB - an improvement

of 14 dB. Improvement in the 3dorder IMD product is even more significant: about 24 dB (from

-43 dB to -67 dB). Since this predistorter is just of 5& order, no improvement in the 7m order

intermodulation is seen from the graphs.

Figure 4.14 Amplifier output spectrum without predistorter (two tones).

PSD (dB1 WWER SPECTRUM

FREQUENCY f/fa (fs=sampling frq.)

Figure 4.15 Amplifier output spectrum with predistorter (two tones).

The simulation results are summarized below in Table 4.4. The Results without the predistorter and

with the predistorter are given for the 16-QAM and the two tone inputs. The improvements obtained

with the predistorter are also shown quantitatively. In each case the amplifier output back-off power

is maintained roughly constant around 5.8 dB.

Table 4.4 Summary of results with and without predistorter.

Amplifier Output Output Stage

IM,(dB) DL(dB) UI,(dB)

No Predistorter 16 QAM 11 -39.0 28.2 Two Tones 24.9 -40.2

With Predistorter 16 QAM 6.7 -50.0 4.3 Two Tones 14 -59.4

Improvement 16 QAM 4.3 11.0 23.9 Two Tones 10.9 19.2

5 CONCLUSIONS

An adaptive predistorter for compensating both AM/AM and AM/PM nonlinearities of power

amplifiers has been studied, and analytical and simulation results are presented. Unlike most other

predistorters found in the literature, it does not have the problem of loop delay or phase shift in its

feedback path which only used for adaptation. This 5'order analogue predistorter is implemented

in the form of a quadrature structure with two amplitude-dependent nonlinearities: one inphase and

the other in quadrature with the input. The inphase and the quadrature nonlinearities are generated

by power series, and their coefficients (the predistorter coefficients) are periodically updated to

make the predistorter adapt to any changes in the amplifier characteristics.

The changes or mismatches between the predistorter and the amplifier are detected and

corrected for using a new technique. The output stage of this system is a part of this technique

whereby a sample of the input is mixed with another sample of the amplifier output. The resulting

signal is low pass filtered to get the output of the output stage. This output is then passed through

a HPF (with cut-off equal to twice the input bandwidth) so that only the components that are the

intermodulation distortion (IMD) products are considered. The power detector measures the average

power in the IMD products, denoted IM, and feeds it into a microprocessor which determines, using

an adaptation algorithm, how the predistorter polynomial coefficients are to be changed in order to

minimize IM. The relationship between the IM and the predistorter coefficients found analytically

is vital to the adaptation algorithm

In the analysis, the amplifier is represented with another quadrature structure in exactly the

same way as done for the predistorter. Assuming the input to be a zero-mean WSS Gaussian sto-

chastic process, the autocorrelation function and PSD of the amplifier output are derived in terms

of the predistorter and the amplifier coefficients. From the PSD, it is clear that IM has a parabolic

shape in the planes of both the 3rdand the S" order predistorter coefficients. Starting from arbitrary

coefficients, a simple adaptation algorithm yields the optimum set of predistorter coefficients

corresponding to the globally minimum IM value. Because it is rather difficult in practice to compute

IM from the amplifier output directly, the output stage is implemented to meet this objective as

described above. The autocorrelation functions (and the PSDs) of the output stage are also derived

and the same dependence of IMD power on the predistorter coefficients is shown. The spectral

characteristic with a two tone input have also been analyzed. The analytical results are reconfirmed

with the simulation.

The quadratic dependence of IM on the predistorter coefficients is also depicted by the

simulation results. An improvement of 11 dB with the predistorter at amplifier back-off of 5.6 dB

is achieved as the simulation results show. Although this much improvement has also been reported

by the other two linearizers, specifically the Feedforward and the LINC transmitter, they are not

adaptive and require periodic manual adjustments. The digital predistorters have the disadvantage

of requiring expensive random-access-memory. The cartesian feedback method suffers from the

problems of loop delay and phase shift in the feedback path which limits the bandwidth and linearity

of the system.

Now that the analytical and the simulation results are quite promising, the next step is to

implement this adaptive predistorter in the hardware. Some work is also required to improve the

adaptation algorithm for updating the predistorter coefficients. In fact this defines the thesis project

of another student (Mr. Flaviu Costescu) who has already started working on it.

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November 1990.