Characterisation, Modelling and Digital Pre-DistortionTechniques for RF Transmitters in Wireless Systems

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Characterisation, Modelling and Digital Pre-Distortion

Techniques for RF Transmitters in Wireless Systems

MAHMOUD ALIZADEH

Doctoral Thesis

Department of Information Science and Engineering

School of Electrical Engineering and Computer Science

KTH

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TRITA-EECS-AVL-2019:9 ISBN 978-91-7873-076-6

KTH, School of Electrical Engineering and Computer Science Department of Information Science and Engineering SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen måndagen den 18 februari 2019 klockan 13.00 i hörsal 99131, Kungsbäcksvägen 47, Gävle. © Mahmoud Alizadeh, February 2019.

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Abstract

Wireless systems have become an inevitable part of modern technologies serving humankind. The rapid growth towards large dimensional systems, e.g. 5th generation (5G) technologies, incurs needs for improving the performance of the systems and considering aspects to make them as far as possible envi-ronmentally friendly in terms of power efficiency, cost, and so on. One of the key parts of every wireless communication system is the radio frequency (RF) power amplifier (PA), which consumes the largest percentage of the total en-ergy. Hence, accurate models of RF PAs can be used to optimize their design and to compensate for signal distortions. This thesis starts with two methods for frequency-domain characterisation to analyse the dynamic behaviour of RF PAs in 3rd-order non-linear systems. Firstly, two-tone signals superim-posed on large-signals are used to analyse the frequency-domain symmetry properties of inter-modulation (IM) distortions and Volterra kernels in differ-ent dynamic regions of RF PAs in a single-input single-output (SISO) system. Secondly, three-tone signals are used to characterise the 3rd-order self- and cross-Volterra kernels of RF PAs in a 3 × 3 multiple-input multiple-output (MIMO) system. The main block structures of the models are determined by analysing the frequency-domain symmetry properties of the Volterra kernels in different three-dimensional (3D) frequency spaces. This approach signifi-cantly simplifies the structure of the 3rd-order non-linear MIMO model.

The following parts of the thesis investigate techniques for behavioural modelling and linearising RF PAs. A piece-wise modelling technique is pro-posed to characterise the dynamic behaviour and to mitigate the impairments of non-linear RF PAs at different operating points (regions). A set of thresh-olds decompose the input signal into several sub-signals that drive the RF PAs at different operating points. At each operating point, the PAs are mod-elled by one sub-model, and hence, the complete model consists of several sub-models. The proposed technique reduces the model errors compared to conventional piece-wise modelling techniques.

A block structure modelling technique is proposed for RF PAs in a MIMO system based on the results of the three-tone characterisation technique. The main structures of the 3rd- and higher-order systems are formulated based on the frequency dependence of each block. Hence, the model can describe more relevant interconnections between the inputs and outputs than conventional polynomial-type models.

This thesis studies the behavioural modelling and compensation tech-niques in both the time and the frequency domains for RF PAs in a 3 × 3 MIMO system. The 3D time-domain technique is an extension of conven-tional 2D generalised memory polynomial (GMP) techniques. To reduce the computational complexity, a frequency-domain technique is proposed that is efficient and feasible for systems with long memory effects. In this technique, the parameters of the model are estimated within narrow sub-bands. Each sub-band requires only a few parameters, and hence the size of the model for each sub-band is reduced.

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Sammanfattning

Trådlösa system har blivit en oundviklig del av modern teknik till nytta för mänskligheten. Den snabba tillväxten mot storskaliga system, t.ex. 5: e ge-nerationens (5G) teknologier, medför större behov av att förbättra systemets prestanda och överväga aspekter som så långt som möjligt gör dem till miljö-vänliga produkter vad gäller effektivitet, kostnad och så vidare. En av nyckel-komponenterna i varje trådlöst kommunikationssystem är radiofrekvens (RF) -förstärkaren (PA), som förbrukar den största andelen av den totala energin. Därför kan exakta modeller av RF-PA användas för att optimera designen och för att kompensera förvrängning av signalen. Denna avhandling börjar med två metoder för frekvensdomänkarakteriseringstekniker för att analysera det dynamiska beteendet hos RF-PA i tredje ordningens icke-linjära system. För det första används tvåtonssignaler som överlagras på stora signaler för att analysera symmetriegenskaper i frekvensdomänen hos intermodulationspro-dukter (IM) och Volterra-kärnor i olika dynamiska områden av RF-PA i ett SISO system (en insignal en utsignal). För det andra används tretonssignaler för att karakterisera 3: e ordningens själv- och kors-Volterra-kärnor av RF-PA i ett 3×3 MIMO-system (flera in och flera utsignaler). De huvudsakliga blockstrukturerna för modellerna bestäms genom att analysera symmetrie-genskaper i frekvensdomänen hos Volterra-kärnorna i olika tredimensionella (3D) frekvensområden. Detta tillvägagångssätt förenklar strukturen hus 3:e ordningen icke-linjärna modell.

De följande delarna av avhandlingen undersöker tekniker för beteende-modellering och linjärisering av RF-PA. En bitvis beteende-modelleringsteknik föreslås för att karakterisera det dynamiska beteendet och för att dämpa försämring-arna orsakade av icke-linjära RF-PA vid olika arbetspunkter (regioner). En uppsättning tröskelvärden delar upp insignalen i flera undersignaler som dri-ver RF-PA:erna vid olika arbetspunkter. Vid varje arbetspunkt modelleras PA:erna av en undermodell, och den fullständiga modellen består därför av flera undermodeller. Den föreslagna tekniken minskar modellfelet jämfört med konventionella bitvisa modelleringstekniker.

En blockstrukturmodellteknik föreslås för RF-PA i ett MIMO-system ba-serat på resultaten av tretonskarakteriseringstekniken. De huvudsakliga block-strukturerna i 3:e och högre ordningens formuleras baserat på frekvensbero-endet för varje block. Modellen kan därför beskriva mer relevanta samman-kopplingar mellan ingångarna och utgångarna än konventionella polynom-typmodeller.

Denna avhandling studerar beteendemodellering och kompensationstek-niker i både tids- och frekvensdomän för RF-PA i ett 3×3 MIMO-system. 3D-tidsdomäntekniken är en förlängning av konventionella 2D generella min-nespolynom (GMP). För att reducera beräkningskomplexiteten föreslås en frekvensdomänsteknik som är effektiv och praktiskt genomförbar för system med långa minneseffekter. I denna teknik uppskattas parametrarna för mo-dellen inom smala delband. Varje delband kräver endast några få parametrar, och därigenom reduceras storleken på modellen för varje delband.

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Acknowledgments

First of all, I would like to thank my supervisors Professor Peter Händel and Professor Daniel Rönnow for their invaluable scientific supports, profound insights, knowledge and great feedback. I also thank all the people of the ATM department at the University of Gävle (HiG) and the information science and engineering department at KTH for providing a creative and inspiring environment.

In addition, I would like to thank the former doktorand colleagues: Efrain Zenteno, Javier Ferrer Coll, Mohamed Hamid, Shoaib Amin, and the current colleagues: Nauman Masud, Rakesh Krishnan, Smruti Ranjan Panigrahi, Vipin Choudhary, Zain Ahmed Khan, as well as my colleagues at the PhD Group at HiG: Amer Jazairy, Arash Jouybari, Hadi Amin, Hanna Andersson, Hossein Khosravi Bakhtiari, Ioana Stefan, Jamila Alieva, Lea Fobbe, Ma-munur Rashid, Mohammad Jahedi, Robert Johansson, Sandra Carlos-Pinedo, Shveta Soam, ... for all our amazing times, friendly discussions and collabo-rations.

I would also like to have this chance to thank Abolfazl, Amir, Arefeh, Chanki, Efrain, Eszter, Jenny, Karl, Mahla, Naeim, Saeid, Shahriar, Taha, and Tim for being friendly, and having lots of fun and nice conversations.

Finally, I am very grateful to my parents and family for their unconditional love and supports.

Mahmoud Alizadeh Gävle, February, 2019.

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List of Tables

4.1 Two types of tones for different test signals . . . 36 5.1 The number of parameters vs. memory length for 3D MP models . . . . 58

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List of Figures

2.1 The measurement setup of the SISO system . . . 11

2.2 The measurement setup of the 2 × 2 MIMO system including input cross-talks . . . 11

2.3 The measurement setup of the 3 × 3 MIMO system including input cross-talks . . . 12

2.4 The measurement setup of the 3 × 3 MIMO system including input and output cross-talks . . . 13

3.1 A K × K MIMO system . . . 17

4.1 Amplitude regions of the input signal . . . 25

4.2 The relative magnitudes of IM distortions of the Doherty amplifier vs. ∆f . . . 26

4.3 The relative phases of IM distortions of the Doherty amplifier vs. ∆f . 27 4.4 The relative magnitudes of IM distortions of the class-AB amplifier vs. ∆f . . . 28

4.5 The relative phases of IM distortions of the class-AB amplifier vs. ∆f . 29 4.6 Upper IM distortions of the class-AB amplifier . . . 30

4.7 Volterra kernels in different regions with a probing-signal 1 dBm . . . . 31

4.8 Volterra kernels in different regions with a probing-signal −4 dBm . . . 32

4.9 Basic block structure of a 3rd-order system . . . 33

4.10 Three different test signals for a 3 × 3 MIMO system . . . 34

4.11 Frequency paths of 1-stepped-2-fixed tones in three regions . . . 37

4.12 Frequency paths of 2-stepped-1-fixed tones in three regions . . . 37

4.13 Magnitudes of the self-kernel H₃(2:2,2,2)along the given first type of fre-quency paths . . . 38

4.14 Magnitudes of the self-kernel H₃(2:2,2,2) along the given second type of frequency paths . . . 39

4.15 Block structure of the self-kernel H₃(2:2,2,2) . . . 39

4.16 Magnitudes of the 2 × 1 cross-kernel H₃(2:2,2,3)along the given first type of frequency paths . . . 40

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List of Figures xi

4.17 Magnitudes of the 2 × 1 cross-kernel H₃(2:2,2,3) along the given second

type of frequency paths . . . 41

4.18 Block structure of the 2 × 1 cross-kernel H₃(2:2,2,3). . . 42

4.19 Magnitudes of the 3 × 1 cross-kernel H₃(2:1,2,3)along the given first type of frequency paths . . . 43

4.20 Magnitudes of the 3 × 1 cross-kernel H₃(2:1,2,3) along the given second type of frequency paths . . . 44

4.21 Block structure of the 3 × 1 cross-kernel H₃(2:1,2,3). . . 45

5.1 The performances of the proposed piece-wise behavioural model . . . 49

5.2 The performances of the proposed piece-wise DPD . . . 50

5.3 Block structures of symmetric 3rd-order Volterra kernels in a 2×2 MIMO system . . . 52

5.4 Block structures of higher-order non-linear 2 × 2 MIMO systems . . . . 53

5.5 The NMSE and ACEPR of the model in a 2 × 2 MIMO system . . . 54

5.6 The PSD of the model errors in a 2 × 2 MIMO system . . . 54

5.7 The NMSE and ACEPR of the time- and frequency-domain 3D models 58 5.8 The PSD of the errors of the time- and frequency-domain 3D models . . 59

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List of Acronyms & Abbreviations

2D Two-Dimensional

3D Three-Dimensional

ACEPR Adjacent Channel Error Power Ratio

ACLR Adjacent Channel Leakage Ratio

ADC Analogue-to-Digital Converter

AM/AM Amplitude-to-Amplitude Modulation

AM/PM Amplitude-to-Phase Modulation

CP Cyclic Prefix

CW Continuous Wave

DAC Digital-to-Analogue Converter

dB Decibel

DFT Discrete Fourier Transform

DPD Digital Pre-Distortion

DSP Digital Signal Processing

DUT Device-Under-Test

FFT Fast Fourier Transform

IDFT Inverse Discrete Fourier Transform

IFFT Inverse Fast Fourier Transform

ILA Indirect Learning Architecture

I/Q In-Phase and Quadrature

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xiv LIST OF ACRONYMS & ABBREVIATIONS

MP Memory Polynomial

GMP Generalised Memory Polynomial

ICI Inter-Carrier Interference

IEEE Institute of Electrical and Electronics Engineers

IF Intermediate Frequency

IM Inter-Modulation

IP Intercept Point

ISI Inter-Symbol Interference

LDMOS Laterally Defused Metal-Oxide Semiconductors

LO Local Oscillator

LSE Least Squares Estimation

MIMO Multiple-Input Multiple-Output

MISO Multiple-Input Single-Output

NMSE Normalised Mean Squared Error

OFDM Orthogonal Frequency Division Multiplexing

PAPR Peak-to-Average Power Ratio

PH Parallel Hammerstein

PSD Power Spectral Density

SISO Single-Input Single-Output

SNR Signal-to-Noise Ratio

SSB Single Side Band

VNA Vector Network Analyser

VSG Vector Signal Generator

VSWR Voltage Standing Wave Ratio

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Part I

Comprehensive summary

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Chapter 1

Introduction

1.1 Background

The wireless communication system is rapidly growing to satisfy the demands for ever increasing capacity, speed, and reliability, leading to the design of transceivers with higher power efficiency, spectral efficiency, and signal quality [1]. Modern wireless communication systems fulfil these requirements by using advanced digital modulation schemes, e.g. orthogonal frequency division multiplexing (OFDM) and wideband code division multiple access (WCDMA) [2]. In these techniques, the dynamic range of the modulated signal may become relatively high, causing a large peak-to-average power ratio (PAPR) at the transmitter side. Consequently, high PAPR signals degrade the performance of radio frequency (RF) power amplifiers (PAs) in terms of linearity, causing spectral deficiency and power dissipation [3].

RF PAs are the pivotal components in wireless systems and are the parts that consume the most energy [4,5]. Higher power efficiencies can be achieved by driving RF PAs close to their saturation regions. However, the non-linear distortion effects increase, which cause a spectral regrowth [3, 6, 7]. In practice, there is a trade-off between linearity and power efficiency [1, 8]. In addition, advanced modulation schemes use wideband signals to achieve higher data rates. Nevertheless, RF PAs show memory effects in the presence of wideband signals [1,3]. A system with mem-ory, which is dispersive in its impulse response or frequency selective in its transfer function [9], may cause inter-symbol interference (ISI) and power dissipation [9,10]. This effect can also cause inter-carrier interference (ICI) in a OFDM modulation scheme if the total memory length (also including the transmission channel disper-sions) is larger than its cyclic prefix (CP) [11]. For such reasons, it has received the most attention from RF engineers for modelling and compensating the impairments of RF PAs at the front-end of transmitters.

There are three main approaches to model RF components: physics based, cir-cuit based, and black-box or behavioural models. A physics-based model describes an RF component by its physical structure, and estimates its performance from the

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4 CHAPTER 1. INTRODUCTION

electromagnetic equations and charge transition analyses [2,12,13]. A circuit-based model describes a system by its electrical elements and characteristics, e.g. resistors, capacitors, inductors, diodes, transistors, etc. [14–16]. A black-box or behavioural model is well-known for describing the input and output relations of a system [1], e.g. Volterra series [17] as a parametric model, and neural networks [18, 19] as non-parametric models. The Volterra series is a powerful mathematical tool to characterise the behaviour of a weakly non-linear system with fading memory [17]. However, the dimensionality and computational complexity of a Volterra model increases dramatically and it shows poor performance for strongly non-linear dy-namic systems [20, 21]. In practice, reduced forms of the Volterra model are used as feasible solutions with adequate accuracy [1]. Memoryless and quasi-memoryless techniques are used for modelling a narrowband system [22–25], whereas models with memory are used for wideband systems [26–30]. An accurate model of RF PAs can be used to modify and optimize amplifier designs. In addition, the inverse behavioural model known as pre-distortion is used to linearise and mitigate the distortions of RF PAs. The pre-distortion techniques can be implemented in the analogue or the digital domain [2, 3].

The digital pre-distortion technique is an elegant approach to overcome the dilemma between linearity and power efficiency, as well as to compensate for the memory effects of RF PAs [4,8,31]. In the digital pre-distortion technique, the input signals are distorted and reshaped in baseband frequency, with the advantages of flexibility and digital signal processing (DSP) capabilities [2].

Modelling and linearising techniques for characterising and mitigating the dis-tortions of RF PAs can be performed either in single-input single-output (SISO) systems [8, 31, 32], or in multiple-input multiple-output (MIMO) systems [33–35]. However, the impairments of cross-talk and coupling effects from transmission lines and antennas in MIMO systems should also be considered in the models. Therefore, additional impairments and multiple PAs increase the dimensionality and compu-tational complexity of the models in MIMO systems [1]. Hence, reducing the num-ber of parameters and using a priori knowledge of the systems, such as models of cross-talk and main block structures, are quite efficient [36]. For instance, a prior knowledge of RF PAs can be achieved by multi-tone characterisation techniques, which are standard approaches to analysing the behaviour of RF PAs and to de-tecting their memory length, which shows to what degree the transfer function of the system is frequency selective [37–39].

1.2 Thesis contribution

This thesis discusses frequency-domain characterisation techniques, behavioural modelling, and linearising of non-linear RF PAs in SISO and MIMO systems.

Several conventional two-tone tests have been reported in the literature to char-acterise the IM distortions and Volterra kernels of RF PAs. However, the Volterra kernels cannot be well characterised using a single dynamic region [40]. In [A], the

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1.2. THESIS CONTRIBUTION 5

IM distortions of RF PAs are characterised using a two-tone signal and a large-signal in SISO systems. The large-large-signal drives the amplifiers to different operating points to analyse the behaviour of the PAs in different dynamic regions. Further, [B] characterises the 3rd-order Volterra kernels of RF PAs in a SISO system. In this work, the 3rd-order Volterra kernels are examined in different dynamic regions and compensated for the contribution of higher-order kernels.

In [C], the 3rd-order Volterra kernels of RF PAs are characterised in a 3 × 3 MIMO system including input cross-talk channels. The cross-talk effects of MIMO systems produce additional distortion, and the self- and cross-Volterra kernels have different symmetry properties. Thus, different test scenarios are considered to anal-yse the kernels in different three-dimensional (3D) frequency volumes.

The Volterra kernels of RF PAs show different behaviour at different operating points (regions). In [D], a new piece-wise modelling technique is proposed to model Volterra kernels in different magnitude regions in a SISO system.

In [E], the main block structures of RF PAs are determined in a 2 × 2 MIMO system including input cross-talk channels. In this technique, the main block struc-tures of the 3rd-order and higher-order systems are formulated corresponding to the analyses of Volterra kernels from [C]. Therefore, the proposed model can describe the interconnections between the input and output signals relevant to the frequency dependence of each block.

A time-domain 3D generalised memory polynomial (GMP) modelling and lin-earising techniques are proposed in [F] for RF PAs in a 3 × 3 MIMO system. The system includes cross-talk at the input and output of the PAs to resemble the ef-fects of cross-channels. A 3D GMP frequency-domain technique is also proposed based on the time-domain technique to reduce the computational complexity of the model.

List of papers included in the thesis

The included conference and journal papers in this thesis are listed below.

[A] M. Alizadeh, and D. Rönnow, A two-tone test for characterizing nonlinear

dynamic effects of radio frequency amplifiers in different amplitude regions,

Elsevier Meas., vol. 89, pp. 273-279, Apr. 2016.

The author of this thesis was involved in part of the planning, developing the models, analysing the data, and writing the manuscript, was also responsible for performing all experimental work and designing the algorithms. The co-author was involved in analysing the data, writing the manuscript, providing feedback, refining the experimental results, and pointing out the focus of the paper.

[B] M. Alizadeh, D. Rönnow, and P. Händel, Characterization of Volterra kernels

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6 CHAPTER 1. INTRODUCTION

Commun. (COMM), Bucharest, Romania, Jun. 2018, pp. 351-356.

The author of this thesis was the main contributor and involved in planning, developing the models, performing all experimental work, designing the algo-rithms, analysing the data, and writing the manuscript. The co-authors were involved in providing feedback, refining the experimental results, and pointing out the focus of the paper.

[C] M. Alizadeh, S. Amin and D. Rönnow, Measurement and analysis of

frequency-domain Volterra kernels of nonlinear dynamic 3×3 MIMO systems, IEEE

Trans. Instrum. Meas., vol. 66, no. 7, pp. 1893-1905, Jul. 2017.

The author of this thesis was involved in part of the planning, of developing the models, analysing the data, and writing the manuscript, was also responsi-ble for designing the algorithms and performing the major experimental work. The co-authors were involved in developing the models, analysing the data, mi-nor experimental work, providing feedback, refining the experimental results, writing the manuscript, and pointing out the focus of the paper.

[D] M. Alizadeh, P. Händel and D. Rönnow, Basis function decomposition approach

in piece-wise modeling for RF power amplifiers, 26th Telecommunications

Fo-rum (TELFOR), Belgrade, Serbia, Nov. 2018 (in press).

The author of this thesis initiated the paper, was the main contributor, and was involved in planning, developing the models, performing all experimental work, designing the algorithms, analysing the data, and writing the manuscript. The co-authors were involved in providing feedback, refining the experimental results, and pointing out the focus of the paper.

[E] M. Alizadeh, D. Rönnow, P. Händel, and M. Isaksson, A new block-structure

modeling technique for RF power amplifiers in a 2×2 MIMO system, 13th

Int. Conf. Adv. Technol. Syst. Services Telecommunications (TELSIKS), Niˇs, Serbia, Oct. 2017, pp. 224-227.

The author of this thesis initiated the paper, was the main contributor, and was involved in developing the models, performing all experimental work, de-signing the algorithms, analysing the data, and writing the manuscript. The co-authors were involved in providing feedback, refining the experimental re-sults, and pointing out the focus of the paper.

[F] M. Alizadeh, P. Händel and D. Rönnow, Behavioural modelling and digital

pre-distortion techniques for RF PAs in a 3×3 MIMO system, Int. J. Microw.

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1.3. THESIS OUTLINE 7

The author of this thesis initiated the paper, was the main contributor, and was involved in developing the models, performing all experimental work, de-signing the algorithms, analysing the data, and writing the manuscript. The co-authors were involved in developing the models, providing feedback, refining the experimental results, and pointing out the focus of the paper.

List of papers not included in the thesis

[G] D. Rönnow, S. Amin, M. Alizadeh, E. Zenteno, Phase noise coherence of two

continuous wave radio frequency signals of different frequency, IET Science,

Meas. and Technol., vol. 11, pp. 77-85, Aug. 2016.

1.3 Thesis outline

The thesis outline is as follows. Chapter 2 briefly covers the measurement setups and devices-under-test (DUTs) for RF PAs in SISO and MIMO systems. Chapter 3 gives a general overview of time- and frequency-domain behavioural modelling tech-niques in SISO and MIMO transmitters. The time-domain real-valued RF band representations of Volterra kernels are considered, as well as the discrete complex baseband models. Model identification and validation metrics are also discussed. Chapter 4 describes a frequency-domain two-tone measurement technique for char-acterising the IM distortions and Volterra kernels of RF PAs in SISO systems, and a three-tone technique to analyse the self- and cross-Volterra kernels of RF PAs in MIMO systems. Chapter 5 proposes a piece-wise linearising technique for RF PAs in SISO systems, a block-structure modelling technique for MIMO systems, and time- and frequency-domain 3D GMP modelling and linearising techniques for MIMO systems. Chapter 6 presents the summaries and conclusions of the thesis and potential avenues for future research.

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Chapter 2

Measurement Systems

This chapter explains the measurement setups required for developing the charac-terisation, modelling and linearisation techniques of RF PAs. Different setups are used in this thesis corresponding to [A]-[F].

The basic approach is to generate complex digital baseband signals in a software, e.g. MATLAB, and to upload them to a vector signal generator (VSG) as a source of real signals to excite a device-under-test (DUT) in the RF bands. Then the output signals of the DUT are down-converted into intermediate frequency (IF) to be digitized by an analogue-to-digital converter (ADC). The digitized signals are transferred to the computer for post-processing and analysis.

Several aspects determine the specification of the measurement instruments, e.g. frequency bandwidth, bit-resolution, dynamic range (linearity), sensitivity, amplitude noise, and phase noise [41, 42]. The bandwidth of a measurement setup is limited to VSGs, PAs, and ADCs. The bit-resolution of digital-to-analogue con-verters (DACs) in VSGs and of ADCs affects the total quality of the measured data. In order to measure the characteristics and behaviour of a DUT, it is necessary that the instruments have accurate dynamic ranges and linearities, e.g. VSGs, mixers, ADCs, which should maintain a linear dynamic range larger than the DUT’s. The sensitivity of measurement setups can be determined by the output signal-to-noise ratio (SNR) of the VSGs, the bit-resolution, phase noise, and amplitude noise. Usu-ally a common reference clock, e.g. 10 MHz (or 100 MHz) is provided to lock the phase of the local oscillator (LO) and baseband clock in VSGs, and the clock for ADCs, which decreases the total phase noise.

2.1 Measurement setups

This section briefly explains the specifications of the devices and components re-quired for the measurement setups in [A]-[F]. The carrier frequency of the trans-mitter in these measurement setups was 2.14 GHz, and the intermediate frequency (IF) for down-converting was 90 MHz.

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10 CHAPTER 2. MEASUREMENT SYSTEMS

• VSG: The maximum RF bandwidth of the VSG (Rohde & Schwarz SMBV 100A) is 120 MHz and its carrier frequency is 6 GHz. The single side band (SSB) phase noise power is < -120 dBc at 2 GHz. The resolution of the DAC for the in-phase and quadrature (I/Q) complex baseband signal is 16-bit. The resolution of the baseband output voltage is 1 mV. The amplitude error of the VSG at RF frequencies < 3 GHz is less than 0.5 dB.

• Mixers: The mixer Mini-Circuits ZX05-42MH-S operates from 5 MHz to 4.2 GHz. The specification of the mixer is given at ∼ 2.2 GHz. The nominal conversion loss is ∼ 8.25 dB. The isolation between LO-RF ports is 32 dB and that between LO-IF ports is 25 dB. The voltage standing wave ratio (VSWR) of the RF is 1.88 and that of the LO port is 1.36.

• Low pass filter: The 3 dB bandwidth of the low-pass filter Mini-Circuits SLP 300+ is 270 MHz, and the nominal VSWR is 1.7. The insertion loss, return loss, and group delay are 0.1 dB, 18.2 dB, and 2.29 ns at 97 MHz, respectively.

• ADC: The SP-device ADQ214 digitizer provides dual channels, 14-bit reso-lution at maximum sampling rate 400 MHz giving a 200 MHz bandwidth of

< 1 dB flatness at the fundamental frequency band. The isolation between

two channels is 115 dB at 79 MHz. The maximum input power is ∼ +13 dBm.

• RF synthesizer: The RF synthesizer Holzworth HS9003A provides a phase coherence between all three integrated channels with optimal channel-to-channel stability. Each channel-to-channel is independently controllable, and gives an accuracy of < ±0.4 degree at 2.048 GHz - 4.096 GHz, and a nominal VSWR 1.3. The SSB phase noise is ≤-127 dBc at 2 GHz with 10 kHz offset. The amplitude error is ± 0.5 dB at 10 MHz - 6GHz.

• RF coaxial switch: The frequency range of the RF miniature coaxial switch SR-2 min-H is DC - 18 GHz. For DC - 4 GHz, the insertion loss < 0.1 dB, the VSWR is 1.2, and the isolation > 80 dB.

• Coherent averaging: A coherent averaging method is used for multiple measurements to increase the SNR, and hence the effective dynamic range is improved by 10 log₁₀(Mav) in dB, where Mav is the number of measurements

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2.1. MEASUREMENT SETUPS 11

SISO system

The measurement setup of the SISO system shown in Fig. 2.1 is used in [A] and [B] to analyse IM3 products and third-order Volterra kernels of DUTs in different amplitude regions. In [D], this setup is also used to evaluate the proposed piece-wise modelling technique. The system includes a VSG, a PA as the DUT, an attenuator, a directional coupler, load, down-converter block, ADC, and the synthesizer to synchronize the different devices.

Figure 2.1: The measurement setup of the SISO system used in [A], [B] and [D].

MIMO system

Fig. 2.2 illustrates the measurement setup of a 2 × 2 MIMO system including input cross-talk channels used in [E]. The two VSGs are synchronized by a reference clock, digital baseband clock, and LO clock. The RF synthesizer is used to cohere

Figure 2.2: The measurement setup of the 2 × 2 MIMO system including input cross-talks used in [E].

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12 CHAPTER 2. MEASUREMENT SYSTEMS

the VSGs, ADC and mixers in phase. The DUT comprises two directional couplers resembling -20 dB input cross-talk effects, and two RF PAs with the same nominal gains and frequency responses. Attenuators are used to protect the components at subsequent steps.

In [C], the Volterra kernels of a non-linear dynamic 3×3 MIMO system including input cross-talk channels are analysed in the frequency domain. The measurement setup of the system is depicted in Fig. 2.3. The three VSGs, mixers, and ADC are synchronized and phase coherent. The DUT consists of three RF amplifiers, and an input coupler. The amplifiers have the same nominal gains and frequency responses. The coupler is designed in microstrip technology using an FR4 substrate (50 Ω). The coupling between the outer and the inner channels is measured using a vector network analyser (VNA) to be -13.5 dB, and between the two outer channels, it is -21.5 dB. The RF switch is used to measure three output signals by a dual channel digitizer.

Figure 2.3: The measurement setup of the 3 × 3 MIMO system including input cross-talks used in [C].

The measurement setup of a 3 × 3 MIMO system including input and output cross-talk channels is illustrated in Fig. 2.4. This setup is used in paper [F] for behavioural modelling and digital pre-distortion techniques of RF PAs in a 3 × 3 MIMO system. The system is phase coherent in VSGs, mixers, and ADC. The input and output couplers are similar in specifications. The couplings between two adjacent channels are -20 dB, and the coupling between two non-adjacent channels are -35 dB measured by the VNA.

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2.2. DEVICES-UNDER-TEST 13

Figure 2.4: The measurement setup of the 3 × 3 MIMO system including input and output cross-talks used in [F].

2.2 Devices-under-test

In [A], two different DUTs are used. The first DUT is an Ericsson AB commercial class-AB amplifier with an LDMOS transistor. The 3 dB bandwidth is ∼ 250 MHz between 2.040-2.290 GHz, and the nominal gain is ∼ 52.2 dB with the 1 dB compression point at the input signal ∼ 4.5 dBm. The second DUT is a Freescale Semiconductor MRF 8521120 HS Doherty amplifier with a typical gain of 15 dB. A pre-amplifier operating in its linear range is used to drive the Doherty amplifier, which its effects are compensated in the post-processing part. In [B] and [D], the same class-AB amplifier is used as in [A].

The RF PAs used in [C], [E] and [F] are medium high power Mini-Circuits ZHL-42. The nominal gains of the PAs are 30 dB (< ±1 dB flatness) within the band 700 MHz - 4.2 GHz. The 1 dB compression points are >+28 at the output. The noise figures (NFs) are 10 dB, and the VSWRs at both input and output are 2.5.

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Chapter 3

Modelling of RF Power Amplifiers

RF PAs have received increasingly attentions for decades, and several methods have been reported in the literatures for their characterisation and modelling in both the time and the frequency domains and to compensate for their deficiencies [3]. These deficiencies include non-linearity distortions when excited by large-signals, and memory effects when excited by wideband signals [4]. The Volterra series pro-vides a mathematically general way to describe the behaviour of weakly non-linear systems with fading memory [17]. Nevertheless, the computational complexity and number of parameters (size) of such a model increase dramatically with an increase in the order of the non-linearity and of the length of the memory, hence performing poorly in practice [20, 21].

In this chapter, the Volterra models are explained for describing and character-ising the behaviour of SISO and MIMO systems in both the time and the frequency domains.

3.1 SISO transmitter

Time-domain Volterra model

The output of a non-linear system can be described by the superposition of linear and non-linear terms

y(t) = y1(t) + y2(t) + · · · + yp(t) + . . . , (3.1)

where y(t) is the output of the system, y1(t) is the linear output and yp(t), p > 1 is the pth-order non-linear output of the system. A time-domain real-valued Volterra

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16 CHAPTER 3. MODELLING OF RF POWER AMPLIFIERS

model of a SISO system is [17]

y(t) = +∞ Z −∞ h1(τ1) x(t − τ1) dτ1 + +∞ Z −∞ +∞ Z −∞ h2(τ1, τ2) x(t − τ1) x(t − τ2) dτ1dτ2 + . . . + +∞ Z −∞ · · · +∞ Z −∞ hp(τ1, . . . , τp) x(t − τ1) . . . x(t − τp) dτ1. . . dτp + . . . , (3.2)

where x(t), and y(t) are the input, and output signals, respectively, τ is a time delay,

hp(τ1, . . . , τp) is the pth-order Volterra kernel, where in a symmetric representation

τ1≤ · · · ≤ τp. The pth-order system in (3.2) is

yp(t) = +∞ Z −∞ · · · +∞ Z −∞ hp(τ1, . . . , τp) x(t − τ1) . . . x(t − τp) dτ1 . . . dτp. (3.3)

In RF applications, the even-order terms produce even harmonics outside of the fundamental frequency band, and hence are suppressed by filtering. The complex-valued discrete baseband form of (3.2) for odd-order terms is [43]

y(n) = +∞ X m1=0 h1(m1) x(n − m1)+ +∞ X m1=0 +∞ X m2=0 +∞ X m3=0 h3(m1, m2, m3) x(n − m1) x(n − m2) x∗(n − m3) + · · · + +∞ X m1=0 · · · +∞ X mp=0 hp(m1, . . . , mp) (p+1)/2 Y i=1 x(n − mi) p Y j=1+(p+1)/2 x∗(n − mj) + . . . . (3.4)

Eq. (3.4) is unbounded due to the order of the non-linearity and the memory length, which makes it infeasible for practical applications [20]. Instead, reduced forms of the Volterra model are preferred, e.g. memory polynomial-type and black-box models [1].

Frequency-domain Volterra model

The frequency-domain representation of the output for the pth-order non-linear system in (3.3) is given [17]

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3.2. MIMO TRANSMITTER 17 Yp(ω) = 1 (2π)p−1 +∞ Z −∞ · · · +∞ Z −∞ Yp(ω − µ1, µ1− µ2, . . . , µp−2− µp−1, µp−1)× dµ1. . . dµp−1, (3.5) where Yp(ω1, . . . , ωp) = +∞ Z −∞ · · · +∞ Z −∞ hp(τ1, . . . , τp) × x(t1− τ1) . . . x(tp− τp) e−j(ω1t1+···+ωptp)dτ1 . . . dτpdt1. . . dtp = Hp(ω1, . . . , ωp) X(ω1) . . . X(ωp), (3.6)

where X(ω) is the Fourier transform of the input x(t), and Hp(ω1, . . . , ωp) is the

pth-order Volterra kernel in the frequency domain

Hp(ω1, . . . , ωp) = +∞ Z −∞ · · · +∞ Z −∞ hp(τ1, . . . , τp) e−j(ω1τ1+···+ωpτp)dτ1 . . . dτp, (3.7)

where for a symmetric pth-order kernel, ω1≤ · · · ≤ ωp.

3.2 MIMO transmitter

Fig. 3.1 shows the block diagram of a K × K MIMO system. In [44, 45], the Volterra theory of a SISO non-linear dynamic system was extended to describe a MIMO system.

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Cross-talk effects

The effects of cross-talk are considered to be deficiencies of the MIMO systems. In general, cross-talk effects occur in a system through non-isolated transmission channels [1]. A MIMO transmitter has different sources of cross-talk effects. One type of cross-talk effect comes from the up-converter LO, cables, and transmission lines before the PAs: this is known as input cross-talk. The other type comes from the output transmission lines, cables and antennas after the PAs: this is known as output cross-talk. For a non-linear amplifier, the input cross-talk contributes to the non-linearity of the system. In contrast, the output cross-talk contributes to the linear behaviour of the system under a matched impedance condition at the output [46]. However, the output cross-talk effects will be a part of the non-linear behaviour of the system under a mismatched impedance condition [47, 48]. In Fig. 3.1, the kth output signal of a K × K MIMO non-linear transmitter, including K power amplifiers, input and output cross-talk effects, is

y(k)(t) =

βk1(t) ∗ f1(α11(t) ∗ x(1)(t) + α12(t) ∗ x(2)(t) + · · · + α1K(t) ∗ x(K)(t))+ βk2(t) ∗ f2(α21(t) ∗ x(1)(t) + α22(t) ∗ x(2)(t) + · · · + α2K(t) ∗ x(K)(t))+

. . . +

βkK(t) ∗ fK(αK1(t) ∗ x(1)(t) + αK2(t) ∗ x(2)(t) + · · · + αKK(t) ∗ x(K)(t)), (3.8) where x(r) for r = 1, . . . , K indicates the rth input signal, y(k) is the kth output signal, fk(·) for k = 1, . . . , K is the non-linear dynamic operator of the kth PA. In general, αk`(t) and βk`(t) describe dynamic models of the input and output cross-talk effects from the `th channel on the kth channel, and the “ ∗ ” denotes the convolution operator.

Time-domain Volterra model

In [44, 45], it was shown that a real-valued model of an R × K MIMO system can be represented by K parallel models of R × 1 multi-input single-output (MISO) systems y(k)(t) = P X p=1 R X r1=1 · · · R X rp=1 Z +∞ −∞ · · · Z +∞ −∞ h(k:r1,...,rp) p (τ1, . . . , τp)× x(r1)_{(t − τ} 1) . . . x(rp)(t − τp) dτ1. . . dτp, (3.9) where y(k)_{(t), k = 1, . . . , K is the kth output, P is the highest order of non-linearity}

in the model, x(r)_{, r = 1, . . . , R is the rth input signal, τ is the time delay, and} h(k:r1,...,rp)

p (τ1, . . . , τp) is the pth-order Volterra kernel of the kth sub-system. The

kernel h(k:r1,...,rp)

p (τ1, . . . , τp) is a self-kernel if r1= · · · = rp, otherwise it is a cross-kernel. In a symmetric representation, r1≤ · · · ≤ rp, and also for rk = r` it holds

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3.2. MIMO TRANSMITTER 19

τi ≤ τj, i ≤ j. The kth output signal of the pth-order sub-system excited by the

input signals x(r1)_{, . . . , x}(rp)_{in (3.9) is} y(k:r1, ..., rp) p (t) = Z +∞ −∞ · · · Z +∞ −∞ h(k:r1,...,rp) p (τ1, . . . , τp)× x(r1)_{(t − τ} 1) . . . x(rp)(t − τp) dτ1. . . dτp. (3.10) The complex baseband form of the Volterra model for an R × K MIMO system is given by y(k)(n) = R X r1=1 +∞ X m1=0 h(k:r1) 1 (m1) x(n − m1)+ R X r1=1 R X r2=1 R X r3=1 +∞ X m1=0 +∞ X m2=0 +∞ X m3=0 h(k:r1, r2, r3) 3 (m1, m2, m3) × x(r1)_{(n − m} 1) x(r2)(n − m2) x(r3) ∗ (n − m3) + · · · + R X r1=1 · · · R X rp=1 +∞ X m1=0 · · · +∞ X mp=0 h(k:r1, ..., rp) p (m1, . . . , mp) × (p+1)/2 Y i=1 x(ri)_{(n − m} i) p Y j=1+(p+1)/2 x(rj)∗_{(n − m} j) + . . . , (3.11)

where the odd-order terms are only included in the model.

Frequency-domain Volterra model

The frequency-domain representation of the kth output y(k:r1, ..., rp)

p (t) is given by Y(k:r1, ..., rp) p (ω) = 1 (2π)p−1 +∞ Z −∞ · · · +∞ Z −∞ × Y(k:r1, ..., rp) p (ω − µ1, µ1− µ2, . . . , µp−2− µp−1, µp−1) dµ1. . . dµp−1, (3.12) where Y(k:r1, ..., rp)

p (ω) is the kth output of the pth-order sub-system, and

Y(k:r1, ..., rp) p (ω1, . . . , ωp) = +∞ Z −∞ · · · +∞ Z −∞ h(k:r1, ..., rp) p (τ1, . . . , τp) × x(r1)_(t 1− τ1) . . . x(rp)(tp− τp) e−j(ω1t1+···+ωptp)dτ1 . . . dτpdt1 . . . dtp = H(k:r1, ..., rp) p (ω1, . . . , ωp) X(r1)(ω1) . . . X(rp)(ωp), (3.13)

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where X(rp)_{(ω) is the Fourier transform of the input x}(rp)_{(t), and the pth-order} frequency-domain Volterra kernel H(k:r1, ..., rp)

p (ω1, . . . , ωp) is H(k:r1, ..., rp) p (ω1, . . . , ωp) = +∞ Z −∞ · · · +∞ Z −∞ h(k:r1, ..., rp) p (τ1, . . . , τp) × e−j(ω1τ1+···+ωpτp)_dτ 1 . . . dτp, (3.14)

where for symmetric pth-order self-kernels, ω1 ≤ · · · ≤ ωp, whereas for symmetric

pth-order cross-kernels ωi≤ ωj, i ≤ j if rk = r` [44, 45].

3.3 System identification

The matrix form of the model in a SISO system is [1]

y = X h (3.15)

where y is the vector of the output signal, X is the matrix of basis functions of the input signal, and h is a vector of parameters.

In a K × K MIMO, the relation between the inputs and outputs is

yk= Xkhk, k = 1, . . . , K, (3.16)

which is described as the kth K × 1 MISO model, where yk _{is the output at the}

kth channel, hk _{is the parameter of the kth MISO model, and X}k _{is the regression}

matrix including all terms corresponding to the self- and cross-kernels of the kth MISO model. In general, Xk _{may vary for a different kth MISO model, e.g. as in} [E]. However, in a symmetric model, Xk _{is the same for all MISO models, e.g. as} in [F]. Eqs. (3.15) and (3.16) show that the model for a SISO system is a special case of a MIMO model, where k = 1.

The memory polynomial-type models are linear in their parameters [20]. There-fore, the parameters of the models in (3.15)-(3.16) are identified by the least squares (LS) method [49] ˆ hk = XkH Xk −1 XkH yk, (3.17)

where ˆhk is the estimated parameter of the model, and XkH is the Hermitian conjugate matrix of Xk.

The LS identification method is used in [D], [E], and [F]. In [A], [B] and [C] the identification of IM3 products and Volterra kernels is performed by taking the discrete Fourier transform (DFT) of the signals and extracting the values of the components at the desired frequencies. The number of DFT points and the input frequencies should be chosen properly to avoid DFT leakage at the fundamental and IM3 frequencies [50, 51].

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3.4. MODEL PERFORMANCE VALIDATION 21

3.4 Model performance validation

Different evaluation metrics have been proposed in the literature to accurately evaluate the performance of a model [43, 52]. In this thesis the metrics: NMSE, ACEPR and ACLR are used.

NMSE

The normalised mean squared error (NMSE) is given by [1]

NMSE = P n|ye(n)|2 P n|yd(n)|2 (3.18) where the numerator is the power of the model errors and the denominator is the power of the desired output signal. The NMSE mainly highlights the performance of a model within the in-band frequency region [43, 52]. The NMSE can also be calculated in the frequency domain, which gives the same results as in the time domain.

ACEPR

The adjacent channel error power ratio (ACEPR) is defined as [19]

ACEPR = R adj.ch.Φe(f ) df R ch.Φd(f ) df (3.19) where Φe(f ) is the power spectral density (PSD) of the model errors over the adjacent channels, and Φd(f ) is the PSD of the desired output signal over the signal channel. Therefore, ACEPR gives the power of the errors within the lower and upper adjacent channels normalised by the power of the in-band desired output signal. ACEPR is suitable for evaluating the performance of behavioural models.

ACLR

The adjacent channel leakage ratio (ACLR) is defined as [4]

ACLR = R adj.ch.Φy(f ) df R ch.Φd(f ) df (3.20) where Φy(f ) is the PSD of the output signal over the adjacent channels. ACLR evaluates the power of the output within the lower and upper adjacent channels normalised by the power of the in-band desired signal. ACLR is a metric for evaluating the performance of digital pre-distortions.

In [E] and [F], a criterion is applied to all MISO models, and then an average is taken representing the value of the corresponding criterion.

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Chapter 4

Characterising the Non-Linear

Dynamic Effects of RF PAs using

Multi-Tone Techniques

Behavioural modelling techniques describe a system based on the measured input and output data. Hence, the identification of the parameters of the model using these techniques is biassed by the measured data. In other words, there is no guarantee of the validity of the model when the system is excited by signals with different characteristics [4]. Multi-tone techniques are alternative approaches for characterising PAs in the frequency domain [1].

Continuous wave (CW) signals have been commonly used for characterising the 1 dB compression point, amplitude-to-amplitude modulation (AM/AM) and amplitude-to-phase modulation (AM/PM) characteristics of PAs by sweeping the power level of the signal [38, 53]. Later on, the conventional two-tone characterisa-tion technique was proposed as a classical standard to find the 3rd intercept point (IP3) and to analyse the dynamic behaviour of PAs [39, 54, 55]. In this method, the excitation signal is a two-tone signal sweeping the frequency space and the measured baseband complex envelope of the output of PAs is analysed. The char-acterisation of PAs is done by analysing the AM/AM and AM/PM characteristics, and the asymmetry of the magnitude and phase of the upper and lower 3rd-order inter-modulation (IM3) products vs. frequency. The asymmetry and imbalance of the magnitude and phase of the upper and lower IM3 products show the frequency dependence and hence, the memory effects of the system [21, 37, 38, 56].

A variety of conventional two-tone measurement techniques have been reported for applications to RF PAs. The AM/AM and AM/PM characteristics of a static non-linear system were analysed in [57]. In [58], the dynamic non-linear behaviour of RF PAs was identified using a two-tone test technique. Furthermore, in [59], a two-tone test was used to characterise the asymmetry of the phase of microwave non-linear systems.

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24

CHAPTER 4. CHARACTERISING THE NON-LINEAR DYNAMIC EFFECTS OF RF PAS USING MULTI-TONE TECHNIQUES In [A], a method is proposed to analyse the frequency-domain asymmetry of the lower and upper IM3 products for class-AB and Doherty amplifiers. The excitation signal consists of a two-tone probing-signal superimposed on different amplitude levels of a large-signal, which resembles the hot scattering parameter of non-linear systems [60, 61] excited by a large-signal and a probing-signal simultaneously. This technique is also used to characterise the 3rd-order Volterra kernels of PAs in [B]. The results of the proposed methods in [A] and [B] can be useful for detecting the memory length of piece-wise modelling techniques [40, 62].

The general Volterra kernel of a 3rd-order system is completely determined by a signal consisting of three or more tones [21, 63]. In [C], a three-tone signal test is used to measure and analyse the frequency-domain Volterra kernels of a non-linear dynamic 3 × 3 MIMO system.

4.1 Two-tone test characterisation in SISO systems

A real-valued probing-signal including two tones in the RF frequency band is given by

x(t) = Re{A1ej(ω1t+φ1)+ A2ej(ω2t+φ2)}, (4.1)

where Re{·} denotes operation of taking the real part. The ith tone of x(t) has amplitude Ai, angular frequency ωi= 2πfi, and phase φi, fcis the carrier frequency,

∆f is the frequency spacing, f1= fc− ∆f /2, and f2= fc+ ∆f /2.

The goal in [A] and [B] is to characterise the dynamic behaviour of a PA in dif-ferent regions by superimposing the probing-signal in (4.1) on a large-signal (carrier signal) with different amplitude levels

xri(t) = Re{A1e

j(ω1t+φ1)_{+ A}

2ej(ω2t+φ2) + Acie

j(ωct+Φi)_}, _(4.2) where xri(t) is the input signal in the ith region. In addition, Aci, ωc = 2πfc, and Φi are the amplitude, angular frequency, and phase of the large-signal in the

ith region, respectively. In the case of a pure two-tone signal Aci = 0, otherwise,

Aci 6= 0. Fig. 4.1 shows an example of the AM/AM characteristic of a non-linear system (left) that is excited by a baseband input signal vs. time in four amplitude regions (right). The AM/AM characteristic of the system shows different non-linearities in different regions: the system is more likely to have a linear behaviour in region 1, whereas it shows a strong non-linearity in region 4.

In [A], the IM distortions produced by the probing-signal are characterised at frequencies 2ω1− ω2 and 2ω2− ω1 (the lower and the upper IM products,

respec-tively) in the presence of a large-signal in different regions. The output signal at frequency 2ω1− ω2 is yri(t) = Re{ [(6/8)H3(ω1, ω1, −ω2) A 2 1A2+ (10/8) H5(ω1, ω1, ω1, −ω1, −ω2) A41A2 + (15/8) H5(ω1, ω1, ω2, −ω2, −ω2) A21A 3 2+ (30/8) H5(ω1, ω1, ωc, −ωc, −ω2) A21A2A2ci + . . . ] e j[(2ω1−ω2)t+2φ1−φ2]_}, _(4.3)

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4.1. TWO-TONE TEST CHARACTERISATION IN SISO SYSTEMS 25 -0.2 0 0.2 0.4 0.6 0.8 1 V_input (V) -0.2 0 0.2 0.4 0.6 0.8 1 Voutput (V) Region 1 Region 2 Region 3 Region 4 0 0.02 0.04 0.06 0.08 0.1 Time -0.2 0 0.2 0.4 0.6 0.8 1 Vinput (V) Region 1 Region 2 Region 3 Region 4

Figure 4.1: Amplitude regions of the input signal. The AM/AM characteristic in different amplitude regions (left). An illustration of the input signal vs. time in different amplitude regions (right).

and at frequency 2ω2− ω1 yri(t) = Re{ [(6/8)H3(ω2, ω2, −ω1) A1A 2 2+ (10/8) H5(ω2, ω2, ω2, −ω2, −ω1) A1A24 + (15/8) H5(ω2, ω2, ω1, −ω1, −ω1) A31A 2 2+ (30/8) H5(ω2, ω2, ωc, −ωc, −ω1) A1A22A 2 ci+ . . . ] e j[(2ω2−ω1)t+2φ2−φ1]_}, _(4.4)

where Hn(·) is the nth-order Volterra kernel in the frequency domain.

In the following, analyses of the amplitude and phase of the IM distortions of the Doherty and class-AB amplifiers in different regions will be given, and then the characterisation of the Volterra kernel for the class-AB amplifier will be explained.

Inter-modulation distortions

Figs. 4.2 and 4.3 show the relative magnitudes and phases of the IM distortions vs. the tone-spacing (∆f ) for the Doherty amplifier. The power of the probing-signal is -30 dBm, and the powers of the large-probing-signal are −∞ dBm, -28.1 dBm, -22 dBm, -20.1 dBm, -18.5 dBm, and -16 dBm, driving the PA to six different operating points (regions), which are indexed from 1 to 6. As shown in Fig. 4.2, the magnitudes of the IM distortions are strongly frequency dependent, indicating there are memory effects in the PA. The dependency and asymmetry of the lower and upper IM distortions are minor for ∆f ≤ 10 kHz in all regions, which means the PA merely shows a static non-linear behaviour. In contrast, the asymmetry of the IM distortions between 100 kHz and 1 MHz is significant, and hence this causes the largest memory effects in the PA. In addition, the PA excited by wideband signals for ∆f ≥ 10 MHz, is increasing the asymmetry of the IM distortions. The details of the IM distortions in different regions show that the PA behaves differently: e.g. the

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26

CHAPTER 4. CHARACTERISING THE NON-LINEAR DYNAMIC EFFECTS OF RF PAS USING MULTI-TONE TECHNIQUES

104 105 106 107 f (Hz) -36 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16 |IM3| (dBc) im3 up, R6 = -16 dBm im3 lo im3 up, R5 = -18.5 dBm im3 lo im3 up, R4 = -20.1 dBm im3 lo im3 up, R3 = -22 dBm im3 lo im3 up, R2 = -28.1 dBm im3 lo im3 up, R1 = - dBm im3 lo

Figure 4.2: The relative magnitudes of IM distortions of the Doherty amplifier vs. ∆f with −30 dBm probing-signal in different regions.

asymmetry in the lowest regions is less significant than in the higher regions. In the highest region, the asymmetry is less pronounced, since the amplifier is operating in strongly non-linear region close to its saturation region.

Fig. 4.3 depicts the relative phases of the IM distortions for the Doherty PA in different regions. The IM distortions in the higher regions are more significantly frequency dependent, indicating stronger memory effects in the PA. The structures of the phase asymmetries in different regions are approximately similar, as also seen in Fig. 4.2. The lowest region, which is a pure two-tone probing-signal, gives the minimum asymmetry or memory effects.

Figs. 4.4 and 4.5 illustrate the relative magnitudes and phases of the IM prod-ucts vs. tone-spacing for the class-AB amplifier. The power of the probing-signal is -13 dBm, whereas the powers of the large signal are −∞ dBm, -9.1 dBm, -3 dBm, -1.1 dBm, 0.5 dBm, and 3 dBm for the regions 1 to 6, respectively. The behaviour of the class-AB PA in both magnitudes and phases vs. frequency is different from that of the Doherty PA. As shown in Fig. 4.4, the asymmetry of the amplitudes

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4.1. TWO-TONE TEST CHARACTERISATION IN SISO SYSTEMS 27 104 105 106 107 f (Hz) -50 -40 -30 -20 -10 0 10 IM3 (Degrees) im3 up,R6=-16 dBm im3 lo im3 up,R5=-18.5 dBm im3 lo im3 up,R4=-20.1 dBm im3 lo im3 up,R3=-22 dBm im3 lo im3 up,R2=-28.1 dBm im3 lo im3 up,R1= - dBm im3 lo

Figure 4.3: The relative phases of IM distortions of the Doherty amplifier vs. ∆f with −30 dBm probing-signal in different regions.

in region 1 (the linear region) is significant for ∆f < 1 MHz, however the IM dis-tortion is strongly frequency dependent for the whole range. Furthermore, the PA increasingly shows a strong memory effect in the highest region for 10 kHz < ∆f < 10 MHz. In other regions, the IM distortions show less of a frequency dependency, although in region 2 and 3, the asymmetry of the higher and upper IM products is significant for ∆f < 100 kHz, and ∆f > 1 MHz, respectively.

The relative phases of the IM distortions in regions 3, . . . , 6 are similar in asym-metry and frequency dependency, which are significant for 10 kHz < ∆f < 10 MHz, as shown in Fig. 4.5. However, in region 1, the IM distortion has the highest asym-metry and a different structure from those in the other regions. The symasym-metry and frequency in region 2 is the least.

By comparing the relative phases between two PAs, the maximum asymmetry in the Doherty PA is ∼ 5 degrees, whereas the class-AB PA gives a maximum ∼ 40 degrees asymmetry in phase. Therefore, the class-AB PA shows stronger memory effects than the Doherty PA. In addition, the class-AB PA gives stronger memory

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28

104 105 106 107 f (Hz) -48 -46 -44 -42 -40 -38 -36 -34 -32 -30 -28 |IM3| (dBc) im3 up, R6 = 3 dBm im3 lo im3 up, R5 = 0.5 dBm im3 lo im3 up, R4 = -1.1 dBm im3 lo im3 up, R3 = -3 dBm im3 lo im3 up, R2 = -9.1 dBm im3 lo im3 up, R1 = - dBm im3 lo

Figure 4.4: The relative magnitudes of IM distortions of the class-AB amplifier vs. ∆f with a −13 dBm probing-signal in different regions.

effects in the lower regions, whereas the Doherty PA shows stronger memory effects in the higher regions.

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4.1. TWO-TONE TEST CHARACTERISATION IN SISO SYSTEMS 29 104 105 106 107 f (Hz) -250 -200 -150 -100 -50 0 50 IM3 (Degrees) im3 up,R6= 3 dBm im3 lo im3 up,R5= 0.5 dBm im3 lo im3 up,R4=-1.1 dBm im3 lo im3 up,R3=-3 dBm im3 lo im3 up,R2=-9.1 dBm im3 lo im3 up,R1= - dBm im3 lo

Figure 4.5: The relative phases of IM distortions of the class-AB amplifier vs. ∆f with a −13 dBm probing-signal in different regions.

Volterra kernel

The output signal at frequencies 2ω1− ω2 and 2ω2− ω1 in (4.3) and (4.4) is

dis-torted not only by the 3rd-order products but also by the higher-order products. Therefore, characterising the 3rd-order Volterra kernel gives more detailed infor-mation about a non-linear dynamic system. The analyses of the IM distortions in the previous part show that the class-AB amplifier gives stronger memory ef-fects and frequency dependency than the Doherty amplifier. Hence, in [B], the characterisation of the 3rd-order Volterra kernel of the class-AB PA is studied.

According to (4.3) and (4.4), the IM products normalised by 6/8A3 are poly-nomial forms in A2_{, where A = A}

1= A2= αAci (α is a constant). Therefore, the estimated 3rd-order Volterra kernel H3at frequencies 2ω1− ω2 and 2ω2− ω1is the

zero-order parameter of the polynomial

Y =H3 + β5H5A2+ · · · + βPHP(A2)

P −3

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30

CHAPTER 4. CHARACTERISING THE NON-LINEAR DYNAMIC EFFECTS OF RF PAS USING MULTI-TONE TECHNIQUES where Y is the normalised output frequency component, P is the maximum order of non-linearity, and βpHp for p = 5, . . . , P , is the parameter of the polynomial. There is a trade-off between the order of P in (4.5) and the noise level. The Volterra kernel is smoother with lower-orders of P , whereas the noise increases with higher-orders of P . Thus, different higher-orders of P should be chosen depending on the power level of the signals.

The powers of the probing-signal are -6.5 dBm, -4 dBm, -1.5 dBm, and 1 dBm, and the powers of the large-signal are −∞ dBm, -6 dBm, -1 dBm, and 4 dBm, driv-ing the PA into regions 1 to 4, respectively. In order to satisfy (4.5) and identify the local Volterra kernel, a set of experimental data in the neighbourhood of each power level is required [64]. In fact, the local kernel in each region is identified by a set of data showing similar frequency dependencies at frequencies 2ω1− ω2 and

2ω2− ω1. To fulfil the requirement, the baseband signal at different power levels

deviates ±1 dB with a step of 0.143 dB. Fig. 4.6 shows the magnitudes and phases of the IM distortions of the output of the class-AB amplifier at frequency 2ω2− ω1,

where the powers of the probing-signal and large-signal are centred at 1 dBm and -1 dBm, respectively.

Fig. 4.7 shows the magnitudes (left) and phases (right) of the 3rd-order Volterra kernels H3(ω1, ω1, −ω2) and H3(−ω1, ω2, ω2) vs. ∆f in four different regions when

104 105 106 107 f [Hz] -22 -21 -20 |Y (2 2 -1 )| [dBm] 104 105 106 107 f [Hz] -190 -180 -170 Y (2 2 -1 ) [Deg.]

Figure 4.6: The magnitudes (right) and the phases (left) of the IM distortions of the class-AB amplifier at frequency 2ω2− ω1vs. ∆f . The power level of the baseband

signal deviates ±1 dBm with the step of 0.143 dB in the neighbourhood of the probing-signal 1 dBm and the large-signal -1 dBm.

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4.1. TWO-TONE TEST CHARACTERISATION IN SISO SYSTEMS 31 104 105 106 107 f [Hz] -4 0 4 8 |H 3 | [dB V -2 ] 104 105 106 107 f [Hz] -150 -100 -50 H 3 [Deg.] H 3( 1, 1,- 2), R4 =4 dBm H 3(- 1, 2, 2), R4 =4 dBm H 3( 1, 1,- 2), R3 =-1 dBm H 3(- 1, 2, 2), R3 =-1 dBm H 3( 1, 1,- 2), R2 =-6 dBm H 3(- 1, 2, 2), R2 =-6 dBm H 3( 1, 1,- 2), R1 =- dBm H 3(- 1, 2, 2), R1 =- dBm

Figure 4.7: The estimated Volterra kernels H3(ω1, ω1, −ω2) (lower) and H3(−ω1, ω2, ω2) (upper). (a) The magnitudes and (b) phases of the kernels vs.

∆f for the class-AB PA in different regions with a probing-signal 1 dBm.

the power of the probing-signal is 1 dBm. The orders of non-linearity P in (4.5) are chosen to be 5, 5, 7, and 11 to estimate the kernels in regions 1 to 4, respectively. The asymmetry of the magnitudes is minor for 10 kHz < ∆f < 100 kHz in the different regions, and increases for ∆f > 100 kHz causing longer memories for wider bandwidth excitation signals.

The asymmetry of the phases between the kernels is minor (≤ 10 degrees) for ∆f < 2 MHz. However, there is a dramatic increase for higher frequencies. The asymmetry and frequency dependency of both the magnitudes and the phases of the kernels are approximately the same in regions 1-3, and have similar structures, indicating that the amplifier shows roughly the same behaviour in different regions. This happens because the power of the probing-signal is relatively high enough to drive the amplifier into a strongly linear region where the effect of the non-linearity is more significant than the memory effects.

As shown in Fig. 4.8 (left), the asymmetry between the magnitudes of the Volterra kernels at the lower and upper frequencies is also minor for 10 kHz < ∆f < 100 kHz in different regions when the power level of the probing-signal is -4 dBm, whereas it increases for ∆f > 100 kHz. The asymmetry and frequency dependence of the kernels in the lower regions are more significant than the higher

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104 105 106 107 f [Hz] -4 0 4 8 |H 3 | [dB V -2 ] 104 105 106 107 f [Hz] -150 -100 -50 H 3 [Deg.] H 3( 1, 1,- 2), R1 =4 dBm H 3(- 1, 2, 2), R1 =4 dBm H 3( 1, 1,- 2), R2 =-1 dBm H 3(- 1, 2, 2), R2 =-1 dBm H 3( 1, 1,- 2), R3 =-6 dBm H 3(- 1, 2, 2), R3 =-6 dBm H 3( 1, 1,- 2), R4 =- dBm H 3(- 1, 2, 2), R4 =- dBm

Figure 4.8: The estimated Volterra kernels H3(ω1, ω1, −ω2) (lower) and H3(−ω1, ω2, ω2) (upper). (a) The magnitudes and (b) phases of the kernels vs.

∆f for the class-AB PA in different regions with a probing-signal −4 dBm.

regions. This is because the power of the probing-signal is low, and different power levels of the large-signal drive the amplifier into different regions. Therefore, in the lower regions, the memory effect is more significant than the effect of non-linearity. However, the kernel in the saturation region still has a similar behaviour as seen in Fig. 4.7 (left). The phase information in Fig. 4.8 (right) does not give distinct differences between different regions compared to the magnitudes in Fig. 4.8 (left). As seen in Figs. 4.7 and 4.8, the class-AB amplifier is driven into the saturation region when the power level of the large-signal is 4 dBm (in region 4). The amplifier shows strong non-linearity in this region, consequently the non-linearity effect is much stronger than the memory effects. Thus, the asymmetry of the kernels is minor (∆f < 2 MHz) regardless of the power levels of the probing-signal. The kernel H3(ω1, ω1, −ω2) is less frequency dependent in magnitude than the kernel H3(−ω1, ω2, ω2). This implies that the amplifier shows lower memory effects at

lower frequencies with respect to the carrier frequency. The class-AB amplifier also shows minor memory effects with respect to the narrower bandwidth excitation signals corresponding to minor asymmetries between the kernels, whereas the wider bandwidth signals impose significant asymmetries on the kernels. Finally, different dynamic behaviours of the amplifier due to different power levels (regions) of the

Characterisation, Modelling and Digital Pre-DistortionTechniques for RF Transmitters in Wireless Systems