Analysis of Digital Predistortion in a Wideband Arbitrary Waveform Generator

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Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Analysis of Digital Predistortion in a

Wideband Arbitrary Waveform Generator

Examensarbete utfört i Kommunikationssystem vid Tekniska högskolan vid Linköpings universitet

av

Marcus Eriksson

LiTH-ISY-EX--15/4909--SE Linköping 2015

Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Linköpings tekniska högskola Linköpings universitet 581 83 Linköping

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Analysis of Digital Predistortion in a

Wideband Arbitrary Waveform Generator

Examensarbete utfört i Kommunikationssystem

vid Tekniska högskolan vid Linköpings universitet

av

Marcus Eriksson

LiTH-ISY-EX--15/4909--SE

Handledare: Per Löwenborg

SP Devices Sweden AB

Anu Kalidas M. Pillai

ISY, Linköpings universitet Examinator: Håkan Johansson

ISY, Linköpings universitet Linköping, 15 december 2015

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Avdelning, Institution

Division, Department

Avdelningen för kommunikationssystem Department of Electrical Engineering SE-581 83 Linköping, Sweden

Datum Date 2015-12-15 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN

Serietitel och serienummer

Title of series, numbering ISSN

—

LiTH-ISY-EX--15/4909--SE —

URL för elektronisk version

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-123410

Titel

Title Analys av digital predistorsion i en bredbandig signalgeneratorAnalysis of Digital Predistortion in a Wideband Arbitrary Waveform Generator

Författare

Author Marcus Eriksson

Sammanfattning

Abstract

Digital predistorsion är en signalbehandlingsteknik som används för att undertrycka oönskade distorsioner orsakade av icke-linjära effekter i elektriska system. Denna metod används i huvudsak för att linjärisera effektförstärkare i kommunikationssys-tem för att erhålla effektiva sändarkedjor men tekniken kan utan större problem även tillämpas på andra typer av icke-linjära system.

Denna uppsats undersöker i vilken utsräckning digital predistorsion kan användas för att undertrycka oönskade signaldistorsioner i en bredbandig signalgenerator. Uppsat-sen preUppsat-senterar en bakgrund som utgår ifrån teorin om icke-linjära systemmodeller, arkiteturer för predistorsion och systemidentifieringsmetoder. En kvantitativ studie i en simuleringsmiljö åtföjs av en utvärdering på ett verkligt system.

Det bästa predistorsionssystemet åstadkommer en fullständig linjärisering i testfallet med en fix tvåtonssignal. Resultaten indikerar även att det existerar ett system som linjäriserar signaler i ett frekvensområde som uppgår till hundratals MHz.

Nyckelord

Keywords Digital Predistortion, Nonlinear Systems, Linearization, Inverse System Estimation

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Sammanfattning

Digital predistorsion är en signalbehandlingsteknik som används för att under-trycka oönskade distorsioner orsakade av icke-linjära effekter i elektriska system. Denna metod används i huvudsak för att linjärisera effektförstärkare i kommu-nikationssystem för att erhålla effektiva sändarkedjor men tekniken kan utan större problem även tillämpas på andra typer av icke-linjära system.

Denna uppsats undersöker i vilken utsräckning digital predistorsion kan an-vändas för att undertrycka oönskade signaldistorsioner i en bredbandig signal-generator. Uppsatsen presenterar en bakgrund som utgår ifrån teorin om icke-linjära systemmodeller, arkiteturer för predistorsion och systemidentifieringsme-toder. En kvantitativ studie i en simuleringsmiljö åtföjs av en utvärdering på ett verkligt system.

Det bästa predistorsionssystemet åstadkommer en fullständig linjärisering i testfallet med en fix tvåtonssignal. Resultaten indikerar även att det existerar ett system som linjäriserar signaler i ett frekvensområde som uppgår till hundratals MHz.

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Abstract

Digital predistortion is a signal processing technique used to remove undesired distortions caused by nonlinear system effects. This method is predominately used to linearize power amplifiers in communication systems in order to achieve effi-cient transmitter circuits. However, the technique can readily be applied to cancel undesired nonlinear behavior in other types of systems.

This thesis investigates the effectiveness of digital predistortion in the context of a wideband arbitrary waveform generator. A theoretical foundation discussing nonlinear system models, predistortion architectures and system identification methods is complemented with a simulation study and followed by verification on a real system.

The best predistorter is able to fully suppress the undesired distortions for any fixed two-tone sinusoidal signal. Furthermore, the results indicate the existence of a wideband predistorter which yield acceptable suppression over a frequency range of several hundred MHz.

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Acknowledgements

I would like to thank Signal Processing Devices Sweden AB and especially Per Löwen-borg for providing the opportunity to undertake this task as a master’s thesis. Fur-thermore, I would like to thank Håkan Johansson and Anu Kalidas M. Pillai at Linköping University for their sensible feedback and advice during the project. Fi-nally, I would like to thank my family and my friends for their unwaivering support during these five years of my life.

Linköping, June 2015 Marcus Eriksson

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List of Abbreviations

Abbreviation Explanation

ADC Analog-to-digital Converter

API Application Programming Interface

AWG Arbitrary Waveform Generator

DAC Digital-to-Analog Converter

DC Direct Current

DPD Digital Predistortion

DLA Direct Learning Architecture

EILA Equalized Indirect Learning Architecture

FD Fractional Delay

FPGA Field Programmable Gate Array

HD Harmonic Distortion

HDL Hardware Description Language

ILA Indirect Learning Architecture

IMD Intermodulation Distortion

LMS Least Mean-Squares

LS Least Squares

LSB Least Significant Bit

LUT Look-Up Table

MLP Memoryless Polynomial

MP Memory Polynomial

PA Power Amplifier

PD Predistortion

PEM Prediction Error Method

PH Parallel Hammerstein

PW Parallel Wiener

PWH Parallel Wiener-Hammerstein

RLS Recursive Least-Squares

SDK Software Development Kit

SDR Software-Defined Radio

SFDR Spurious-Free Dynamic Range

SPD Signal Processing Devices Sweden AB (company)

THD Total Harmonic Distortion

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1

Introduction

This document presents a master’s thesis on the linearizaion of a wideband arbi-trary waveform generator (AWG) through the use of digital predistortion (DPD). The AWG constitutes the output channel of a high-speed software-defined radio (SDR) platform developed by Signal Processing Devices Sweden AB (SPD).

The research field of DPD is strongly driven by the interest to linearize power amplifiers (PAs). These devices are a crucial part of radio communication systems and inherently nonlinear. The available academic resources naturally investigate DPD in the context of PAs, allowing certain assumptions to be made as a result of the intended application.

However, this thesis aims to investigate the extent to which digital predistortion can be utilized to remove any undesired harmonic distortions without assump-tions on the intended application of the system. That is, ideally, the output signal should be distortion-free.

This chapter will introduce basic theory of nonlinear systems and explain the concept of DPD. Furthermore, the approach to solving the problem will be dis-cussed along with the outline of the thesis.

1.1 Background

Nonlinear systems and their effects have been the subject of much research over the years. Due to this high level of interest, a strong general theory on nonlinear

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2 Chapter 1. Introduction

systems exists and have aided in the development of methods to deal with their often undesired effects in, for example, electrical circuits. However, in spite of or-ganized efforts to remove the nonlinear behavior, it remains a serious problem in many applications because of inevitable physical effects. Often, it is more efficient to correct the signal after the nonlinear system, using additional circuitry, than it is to design a more linear system.

To develop such correcting methods, accurate models of nonlinear systems are needed. One idea is to base a model on what is known, i.e. using information spe-cific to the system to define nonlinear expressions relating the input and output. However, this may prove difficult in practice. It might be that no such expression exists on closed form, or that any means of computing such an expression in real time will require a disproportionately large amount of resources. A more robust way is to use behavioral modeling which is a black box approach [1, p. 3,p. 9], i.e. no information on its inner structure is required. The black box is modeled by a general nonlinear expression, which is fitted to the actual system by observing input and output values and through these, determining the parameters of the ex-pression. Hence, the problem becomes finding suitable nonlinear expressions to represent the black box.

1.1.1 Basic Nonlinear System Theory

Consider a simple nonlinear system where the input-output relation is given by y(t) ≈ α1x(t) + α2x2(t) + α3x3(t). (1.1)

If a single-tone sinusoid x(t) = A cosωt is applied, y(t) can be simplified to y(t) =α2A 2 2 + µ α1A +3α3A 3 4 ¶ cos(ωt) +α2A 2 2 cos(2ωt) + α3A3 4 cos(3ωt), (1.2) using standard trigonometric identities [2, p. 14]. The first term is a constant (DC component), the second term is the scaled fundamental signal and the remaining terms represent frequency components which are integer multiples of the input frequency. These terms are called harmonic distortions (HDs) or harmonics. A common convention is to use the labeling ‘HD2’ to denote the second-order har-monic distortion and ‘HD3’ to denote the distortions due to third-order effects and so on.

Applying a more complex input signal reveals additional sources of spectral distortion. Consider, for example, x(t) according to

x(t) = A1cos(ω1t) + A2cos(ω2t). (1.3)

The resulting output will not only contain harmonic distortions but intermodula-tion (IM) products as well. These contribute to the overall distorintermodula-tion as addiintermodula-tional

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1.1. Background 3 Fund. HD3 Fund. HD3 IM2 HD2 HD2

IM2 IM3 IM3 HD3IM3 IM3HD3

ω1 + ω2 ω2 − ω1 2ω 1 − ω2 2ω 2 − ω1 ω1 ω2 _3ω1 _3ω2 2ω 1 + ω2 2ω 2 + ω1 2ω 1 2ω 2 ω |Y |

Figure 1.1: Spectrum of the output signal y(t) in (1.1) following a two-tone test. (Input

signal as in (1.3) with A1= A2= A.)

frequency components are added to the output signal, causing the complete out-put spectrum to appear as shown in Fig. 1.1.

Applying the signal defined in (1.3) with A1= A2= A is called a two-tone test [2,

p. 25] and is a common tool used to characterize nonlinear systems.

1.1.2 Discrete-Time Representation and Spectral Folding

Finding a discrete-time representation of a continuous signal is achieved through sampling. The sampling frequency fs(sampling period T = 1/fs) relates the discrete-time signal x(n) to its analog source xa(t) as

x(n) = xa(nT ), n ∈ {0,1,2,...}. (1.4)

Central to the theory of discrete-time signals is the Nyquist-Shannon sampling theorem [3, p. 27]. The theorem makes it natural to define fs/2 as the Nyquist

frequency. If the analog signal is bandlimited to this frequency, an unambiguous representation in x(n) is possible [3, p. 27], otherwise, information is lost during the sampling process.

Figure 1.2 illustrates spectral folding by presenting the spectra for an analog signal xa(t) and its discrete-time representation x(n). The analog signal is the out-put from a second-order nonlinear system fed with a single-tone sinusoid with fundamental frequency ω0. The sampling frequency fsis sufficient to observe the undistorted signal. However, the harmonic doubles the bandwidth, resulting in a violation of the sampling theorem. This causes the second-order harmonic to fold into the observable band, creating a frequency component at π − 2ω0T . If

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the bandwidth of xa(t) exceeds the Nyquist frequency, x(n) is said to be

under-sampled. Conversely, an unambiguous discrete-time representation is obtained through oversampling or critical sampling of the analog signal.

2π T −2π T π T −π T ω Xa(j ω) ω0 − ω0 _2ω0 − 2ω 0 First Nyquist Band Second Nyquist Band 2π −2π −π π ωT X (ej ωT₎ Folded HD2

Figure 1.2: Undersampling and spectral folding.

In the case of digital predistortion, it is recommended in literature that the harmonic distortions are kept in the first Nyquist band by selecting a sampling frequency of five to seven times the signal bandwidth [4], [5]. This places signifi-cant requirements on the observation loop, most importantly the analog-to-digital converter (ADC), which creates the sequence x(n). Since high speed ADCs are ex-pensive [5], any wideband DPD scheme for high-order nonlinear systems comes at a considerable cost. However, as will be demonstrated later in this thesis, it is possible to achieve successful DPD using undersampled training signals. There-fore, the work presented herein will not take into consideration the requirement of oversampling the output signal in order to better investigate the limits of DPD.

1.2 Digital Predistortion

Predistortion is an attractive linearization method due to its power efficiency and ability to handle wideband signals [6, pp. 46-47]. Even though [6] deals with the distortion in radio frequency power amplifiers, the authors’ reasoning with respect to linearization techniques holds true for any nonlinear system.

Implementing the predistortion in the digital domain, i.e. digital predistortion, has the additional benefit of flexibility in the model. This is highly valued since, in addition to being one of the most cost effective methods of linearization [7],

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1.2. Digital Predistortion 5

the nonlinear system may drift from its initial state over time, whereby the model parameters require adjustment in order to maintain the overall linear behavior [8]. Theoretically, the predistorter mimics the inverse characteristic of the target nonlinear system. In this way, the overall system is perceived as a true linear sys-tem. Figure 1.3 illustrates the concept further.

Input Linearized output

PD Characteristic NL Characteristic

Predistorter Nonlinear_System

Figure 1.3: The concept of predistortion of a nonlinear system.

1.2.1 Simple Example of Digital Predistortion

Consider Fig. 1.4(a). The nonlinear system adds a disturbance e[x(n)], which is input dependent. The output signal is the sum of the disturbance and the input signal. Finding the inverse of this system is equivalent to estimating the distur-bance, which is simply carried out by computing

e[x(n)] = y(n) − x(n). (1.5)

Now, take the estimated disturbance with a negative sign as the correction signal added by the predistorter, eDPD[x(n)], leading to

xDPD(n) = x(n) + eDPD[x(n)] = x(n) − e[x(n)], (1.6)

in Fig. 1.4(b). In order to continue, a necessary assumption is that x(n) À e(n), i.e. that the target nonlinear system is weak. Thus, the disturbance added when the nonlinear system is fed with the predistorted signal, e[xDPD(n)], will be

approx-imately the same as before (Fig.1.4(a)). This sets up the cancellation of the distur-bance in the output signal as

yDPD(n) = xDPD(n) + e[xDPD(n)] = ± e[xDPD(n)] ≈ e[x(n)] ± = = x(n) − e[x(n)] + e[x(n)] = x(n), (1.7)

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6 Chapter 1. Introduction Σ e[x(n)] x(n) y(n) Σ eDPD[x(n)] Σ e[xDPD(n)] x(n) xDPD(n) yDPD(n)

Predistorter Nonlinear System Nonlinear System

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Figure 1.4: Predistortion of a weakly nonlinear system.

1.2.2 Why Is Digital Predistortion Needed?

From a technical point of view, the advantages of DPD are substantial. Since the behavior of any practical system deviates from the ideal, methods of correcting the undesired characteristics, thereby improving the system performance, is highly desirable.

As briefly mentioned in the introduction to this chapter, a proven area of ap-plication is in the transmitter chain in modern radio communication systems. The final system component, a power amplifier, is an inherently nonlinear device for which the trade-off is between output power and signal distortion [2, pp. 14-19]. A high output power is needed since the inverse-square law causes the signal power to drop off as ∼ _d12, where d is the distance traveled. However, this brings about

large distortions in the received signal, complicating the information decoding process. Hence, it is easy to see the technical benefits of DPD applied to these systems.

Ethical Aspects

From a societal point of view, DPD enables a more efficient use of the energy con-sumed by, for example, the base stations in any large-scale communication net-work. Furthermore, it is reasonable to consider DPD to be free from ethical aspects as long as the systems are used for nonviolent purposes.

1.3 Scope of the Thesis

This thesis will concern studies on the linearization method ‘digital predistortion’ in the context of a wideband AWG. Therefore, no additional linearization tech-niques will be studied to compare performance.

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1.4. Method 7

1.3.1 Models for Nonlinear Systems

There are many ways to model nonlinear systems, further discussed in Chapter 4. Some are purely theoretical with minimal practical application while other mod-els enjoy the benefit of practical application at the expense of accuracy in their description of the target nonlinear system.

In this thesis, several models will be investigated theoretically. However, some will be deemed more promising than others in relation to the task at hand. As such, not all of them will be evaluated using real input data.

Additionally, in modeling the nonlinear system, no case-specific approaches will be considered, i.e. the system will always be treated as a black box.

1.3.2 Predistortion Architectures

Regarding predistortion architectures, a theoretical discussion on the two promi-nent suggestions in literature will be held. As a consequence of this discussion, only the indirect learning architecture (ILA) will be evaluated in practice. See Chap-ter 5 for further details.

1.3.3 Parameter Estimation Algorithms

Once a nonlinear expression to use is decided, the parameters need to be esti-mated using data from the target nonlinear system. There are several well-known algorithms that can be employed for this purpose. Furthermore, in the interest of future work, it is valuable to evaluate both online and offline approaches to param-eter estimation. Chapter 6 contains derivations and discussions on the estimation methods used in this thesis.

1.4 Method

Based on the available literature, several DPD techniques will be suggested and their theoretical benefits and drawbacks discussed. Attractive predistorter models are evaluated in a simulation environment usingMATLAB.

Once this analysis has yielded a set of DPD approaches which warrants further investigation, a quantitative analysis on the real system will be performed. Algo-rithms which can readily be converted into software running on the target system will be formulated and tested using SPDs custom application programming inter-face (API). The API allows for simplified hardware access directly fromMATLAB,

which results in a fast work flow and easy signal manipulation.

The data will be interpreted employing commonly used performance metrics which facilitate the comparison of the different DPD methods.

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1.5 Thesis Outline

The outline of the thesis is as follows:

• Chapter 2 presents the performance metrics used in this thesis to facilitate the comparison of the DPD methods.

• Chapter 3 presents a high-level description of the target system SDR14 and discusses important considerations in relation to DPD.

• Chapter 4 presents an overview of nonlinear system models and provides reasoning for the choice of evaluated models.

• Chapter 5 presents the two major predistortion architectures used in research. A discussion on the two architectures and motivation behind the choice of the ILA is provided.

• Chapter 6 presents the parameter estimation methods. A derivation of two online iterative algorithms is provided along with goal functions for con-strained optimization.

• Chapter 7 combines the parameter estimation methods with the chosen pre-distorter models in order to describe the system identification procedure. • Chapter 8 presents the results from evaluating the chosen predistorter

mod-els in a simulation environment inMATLAB.

• Chapter 9 presents the results from evaluating the chosen predistorter mod-els on the SDR14 system.

• Chapter 10 summarizes the results in this thesis and provides conclusions on DPD in the context of the SDR14 device. Some thoughts on future work are also provided.

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2

Performance Metrics

In its most general sense, performance metrics are measures used to qualitatively compare two items that perform the same task. In the case of digital predistor-tion, the main objective is to achieve an output signal free from spectral distortion. Therefore, relevant performance metrics focus on the quality of the output signal. In this chapter, x(n) is a uniformly sampled discrete-time signal with sampling period T , that is

x(n) = xa(nT ), (2.1)

where xa(nT ) is continuous. The Fourier transform of this signal is

X (ej ωT) = X∞ n=−∞x(n)e

−j nωT_. _(2.2)

2.1 Spurious-Free Dynamic Range

The spurious-free dynamic range (SFDR) is a measure of the signal quality ob-tained through spectral analysis. It is a measure of the difference in energy be-tween the fundamental tone and the second most prominent spectral component, either caused by noise or spectral distortion. Hence, the unit is [dBc], meaning decibels below carrier. Figure 2.1 illustrates the measure for a third-order nonlin-ear system excited by a single-tone input signal and the SFDR is given by

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10 Chapter 2. Performance Metrics f0 2f0 3f0 f |X | [dB] HD2 HD3 Fundamental tone SFDR [dBc]

Figure 2.1: Illustration of the SFDR performance metric. A single-tone sinusoid is passed

through a third-order nonlinear system.

SFDR = 20log10|X (ej 2πf0T)| − max_{f 6=f}

0

h

20log₁₀|X (ej 2πf T)|i. [dBc] (2.3) The SFDR represents a value of the effective signal quality. A value of 0 dBc signifies that the fundamental tone is unable to be distinguished from other dis-tortions in the signal. The value represent the ‘worst case’ assumption which is important from a design perspective.

2.2 Total Harmonic Distortion

The total harmonic distortion (THD) is a measure of the energy in the harmonic distortions in relation to the energy in the fundamental tone. The measure can be interpreted as demonstrating how closely a signal resembles a pure sine save. Since it describes the relation between two values, the unit commonly used is deci-bels [dB].

Equation (2.4) presents the general expression where the sum of the root mean-square (RMS) values of the harmonic distortions are compared to the RMS value of the fundamental tone.

THD = v u u tP∞n=2VRMS,n2 V2 RMS,1 (2.4) The THD is used to characterize nonlinear systems and provides a means to compare performance between systems. Although not the most relevant parame-ter in many cases [9, p. 14], it is often specified.

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2.3. Mean-Square Error 11

2.3 Mean-Square Error

Comparing two uniformly sampled signals in the time-domain is commonly car-ried out by computing the sample-wise error and then forming the mean-square error (MSE).

Let x(n) and y(n) be two discrete-time signals. If N samples are observed, the MSE is computed as

MSE =_N1 N −1X

n=0(x(n) − y(n))

2 _(2.5)

2.3.1 Normalized Mean-Square Error

If multiple tests are carried out using different x(n), problems arise when the indi-vidual MSE values are compared since the value does not contain any information on the input signal specifically. The normalized mean-square error (NMSE) ex-ists to address this problem. The error signal energy is normalized using the input signal energy, yielding a value suitable for comparison between different input sig-nals. NMSE = PN −1 n=0(x(n) − y(n))2 PN −1 n=0x2(n) (2.6) Note that the_N1-factors from the mean-computations cancel.

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3

Target System Description

This chapter will give an overview of the SDR14 and SDR14-DC platforms devel-oped by SPD. It is the purpose of this thesis to investigate the potential of digital predistortion in relation to this device family. The general characteristics of the system will be discussed along with specific considerations.

3.1 SDR14

The SDR14 is a device with a wide range of applications. It is suitable for software-defined radio (SDR), wireless communication or as a test and measurement equip-ment.

The platform combines a 14-bit channel digitizer (input) with 14-bit dual-channel signal generation (output). The arbitrary waveform generator (AWG) sists of a 500 MB waveform memory, control logic and a digital-to-analog con-verter (DAC) operating at 1600 MS/s. Each input channel consists of an analog-to-digital converter (ADC) operating at 800 MS/s with an input stage bandwidth of 550 MHz. The captured waveform is stored in another dedicated waveform mem-ory of 500 MB. The system is controlled by an FPGA which is tasked with directing the data flow and applying both rudimentary and advanced signal processing.

Figure 3.1 presents a simplified block diagram of the system containing all parts which are of significance to this thesis. The figure shows one of the two out-put channels connected to one of the two inout-put channels by means of a cable. This

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14 Chapter 3. Target System Description

is necessary because there are no feedback paths built in to the system that allow observation of the output waveform. Hence, to create an observation loop, the output channel to be linearized is physically connected to one of the input chan-nels. Waveform Memory DAC 1600 MS/s 14-bit Waveform Memory ADC 800 MS/s 14-bit M A T L A B Output Channel Input Channel Custom API Control Logic P h ys ic al Fe ed b ac k C o n n ec ti o n

Figure 3.1: Block diagram of the SDR14 platform showing one output channel and one input

channel.

A custom application programming interface (API) is provided by SPD which have support for C/C++ andMATLAB. The latter allows for rapid implementation of

algorithms and easy data manipulation and will therefore be utilized for the work in this thesis. The performance of the specific unit used for the experimental veri-fication is presented in Section 9.1.

3.2 SDR14-DC

There exists a modified version of the SDR14 platform with a DC-coupled output which includes a power amplifier to buffer the output signal and increase the out-put power. Except for the extended outout-put stage which add stronger nonlinear effects, it operates in an identical manner to the default unit: SDR14. Figure 3.2 illustrates a simplified block diagram of one of the output channel and one input channel, showing only the parts significant to this thesis.

3.3 Discussion

Digital predistortion applies to the output channel of the SDR14 and SDR14-DC systems. However, since the output has to be observed to gain information on the nonlinear behavior present at the output, the act of observing may introduce ad-ditional disturbances. Figure 3.3 illustrates the inevitable effects of observation.

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3.3. Discussion 15 Waveform Memory DAC 1600 MS/s 14-bit PA Waveform Memory ADC 800 MS/s 14-bit M A T L A B Output Channel Input Channel Custom API Control Logic P h ys ic al Fe ed b ac k C o n n ec ti o n

Figure 3.2: Block diagram of the SDR14-DC platform. A power amplifier is connected in

series with the DAC, forming the output stage.

Since the goal is to achieve a linearized output waveform, ideally, the system to observe only consists of the output stage. When measuring equipment is intro-duced (cable and input channel), the black box boundary is inevitably extended and the observed output is effectively redefined to the ADC output. While this may seem like a critical defect, the assumption is generally that the input channel is near-perfect [10, pp. 89–90], placing high linearity requirements on the ADC. In Chapter 9, when the SDR14 device used in this thesis is characterized, it will be demonstrated that the nonlinear effects caused by the DAC dominates the overall behavior.

The system architecture places certain hard requirements on the predistortion scheme. Noting that the feedback path is created manually and will not remain during normal operation, it is clear that the predistorted system must run in open-loop. The implications of this is further discussed in Section 5.3.

The predistorter can be placed in different locations in the output channel data path. Three cases are presented in Fig. 3.4. One method is to store the manually predistorted waveform in memory (Fig. 3.4(a)), completely avoiding the need of modifying the system or the tools. However, this is clearly not a viable long-term solution because of the work involved. To address this issue, two suggestions more suitable for permanent implementations are given in Fig. 3.4(b)-(c). The main dif-ference between the two methods is that one predistorts the data in software, by extending the already existing API, and one in hardware, synthesizing a dedicated DPD block into the FPGA design. Clearly, the software approach will be more flex-ible by nature, allowing the use of scalable models without penalty. Any hardware approach will require a predetermined model, where even supporting dynamically updated coefficients may prove costly. The benefits of a hardware approach is that the signal processing is carried out on a sample-per-sample basis. This is

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advan-16 Chapter 3. Target System Description Waveform Memory DAC 1600 MS/s 14-bit Waveform Memory ADC 800 MS/s 14-bit M A T L A B Output Channel Input Channel Custom API Control Logic P h ys ic al Fe ed b ac k C o n n ec ti o n Target system

Actually observed system

Figure 3.3: Block diagram of the SDR14 platform detailing the desired nonlinear system to

observe from a DPD perspective and the actually observed system.

tageous since the software approach would process the entire waveform before sending any data to the device memory, thereby increasing the system delay no-ticeably for large waveforms.

An important observation is the fact that the output data rate and the input data rate differ by a factor of two. Because of the predistortion architectures em-ployed (further detailed in Chapter 5), the predistorter will inherently operate at the lower sampling rate of 800 MS/s. When this predistorter is tasked with han-dling the data at a higher sampling rate, there are a few options to consider. One option is to generate the input waveform at the lower sampling rate, predistort and subsequently interpolate the resulting signal by a factor of two. Another option is to generate the input waveform at the higher sampling rate and use two interleaved predistorters in order to construct the predistortion signal at 1600 MS/s.

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3.3. Discussion 17 Waveform Memory DAC 1600 MS/s 14-bit Predistorter MATLAB Custom API Control Logic Waveform Memory DAC 1600 MS/s 14-bit Predistorter

MATLAB Custom API

Predistorter DAC 1600 MS/s 14-bit Waveform Memory MATLAB Custom API (a) (b) (c)

Figure 3.4: Block diagram showing different placements of the predistorter in the output

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4

Nonlinear System Models

Nonlinear system models are essential in studying and working with systems demo-nstrating nonlinear behavior. By identifying a nonlinear system model, its parame-ters are determined in such a way that the input-output characteristic mimics that of a target system, for example a DAC or a PA. The approximated system lends it-self more naturally to theoretical investigation since the internal structure, i.e. the input-output transfer function, is known. In this thesis, nonlinear system models are an integral part since they are used to approximate the inverse characteristic of the target nonlinear system. In other words, the models are the ‘predistorters’.

Throughout the figures in this section symbols such as (·)kand | · | will be used to denote the mathematical operations performed by a block on its input signal. For example, | · | represent the absolute value of the input signal and (·)kthe input signal raised to the power k.

4.1 Volterra Systems

The most general expression for nonlinear systems is the Volterra series, where the discrete-time output y(n) is determined as

y(n) =X∞

k=1

Hk{x(n)}, (4.1)

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20 Chapter 4. Nonlinear System Models where Hk{x(n)} = ∞ X q1=−∞ ··· X∞ qk=−∞ hk(q1, q2,··· ,qk)x(n − q1)x(n − q2)···x(n − qk). (4.2) The series can be seen as a generalization of the standard theory of linear sys-tems. The functions hk(q) are called Volterra kernels and can be interpreted as

k-dimensional discrete-time impulse responses of the system [9, p. 64]. For exam-ple, H1{x(n)} = ∞ X q1=−∞ h1(q1)x(n − q1) (4.3)

is the first-order response of the system, which is easily recognizable as the stan-dard convolution sum from the theory of linear systems. The second-order re-sponse of the system is

H2{x(n)} = ∞ X q1=−∞ ∞ X q2−∞ h2(q1, q2)x(n − q1)x(n − q2), (4.4) and so on.

Due to the infinite nature of the Volterra series, it enables arbitrary systems to be modeled with high precision, though some restrictions apply [9, p. 65]. Clearly, this general theory is challenging to use in practice, mainly because of the cum-bersome task of estimating the Volterra kernels. Therefore, there exist many alter-native models, all of which make efforts to reduce the complexity of the Volterra series at the cost of generality.

4.2 Concept of Memory

The concept of memory is important in nonlinear systems. A system is said to be memoryless if the present output signal value is independent of previous input signal values. Consequently, for systems with memory, the current output signal value is dependent of previous input signal values. It is with respect to this prop-erty, among others, the different simplified models reduce their complexity.

4.2.1 Memoryless Polynomial

The memoryless polynomial (MLP) is a straightforward construction of a nonlin-ear expression without memory. It is defined according as

y(n) = α1x(n) + α2x2(n) + ··· + αKxK(n) = K X k=1

αkxk(n), (4.5) where αkare coefficients, x(n) is the input and y(n) is the output. Clearly, this ex-pression only requires the current value of the input signal to compute the output

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4.3. Hammerstein Systems 21

at any given time. Hence, it is memoryless. To allow compact notation throughout this thesis, (4.6) provides a convenient definition.

ΓK{x(n)},        x(n) x2(n) .. . xK(n)        (4.6)

Due to its lack of memory terms, the MLP may prove insufficient when used as a model for general nonlinear systems. Even though it may give sufficient perfor-mance in instances where memory effects are less prominent, its use cases are sig-nificantly increased when coupled with a simple system with memory, for example an LTI-system.

4.2.2 Systems With Memory

In systems with memory, the output at any given time instance depends on the current as well as previous input samples. The number of previous input samples required to compute the output is known as the memory depth of the system. Dig-ital filters are good examples of well behaved systems with memory with arbitrary depth, determined by the filter order.

4.3 Hammerstein Systems

Hammerstein systems is a class of nonlinear system models consisting of two blocks: a nonlinear memoryless system and a linear time-invariant (LTI) system with mem-ory, connected in series, as shown in Fig. 4.1 [11, p. 19].

Nonlinear

System LTI System

Memoryless With memory

Input Output

Figure 4.1: The Hammerstein nonlinear system model. A nonlinear memoryless system is

followed by an LTI system with memory.

Any system meeting the criteria of any of the two blocks can be used to con-struct a Hammerstein system. From an implementation standpoint, low complex-ity is desired, narrowing the choice of actual subsystems to a common few. Popular choices are simple memoryless polynomials for the first sub-system and digital fil-ters (mostly FIR-filfil-ters) for the LTI system.

One of the main benefits of the Hammerstein system model is that the model parameters (coefficients of the two sub-blocks) appear linearly at the output. This

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22 Chapter 4. Nonlinear System Models

is a desired property since it allows a broader class of estimation algorithms to be used to determine their values.

4.3.1 Single-Branch Hammerstein System

A single-branch structure is the simplest form of the system model. Figure 4.2 presents a block diagram of the model using a memoryless polynomial of order K to represent the memoryless nonlinear system. An FIR system of length Q is used to represent the LTI system.

P kαk(·)k h(q) Order K MLP Length Q FIR filter x(n) z(n) y(n)

Figure 4.2: A single-branch Hammerstein system consisting of a memoryless polynomial

followed by an FIR filter.

In total there are K +Q parameters in the model and by the definitions made in Fig. 4.2 the output can be expressed in terms of the model parameters and the input as follows: y(n) =Q−1X q=0h(q)z(n − q) = ³ h(0) h(1) ··· h(Q − 1)´ | {z } ,hT        z(n) z(n − 1) .. . z(n −Q + 1)        (4.7) z(n) =XK k=1 αkxk(n) = ³ α1 α2 ··· αK ´ | {z } ,αT        x(n) x2(n) .. . xK(n)        =αTΓK{x(n)} (4.8) Combining (4.7) and (4.8) we have the following input-output relation

y(n) =hT       αTΓK{x(n)} αTΓK{x(n − 1)} ... αTΓK©x(n −Q + 1)ª      , (4.9)

from which it is clear that the system is linear in the model parametershandα. This allows the system model to be restructured into a one-block model with KQ

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4.3. Hammerstein Systems 23

coefficients. This implies that although the resulting system performs the same task, its inner structure does not have the same identifiable system blocks (e.g. digital filters) as the non-combined model, which may limit effective implementa-tion.

4.3.2 Parallel Hammerstein Systems

Generalizing the single-branch Hammerstein model results in the parallel merstein model. Figure 4.3 presents a block diagram where M single-branch Ham-merstein models are combined.

Nonlinear

System LTI System

x(n) y1(n)

Nonlinear

System LTI System

y2(n)

.. .

Nonlinear

System LTI System

yK(n)

y(n)

Σ Σ

Figure 4.3: A parallel Hammerstein structure consisting of M branches.

If the MLP model is used as the nonlinear system in each branch, redundancy becomes a problem. Since each nonlinear block may have an arbitrary order, sev-eral branches may act on the same input data, only to sum the final result. The same end result could easily be achieved using a single branch, which is why a more efficient approach is to reduce the MLP to a single term, unique to that branch. The resulting structure effectively separates the terms in the MLP and passes each one through a unique LTI system. A model using such reductions is presented in Fig. 4.4. and will be referred to as the parallel Hammerstein model from here on.

In the reduced model, the linear branch is left undisturbed, which means that the model is unable to compensate for any linear errors. However, adding a filter to the branch will extend the predistorter’s functionality to further include com-pensation of linear errors and even strong memory effects. This filter will not be considered in this thesis, as the purpose is to cancel harmonic and intermodula-tion distorintermodula-tion and leave the fundamental signal unaffected.

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24 Chapter 4. Nonlinear System Models x(n) (·)2 _h₂_(q) Length Q2 z2(n) y2(n) (·)K _h_K_(q) Length QK zK(n) yK(n) y(n) Σ Σ

Figure 4.4: A parallel Hammerstein structure consisting of K branches, each one composed

of a unique nonlinear block followed by an FIR filter of length Q_kwith impulse response

h_k(q).

The model consists of PK_k=2Qk parameters, where Qk is the filter length in branch k. The input-output relation is given by

y(n) = x(n) +XK k=2 yk(n), (4.10) yk(n) = Q_Xk−1 q=0 hk(q)xk(n − q), k 6= 1. (4.11) Another nonlinear system model which make additional simplifications on the parallel Hammerstein structure is the memory polynomial system model [12], pre-sented in Section 4.6.

4.4 Wiener Systems

Wiener systems is a class of nonlinear system models consisting of the same two fundamental blocks as the Hammerstein system, discussed in Section 4.3, although in this case, the order of the sub-systems are reversed [11, p. 19], as presented in Fig. 4.5

Just as with Hammerstein systems, Wiener systems can be constructed using any sub-system meeting the criteria. In practical applications, where low complex-ity is desired, popular choices are MLPs to represent the nonlinear memoryless system and various digital filters to represent the LTI system.

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4.4. Wiener Systems 25

LTI System Nonlinear

System

With memory Memoryless

Input Output

Figure 4.5: The Wiener nonlinear system model. An LTI system with memory is followed by

a nonlinear memoryless system.

As an intuitive opposite to the Hammerstein system, an unsurprising aspect of the Wiener system model is that the model parameters no longer appear linearly at the output, reducing the flexibility in which extraction methods are applicable. However, algorithms which achieve comparable performance still exist, which will be shown in later chapters.

4.4.1 Single-Branch Wiener Systems

The obvious way to use the Wiener model is in a single-branch structure. Here, an FIR filter of length Q is chosen to represent the LTI system and an MLP of order K is chosen to represent the nonlinear memoryless system. Figure 4.6 presents the refined model. w(q) P_kαk(·)k Length Q FIR filter Order K MLP x(n) z(n) y(n)

Figure 4.6: A single-branch Wiener system consisting of an FIR filter of length Q in series

with a memoryless polynomial of order K .

The system model contains K +Q model parameters and the input-output re-lation is given by y(n) =XK k=1 αkzk(n) = ³ α1 α2 ··· αK ´ | {z } ,αT        z(n) z2_(n) .. . zK(n)        =αTΓK{z(n)}, (4.12)

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26 Chapter 4. Nonlinear System Models z(n) =Q−1X q=0w(q)x(n − q) = ³ w(0) w(1) ··· w(Q − 1)´ | {z } ,wT        x(n) x(n − 1) .. . x(n −Q + 1)        . (4.13)

Combining (4.12) and (4.13) yields

y(n) =αTΓK              wT        x(n) x(n − 1) .. . x(n −Q + 1)                     . (4.14)

Since the Γ-function is nonlinear by definition (see (4.6)) it is clear that the FIR filter coefficientswappear in a nonlinear manner at the output.

4.4.2 Parallel Wiener Systems

Generalizing the standard single-branch Wiener model results in the parallel Wiener system model. Figure 4.7 presents a block diagram of the structure combining M single-branch Wiener models.

System

x(n) y1(n)

LTI System Nonlinear_System y2(n)

.. .

LTI System Nonlinear_System yK(n)

y(n)

Σ Σ

Figure 4.7: A parallel Wiener system model with M branches.

Similar to the case of parallel Hammerstein system models, using MLPs for the nonlinear expressions causes a redundancy issue since it is possible for several branches to act on the same input data. To avoid this, the strategy of pruning the

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4.5. Wiener-Hammerstein Systems 27 x(n) w2(q) α2(·)2 Length Q2 z2(n) y2(n) wK(q) αK(·)K Length QK zK(n) yK(n) y(n) Σ Σ

Figure 4.8: A parallel Wiener system model with K branches. An FIR filter of length Q_kand

a branch-specific nonlinear term constitute each branch k ∈ {2,...,K }. The linear branch is left unaffected.

MLPs such that each branch contains a unique nonlinear term is used once more, resulting in the architecture in Fig. 4.8.

The model consists ofPK_k=2Qk parameters, where Qk is the filter length in branch k. The input-output relation is given by

y(n) = x(n) + K X k=2 yk(n), (4.15) yk(n) = αk Ã_Q k−1 X q=0wk(q)x(n − q) !k , k 6= 1. (4.16)

4.5 Wiener-Hammerstein Systems

Wiener-Hammerstein system models is another class of nonlinear system mod-els consisting of a Wiener-type front-end together with a Hammerstein-type back-end, presented in Fig. 4.9.

Wiener-Hammerstein systems can be constructed using any sub-systems meet-ing the criteria (e.g. FIR filters or MLPs). Due to the nonlinear system followmeet-ing the initial LTI system, the model parameters of the latter will appear nonlinearly at the output, reducing the flexibility of parameter extraction methods, comparable to the case for Wiener systems.

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System LTI System

With memory Memoryless With memory

Input Output

Figure 4.9: The Wiener-Hammerstein system model.

4.5.1 Single-Branch Wiener-Hammerstein Systems

The straight-forward way of utilizing the Wiener-Hammerstein system model is in a single-branch structure. In Fig. 4.10, two FIR filters of lengths QW and QH are chosen to represent the two LTI systems. The filters encapsulate an MLP of order K , used to represent the memoryless nonlinear system. In total, the model con-tains K +QW+QHmodel parameters which relate the input to the output through (4.17) – (4.19). w(q) P_kαk(·)k h(q) Length QW FIR filter Order K MLP Length QH FIR filter x(n) z1(n) z2(n) y(n)

Figure 4.10: The Wiener-Hammerstein system model in a single-branch structure.

y(n) = Q_XH−1 q=0 h(q)z2(n −q) = ³ h(0) h(1) ··· h(QH− 1) ´ | {z } ,hT        z2(n) z2(n − 1) .. . z2(n −QH+ 1)        (4.17) z2(n) = K X k=1 αkzk1(n) = ³ α1 α2 ··· αK ´ | {z } ,αT        z1(n) z2₁(n) .. . zK 1(n)        =αTΓK{z1(n)} (4.18) z1(n) = QW_X−1 q=0 w(q)x(n − q) = ³ w(0) w(1) ··· w(QW− 1) ´ | {z } ,wT        x(n) x(n − 1) .. . x(n −QW+ 1)        (4.19)

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4.5. Wiener-Hammerstein Systems 29

Combining (4.17) – (4.19), the output can be expressed in terms of the input. However, to keep the expressions presented in this thesis on a compact form, it is not given explicitly.

4.5.2 Parallel Wiener-Hammerstein Systems

In the same manner as for the Hammerstein and Wiener system models, the single-branch Wiener-Hammerstein model can be generalized as well. Using K single-branches and restricting the nonlinear blocks to a branch-specific term from the memory-less polynomial results in the the architecture in Fig. 4.11.

x(n) w2(q) _(·)2 _h₂_(q) Length QW,2 FIR filter Length QH,2 FIR filter z1,2(n) z2,2(n) y2(n) wM(q) (·)M _h_M_(q) Length QW,M FIR filter Length QH,M FIR filter z1,M(n) z2,M(n) yM(n) y(n) Σ Σ

Figure 4.11: The parallel Wiener-Hammerstein system model used in this thesis. Each

non-linear branch consist of two FIR filters of arbitrary lengths encapsulating a branch-specific nonlinear term.

The model consists ofPK_k=2(QW,k+QH,k) parameters where QW,kand QH,kare the filter lengths in branch k, as shown in Fig. 4.11. Any scaling parameter placed after the nonlinear term (similar to αkin the parallel Wiener system model) can be seen as a simple scaling to the filter coefficients hkand therefore be ‘absorbed’ by the filter. The parameters relate the output y(n) to the input x(n) through

y(n) = x(n) +XK k=2 yk(n), (4.20) yk(n) = QH,k_X−1 qh=0 hk(qh) Ã_Q_W,k₋₁ X qw=0 wk(qw)x(n − qw− qh) !k , k 6= 1. (4.21)

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4.6 Memory Polynomial Systems

The memory polynomial system model was proposed in [12] and has since be-come a popular model for use in digital predistortion (a continued discussion can be found in Section 4.7). It is similar to the parallel Hammerstein model, seen in Fig. 4.3, in its structure, differing only in the nonlinear expressions.

x(n) α1(·) h1(q) Length Q1 FIR filter α2(·)|·| h2(q) Length Q2 FIR filter z2(n) y2(n) αK(·)|·|K −1 hK(q) FIR filter Length QK zK(n) yK(n) y(n) Σ Σ

Figure 4.12: The memory polynomial system model. Each branch consist of a unique

non-linear term, followed by an FIR filter of length Q.

Starting from Fig. 4.12, it is possible to derive the output y(n) expressed in terms of the input x(n) as

y(n) = PK k=1, PQk−1 q=0 ,αkhk(q) | {z } ,gk(q) x(n − q)|x(n − q)|k−1 = PK_k=1PQk−1 q=0 gk(q)x(n − q)|x(n − q)|k−1. (4.22)

Two aspects of the model are worthy of noting. Firstly, since the model assumes a Hammerstein structure, the output is linear in the model parameters, denoted by gk(q) in (4.22). It is common practice to let the scaling parameter, αk, per-taining to the nonlinear expression in branch k scale the filter impulse response directly to reduce the number of parameters in the model without any loss of

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gen-4.7. Discussion 31

erality [13] [14]. In total, the model will consist of KQ parameters representing the filter coefficients.

Secondly, because of the specific nonlinear expression used, the model is un-able to recreate even-order harmonics and thus unsuitun-able for predistortion pur-poses if the goal is to indiscriminately cancel all harmonic and intermodulation distortions. Why this is not seen as a critical problem in literature is discussed in Section 4.7. However, in the context of this thesis, this makes the memory polyno-mial model highly unattractive.

4.7 Discussion

To determine which predistorter model is best suitable for DPD, it is important to first gain an understanding of the typical behavior of the system which is to be linearized. Later it will be shown that this step is crucial to avoid overdesigning the predistorter, either in terms of order or memory depth, any of which could lead to an overall system with higher distortion than before.

An intuitive result in choosing a predistorter model is the fact that the Ham-merstein and Wiener model acts as each other’s inverse. That is, a Wiener-type system is correctly predistorted by a Hammerstein predistorter and vice versa, as shown in Fig. 4.13 [15, pp. 82-83].

Nonlinear

System LTI System LTI System

Nonlinear System Hammerstein Predistorter Wiener System

Input Output

LTI System Nonlinear System

Nonlinear

System LTI System Wiener Predistorter Hammerstein System

Input Output

Figure 4.13: Illustration of correct predistorter choice.

However, real systems that are interesting from a predistortion perspective are rarely true Wiener or Hammerstein systems, for example DACs or PAs. Their be-havior can, on the other hand, be mimicked with varying results using the nonlin-ear models presented in this chapter. Obviously, the model which is able to best approximate the system behavior will be chosen to characterize the system from an engineering standpoint.

Intuitively, the existance of a single set of filter coefficients able to correct all types of undesired system behavior in practice, is unlikely. A more reasonable ap-proach is to employ a specific filter for each nonlinear term. This is the major benefit of the system models with parallel structures over the single-branch coun-terparts. Hence, this thesis will focus on the parallel Hammerstein, Wiener and Wiener-Hammerstein system models.

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As with all engineering disciplines, there are trade-offs to consider. Depend-ing on how much information is known on the intended use cases, system com-ponents and overall distortion behavior, it is possible to optimize the predistorter model with respect to performance and computational complexity. The cost of this is, unsurprisingly, loss of generality since the predistortion structure is then specif-ically designed for a narrow set of use cases. Such optimizations may be achieved by choosing the model depending on information on the signal (e.g. bandwidth), information on the system (e.g. order of dominant nonlinear behavior) or infor-mation on intended use (e.g. communication systems).

Since highly linear PAs are essentially a requirement in today’s communica-tion systems [14], [16], finding efficient linearizacommunica-tion methods is the major driving force behind field of DPD. Therefore, the main focus is reducing the in-band distor-tion as well as any distordistor-tion interfering with adjacent channels. These distordistor-tion components are largely caused by odd-order nonlinear behavior, often resulting in models ignoring the even-order terms [17], [18], [19]. Additionally, the fact that the signal processing in modern radio communication systems operates with a base-band representation of the signals [20, p. 5], is the reason behind the popularity of the memory polynomial model.

However, this thesis is concerned with investigating to what extent a black-box predistortion scheme can be used to suppress all spectral distortions in the output signal without making assumptions on the signal or the intended use of the system. Therefore, the memory polynomial model will not be investigated further, in spite of its strong presence in the field of DPD.

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5

Predistortion Architectures

To construct the predistorted system once the model has been chosen, there are two widely used architectures to consider: the indirect learning architecture (ILA, Section 5.1) or the direct learning architecture (DLA, Section 5.2) [8]. Both of these methods have advantages and drawbacks which should be weighed against each other when deciding which is most suitable for predistorting the target system.

5.1 Indirect Learning Architecture

The indirect learning architecture is commonly used in DPD systems because of its simplicity in estimating the model parameters [21], [22]. Figure 5.1 presents a schematic view of the architecture.

The training phase of the ILA works by feeding the unchanged input signal x(n) through the target nonlinear system, i.e. bypassing the predistorter. The obtained output is fed through the inverse system model and an estimated inverse ˆxDPD(n)

is computed. The parameters in the inverse system model are iteratively updated using a suitable estimation algorithm until the error signal e(n) converges, ideally towards zero.

Once the model parameters have been estimated, the training branch is shut down [8] and the parameter set is copied to the predistorter, which has an iden-tical inner structure. This means that once the training phase is completed, the predistorted system is run in open loop.

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34 Chapter 5. Predistortion Architectures x(n) xDPD(n) ˆxDPD(n) y(n) e(n) −

Predistortion Nonlinear_System

Inv. System Model Estimation Algorithm Σ b b

Training Phase Bypass

Figure 5.1: The indirect learning architecture for digital predistortion.

The ILA has the advantage of lower computational complexity in relation to its counterpart, the DLA [23]. The architecture also proves exceedingly simple to implement in real systems without hardware modification because of its open-loop nature.

In theory, the ILA have drawbacks that may greatly undermine successful pre-distortion, as explained below. However, in practice the impact of these drawbacks depend largely on the characteristics of the target nonlinear system, shown in later chapters.

Most importantly, since the inverse is estimated using the output y(n) as input to the inverse system model and the error is computed using the input x(n) (dur-ing the train(dur-ing phase), effectively, the post-inverse is estimated. The fundamental idea behind DPD is to use an inverse system model placed before the target system in the signal path. Hence, the desired inverse is ideally the pre-inverse. By copying the parameters from the inverse system model to the predistorter the assumption that the post-inverse is equal to the pre-inverse is made. Concerning nonlinear systems, commutation is not guaranteed, which is why a post-inverse model used for predistortion purposes may give inadequate performance [22].

Since the output signal will be noisy in practice, yet another undesired effect caused by using y(n) as the input to the inverse system model is the presence of a bias in the estimated parameter values [23]. This noise bias is both dependent on the DPD model as well as on the noise performance of the measurement setup [24] as well as the noise conditions during the experimental measurement [23].

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5.2. Direct Learning Architecture 35

5.2 Direct Learning Architecture

The direct learning architecture is less widespread in DPD systems than its coun-terpart due to one crucial implication concerning the parameter estimation, dis-cussed later in this section. First, consider the schematic view of the architecture, presented in Fig. 5.2, and its operation.

x(n) xDPD(n) y(n)

r (n) e(n)

Predistortion Nonlinear_System

Estimation

Algorithm Σ ₋

Figure 5.2: The direct learning architecture for digital predistortion.

The DLA works by feeding the predistorted input samples through the non-linear system and computing an error with the help of a reference signal. This signal represents what the desired output should be. For example, if a unity-gain distortion-free system is desired, r (n) is set to the input signal x(n).

The parameters in the predistorter are iteratively updated using a suitable es-timation algorithm until the error signal converges, ideally towards zero. This im-plies that the output sample at time instance (n + 1)T is subject to the parameter update at time instance nT , which may be problematic in practice, as discussed in Section 5.3. Additionally, inherent to the architecture is the fact that the predistor-tion is running in closed-loop since the training branch is never disabled.

One of several advantages of the DLA is that the actual pre-inverse is estimated. Furthermore, in practical applications, the DLA offers yet another advantage be-cause both the estimation algorithm and predistorter are fed with the input signal x(n), which can be considered noise-free, and will therefore not bias the final val-ues.

The drawbacks of the DLA are centered around the need of a fully identified target system before any inverse can be estimated [15, p. 90,p. 99]. That is, the nonlinear system will first need to be approximated using a suitable model. Once the parameter set is estimated, it is possible to apply the DLA to estimate the pre-inverse of the said system.

To summarize, in practice, the DLA will find the pre-inverse to an approxima-tion of an actual system. Therefore, the first step in designing a predistorter using a DLA approach is to first identify a sufficiently accurate forward model of the target system. Once complete, the pre-inverse is estimated and used as the predistorter to the actual system. These steps are required because the error signal e(n) de-pends directly on the predistortion parameters. Without making assumptions on

Analysis of Digital Predistortion in a Wideband Arbitrary Waveform Generator

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Analysis of Digital Predistortion in a

Wideband Arbitrary Waveform Generator

Analysis of Digital Predistortion in a

Wideband Arbitrary Waveform Generator

Examensarbete utfört i Kommunikationssystem

vid Tekniska högskolan vid Linköpings universitet

av

Sammanfattning

Abstract

Acknowledgements

Contents

List of Abbreviations

1

Introduction

1.1

Background

1.1.1

Basic Nonlinear System Theory

1.1.2

Discrete-Time Representation and Spectral Folding

1.2

Digital Predistortion

1.2.1

Simple Example of Digital Predistortion

1.2.2

Why Is Digital Predistortion Needed?

1.3

Scope of the Thesis

1.3.1

Models for Nonlinear Systems

1.3.2

Predistortion Architectures

1.3.3

Parameter Estimation Algorithms

1.4

Method

1.5

Thesis Outline

2

Performance Metrics

2.1

Spurious-Free Dynamic Range

2.2

Total Harmonic Distortion

2.3

Mean-Square Error

2.3.1

Normalized Mean-Square Error

3

Target System Description

3.1

SDR14

3.2

SDR14-DC

3.3

Discussion

4

Nonlinear System Models

4.1

Volterra Systems

4.2

Concept of Memory

4.2.1

Memoryless Polynomial

4.2.2

Systems With Memory

4.3

Hammerstein Systems

4.3.1

Single-Branch Hammerstein System

4.3.2

Parallel Hammerstein Systems

4.4

Wiener Systems

4.4.1

Single-Branch Wiener Systems