Phase Noise in Multi-Carrier Systems by Gokul Sridharan A ...

Phase Noise in Multi-Carrier Systems

by

Gokul Sridharan

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

Copyright © 2010 by Gokul Sridharan

Abstract

Phase Noise in Multi-Carrier Systems

Gokul Sridharan

Master of Applied Science

Graduate Department of Electrical and Computer Engineering

University of Toronto

2010

This thesis concerns the effect of phase noise (PHN) on multi-carrier systems such as

OFDM and the detection of multi-carrier symbols affected by PHN. It is known that PHN

causes mixing between sub-carriers resulting in inter-carrier interference (ICI) and rotates

symbols on every sub-carrier by a certain angle called the common phase error (CPE).

We explore how these two effects arise and show that these two effects are coupled to

each other. We also note that higher order M-QAM constellations like 64-QAM are more

sensitive to CPE than smaller constellations like 4-QAM. Based on our observations on

CPE, we propose a blind CPE estimation algorithm. The key idea behind this algorithm

is the realization that not all symbols of a constellation are equally prone to errors due to

CPE. We then address the issue of ICI and propose a turbo receiver design to mitigate it.

The turbo receiver is based on a soft-in soft-out detection algorithm that we develop using

the principles of variational inference. The algorithm jointly detects the PHN sequence

and the transmitted bits. The CPE estimation algorithm and the variational inference

based turbo receiver together constitute a complete solution to detection under PHN

for multi-carrier systems. Simulations show that using the proposed algorithms we can

significantly improve the performance. In addition to the analysis for OFDM systems,

we also analyze the effect of PHN on Single-Carrier Frequency Division Multiple Access

(SC-FDMA)– a multiple access scheme proposed recently. We draw parallels to the effect

on OFDM and provide suggestions on detection strategies to alleviate the effects of PHN

on SC-FDMA.

ii

Acknowledgements

This work is a culmination of contributions, sacrifices and support, both directly and

indirectly by many people over the last two years. First and foremost in that long list is

my advisor Prof. Teng Joon Lim. Starting from the weekly group meetings, to nudging

me to take a closer look at the problem that eventually defined my Master’s work, and

in pushing me to adopt a more lucid writing style for this thesis, I have had valuable

inputs from him at every stage. I would like to thank him for his insights and comments

during our weekly meetings and for his patience in editing many of my technical papers.

I greatly appreciate him for accomodating numerous requests for meetings at short notice

and for taking pains to answer e-mail queries promptly and to great depth.

I would also like to thank my thesis examiners, Prof. Frank Kschischang and Prof.

Wei Yu for taking time off to read this thesis and for making some insighful suggestions

and comments on this work.

It was a pleasure working in BA7114, thanks to all my colleagues who made it a very

enjoyable work environment. I have greatly benefited from discussions, both technical

and otherwise, with my colleagues and for this I am very thankful to them. In this

regard, I would like to especially thank Adam Tenenbaum, K.V. Srinivas, Taiwen Teng,

Ali Khanafer, Kianoush Hosseini, Sachin Kadloor, Siddarth Hari, Muhammad Nazmul

Islam, Ehsan Karamad, Hassan Masoon, Willian Chou, Amir Aghaei, Yashar Ghiassi,

Sanam Sadr, Wael Louis and Muhammad Mohanta.

While my wingmates from IITM and friends from India haven’t been around in

Toronto, I thank them for their constant support and reassurance via e-mail. In par-

ticular, I would like to thank my close friend Harini Eavani who has been a key source

of support, comfort and encouragement. Finally, I would like to thank my parents and

my sister for supporting me in all my endeavours all through my life.

iii

Contents

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Phase Noise: Characteristics, Effects and Consequences 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Sample Mean Statistics . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Receiver Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Transmitter Phase Noise . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.3 Multiple Antenna Scenario . . . . . . . . . . . . . . . . . . . . . . 21

2.3.4 Multi-User Scenario : Uplink/Downlink . . . . . . . . . . . . . . . 23

2.4 Relationship Between CPE and ICI . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Probability of High CPE Events . . . . . . . . . . . . . . . . . . . 29

2.4.2 Energy Split Between CPE and Higher Order Terms . . . . . . . 32

2.5 Phase Noise Estimation and Mitigation: Literature Survey . . . . . . . . 35

2.5.1 Preliminary Analysis : Early Papers . . . . . . . . . . . . . . . . 37

iv

2.5.2 Phase Noise Estimation and Compensation . . . . . . . . . . . . . 39

2.5.3 Key Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.4 Focus of Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Blind CPE Estimation 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Received Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Existing Pilot-Based Approaches to CPE Estimation . . . . . . . 47

3.3 Blind CPE Estimation Under Detection Uncertainty . . . . . . . . . . . . 49

3.3.1 Key Idea– Auxiliary Variable . . . . . . . . . . . . . . . . . . . . 49

3.3.2 Computing p(uk|dk) . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Turbo Receiver Design for ICI Mitigation 63

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Received Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.1 Gray mapping : bits to M-QAM symbols ( from [6] ) . . . . . . . 67

4.2.2 The Received Signal . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.3 Actual and Postulated Posterior Distributions . . . . . . . . . . . 68

4.3 The Bit-Level Variational Inference Algorithm . . . . . . . . . . . . . . . 71

4.3.1 Free Energy Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.2 Free Energy Minimization . . . . . . . . . . . . . . . . . . . . . . 72

4.3.3 The Variational Inference Algorithm . . . . . . . . . . . . . . . . 73

4.3.4 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

v

5 Effect of Phase Noise on SC-FDMA 81

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.1 ZF Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.2 MMSE Equalization . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Detection in the Presence of Phase Noise . . . . . . . . . . . . . . . . . . 87

5.3.1 Received Signal Model . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.2 Effect on Linear Receivers . . . . . . . . . . . . . . . . . . . . . . 88

5.3.3 Self Interference vs. Multi-User Interference . . . . . . . . . . . . 93

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Conclusion 97

Appendices 100

A Evaluating the Free Energy Expression 100

B Computing the Gradient 102

B.1 Gradient w.r.t Sθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

B.2 Gradient w.r.t mθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.3 Gradient w.r.t brk and bik . . . . . . . . . . . . . . . . . . . . . . . . . . 104

C Properties of the Permutation Matrix Tk 106

D SC-FDMA Received Signal in the Presence of PHN 110

Bibliography 111

vi

List of Tables

2.1 Table listing number of errors that are likely to occur due to CPE while

using 16/64/256-QAM constellations. . . . . . . . . . . . . . . . . . . . . 26

2.2 Table listing CPE thresholds for different M-QAM constellations. . . . . 29

3.1 Parameters used for simulations to test the proposed estimation algorithm. 58

5.1 Table listing percentage of energy in self interference for different number

of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

vii

List of Figures

1.1 Block diagram representing the proposed detection scheme for a CPE sen-

sitive constellation in a CPE-dominant regime. . . . . . . . . . . . . . . . 5

1.2 Block diagram representing the proposed detection strategy in an ICI-

dominant regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Block diagram representing the SC-FDMA scheme. Note that M > N ,

and usually, MN

= K, an integer representing the number of users in the

uplink. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Diagram illustrating the basic building blocks of a PLL frequency Synthe-

sizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Figure illustrating contributions to phase noise spectrum from various

sources [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Measured phase noise spectra of PLL frequency synthesizers generating a

signal of 2.5 GHz from a reference frequency of 10 MHz and having a loop

bandwidth of 5 kHz in (a) and 10 kHz in (b) [8]. . . . . . . . . . . . . . . 13

2.4 Figure illustrating transmitted and received vectors in an Nr ×Nt MIMO

link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Plot illustrating the effect of the CPE and ICI on bit error probability of

a coded-OFDM system in a frequency selective fading channel. . . . . . . 28

2.6 Plot illustrating the effect of sampling rate on the probability of high CPE

events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

viii

2.7 Plot illustrating the effect of the RMS value of phase noise on the proba-

bility of high CPE events . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Plot illustrating the effect of the loop bandwidth Ωo on the probability of

high CPE events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.9 Percentage of phase noise energy concentrated in the CPE term as a func-

tion of subcarrier spacing and loop bandwidth. . . . . . . . . . . . . . . . 34

2.10 SINR (Signal to Interference and Noise Ratio) as a function of sub-carrier

spacing at SNR=20dB. Loop bandwidth was set to 20kHz. . . . . . . . 35

2.11 A chart on detection strategies to be adopted under different operation

conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1 Figure showing that not all symbols are equally error prone in the presence

of CPE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Phase noise tolerance of a symbol with a square decision region. . . . . . 53

3.3 Effect of PHN and ADN on a symbol . . . . . . . . . . . . . . . . . . . . 54

3.4 Figure illustrating the effect of CPE on a type I symbol. . . . . . . . . . 55

3.5 Figure illustrating the effect of CPE on a type II symbol. . . . . . . . . . 55

3.6 Simulation 1: BER plots for different CPE estimation schemes. . . . . . . 59

3.7 Simulation 2: BER plots for different CPE estimation schemes. . . . . . . 60

3.8 Simulation 1: BER plot of coded-OFDM with different CPE compensation

schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.9 Simulation 2: BER plot of coded-OFDM with different CPE compensation

schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1 Figure illustrating the two different settings in which the bit-level detection

algorithm is tested. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 FER plot of the turbo-VI receiver. . . . . . . . . . . . . . . . . . . . . . 75

ix

4.3 Figure illustrating the overall receiver design used to comprehensively sup-

press the effects of phase noise. . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 FER plot of the turbo-VI receiver with CPE compensation. . . . . . . . . 77

4.5 FER plot of the turbo-VI receiver with perfect CPE compensation. . . . 79

4.6 FER plot comparing the performance of turbo-VI and DD-LMMSE re-

ceivers under different settings. . . . . . . . . . . . . . . . . . . . . . . . 80

5.1 Block diagram representing the SC-FDMA scheme and the use of a linear

receiver. Note that N > M , and usually, MN

= K, an integer representing

the number of users in the uplink. . . . . . . . . . . . . . . . . . . . . . . 82

5.2 Plot comparing the performance of interleaved and localized OFDMA/SC-

FDMA while using MMSE channel equalization. . . . . . . . . . . . . . . 86

5.3 Plot illustrating the performance of interleaved and localized SC-FDMA

when all channel coefficients are generated independently and when phase

noise is uncorrelated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.4 Plot illustrating the improvement in performance of an SC-FDMA system

affected by phase noise by cancelling self interference alone. . . . . . . . . 95

x

Chapter 1

Introduction

1.1 Introduction

Recent advances in wireless technologies such as multi-antenna multi-carrier systems,

coupled with a spurt in demand for wireless applications have led to significant increases

in data rates and transmission bandwidths. An immediate consequence of operating at

high data rates and high spectral efficiency is that receiver non-idealities such as phase

noise, frequency offset and IQ (In-Phase and Quadrature) imbalance effects, which were

either marginal or could be neglected previously, become significant and need to be

addressed. In this work we focus on the effects of phase noise on multi-carrier systems.

Phase noise (PHN) arises because of imperfections in the carrier frequency synthesizer.

While the output of an ideal frequency generator is given by

s(t) = A sin(ωt), (1.1)

the output of an actual generator is noisy and can be written as

s(t) = (A+ a(t)) sin(ωt+ θ(t)), (1.2)

where a(t) and θ(t) are amplitude and phase fluctuations. Amplitude noise does not

affect the zero crossings while phase noise does not affect the peak amplitude. Well

1

Chapter 1. Introduction 2

designed signal sources have small amplitude noise as it can be kept in check using an

automatic gain controller. Phase noise, on the other hand, is very difficult to remove

and can have a major impact on the system performance. We will concern ourselves

only with the issue of phase noise. In digital communication links, phase noise results

in erroneous data sampling leaving behind residual phase offsets that differ from symbol

to symbol. In order to factor in the effects of phase noise, a simple discrete time signal

model such as

y[k] = x[k] + n[k], (1.3)

gets altered to

y[k] = ejθ[k]x[k] + n[k], (1.4)

where y[k] are discrete time samples of the received signal, θ[k] are discrete time samples

of the continuous time phase noise process θ(t), x[k] and n[k] are the transmitted symbols

and the additive noise respectively.

Phase noise is a cause of concern in both single carrier and multi-carrier systems.

In [1] it was established that multi-carrier systems such as those that employ OFDM

(Orthogonal Frequency Division Modulation) suffer a much higher loss in SNR due to

phase noise than single carrier systems. The higher loss in SNR is a result of the longer

duration of a multi-carrier symbol and the loss of orthogonality between the sub-carriers.

While there are many transceiver non-idealities that are to be taken into account

when designing a communication system, there is a good reason for focusing specifically

on the issue of phase noise. While non-idealities such as IQ imbalance or frequency offsets

are impairments that remain constant over time, phase noise is time varying in nature.

Hence, while one could measure frequency offset and IQ imbalance in the training phase,

and then compensate for them while detecting symbols in the data transmission stage,

such an approach cannot be adopted in the case of phase noise. As a result, in the

data transmission stage, phase noise is likely to be the most dominant impairment that

determines system performance.


Characteristics of phase noise depend on the type of frequency generator that is being

used. Prior knowledge of the characteristics can play a vital role in mitigating the effects

of phase noise. Of the many possible ways of generating a reliable carrier signal, phase

locked loop (PLL) based frequency synthesizers are very widely used in most wireless

transceivers. PLL frequency synthesizers offer many advantages over the use of other

forms of local oscillator such as high levels of stability, ease of control through digital

circuitry and higher accuracy. In our work we assume that a PLL based frequency

synthesizer is used to generate the required carrier frequency and will restrict ourselves

to the phase noise characteristics corresponding to such a signal source.

Addressing the practical concerns that arise while designing a transceiver is vital

in ensuring optimal performance of the communication link. While better hardware

designs can alleviate these non-idealities to some extent, it is still important to develop

schemes that can mitigate these effects in the digital domain. Given that phase noise

is an unavoidable phenomenon, it is important to have a thorough understanding of its

effects on the performance of a transceiver. Developing better detection schemes and

computationally efficient algorithms that can estimate and compensate for the effects of

phase noise can have a significant impact on the link performance and can also relax

stringent hardware design constraints.

The next section outlines the various aspects of phase noise, its effects on system

performance and strategies to mitigate these effects that have been addressed in this

thesis.

1.2 Thesis Overview

Phase noise in OFDM/OFDMA systems has received widespread attention over the last

few years. While many researchers have identified the two major consequences of phase

noise as being the rotation of transmitted symbols by a small angle called the common


phase error (CPE) and the mixing of signals on adjacent sub-carriers resulting in inter-

carrier interference (ICI), very little attention has been paid to identify which of the two

effects is likely to dominate under different operating conditions. This aspect has a direct

influence on the choice of algorithms to be adopted for symbol detection. This is the

first issue that we address in our work. We identify circumstances under which effects

of CPE may dominate over ICI and vice versa, and provide guidelines with respect to

the choice of parameters and the set of algorithms that one needs to adopt while dealing

with phase noise.

Traditionally, it was common to embed pilot symbols amongst the data symbols

within a single OFDM symbol. This was done so as to aid channel estimation of a time

varying channel. Many works in the past have taken advantage of these pilot symbols to

aid detection of data symbols in the presence of phase noise. But recently, new wireless

standards such as WiMAX (Worldwide Interoperability for Microwave Access, IEEE

802.16) and LTE (3rd Generation Partnership Project Long Term Evolution, put forth

by European Telecommunications Standards Institute) have chosen to adopt a different

distribution of pilot symbols. While WiMAX embeds pilots only in two out of every

three consecutive OFDM symbols, in LTE one full OFDM symbol is devoted to channel

estimation once every seven time slots. Thus, there arise situations where we need to

consider symbol detection without the aid of pilot symbols in each OFDM symbol. In

our work, we exclusively focus on the situation where there are no pilot subcarriers in

an OFDM symbol. Since the case of OFDM symbols with pilots can be very easily

incorporated into algorithms we develop for data detection and phase noise estimation,

the scenario we focus on subsumes all possible distributions of pilots.

As stated previously, depending on system design and the operating conditions, one

could be in either a CPE-dominant regime or in an ICI-dominant regime. Each calls for

a very different approach to detection. In the CPE-dominant regime, when using a CPE

sensitive constellation (explained in Chapter 2) a large number of errors are likely to result


due to uncompensated CPE. Hence, it is very important to estimate and compensate

for CPE before detecting symbols. While estimation of CPE is straightforward in the

presence of pilots, blind estimation poses some challenges. We propose a blind CPE

estimation algorithm whose operating range, i.e., the ability to correctly estimate high

CPEs, is much better than existing algorithms. This algorithm stems from the realization

that not all sub-carrier symbols in an OFDM symbol are equally affected by CPE. In

the CPE-dominant regime, interference is likely to be limited to only a few adjacent

sub-carriers and does not significantly affect performance. It suffices to adopt a simple

ICI mitigation scheme such as that suggested in [2,3]. The proposed detection strategy is

illustrated in Figure 1.1. A detailed analysis on the effects and behaviour of CPE forms

the second part of the thesis.

FEC DecoderSymbol detector

(with ICI cancellation)

Robust CPE estimation

and compensation

(see Chapter 3)

Figure 1.1: Block diagram representing the proposed detection scheme for a CPE sensitive

constellation in a CPE-dominant regime.

In the ICI-dominant regime, one needs to adopt an approach that addresses inter-

ference comprehensively. While standard interference cancellation schemes are one op-

tion [3, 4], the effect of ICI can be so severe as to warrant the need to design a turbo

receiver. With this need in mind, we use an approximate inference technique called vari-

ational inference (VI) to develop a bit-level soft-in soft-out detection algorithm. Varia-

tional Inference is an approximate probabilistic inference technique associated with the

minimization of an objective function called the Variational Free Energy. In [5, 6] a

probabilistic framework for near-optimal OFDM detection in the presence of phase noise

was developed using variational inference. The framework can jointly estimate the phase

noise sequence along with the data symbol sequence. We adopt this framework to develop


a bit-level SISO algorithm and set up a joint iterative detection-decoding procedure that

is very effective in nullifying the interference resulting due to phase noise. The proposed

detection scheme is illustrated in Figure 1.2. Although CPE does not dominate in an ICI-

dominant regime, it cannot be completely neglected and hence, a CPE pre-compensation

block is included in the detection scheme. The choice of CPE compensation algorithm

depends on the symbol constellation being used, this is discussed in Chapter 2. Develop-

ment of the SISO detection algorithm and setting up the turbo receiver forms the third

part of the thesis.

FEC DecoderSignal Vector

ReceivedCPE Estimation

and Compensation

([15]/Chapter 3)

Soft−in Soft−out Symbol

Detector (Chapter 4)

Figure 1.2: Block diagram representing the proposed detection strategy in an ICI-

dominant regime.

A major drawback with conventional OFDMA (Orthogonal Frequency Division Mul-

tiple Access) based multiple access systems is that the resulting signal has a high peak to

average power ratio (PAPR) thus requiring power amplifiers to operate with a large back

off from their peak power. Single-carrier FDMA (Frequency Division Multiple Access)

is a multiple access scheme that addresses the issue of high PAPR, while also lending

itself to frequency domain equalization– one of the advantages of using OFDMA. This

advantage of SC-FDMA has led to it being adopted as the multiple access scheme for

uplink in the LTE standard.

SC-FDMA is essentially a linear precoded OFDMA scheme, with the linear precoding

consisting of a DFT (Discrete Fourier Transform) block. The transmitter and receiver

structure of a DFT-OFDMA/SC-FDMA system is shown in Fig. 1.3.

In an ideal scenario, there exists no multiple access interference as different users are

assigned orthogonal sub-carriers and the receiver segregates the user streams based on the


N point

DFT

DFT

M point

M point ADC/

Mapping

DAC/

RF

RF

P to

S

S to

P

Sub carrier

De−mapping

Symbol

Stream IDFT

Sub carrier

S to

P

CHANNEL

Channel

Equalization

N point

IDFTDETECTOR

Figure 1.3: Block diagram representing the SC-FDMA scheme. Note that M > N , and

usually, MN

= K, an integer representing the number of users in the uplink.

sub-carrier allocation. In the presence of phase noise, the orthogonality between the sub-

carriers is lost, and this leads to multiple access interference. We use the term multiple

access interference to include the inter-symbol interference within a user’s stream as well

as inter-symbol interference across other active users.

While the effect of phase noise on OFDM/OFDMA has received a lot of attention,

its effect on SC-FDMA is not completely understood. In order to better understand the

effect of phase noise on SC-FDMA, we develop a signal model to analyze the effect of

phase noise on SC-FDMA and characterize the effects along with suggestions on strategies

to adopt while dealing with the detection of an SC-FDMA symbol affected by phase noise.

This forms the last part of the thesis.

1.3 Notation

We adopt the following notation all through this thesis:

All vectors and matrices are represented in bold font, e.g. x, M. Vectors are

represented using lower case while matrices are represented using upper case. The

notation x(1:k) is used to represent the length-k vector formed using the first k


entries of x. The notation M(1:k, 1:k) represents the submatrix formed using the

first k rows and first k columns of M .

Discrete time processes are represented using the name of the process followed

by the index in square brackets, e.g. x[n], y[n]. Continuous time processes are

represented using the name of the process followed by round parentheses, e.g. x(t),

y(t).

If M is a matrix, MT and MH represent the transpose and the Hermitian transpose

of the matrix, respectively; tr(M) represents the trace of the matrix and diag(M)

represents the diagonal matrix formed using the diagonal entries of M. <(M) and

=(M) represent the matrices formed using the real and imaginary parts of M

respectively.

If x is a vector, <(x) and =(x) represent the vectors formed using the real and imag-

inary parts of x respectively, and diag(x) represents the diagonal matrix formed

using the vector x.

The identity matrix is denoted as I.

The expectation of a random variable x is denoted as E[x].

N (µ, σ2) is used to denote the Gaussian distribution and CN (µ, σ2) is used to

denote the circular symmetrix complex Gaussian distribution. σ2 in the second

case donotes the expectation E[|x-E[x]|2], where x is a circular symmetrix complex

Gaussian distributed random variable.

Chapter 2

Phase Noise: Characteristics, Effects

and Consequences

This chapter deals with various aspects of phase noise. We briefly discuss how phase

noise arises in a frequency synthesizer and characterize its power spectral density (PSD).

Based on the observations made, we introduce a commonly used model for generating

phase noise and contrast it with other models. We then take note of the properties of a

discrete time phase noise sequence, such as, the auto-correlation function, sample mean

statistics etc. We present the signal model of a multi-carrier system affected by phase

noise at the transmitter as well as the receiver. We further investigate the scenario where

the signal is corrupted only by receiver side phase noise and break down the effects of

phase noise into rotation of symbols because of common phase error and inter-carrier

interference resulting from the higher order components of phase noise. Finally, we

proceed to identify scenarios where the effect of CPE dominates over the effect of higher

order components and lay down some guiding principles that aid system design.

New contributions in this chapter include an in-depth analysis of the behaviour of

CPE as a function of various system and hardware parameters, identification of issues

that arise due to high CPE events and computing the frequency of such events. Based on

9

Chapter 2. Phase Noise: Characteristics, Effects and Consequences 10

this analysis certain guidelines for phase noise estimation and mitigation are established.

2.1 Introduction

To cater to demands for higher data rates within a limited bandwidth, many new stan-

dards have adopted higher order modulation schemes such as 64-QAM and also techniques

such as OFDM to transmit effectively over a frequency selective channel. Use of higher-

order constellations along with multi-carrier systems brings to the forefront the issue of

phase noise that can severely limit the system performance. Phase noise is generated

at the frequency signal source which can either be a simple free running oscillator or a

frequency synthesizer. Since the power spectral density of phase noise directly affects the

amount of ICI1, the PSD of phase noise has received a lot of attention. The noise sources

in all building blocks of a signal source contribute towards the PSD of phase noise. In

many works, the phase noise spectrum of a free running oscillator was assumed to fall off

as 1/f 2 from the central frequency, but this assumption was shown to be not accurate

in [7].

In most wireless applications, PLL frequency synthesizers are the preferred form of

signal source primarily because of higher levels of stability and easy control through

digital circuitry. Secondly, it allows one to generate a wide range of very high carrier

frequencies from oscillators tuned to much lower frequencies. The major building blocks

of a PLL frequency synthesizer include a reference signal source, a phase detector, a

low pass filter, a VCO and frequency dividers and multipliers. As shown in Figure 2.1,

the phase detector compares two input signals and produces an error signal which is

proportional to their phase difference. This signal is passed through the low pass filter

(LPF) and used to drive a voltage controlled oscillator which creates the output frequency.

The output frequency is fed through a frequency divider back to the input of the system.

1Larger bandwidth phase noise causes more ICI, as shown later.


If the output frequency drifts, the error signal will increase, driving the VCO frequency

in the opposite direction so as to reduce the error. Thus the output is locked to the

frequency at the other input. This input is called the reference and is usually derived

from a crystal oscillator. This explains the functioning of a PLL frequency synthesizer

in brief.

Phase

Detector

Voltage

Controlled

Oscillator

Low

Pass

Filter

Frequency

Divider

Reference

Signal

SourceCarrier Output

Signal

Figure 2.1: Diagram illustrating the basic building blocks of a PLL frequency Synthesizer.

Every building block listed above is a potential noise source and needs to be taken

into account when predicting the output phase noise spectrum. In [8] it is shown that

the output phase noise is mainly influenced by the VCO phase noise at high offset fre-

quencies from the central frequency and by phase noise of the reference signal source at

low offset frequencies. Noise from the phase detector, the low pass filter and frequency

divider contribute towards output phase noise mainly in the in-band region, i.e., within

the bandwidth of the LPF, known as the loop bandwidth. Figure 2.2, taken from [8],

illustrates the contributions of various noise sources to the PSD of output phase noise.

Note that the level of phase noise remains almost a constant within the loop bandwidth

of the PLL, flaring a little near the central frequency due to noise from the reference

oscillator. Figure 2.3, taken from [8], shows measured phase noise spectra of PLL fre-

quency synthesizers generating a signal of 2.5 GHz from a reference frequency of 10 MHz

and having a loop bandwidth of 5 kHz and 10 kHz each. It can be noted from the figure

that the PSD falls off at about 30dB/dec beyond the loop bandwidth.

Phase noise from the frequency synthesizer gets mixed with the transmit signal when

the baseband signal is up-converted to the carrier frequency. The same happens when


Figure 2.2: Figure illustrating contributions to phase noise spectrum from various sources

[8].

the received signal gets down-converted to the baseband at the receiver. In addition to

down-conversion, the carrier recovery loop that tries to correct for the residual frequency

offset and phase offset is also a potential source of phase noise. Carrier recovery loops

also employ PLLs for removing the residual frequency and phase mismatch. In general,

this is a decision-directed approach in which, based on the decisions made on the received

signal, the phase detector computes the error signal that drives a VCO which generates

the residual offset frequency that is used to compensate the baseband signal. Thus, while

the profile of the final output phase noise at the receiver is still likely to be the same as

that in Figure 2.3, the bandwidth of the phase noise is determined by the low pass filter

in the carrier recovery loop. In fact, phase noise models adopted in standards such as

IEEE 802.11g, are based on the VCO characteristics and the PLL response of the carrier

recovery loop.

The output phase noise spectrum is the starting point for modelling the phase noise

process. A constant gain within the loop bandwidth and a 30db/dec roll off outside the


Figure 2.3: Measured phase noise spectra of PLL frequency synthesizers generating a

signal of 2.5 GHz from a reference frequency of 10 MHz and having a loop bandwidth of

5 kHz in (a) and 10 kHz in (b) [8].

loop bandwidth motivates the use of a simple single pole Butterworth filter driven by

a Gaussian white noise process to model phase noise. Such a model assumes the phase

noise process to be a stationary process and has been adopted in IEEE standards such

as 802.11g. In the continuous domain, the input-output relation can be written as

θ(t) +

(1

2πΩo

)dθ(t)

dt= n(t), (2.1)

where Ωo is the one sided 3-dB loop bandwidth and n(t) is a Gaussian white noise process

with a certain variance. Such a process can be shown to have an auto-correlation function


of the form

Rθ(t) = Rθ(0)e−2πΩo|t|. (2.2)

In the discrete domain, such a model corresponds to a first order auto-regressive

process that can be represented as

θ[k] = (1− a)θ[k − 1] + (a)n[k], (2.3)

which has an an autocorrelation function of the form

Rθ[k −m] = Rθ[0](1− a)|k−m|. (2.4)

The discrete time model and the continuous time model are related through the

equation

1

2πΩoTs=

1− aa

, (2.5)

where we have assumed the sampling rate to be 1/Ts. If we solve for ‘a’ using (2.5) and

substitute in (2.4), we get

Rθ[k −m] = Rθ[0]

(1

1 + 2πΩoTs

)|k−m|. (2.6)

To reconcile the difference in (2.2) and (2.6), note that, in general, the loop bandwidth

tends to be in the range of tens of kHz and the sampling rate Ts in the range of a few MHz.

As a result, we have 2πΩoTs 1, which enables us to write e−2πΩo|t| ≈(

11+2πΩoTs

)t/Ts,

this establishes the equivalence between the two auto-correlation functions.

In [5], the discrete time auto-correlation function of phase noise was assumed to be

of the form

Rθ[k] = Rθ[0]e−2πΩoTs|k|, (2.7)

which is essentially a sampled version of (2.2). Having established the equivalence be-

tween the continuous and the discrete time models, it should make no difference whether

one uses (2.6) or (2.7) to represent the discrete time autocorrelation function. We use

(2.7) as the auto-correlation function of phase noise all through our work. We however


note that phase noise samples generated through a computer simulation will have an

auto-correlation function that is dependent on the exact type of digital Butterworth fil-

ter used to generate phase noise. While in most designs the auto-correlation function will

very closely resemble (2.7), there is bound to be some difference depending on the exact

digital filter used. The digital Butterworth filter we used had one pole and one zero.

In practice, most PLL frequency synthesizers come with design specifications that

include the root-mean-square (RMS) value of the output phase noise process, this gives

us the value of Rθ[0]. Denoting the RMS value in radians as σθ, we have Rθ[0] = σ2θ .

Even if the RMS value is unknown, simple measurements on the output signal of the

synthesizer can be made to measure the actual RMS value, as outlined in [9]. Typically,

the RMS value is between 1-3.

At this stage, we would like to point out that most of the existing work in addressing

phase noise assumes a Wiener model (Random walk) for phase noise. Such a model is

only appropriate when the carrier signal is generated from a stand alone local oscillator.

Such a design is almost never adopted as a free running oscillator can slowly drift away

from the required phase and frequency. Such a model corresponds to a non-stationary

process whose variance increases linearly with time.

Another assumption made, which was probably necessitated by the use of the Wiener

model, is that perfect phase synchronization exists at the start of every multi-carrier

symbol. Without this assumption there is the possibility of seeing very large drifts in the

phase of the carrier signal, which can make it impossible to detect the transmitted data.

Such an assumption is impractical since frequency and phase matching is done at the

start of the transmission of a payload or packet, and as a result perfect synchronization

at the start of every symbol cannot be guaranteed. The use of a stationary phase noise

model does not force this assumption, thus catering better to the practical scenario. The

next section discusses some statistical properties of a phase noise sequence generated

using the proposed model.


2.2 Statistical Properties

We are interested in the statistical properties of a discrete time phase noise sequence of

length N , obtained by sampling the continuous time phase noise at the rate of 1/Ts. It

is easy to see from (2.3) that since n[k] is a zero mean Gaussian white noise process, the

resulting phase noise sequence is also zero mean.

2.2.1 Covariance Matrix

The covariance matrix of a phase noise sequence of length N can be written as

Φ = σ2θ

1 p p2 . . . pN−1

p 1 p . . . pN−2

......

. . . . . .

pN−1 pN−2 pN−3 . . . 1

(2.8)

where, p = e−2πΩoTs . This follows from (2.7).

2.2.2 Sample Mean Statistics

Since the CPE plays a critical role in the detection of symbols affected by phase noise,

we take a closer look at the sample mean of a length-N phase noise sequence, which

constitutes the CPE within an OFDM symbol. We assume phase noise to lie in the

interval [−π π) and since the variance of the phase noise process is very low, no wrap

around effects are expected. With this in mind, suppose we let θ denote the sample mean

of a length-N sequence of the phase noise process, we have

θ =1

N

N∑k=1

θ[k]. (2.9)

Since the statistics of the phase noise process are known, it can be shown that θ is a

zero mean Gaussian random variable with variance 1TΦ1/N2. One can also write the

variance as


σ2θ = 1TΦ1/N2 =

σ2θ

N2

(1 + p

1− pN −2p(1− pN)

(1− p)2

)(2.10)

2.3 Signal Model

In this section we show how the presence of phase noise affects the signal model of an

OFDM symbol transmitted through a frequency selective channel. We first consider

a single antenna, point to point link with a frequency selective channel between the

transmitter and the receiver. The frequency selective channel is assumed to vary slowly

and remain constant over the duration of an OFDM symbol. We assume no residual

frequency offsets to be present in the received signal. By estimating the residual frequency

offset in the training phase and compensating for it in the data transmission phase, we

can ensure that almost no residual frequency offsets are seen. We also assume that

current channel conditions have been estimated during the training phase and that the

channel state information is available at the receiver. Algorithms that can estimate the

channel in the presence of phase noise and carrier frequency offset have been presented

in [9, 10]. Since phase noise is an impairment at both the transmitter and the receiver,

two immediate cases arise, both of which are discussed in detail below.

2.3.1 Receiver Phase Noise

We first consider the case where there is only receiver phase noise affecting the signal.

Let N be the number of sub-carriers and let the channel have L taps. Let F be the

N × N DFT matrix with the (l,m)th entry given by (1/√N)e−(2πjlm/N), with indices l

and m going from 0 to N − 1. Let d = [d0 d1 . . . dN−1]T be the data vector and let

s = [s0 s1 . . . sN+L−1]T with [s0 s1 . . . sL−1]T = [sN sN+1 . . . sN+L−1]T , be the vector to

be transmitted after appending the cyclic prefix. The vector s without the cyclic prefix

is simply the IDFT of the data vector d i.e., FHd.


Let θr = [θr0 θr1 . . . θrN+L−1]T be the phase noise sequence affecting the received signal

and let pr = [ejθr0 ejθ

r1 . . . ejθ

rN+L−1 ]T . Let the vector g = [g0 g1 . . . gL−1]T represent the

impulse response of the time domain channel. The received signal in discrete time after

appropriate sampling can be represented as

r = diag(pr)Gs + n, (2.11)

where n is complex white Gaussian noise with variance σ2 per dimension and the matrix

G is of the form

G =

g0 0 0 0 . . . 0 . . . 0 0 . . . 0 0

g1 g0 0 0 . . . 0 . . . 0 0 . . . 0 0

......

......

......

......

......

......

gL−1 gL−2 gL−3 . . . g0 0 . . . 0 0 . . . 0 0

......

......

......

......

......

......

0 0 0 0 . . . 0 . . . 0 gL−1 . . . g1 g0

. (2.12)

After the removal of cyclic prefix, one can write,

r(L:N+L−1) = diag(pr(L:N+L−1))Gs(L:N+L−1) + n(L:N+L−1), (2.13)

where G is the circulant matrix given by

G =

g0 0 0 . . . 0 0 gL−1 gL−2 . . . g2 g1

g1 g0 0 . . . 0 0 0 gL−1 . . . g3 g2

......

......

......

......

......

...

0 0 0 . . . 0 gL−1 gL−2 gL−3 . . . g1 g0

. (2.14)

Denoting the vector FH [g0 0 0 0 . . . 0 gL−1 . . . g2 g1]T as h, we note that


H=diag(h)= FGFH . Now, the DFT of the received signal is given by

Fr(L:N+L−1) =Fdiag(pr(L:N+L−1))GFHd + Fn(L:N+L−1) (2.15)

=Fdiag(pr(L:N+L−1))FHFGFHd + Fn(L:N+L−1) (2.16)

=Fdiag(pr(L:N+L−1))FHHd + Fn(L:N+L−1) (2.17)

=QrHd + Fn(L:N+L−1) (2.18)

where, Fdiag(pr(L:N+L−1))FH is a circulant matrix and is denoted by Qr. It can be shown

that Fn(L:N+L−1) is an uncorrelated white noise vector distributed as CN (0, 2σ2I). The

rows of Qr are circular shifts of the frequency domain representation of the phase noise

vector given by 1√N

FHpr(L:N+L−1). We define the vector c=[c0 c1 . . . cN−1]T to be

c =1√N

FHpr(L:N+L−1), (2.19)

and note that the lower order frequency components of phase noise are given by c1, cN−1,

c2, cN−2 etc. Using the vector c, the kth component of (2.18) can be written as

(Fr(L:N+L−1)

)k

= c0dkhk +N−1∑

l=0,l 6=k

dlhlc(l−k)mod N +(Fn(L:N+L−1)

)k. (2.20)

Equation (2.20) clearly illustrates how phase noise affects the received signal. Note

that c0 is the common phase error given by

1

N

k=N+L−1∑k=L

ejθk ≈ 1 +1

N

∑θk = 1 + jθ. (2.21)

Its effect is to rotate every received symbol by the average phase angle θ. The other

effect is the inter-carrier interference resulting due to the higher order components of

phase noise. We collectively refer to the components c1, c2, . . . , cN−1 as the higher

order components of phase noise. Another pertinent observation to be made is that the

phase noise sequence affecting the cyclic prefix does not play a role in detection. This is

not the case with transmitter phase noise as we shall see next.


2.3.2 Transmitter Phase Noise

Let θt = [θt0 θt1 . . . θtN+L−1]T be the phase noise sequence affecting the transmitted

signal and let pt = [ejθt0 ejθ

t1 . . . ejθ

tN+L−1 ]T . Then, the transmitted signal is given by

diag(pt)s, and it can be noted right away that the transmitted signal is no longer cyclic

and as a result the received signal will not be completely ICI free. Defining ptcyc =

[ejθtN ejθ

tN+1 . . . eθ

tN+L−1 eθ

tL eθ

tL+1 . . . eθ

tN+L−1 ]T , where we have replaced the first L

components of pt with its last L components, the received signal can be written as

r =Gdiag(pt)s + n (2.22)

=Gdiag(ptcyc)s + Gdiag(pt − ptcyc)s + n (2.23)

After removing the cyclic prefix and taking the DFT of the received signal, we get

Fr(L:N+L−1) =FGFHFdiag(pt(L:N+L−1))FHd (2.24)

+ FG(L:N+L−1,0:N+L−1)diag(pt − ptcyc)s + Fn(L:N+L−1)

=HQtd + FG(L:N+L−1,0:N+L−1)diag(pt − ptcyc)s + Fn(L:N+L−1) (2.25)

=HQtd + FG(L:N+L−1,0:L−1)diag((pt − ptcyc)(0:L−1))s(0:L−1) + Fn(L:N+L−1),

(2.26)

where Qt is Fdiag(pt(L:N+L−1))FH . As one can see from (2.26), the effect of transmitter

phase noise on the received signal is a little different when compared to that of receiver

phase noise. While the effect of CPE is still the same, ICI results not only from the higher

order components of phase noise but also because of the loss of the cyclic nature of the

transmitted signal, which manifests itself as the second term in (2.26). Further, one can

see that the phase noise affecting the cyclic prefix also contributes towards ICI and hence

any scheme that looks at mitigating ICI needs to consider the whole phase noise sequence

of length N+L as opposed to considering just the length-N phase noise sequence as in

the case of receiver phase noise. The additional ICI term has been erroneously left out

in many papers that try to address the issue of transmitter phase noise.


Suppose one were to ignore the second term in (2.26), the DFT of the received signal

in the presence of both transmitter and receiver phase noise can be written as

Fr(L:N+L−1) = QrHQtd + Fn(L:N+L−1). (2.27)

Letting ct0 represent the CPE of transmit side phase noise and cr0 represent the receive

side CPE, and denoting (Qr − cr0I) as Qr

and (Qt − ct0I) as Qt, we can write (2.27) as

Fr(L:N+L−1) =(Qr

+ cr0I)H(Qt+ ct0I)d + Fn(L:N+L−1) (2.28)

=(cr0I)H(ct0I)d + QrH(ct0I)d + (cr0I)HQ

td + Fn(L:N+L−1) (2.29)

=(cr0ct0)Hd + ct0Q

rHd + cr0HQ

td + Q

rHQ

td + Fn(L:N+L−1). (2.30)

This equation explicitly shows the combined effect of transmitter and receiver CPE

on the received signal. Since transmitter and receiver CPEs jointly rotate the received

symbols, we can use a single CPE estimation scheme to estimate both the CPEs. While

the algorithm developed in Chapter 3 is used to compensate for receive side CPE, the

same algorithm can be used to jointly estimate the effective CPE by taking into the

account the statistics of both the CPEs.

2.3.3 Multiple Antenna Scenario

In the case of a point to point MIMO scenario, we assume that a single frequency synthe-

sizer generates the carrier signal for all the RF (radio frequency) chains. Thus, signals

transmitted from/received at all the antennas are affected by the same phase noise se-

quence. Suppose a MIMO link consists of Nt transmit antennas and Nr receive antennas

with Nt > Nr, we split an (NNt) length data vector d into Nt vectors of length N each,

as shown in 2.31, where each dk denotes the data vector for the kth antenna. The IDFT

of the data vector is given by


[d11 d12 . . . d1N ]→[s11 s12 . . . s1N ]

[d21 d22 . . . d2N ]→[s21 s22 . . . s2N ]

[dNt1 dNt2 . . . dNtN ]→[sNt1 sNt2 . . . sNtN ]

[r11 r12 . . . r1N ]

[r21 r22 . . . r2N ]

[rNr1 rNr2 . . . rNrN ]

Figure 2.4: Figure illustrating transmitted and received vectors in an Nr × Nt MIMO

link.

s = [sT1 sT2 . . . sTNt ]T = (INt ⊗ FH)[dT1 dT2 . . .dTNt ]

T = (INt ⊗ FH)d, (2.31)

where ⊗ denotes the Kronecker product and INt represents the Nt ×Nt identity matrix.

The vectors s1, s2 to sNt are transmitted on antennas 1 to Nt as shown in Figure 2.4.

Let G denote the (NNr)× (NNt) time domain channel matrix where the (i, j)th N ×Nsub-block represents the circulant channel matrix between the jth transmit antenna and

the ith receive antenna. Denoting the N × 1 (after removing the cyclic prefix) received

vector at the kth receiver antenna as rk, we define the concatenated received signal to be

the vector r given by [rT1 rT2 . . . rTNr

]T . We can write the DFT of the received signal r as

(INr ⊗ F)r =(INr ⊗ F)G(INt ⊗ FH)d + n (2.32)

=Hd + n, (2.33)

where (INr ⊗ FH)G(INt ⊗ FH) can be shown to consist of blocks of N × N diagonal

matrices and is denoted as H.


In the presence of receiver phase noise pr 2, the expression for the received signal can

be written as

(INr ⊗ F)r =(INr ⊗ F)(INr ⊗ diag(pr))G(INt ⊗ FH)d + n (2.34)

=(INr ⊗ F)(INr ⊗ diag(pr))G(INt ⊗ FH)Hd + n, (2.35)

Using the identity (A⊗B)(C⊗D) = (AC⊗BD), we can write (2.35) as

(INr ⊗ F)r = (INr ⊗Qr)Hd + n, (2.36)

where Qr is as defined before. It is clear that even in the case of multiple antennas,

the effect of phase noise on the received signal is the same. CPE rotates the symbols

of interest by a certain angle while the higher order components of phase noise result in

ICI.

In the case of transmitter phase noise, it can be shown that the received signal can

be written as

(INr ⊗ F)r = H(INr ⊗Qt)d + n, (2.37)

while ignoring the additional interference term that results due to the loss of the cyclic

nature of the transmitted signal. An additional interference term very similar to that in

(2.26) results if one were to take the loss of cyclic nature of the transmitted signal into

account.

2.3.4 Multi-User Scenario : Uplink/Downlink

Since we have established the similarity between the effects phase noise has on SISO and

MIMO scenarios for a point to point link, we only discuss the SISO Multi-User scenario

here. It is easy to see that the cases of uplink and downlink OFDMA with receiver

phase noise, and downlink OFDMA with transmitter phase noise are all equivalent to

2pr is an N × 1 vector, we do not consider the phase noise affecting the cyclic prefix.


single user scenarios because there is only one phase noise sequence affecting the received

signal. The case of uplink OFDMA with transmitter phase noise is quite different and

needs more attention. We discuss this particular case in more detail.

Let there be K users each of whom has been allocated M sub-carriers out of a total

of N sub-carriers. Denote the M × 1 symbol vector of the kth user as dk and the

concatenated symbol vector [dT1 dT2 . . .dTk ]T as d. The transmit vector (without the cyclic

prefix) corresponding to the kth user is given by

sk = Tkdk, (2.38)

where Tk is a N × M sub-carrier mapping matrix that assigns the M data symbols

to one of the M sub-carriers allocated to the user. Let Hk be the frequency domain

channel matrix corresponding to the channel between the kth user and the base station.

The overall received vector in the frequency domain, after the N -DFT operation can be

written as

Fr =K∑k=1

HkTkdk + n. (2.39)

In the presence of transmitter phase noise, the DFT of the received signal can be written

as

Fr =K∑k=1

HkQtkTkdk + n, (2.40)

where Qtk = Fdiag(ptk)F

H , ptk is the phase noise sequence corrupting the transmitted

signal at the kth user. In (2.40) we have ignored the additional ICI term that results due

to non-cyclic nature of the transmitted signal. It can be observed from (2.40) that symbols

corresponding to different users undergo a rotation corresponding to the CPE affecting

that user. In addition to CPE, ICI results from the mixing of data streams corresponding

to different users. Two immediate questions arise from the signal model. One is to find

out which user is likely to be the most disruptive in terms of the interference caused by

that particular user and two, to find out which user is most affected by a disruptive user.


The kth user is likely to be disruptive if the off-diagonal terms in Qtk are significant, i.e.,

N−1∑l=1

E[|(ck)l|2] > N−1∑

l=1

E[|(cj)l|2] , for j = 1 to K, j 6= k. (2.41)

If a user is operating in a regime where the CPE term contains most of the energy of

the phase noise, then the higher order terms tend to be small. Since these higher order

components occupy the off-diagonal positions in Qtk, such a user is likely to cause very

little interference. On the other hand, if the higher order components of phase contain

most of the energy, off-diagonal entries in Qtk become significant and cause larger ICI.

Since a higher variance of the phase noise process can also lead to more energy in the

off-diagonal entries, a user suffering from severe phase noise is also likely to cause higher

interference. As to the question of which user suffers most because of the resulting ICI,

since the PSD of phase noise tapers off rapidly beyond the loop bandwidth, most of the

energy in a phase noise sequence is contained in the frequency components corresponding

to the first few orders. Hence, the largest contribution to interference on a particular

sub-carrier is likely to come from users occupying adjacent sub-carriers. As a result, users

occupying sub-carriers adjacent to a disruptive user are likely to suffer the most.

2.4 Relationship Between CPE and ICI

Having drawn attention to the two issues of rotation due to CPE and ICI due to the

higher order components, we now discuss the interplay between these two effects. Recall

that the average power of the phase noise sequence and in turn its frequency domain

representation is given by σ2θ , the square of the RMS value of the process. In the vector

c defined in (2.19), the first term c0 corresponds to CPE while the rest of the terms from

c1 to cN−1 contribute towards ICI. Since the total power is fixed, this means that the

two effects are coupled to each other i.e., if the variance of the CPE term is high, it in

turn means less ICI, and similarly more ICI means lesser variance in CPE. Before we go

any further, we would like to point out that hence forth the signal model that we will


investigate in detail is the one corresponding to SISO point-to-point link with receiver

phase noise, as given in (2.20).

Having identified that the two effects are coupled to each other, we explore how

different parameters affect the energy split. We also try to define what a CPE-dominant

regime and an ICI-dominant regime are.

16-QAM

CPE % of errors

16 0% (0/16)

17 25% (4/16)

18 25% (4/16)

19 50% (8/16)

20 50% (8/16)

21 75% (12/16)

64-QAM

CPE % of errors

7 0% (0/64)

8 18.8% (12/64)

9 43.8% (28/64)

10 56.3% (36/64)

11 68.8% (44/64)

12 75% (48/64)

13 75% (48/64)

14 75% (48/64)

256-QAM

CPE % of errors

3 0% (0/256)

4 23.4% (60/256)

5 45.3% (116/256)

6 60.9% (156/256)

7 75% (192/256)

Table 2.1: Table listing number of errors that are likely to occur due to CPE while using

16/64/256-QAM constellations.

We first make the important note that the nature of errors that result due to CPE

are different from those resulting due to ICI. Further, while the effect of ICI is the same

on any constellation, this is not the case with CPE. Constellations of small size such as

BPSK and QPSK are highly tolerant to rotation by CPE. As an example, for a QPSK

symbol to be detected in error exclusively because of CPE, CPE needs to be higher than

45, clearly an improbable event. Such is not the case for higher order constellations such

as 64-QAM, 256 QAM. In fact, Table 2.1 shows the number of symbols of a constellation

that are likely to be in error for different degrees of rotation, while ignoring the effects of

ICI and additive noise. It can be seen from Table 2.1 that an instance of a phase noise


sequence with CPE greater than 9 will cause a majority of the symbols of the 64-QAM

constellation to be detected in error. Thus, an instance of high CPE can be a potentially

catastrophic event where there could be lots of errors and recovery from such an event,

even with an error correcting code, might not be possible. This is what sets apart the

effect of CPE from that of ICI. While ICI behaves more like additive noise resulting in

a few random errors in every OFDM symbol, certain CPE events tend to cause errors in

bursts that affect the whole OFDM symbol. Since it can happen that the errors resulting

because of a high CPE event cannot be corrected even with an error correcting code,

these events determine the performance at high SNR and the error floor characteristics.

The key takeaway from this is that as far as CPE is concerned, close attention must be

paid to the frequency of high CPE events, where ‘high CPE’ is a constellation dependent

definition.

Another cause of concern is that any blind CPE estimation scheme involves an initial

step where the symbols are estimated while neglecting the presence of phase noise and

it is very important that a good fraction of symbols detected this way are not in error.

This step is key to blind CPE estimation and most existing algorithms are incapable of

reliably estimating CPE when this initial step contains many errors.

To illustrate the point, Figure 2.5 shows the performance of a coded-OFDM system

affected by phase noise. We compare the performance under four scenarios: (a) no phase

noise in the system, (b) detection ignoring the presence of phase noise, (c) genie aided

ICI compensation in the presence of phase noise and (d) genie aided CPE compensation

in the presence of phase noise. 64-QAM, a CPE sensitive constellation was chosen to

map the bits to symbols and the other parameters of the system were chosen so as to

simulate scenarios where high CPE events (defined later) occur frequently (approximately

once every 3500 OFDM symbols). It is clear from the plot that in such a scenario,

uncompensated CPE leads to a premature error floor.

We define a high CPE event to be one where the CPE exceeds a constellation depen-


12 13 14 15 16 17 18 19 2010

−7

10−6

10−5

10−4

10−3

10−2

Eb/No in dB

BE

R

Phase noise ignoredonly CPE compensatedonly ICI compensatedno Phase noise

Figure 2.5: Plot illustrating the effect of the CPE and ICI on bit error probability of a

coded-OFDM system in a frequency selective fading channel.

dent threshold θth. This threshold is dependent only on the nature of the constellation,

and for smaller constellations such as 16-QAM it tends to be much higher. One way

to set the threshold is to pick an angle such that rotation of symbols above that angle

results in more than 40% of the symbols being detected in error. This way of defining the

threshold was motivated by simulations that showed blind estimation of CPE through

decision directed techniques such as [11] are likely to fail when a majority of the symbols

are detected in error. For the 64-QAM constellation that we use in all our simulations,

we define a high CPE event to be one where the CPE is greater than 9. Thresholds for

other M-QAM constellations are given in 2.2. A constellation with a very high threshold

suggests that issues surrounding high CPE events are not likely to arise in the case of


Constellation CPE Threshold Comments

4-QAM 45 not sensitive to CPE

16-QAM 19 not sensitive to CPE

64-QAM 9 sensitive to CPE

256-QAM 5 very sensitive to CPE

Table 2.2: Table listing CPE thresholds for different M-QAM constellations.

that constellation.

We now turn our focus to the frequency of such events. A low frequency of such

events, say once every 105 OFDM symbols, does not merit any special focus on high

CPE events, but suppose they occur once every 103 OFDM symbols, then we need to

have a robust CPE estimation-compensation scheme that is capable of handling such

scenarios. Acceptable frequency of such events is application specific and is dependent

on the target frame error rate. Since we were interested in bit error rates around 10−6,

we felt any frequency of such events greater than one in 105 warranted special attention.

We say a system to be operating in a CPE-dominant regime if the frequency of high

CPE events is higher than what can be tolerated. The next section explores how various

parameters affect the frequency of such events.

2.4.1 Probability of High CPE Events

In the following analysis, we restrict ourselves to the 64-QAM constellation. While the

analysis easily extends to other constellations by just changing the threshold, it is only

meaningful for constellations that are sensitive to CPE.

Given the CPE distribution, the tail probability is reflective of the frequency of high

CPE events. Since CPE is zero mean Gaussian in nature, its variance has a direct bearing

on the tail probability. The CPE variance is given by (2.10). Clearly, it is a function of

the oscillator parameters σ2θ and Ωo as well as the system parameters N and fs. Nominal


values for phase noise RMS and loop bandwidth are around 3 and around tens of kHz

respectively; fs/N ratio i.e. the sub-carrier spacing is generally around tens of kHz. Our

parameter of interest here is the log-probability of a high CPE event, which is given by

log(P(high CPE event)) = log(|θ| > θth) = log(2Q(θth/σθ)

), (2.42)

where Q() represents the tail probability of the standard Gaussian distribution. We now

look at the effect of each of the above parameters on the probability of a high CPE event.

0 10 20 30 40 50 60 70 80 90 100−16

−14

−12

−10

−8

−6

−4

−2

Sampling rate in MHz

Log−

Pro

babi

lity

of h

igh

CP

E

N=64 N=128 N=256

N=512

N=1024

N=2048

Figure 2.6: Plot illustrating the effect of sampling rate on the probability of high CPE

events

Figure 2.6 illustrates the effects of the system parameters N and fs. We set the

oscillator parameters to nominal values of Ωo = 20kHz and σθ = 3 and look at the

probability of seeing high CPE events as we vary fs and N . Clearly small N and high fs

seem to be settings when one should be careful about these events as they start to occur


more frequently. A careful observation also points to a dependence on the ratio fs/N ,

which represents sub-carrier spacing, rather than individual dependence on N and fs.

This observation is in fact substantiated by the fact that we can approximate the CPE

variance as

σ2θ =

σ2θ

N2

(1 + p

1− pN −2p(1− pN)

(1− p)2

)(2.43)

=σ2θ

N2

(1 + e−ζ/N

1− e−ζ/NN −2e−ζ/N(1− e−ζ/N)N

(1− e−ζ/N)2

)(2.44)

≈ σ2θ

N2

(1 + (1− ζ/N)

1− (1− ζ/N)N − 2(1− ζ/N)(1− eζ)

(1− (1− ζ/N))2

)(2.45)

σ2θ =σ2

θ

(2− ζ/N

ζ− 2(1− ζ/N)(1− eζ)

(ζ/N)2

)(2.46)

=σ2θ

(2

ζ− 2(1− e−ζ)

ζ2

)+σ2θ

N

(2(1− e−ζ)

ζ− 1

)(2.47)

≈σ2θ

(2

ζ− 2(1− e−ζ)

ζ2

), (2.48)

where ∆fsubc = fs/N and ζ = 2πΩo∆fsubc

. The first approximation is due to the fact that ζ/N

is very small and goes to zero as N increases. The second approximation neglects terms

dependent on N as they are significantly smaller than terms independent of N , for large

N . The plot illustrates the fact that systems with very large sub-carrier bandwidth are

the most susceptible to high CPE events.

In Figure 2.7, we fix the system parameters and look at the effect of RMS value on

the frequency of high CPE events. The loop bandwidth was set to 20 kHz. Since it is

very probable that the RMS value of the oscillator changes with variation in the physical

conditions surrounding the receiver (temperature etc.) changes, it is important for us

to see how sensitive the probability of high CPE events is to the RMS value. In Figure

2.7, we plot the log-probability curves for different settings of N and subcarrier spacing

fsubc. It is clear that the probability is only dependent on fsubc and that it can change

quite rapidly as the RMS value changes by a few degrees. Hence, even though the system

design might suggest a low probability of a high CPE event, it is better to have some


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−16

−14

−12

−10

−8

−6

−4

−2

0

Phase noise rms value (in degrees)

Log−

Pro

babi

lity

of h

igh

CP

Efsubc

=300kHz

N=(64, 128, ...,2048)

fsubc

=100kHz

N=(64, 128, ...,2048)

fsubc

=75kHz

N=(64, 128, ...,2048)

fsubc

=50kHz

N=(64, 128, ...,2048)

fsubc

=30kHz

N=(64, 128, ...,2048)

fsubc

=15kHz

N=(64, 128, ...,2048)

Figure 2.7: Plot illustrating the effect of the RMS value of phase noise on the probability

of high CPE events

contingency measures.

In Figure 2.8, we look at the effect of the loop bandwidth on the probability of high

CPE events. The plot shows that smaller loop bandwidth results in a higher probability

of seeing high CPE events. This makes intuitive sense because with decreasing loop

bandwidth higher frequencies get attenuated, resulting in phase noise sequences that

have a larger DC component (CPE).

2.4.2 Energy Split Between CPE and Higher Order Terms

The analysis in the previous section provided some insight into high CPE events that are

critical determinants of performance while using a CPE-sensitive constellation. For con-

stellations that are not sensitive to CPE, a different analysis is required. An appropriate


0 10 20 30 40 50 60 70 80 90 100−16

−14

−12

−10

−8

−6

−4

−2

Oscillator bandwidth in KHz

Log−

Pro

babi

lity

of h

igh

CP

E

fsubc

=15kHz

fsubc

=50kHz

fsubc

=100kHz

fsubc

=30kHz

fsubc

=300kHz

Figure 2.8: Plot illustrating the effect of the loop bandwidth Ωo on the probability of

high CPE events

analysis for such constellations is to directly look at the way the energy in a phase noise

process gets split between CPE and rest of the frequency components.

In Figure 2.9, we plot the split in power between CPE and higher order components

as a function of sub-carrier spacing. It is quite clear that as one increases the sub-carrier

spacing, CPE variance grows larger, while increasing loop bandwidth decreases CPE

variance. In relative terms, if a high percentage of energy is concentrated in CPE, a CPE

estimation-compensation scheme is likely to offer better performance gain. Similarly,

when a high percentage of energy is concentrated in the higher order terms, we should

focus on effective interference mitigation. Note that we are only comparing relative

energies, in absolute terms even 20% energy in higher order terms can cause severe

performance loss. To get a grasp on the magnitude of ICI, we look at the SNR degradation

due to phase noise. Such an analysis has been done in [2] for Wiener phase noise. We


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

80

90

100

sub−carrier spacing in kHz

perc

enta

ge o

f ene

rgy

in C

PE

Ωo=10kHz

Ωo=20kHz

Ωo=50kHz

Ωo=5kHz

Figure 2.9: Percentage of phase noise energy concentrated in the CPE term as a function

of subcarrier spacing and loop bandwidth.

briefly present some observations here.

In Figure 2.10, SINR as a function of sub-carrier spacing is plotted for a fixed SNR

of 20dB. The loop bandwidth was set to 20kHz and RMS of phase noise was varied from

1 to 5. It can be seen that phase noise with an RMS of upto 2 − 3 does not cause a

major degradation in SNR even when sub-carrier spacing is very small. ICI effects are

more severe when RMS of phase noise is higher than 3 and when sub-carrier spacing

is lesser than 50 kHz. Such a combination of parameters that lead to a significant drop

in SNR is considered to be an ICI-dominant regime. To operate efficiently in such a

regime, effective ICI cancellation schemes including a turbo receiver might be required.

In Chapter 4, we develop a soft-in soft-out detection algorithm that can comprehensively

mitigate the effects of phase noise and use this algorithm to design a turbo receiver

Figure 2.11 summarizes most of the points made in this section. The diagram shows

the distinction between constellations based on their CPE sensitivity and distinction we


0 50 100 150 200 250 30017.5

18

18.5

19

19.5

20

sub−carrier spacing in kHz

SIN

R

σθ2=1o

σθ2=2o

σθ2=3o

σθ2=4o

σθ2=5o

Figure 2.10: SINR (Signal to Interference and Noise Ratio) as a function of sub-carrier

spacing at SNR=20dB. Loop bandwidth was set to 20kHz.

make in the operating conditions based on the split in energy between CPE and higher

order components. A combination of the above two gives rise to four scenarios, each

requiring a slightly different approach to detection as shown in the diagram.

2.5 Phase Noise Estimation and Mitigation: Litera-

ture Survey

We present a detailed survey of the existing analyses on phase noise and the various

techniques that have been proposed to estimate and compensate phase noise.

We first discuss papers that have addressed phase noise in the SISO point to point

scenario. Except a few papers, almost all of them concern themselves with receiver phase

noise alone. In this section, unless specifically stated, phase noise will implicitly mean


Typ

e of

con

stel

latio

n

Not

ver

y se

nsiti

ve to

CP

ELa

rge

angu

lar

sep.

am

ongs

t sym

b.S

mal

l ang

ular

sep

. am

ongs

t sym

b.

high

CP

E e

rror

eve

nts

prob

able

Sen

sitiv

e to

CP

E

Hig

h C

PE

err

or e

vent

s im

prob

able

eg. 6

4 Q

AM

, 256

QA

Meg

. 4 Q

AM

, 16Q

AM

Mos

t err

ors

due

to IC

I.S

tron

g IC

I can

cella

tion

reqd

.

Use

sim

ple

CP

E e

stm

. eg.

[15]

Use

rob

ust C

PE

est

m. (

see

Cha

p. 3

)N

o ne

ed fo

r tu

rbo

rece

iver

Use

sim

ple

det.

sche

me

eg. [

34,3

6]

Pha

se n

oise

cha

ract

eris

tics

Sm

all s

ub−

carr

ier

spac

ing/

Lar

ge lo

op B

W.

low

CP

E v

aria

nce

Ene

rgy

conc

entr

ated

in h

ighe

r or

der

term

s C

an r

esul

t in

sign

ifica

nt IC

I

Larg

e su

b−ca

rrie

r sp

acin

g/ S

mal

l loo

p B

W.

Ene

rgy

conc

entr

ated

in C

PE

Hig

h C

PE

var

ianc

eS

mal

ler

high

er o

rder

term

sIn

sign

ifica

nt IC

I

No

need

for

turb

o re

ceiv

erU

se s

impl

e de

t. sc

hem

e eg

. [34

,36]

Use

sim

ple

CP

E e

stm

. eg.

[15]

Mos

t err

ors

due

to IC

I.S

tron

g IC

I can

cella

tion

reqd

.

Use

rob

ust C

PE

est

m. a

s a

safe

guar

dU

se tu

rbo

rece

iver

(se

e C

hap.

4)

Use

turb

o re

ceiv

er (

see

Cha

p. 4

)

Hig

hLo

w

Det

ectio

n S

trat

egie

s

θ th

θ th

Fig

ure

2.11

:A

char

ton

det

ecti

onst

rate

gies

tob

ead

opte

dunder

diff

eren

top

erat

ion

condit

ions.


receiver phase noise.

2.5.1 Preliminary Analysis : Early Papers

Pollet et al. [1] were one of the very first researchers to investigate the effect of phase

noise on OFDM in which it was established that multi-carrier systems are orders of

magnitude more sensitive to phase noise than single-carrier systems. In [12] a method

to determine the error probability of an OFDM system in the presence of phase noise is

discussed. Both [1] and [12] considered Wiener phase noise i.e. phase noise modelled as a

Wiener (Random Walk) process. Robertson et al. [13] used a model for phase noise that

was suggested in [14], which is based on [15], this model was based on the phase noise

characteristics of PLL systems and very closely resembles the model described in earlier

sections. The behaviour of CPE variance as a function of N and its auto-correlation

across OFDM symbols is studied.

An important observation is that the analysis in [13] fixes the bandwidth of the OFDM

symbol as the number of sub-carriers is varied. As a result, the overall sampling rate

remains the same, the duration of the OFDM symbol grows with N , and the sub-carrier

spacing decreases with increasing N . Since the duration of the OFDM symbol changes

with N , autocorrelation properties of CPE are bound to vary with N . Such is not the

case in our analysis, where we keep the sub-carrier spacing constant as N varies. As a

result, the duration of the OFDM symbol is independent of N , and only the sampling

rate varies. A major consequence of this is that, the statistics of CPE are invariant to N .

In the former case, since CPE is averaged over varying durations, its statistics vary with

N , and one can easily see that CPE related effects are expected to be the most severe

for OFDM symbols of very short duration.

El-Tanany et al. [16] present a generic analysis on phase noise for any given phase

noise mask and provide a phase noise estimation scheme based on an isolated pilot tone in

the OFDM symbol. Armada et al. [17] provide results on SNR degradation due to phase


noise and presents a CPE estimation scheme based on pilots embedded in the OFDM

symbol. This work also also makes mention of scenarios where CPE can dominate over

the higher order terms, but provides no quantitative results. In [11], non-pilot aided

estimation of CPE is investigated and some decision-directed mechanisms are suggested.

The authors make the observation that not all symbols in a constellation are equally

prone to errors because of CPE. It is this key observation that we exploit in chapter 3.

Before we go any further, we would like to recall the signal model in (2.18), which

can be written as

Fy =QrHd + Fz, (2.49)

where we have denoted r(L:N+L−1) as y and n(L:N+L−1) as z. We also have the equivalent

representation of (2.49) given by

Fy =Dc + Fz, (2.50)

where D is a left circulant matrix, consisting of circular left shifts of the vector Hd. The

vector c is the frequency domain representation of phase noise, as defined before. If one

were to look at just the received signal, one can write

y =PrFHHd + z, (2.51)

where P is a diagonal matrix consisting of the the vector pr(L:N+L−1). Equation (2.51)

can also be written as

y =diag(FHHd)pr(L:N+L−1) + z. (2.52)

These four equations form the basis for most of the algorithms that try to jointly

estimate data symbols and phase noise. Equations (2.51) and (2.52) are the basis for

algorithms that look at estimating the phase noise sequence in time domain, while (2.49)

and (2.50) form the crux of algorithms that look at estimating phase noise in the frequency


domain. While one can estimate phase noise in either domain, estimation in the frequency

domain has the advantage that it suffices to estimate only the first few components to

compensate for a large percentage of ICI. Due to the small bandwidth and the exponential

decay of the PSD, the first few components account for almost all the energy in the phase

noise process, thus enabling easier ICI mitigation.

2.5.2 Phase Noise Estimation and Compensation

Extensive research has been carried out on the effect of Wiener phase noise on OFDM.

Since CPE shows strong correlation in the case of Wiener phase noise, [18] and [19] use

a Kalman filter to predict and pre-compensate for CPE. In [20], the Kalman filter based

CPE predictor and a decision directed mechanism to evaluate the first few significant

frequency components of phase noise are used to compensate for phase noise. Petrovic et

al. [21] make the case for evaluating all the components in a frequency selective setting

as individual components contributing to ICI get scaled according to the channel gains

corresponding to different sub-carriers. Petrovic et al. [22] point out that certain phase

noise realizations lead to bursts of errors, and suggests interleaving and coding across

multiple OFDM symbols to randomize these errors. The effect of phase noise on coded

OFDM transmission is studied in [23] and an iterative procedure to estimate higher order

frequency components of phase noise using the bit LLRs (log-likelihood ratios) computed

by the decoder is discussed in [4].

Another extensive analysis on Wiener phase noise includes [24], which discusses

MMSE (Minimum Mean Square Error) detection of OFDM symbols affected by phase

noise by considering the ICI term as additional noise. It also discusses MMSE estima-

tion of CPE using pilot symbols. In [2] SINR (Signal to Interference and Noise Ratio)

expression for an OFDM symbol is computed without assuming small phase noise. [3]

discusses a decision directed approach to estimate phase noise and data symbols iter-

atively. Both Maximum Likelihood (ML) and LMMSE (Linear MMSE) techniques of


estimating the frequency domain components of the phase noise sequence are discussed.

Two compensation schemes, namely, a decorrelator and an interference canceler are sug-

gested to mitigate phase noise. The LMMSE scheme suggested here is compared against

the scheme we propose in Chapter 3.

There have also been more sophisticated techniques that have been pursued to counter

phase noise. Septier et al. [25] jointly tackle the issue of estimating phase noise, carrier

frequency offset and data symbols. The proposed scheme is non-pilot aided and is based

on Bayesian estimation using sequential Monte Carlo filtering, also referred to as particle

filtering. Merli et al. [26] provide an iterative solution for the joint estimation of phase

noise and carrier frequency offset through the application of the sum-product algorithm

to a factor graph representing the joint aposteriori probability density function. A parti-

cle filtering based strategy is adopted to estimate parameters in the continuous domain.

In [5,6] a soft-in soft-out algorithm for near-optimal OFDM detection in the presence of

phase noise was developed using a probabilistic framework. The framework can jointly

estimate the phase noise sequence along with the data symbol sequence. The theoretical

basis for the algorithm stems from variational inference (VI), which is an approximate

probabilistic inference technique associated with the minimization of an objective func-

tion called variational free energy.

Transmitter phase noise and its effects on the received signal has received very little

attention. Liu et al. [27] have shown an equivalence between transmitter and receiver

phase noise i.e, the transmitter phase noise process can be replaced with an equivalent

process on the receive side. Such an equivalence is shown to hold only under certain

constraints on the transmitter phase noise process, these constraints are equivalent to

operating in a CPE-dominant regime, where the phase noise process can be approximated

by just its CPE component, which preserves the cyclic nature of the transmitted phase

noise. Peng et al. [28] point out the exact effect transmitter phase noise has on the

received signal and propose a compensation scheme at the transmitter based on feeding


back the RF signal. In [29], an iterative LMMSE based joint estimation of transmitter

and receiver phase noise components is presented. The algorithm estimates only the first

few frequency components and does not attempt to separately estimate the transmitter

and receiver phase noise components.

The influence of phase noise on MIMO OFDM was studied in [30] and [31], where

the effects are shown to be the same as that in the SISO scenario. An ML and a least

squares (LS) estimate of the CPE is derived using pilot signals. Empirical results in [32]

show that the ICI in the MIMO OFDM case does not exhibit Gaussian behaviour.

2.5.3 Key Observations

From the point of view of complexity, CPE estimation schemes are fairly straightforward

with just one unknown parameter to estimate and compensate for. The complexity of

estimation is linear in the number of pilots/symbols being used to estimate CPE. To

compensate, every sub-carrier needs to be derotated by the estimated CPE and hence

scales linearly with the number of sub-carriers. Schemes that involve estimating all the

components of phase noise, such as the ML or the LMMSE approach can be implemented

in a computationally efficient way with complexity growing as O(N logN) and O(N3)

respectively [3]. The methods used in [25], [26] and [5] are certainly more complex, with

the method in [26] being an order higher in complexity than the scheme in [3]. The

method in [5] is O(N logN) in complexity, which is the same order of complexity as the

algorithm suggested in [3], but it requires more number of computations.

Another distinction to be made amongst the suggested algorithms is on their de-

pendence on pilots. Almost all CPE estimation-compensation schemes that have been

suggested rely on pilots embedded in the OFDM symbol, [11] being an exception. In

the case of estimating the complete phase noise sequence along with the data vector,

some papers such as [16], embed isolated pilot tones or specific pilot distributions to aid

estimation while most other papers assume blind detection.


For reasons stated in Section 2.5.1, frequency domain estimation seems to have been

preferred over time domain estimation [20], [2], [3]. Time domain approaches include [26],

[25] and [5]. Time domain approaches, with the exception of [5] are all more complex

than the frequency domain approaches.

Most of the papers have chosen to model phase noise as a Wiener process, which has

led to quite a few issues. Firstly, small phase noise assumption does not hold while using

this model, hence, the approximation ejθ ≈ 1 + jθ cannot be used. CPE gets strongly

correlated across OFDM symbols, leading to schemes that suggest pre-compensation for

CPE. Since phase noise need not be ‘small’ anymore, one is forced to assume synchro-

nization at the beginning of every OFDM symbol.

2.5.4 Focus of Our Work

Our efforts are focused on the stationary model of phase noise, which leads to a very

clear understanding of the two primary effects. We do not rely on any kind of pilot

distribution in the system. We exclusively focus on the blind scenario, where there are

no pilots in the OFDM symbol. While in our work we focus exclusively on receiver phase

noise, methods we develop to counter CPE are also applicable to scenarios where there

is transmitter phase noise as well. Our analysis on the energy split between CPE and

ICI also applies to transmitter phase noise.

Our approach to estimating CPE has a much wider operating range i.e., it can cor-

rectly estimate CPE from a much larger interval than most other existing algorithms.

The effective use of prior statistics of CPE and the current channel state sets it apart

from other algorithms.

Most algorithms suggested to counter ICI involve a decision directed process where

the data symbols and phase noise are estimated iteratively [3,20]. These are hard decision

algorithms that do not use any available priors and do not provide any soft information

on the decisions made. Lin et al. [5] developed algorithms for PHN mitigation using vari-


ational inference that generated soft symbol estimates using a Gaussian approximation.

We extend this framework to design a soft detection algorithm that computes soft bit

estimates of the transmitted bits using the variational inference framework and a discrete

distribution for the bits. The algorithm is capable of incorporating prior information on

bits and hence fits in naturally with a soft decoding algorithm for the Forward error

correcting (FEC) code. The algorithm uses a turbo receiver setup to effectively counter

ICI.

Thus with an effective algorithm for CPE estimation and a turbo receiver design for

countering ICI, we present a complete set of solutions for an OFDM system affected by

phase noise.

2.6 Summary

This chapter provided a brief introduction to the phase noise process and the various

issues surrounding its modelling. We focused on how phase noise affects the signal model

under different scenarios and identified its two main effects. We classified constella-

tions based on their CPE sensitivity and studied how CPE behaves as a function of

system/hardware parameters. We identified regimes where the effect of CPE is likely to

dominate ICI, and vice versa. We suggested possible detection strategies one could adopt

under different regimes. This was followed by a detailed survey of all the earlier works

that have focused on phase noise in multi-carrier systems. We distinguished the different

categories/classes of algorithms that have been proposed and elaborated a little on the

specific case we chose to focus on in this thesis.

New contributions in this chapter:

Identification of issues surrounding high CPE events while using CPE sensitive

constellations. Analysis on the frequency of such events and its relation to various

systems and hardware parameters.


Analysis on the energy split between CPE and other higher components of phase

noise and the consequences thereof on detection strategies.

Chapter 3

Blind CPE Estimation

This chapter presents a new algorithm for blind CPE estimation. We first describe

common approaches to CPE estimation using pilots embedded in the OFDM symbols

and then discuss techniques that have been proposed to estimate CPE blindly. We

highlight issues that arise while using these techniques to estimate high CPE values.

We then present our approach to blind estimation and show that by sorting detected

symbols according to their reliability we can avoid estimation errors even under high

CPE situations. We finally present simulation results that establish the advantages of

using the new algorithm.

3.1 Introduction

Most algorithms suggested for blind CPE estimation in OFDM systems involve a decision

directed process where the data symbols are first estimated ignoring PHN and these

decisions are fed back to estimate CPE. These algorithms treat all detected symbols as

correct. As a result, crucial to all these algorithms is the success of the initial detection

of symbols. But, in reality, a lot of errors can result in this first step whenever CPE

is beyond a threshold, details of which were dealt with in Chapter 2. This requirement

greatly restricts the ability of these algorithms to correctly estimate high CPE values.

45

Chapter 3. Blind CPE Estimation 46

s1 s2

Figure 3.1: Figure showing that not all symbols are equally error prone in the presence

of CPE.

One key observation that these algorithms do not take advantage of is that not all

symbols are equally prone to errors. Symbols on subcarriers that see a better channel

are less susceptible to errors due to additive noise while symbols with smaller magnitude

(for M-QAM constellations) are less susceptible to errors resulting from phase noise. To

illustrate the point, Figure 3.1 shows a part of the 16-QAM constellation with the dashed

lines representing the decision boundaries. It is clear from the figure that symbol s2 is

more sensitive to rotation by CPE than s1. Thus, by ordering sub-carriers according to

the probability of a correct decision being made on the symbols transmitted on them, we

can identify the most reliable symbols and use them as virtual pilots in the estimation

of CPE.

3.2 Received Signal Model

The signal model that we focus on in this chapter is the one given in (2.49) for a SISO

point to point link with receiver phase noise. The equation is repeated here for conve-

nience.

r = FPFHHd + n (3.1)


The matrix P is a diagonal matrix given by diag(ejθr) ≈ diag(1 + jθ), where θ is the

PHN sequence affecting the signal after removing the cyclic prefix. n is complex white

Gaussian noise with variance σ2 each in the in-phase and quadrature dimensions. The

phase noise is modelled as an AR(1) process, as detailed in Chapter 2.

3.2.1 Existing Pilot-Based Approaches to CPE Estimation

Pilot aided CPE estimation only considers sub-carriers with pilots. Let κ represent the set

of indices corresponding to the pilots in an OFDM symbol. Let rκ denote the frequency

domain representation of the received signal corresponding to the sub-carriers with pilots.

Define Hκ, dκ and nκ analogously. We will treat the ICI resulting from phase noise as

just another noise term with a variance of σ2ICI per dimension. From (2.20) we can write

the variance as

2σ2ICI =E[|d|2]E[|h|2]

N−1∑k=1,

E[|ck|2] (3.2)

≈E[|d|2]E[|h|2](σ2θ − σ2

θ), (3.3)

where E[|d|2] is the average symbol energy and E[|h|2] is the average energy contributed

by the channel per sub-carrier. The approximation holds only when the small angle

approximation is valid. By treating the ICI as additional noise, the received signal

corresponding to the kth sub-carrier can be written as

rk = cohkdk + nICI + n. (3.4)

Using this simplified model, the ML estimate of CPE is given by [11]


As one would expect, the mean squared errors go down with increasing number of

pilots, but the decrease is dependent on the channel conditions. One drawback of using

pilots to estimate CPE is that the estimates can be unreliable if the pilot signal energy

is too low, due to fading at the pilot tones, an inadequate number of pilot symbols,

or insufficient transmitted pilot energy. This is particularly an issue in coded OFDM

systems as they operate at much lower SNRs.

If there are no pilots embedded in an OFDM symbol, then one can make a preliminary

estimate of the symbols while ignoring the PHN and use these symbol estimates to

compute the CPE. In general, all the symbol decisions are used while estimating CPE.

Denoting the tentative hard decisions on the symbols as d, under the assumption that

no errors have been made, the ML and MAP estimates can be written as

ˆθML ==((Hd)Hr)

|Hd|2 (3.17)

ˆθMAP =σ2θ=((Hd)Hr)

σ2 + σ2ICI + σ2

θ|Hd|2 (3.18)

However, due to the possibility of making errors in the initial symbol decisions the

CPE estimate is not very reliable and hence an iterative procedure must be adopted to

get a better estimate.

3.3 Blind CPE Estimation Under Detection Uncer-

tainty

3.3.1 Key Idea– Auxiliary Variable

Clearly, not all symbols are detected correctly in a practical scenario. Even in the high

SNR regime, if the CPE exceeds a certain threshold, detection errors are very likely to


be made. The estimates in the previous subsection do not take this into consideration.

In this section we present a more careful formulation of the CPE estimation problem.

We first introduce an auxiliary binary random vector u. uk, the kth entry of the vector

u is one if the kth symbol has been detected correctly and is zero otherwise. Rather than

looking at estimation of θ in isolation, we look at jointly estimating θ and u as follows:

(ˆθ, u) = arg maxθ,u

p(θ,u|r, d) (3.19)

= arg maxθ,u

p(r|θ, d,u)p(u, θ|d) (3.20)

= arg maxθ,u

p(r|θ, d,u)p(u|θ, d)p(θ) (3.21)

≈ arg maxθ,u

p(r|θ, d,u)p(u|d)p(θ) (3.22)

= arg maxθ,u

p(r|θ, d,u)p(θ)N∏k=1

p(uk|dk) (3.23)

Equation (3.22) follows from (3.21) if we treat θ and u as independent variables.

While this is clearly not true, it simplifies the maximization greatly. For the moment,

we just assume that for any dk, the value of p(uk|dk) is known to us. Now, (3.23) can be

written as

(ˆθ, u) = arg maxθ,u

(− wu ln(2πσ2) +

N∑k=1

ln(p(uk|dk))− 1

2σ2θ

(θ)2

− 1

2σ2

∑k:uk=1

|yk − (1 + jθ)hkdk|2)

(3.24)

= arg maxθ,u

(au(θ)2 + bu(θ) + cu

)(3.25)


where,

au =

[− 1

2σ2θ

−N∑k=1

1

2σ2uk|hkdk|2

](3.26)

bu =

[N∑k=1

1

σ2uk=(yk − hkdk)(hkdk)∗

](3.27)

cu =N∑k=1

uk

− ln(2πσ2)− 1

2σ2|yk − hkdk|2+

ln

(P (uk = 1|dk)P (uk = 0|dk)

)+

N∑k=1

ln(P (uk = 0|dk)

). (3.28)

In (3.24), wu is the weight of the vector u. The maximization in (3.25) involves a

search over the continuous parameter θ and all possible N -tuples of the random vector

u. This is an instance of a Mixed Integer Non Linear Problem (MINLP) and solving

for the optimum in such cases is not easy. In this particular case, since the problem

is convex in the continuous parameter θ when the binary vector is held constant, there

are methods which when given enough time and computational resources can find the

optimal solution [33]. But, as pointed out in [33], such methods are computationally

intensive and are not suited for real time applications. With simplicity of optimization

in mind and the fact that it is possible to initialize the optimization at a very good initial

point (shown later), we adopt a likelihood ascent search (LAS) algorithm along the lines

of that suggested in [34]. The algorithm presented here is a sequential likelihood ascent

search (SLAS) with one binary variable being updated at each stage. The algorithm can

be broken down into the following steps.

1. Sort the subcarriers in decreasing order of the tolerance to CPE of the symbols dk.

Computing the CPE tolerance of a symbol is elaborated upon in the next section.

2. Assume that the first l0 subcarriers after sorting have been detected correctly.

Simulations show that l0 can typically be set to a value between 4 to 8. This forces

the first l0 entries in u to be one. Set all other entries in u to zero.


3. Find the θ that maximizes the likelihood function in (3.24). Toggle the (l0 + 1)th

entry in u to one and find the θ that maximizes (3.24). If the new maximum of the

likelihood function is greater than the previous maximum, fix the (l0 + 1)th entry

to one else set it to zero.

4. Proceed as in step 3 for all the remaining subcarriers in a sequential manner until

all subcarriers are exhausted.

At the end of the above procedure, we end up with a vector u that gives a list of all

the subcarriers that have been taken into consideration for the estimation of θ, and an

estimate of θ obtained through a local search. While this estimate of θ might not be the

optimal estimate, it is hoped that the initialization ensures that the process converges to

a point very close to the global optimum.

3.3.2 Computing p(uk|dk)

In this section we discuss how we compute the conditional probability p(uk|dk).One basic assumption we make is that |dk| = |dk|. Since phase noise only alters

the phase of the transmitted symbol, this assumption holds in the regime where errors

due to CPE dominate over those due to additive noise (ADN). Also, the discussion that

follows only applies to constellations with square decision boundaries such as 16-QAM,

64-QAM etc. Under the above assumptions, the tolerance to CPE of a symbol dk that is

|dk| distance away from the origin with the decision boundary B surrounding the symbol

forming a square of side ‘s’ is illustrated in Figure 3.2.

We define the tolerance to CPE θtol to be the maximum angle of rotation that a

symbol can tolerate before it falls outside the decision boundary. It is easy to see from

Figure 3.2 that the tolerance to CPE of such a symbol lies between the angle subtended

by the incircle (θin) and the circumcircle (θcir) of the decision region B at the origin. It is

equal to the angle subtended by the incircle when the symbol is very close to one of the


θcirθtolθin

dk

B

|dk|

Figure 3.2: Phase noise tolerance of a symbol with a square decision region.

axes and is equal to the angle subtended by the circumcircle when the symbol is close to

the 45/135 lines. Hence, for a symbol that is |dk| away, one can write

sin−1

(s

2|dk|

)≤ |θtol| ≤ sin−1

(s√

2|dk|

). (3.29)

Note that this expression holds regardless of the value of the fading coefficient at

each sub-carrier, and hence for the entire OFDM symbol. Since the effect of the chan-

nel parameter is to either shrink or expand the constellation (rotation can always be

compensated for without any consequences), this has the same effect on both ‘s’ as well

as the distance of the symbols from the origin and hence, the CPE tolerance remains

unchanged.

Now, with regard to symbol detection, the combined effect of PHN and ADN leads

to four scenarios. Scenarios that lead to a correct decision include the situation where

the CPE rotation is small enough to keep the transmitted symbol within the decision

boundary and the ADN too is small enough to ensure correct decision. The other scenario

that leads to correct decision is when the CPE is high enough to rotate the symbol to

a point outside the decision boundary, but the ADN is such that it brings the received

symbol back within the decision region. Let us call the former event as E1 and the latter

as E2. Computing the probabilities P(E1) and P(E2) gives us the probability of correct


decision. Clearly, P(E2) is negligible in comparison to P(E1) and hence we focus on

2|h|

2|h||h|dejθ|h|d

B

Figure 3.3: Effect of PHN and ADN on a symbol

computing P(E1) in the rest of this section. We introduce two classes of symbols for the

convenience of the discussion that is to follow. We define a symbol to be of type I if it is

on either the real or imaginary axis and type II if it is on the 45/135 lines.

Assume that a symbol d was transmitted on a subcarrier with channel parameter h

associated with it. Let Ω represent the set of all θ that keep |h|d within the decision

region B, then Ω = [−θtol θtol]. Let Nrθ and Niθ represent the set of values of the additive

noise in the in-phase and quadrature directions that keep ejθ|h|d within B (refer to Figure

(3.3)). We can write

P (E1) =

∫θ∈Ω

P (Nrθ)P (Niθ)p(θ)dθ (3.30)

=2

∫ θtol

0

P (Nrθ)P (Niθ)p(θ)dθ, (3.31)

where P(.) denoted the probability of a certain event. The trajectory taken by a symbol

as θ increases until it leaves the decision region B is dependent on the exact value of

the symbol and not just its magnitude. Since the knowledge of the exact symbol is not

available at the receiver, we consider two extreme trajectories, the one along a diagonal

of B (see Figure 3.5) and the other along a line parallel to one of the sides and passing

through the center of B (see Figure 3.4) and treat P(E1) to be the average of the integral


over these two trajectories. The first trajectory corresponds to type II symbols while

the second trajectory corresponds to type I symbols. For type I symbols we set θtol =

sin−1(

s

2|dk|

)and for type II symbols, we set θtol = sin−1

(s√

2|dk|

).

2|h|

2|h||h|dejθ|h|d

B

Figure 3.4: Figure illustrating the effect

of CPE on a type I symbol.

2|h|

2|h||h|d

ejθ|h|d

B

Figure 3.5: Figure illustrating the effect of

CPE on a type II symbol.

Since an explicit computation of the integral is not possible along either trajec-

tory, we approximate P (Nrθ)P (Niθ) using linear functions. But first, we make some

observations on the product P (Nrθ)P (Niθ). Note that for any symbol, the product

P (Nrθ)P (Niθ) monotonically decreases as θ increases. It takes the highest value of

K0 =(

1− 2Q(|h|σ

))2

at θ = 0 and decreases to approximately K2 =(

0.5−Q(

2|h|σ

))2

at θ = θtol for type II symbols and decreases to K1 =(

0.5−Q(

2|h|σ

))(1− 2Q

(|h|σ

))at θ = θtol for a type I symbol.

Note that for a type II symbol, both, P (Nrθ) and P (Niθ) are of the form

P (Nrθ) =

∫ 1σ

(|h|+ 1√2|hd| sin(θ))

− 1σ

(|h|− 1√2|hd| sin(θ))

g(x)dx (3.32)

where g(x) is the standard Gaussian distribution. For type I symbols, one of P (Nrθ) or

P (Niθ) is of the above form while the other is a constant and is equal to

P (Nrθ) =

∫ 1σ|h|

− 1σ|h|g(x)dx. (3.33)


Further, note that for large |h| the RHS of (3.32) remains almost constant over a long

range of θ until θ equals a certain critical value of θc. To quantify this better we define

θc to be that angle below which the absolute value of the lower limit in the integral in

(3.32) is greater than 3 standard deviations. Thus,

θc = sin−1

√2

|d|(

1− 3σ

|h|)

. (3.34)

Since a negative θc does not makes sense, the critical angle is defined only when |h| > 3σ.

Thus whenever |h| > 3σ, we can split the integral in (3.30) into two parts:

P (E1) =

∫θ∈Ω

P (Nrθ)P (Niθ)p(θ)dθ

=2

∫ θc

0

P (Nrθ)P (Niθ)p(θ)dθ + 2

∫ θtol

θc

P (Nrθ)P (Niθ)p(θ)dθ (3.35)

Using a linear approximation to the product P (Nrθ)P (Niθ) in the second integral and

treating the product as a constant in the first integral, for type II symbols we get,

P (E1) = 2

∫ θc

0

K0p(θ)dθ + 2

∫ θtol2

θc

[K0 + K2−K0

θtol2θ]p(θ)dθ (3.36)

= 2K0

[1−Q

(θtol2σθ

)]+

2σθ(K2−K0)√2π(θtol2−θc)

[e−

θ2c

2 − e−θ2tol2

2

], (3.37)

where θtol2 denotes the CPE tolerance of type II symbols. θtol1 is defined similarly. A

similar computation for type I symbols yields

P (E1) = 2K0

[1−Q

(θtol1σθ

)]+

2σθ(K1−K0)√2π(θtol1−θc)

[e−

θ2c

2 − e−θ2tol1

2

](3.38)

(3.39)

Taking the average of the two expressions, one gets

P (E1) = K0

[2−Q

(θtol1σθ

)−Q

(θtol2σθ

)]+∑

j=1,2

σθ(Kj−K0)√2π(θtolj−θc)

[e−

θ2c

2 − e−θ2tolj

2

](3.40)

For scenarios where |h| ≤ 3σ, we do not split the integral and use a simple linear approx-

imation over the whole interval. This gives us, for type II symbols,


P (E1) = 2K0

[1−Q

(θtol2σθ

)]+

2σθ(K2−K0)√2πθtol2

[1− e−

θ2tol2

2

](3.41)

and for type I symbols,

P (E1) = 2K0

[1−Q

(θtol1σθ

)]+

2σθ(K1−K0)√2πθtol1

[1− e−

θ2tol1

2

](3.42)

And the average of the expressions in (3.41) and (3.42) gives,

P (E1) = K0

[2−Q

(θtol1σθ

)−Q

(θtol2σθ

)]+∑

j=1,2

σθ(Kj−K0)√2πθtolj

[1− e−

θ2tolj

2

](3.43)

Thus, depending on the channel parameter corresponding to the subcarrier, we use either

(3.40) or (3.43) to compute the probability of a correct decision being made. While this

is clearly a crude approximation to the actual probability, this level of accuracy proves

to be sufficient for the application in hand.

3.4 Simulation Results

To test the performance of the suggested algorithm we set up the following simulations.

We simulated a link using 64-QAM constellation and OFDM with 64 sub-carriers over a

frequency selective channel. The channel was assumed to be a Rayleigh multipath fading

channel with 10 taps and an exponential power delay profile. To illustrate the difference

between the scenarios where high-CPE events are frequent and where they are rare, we

chose two sets of parameters as given in Table 3.1. Setting the high-CPE threshold

at 9, for the first set of parameters with small sub-carrier spacing and a large loop

bandwidth, high-CPE events occur once every 4 × 104 OFDM symbols. For the second

set of parameters, because of large sub-carrier spacing and a small loop bandwidth, high-

CPE events occur once every 500 OFDM symbols. The MATLAB code presented in [35]

was used to generate the PHN sequences.


Parameters Simulation 1 Simulation 2

sub-carrier spacing 50 kHz 300 kHz

Loop bandwidth 20 kHz 10 kHz

Phase noise RMS 3 3

Table 3.1: Parameters used for simulations to test the proposed estimation algorithm.

The algorithm proposed earlier was tested along with decision-directed ML/MAP

(DD-ML/DD-MAP) CPE estimation-compensation algorithms that first detect symbols

ignoring the effects of phase noise, assume no uncertainty in the symbol decisions and use

them to estimate CPE. The parameter l0 in Section 3.3 was set to 8 as it was found from

previous simulations that a pilot driven CPE estimation with at least 8 pilots gives a good

estimate of the CPE. As seen in Figure 3.6, it is clear that there is no difference in the

performance of the proposed algorithm and the DD-ML/DD-MAP estimation schemes

in settings where high-CPE events are infrequent. On the other hand, for the settings

in the second simulation, from Figure 3.7 it is clear that there is an advantage in using

the proposed algorithm. In fact, the curve for the proposed method is indistinguishable

from the one where genie-aided CPE compensation is used.1.

In addition to the above simulations, the new scheme was also tested in a coded

system. Since the SNRs of interest are much lower for a coded system it is important to

test the performance of the algorithm at these SNRs, especially because the assumption

in (3.3.2) that the magnitude of the decoded symbol and the transmitted symbol are the

same need not hold at lower SNRs because the noise may be large enough to bring the

decision statistic into the decision region of a symbol with a different magnitude from

the transmitted symbol. Secondly, because the code is, in general, able to correct errors

due to additive noise or ICI that are random in nature, frame errors are likely to arise

only when the CPE estimation is not perfect and has resulted in an error burst and this

1Note that there is still ICI due to the time-varying part of PHN.


18 20 22 24 26 28 30 32 34 3610

−3

10−2

10−1

Eb/No (in dB)

BE

R

no CPE compensationDD−ML estimationDD−MAP estimationProposed CPE estmationPerfect CPE compensation

Figure 3.6: Simulation 1: BER plots for different CPE estimation schemes.

gives us a better picture of the effectiveness of the proposed algorithm. When no coding

is employed, errors that occur even without PHN tend to dominate errors that occur

because of the combined presence of PHN and ADN, and in such a situation, the efficacy

of estimation methods that address the issue of CPE resulting due to PHN is difficult to

measure.

In these simulations, we used an LDPC code of rate 3/4 and length 2304, taken from

the WiMAX standard. On the transmit side, the message bits were encoded, interleaved

and mapped to symbols from the 64-QAM constellation. The length of the outer code

was chosen so as to span 6 OFDM symbols. On the receiver side, after the reception of

the 6 OFDM symbols, each was compensated for CPE and soft information was passed

to the LDPC decoder. The LDPC decoder was run for 18 iterations. Figures 3.8 and 3.9,

corresponding to the first and the second simulations clearly illustrate scenarios where the

new algorithm is likely to outperform the DD-ML/DD-MAP approaches. Since imperfect


18 20 22 24 26 28 30 32 34 3610

−4

10−3

10−2

10−1

Eb/No (in dB)

BE

R

no CPE estimationDD−ML estimationDD−MAP estimationProposed CPE estimationPerfect CPE estimation

Figure 3.7: Simulation 2: BER plots for different CPE estimation schemes.

CPE estimation affects any detection stage that follows, it is very important to assess

the level of CPE one is likely to encounter and accordingly choose the CPE estimation

scheme. As a safeguard, when using CPE sensitive constellations, it is best to use the

algorithm we have proposed so that irrespective of the operating conditions, high-CPE

events do not affect the performance.

3.5 Summary

In this chapter we developed a new algorithm to estimate CPE that takes into account the

reliability of symbol detection in each sub-carrier. Through a novel use of the likelihood

ascent search algorithm, we identify the most reliable sub-carriers and effectively turn

them into virtual pilots. Through simulations we have established the performance gains

that can be achieved using the new scheme.


12 13 14 15 16 17 18 19 2010

−6

10−5

10−4

10−3

10−2

Eb/No (in dB)

BE

R

no CPE compensationDD−ML estimationDD−MAP estimationProposed CPE estmationPerfect CPE compensation

Figure 3.8: Simulation 1: BER plot of coded-OFDM with different CPE compensation

schemes.


12 13 14 15 16 17 18 19 2010

−6

10−5

10−4

10−3

10−2

Eb/No (in dB)

BE

R

No CPE compensationDD−ML estimationDD−MAP estimationProposed CPE estm.Perfect CPE compensation

Figure 3.9: Simulation 2: BER plot of coded-OFDM with different CPE compensation

schemes.

Chapter 4

Turbo Receiver Design for ICI

Mitigation

This chapter addresses the issue of inter-carrier interference (ICI) due to phase noise.

Rather than adopting a standard ICI cancellation scheme, we adopt a more sophisticated

approach that jointly estimates the transmitted symbols and the phase noise sequence

corrupting the received signal. We use an approximate probabilistic framework called

variational inference (VI), using which we develop a soft-in soft-out (SISO) algorithm

that generates posterior bit-level soft estimates while taking into account the effect of

phase noise. The algorithm also provides an estimate of the phase noise sequence. Using

this SISO algorithm, a turbo receiver is designed by passing soft information between

the SISO detector and an outer forward error correcting (FEC) decoder that uses a

soft decoding algorithm. We compare the performance of the turbo receiver against a

single-pass detection-decoding setup, both with and without the algorithm presented in

Chapter 3 is used to estimate and compensate for CPE.

63

Chapter 4. Turbo Receiver Design for ICI Mitigation 64

4.1 Introduction

As outlined in Chapter 2, certain operating conditions such as large loop bandwidth

and small sub-carrier spacing along with high phase noise RMS give rise to situations

with severe interference, resulting is significant degradation in SNR and consequently,

a much higher bit error rate. Under such circumstances, a comprehensive approach to

phase noise induced ICI mitigation is necessary. In this chapter, we develop a detection

algorithm that jointly detects the transmitted bits and the phase noise sequence affecting

the OFDM symbol. So as to completely alleviate the effects of phase noise, we use this

algorithm to set up a turbo receiver that iterates soft information between the detector

and the decoder.

Joint data and phase noise estimation for single carrier systems and multi carrier

systems such as OFDM has been studied extensively [24], [5,10,36–38]. While for single

carrier systems it is possible to construct the factor graph and adopt a message passing

algorithm [38] such an approach becomes prohibitively complicated for OFDM systems

due to inter-connections between all the transmitted bits. Hence, there is a need to

adopt approximate inference techniques that can make the best use of the available

priors in computing the joint estimate. One such technique that lets us compute soft

estimates in such a scenario is variational inference. For the single carrier case, Nissila [39]

considered the use of variational inference for detection of symbols transmitted over

an AWGN channel in the presence of phase uncertainty. In the case of multi-carrier

systems, Lin and Lim [5] developed algorithms for PHN mitigation using variational

inference that generated soft symbol and phase noise estimates. Variational inference

is an approximate probabilistic inference technique associated with the minimization

of an objective function called variational free energy. Variational inference in essence

computes an approximation to the joint posterior distribution of all unknown variables.

In the case of detection in phase noise, these are the transmitted symbols and the phase

noise sequence. The input to the algorithm are the received signal and priors on the


transmitted symbols and the phase noise sequence. In [5], Gaussian priors were used for

both the unknown vectors. The algorithm outputs a Gaussian posterior distribution for

the transmitted symbols as well as the phase noise sequence.

With the design of a turbo receiver in mind, we note that, typically, an FEC decoder

employs a bit level soft decoding algorithm such as the sum-product algorithm for de-

coding LDPC codes. An FEC decoder working at a bit level and a detection algorithm

working at a symbol level pose a mismatch that needs to be bridged. While one can com-

pute the individual bit priors (log likelihood ratios) using the mean and variance of the

posterior distribution of a symbol, computing the prior mean and variance of the sym-

bols using the extrinsic bit LLRs computed by the FEC decoder is not straightforward.

Hence, keeping the compatibility of the detection and decoding algorithms in mind, we

develop a bit level detection algorithm, using variational inference. The algorithm is

capable of accepting prior information on the transmitted bits and outputs the posterior

distribution of the bits after transmission, while taking the current channel state, phase

noise and additive noise into consideration.

Two advantages arise in developing a bit level detection algorithm. One, there is no

need to impose an artificial Gaussian prior on the symbols. We adopt a more natural

Bernoulli distribution, with zero mean as the prior for transmitted bits. Second, the

symbol level detection algorithm requires an initialization step that necessitates hard

decisions on symbols while ignoring phase noise. In the algorithm that we develop, the

algorithm is initialized to uniform priors on all the bits.

The rest of the chapter is organized as follows. We first introduce the function used to

map bits to symbols, this is a multi-linear function and plays a key role in our derivation.

We then introduce the signal model that we will focus on. Next, we discuss the actual

posterior distribution of the symbols and phase noise, and introduce the concept of

variational inference. We then present the algorithm that uses variational inference to

compute an approximation of the posterior distribution. This approximate posterior


distribution gives the soft estimates of the bits as well as the phase noise sequence. We

finally test this algorithm under different scenarios and establish the performance gains.

We compare the proposed algorithm against an algorithm that involves a decision

directed process. In this algorithm, the data symbols are estimated ignoring PHN and

these decisions are fed back to estimate the frequency domain components of PHN [3].

Using the estimated phase noise, the interference is cancelled out and the symbols are

detected again. This process is iterated either for a fixed number of times or until

convergence. This is a hard decision algorithm, in that no soft information is exchanged

between iterations.

FEC Decoder

Soft−in Soft−outBit−Level VI

DetectorSignal

Received

FEC Decoder


DetectorSignal

Received

(a) Separate detection and decoding

(b) Turbo receiver: joint detection and decoding

Figure 4.1: Figure illustrating the two different settings in which the bit-level detection

algorithm is tested.

For simulating the performance of the proposed algorithm, we consider using the de-

tection algorithm in two ways. One, for separate detection and decoding where messages

are not iterated between the detector and the decoder and two, use it to setup a turbo

receiver as shown in Figure 4.1. We expect the turbo receiver to outperform the former.

The proposed detection algorithm as well as the algorithm in [3] inherently compute

the CPE. Thus, both algorithms by themselves are capable of handling both the effects

of phase noise. Simulations show that this inherent CPE estimation is not capable of


handling high-CPE events and necessitates an explicit CPE estimation-compensation

scheme as a pre-processing step. This aspect is elaborated further in Section (4.4).

4.2 Received Signal Model

4.2.1 Gray mapping : bits to M-QAM symbols ( from [6] )

We assume M to be an integer of the form 22k, where k is a positive integer. Let L

denote log2M and let B ∈ −1, 1N×L represent a matrix of N × L bits that need to be

mapped to N M-QAM symbols. Denote the columns of B as br1, br2, . . . brL/2, bi1, bi2

. . . biL/2. If the resulting vector of M-QAM symbols is represented as d, then, from [6],

we have the following relation :

d =

L/2∑l=1

2l−1

p=L/2

6∏p=1

brp + jL∑

l=L/2+1

2l−1

p=L/2

6∏p=1

bip (4.1)

Here, 6∏

represents element wise product of the vectors. Let the function f denote

the mapping from bits to symbols, so that d = f (B). Further, denote the matrix [br1

br2 . . . brL/2] as Br, and the matrix [bi1 bi2 . . . biL/2] as Bi and define the functions f r

and f i as :

f i(Bi) = j.=(f (B)) (4.2)

f r(Br) = <(f (B)) (4.3)

Hence, d = f (B) = f r(Br) + f i(Bi).

4.2.2 The Received Signal

The signal model that we focus on in this chapter is the one given in (2.49) for a SISO

point to point link with receiver phase noise. Using the representation in Section 4.2.1,


the received signal in the frequency domain can be written as

r = PFHHd + n = PFHHf (B) + n (4.4)

The matrix P is a diagonal matrix given by diag(ejθr) ≈ diag(1 + jθ), where θ is the

PHN sequence affecting the signal after removing the cyclic prefix. The phase noise is

modelled as an AR(1) process, as detailed in Chapter 2. H = diag(h) is the channel

matrix in the frequency domain and the N×L binary matrix B contains the transmitted

bit sequence (d is the corresponding symbol sequence). n is complex white Gaussian

noise with variance σ2 per dimension.

4.2.3 Actual and Postulated Posterior Distributions

From [5], we note that by approximating ejθ to 1+ jθ under the small angle assumption,

one can write the conditional distribution of r given B and θ to be :

p(r|B,θ) = CN (diag(1 + jθ)FHHf (B), 2σ2I)

(4.5)

Further, in [5] it was shown that the optimal detector (i.e. ML estimate of d) for

such a signal model, given the prior distribution of the phase noise sequence θ, has an

exponential complexity in N . Given that the distribution p(r|B) and consequently the

posterior distribution p(B|r) do not lend themselves to efficient optimal detection, we look

at an approximation to p(B|r), such that, computing the optimal estimate corresponding

to the approximated distribution is straightforward. To this end, we note that p(B|r) is

computed by marginalizing p(B,θ|r) with respect to θ, i.e.

p(B|r) =

∫p(B,θ|r)dθ (4.6)

The variational inference approach approximates p(B,θ|r) with a function Q(B,θ) of the

form QB(B)Qθ(θ) such that

QB(B)Qθ(θ) ≈ p(B,θ|r) (4.7)


This kind of an approximation is equivalent to assuming that B and θ are independent

conditioned on r and this in turn implies the maximizer of the distribution QB(B) is the

optimal estimate of B. We further assume that the distribution QB(B) can be factorized

into∏N

n=1

∏Ll=1 Qb(bnl). Such a factorization for the postulated posterior distribution of

the bits, where the distribution is assumed to be independent over n and l, is commonly

known as the mean field approximation. In this paper, we assume Qb(bnl) to be a Bernoulli

distribution with parameter λnl. We postulate the posterior conditional distribution of

phase noise to be a multi-variate Gaussian distribution with mean mθ and covariance

Sθ. Thus, we have

Q(B,θ) = QB(B)Qθ(θ)

=

[N∏n=1

L∏l=1

(λnl)1+bnl

2 (1− λnl)1−bnl

2

]N (mθ,Sθ) (4.8)

=

[N∏n=1

L∏l=1

(1+bnl

2

) 1+bnl2(

1−bnl2

)1−bnl2

]N (mθ,Sθ) (4.9)

where, bnl = 2λnl − 1, is the mean of the postulated posterior distribution Qb(bnl). Note

that inspite of prior knowledge that the phase noise is generated from an AR(1) process,

we do not include this aspect in the postulated posterior distribution of phase noise. We

need the distibution Qθ(θ) to closely resemble the distribution p(θ|r), which is given by

p(θ|r) =

∫p(B,θ|r)dB. (4.10)

It is clear from the above equation that the marginal posterior distribution for θ need

not have the AR(1) property and hence such a structure must not be imposed on the

postulated distribution.

Having fixed the structure of the postulated posterior distribution, it remains now

to compute the parameters of this distribution such that it closely approximates the

actual posterior distribution. The variational inference approach introduces the concept

of variational free energy as a measure of similarity between two distributions. The


variational free energy between the two distributions of interest here is given by

F (Q, p) =

∫B,θ

Q(B,θ) logQ(B,θ)

p(B,θ, r)dBdθ. (4.11)

We use the distribution p(B,θ, r) instead of p(B,θ|r) as they are related simply

through a constant of proportionality. It is to be noted that the free energy expression is

exactly equal to the Kullback-Leibler divergence between Q(B,θ) and p(B,θ|r) to within

an additive constant. The variational inference approach involves the minimization of

the free energy over the parameters of Q(B,θ) so that the resulting distribution closely

approximates the actual posterior distribution.

The actual posterior distribution can be computed by conditioning over the unknowns

as follows:

p(B,θ, r) =p(r|B,θ)p(B)p(θ) (4.12)

Using (4.5), and assuming independent Bernoulli distributed priors on bits, with the lth

bit of the nth symbol having mean µnl (set µnl to 0 if no prior information is available)

and assuming the prior distribution of PHN to be a multi-variate Gaussian distribution

with mean µθ and covariance matrix φθ (if no prior information is available, set mean to

0 and covariance to Φ as given in (2.8)), we can write the posterior distribution as

p(B,θ, r) =CN (diag(1 + jθ)FHHf (B), 2σ2I)×[

N∏n=1

L∏l=1

(1+µnl

2

)1+bnl2(

1−µnl2

)1−bnl2

]N (µθ,φθ). (4.13)

The next section discusses the computation and minimization of the free energy ex-

pression.


4.3 The Bit-Level Variational Inference Algorithm

4.3.1 Free Energy Evaluation

The free energy expression given in (4.11) can be written as the summation of five terms

as shown in (4.14).

F (Q, p) =−∫

B

Q(θ) log p(B)dB︸︷︷︸i

−∫

θ

Q(θ) log p(θ)dθ︸︷︷︸ii

+

∫B

Q(B) log p(B)dB︸︷︷︸iii

+

∫θ

Q(θ) logQ(θ)dθ︸︷︷︸iv

−∫

B,θ

Q(B)Q(θ) log(p(r|B,θ))dBdθ︸︷︷︸v

(4.14)

The exact computation of the free energy expression is given in Appendix A and the

final expression is presented here. Note that the free energy expression is parameterized

by the mean and the covariance matrix of the PHN sequence and the mean value of

the posterior distribution of the bits. Note that when treating the unknown B as a

matrix of Bernoulli random variables, we denote B as the matrix of the means (bnl) of

the corresponding random variables in B. We define Br, Bi, brj and brk in a similar

fashion.

F (mθ,Sθ, B) =

−N∑n=1

L∑l=1

[(1+bnl

2

)log(

1+µnl2

)+(

1−bnl2

)log(

1−µnl2

)]+

1

2tr(φ−1

θ Sθ) +1

2mT

θ φ−1θ mθ

− µTθ φ−1

θ mθ +N∑n=1

L∑l=1

[(1+bnl

2

)log(

1+bnl2

)+(

1−bnl2

)log(

1−bnl2

)]− 1

2log |Sθ|

+1

2σ2

[− rHZf (B)− f (B)

HZHr + f r(Br)

HM0f i(Bi) + f i(Bi)

HM0f r(Br)

+ f r(Br)H

M1f r(Br) + f i(Bi)H

M1f i(Bi) + 1HM2νr + 1HM2νi

](4.15)


4.3.2 Free Energy Minimization

Clearly, closed form expressions for the optimal parameters that minimize the free energy

expression cannot be computed. Hence, we adopt a coordinate-descent approach to

minimization, wherein one parameter is updated while keeping the others constant. Such

an approach will converge to a local minimum. The update to each parameter is obtained

by computing the gradient of the free energy expression given in (4.15) w.r.t the parameter

and setting it to zero. This leads to three update equations, one each for the mean and

covariance matrix of the phase noise and one for updating the mean corresponding to

the bits. The update equations are :

STθ =

[φ−1θ +

1

σ2XmXH

m +1

σ2

(diag

(FHHdiag(νr + νi − |f (B)|2)HHF

))]−1

(4.16)

mθ =Sθ

[− 1

σ2Im(rHXm)T + φ−1

θ µθ

](4.17)

tbrk =tµrk +1

σ2

[diag(αk)<ZHr − diag(δk)M

T1 1

+ diag(αk)(j=M0f i(Bi)−<MT

2 f r(Br))]

(4.18)

tbik =tµik +1

σ2

[diag(βk)=ZHr − diag(Ωk)M

T1 1

+ diag(βk)(=M0f r(Br) + j<MT

2 f i(Bi))]

(4.19)

In the above equations, tbrk and tbik represent tanh−1(brk) and tanh−1(bik) (computed

element wise) respectively, while tµrk and tµik represent tanh−1(µrk) and tanh−1(µik)

respectively, with µrk and µik defined analogous to brk and bik but with prior means.

Further, the derivatives ∂f r(Br)

∂brkand ∂f i(Bi)

∂bikare denoted as diag(αk) and diag(βk). The

derivatives ∂νr∂brk

and ∂νi∂bik

are denoted as diag(δk) and diag(Ωk). Detailed derivation of

the update equations is given in Appendix B.


4.3.3 The Variational Inference Algorithm

Algorithm 1 gives the pseudo code for the steps involved in the VI based soft bit detec-

tion algorithm. We note that because of the coordinate descent approach to free energy

minimization, the algorithm converges to a local minimum and not the global minimum.

Simulations indicated a quick convergence within 3-5 iterations. However, since conver-

gence is only to the local minimum, this is likely to be an issue in high-CPE events and is

actually seen as an error floor in the performance of the stand alone turbo receiver shown

in (4.1). The bit LLRs given by 2(tbrk − tµrk) and 2(tbik − tµik) are the soft messages

passed to the FEC decoder. The SISO FEC decoder is standard for many codes such

as LDPC and turbo and is not described here. For the turbo receiver setup, the phase

noise mean and covariance are re-initialized to 0 and Φ after every iteration between the

detector and the decoder.

4.3.4 Complexity Analysis

In this section we analyze the order of complexity of each step in the pseudocode presented

in Algorithm 1. The first step involves the computation of the vectors νr, νi, αk, δk,

Xm, each of which involve only element wise products of vectors and can be computed in

O(N) time. Next, the update equation of the covariance matrix Sθ involves computing

the inverse of a tridiagonal matrix, with each of the off-diagonal vectors having constant

entries. There exist algorithms that can invert such a matrix in O(N logN) time [40].

Updating mθ is equivalent to solving a tridiagonal system of equations, which is of O(N)

complexity [40]. The bit update equations involve computing four terms. Since the

only dense matrix involved in any of the terms is the DFT matrix, each of the terms

can be computed in O(N logN) time. Thus the overall complexity of the algorithm is

O(N logN).


Algorithm 1 Bit level Variational Inference Algorithm

1: Initialize. µnl ← 0 or given priors; µθ ← 0; φθ ← Φ; mnl ← 0; mθ ← 0; Sθ ← 0;

2: Compute tµrk and tµik for k ∈ 1, 2, . . . L2.

3: for s=1:num iter do

4: Compute νr, νi, αk, βk, δk, Ωk, Xm.

5: Update Sθ using (4.16).

6: Update mθ using (4.17).

7: for k=L/2:1 do

8: Compute tbrk and tbik using (4.18) and (4.19).

9: for p=1:L/2 do

10: Update B, αp, βp, δp, Ωp.

11: end for

12: end for

13: end for

14: Return 2(tbrk − tµrk) & 2(tbik − tµik) for k ∈ 1, 2, . . . L2.

4.4 Simulation Results

To test the performance of the suggested algorithm we set up the following simulation.

We simulated a link using 64-QAM constellation and OFDM with 64 sub-carriers over a

frequency selective channel. The channel was assumed to be a Rayleigh multipath fading

channel with 10 taps and an exponential power delay profile. The sub-carrier spacing was

set to 30 kHz. The loop bandwidth was set to 20 kHz and the standard deviation σθ was

set to 4. The parameters chosen were so as to simulate an ICI dominant regime, with

64% of energy in the phase noise process being contained in the higher order components

and resulting in an SINR of 22 dB when the SNR is set to 25 dB (Eb/No = 20 dB). With

36% of the energy concentrated in CPE, high CPE events occur at a frequency of once

every 5000 OFDM symbols. The MATLAB code presented in [35] was used to generate


the PHN sequences.

12 13 14 15 16 17 18 19 2010

−5

10−4

10−3

10−2

10−1

100

Eb/No (in dB)

Fra

me

Err

or R

ate

(FE

R)

turbo−VI iter1turbo−VI iter2turbo−VI iter3turbo−VI iter4turbo−VI iter5No PHNIgnoring PHNOnepass−VI

Figure 4.2: FER plot of the turbo-VI receiver.

For the turbo receiver setup, an outer LDPC code of rate 3/4 and length 1152 was

used. On the transmitter side, the message bits were encoded, interleaved and mapped to

symbols from the 64-QAM constellation. The length of the outer code was chosen so as to

span 3 OFDM symbols. On the receiver side, after the reception of the 3 OFDM symbols,

each was passed through the bit level detection algorithm and the extrinsic information so

obtained was passed to the soft decoder after deinterleaving. The detector was initialized

to uniform priors at the beginning. In subsequent iterations, the extrinsic information

obtained from the decoder was used as priors. For this particular simulation, we used 5

outer iterations with 6 iterations for the decoder and 5 iterations for the detector.

Fig. 4.2 presents the results of this simulation and the turbo receiver (denoted as

turbo-VI) shows a clear performance gain over a single-pass setup (denoted as Onepass-


VI), where the detector and decoder function separately, with no messages passed from

the decoder to the detector (see Figure 4.1). The receiver performance is compared

against scenarios where there is no PHN and where PHN is completely ignored. For both

these cases and for the one pass setup, the FEC decoder was run for 18 iterations. In

order to reduce the simulation time, the turbo-receiver was invoked only for instances

where a simple receiver consisting of a zero-forcing (ZF) detector followed by the decoder

failed to converge to the correct transmitted bit sequence.

Investigations on the error floor seen in the performance of the turbo receiver revealed

convergence to an incorrect phase noise sequence in a high-CPE event. Since we use the

64-QAM constellation in our simulations, without pilots, it is challenging to distinguish

between the actual symbol being transmitted with a large phase error, and the neigh-

boring symbol being transmitted with a small phase error. Since we initialize the SISO

detection algorithm with phase noise prior set to a zero mean Gaussian distribution, it

is possible that the algorithm does not converge to the right phase noise sequence under

high CPE scenarios. In order to fix this problem, we simulated a setup where CPE is

estimated and compensated for before passing the received signal to the turbo-receiver as

shown in Figure 4.3. Although the parameter settings in our simulations result in signif-

icant ICI, our use of a CPE sensitive constellation results in high CPE events occurring

with a probability of 2× 10−4. Hence, high CPE events cannot be completely neglected.

This setup is expected to outperform the stand alone turbo receiver for frame error rates

below 10−3. For CPE compensation, a genie aided scheme as well as the algorithm

proposed in Chapter 3 are used. The key observation here is that while using a CPE

sensitive constellation, ICI-dominant regime and CPE-dominant regime are not mutually

exclusive. Under certain settings such as ours, one can see significant degradation due to

both these effects simultaneously.

We first simulated the turbo receiver setup along with the CPE compensation scheme

presented in Chapter 3. As one can see in Figure 4.4, the error floor of the turbo receiver


FEC Decoder


Detector

CPE

Compensation

Received

Signal

Figure 4.3: Figure illustrating the overall receiver design used to comprehensively sup-

press the effects of phase noise.

in the high-SNR regime is slightly lowered but more significantly, the single pass receiver’s

performance is significantly better. Since high-CPE events occur only with a probability

of 2 × 10−4, the role of the CPE estimation-compensation scheme is only seen in the

high SNR regime where frame errors rates are lower than 10−3. In Figure 4.4, the CPE

compensation scheme is denoted as CC(PDD)– CPE Compensation (Partially Decision

Directed).

12 13 14 15 16 17 18 19 2010

−5

10−4

10−3

10−2

10−1

100

Eb/No (in dB)

Fra

me

Err

or R

ate

(FE

R)

CC(PDD)+turbo−VI iter1CC(PDD)+turbo−VI iter2CC(PDD)+turbo−VI iter3CC(PDD)+turbo−VI iter4CC(PDD)+turbo−VI iter5Ignoring PHNCC(PDD)+Onepass−VINo PHN

Figure 4.4: FER plot of the turbo-VI receiver with CPE compensation.


The slight widening of the performance gap at high SNR in Figure 4.4 is indicative

of an error floor seen at much lower frame error rates and higher SNRs. This error floor

can arise because of two reasons– either the CPE compensation algorithm has failed to

correctly estimate the CPE in some high-CPE events or the turbo receiver has failed to

converge to the right phase noise sequence in spite of correct CPE estimation and com-

pensation. To determine which of the two is the reason for the error floor, we simulated

the turbo receiver along with genie-aided CPE compensation scheme that completely

compensates for CPE. In genie-aided CPE compensation, the CPE was computed from

the actual phase noise sequence affecting the signal and was used to compensate for CPE

in the received signal. In Figure 4.5, this scheme is denoted as PCC– Perfect CPE Com-

pensation. As one can see from Figure 4.5, there is no error floor seen in this setup. This

shows that once CPE has been correctly compensated for, the detection algorithm is able

to converge to the right phase noise sequence. This also indicates that the error floor seen

in Figure 4.4 was because of imperfect CPE estimation, rather than non-convergence of

the detection algorithm.

Finally, we compare the turbo receiver setup using the proposed detection algorithm

(VI algorithm) against another turbo receiver setup using a decision-directed detection

algorithm as suggested in [3]. The algorithm suggested in [3] iteratively estimates sym-

bols and phase noise in the frequency domain using the equations (2.49) and (2.50). The

algorithm first estimates the symbols while ignoring phase noise and then computes a

linear minimum mean square (LMMSE) estimate of the phase noise sequence. This esti-

mate is used to cancel out the interference from the received signal. To set up the turbo

receiver, soft information on bits is computed from the received signal after complete re-

moval of ICI and passed to the decoder. After a few iterations of the decoding algorithm,

the soft information on bits is used to re-estimate the transmitted symbols and is passed

back to the detection algorithm. The detection algorithm assumes this to be the actual

transmitted symbol vector and proceeds to estimate the phase noise sequence. This pro-


12 13 14 15 16 17 18 19 2010

−5

10−4

10−3

10−2

10−1

100

Eb/No (in dB)

Fra

me

Err

or R

ate

PCC+turbo−VI iter1PCC+turbo−VI iter2PCC+turbo−VI iter3PCC+turbo−VI iter4PCC+turbo−VI iter5No PHNIgnoring PHNPCC+Onepass−VI

Figure 4.5: FER plot of the turbo-VI receiver with perfect CPE compensation.

cess is repeated several times. All simulation parameters for this setup are identical to

the turbo receiver discussed previously.

Just as in the the previous scenario, the algorithm was tested under three scenarios:

no CPE compensation, perfect CPE compensation and CPE compensation using the

algorithm in Chapter 3. The results are compared against the turbo receiver set up using

the VI algorithm and the comparative plot is presented in Figure 4.6. The plot shows

performance curves after 5 outer iterations of the turbo receiver. The plot establishes a

clear gain in performance using the VI algorithm. But it must be pointed out that this

improvement in performance comes at the expense of complexity.


12 13 14 15 16 17 18 19 2010

−5

10−4

10−3

10−2

10−1

100

Eb/No (in dB)

Fra

me

Err

or R

ate

No PHNIgnoring PHNturbo−VICC(PDD)+turbp−VIPCC+turbo−VIturbo−LMMSEPCC+turbo−DD−LMMSECC(PDD)+turbo−DD−LMMSE

Figure 4.6: FER plot comparing the performance of turbo-VI and DD-LMMSE receivers

under different settings.

4.5 Summary

In this chapter we presented a comprehensive solution to address the issue of phase noise

induced ICI in OFDM. Using the principle of variational inference, we developed a soft-

in soft-out bit-level detection algorithm. This algorithm generates soft estimates on the

transmitted bits and phase noise, and was used to build a turbo receiver. The turbo

receiver along with the CPE compensation scheme suggested in Chapter 3 was shown

to substantially alleviate all the effects of phase noise and result in close to optimal

performance.

Chapter 5

Effect of Phase Noise on SC-FDMA

In this chapter we study the effect of receiver phase noise on SC-FDMA, which has been

adopted as the uplink modulation technique in LTE. In particular, we study the effect of

phase noise on the performance of linear receivers adopted in the detection of SC-FDMA

symbols. It is shown that CPE affects SC-FDMA symbols in the same manner as OFDM

symbols. While the higher order components result in ICI, the nature of ICI affecting the

sub-carriers depends on whether an interleaved or a localized allocation of sub-carriers

has been adopted. Its effect on both localized and interleaved SC-FDMA is discussed

along with some suggestions for mitigating the ICI from phase noise.

5.1 Introduction

In order to address the issue of peak-to-average power ratio (PAPR) encountered in uplink

OFDMA, SC-FDMA has been adopted for uplink transmission in the LTE standard. SC-

FDMA can be viewed as a DFT precoded OFDMA scheme. Two common approaches to

allocating sub-carriers to users include localized SC-FDMA and interleaved SC-FDMA.

In localized SC-FDMA each user is allocated a set of contiguous sub-carriers for uplink

transmission. In interleaved SC-FDMA, the users are allocated sub-carriers spread evenly

over the transmission bandwidth. Interleaved SC-FDMA offers some diversity benefits

81

Chapter 5. Effect of Phase Noise on SC-FDMA 82

over localized SC-FDMA. Before analyzing the effects of phase noise on SC-FDMA, we

first set up the signal model corresponding to a multi-user uplink transmission using SC-

FDMA and take a closer look at the linear receivers (Zero Forcing (ZF) and Minimum

Mean Square Error (MMSE)) that are used to detect SC-FDMA signals. We then study

how the signal models corresponding to these linear receivers change in the presence of

phase noise.

5.2 Signal Model

In this section we derive the received signal model for SC-FDMA in the absence of phase

noise. Let there be K users, each of whom has been allocated M sub-carriers out of a

total of N sub-carriers. Each user performs an M-DFT precoding of the symbols,maps the

M-DFT outputs to its assigned sub-carriers, and then performs multi-carrier modulation

with an N-IDFT. Denote the symbol vector of the kth user as dk and the concatenated

symbol vector [dT1 dT2 . . .dTk ]T as d.

ADC/

RF

DAC/

RF

P to

SN point

IDFT

Sub carrier

Mapping

M point

DFT

S to

P

Symbol

Stream

CHANNEL

N point

DFT

Sub carrier

De−mapping

sub−carrier

equalization

channel based

(ZF/MMSE)

M point

IDFT S to

P

P to

S

StreamSlicer

Symbol

Figure 5.1: Block diagram representing the SC-FDMA scheme and the use of a linear

receiver. Note that N > M , and usually, MN

= K, an integer representing the number of

users in the uplink.

The time domain transmit vector (without the cyclic prefix) corresponding to the kth


user is given by

xk = FNTkFMdk (5.1)

where, Tk is a N ×M sub-carrier mapping/permutation matrix and FM is the M ×MDFT matrix. The structure of Tk and some properties associated with operations that

involve these permutation matrices are discussed in Appendix C. Let the channel between

the kth user and the base station be represented as hk in the frequency domain and let

Hk = diag(hk). Then the received signal corresponding to the kth user at the output of

the N-DFT front end is given by (ignoring noise)

yk = HkTkFMdk (5.2)

The overall received vector corresponding to the signals received from all the K users is

given by

y =K∑k=1

HkTkFMdk + n (5.3)

Figure 5.1 shows the basic operations involved in the transmission and reception of

an SC-FDMA symbol. The figure corresponds in particular to a linear receiver, where

frequency domain channel equalization is carried out on a sub-carrier basis. In this

chapter we discuss in detail the effect of phase noise on linear receivers used to detect

SC-FDMA symbols. In Figure 5.1, ‘P to S’ denotes parallel to serial conversion, ‘S to P’

denotes serial to parallel conversion. DAC and ADC refer to digital to analog conversion

and analog to digital conversion respectively. RF represents all radio frequency circuitry

required for up-conversion to carrier frequency.

5.2.1 ZF Equalization

Assuming we have a frequency domain ZF equalizer at the receiver, we extract the infor-

mation corresponding to the kth user from the vector y by first forming an M × 1 vector

consisting of all the sub-carriers corresponding to the kth user and then compensating


for the channel. To aggregate all the sub-carriers allocated to the kth user, we multiply

y with TTk to get

TTk y = TT

kHkTkFMdk +

p=K∑p=1,p 6=k

TTkHpTpFMdp + TT

kn. (5.4)

We then compensate for the channel by multiplying the above with (TTkHkTk)

−1, which

can be shown to be equivalent to TTkHkTk using Property C.7. After this operation, the

received signal can be written as

(TTkH−1

k Tk)TTk y = (TT

kH−1k Tk)T

TkHkTkFMdk +

p=K∑p=1,p 6=k

(TTkH−1

k Tk)TTkHpTpFMdp

+ (TTkHkTk)

−1TTkn (5.5)

= FMdk + TTkH−1

k n, (5.6)

where we have used Property C.5 and Property C.6 to simplify (5.5) to obtain (5.6).

Thus, after collating the sub-carriers and compensating for the channel, all infor-

mation regarding the symbols transmitted by the kth user is contained in the vector

(TTkH−1

k Tk)TTk y, which we denote as vk. Note that the noise vector is still uncorrelated

but the individual components get scaled differently due to channel equalization. The

final step in detection involves computing the M-IDFT of the vector vk, which gives

FHMvk = dk + FH

MTTkH−1

k n. (5.7)

Individual symbols are independently detected after this step. This summarizes the

general approach to detecting SC-FDMA signals. In the next section we use an MMSE

equalizer, where it will be seen that symbols corresponding to a particular user interfere

with each other. This type of interference has been termed self-interference.

5.2.2 MMSE Equalization

When an MMSE equalizer is used, the channel corresponding to a sub-carrier allocated to

the kth user is equalized in the frequency domain by multiplying the signal corresponding


to that sub-carrier by(hkp)∗

|hkp|2+σ2n

σ2d

, where p is the sub-carrier index. σ2d and σ2

n are the

average symbol energy and the noise variance respectively. For the kth user, we define

the matrix Hk to represent the diagonal matrix used for channel equalization. The pth

entry along the diagonal is given by(hkp)∗

|hkp|2+σ2n

σ2d

. When MMSE equalization is employed,

we have

(TTk H

−1

k Tk)TTk y = (TT

k HkTk)TTkHkTkFMdk +

p=K∑p=1,p 6=k

(TTk HkTk)T

TkHpTpFMdp

+ (TTk HkTk)T

Tkn (5.8)

= (TTk HkTk)T

TkHkTkFMdk + (TT

k HkTk)TTkn (5.9)

= (TTk HkHkTk)FMdk + (TT

k HkTk)TTkn (5.10)

We have used Property C.5 and Property C.6 to simplify (5.8) to (5.9). Defining Rk to

be the diagonal matrix with entriesσ2n/σ

2d

|hkp|2+σ2n/σ

2d

for p = 1 to N , we get

(TTk H

−1

k Tk)TTk y = FMdk − (TT

kRkTk)FMdk + (TTk HkTk)T

Tkn (5.11)

Finally, we pass the signal in (5.11) through an M-IDFT block to obtain

FHM(TT

k H−1

k Tk)TTk y = dk − FM(TT

kRkTk)FMdk + FM(TTk HkTk)T

Tkn. (5.12)

The second term in the above expression is termed self interference, where symbols

corresponding to the kth user interfere with each other. Note that we do not see such a

phenomenon in OFDMA while using an MMSE equalizer.

Figure 5.2 compares the performance of OFDMA and SC-FDMA while using an

MMSE equalizer that is oblivious to the presence of phase noise i.e. the same filter is

used whether phase noise is present or not. The figure consists of two sets of plots– one

corresponding to the scenario where there is no phase noise and one where there is phase

noise at the receiver. The system simulated consisted of 8 users equally sharing a total


18 20 22 24 26 28 30 32 34 3610

−4

10−3

10−2

10−1

Eb/No (in dB)

BE

R

Local OFDMA, no PHNIntrlv. OFDMA, no PHNIntrlv. SCFDMA, no PHNLocal SCFDMA, no PHNLocal OFDMA, PHNLocal SCFDMA, PHNIntrlv. OFDMA, PHNIntrlv. SCFDMA, PHN

Figure 5.2: Plot comparing the performance of interleaved and localized OFDMA/SC-

FDMA while using MMSE channel equalization.

of 256 sub-carriers. All the users were assumed to use 64-QAM constellation. A 10 tap

frequency selective channel corresponding to every user was independently generated.

Phase noise characteristics at the receiver were set to 3 RMS value and 10 kHz loop

bandwidth. The sub-carrier spacing was set to 50 kHz.

From Figure 5.2 it is seen that in the case where there is no phase noise, SC-FDMA

and OFDMA perform similarly, with interleaved SC-FDMA showing slightly better per-

formance due to better diversity gains. For the scenario where there is phase noise, we

assume that the receiver is ignorant of the presence of phase noise and same MMSE

equalizer as before was used. From the figure it can be observed that in such a scenario,

for a given sub-carrier allocation, OFDMA has a slight performance gain over SC-FDMA.

Further, for both OFDMA and SC-FDMA, localized sub-carrier allocation seems to per-


form much better than the corresponding interleaved sub-carrier allocation. This is a

surprising result since interleaved SC-FDMA performs better than localized SC-FDMA

in the absence of phase noise. We explore the reasons behind such behaviour in the next

few sections. The next section analyzes effect of phase noise on the received signal model.

5.3 Detection in the Presence of Phase Noise

5.3.1 Received Signal Model

As defined in Chapter 2, let θr be the phase noise sequence affecting an SC-FDMA

symbol, let pr = ejθr

and let Q be the circulant matrix given by FNdiag(pr)FHN . The

rows of Q are circular shifts of the frequency domain components of phase noise given

by 1√N

FHNpr = [c0 c1 c2 . . . cN−1]T . In the presence of receiver phase noise, the received

signal after the N-DFT front end can be written as

y = QK∑k=1

HkTkFMdk + n. (5.13)

Aggregating the sub-carriers corresponding to the kth user by multiplying with TTk

gives

TTk y = TT

kQK∑p=1

HpTpFMdp + TTkn. (5.14)

In Appendix D, we further manipulate this expression to obtain

TTk y =c0(TT

kHkTk)FMdk + (TTk (Q− c0I)Tk)(T

TkHkTk)FMdk

+

p=K∑p=1,p 6=k

(TTkQTp)(T

Tp HpTp)FMdp + TT

kn (5.15)

In (5.15), the first term is the same as the received signal in the absence of phase

noise except for the rotation by CPE. Its seen that the effect of CPE on an SC-FDMA

symbol is the same as that on an OFDM/OFDMA symbol. The second term repre-

sents the self-interference (interference within a users data stream) that results due to


phase noise. The third term represents the multi-user interference that results from

data streams corresponding to other users. Further discussion on self interference and

multi-user interference is presented in Section 5.3.3.

5.3.2 Effect on Linear Receivers

Our analysis so far has been focused on the basic received signal model and is independent

of the type of receiver being used. We now turn our attention to the effect phase noise has

on linear receivers (ZF/MMSE). In the absence of impairments such as phase noise, it is

optimal to treat every sub-carrier to be independent of each other and adopt frequency

domain equalization as shown in Section 5.2.2. Due to their low complexity and ease

of implementation, such receivers are very commonly used in practice and hence it is

important for us to assess the impact phase noise is likely to have on such receivers.

Through this analysis we also try to explain why the impact of phase noise on interleaved

and localized SC-FDMA is different while using a linear receiver as seen in Figure 5.2.

To keep the analysis simple, we consider a ZF equalizer here. Arguments made here also

hold for MMSE equalizers at high SNR.

Given a ZF equalizer at the receiver, from (5.15), the signal after channel equalization

can be written as

(TTkH−1

k Tk)TTk y =c0(TT

kH−1k Tk)(T

TkHkTk)FMdk

+ (TTkH−1

k Tk)(TTk (Q− c0I)Tk)(T

TkHkTk)FMdk

+

p=K∑p=1,p 6=k

(TTkH−1

k Tk)(TTkQTp)(T

Tp HpTp)FMdp + (TT

kH−1k Tk)T

Tkn

(5.16)


(TTkH−1

k Tk)TTk y =c0FMdk + (TT

kH−1k Tk)(T

Tk (Q− c0I)Tk)(T

TkHkTk)FMdk

+

p=K∑p=1,p 6=k

(TTkH−1

k Tk)(TTkQTp)(T

Tp HpTp)FMdp + (TT

kH−1k Tk)T

Tkn.

(5.17)

After the final M-IDFT operation, we can write

FHM(TT

kH−1k Tk)T

Tk y =c0dk + FH

M(TTkH−1

k Tk)(TTk (Q− c0I)Tk)(T

TkHkTk)FMdk

+

p=K∑p=1,p 6=k

FHM(TT

kH−1k Tk)(T

TkQTp)(T

Tp HpTp)FMdp

+ FHM(TT

kH−1k Tk)T

Tkn. (5.18)

While CPE has the same effect as before, quantifying the resulting ICI is not straight-

forward. We first take a closer look at the second term in (5.18). The second term in

(5.18) representing self interference is an M × 1 vector that is essentially the product of

five matrices – FHM , (TT

kH−1k Tk), (TT

k (Q − c0I)Tk), (TTkHkTk), FM and the vector dk.

Any entry in the resulting M × 1 vector is a summation of many terms having a generic

representation given by

e1 = αh−1i cxhjβdl, (5.19)

where α and β are any two entries from FHM and FM respectively 1, h−1

i and hj are entries

from the matrix (TTkH−1

k Tk) and (TTkHkTk) respectively, and cx is an entry from the

matrix TTk (Q − c0I)Tk

2. Similarly, any entry in the resulting M × 1 vector from the

third term in (5.18) representing multi-user interference, is a summation of many terms

having a generic representation given by

e2 = αh−1i czhjβdl, (5.20)

1Note that |α| = |β| = 12By Properties C.8 and C.9, this matrix is independent of k, hence the argument is valid for any user


where α and β are as defined before, while h−1i and hj are entries from the matrix

(TTkH−1

k Tk) and (TTp HpTp) respectively, with k 6= p, and cz is an entry from the matrix

(TTkQTp).

While e1 and e2 look similar, the key difference arises from the fact that hi and hj

can be strongly correlated in the case of e1, while this is never the case with e2, where

hi and hj come from two different users. When hi and hj are either identical or strongly

correlated, the variance of e1 can be written as

V ar(e1) ≈ E[|dl|2]E[|cx|2], (5.21)

and in the case of e2, since hi and hj are independent of each other, the variance of e2

can be written as

V ar(e2) = E[|dl|2]E[|cz|2]E

[( |hj||hi|)2]. (5.22)

The variance of e2 contains a term involving the second moment of the ratio of two

Rayleigh random variables (or the mean of the ratio of two Chi-square random variables

with two degrees of freedom). The distribution of the ratio of two Rayleigh random

variables is called the Fisher-Snedecor distribution and it is known that the variance of

such a ratio diverges [41], suggesting that the interference resulting from the third term

in (5.18) can be significant provided E[|cz|2] does not go to zero. This difference between

e1 and e2 differentiates localized and interleaved SC-FDMA, but the argument is not yet

complete.

In the case of localized SC-FDMA, the significant frequency components of phase

noise that contain most of the energy of the phase noise process are contained in the sub-

matrix (TTk (Q−c0I)Tk) and contribute towards self interference. As a result, the terms in

multi-user interference, of the form e2, have very small phase noise frequency components

associated with them, rendering terms resembling e2 insignificant. The coming together

of this fact and the high correlation amongst the channel coefficients corresponding to


a user as a result of contiguous sub-carrier allocation, that most interference terms in

localized SC-FDMA comes from terms resembling e1, whose variance remains bounded.

On the other hand, in the case of interleaved SC-FDMA, not all significant frequency

components are contained in (TTk (Q− c0I)Tk). As a result, among the terms that con-

tribute towards multi-user interference, there are terms where E[|cz|2] can be significant.

Since the channel coefficients of such terms, with the generic representation given by e2,

are completely independent of each other, the variance of these terms becomes unbounded

and can cause significant interference. We believe this to be reason behind interleaved

SC-FDMA being affected by phase noise much more than localized SC-FDMA while

using a linear receiver.

In order to test our hypothesis, we ran two sets of simulations. Our hypothesis hinges

on two things: channel coefficient correlation in the case of localized SC-FDMA and phase

noise energy concentration in the lower order frequency components. Removal of either

property must render the effects on interleaved and localized SC-FDMA equivalent. This

is what we intended to check.

In the first set of simulations, rather than generating the frequency selective channel

in the time domain and then computing its frequency domain coefficients, we directly

generated independent channel coefficients corresponding to different sub-carriers in the

frequency domain. As a consequence of this, if our hypothesis is right, localized SC-

FDMA will now not have the advantage of correlated channel coefficients being associated

with dominant phase noise frequency components and we expect interleaved and localized

SC-FDMA to be equally affected by phase noise. This is verified through simulations in

Figure 5.3, where it is seen that interleaved and localized SC-FDMA have very similar

performance curves in the presence of phase noise.

In the second test, rather than modelling phase noise as a correlated process, we

modelled it as an uncorrelated process. As a result, the energy in the phase noise process

is equally spread amongst all the frequency components. As a consequence of this, local-


18 20 22 24 26 28 30 32 34 3610

−3

10−2

10−1

Eb/No (in dB)

BER

Local OFDMA Ind. Channel Coeff.Intrlv OFDMA Ind. Channel Coeff.Local SCFDMA Ind. Channel Coeff.Intrlv SCFDMA Ind. Channel Coeff.Local OFDMA Uncorrelated PHNIntrlv OFDMA Uncorrelated PHNLocal SCFDMA Uncorrelated PHNIntrlv SCFDMA Uncorrelated PHN

Figure 5.3: Plot illustrating the performance of interleaved and localized SC-FDMA when

all channel coefficients are generated independently and when phase noise is uncorrelated.

ized SC-FDMA will not have the advantage of having dominant frequency components

being associated with correlated channel coefficients and we once again expect localized

and interleaved SC-FDMA to be equally affected by phase noise. Figure 5.3 shows the

performance of both the SC-FDMA schemes to be similar when affected by uncorrelated

phase noise.

All the above arguments also hold for interleaved and localized OFDMA. This is

easily seen by simply substituting every occurrence of FM in the discussion above with

an identity matrix.


5.3.3 Self Interference vs. Multi-User Interference

In Section 5.3.1 we studied the SC-FDMA received signal model immediately after the

N-DFT operation at the receiver and noted in (5.15) that the total interference consists

of two terms – self interference and multi-user interference.

A point of interest here is to find out which of the two interferences is likely to

dominate and under what circumstances. Suppose one finds self-interference to be dom-

inant, then interference cancellation schemes that work exclusively on individual user

data streams will suffice to substantially reduce the effects of phase noise. This is of

particular advantage when different users use different constellations for modulation. In

such a scenario, we can pick a user using a smaller constellation (4-QAM/16-QAM) that

is not CPE sensitive and more robust to ICI and additive noise to estimate the significant

frequency components of phase noise that contribute towards ICI and use these estimates

to cancel ICI in other users’ data streams. On the other hand, if multi-user interference

is likely to dominate, then we cannot take recourse to interference cancellers that work

on individual user data streams, and we have to look at joint interference cancellation

across all the data streams. This is clearly much more complicated than the case where

self-interference tends to dominate.

From (5.15), it is seen that the amount of self-interference is directly related to

the amount of energy contained in TTk (Q − c0I)Tk, which is given by E[tr(TT

k (Q −c0I)Tk)(T

Tk (Q− c0I)Tk)

H]. Note that we are essentially comparing the average energy

contained in the vector given by the second term in (5.15) to the average energy con-

tained in the vector given by the third term in (5.15) which consists of a summation

of contributions from (K-1) other users. Denoting the two average energies as ESI and

EMUI , and assuming all channel gains to be independent with variance Eh and average

symbol energy to be Ed, we get

ESI = EhEdE[tr

(TTk (Q− c0I)Tk)(T

Tk (Q− c0I)Tk)

H]. (5.23)


Number Localized SC-FDMA Interleaved SC-FDMA

of users ESI/EICI (%) ESI/EICI (%)

4 94.5 6.5

8 90.2 1.6

16 83.0 0.4

32 71.5 0.1

64 54.1 0.02

Table 5.1: Table listing percentage of energy in self interference for different number of

users.

A similar computation for multi-user interference gives

EMUI = EhEd

K∑p=1,p 6=k

E[tr

(TTkQTp)(T

TkQTp)

H]. (5.24)

We can further simplify the above equation to get

EMUI = EhEd(M(σ2

θ − σ2θ)− E

[tr

(TTk (Q− c0I)Tk)(T

Tk (Q− c0I)Tk)

H])

(5.25)

= EhEd(M(σ2

θ − σ2θ))− ESI . (5.26)

Now, because the phase noise PSD tapers off rapidly beyond the loop bandwidth, most

of the energy is contained in the first few lower order frequency components i.e. c1, cN−1,

c2, cN−2 etc. For localized sub-carrier allocation, using the Property C.3, (TTkQTk) is the

sub-matrix of Q formed by the first M rows and columns of Q. Using this observation,

it can be shown that the total energy in self-interference for localized SC-FDMA is given

by

ESI = EhEdE

[M−1∑j=1

(M − j) (|cj|2 + |cN−j|2)]

(5.27)

Table 5.1 lists the percentage of energy in self interference for varying number of users.

The parameters corresponding to the phase noise process were set to 3 RMS and 10 kHz

loop bandwidth. The sub-carrier spacing was set to 50 kHz. It is clear that in the case


of localized SC-FDMA, a large fraction of interference results from the self interference

term, suggesting that cancelling just the self interference term is likely to improve the

performance significantly. In Figure 5.4, we show the improvement in performance using a

genie aided self interference cancellation scheme. The self interference term was computed

using the phase noise sequence affecting an SC-FDMA symbols and was compensated

for before the channel equalization step. In addition to self interference, CPE was also

compensated for in a similar manner. Parameters used in the simulation are the same as

those used to generate Figure 5.2.

18 20 22 24 26 28 30 32 34 3610

−4

10−3

10−2

10−1

Eb/No (in dB)

BE

R

Local SCFDMA, no PHNLocal SCFDMA, PHNLocal SC−FDMA, SI cancellationIntrlv SC−FDMA, SI cancellationIntrlv SC−FDMA, PHNIntrlv SC−FDMA, no PHN

Figure 5.4: Plot illustrating the improvement in performance of an SC-FDMA system

affected by phase noise by cancelling self interference alone.


5.4 Summary

In this chapter we analyzed the effect of phase noise on SC-FDMA. We noted that

when linear receivers are used, the effect of phase noise on interleaved and localized SC-

FDMA can be quite different. We provided a qualitative argument for the difference in

performance of the two SC-FDMA schemes and tested our hypothesis under two different

scenarios. We also studied the general received signal model of a system employing SC-

FDMA and suggested some detection strategies that one could adopt while detecting

SC-FDMA symbols affected by phase noise.

Chapter 6

Conclusion

In the first part of this thesis we focused on the effect phase noise has on OFDM/OFDMA

systems. We focused on how phase noise affects the signal model under different scenarios

and identified its two main effects. We separately studied the behaviour of CPE and the

higher order components and identified the main issues surrounding each of them. To be

specific, we classified constellations based on their CPE sensitivity and highlighted the

effect high CPE can have on blind/decision-directed algorithms.

We then developed an algorithm to blindly estimate CPE that is robust to decision-

directed errors. The algorithm is able to side step the issues surrounding erroneous

detection by taking into account the reliability of symbol detection in each sub-carrier.

Through a novel use of the likelihood ascent search algorithm, we identify the most

reliable sub-carriers and effectively turn them into virtual pilots. The algorithm was

shown to provide accurate CPE estimates even in a high CPE scenario.

We then turned our attention to the issue of ICI mitigation in OFDM systems affected

by phase noise. Using the principle of variational inference, we developed a soft-in soft-

out bit-level detection algorithm. This algorithm jointly estimates the transmitted bits

and the phase noise sequence and generates soft estimates on the transmitted bits and

phase noise. This algorithm was used in three different receiver setups including its use in

97

Chapter 6. Conclusion 98

building a turbo receiver. The turbo receiver along with the CPE compensation scheme

suggested in Chapter 3 was shown to substantially alleviate all the effects of phase noise

and result in close to optimal performance.

Our focus in this thesis was exclusively on receiver phase noise. Future work in this

area must first separately account for the effect of transmitter phase noise and then

consider the joint effects of transmitter and receiver phase noise. Not all phase noise

parameters can be individually estimated in such a scenario and one will have to consider

estimating the effective components that contribute towards interference. In our work

we assumed the channel coefficients to be perfectly known. This is not case in a practical

scenario and there is a need to investigate the performance of the suggested algorithms

under channel uncertainty.

Finally, we analyzed the effect of phase noise on SC-FDMA. We took note of the

surprising result that sub-carrier mapping plays a crucial role in determining the perfor-

mance of the system while using a linear receiver. In particular, we observed that in-

terleaved SC-FDMA was much more sensitive to phase noise than localized SC-FDMA.

We provided a qualitative argument to explain the difference in performance between

the two sub-carrier allocation schemes and substantiated our argument by testing the

proposed hypothesis under two different scenarios. We also studied the general received

signal model of a system employing SC-FDMA and suggested some detection strategies

that one could adopt while detecting SC-FDMA symbols affected by phase noise.

Future work in this area could include providing a clear quantitative explanation

for the difference in performance between interleaved and localized SC-FDMA. Interfer-

ence mitigation in a phase noise affected SC-FDMA system by estimating the dominant

frequency components of phase noise from the received signal corresponding to the sub-

carriers allocated to a user using a constellation robust to CPE can also be explored.

Appendices

99

Appendix A

Evaluating the Free Energy

Expression

As shown in (4.14), the free energy expression can be written as the sum of five terms.

Since we have adopted a discrete distribution for QB(B), it is straightforward to compute

the first and third terms. The second term is the Kullback-Liebler divergence between

two multi-variate Gaussian distributions and can be computed easily. The fourth term is

simply the differential entropy of a multi-variate Gaussian distribution. We discuss the

computation of the fifth term (v) here. Now,

−2σ2(v) =const+

∫B,θ

[Q(B)Q(θ)(r−PFHHf (B))H(r−PFHHf (B))

]dBdθ (A.1)

Proceeding exactly as in [5], we can simplify (A.1) to

−2σ2(v) =const+

∫B

Q(B)[f (B)HZHZf (B)− rHZf (B)

− f (B)HZHr + f (B)HΨf (B)]dB, (A.2)

100

Appendix A. Evaluating the Free Energy Expression 101

where, we have defined Z = diag(1 + jmθ)FHH and Ψ = HHFdiag(Sθ)F

HH). Using

Lemmas 5 and 6 from [6], we can simplify (A.2) to

−2σ2(v) =const− rHZf (B)− f (B)H

ZHr + 1H(M1)(νr + νr)

+ f r(Br)H

(M0)f i(Bi) + f i(Bi)H

(M0)f r(Br)

+ f r(Br)H

(M2)f r(Br) + f i(Bi)H

(M2)f i(Bi) (A.3)

where, M0 =(Ψ + ZHZ) (A.4)

M1 =Ψ + ZHZ− diag(Ψ + ZHZ) (A.5)

M2 = diag(Ψ + ZHZ) (A.6)

νr =∑

0<i≤j<(L/2)

2i+jp=j

6∏p=i

bp +

L/2∑i=1

22(i−1) (A.7)

νi =∑

0<i≤j<(L/2)

2i+jp=j

6∏p=i

bp+(L/2) +

L/2∑i=1

22(i−1). (A.8)

Appendix B

Computing the Gradient

B.1 Gradient w.r.t Sθ

Using (4.11) and (4.15), we can write

∂F

∂ S−1θ

=∂(−i− ii+ iii+ iv − v)

∂S−1θ

(B.1)

One can compute the terms above to be :

∂(i)

∂S−1θ

= 0;∂(ii)

∂S−1θ

=1

2STθ φ−1

θ STθ ;∂(iii)

∂S−1θ

= 0;∂(iv)

∂S−1θ

=1

2STθ (B.2)

To compute ∂(v)

∂S−1θ

, note the following two results.

(a)∂(xHdiag(Sθ)y)

∂S−1θ

=∂(tr(diag(xH)Sθdiag(y))

)∂S−1

θ

=∂(tr(diag(y)diag(xH)Sθ)

)∂S−1

θ

=− STθ(diag(y)diag(xH)

)Sθ (B.3)

(b)∂ (tr(Xdiag(Sθ)))

∂S−1θ

=∂ (tr(diag(X)Sθ))

∂S−1θ

=− STθ (diag(X)) STθ (B.4)

102

Appendix B. Computing the Gradient 103

Using (B.3) and (B.4) we can compute ∂(v)

∂S−1θ

to be

∂(v)

∂S−1θ

=−1

2σ2

[− STθ (XmXH

m)STθ

− STθ diag(FHHdiag(νr + νi − |f (B)|2)HHF)STθ

], (B.5)

where Xm = diag(FHHf(B)). Using (B.2) and (B.5) it is straightforward to compute

(4.16).

B.2 Gradient w.r.t mθ

Using (4.14) and (4.15), we can write

∂F

∂ mθ

=∂(−i− ii+ iii+ iv − v)

∂mθ

. (B.6)

One can compute the terms above to be :

∂(i)

∂mθ

= 0;∂(ii)

∂mθ

= φ−Tθ µθ − φ−1θ mθ;

∂(iii)

∂mθ

= 0;∂(iv)

∂mθ

= 0; (B.7)

To compute ∂(v)∂mθ

, note the following two results.

(a)∂(xH(I+jdiag(mθ))

H(I+jdiag(mθ))y)

∂mθ

=∂((1+jmθ)

Hdiag(xH)diag(y)(1+jmθ))

∂mθ

=− 2diag(xH)diag(y)mθ (B.8)

(b)∂(xHdiag(ZHZ)y

)∂mθ

=∂(tr(diag(xH)ZHZdiag(y))

)∂mθ

=∂((1 + jmθ)

TM(1− jmθ))

∂mθ(where M = diag(FHHdiag(y)diag(xH)HHF)

)= 2diag(FHHdiag(y)diag(xH)HHF)mθ (B.9)

Using (B.8) and (B.9), we can compute ∂(v)∂mθ

to be :

∂(v)

∂mθ

=−1

σ2

[XHmXmmθ + =(rHXm)T

+ diag(FHHdiag(νr + νi − |f (B)|2)

)mθ

](B.10)


Using (B.7), (B.10) and the update equation for Sθ given in (4.16), we can compute the

update equation for mθ to be as given in (4.17).

B.3 Gradient w.r.t brk and bik

To compute∂F

∂brk, consider the term (i) from (4.11):

(i) =N∑n=1

L∑l=1

[(1 +mnl

2

)log

(1 + µnl

2

)+

(1−mnl

2

)log

(1− µnl

2

)](B.11)

(ii) =N∑n=1

L∑l=1

[(1 +mnl

2

)log

(1 +mnl

2

)+

(1−mnl

2

)log

(1−mnl

2

)](B.12)

Differentiating (B.11) w.r.t to mnl gives tanh−1(µnl). Denote ∂(i)

∂blkas tµlk for l ∈ r, i

and k = 1, 2, . . . , L/2. Similarly, differentiating term (ii) (given in (B.12)) w.r.t to mnl

gives tanh−1(mnl). Denote ∂(ii)

∂blkas tmlk for l ∈ r, i and k = 1, 2, . . . , L/2.

Further, note the following :

(a)∂f r(Br)

∂brk= diag(αk), where αk = I

(k = L/2

)2k−1.1 +

k∑l=1

2l−1

L/2

6∏

p=l,p6=k

brp (B.13)

(b)∂f i(Bi)

∂bik= diag(βk), where βk = j.

(I(k = L/2

)2k−1.1 +

k∑l=1

2l−1

L/2

6∏

p=l,p 6=k

bip

)(B.14)

(c)∂f r(Br)

HMf r(Br)

∂brk=∂f r(Br)

HMf r(Br)

∂f r(Br)

∂f r(Br)

∂brk= 2[<f r(Br)

HMdiag(αrk)]T

(B.15)

(d)∂f i(Bi)

HMf i(Bi)

∂bik=∂f i(Bi)

HMf i(Bi)

∂f i(Bi)

∂f i(Bi)

∂bik= 2[=f i(Bi)

HMdiag(βik)]T(B.16)


(e)∂νr

∂brk=diag

(I(0 < k < L/2

).22k1 +

∑0<i≤k≤j<L/2

i 6=j

j

6∏p=ip 6=k

brp

), diag(δk) (B.17)

(f)∂νi

∂bik=diag

(I(0 < k < L/2

).22k1 +

∑0<i≤k≤j<L/2

i 6=j

j

6∏p=ip 6=k

bip

), diag(Ωk) (B.18)

(g)∂(1TMνr)

∂brk=MT1diag(δk) (B.19)

(h)∂(1TMνi)

∂bik=MT1diag(Ωk) (B.20)

In (B.17) and (B.18), I(·) represents the indicator function. Using all of the results from

equations (B.13) to (B.20), we can compute the gradient to be :

∂F

∂brk= tµrk − tmrk +

1

σ2

[diag(αk)

(<ZHr − <MT

2 f r(Br))

+ jdiag(αk)=M0f i(Bi)− diag(δk)MT1 1

](B.21)

∂F

∂bik= tµik − tmik +

1

σ2

[diag(βk)

(=ZHr+ =M0f r(Br))

+ j.diag(βk)<MT2 f i(Bi)− diag(Ωk)M

T1 1

](B.22)

Appendix C

Properties of the Permutation

Matrix Tk

The matrix Tk maps M data symbols of the kth user to M out of the N available sub-

carriers. Assuming the number of users to be K, we have MK = N . Let Uk represent the

set of indices of the sub-carriers allocated to the kth user. Suppose the ith symbol in the

data vector dk is mapped to the jth sub-carrier then, Tk(i,j) = 1, and is zero otherwise.

Of the N rows of Tk, N −M rows are all zeros and the remaining M rows consist of

the M unit vectors of the standard basis of an M-dimensional space. As a result, TkTTk

is an N × N diagonal matrix with the jth entry along the diagonal being one if the jth

row of Tk is non-zero and is zero otherwise. Further, the columns of Tk form a size-M

subset of the unit vectors of the standard basis of an N -dimensional space. As a result,

TTkTk is the M ×M identity matrix. We denote the concatenated matrix [T1T2 . . .TK ]

as T. Since a sub-carrier is assigned exclusively to a user, T is a permutation of the

identity matrix and is orthogonal i.e. TTT = I. We observe the following properties of

a permutation matrix Tk.

Property C.1. Given an N × N matrix R, TTkR is an M × N matrix consisting of a

subset of M rows from R whose indices are in Uk.

106

Appendix C. Properties of the Permutation Matrix Tk 107

Proof. Since every row of TTk is a unit vector from the standard basis of an N -dimensional

space and no two rows are identical, every row in TTk picks a row from R. Specifically, if

the ith row in TTk has a non-zero entry in the jth location, then the ith row in the resulting

matrix is simply the jth row of R.

Property C.2. Given an N × N matrix R, RTk is an N ×M matrix consisting of a

subset of M columns from R whose indices are in Uk.

Proof. This immediately follows from Property C.1, once we note that RTk is (TTkRT )T .

Property C.3. Given an N × N matrix R, TTkRTk is an M × M sub-matrix of R

formed by M columns and M rows of R, sharing the same set of indices Uk.

Proof. This property follows immediately from the Property C.1 and Property C.2.

Property C.4. Suppose R is an N × N diagonal matrix then TTkRTk is a M × M

diagonal matrix with entries picked from diag(R) such that the indices of the chosen

entries belong to Uk.

Proof. This property follows from Property C.3.

Property C.5. If R and S are two N × N diagonal matrices, then TTkRTkT

TkSTk is

equal to TTkRSTk.

Proof. We first note that TkTTk is a diagonal matrix. Since R and S are also diagonal

matrices, we can change the order of the product. Hence, we have,

(TTkRTk)(T

TkSTk) = TT

k (TkTTk )RSTk (C.1)

= (TTkTk)(T

TkRSTk) (C.2)

= TTkRSTk, (C.3)

where the last equality follows from the fact that (TTkTk) is the M ×M identity matrix.


Property C.6. If R is an N ×N diagonal matrix, then, TTkRTp = 0 for p 6= k.

Proof. Using Property C.1 and Property C.2, we note that the set of indices of the rows

and columns that form the resulting matrix belong to the sets Uk and Up respectively.

Since users do not share sub-carriers, Uk ∩ Up = ∅ and hence no chosen row and chosen

column share the same index. Since all the non-zero entries in R lie along the diagonal,

the result follows.

Property C.7. If R is an N × N invertible diagonal matrix, then, (TTkRTk)

−1 =

TTkR−1Tk.

Proof. We first note that TTkRTk is an M ×M diagonal matrix. Specifically, if the ith

row in TTk has a non-zero entry in the jth location, then the (i, i)th element in TT

kRTk

is given by R(j,j). Since the resulting matrix is a diagonal matrix, inverting TTkRTk

is equivalent to inverting the individual components along the diagonal of the resulting

matrix. Since these individual components are entries along the diagonal in R, to get

the required inverse, we first invert R and then compute the product TTkR−1Tk.

All the above properties are true for any general sub-carrier allocation. We now focus

on properties of permutation matrices specific to localized and interleaved sub-carrier

mapping. In the case of localized SC-FDMA, the first user is allocated the first M sub-

carriers, thus T1 (1:M, 1:M) = I. Similarly, for the kth user, Tk (1+(kM):(k+1)M, 1:M) = I.

Property C.8. For the case of localized sub-carrier allocation, if R is an N×N circulant

matrix, then TTkRTk = R(1+(kM):(k+1)M, 1+(kM):(k+1)M) = R(1:M, 1:M) and is independent

of k.

Proof. This property follows from Property C.3 and the observation that all M × M

submatrices of R formed from a set of M contiguous rows and columns that share the

same set of indices are equivalent.


For interleaved sub-carrier mapping, the data symbols corresponding to the kth user

are mapped to sub-carriers with indices k,M + k, 2M + k, . . .KM + k. As a result,

the matrices Tk are row shifted copies of each other.

Property C.9. For the case of localized sub-carrier allocation, if R is an N×N circulant

matrix, then TTkRTk is independent of k.

Proof. Let the first row of R be denoted as r. Then R(i,j) = v((j−i) mod N)+1. Now, any

entry in the sub-matrix TTkRTk belong to a row in R whose index is given by pM + k

for some p ∈ 0, 1, . . . (N − 1) and a column in R whose index is given by qM + k

for some q ∈ 0, 1, . . . (N − 1). We now note that such an element in R is given by

R(pM+k,qM+k) = v((qM+k−pM−k) mod N)+1 = v((qM−pM) mod N)+1, which is independent of

k.

Appendix D

SC-FDMA Received Signal in the

Presence of PHN

In this Appendix we show how we manipulate (5.14) to get (5.15). From (5.14), we have

TTk y =TT

kQK∑p=1

HpTpFMdp + TTkn (D.1)

=TTkQTTT

K∑p=1

HpTpFMdp + TTkn (D.2)

=TTkQT

K∑p=1

TTHpTpFMdp + TTkn (D.3)

=[TTkQT1 TT

kQT2 . . .TTkQTK

] K∑p=1

TTHpTpFMdp + TTkn (D.4)

=[TTkQT1 TT

kQT2 . . .TTkQTK

]×K∑p=1

[(TT

1 HpTpFMdp)T (TT

2 HpTpFMdp)T . . . (TT

KHpTpFMdp)T]T

+ TTkn (D.5)

=i=K∑i=1

p=K∑p=1

(TTkQTi)(T

Ti HpTp)FMdp + TT

kn (D.6)

110

Appendix D. SC-FDMA Received Signal in the Presence of PHN 111

Now, we use Property C.6 to simplify the previous equation to

TTk y =(TT

kQTk)(TTkHkTk)FMdk +

p=K∑p=1,p 6=k

(TTkQTp)(T

Tp HpTp)FMdp + TT

kn (D.7)

=c0(TTkHkTk)FMdk + (TT

k (Q− c0I)Tk)(TTkHkTk)FMdk

+

p=K∑p=1,p 6=k

(TTkQTp)(T

Tp HpTp)FMdp + TT

kn (D.8)

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