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School of Electrical, Electronic, and Computer Engineering

Impulsive Noise Detection and Mitigation in

OFDM-based Power-Line Communications

Filbert Hilman Juwono

This thesis is presented for the degree of

Doctor of Philosophy

of

The University of Western Australia

December 2016

Declaration

To the best of my knowledge and belief this thesis contains no material previously pub-

lished by any other person except where due acknowledgement has been made.

This thesis contains no material which has been accepted for the award of any other

degree or diploma in any university.

Signature: . . . . . . . . . . . . . . . . . . . . . . . . . . .

Date: . . . . . . . . . . . . . . . . . . . . .

To

my beloved parents

Acknowledgements

All you need is the plan, the road map, and the courage to press on to your destination.

– Earl Nightingale

I have just three things to teach: simplicity, patience, compassion. These three are

your greatest treasures.

– Lao Tzu

First of all, I would like to thank Jesus for His blessings during my PhD

study.

I would like to express my sincere gratitude to my supervisors, Prof.

Defeng (David) Huang and Prof. Kit Po Wong. Prof. Huang has supported

me over the past years and taught me countless valuable lessons, especially in

conducting fruitful research. I owe him for his constant assistance, guidance,

and encouragement. Prof. Wong, with his broad experience in academics

and research, has given useful and constructive suggestions during my study.

I am grateful to Dr. Qinghua Guo and Dr. Lu Xu for their assistance

during my study, in which I benefited a lot. I would like to thank all members

of Signal Processing and Wireless Communications Laboratory (SPWCL) at

The University of Western Australia. I extend my gratitude to Prof. Yifan

Chen for providing me an opportunity to visit his laboratory at Southern

University of Science and Technology. I also would like to recognise Regina

Reine for some useful discussions.

I acknowledge the Australian Awards Scholarships for providing me such

a prestigious scholarship which make this research possible. In particular,

I would like to thank the AusAID representatives Deborah Pyatt, Debra

Basanovic, Kristine Marginis, and Celia Seah from the International Centre

at The University of Western Australia. They always had time for assisting

me during my study.

Last but not least, I thank my parents who always encourage and sup-

port me in this journey.

Abstract

Smart grid can be a solution for reducing greenhouse gas (GHG) emissions.

To perform its task, smart grid needs a communication link to communi-

cate between nodes or devices. Using power-line communications (PLC)

is beneficial as the power-lines infrastructure has been available. However,

in PLC the transmitted signal undergoes multipath channels and impulsive

noise corruption. Meanwhile, orthogonal frequency-division multiplexing

(OFDM) is the well-known modulation used in PLC. Although OFDM is

able to spread the energy of the impulsive noise, it still degrades the system

performance. Therefore, detecting and mitigating the impulsive noise are

required to improve the reliability of the OFDM-based PLC.

This thesis focuses on impulsive noise mitigation methods in the broad-

band PLC systems. First, a generic nonlinear preprocessor, which is called

deep clipping, is presented. Deep clipping is characterised by two param-

eters, namely, threshold and depth factor. The expression for the output

signal-to-noise ratio (SNR) of the deep clipping is derived. By optimising

the parameters, it can be found that the performance of deep clipping out-

performs the other nonlinear preprocessors, such as blanking, conventional

clipping, and joint blanking/clipping. In addition, it can be found that by

setting the two parameters of the deep clipping properly, the expressions for

other nonlinear preprocessors can be obtained.

Next, a linear combination of nonlinear preprocessors is proposed to fur-

ther improve the performance of the OFDM-based PLC. The combination

of blanking and conventional clipping and the combination of conventional

clipping and joint blanking/clipping are analysed. The output SNR mathe-

matical expressions for both combinations are also derived. The simulation

results show that the output SNR of the linear combination method outper-

forms the output SNR of the individual method.

Then, as OFDM systems suffer from high peak-to-average power ratio

(PAPR), the nonlinear preprocessor can falsely detect the impulsive noise.

As a result, if the PAPR is reduced, the performance of the system is im-

proved. The use of Guel’s clipping method as a PAPR reduction method is

proposed as this clipping method does not impact the bit error rate (BER)

performance. From the simulation results, it can be seen that our proposed

method improves the BER performance of OFDM-based PLC over impulsive

noise channel.

Finally, considering that impulsive noise is sparse and OFDM systems

usually employ null subcarriers, the smoothed ℓ0-norm method is proposed

to detect impulsive noise. The proposed method is compared with the con-

ventional ℓ1-norm detection. The simulation results show that the proposed

method has lower complexity than that of the conventional ℓ1-norm while

having better (or sometimes the same) performance.

Publications

Parts of this thesis and concepts from it have been previously published in the following

journal and/or conference papers.

[1] F. H. Juwono, Q. Guo, Y. Chen, L. Xu, D. Huang, and K. P. Wong, ”Linear com-

bining of nonlinear preprocessors for OFDM-based power-line communications,”

IEEE Trans. Smart Grid, vol. 7, no. 1, Jun., pp. 253-260, 2016.

[2] F. H. Juwono, Q. Guo, D. Huang, and K. P. Wong, ”Deep clipping for impul-

sive noise mitigation in OFDM-based power-line communications,” IEEE Trans.

Power Del., vol. 29, no. 3, pp. 1335-1343, Jan., 2014.

[3] F. H. Juwono, Q. Guo, D. Huang, K. P. Wong, and L. Xu, ”Impulsive noise

detection in PLC with smooth ℓ0-norm,” Proc. Int. Conf. Acoustics, Speech, and

Signal Proces. (ICASSP), Brisbane, Australia, 2015.

[4] F. H. Juwono, Q. Guo, D. Huang, and K. P. Wong, ”Joint peak amplitude and

impulsive noise clippings in OFDM-based power line communications,” Proc. Asia

Pacific Conf. Commun. (APCC), Bali, Indonesia, 2013.

Contents

Publications vii

List of Figures xiii

List of Tables xv

List of Acronyms xvii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 PLC System and Channel 7

2.1 OFDM Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 PLC Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Impulsive Noise Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Deep Clipping for Impulsive Noise Mitigation in OFDM-based PLC 11

3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Deep Clipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.2 Conventional Clipping, Blanking, and Blanking/Clipping . . . . 17

3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Mathematical Expressions Validation . . . . . . . . . . . . . . . . 18

3.3.2 Threshold Optimization . . . . . . . . . . . . . . . . . . . . . . . 19

x CONTENTS

3.3.3 Maximum γ with Varying SNR . . . . . . . . . . . . . . . . . . . 23

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Linear Combining of Nonlinear Preprocessors for OFDM-based PLC 25

4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Combining Conventional Clipping and Blanking . . . . . . . . . 29

4.2.2 Combining Conventional Clipping and Joint Blanking/Clipping . 31

4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Maximum γ with Varying SINR . . . . . . . . . . . . . . . . . . 33

4.3.2 Maximum γ with Varying SNR . . . . . . . . . . . . . . . . . . . 35

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Joint Peak Amplitude and Impulsive Noise Nonlinear Preprocessors

in OFDM-based PLC 41

5.1 System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.1 Clipping at the Transmitter using Guel’s Method . . . . . . . . . 43

5.1.2 Impulsive Noise and Blanking at the Receiver . . . . . . . . . . . 45

5.1.3 PLC Multipath Channel with Impulsive Noise . . . . . . . . . . 45

5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Impulsive Noise Detection in PLC with Smoothed ℓ0-norm 53

6.1 Impulsive Noise Detection Algorithm . . . . . . . . . . . . . . . . . . . . 54

6.1.1 System Model and Problem Formulation . . . . . . . . . . . . . . 54

6.1.2 The Smoothed ℓ0-Norm and Minimisation Algorithm . . . . . . . 55

6.1.3 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7 Conclusions and Future Works 63

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A Derivation of E[(T + µT )ejϕks∗k

∣∣C2, I

]and E

[µ|rk|ejϕks∗k

∣∣C2, I

]65

CONTENTS xi

B Derivation of E[|yk|2

∣∣C2, I

]69

Bibliography 71

List of Figures

2.1 An OFDM-based PLC system . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 OFDM-based PLC system model with a nonlinear preprocessor at the

receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 α with varying threshold, p = 0.01 . . . . . . . . . . . . . . . . . . . . . 19

3.3 γ with varying threshold and p . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 γ with varying depth factor and p . . . . . . . . . . . . . . . . . . . . . 20

3.5 Optimum thresholds, SNR = 25 dB, p = 0.01 . . . . . . . . . . . . . . . 21

3.6 Maximum γ, SNR = 25 dB, p = 0.01 . . . . . . . . . . . . . . . . . . . . 22

3.7 Maximum γ, deep clipping uses optimum threshold and depth factor,

SNR = 25 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.8 Maximum γ with varying SNR, SINR = −7 dB . . . . . . . . . . . . . . 23

4.1 OFDM-based PLC system model with combination of nonlinear prepro-

cessors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Maximum γ vs. SINR for combining conventional clipping and blanking,

SNR = 25 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Maximum γ vs. SINR for combining conventional clipping and joint

blanking/clipping, SNR = 25 dB. . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Weighting factor vs. SINR for Fig. 4.2. . . . . . . . . . . . . . . . . . . 36

4.5 SER vs. SINR for combining conventional clipping and blanking, un-

coded, p = 0.005, SNR = 25 dB . . . . . . . . . . . . . . . . . . . . . . 36

4.6 BER vs. SINR for combining conventional clipping and blanking, convolutional-

coded, p = 0.005, SNR = 25 dB . . . . . . . . . . . . . . . . . . . . . . 37

4.7 Maximum γ vs. SNR for combining conventional clipping and blanking,

SINR = −10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xiv LIST OF FIGURES

4.8 Maximum γ vs. SNR for combining conventional clipping and joint blank-

ing/clipping, SINR = −10 dB. . . . . . . . . . . . . . . . . . . . . . . . 38

4.9 SER vs. SNR for combining conventional clipping and blanking, uncoded,

p = 0.005, SINR = −10 dB . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1 Example of false alarm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 System block diagram without multipath channel (scenario 1) . . . . . . 43

5.3 Peak amplitude clipping block . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 System block diagram with multipath channel (scenario 2) . . . . . . . . 46

5.5 One-branch PLC network . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.6 PAPR reduction with optimum A . . . . . . . . . . . . . . . . . . . . . . 49

5.7 BER for scenario 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.8 BER for scenario 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.1 System model with ℓ0-norm impulsive noise detection block. . . . . . . . 54

6.2 Recovery of impulsive noise using (P0) and (P1) algorithms. . . . . . . . 60

6.3 Performance of ηavg vs. the number of null subcarriers. . . . . . . . . . . 62

List of Tables

5.1 Philipps’ model representation for Fig.5.5 . . . . . . . . . . . . . . . . . 47

5.2 Channel Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Mean Peak Amplitude after Peak Amplitude Clipping . . . . . . . . . . 49

6.1 Statistical performance comparison of (P0) compared to (P1) minimisation. 61

6.2 F-measure F , precision of recovery P , and recall of support R perfor-

mance comparison of (P0) compared to (P1) minimization. . . . . . . . . 61

List of Acronyms

AMI advanced metering infrastructure

AMR automatic meter reading

AWGN additive white Gaussian noise

BB broadband

BER bit error rate

BFGS Broydon-Fletcher-Goldfarb-Shanno

CCDF complementary cumulative distributive function

CIR channel impulse response

DFT discrete Fourier transform

DR demand response

HAN home area networks

IBI interblock interference

IDFT inverse discrete Fourier transform

ISI intersymbol interference

LS least square

MMSE minimum mean-square error

NB narrowband

OFDM orthogonal frequency-division multiplexing

xviii LIST OF ACRONYMS

PAPR peak-to-average power ratio

PDF probability density function

PLC power-line communications

PRIME Powerline Intelligent Metering Evolution

QAM quadrature amplitude modulation

QPSK quadrature phase shift keying

SER symbol error rate

SINR signal-to-impulsive-noise ratio

SNR signal-to-noise ratio

TWACS two-way automatic communication system

UNB ultra narrowband

Chapter 1

Introduction

1.1 Background

A climate change study reported that 31% of greenhouse gas (GHG) emissions in the

U.S. in 2013 came from the electricity sector [1], which is the largest among other sources.

Meanwhile, the same condition also occurs in Australia where electricity contributed

about 34% of GHG emissions in 2014 - 2015 [2]. This is because the power systems

mainly use fossil fuel for power generation [3]. In particular, another report said that

global average temperature has risen 6.4C during the 21th century [4]. In fact, 40 -

70% of the species will become extinct if the temperature rises by 3.5C. Moreover, the

sea level is also predicted to be 59 cm higher by the end of the 21st century. Thereby,

floods will occur in more areas.

Some actions must be taken to prevent further effects. One solution to reduce GHG

emissions from electricity is to use smart grid which is able to control and maximise

operational efficiency of the production, transmission, and distribution of electricity

over power-lines [3, 5]. Smart grid is a merging technology between power systems and

communications technology [6]. The use of smart grid is expected to reduce about 30%

of GHG emissions [4]. To perform its task, smart grid needs a communication link

to communicate between nodes/devices. The communication link can be either wired

(such as power-line and optical fiber) or wireless. Using power-line communications

(PLC) is more beneficial as the power-lines infrastructure has been available so that

the deployment cost is low [5]. Specifically, high-rise apartments and buildings in urban

areas typically place the electric meters in underground locations where the access of

wireless communications is poor [3].

2 Chapter 1. Introduction

Historically, PLC has been in existence since 1918 and its first application was

for reading meters at remote locations [7]. Afterwards, PLC gained much attention.

Its research activities have brought PLC to many applications, including the above

mentioned smart grid systems and broadband PLC systems as a last mile technology

[8]. The applications of PLC, in general, fall into three bandwidth categories [5, 9, 10]:

1. Ultra Narrowband (UNB) PLC: It operates in the 0.3 - 3.0 kHz band to provide

about 100 bps up to 150 km. Its main examples are the Turtle System which is

used for automatic meter reading (AMR) and two-way automatic communication

system (TWACS) that can be used for several applications such as advanced

metering infrastructure (AMI), distribution automation, and demand response

(DR) application;

2. Narrowband (NB) PLC: It operates in the 3 - 500 kHz band to provide a few kbps

in the single carrier case and up to 800 kbps in the multicarrier case. Particularly,

the single carrier systems are called Low Data Rate (LDR) NB PLC while the

multicarrier ones are called High Data Rate (HDR) NB PLC. The LDR NB PLC

examples are ISO/IEC 14908-3 (LonWorks), ISO/IEC 14543-3-5 (KNX), CEA-

600.31 (CEBus), IC 61334-3-1, IEC 61334-5 (FSK and Spread-FSK), Insteon,

X10, HomePlug C&C, SITRED, Ariane Controls, and BacNet. Some examples of

HDR NB PLC are ITU-T G.hnem, IEEE 1901.2, Powerline Intelligent Metering

Evolution (PRIME), and G3-PLC;

3. Broadband (BB) PLC: It operates in the 1.8 - 250 MHz band and achieves data

rate up to 200 Mbps for home area networks (HAN) and Audio/Video applica-

tions. Some examples are TIA-1113 (HomePlug 1.0), IEEE 1901, ITU-T G.hn

(G.9960/G.9961), HomePlug AV/Extended, HomePlug Green PHY, HD-PLC,

UPA Powermax, and Gigle MediaXtreme.

As a power-line is basically designed for transmitting electricity power rather than

communications data, it has severe channel impairments when employed as a commu-

nication channel [11]. The multipath effects and noise are two regular issues for PLC.

PLC channel may be connected to various loads, so that the impedance of the line may

not be the same as the impedance of the load. The impedance mismatch causes some

portions of the signals to be reflected back. This phenomenon leads to multipath signal

propagation with frequency selective fading [12]. This thesis deals with noise as it is

the major impairment in PLC.

1.2. Previous Works 3

Noise in PLC is categorised into background and impulsive noise [11, 13]. Back-

ground noise comprises 1) colored background noise; 2) narrowband noise. Background

noise has slow varying root mean square (rms) amplitude over time and can be mod-

elled as additive white Gaussian noise (AWGN) [11, 14, 15]. Impulsive noise comprises

1) periodic impulsive noise asynchronous to the mains frequency; 2) periodic impulsive

noise synchronous to the mains frequency; 3) asynchronous impulsive noise. Different

from background noise, the rms amplitude of impulsive noise fluctuates fast over time,

ranging from millisecond to microsecond. Particularly, asynchronous impulsive noise

has random occurrence and is much stronger than the other impulsive noise types.

Hence, asynchronous impulsive noise is the major concern when dealing with impulsive

noise problems [13], especially in BB-PLC [16, 17]. Asynchronous impulsive noise (or

simply ”impulsive noise”) is often modelled as a Bernoulli-Gaussian random process

[14, 15, 18–20].

There are at least three modulation technique candidates for PLC [21, 22]. The

first candidate is single-carrier modulation. However, PLC channels have significant

intersymbol interference (ISI) which must be overcome through expensive and complex

channel equalizers. The second candidate is spread spectrum modulation. Although

it is robust against multipath fading and narrowband interference and also has lower

power spectral density, it has a problem regarding processing gain and number of users.

Processing gain is the ratio of the bandwidth of the transmitted signal and the message

bandwidth after conventional modulation which should be between 10 and 100. The

number of users, on the other hand, must be below the processing gain to maintain

the robustness against the interference. Moreover, it has a low spectral efficiency and

low-pass characteristics [23]. The third candidate is orthogonal frequency-division mul-

tiplexing (OFDM) which apparently is the most popular modulation technique for PLC,

especially for obtaining a high data rate as in BB PLC [11, 24]. In case of impulsive

noise, although OFDM spreads the power of impulsive noise among multiple subcar-

riers, impulsive noise can still degrade system performance [25]. Therefore, impulsive

noise mitigation in OFDM-based PLC is needed.

1.2 Previous Works

The simplest and most efficient method for mitigating impulsive noise in OFDM-based

PLC systems is to apply a nonlinear pre-processor at the receiver [24]. Nonlinear prepro-


cessors use a threshold to detect the impulsive noise. When the signal exceeds a given

threshold, the signal is assumed to be contaminated by the impulsive noise. Then, the

contaminated signal is treated based on the characteristic of the nonlinear preprocessor,

e.g. the signal can be nulled or clipped. As a result, an optimum threshold, that results

in the optimum performance, is required.

Three nonlinear pre-processors have been discussed and analyzed in [26] and [27].

Those methods are conventional clipping, blanking, and joint blanking/clipping. The

optimum threshold Blanking nonlinear preprocessor was analyzed thoroughly in [27].

The conventional clipping and joint blanking/clipping were analyzed in [26]. The math-

ematical expressions regarding the output signal-to-noise ratio (SNR) of the above men-

tioned nonlinear preprocessors were derived. In addition, the maximum SNR achieved

by optimising the threshold was analysed. In [28], clipping was performed separately

in the real and imaginary parts of the signal rather than in the amplitude of complex

OFDM baseband signal. The separate real/imaginary clipping is simpler to implement

than the amplitude blanker. However, the performance is worse.

OFDM suffers from large peak-to-average power ratio (PAPR) which increases the

probability of impulsive noise false detection of the nonlinear preprocessor [29]. To im-

prove the performance of the nonlinear preprocessor, it is desirable to reduce the PAPR.

In [19], the authors used selective mapping (SLM) as the PAPR reduction technique at

the transmitter to improve the blanking performance in an OFDM-based PLC under

impulsive noise. The proposed method minimised the blanking error and increased the

output SNR of the blanking nonlinear preprocessor.

OFDM systems often use some null subcarriers that do not carry information. With

the aid of null subcarriers and the fact that impulsive noise is sparse, in [25], principles

of compressive sensing [30–32] were used to reconstruct, i.e. detect and estimate, the

impulsive noise which can be modeled as Bernoulli-Gaussian. An extension to bursty

impulsive noise, which in the form of block-based compressive sensing, was proposed in

[33]. It was shown that the proposed method could detect the bursts and estimate the

impulsive noise.

1.3 Thesis Contributions

The contributions of this thesis are as follows. This thesis focuses on impulsive noise

detection and mitigation in the BB-PLC systems. Chapters 3-5 focus on the perfor-

1.3. Thesis Contributions 5

mance improvement of the nonlinear preprocessors as the impulsive noise mitigation

technique for OFDM-based PLC. Chapters 3 and 4 propose new variants of nonlinear

preprocessor and analyse the performance of the proposed nonlinear preprocessors in

terms of the output SNR. In order to obtain the optimum threshold, the expressions of

the output SNR is required. Chapter 5 improves the performance of the nonlinear pre-

processor by adding a PAPR reduction block at the transmitter due to the high PAPR

of the OFDM signal. Chapter 6 proposes the use of the smoothed ℓ0-norm which has

low complexity to detect the impulsive noise. Once detected the impulsive noise can be

treated accordingly. The organisation of this thesis is as follows.

Chapter 3 presents a new variant of nonlinear preprocessor, namely deep clipping,

to mitigate impulsive noise. Deep clipping is characterised by two thresholds, with one

of them controlled by a parameter called depth factor. In addition, some mathematical

expressions for deep clipping are presented. It will be shown that deep clipping can

be considered as a generic form of the widely used nonlinear preprocessors such as

blanking, conventional clipping, and joint blanking/clipping. Finally, it is found that,

by optimising both thresholds, the performance of deep clipping in terms of output SNR

outperforms the other nonlinear preprocessors.

In chapter 4, a new variant of nonlinear preprocessor in the form of combination

of two nonlinear preprocessors is proposed. This method is able to further improve

the output SNR of a single nonlinear preprocessor. A general mathematical expression

of the output SNR for the proposed method is derived. The mathematical expression

depends on the type of the nonlinear preprocessor used. Two examples of the proposed

method along with the mathematical expressions of the output SNR are analysed. It

can be shown that both analytical and simulation results agree.

Chapter 5 shows that the performance of nonlinear preprocessors in an OFDM-based

PLC can be improved by reducing the PAPR. High PAPR results in false detection of

impulsive noise in the nonlinear preprocessor. Moreover, Guel’s clipping method as

a PAPR reduction technique is used as it does not decrease the performance of the

system in terms of the bit error rate (BER). The proposed system is analysed in two

scenarios, with and without the multipath channels. The simulation results show that

the proposed method can significantly improve the BER performance.

In chapter 6, a non-parametric impulsive noise mitigation method is presented. Due

to the fact that the impulsive noise is sparse and the null subcarriers are usually used


in an OFDM system, a compressive sensing method can be used to detect the impul-

sive noise. Investigation of the smoothed ℓ0-norm minimisation algorithm to detect the

impulsive noise is proposed. Smoothed ℓ0-norm minimisation method has low complex-

ity compared with the conventional ℓ1-norm minimisation method. Simulation results

show that this approach is promising as it achieves comparable performance as the con-

ventional ℓ1-norm minimisation but with much lower complexity, i.e. lower processing

time.

Chapter 7 concludes this thesis and also presents some future works in this field.

Chapter 2

PLC System and Channel

2.1 OFDM Modulation

OFDM modulation belongs to multicarrier modulation in which the total carrier is

divided into some narrowband subcarriers [34]. The analog OFDM modulated signal is

given by

x(t) =1

N

N−1∑

k=0

Xk exp(j2πkt/Ts) for 0 ≤ t ≤ Ts, (2.1)

where Xk is the complex modulated symbol, N is the number of subcarriers, j =√−1,

and Ts is symbol duration. A cyclic prefix is then appended to overcome interblock

interference (IBI).

In digital communication systems, we sample the OFDM signal in (2.1) at Tsamp =

Ts/N . The n-th sample of the discrete OFDM signal is given by

xn =

N−1∑

k=0

Xk exp (j2πkn/N) = IDFTXk, (2.2)

where IDFT is the inverse discrete Fourier transform. Therefore, in practice, OFDM

generation (modulation) can be done by using IDFT. At the receiver, the discrete Fourier

transform (DFT) is implemented as the OFDM demodulator.

2.2 System Model

This thesis considers a generic OFDM-based PLC system as shown in Fig. 2.1. Data

symbols after QPSK or QAM modulation, XkN−1k=0 , are passed to an OFDM modulator

8 Chapter 2. PLC System and Channel

OFDM

Mod

(IDFT)

OFDM

Demod

(DFT)+ +

PLC channelPAPR

Reduction

Impulsive Noise

Mitigation

Figure 2.1: An OFDM-based PLC system

to form OFDM signal, xkN−1k=0 , where N is the number of subcarriers. As OFDM

signal suffers high peak-to-average power ratio (PAPR), a PAPR reduction block may

be optionally used. The OFDM signal is then appended a cyclic prefix (not shown

in the figure) and passed to the PLC channel and contaminated by background and

impulsive noise. Without loss of generality, the power of the OFDM signal is normalised.

Assuming that the cyclic prefix is long enough to overcome the IBI, the received signal

is given by

rk =

L−1∑

n=0

hnx(k−n)modN + wk + ik

= sk + nk, (2.3)

where sk =∑L−1

n=0 hnx(k−n)modN = hk ⊛ xk and ⊛ denotes circular convolution, hk is

the normalised discrete-time channel impulse response, i.e.∑ |hk|2 = 1, with length L,

wk is AWGN with mean 0 and variance 2σ2w, ik is the impulsive noise, and nk is the

total noise. Keep in mind that sk is also Gaussian distributed with the same mean and

variance (2σ2s ) as xk.

The received signal is then processed by an impulsive noise mitigation block, which

the focus in this thesis. The output from the impulsive noise mitigation block is then

passed to an OFDM demodulator. Note that the impulsive mitigation technique is

generally categorised into parametric and non-parametric methods. Parametric methods

need an assumption of the noise model and information of its parameter values. In

contrast, non-parametric methods do not require such things [16]. In this thesis, both

methods in the form of nonlinear preprocessors (parametric method) and smoothed

ℓ0-norm minimisation approach (non-parametric method) will be discussed.

2.3 PLC Channel

PLC channel can be modeled by using either top-down approach or bottom-up approach.

The top-down approach considers the communication channel as a black box and uses

./Chapter1/fig_system_model.eps

2.4. Impulsive Noise Models 9

a frequency response to model the channel [17]. The parametric model for typical

power-line channel with frequency range from 500 kHz to 20 MHz is given by [17]

H(f) =P∑

v=1

gve−(a0+a1fη)dve−j2πf(dv/vp) (2.4)

where P is the number of path, a0 and a1 are the attenuation parameters, η is the

exponent of the attenuation factor (between 0.5 and 1), gv is the weighting factor for

path v, dv is the length of path v, τv = dv√ǫr/c0 = dv/vp is the delay of path v, vp is

the phase velocity, ǫr is the dielectric constant, and c0 is the speed of light.

In the bottom-up approach, the channel impulse response/frequency response is

obtained from the channel topology. One of the model of this approach is the echo

model [35]. The channel impulse response of a channel with P path is given by

h(t) =P∑

v=1

|ρv|ejϕvδ(t− τv) (2.5)

where ρv = |ρv|ejϕv is the complex attenuation factor with ϕv = arctan(ℑ(ρv)ℜ(ρv)

)

.

2.4 Impulsive Noise Models

There are three (memoryless) impulsive noise models commonly used in the literature:

Middleton Class A (MCA), Bernoulli-Gaussian (BG), and Symmetric α-stable (SαS). In

the following, the three impulsive noise models will be discussed briefly and compared.

Suppose the complex-valued MCA noise is denoted by nk,MCA, where k is the

discrete-time sample index, the probability density function (PDF) is given by

fnk(nk,MCA) =

∞∑

m=0

e−AAm

m!

1√

2πσ2mexp

(

−|nk,MCA|22σ2m

)

(2.6)

where

σ2m =

(

1 +1

Γ

)(m/A+ Γ

1 + Γ

)

σ2b , (2.7)

σ2b is the background noise variance, A is the impulsive index, and Γ is the background-

to-impulsive noise ratio. Note that, in practice, the occurrence of impulsive noise will

be rare, i.e. small A. For small A, the terms of the model in (2.6) can be truncated. It

has been shown that it is sufficient to use two terms only [36, 37].

The BG impulsive noise is modeled as [38]

ik = bk · gk, (2.8)

10 Chapter 2. PLC System and Channel

where bk is a Bernoulli sequence of 1 and 0, with probabilities p and 1− p, respectively,and gk is a random variable with Gaussian distribution of mean 0 and variance 2σ2g .

Here, bk denotes the occurrence of the impulsive noise while gk denotes its amplitude.

The total noise nk,BG = wk + ik, where wk is the background noise has PDF

fnk(nk,R, nk,I) = (1−p)G(nk,R, 0, σ2w)G(nk,I , 0, σ2w)+pG(nk,R, 0, σ2w+σ2g)G(nk,I , 0, σ2w+σ2g)

(2.9)

where nk,R and nk,I are the real and imaginary parts of nk,BG, respectively andG(x, 0, σ2) ,

1/√2πσ2 exp−(x2/(2σ2)). Note that BG model is accurate for modeling many natural

impulsive noise sources [36].

On the other hand, SαS model does not have a closed form expression and does

not follow Gaussian distribution [39]. This makes it difficult to obtain closed-form

expressions of the output SNR of the nonlinear preprocessor.

Chapter 3

Deep Clipping for Impulsive

Noise Mitigation in OFDM-based

PLC

The simplest and most efficient method for mitigating impulsive noise in OFDM-based

PLC systems is nonlinear preprocessors at the receiver [24]. Three nonlinear preproces-

sors have been discussed and analysed in [26] and [27]. Those methods were conventional

clipping, blanking, and joint blanking/clipping. Blanking was analysed thoroughly in

[27]. The conventional clipping and joint blanking/clipping were analysed in [26]. In

[28], clipping was performed separately in the real and imaginary parts of the signal

rather than in the amplitude of complex OFDM baseband signal.

In OFDM sytems, nonlinear preprocessors are also commonly used to reduce peak-

to-average power ratio (PAPR) [40–42]. A new variant of nonlinear pre-processor as

PAPR reduction technique, called deep clipping, was proposed in [43] to reduce the

effect of peak regrowth due to filtering. Different from clipping and/or blanking, deep

clipping cuts off the amplitude of the signal linearly (characterised by the linear clipping

slope or depth factor) until a certain threshold and then nulls the signal that exceeds

the threshold. Since high amplitude of impulsive noise leads to high amplitude of the

received signal, we conceive that the received signal with higher amplitude should be

clipped more, and deep clipping can be used as an effective impulsive noise mitigation

technique as well. Moreover, deep clipping can be seen as an intermediate form between

conventional clipping and blanking. When the clipping slope approaches zero, the deep

12

Chapter 3. Deep Clipping for Impulsive Noise Mitigation in OFDM-based

PLC

clipping becomes conventional clipping while when the clipping slope approaches infinity,

it becomes blanking.

There are at least two differences between using nonlinear preprocessor for reducing

PAPR and mitigating noise. First, for reducing PAPR nonlinear preprocessor is placed

at the transmitter while for mitigating impulsive noise it is at the receiver for mitigating

impulsive noise. Second, when the nonlinear preprocessor is used to reduce PAPR, it is

not expected to cut the signal as much as possible since it contains signal information

and cutting the signal too much can degrade the system performance. However, when

the nonlinear preprocessor is used to mitigate impulsive noise, it is desirable to cut the

noise as much as possible since noise is unwanted.

In this chapter, the characteristics of deep clipping will be analyzed and it will be

compared with the three nonlinear preprocessors discussed in [26]. With optimal pa-

rameters, deep clipping outperforms the other nonlinear preprocessors. In particular,

compared with using deep clipping for PAPR reduction in [43], it will be shown that the

value of the depth factor is larger in impulsive noise mitigation to cut the noise as much

as possible. Moreover, a mathematical expression regarding the SNR performance of the

deep clipping nonlinearity will be derived. The SNR expressions of conventional clip-

ping, blanking, and joint blanking/clipping can be easily obtained using the analytical

methods.

3.1 System Model

A system shown in Fig. 3.1 is considered. The system model is similar as in Fig. 2.1

except that a nonlinear preprocessor is used as the impulsive noise mitigation block at

the receiver. The received signal after sampling is given by [44]

rk =L−1∑

k=0

hkx(n−k)modN + wk + ik

= sk + wk + ik (3.1)

where xk = x(kTs/N) is the sampled OFDM signal in time domain with variance 2σ2x,

sk is the signal component of the channel output with variance 2σ2s (=1), wk is the

AWGN with mean 0 and variance 2σ2w, and ik is the impulsive noise. It is then passed

to the nonlinear pre-processor. The output of the pre-processor, yk, is then processed

by the OFDM demodulator before QAM demodulation and detection.

3.1. System Model 13

OFDM

Mod

(IDFT)

OFDM

Demod

(DFT)

Nonlinear

Preprocessor+ +PLC channel

Figure 3.1: OFDM-based PLC system model with a nonlinear preprocessor at the receiver.

The impulsive noise is modeled as Bernoulli-Gaussian impulsive noise [38] as follows:

ik = bk · gk (3.2)

The Bernoulli sequence bk of 1 and 0, with probability p and 1 − p, respectively, de-

notes the occurence of impulsive noise while gk is the random variable with Gaussian

distribution of mean 0 and variance 2σ2g that denotes its amplitude. The noisy chan-

nel is, thus, characterized by two parameters: signal-to-(background)-noise ratio (SNR)

and signal-to-impulsive-noise ratio (SINR) which are given by SNR = 10 log (1/2σ2w)

and SINR = 10 log (1/2σ2g ), respectively. The nonlinear preprocessors discussed in this

chapter are as follows:

1. Deep clipping [43]

yk =

rk, |rk| ≤ T,T − µ(|rk| − T )ejϕk , T < |rk| ≤ βT,0, |rk| > βT,

(3.3)

where T is clipping threshold, µ is depth factor or clipping slope, ϕk = arg(rk),

and β is a constant which is set such that yk(βT ) = 0, i.e. β = (1 + µ)/µ.

2. Conventional clipping [26]

yk =

rk, |rk| ≤ T,Tejϕk , |rk| > T.

(3.4)

3. Blanking [26, 27]

yk =

rk, |rk| ≤ T,0, |rk| > T.

(3.5)

4. Joint blanking/clipping [26]

yk =

rk, |rk| ≤ T1,T1e

jϕk , T1 < |rk| ≤ T2,0, |rk| > T2,

(3.6)

where T1 and T2 (T1 < T2) denote the thresholds for clipping and blanking, re-

spectively.

./Chapter2/fig_system_model.eps

14


PLC

It is assumed that the number of subcarriers is large enough so that the real and

imaginary parts of OFDM signal, xk, can be modelled as Gaussian random variable. As

stated previously, if the channel impulse response is normalised such that∑ |hk|2 = 1,

the output of the channel, sk, is also Gaussian distributed with the same variance

(also the same mean 0). As a result, to analyse the nonlinear preprocessors, Bussgang’s

theorem can be used[26, 27, 45]. The theorem says that for the SNR analysis the output

of a nonlinear system is equivalent to a scaled version of the useful signal, sk plus the

total distortion, dk or

yk = α · sk + dk, (3.7)

where α is a real-valued constant. To satisfy this theorem, it is required to choose α

such that E[dks∗k] = 0, where E[·] denotes expectation and * denotes complex conjugate.

Therefore,

α =E[yks

∗k]

E[sks∗k]

=1

E[|sk|2]E[yks

∗k]. (3.8)

In order to analyse the performance, the output SNR after the nonlinear preproces-

sor, γ, is evaluated by using

γ =E[|αsk|2]

E[|yk − αsk|2]=

(E[|yk|2]α2E[|sk|2]

− 1

)−1

. (3.9)

Note that the output SNR before and after DFT in OFDM demodulator are the same

[26].

3.2 Performance Analysis

As E[|sk|2] = 1, to use (3.9) to calculate γ, finding the mathematical expessions for

α and E[|yk|2] for each nonlinear preprocessor is needed. two events are considered

that are associated with impulsive noise occurrence: let I be the event that impulsive

noise occurs with probability P (I) = p and I be the complement, with probability

P (I) = 1 − p. Therefore, for event I, the received signal is rk = sk + wk + ik and the

total variance is 2σ2I = 2σ2s +2σ2w +2σ2g . For event I the received signal is rk = sk +wk

with total variance 2σ2I= 2σ2s + 2σ2w.

16


PLC

The complete derivation of the two terms in (3.13) can be found in Appendix A. In

particular, E[(T + µT )ejϕks∗k|C2, I] and E[µ|rk|ejϕks∗k|C2, I] are given by (A.6) and

(A.11), respectively. To obtain E[yks∗k|C2, I]P (C2, I), P (C2, I) has to be calculated

first. Using Bayes’ rule it can be written

P (C2, I) =

(

e− T2

2σ2I − e

−β2T2

2σ2I

)

(1− p). (3.14)

As a result E[yks∗k|C2, I ]P (C2, I) is given by (3.15). In (3.15), Q(·) denotes Gaussian

Q-function.

E[yks∗k|C2, I ]P (C2, I) =

√

2πσ2s(T + µT )√

σ2I

Q

T√

σ2I

−Q

βT√

σ2I

+σ2sT

2

σ2I

e− T2

2(σ2I) − 2σ2sµ

(

e− T2

2(σ2I) − e

−β2T2

2(σ2I)

)

(1− p).

(3.15)

The term E[yks∗k|C2, I]P (C2, I) can be obtained using (3.15) by changing σ2

I→ σ2I and

(1− p)→ p. As a result, the mathematical expression for E[yks∗k] is shown in (3.16).

E[yks∗k] = 2σ2s −

∑

i∈I,I

2σ2s

e− T2

2σ2i −

√π

2

(T + µT )√

σ2i

Q

T√

σ2i

−Q

βT√

σ2i

+ µ

(

e− T2

2σ2i − e

−β2T2

2σ2i

)

P (i). (3.16)

Consequently, using (3.8) the expression of α for deep clipping can be obtained as given

in (3.17).

αdc = 1−∑

i∈I,I

e− T2

2σ2i −

√π

2

(T + µT )√

σ2i

Q

T√

σ2i

−Q

βT√

σ2i

+ µ

(

e− T2

2σ2i − e

−β2T2

2σ2i

)

− [µ(β − 1)− 1]βT 2

2σ2ie−β2T2

2σ2i

P (i). (3.17)

To get the mathematical formula for γ, the formula for E[|yk|2] should be derived.

It is straightforward to express

E[|yk|2] = E[|yk|2|C1, I]P (C1, I) + E[|yk|2|C1, I]P (C1, I)

+ E[|yk|2|C2, I]P (C2, I) + E[|yk|2|C2, I]P (C2, I)

+ E[|yk|2|C3, I]P (C3, I) + E[|yk|2|C3, I]P (C3, I). (3.18)


Again, the third line of (3.18) equals zero. The term E[|yk|2|C1, I] is given by (see [27]

for the derivation)

E[|yk|2

∣∣C1, I

]= 2σ2I −

T 2

eT2

2σ2I − 1

. (3.19)

Meanwhile, the term E[|yk|2|C2, I] is given by (B.1) in Appendix B. Then, E[|yk|2|C1, I]

and E[|yk|2|C2, I] can be obtained easily by replacing σ2Iin (3.19), (A.8), and (B.3) with

σ2I . As a result, the formula for E[|yk|2] is shown in (3.20).

E[|yk|2]dc =∑

i∈I,I

2σ2i + 2σ2i (µ2 − 1)e

− T2

2σ2i + ([µβ(1 + µ)− (1 + µ)2]T 2 − 2µ2σ2i )e

−β2T2

2σ2i

− µ√

2πσ2i (T + µT )

Q

T√

σ2i

−Q

βT√

σ2i

P (i). (3.20)

Substituting (3.17) and (3.20) into (3.9) yields the mathematical equation for γdc.

Note that for deep clipping the terms µ(β − 1) − 1 and µβ(1 + µ) − (1 + µ)2 in (3.17)

and (3.20) have a value of zero. Those terms are kept on purpose, as for conventional

clipping, blanking, and blanking/clipping the terms are not zero.

3.2.2 Conventional Clipping, Blanking, and Blanking/Clipping

Using the equations in Section 3.2.1, the results for conventional clipping, blanking,

and blanking/clipping in [26] can be easily obtained. As explained before, deep clipping

transfer function can be seen as the intermediate form between conventional clipping

and blanking. When µ = 0, it becomes conventional clipping, that means β =∞. The

parameters are substituted into (3.17) and (3.20) for conventional clipping to get

αcc = 1−∑

i∈I,I

e− T2

2σ2i −

√π

2

T√

σ2i

Q

T√

σ2i

P (i), (3.21)

and

E[|yk|2]cc =∑

i∈I,I

2σ2i − 2σ2i e− T2

2σ2i

P (i). (3.22)

When µ = ∞, it results in β = 1. Substituting β = 1 into (3.17) and (3.20) yields

blanking nonlinearity formulae as given in (3.23) and (3.24).

αb = 1−∑

i∈I,I

(

1 +T 2

2σ2i

)

e− T2

2σ2i

P (i). (3.23)

18


PLC

E[|yk|2]b =∑

i∈I,I

2σ2i −[2σ2i + T 2

]e− T2

2σ2i

P (i). (3.24)

The blanking/clipping method is a joint method between classical clipping and

blanking. It is needed to set µ = 0 and change T → T1 and βT → T2 in (3.17)

and (3.20) to yield (3.25) and (3.26) as follows

αbc = 1−∑

i∈I,I

e−

T21

2σ2i +

T1T22σ2i

e−

T22

2σ2i

−√π

2

T1√

σ2i

Q

T1√

σ2i

−Q

T2√

σ2i

P (i). (3.25)

E[|yk|2]bc =∑

i∈I,I

2σ2i − 2σ2i e−

T21

2σ2i − T 2

1 e−

T22

2σ2i

P (i).

(3.26)

3.3 Numerical Results

3.3.1 Mathematical Expressions Validation

An OFDM system with 16-QAM modulation and 512 subcarriers is simulated. In this

subsection the SNR and SINR are set to be 25 dB and -10 dB, respectively, and depth

factor is µ = 1. The agreement between simulation and (3.17) is shown in Fig. 3.2.

The validation of (3.9) with the use of (3.17) and (3.20) appears in Fig. 3.3. It shows

the SNR after nonlinear pre-processing, γ, with varying threshold and impulsive noise

occurence probability, p. It shows that for every p, there is an optimum threshold that

maximises γ. The optimum threshold value should be chosen so that the useful signal is

kept as much as possible and at the same time the impulsive noise is mitigated as much

as possible. In this case, the α should be α ≈ 1 [48]. From the figure it is clear that

the (maximum) γ values are different for different p. When p = 0.001, the maximum

γ nearly approaches SNR value, that is about 25 dB. That is because the number of

signals exceeding the threshold is few, so that when the clipper removes the impulsive

noise, most of the signal samples are not corrupted. However, when the impulsive noise

occurence is large, e.g. p = 0.1, the maximum γ is only about 7 dB.

3.3. Numerical Results 19

0 5 10 150

0.2

0.4

0.6

0.8

1

Threshold, T

Bus

sgan

gs c

onst

ant,

α

simulationtheory

Figure 3.2: α with varying threshold, p = 0.01

From Fig. 3.2, the thresholds above 2.5 satisfy the requirement that α ≈ 1. However,

the threshold cannot be chosen too large. Fig. 3.3 gives additional information for

optimum threshold selection. The maximum γ that yields optimum threshold, occurs

when T is about 2.5 for p = 0.01. The optimum threshold selection is discussed in the

next subsection.

When the threshold is approaching infinity, T → ∞, the γ values reach a fixed

value because the nonlinear pre-processor will not perform significant clipping. In this

case, the SNR approaches a certain fixed value that depends only on the variances and

probability of the impulsive noise occurrence as follows:

limT→∞

γdc =1

2(σ2w + pσ2g). (3.27)

In this subsection, only (3.17) and (3.20) are validated since the validations of

mathematical expressions (3.21) - (3.26) for conventional clipping, blanking, and blank-

ing/clipping have been provided in [26]. In addition, the plot of γ with varying depth

factor value for fixed threshold, T = 2 for various p is shown in Fig. 3.4.

3.3.2 Threshold Optimization

Optimising the output SNR after clipping, γ, is necessary. To obtain maximum value

of γ, (3.9) should be maximised by minimising E[|yk|2]/α2.

The optimum threshold that optimises γ is calculated by

minT

(E[|yk|2]/α2). (3.28)

./Chapter2/fig2.eps

20


PLC

0 5 10 15−10

−5

0

5

10

15

20

25

Threshold, T

Out

put S

NR

, γ (

dB)

p = 0.001

p = 0.01

p = 0.1

T → ∞

T → ∞

T → ∞

Figure 3.3: γ with varying threshold and p

0 1 2 3 4 54

6

8

10

12

14

16

18

20

22

Depth factor, µ

Out

put S

NR

, γ (

dB)

simulationtheory

p = 0.001

p = 0.01

p = 0.1

Figure 3.4: γ with varying depth factor and p

./Chapter2/fig3.eps

./Chapter2/fig4.eps


−30 −25 −20 −15 −10 −5 01

1.5

2

2.5

3

3.5

4

4.5

5

SINR(dB)

Opt

imum

thre

shol

d

conventional clippingblankingblanking/clippingdeep clipping

Figure 3.5: Optimum thresholds, SNR = 25 dB, p = 0.01

One of the most popular optimisation techniques is the quasi-Newton method [49, p.

350]. The mathematical expressions of E[|yk|2] and α are substituted into (3.28) and

then the quasi-Newton method with Broydon-Fletcher-Goldfarb-Shanno (BFGS) up-

date formula [49] is used to optimise (3.28). The optimum thresholds of the four nonlin-

ear pre-processors are shown in Fig. 3.5. The SNR and p are 25 dB and 0.01, respectively.

For blanking/clipping technique, T2 = 1.4T1 is used as the optimum one [26].

Using the optimum threshold, the maximum γ can be obtained as shown in Fig. 3.6.

The depth factor for deep clipping is set to be 1. From the figure, it is clear that for

SINR below about −12 dB, deep clipping technique yields better SNR. For SINR lower

than −12 dB, blanking/clipping technique has the best performance.

When optimising both the depth factor and threshold for deep clipping technique,

the optimisation problem is modified to be

minT,µ

(E[|yk|2]/α2). (3.29)

The optimum depth factor and threshold are obtained by using the same procedure as

before. They are then used to determine the maximum γ. The results are shown in

Fig. 3.7, for p = 0.01 and p = 0.1. It clearly shows that deep clipping has the best

performance among the four techniques when both µ and T are optimised, especially

when the impulsive noise is harsh.

./Chapter2/fig5.eps

22


PLC

−30 −25 −20 −15 −10 −5 012

13

14

15

16

17

18

19

SINR(dB)

Max

imum

γ(d

B)

conventional clipping, theoryblanking, theoryblanking/clipping, theorydeep clipping, theorysimulation

Figure 3.6: Maximum γ, SNR = 25 dB, p = 0.01

−30 −25 −20 −15 −10 −5 04

6

8

10

12

14

16

18

20

SINR(dB)

Max

imum

γ(d

B)

conventional clipping, theoryblanking, theoryblanking/clipping, theorydeep clipping, theorydeep clipping, simulation

p = 0.01

p = 0.1

Figure 3.7: Maximum γ, deep clipping uses optimum threshold and depth factor, SNR =

25 dB

./Chapter2/fig6.eps

./Chapter2/fig7.eps

3.4. Conclusions 23

3.3.3 Maximum γ with Varying SNR

The maximum γ with varying SNR is shown in Fig. 3.8. The SINR is kept constant

at −7 dB and p is set to be 0.01 and 0.1. Numerical results in Fig. 3.8 (with optimum

thresholds for the four clipping techniques and also optimum depth factor for deep

clipping), clearly show that the maximum γ for deep clipping outperforms the others,

especially when p is large.

0 5 10 15 20 25 30−4

−2

0

2

4

6

8

10

12

14

16

SINR(dB)

Max

imum

γ(d

B)

conv clip, theoryblanking, theoryblanking/clipping, theorydeep clipping, theorydeep clipping, simulation

p = 0.01

p = 0.1

Figure 3.8: Maximum γ with varying SNR, SINR = −7 dB

3.4 Conclusions

In this chapter, a deep clipping technique to mitigate impulsive noise in PLC systems

has been proposed and analysed. Deep clipping is characterised by two parameters, i.e.

a threshold and a depth factor. Moreover, an expression regarding the output SNR of

the deep clipping has been derived and from the expression, the ouput SNR expres-

sions of the other three nonlinear preprocessors can be obtained in a straightforward

manner. The performance of deep clipping, in terms of ouput SNR, has been compared

with the three existing techniques, i.e. conventional clipping, blanking, and joint blank-

ing/clipping. Numerical results have shown that deep clipping outperforms the other

techniques when the deep clipping parameters, T and µ, are optimised.

./Chapter2/fig8.eps

Chapter 4

Linear Combining of Nonlinear

Preprocessors for OFDM-based

PLC

It has been discussed that the simplest method to mitigate impulsive noise in OFDM

is by applying a nonlinear preprocessor at the receiver. The nonlinear preprocessor

can be one of the following: conventional clipping, blanking, joint blanking/clipping,

and deep clipping [26, 27, 50]. A nonlinear preprocessor is often characterised by one

threshold (such as conventional clipping and blanking) or two thresholds (such as joint

blanking/clipping and deep clipping). An optimum output SNR is then achieved by

optimising those thresholds. Thorough discussions of those preprocessors, including the

mathematical expressions regarding the output SNR, can be found in [26, 50].

In wireless communications, on the other hand, a well known diversity technique

has been commonly employed to improve system performance over fading channels [44].

Using the diversity technique, a transmitter sends the signal over some independent

fading channels and then a receiver weighs and combines the received signals so that

the effect of the fading channel is minimised. Motivated by the diversity concept,

a linear combining method by utilising more than one nonlinear preprocessor at one

time is proposed. For the sake of simplicity, a PLC system employing two nonlinear

preprocessors simultaneously is discussed in this chapter. Note that in [26, 50], only one

of the above mentioned nonlinear preprocessors was used at the receiver at one time.

The output SNR for the proposed method can be derived by using Bussgang’s theorem

26

Chapter 4. Linear Combining of Nonlinear Preprocessors for OFDM-based

PLC

OFDM

Mod

(IDFT)

OFDM

Demod

(DFT)+ +

Weighting

&

Combining

PLC

ChannelMMSE

Equalizer

Figure 4.1: OFDM-based PLC system model with combination of nonlinear preprocessors.

[26, 27, 45, 50]. Both numerical and simulation results show that the proposed method

yields higher output SNR and better symbol/bit error rate (SER/BER) compared with

using each preprocessor separately.

4.1 System Model

The proposed system is shown in Fig. 4.1. The system model is similar as in Chapter 1

except that a combination of nonlinear preprocessors as the impulsive noise mitigation

method is used at the receiver. The received signal after sampling is given by [44]

rk =

L−1∑

k=0

hkx(n−k)modN + wk + ik

= sk + wk + ik (4.1)

where xk = x(kTs/N) is the sampled OFDM signal in time domain with variance 2σ2x,

sk is the signal component of the channel output with variance 2σ2s (=1), wk is the

AWGN with mean 0 and variance 2σ2w, and ik is the impulsive noise. It is then passed

to the nonlinear pre-processor. The output of the pre-processor, yk, is then processed

by the OFDM demodulator before QAM demodulation and detection.

The impulsive noise is modeled as Bernoulli-Gaussian impulsive noise [38] as follows:

ik = bk · gk (4.2)

The Bernoulli sequence bk of 1 and 0, with probability p and 1−p, respectively, denotes

the occurence of impulsive noise while gk is the random variable with Gaussian distri-

bution of mean 0 and variance 2σ2g that denotes its amplitude. The noisy channel is,

thus, characterized by two parameters: signal-to-(background)-noise ratio (SNR) and

signal-to-impulsive-noise ratio (SINR) which are given by SNR = 10 log (1/2σ2w) and

SINR = 10 log (1/2σ2g), respectively.

./Chapter3/system_model.eps


From Fig. 4.1, it can be seen that the received signal is passed to two nonlinear

preprocessor subsystems f1(rk) and f2(rk). The output from the two subsystems, yk,1

and yk,2, are then weighted and combined to yield yk,comb as

yk,comb = qyk,1 + (1− q)yk,2, (4.3)

where q is the weighting factor, 0 ≤ q ≤ 1. The nonlinear processor output signal is

then processed by an OFDM demodulator and a minimum mean-square error (MMSE)

equalizer.

Note that an MMSE equalizer is commonly employed to compensate the channel ef-

fect. The one tap MMSE equalizer coefficients are given byWk = H∗k/(|Hk|2+SNR−1),

where Hk is channel frequency response and * denotes complex conjugate operator [51].

Finally, Xk = WkYk,comb, where Yk,comb is the output of OFDM demodulator, can be

obtained.

4.2 Performance Analysis

To analyse the nonlinear preprocessor, the well known Bussgang’s theorem is used as

given by [26, 27, 50]

yk,comb = αcomb · sk + dk,comb. (4.4)

where αcomb is a real-valued constant that is chosen such that E[dk,combs∗k] = 0. Thus,

αcomb is given by (4.5).

αcomb =E[yk,combs

∗k]

E[sks∗k]

=E[yk,combs

∗k]

E[|sk|2]= E[yk,combs

∗k]. (4.5)

Note that E[|sk|2] = 1.

The term E[yk,combs∗k] can be expressed by

E[yk,combs∗k] = qE[yk,1s

∗k] + (1− q)E[yk,2s

∗k]. (4.6)

Therefore, (4.5) can be written as

αcomb = qα1 + (1− q)α2, (4.7)

where α1 and α2 are the constant factors corresponding to f1(rk) and f2(rk), respec-

tively. The two nonlinear preprocessors also satisfy the Bussgang’s theorem as given by

(4.8) and (4.9), respectively.

yk,1 = α1 · sk + dk,1. (4.8)

28


PLC

yk,2 = α2 · sk + dk,2. (4.9)

where

α1 =E[yk,1s

∗k]

E[|sk|2]= E[yk,1s

∗k]. (4.10)

α2 =E[yk,2s

∗k]

E[|sk|2]= E[yk,2s

∗k]. (4.11)

From (4.4), the output SNR from the nonlinear preprocessor can be calculated by

using

γcomb =E[|αcombs∗k|2]

E[|yk,comb − αcombsk|2]

=

(E[|yk,comb|2]α2combE[|sk|2]

− 1

)−1

=

(E[|yk,comb|2]

α2comb

− 1

)−1

, (4.12)

where αcomb is given in (4.7) and E[|yk,comb|2] is given by

E[|yk,comb|2] = E[yk,comby∗k,comb]

= E[(qyk,1 + (1− q)yk,2)(qyk,1 + (1− q)yk,2)∗]

= q2(E[|yk,1|2]− 2E[yk,1y∗k,2] + E[|yk,2|2])

+ 2q(E[yk,1y∗k,2]− E[|yk,2|2]) + E[|yk,2|2]. (4.13)

then (4.12) can be maximised by choosing the weighting factor, q, using

q = argmaxq

(γcomb). (4.14)

For optimum performance, γcomb in (4.12) is a function of optimum thresholds (see

4.2.1 and 4.2.2). The optimum threshold of a nonlinear preprocessor, however, is diffi-

cult to be expressed in a closed-form and can be solved numerically(e.g. using quasi-

Newton method with BFGS update [49]) [27, 50]. Once the optimum thresholds are

obtained1,α1, α2, E[|yk,1|2], E[yk,1y∗k,2], and E[|yk,2|2] can be calculated and q can be

obtained by setting dγcomb/dq = 0. As a result, a closed-form expression for q can be

obtained as follows

q =−(t1 − t2)−

√

(t1 − t2)2 − 4(t3 − t4)(t5 − t6)2(t3 − t4)

(4.15)

1All the noise parameters (2σ2w , 2σ

2g , and p) are assumed to be known.


where

t1 = 2(α1 − α2)2E[|yk,2|2],

t2 = 2α22(E[|yk,1|2]− 2E[yk,1y

∗k,2 + E[|yk,2|2]]),

t3 = 2(α1 − α2)2(E[yk,1y

∗k,2]− E[|yk,2|2]),

t4 = 2(α1α2 − α22)(E[|yk,1|2]− 2E[yk,1y

∗k,2] + E[|yk,2|2]),

t5 = 2(α1α2 − α22)E[|yk,2|2], and

t6 = 2α22(E[yk,1y

∗k,2]− E[|yk,2|2]).

The mathematical expressions for α1 and α2 in (4.7) and E[|yk,1|2], E[|yk,2|2], andE[yk,1y

∗k,2] in (4.13) depend on f1(rk) and f2(rk). In the following subsection, two

examples of the proposed method are presented and relevant mathematical expressions

are derived.

Before going through the examples, let I be an event that impulsive noise occurs with

probability P (I) = p and I be the complement of event I with probability P (I) = 1−p.From now on, subscripts I and I are used to indicate any quantities corresponding to

events I and I, respectively. As a result, the two received signals are rk,I = sk+wk+ ik

with total variance 2σ2I = 2σ2s + 2σ2w + 2σ2g = 1 + 2σ2w + 2σ2g and rk,I = sk + wk with

total variance 2σ2I= 2σ2s + 2σ2w = 1 + 2σ2w.

4.2.1 Combining Conventional Clipping and Blanking

The first example is combining optimum conventional clipping [as f1(rk)] and optimum

blanking [as f2(rk)]. The optimum conventional clipping is given by [26, 50]

yk,1 = yk,c =

rk, |rk| ≤ Tc,opt,Tc,opte

jϕk , |rk| > Tc,opt.(4.16)

and optimum blanking is given by [26, 50]

yk,2 = yk,b =

rk, |rk| ≤ Tb,opt,0, |rk| > Tb,opt.

(4.17)

where Tc,opt and Tb,opt are the thresholds that yield optimum output SNR for con-

ventional clipping and blanking, respectively and ϕk = arg (rk). The terms α1 = αc,

α2 = αb, E[|yk,1|2] = E[|yk,c|2], and E[|yk,2|2] = E[|yk,b|2] have been derived and verified

in [50]. For convenience, the expressions are represented here in (4.18), (4.19), (4.20),

(4.21).

αc = 1−∑

i∈I,I

e−

T2c,opt

2σ2i −

√π

2

Tc,opt√

σ2i

Q

Tc,opt√

σ2i

P (i). (4.18)

30


PLC

E[|yk,c|2] =∑

i∈I,I

2σ2i − 2σ2i e

−T2c,opt

2σ2i

P (i). (4.19)

αb = 1−∑

i∈I,I

(

1 +T 2b,opt

2σ2i

)

e−

T2b,opt

2σ2i

P (i). (4.20)

E[|yk,b|2] =∑

i∈I,I

2σ2i − (2σ2i + T 2

b,opt)e−

T2b,opt

2σ2i

P (i). (4.21)

In (4.18), Q(·) denotes the Q-function [46].

From [50], we notice that the optimum threshold for blanking is always higher than

that for conventional clipping, Tb,opt > Tc,opt. Therefore, (4.22) for yk,cy∗k,b is obtained.

yk,cy∗k,b =

|rk|2, |rk| ≤ Tc,opt,Tc,opt|rk|, Tc,opt ≤ |rk| ≤ Tb,opt,0, |rk| ≥ Tb,opt.

(4.22)

It can be seen that (4.22) has three events C1, C2, and C3 corresponding to |rk| ≤ Tc,opt,Tc,opt ≤ |rk| ≤ Tb,opt, and |rk| ≥ Tb,opt, respectively. E[yk,cy

∗k,b] can be written as

E[yk,cy∗k,b] = E[yk,cy

∗k,b|C1, I]P (C1, I) + E[yk,cy

∗k,b|C1, I ]P (C1, I)

+ E[yk,cy∗k,b|C2, I]P (C2, I) +E[yk,cy

∗k,b|C2, I ]P (C2, I)

+ E[yk,cy∗k,b|C3, I]P (C3, I) +E[yk,cy

∗k,b|C3, I ]P (C3, I) (4.23)

The third line of (4.23) is zero. Each term in (4.23) will be calculated. The term

E[yk,cy∗k,b|C1, I]P (C1, I) can be calculated using P (C1, I) = P (C1|I)P (I) as follows

E[yk,cy∗k,b|C1, I]P (C1, I) = E[|rk|2

∣∣C1, I]P (C1|I)P (I)

=

∫ Tc,opt

0|rk|2 ·

|rk|σ2I

e−

|rk|2

2σ2I d|rk|

P (I)

=

2σ2I − 2σ2Ie

−T2c,opt

2σ2I − T 2

c,opte−

T2c,opt

2σ2I

P (I), (4.24)


E[yk,cy∗k,b|C2, I]P (C2, I) = Tc,optE[|rk|

∣∣C2, I]P (C2|I)P (I)

=

Tc,opt

∫ Tb,opt

Tc,opt

|rk| ·|rk|σ2I

e−

|rk|2

2σ2I d|rk|

P (I)

=

Tc,opt

√

π · 2σ2I

Q

Tc,opt√

σ2I

−Q

Tb,opt√

σ2I

+ T 2c,opte

−T2c,opt

2σ2I − Tc,optTb,opte

−T2b,opt

2σ2I

P (I). (4.25)

The terms E[yk,cy∗k,b|C1, I]P (C1, I) and E[yk,cy

∗k,b|C2, I]P (C2, I) can be obtained by

changing σ2I→ σ2I and P (I)→ P (I) in (4.24) and (4.25), respectively. Substituting all

terms into (4.23) yields (4.26).

E[yk,cy∗k,b] =

∑

i∈I,I

2σ2i − 2σ2i e

−T2c,opt

2σ2i + Tc,opt

√

π · 2σ2i

×

Q

Tc,opt√

σ2i

−Q

Tb,opt√

σ2i

− Tc,optTb,opte−

T2b,opt

2σ2i

P (i). (4.26)

4.2.2 Combining Conventional Clipping and Joint Blanking/Clipping

In the second example, the combination of the optimum conventional clipping [as f1(rk)]

and optimum joint blanking/clipping [as f2(rk)] is analysed. Optimum joint blank-

ing/clipping is given by [26, 50].

yk,2 = yk,bc =

rk, |rk| ≤ Tbc,opt,Tbc,opte

jϕk , Tbc,opt < |rk| ≤ 1.4Tbc,opt,

0, |rk| > 1.4Tbc,opt,

(4.27)

where Tbc,opt is the optimum threshold for joint blanking/clipping. The expressions

of α1 and E[|yk,1|2] for optimum conventional clipping are given by (4.18) and (4.19).

The expressions of α2 = αbc and E[|yk,2|2] = E[|yk,bc|2] are given by (4.28) and (4.29),


The term E[yk,cy∗k,bc|C3, I ]P (C3, I) can be calculated as

E[yk,cy∗k,bc|C3, I ]P (C3, I) = E[Tc,optTbc,opt|C2, I]P (C3|I)P (I)

=

Tc,optTbc,opt

∫ 1.4Tbc,opt

Tbc,opt

|rk|σ2I

e−

|rk|2

2σ2I d|rk|

P (I)

=

Tc,optTbc,opt

e−

T2bc,opt

2σ2I − e

−(1.4Tbc,opt)

2

2σ2I

P (I).

(4.32)

Note that all terms for event I can be obtained from the corresponding terms for event

I by changing σ2I→ σ2I and P (I)→ P (I). Substituting all the terms into (4.31) yields

E[yk,cy∗k,bc] =

∑

i∈I,I

2σ2i − 2σ2i e

−T2c,opt

2σ2i + Tc,opt

√

π · 2σ2i

×

Q

Tc,opt√

σ2i

−Q

Tbc,opt√

σ2i

− Tc,optTbc,opte−

(1.4Tbc,opt)2

2σ2i

P (i). (4.33)

4.3 Numerical Results

An OFDM system with 512 subcarriers and 16-QAM modulation is simulated. The

multipath PLC channel model as in [12] is used in the simulations. The channel model

considers many factors of power-lines including reflection, transmission, attenuation,

and delay. Moreover, the parameters of the real world networks in [12] are used to

reflect the practical PLC environment. For details refer to [12].

4.3.1 Maximum γ with Varying SINR

Fig. 4.2 and Fig. 4.3 show the theoretical and simulated maximum γcomb with varying

SINR for combination of conventional clipping and blanking and combination of con-

ventional clipping and joint blanking/clipping, respectively. The figures also present the

related theoretical maximum γ for the individual nonlinear preprocessors. The SNR is

set to be SNR = 25 dB [27, 52, 53] and the probabilities of impulsive noise occurrence

34


PLC

−20 −15 −10 −5 06

8

10

12

14

16

18

20

22

24

SINR(dB)

Max

imum

γ (

dB)

conv clip + blank, theoryconv clip + blank, simulationconv clipblank

p = 0.005

p = 0.01

p = 0.05

Figure 4.2: Maximum γ vs. SINR for combining conventional clipping and blanking,

SNR = 25 dB.

is set to be p = 0.005, p = 0.01, and p = 0.05 [18, 26, 27]. Those probabilities are used

to represent low, mild, and severe impulsive noise environment, respectively.

From Figs. 4.2 and 4.3 it can be seen that the simulation results for the proposed

method present good agreements with the theoretical expressions for all p’s. Further-

more, it is clear that the proposed method yields an improved output SNR than the

individual nonlinear preprocessors. Significant improvement is achieved in the severe

impulsive noise environment. Fig. 4.3 compares the performance of the output SNR

with that of deep clipping with optimum threshold and depth factor. Note that deep

clipping is characterised by two thresholds (the second threshold is determined by a

depth factor) and is the intermediate form between conventional clipping and blank-

ing [50]. From Fig. 4.3 it can be seen that the proposed method results in slightly

higher output SNR than the deep clipping for all SINR values. Note that deep clip-

ping method involves joint optimisation of two parameters (i.e. clipping threshold and

depth factor) leading to higher complexity.

From (4.3), it can be seen that for some SINR values, the weight, q, may have value

of either 1 or 0. It implies that the proposed method, at least, is able to perform as

well as either of the individual nonlinear preprocessor. For example, Fig. 4.4 shows the

optimum q values for Fig. 4.2. It can be inferred that q → 0 when SINR approaches

./Chapter3/SINR_vs_max_gamma_clip_and_blank_all_2.eps


−20 −15 −10 −5 06

8

10

12

14

16

18

20

22

24

SINR(dB)

Max

imum

γ (

dB)

conv clip + blank/clip, theoryconv clip + blank/clip, simulationconv clip

blank/clipdeep clip

p = 0.005

p = 0.01

p = 0.05

Figure 4.3: Maximum γ vs. SINR for combining conventional clipping and joint blank-

ing/clipping, SNR = 25 dB.

lower values and q → 1 when SINR values are higher. It is consistent with Fig. 4.2 in

which the proposed preprocessor performs as a conventional clipper when SINR > −5dB and when SINR < −15 dB, the proposed preprocessor tends to act as a blanker.

The SER performance for combination of conventional clipping and blanking with

p = 0.005 is illustrated in Fig. 4.5. The results also show that the combination of

two nonlinear preprocessors has better SER than the individual ones. Moreover, the

BER performance of the proposed system with convolutional code for combination of

conventional clipping and blanking with p = 0.005 is also evaluated as depicted in Fig.

4.6. A rate 1/2 code with constraint length K = 3 and generator polynomial G = [5, 7]8

[46] is used. The performance is similar to the uncoded cases.

4.3.2 Maximum γ with Varying SNR

The maximum γ vs. SNR in Figs. 4.7 and 4.8 for p = 0.005, 0.01, and 0.05 are pre-

sented. In this case, the SINR is kept at −10 dB. It can be seen that the simulation

results agree with theoretical results. Clearly, they also show that the performance of

the proposed method outperforms the individual preprocessors.

To complete our results, Fig. 4.9 shows SER performance vs. SNR for combination

./Chapter3/SINR_vs_max_gamma_clip_and_blankclip_all_2.eps

36


PLC

−20 −15 −10 −5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR (dB)

Opt

imum

wei

ghtin

g fa

ctor

, q

p = 0.005p = 0.01p = 0.05

Figure 4.4: Weighting factor vs. SINR for Fig. 4.2.

−20 −15 −10 −5 010

−2

10−1

100

SINR (dB)

SE

R

conv clip + blankconv clipblank

Figure 4.5: SER vs. SINR for combining conventional clipping and blanking, uncoded,

p = 0.005, SNR = 25 dB

./Chapter3/weighting_factor_blank_clip_001a.eps

./Chapter3/SINR_vs_SER_clip_and_blank0005.eps


−20 −15 −10 −5 010

−3

10−2

10−1

SINR (dB)

BE

R


Figure 4.6: BER vs. SINR for combining conventional clipping and blanking,

convolutional-coded, p = 0.005, SNR = 25 dB

0 5 10 15 20 25 30 35 40−5

0

5

10

15

20

SNR(dB)

Max

imum

γ (

dB)

conv clip + blank, theoryconv clip + blank, simulationconv clipblank

p = 0.005

p = 0.01

p = 0.05

Figure 4.7: Maximum γ vs. SNR for combining conventional clipping and blanking,

SINR = −10 dB.

./Chapter3/SINR_vs_BER_CC_clip_and_blank0005.eps

./Chapter3/SNR_vs_max_gamma_clip_and_blank_all.eps

38


PLC

0 5 10 15 20 25 30 35 40−5

0

5

10

15

20

SNR(dB)

Max

imum

γ (

dB)

conv clip + blank/clip, theoryconv clip + blank/clip, simulationconv clipblank/clipdeep clip

p = 0.005

p = 0.01

p = 0.05

Figure 4.8: Maximum γ vs. SNR for combining conventional clipping and joint blank-

ing/clipping, SINR = −10 dB.

0 5 10 15 20 25 30 3510

−2

10−1

100

SNR (dB)

SE

R


Figure 4.9: SER vs. SNR for combining conventional clipping and blanking, uncoded,

p = 0.005, SINR = −10 dB

of conventional clipping and blanking with p = 0.005. The SNR is set to be SINR =

−10 dB. It can be seen that the SER for the combination of conventional clipping

and blanking yields better performance than the conventional clipping and blanking

separately for high SNR values.

./Chapter3/SNR_vs_max_gamma_clip_and_blankclip_all.eps

./Chapter3/SNR_vs_SER_clip_and_blank0005.eps

4.4. Conclusions 39

4.4 Conclusions

In this chapter, a linear combination of two nonlinear preprocessors has been proposed.

The proposed technique further improves the output SNR of the nonlinear preprocessor

in PLC systems impaired by the impulsive noise. The generic output SNR expression

for the proposed method has been derived. The expression depends on the types of the

nonlinear preprocessor employed. Two examples, i.e. the combination of conventional

clipping and blanking and the combination of conventional clipping and joint blank-

ing/clipping, have been also presented. It has been shown that both numerical and

simulation results agree and the proposed method yields higher output SNR than the

individual ones.

Chapter 5

Joint Peak Amplitude and

Impulsive Noise Nonlinear

Preprocessors in OFDM-based

PLC

Nonlinear preprocessors are preferred as the most popular method for mitigating im-

pulsive noise because of their simplicity [27, 54]. They include conventional clipping,

blanking, joint blanking/clipping, and deep clipping [55], where a signal whose ampli-

tude exceeds a certain threshold, T , will be assumed that it is contaminated by impulsive

noise and will be handled according to the processing rule of the nonlinear preprocessor

employed.

However, OFDM systems result in some high peak amplitude signals because of the

use of the IDFT at the transmitter. Therefore, OFDM systems have large PAPR. As a

result, OFDM signals amplitude can be high enough to be falsely considered as impulsive

noise contaminated signals by the clipper at the receiver [29]. As an example, Fig. 5.1

shows the signals at the output of the IDFT that are falsely clipped by impulsive noise

clipper.

Therefore, a joint peak amplitude and impulsive noise nonlinear preprocessors is

42

Chapter 5. Joint Peak Amplitude and Impulsive Noise Nonlinear

Preprocessors in OFDM-based PLC

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

sample index

ampl

itude

contaminatedby impulsive noise?

impulsive noiseclipping threshold

Figure 5.1: Example of false alarm

proposed to improve the system performance. The peak amplitude nonlinear prepro-

cessor is placed at the transmitter while the impulsive noise one is at the receiver. Peak

amplitude nonlinear preprocessor aims to limit the peak amplitude so that increasing

the probability of the impulsive noise nonlinear preprocessor to cut only the ”real”

impulsive noise. The idea of Guel’s method for the peak amplitude nonlinear prepro-

cessor, which is called peak amplitude clipping [56], is used. Basically, in this method

conventional clipping with threshold A is used. To maintain the system’s bit error rate

(BER) performance, the unused subcarriers (i.e. null subcarriers) are employed through

a filtering process. However, the filtering process may result in peak regrowth. Conse-

quently, a few iterations to suppress the peak regrowth can be performed. Then, both

optimal thresholds will be obtained through extensive simulations.

In this chapter, two scenarios are studied. In the first scenario, the PLC multipath

effects are ignored (i.e. the PLC channel is assumed to be perfect). It can be found that

the peak amplitude clipping will improve BER significantly. In the second scenario,

the PLC multipath effects are included. The multipath channel and impulsive noise

occurrence make the system performance degrade. The convolution process between

the signals and the channel results in peak regrowth that leads to the false alarm at

the impulsive noise clipper. However, by using peak amplitude clipping, a noticeable

performance improvement can be obtained. In this section, Phillipps’ echo model is used

./Chapter4/false_alarm_ok.eps

5.1. System Models 43

OFDM

Mod

(IDFT)

OFDM

Demod

(DFT)

Blanking+ +Peak Amplitude

Clipping

Figure 5.2: System block diagram without multipath channel (scenario 1)

as the PLC channel model and the one-branch PLC network model is implemented. In

addition, blanking is used as the impulsive noise nonlinear preprocessor.

5.1 System Models

5.1.1 Clipping at the Transmitter using Guel’s Method

First, consider a system model as shown in Fig. 5.2, where the multipath effects are

ignored. This is considered as scenario 1. The QAM-modulated data are passed to the

IDFT block to form OFDM signal samples. In practice, there usually exists some null

subcarriers. For this matter, let N be the set of locations of the null subcarriers in an

OFDM block and Nc is the complement.

Throughout this section, the discrete system representation is obtained by sampling

it at frequency fs = NJ/Ts, where J is the oversampling factor. Therefore, as an

example, the discrete OFDM signal is given by xk = x(kTsNJ ). The discrete OFDM

signals can be also obtained by zero-padding Xk with NJ − N zeros and taking NJ-

point IDFT [57]. So that Xk can be expressed as [58]

X = X0,X1, · · · ,XN−1, 0, 0, · · · , 0︸︷︷︸

(J−1)N

(5.1)

The OFDM signal is then passed to the peak amplitude clipping block which is

shown in Fig. 5.3. The conventional clipping is formulated as follows:

xk =

xk, |xk| ≤ A,

Aej·arg(xk), |xk| > A,

(5.2)

where A is the clipping threshold.

./Chapter4/system_awgn.eps

44



FFT IFFTFilternal

Clipping

Figure 5.3: Peak amplitude clipping block

The clipping noise (which is the differences between xk and xk), ck, is transformed

to the frequency domain (denoted by Cn) by using FFT as follows

C = C0, C1, · · · , CN−1, CN , · · · , CNL−1︸︷︷︸

out-of-band

(5.3)

Let O be the set of out-of-band signals locations. In order to clip the signal at null

subcarriers, the following filtering process is performed

Cn =

Cn, n ∈ N,

0, n ∈ (Nc ∪ O).

(5.4)

Next, Cn, n = 0, 1, · · · , NL − 1 is transform back to the time domain (i.e. ck in Fig.

5.3) by performing IFFT. The resulting signals, ck is then summed with xk that yield

sk, which does not degrade the BER because N ∩ Nc = ∅ [56]. The filtering process

of (5.4) results in peak regrowth. Consequently, a few iterations can be performed to

suppress the peak regrowth.

The output power after peak amplitude clipping is obviously lower than the power

before peak amplitude clipping. Let the power after peak amplitude clipping be Po.

The output signals after peak amplitude clipping are normalised by a factor of 1/√Po.

The PAPR is defined as

PAPR =

maxk∈[0,JL]

|sk|2

Ps(5.5)

where Ps is the average power of sk. Because the power has been normalised, therefore,

in this case the PAPR can be written as PAPR = maxk∈[0,JL] |sk|2.

./Chapter4/papr_red_block.eps

5.1. System Models 45

5.1.2 Impulsive Noise and Blanking at the Receiver

The output signal from the peak amplitude clipping block is then passed to the noisy

channel. At the receiver, the received signal is expressed as

rk = sk + wk + ik (5.6)

where wk is AWGN with variance 2σ2w and ik is the impulsive noise that is modelled by

a Bernoulli-Gaussian random process.

At the receiver, blanking is used as the nonlinear preprocessor as follows

yk =

rk, |rk| ≤ T,

0, |rk| > T,

(5.7)

where T is the blanking threshold.

5.1.3 PLC Multipath Channel with Impulsive Noise

In this subsection, the PLC multipath effects are taken into attention. This is referred

to as scenario 2. The system model is shown in Fig. 5.4. Most of the blocks are the

same as Fig. 5.2 except that the frequency domain MMSE equalizer is included at

the receiver to compensate for the multipath effects. The multipath channel impulse

response (CIR) , hk, with length L performs convolution with the transmitted signal. A

sufficient length of cyclic prefix (not shown in the figure) is appended in the symbols to

avoid IBI and then dropped at the receiver. Thus, the received signal can be expressed

as

rk =

L−1∑

m=0

hms(k−m)modN + wk + ik (5.8)

The CIR can be obtained from the network configuration and also cable parameters.

In this chapter, Phillipps’ echo model is used as follows [35]

h(t) =

M∑

v=1

|ρv|ejψvδ(t− τv) (5.9)

where M is number of path, τv, ρv, and ψv are delay time, complex attenuation factor,

and phase shift on path v, respectively. The time delay is given by τv = dv/c, where

46



OFDM

Mod

(IDFT)

OFDM

+ +Peak Amplitude

Clipping

Blanking

nnel

Figure 5.4: System block diagram with multipath channel (scenario 2)

tter

Figure 5.5: One-branch PLC network

dv is the length of the v-path, c = c0/√ǫr, ǫr is the relative dielectric constant of the

cable, and c0 is the velocity of light that equals 3 · 108 m/s.

In digital communications, transmit and receive filters are used. The discrete CIR,

hk, is obtained by sampling the resulting pulse of double convolution involving the

impulse response of the transmit filter, h(t), and the receive filter [59].

In this paper, a one-branch network configuration is used as shown in Fig. 5.5 [60,

pp. 23–25]. The network consists of three lines, which are AB, BC, and BD , and ZL

as the load at line 2. The channel parameters using Phillipps’ model for M = 4 can

be calculated by using the formulae shown in Table 5.1. The parameters in Table 5.1

are explained as follows:

Γ13 = transmission factor from line 1 to 3 = 1 + ρ13,



ρ13 = reflection factor from line 1 to 3 = Z2//Z3−Z1

Z2//Z3+Z1,


Z2//Z3+Z1,


Z3//Z1+Z2,

./Chapter4/system_chan_fd_post.eps

./Chapter4/plc_net.eps

5.2. Simulation Results 47

Table 5.1: Philipps’ model representation for Fig.5.5

Path,v Path Direction ρv τv

1 ABD Γ13 (d1 + d3)/c

2 ABCBD Γ12ρ2cΓ23 (d1 + 2d2 + d3)/c

3 ABCBCBD Γ12ρ22cρ2BΓ23 (d1 + 4d2 + d3)/c

4 ABCBCBCBD Γ12ρ32cρ

22BΓ23 (d1 + 6d2 + d3)/c

ρ2c = reflection factor at point C, line 2 = ZL//Z2

ZL+Z2,

ρ2B = ρ23 = ρ21,

ρ1B = ρ13 = ρ12,

and

Z =

√

Rl + jωLlGl + jωCl

(5.10)

where Rl, Ll, Gl, and Cl are the resistance, inductance, conductance, and capacitance

per unit length, respectively, and ω is the angular frequency of the propagating signal.

Note that the proposed method is general and may be applied to more complex network

topologies.

As stated before, to compensate for the channel effect, the MMSE equalizer is used.

The one tap MMSE equalizer coefficients are given as [51]

Wk =H∗k

|Hk|2 + SNR−1(5.11)

where Hk is channel frequency response. Therefore, Xk can be written as Xk =WkYk.

5.2 Simulation Results

The parameters for simulation are as follows: 16-QAM modulation, number of subcar-

riers 128, where 12 of them are used as null subcarriers, L = 4, Ts = 640 µs, CP length

= 60 µs, p = 0.01, SNR = 25 dB, and SINR = −10 dB.

In scenario 2, the cable parameters used are as follows: Rl = 0.0068 Ω/m, Ll = 0.53

µH/m, Cl = 47 pF/m, and Gl is negligible. By using (5.10) the characteristic impedance

48



Table 5.2: Channel Impulse Response

Path,v Path Direction ρv τv (µs)

1 ABD 0.6667 1.25

2 ABCBD −0.2221 + j0.0839 5

3 ABCBCBD −0.0317 + j0.0279 8.75

4 ABCBCBCBD −0.0035 + j0.0066 12.5

of the cable can be calculated which results in Z = 125− j66 Ω. It is assumed that AB,

BC, and BD are the same cables, therefore they have the same characteristic impedance.

The cable insulator is assumed to be polyethylene with ǫr = 2.25. The length of each

cable is d1 = 100 m, d2 = 375 m, and d3 = 250 m. Line 2 is terminated by 50 Ω load.

The formulae in Table 5.1 can be used to calculate the discrete CIR and the result is

shown in Table 5.2. The resulting CIR is normalised such that∑ |hk|2 = 1.

First, it is required to find the optimum value of peak amplitude clipping threshold,

A. The peak amplitude for every OFDM signal block is observed and then for a given

value of A the mean of the peak amplitude is calculated for all the total blocks simulated.

The results are shown in Table 5.3. A is examined for one and five iterations. The table

can be used to find the optimum A that results in minimum mean peak amplitude,

which are A = 1.2 and A = 1.4 for peak amplitude clipping with one and five iterations,

respectively. Therefore, those thresholds will be used throughout the simulations.

The distribution of PAPR as a result of the peak amplitude clipping is shown in Fig.

5.6. The distribution of PAPR is usually analysed by using a statistical parameter called

complementary cumulative distribution function (CCDF) . CCDF shows the probability

of PAPR exceeds a certain threshold. From that figure it can be found that the PAPR

(and also the peak amplitude) is significantly reduced. The PAPR reductions are about

0.6 dB and 1.1 dB at probability of 10−3 for one and five iterations, respectively.

The simulation results for BER performance with varying impulsive noise blanking

threshold for scenario 1 and 2 are shown in Fig. 5.7 and Fig. 5.8, respectively. In


Table 5.3: Mean Peak Amplitude after Peak Amplitude Clipping

ANumber of Iterations

1 5

1 2.3943 2.3221

1.1 2.3910 2.3038

1.2 2.3904 2.2902

1.3 2.3923 2.2822

1.4 2.3968 2.2803

1.5 2.4033 2.2848

1.6 2.4117 2.2958

1.7 2.4214 2.3127

1.8 2.4320 2.3342

0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

PAPR, x dB

Pro

babi

lity,

PA

PR

≥ x

no peak amplitude clippingwith peak amplitude clipping, iter = 1with peak amplitude clipping, iter = 5

Figure 5.6: PAPR reduction with optimum A

./Chapter4/ccdfnew.eps

50



1 1.5 2 2.5 3 3.5 410

−6

10−5

10−4

10−3

10−2

10−1

100

Threshold, T

BE

R

no peak amplitude clippingwith peak amplitude clipping, iter=1with peak amplitude clipping, iter=5

Figure 5.7: BER for scenario 1

1 1.5 2 2.5 3 3.5 410

−5

10−4

10−3

10−2

10−1

100

Threshold, T

BE

R

no peak amplitude clippingwith peak amplitude clipping, iter=1with peak amplitude clipping, iter=5

Figure 5.8: BER for scenario 2

./Chapter4/ber_awgn_ok.eps

./Chapter4/ber_chan_ok.eps

5.3. Conclusions 51

scenario 1, it is observed that there is significant minimum BER improvement from

1.12× 10−5 to 7.75× 10−6 by using five iterations peak amplitude clipping. In scenario

2, it is clear that the peak amplitude clipping improves the minimum BER performance

although it is lower than that in scenario 1. Without peak amplitude clipping the

BER is 1.67 × 10−4 and by using five iterations peak amplitude clipping the BER is

8.71× 10−5. The multipath channel performs convolution with the transmitted signals,

so that it causes peak regrowth. As a consequence, the blanker may cut some signals

falsely.

5.3 Conclusions

A new method for improving the clipping nonlinearities performance for mitigating

impulsive noise in PLC has been proposed. The method jointly uses peak amplitude

and impulsive noise nonlinear preprocessors to minimalise false clipping at the impul-

sive noise detector. The false clipping occurs due to the high PAPR of the OFDM

systems. The peak amplitude nonlinear preprocessor is used to reduce the PAPR so

that the impulsive noise can detect the impulsive noise more accurately. It has also

been shown two scenarios where in the first scenario the PLC channel is ignored and

in the second scenario the effect of the PLC channel is considered. In the simulation,

the threshold for the peak amplitude nonlinear preprocessor is obtained from extensive

simulation. The simulation results have shown that the proposed method improves the

BER performance.

Chapter 6

Impulsive Noise Detection in

PLC with Smoothed ℓ0-norm

Impulsive noise mitigation in PLC can be conducted by first detecting the impulsive

noise and if detected, further processing can be performed. The simplest impulsive noise

detection is by using a threshold. If a sample exceeds a certain threshold, the sample

is assumed to be contaminated by impulsive noise. Then, an appropriate nonlinear

preprocessing (such as clipping) is employed [26, 27, 55]. However, this technique is

prone to false alarm since OFDM has large peak-to-average power ratio (PAPR) [29,

55, 61].

On the other hand, OFDM systems often use some null subcarriers that do not

carry information. In particular, more than half of the subcarriers are occupied by null

subcarriers in some modern PLC standards [62]. For example, the PRIME standard

uses 158 null subcarriers out of 256 subcarriers. With the aid of null subcarriers and the

fact that impulsive noise is sparse, in [25], principles of compressive sensing [30–32] were

used to detect and estimate the impulsive noise modelled as Bernoulli-Gaussian. An

extension to bursty impulsive noise was proposed in [33]. The basic idea was to estimate

the number of impulsive noise samples by minimising the ℓ0-norm. However, working

on ℓ0-norm directly is not easy as minimisation of ℓ0-norm is NP-hard. As a result,

some previous works relaxed the minimisation model by using the convex programming

algorithm with ℓ1-norm minimisation.

54 Chapter 6. Impulsive Noise Detection in PLC with Smoothed ℓ0-norm

OFDM

Mod

(IDFT)

OFDM

Demod

(DFT)+ +

sive noise

detection

Figure 6.1: System model with ℓ0-norm impulsive noise detection block.

In [63], an approximation of ℓ0-norm, called smoothed ℓ0-norm, along with its min-

imisation algorithm were proposed. Different from the estimation using ℓ1-norm min-

imisation, using the minimisation of smoothed ℓ0-norm algorithm yielded a lower com-

plexity while having the same (or better) accuracy. Its lower complexity motivates us

to investigate its potential for impulsive noise detection in power-line communications.

In this chapter, the use of the smoothed ℓ0-norm minimisation algorithm for de-

tecting impulsive noise in power-line communications is proposed. In particular, the

impulsive noise detection performance of the proposed smoothed ℓ0-norm minimisa-

tion is compared with the conventional one using ℓ1-norm minimisation. The proposed

method yields lower complexity and simulation results show that similar (or better)

accuracy to the conventional detection method can be achieved. In addition, the effect

of the number of null subcarriers on the detection performance for smoothed ℓ0 and

ℓ1 algorithms is presented. Simulation results show that the larger the number of null

subcarriers, the better the detection accuracy.

6.1 Impulsive Noise Detection Algorithm

6.1.1 System Model and Problem Formulation

Fig. 6.1 shows the equivalent complex baseband PLC system model. The system is

similar as in Chapter 1 except that an ℓ0-norm impulsive noise detection block is used.

For the sake of simplicity, in this chapter matrix notations are used. The received signal

after dropping the cyclic prefix is given by

r = Hx+w + i, (6.1)

where H is an N × N column circulant channel matrix with the first column being

normalised discrete-time channel impulse response [33], w is the AWGN with variance

./Chapter5/system_model2.eps

6.1. Impulsive Noise Detection Algorithm 55

2σ2w, and i is the Bernoulli-Gaussian impulsive noise.

The impulsive noise detection block is discussed in the following sentences. Let N

be the set of indices of null subcarriers and NC its complement. An (N −K)×N parity

matrix F1 can be constructed. The (N −K) rows are obtained from F that are in N,

i.e. F1 = F(N, :). It can be obtained

S = F1r

= F1s+F1w + F1i

= F1i+ w, (6.2)

where s = Hx, w = F1w, and F1s = 0. Note that w is also AWGN and has the same

mean and variance as w. The objective is to find the sparsest solution of i as follows

(P0) ip = argmini‖i‖0 subject to ‖F1i− S‖2 ≤ ǫ, (6.3)

where ‖i‖0 is the smoothed ℓ0-norm of i and ǫ is any positive number. Note that (P0)

is used to denote the smoothed ℓ0-norm algorithm. The amplitude of the estimated

impulsive noise, ip, is then refined to get a better estimate. The refined estimated

impulsive noise, ip, is then subtracted from the received signal to get the clean signal,

i.e. x = r− ip. The signal is then passed to the OFDM demodulator (i.e. the DFT).

6.1.2 The Smoothed ℓ0-Norm and Minimisation Algorithm

Let us consider how to find the sparsest solution for Ay = z by minimising the ‖y‖0. Incontrast to [63] where only real numbers are considered, this section deals with complex

numbers so the smoothed ℓ0-norm for complex numbers is used [64].

The ℓ0-norm of y = [y0, y1, · · · , yn−1]T is defined as the number of nonzero elements

of y or

‖y‖0 =

n−1∑

i=0

ν(yi), (6.4)

where

ν(y) =

1, |y| 6= 0,

0, |y| = 0.

(6.5)


An approximation of (6.5) can be developed by using a continuous (smoothed) function,

such as

fσ(y) = e−|y|2

2σ2 . (6.6)

As a result, it can be obtained

limσ→0

fσ(y) =

1, |y| = 0,

0, |y| 6= 0.

(6.7)

or [65]

fσ(y) ≈

1, |y| ≪ σ,

0, |y| ≫ σ.

(6.8)

It is defined

Fσ(y) =

n−1∑

i=0

fσ(yi). (6.9)

Using (6.9),(6.4) can be rewritten as

‖y‖0 ≈ n− Fσ(y) (6.10)

for small σ. Note that when σ → 0, ‖y‖0 is close to the true solution (6.4). From (6.10),

‖y‖0 can be minimised by maximising Fσ(y) subject to Ay = z.

A small value σ results in a lot of local maxima. When σ is large, the function

becomes smoother and contains less local maxima, thereby easier to solve. As the

algorithm requires σ → 0, one strategy is to use a decreasing σ sequence, e.g. by using

a gradient algorithm. The algorithm finds y0, i.e. the solution for σ →∞. This solution

has been given in [63–65], as follows

Theorem 1 The solution of the problem

maxFσ(y) subject to Ay = z, (6.11)

where σ → ∞ is the minimum ℓ2-norm solution of Ay = z, that is, y = A†z, where

A† = AH(AAH)−1 is the pseudo-inverse of A.

Next step is to choose a sequence of σ, [σ1, σ2, · · · , σJ ], where σ1 can be chosen two to

four times of maxi |yi| [65]. The next values of σ can be calculated by σj = cσj−1, where

6.1. Impulsive Noise Detection Algorithm 57

c is the σ decreasing factor and j = 2, 3, · · · , J . For each value of σj, Fσ is maximised

on Y = y|Ay = z by using M iterations of the steepest ascent algorithm. For every

iteration, y ← y + (µσ2)∇Fσ = y − µδ can be calculated, where µ is a decreasing

step-size parameter and δ , −σ2∇Fσ = [y0e−

|y0|2

2σ2 , y1e−

|y1|2

2σ2 , · · · , yn−1e−

|yn−1|2

2σ2 ]T .

The last step is to project back y to the feasible set Y, i.e. by calculating y ←y −A†(Ay − z). The final answer is y0 = yJ .

The above algorithm is summarised as follows:

1. Initialisation

(a) Choose a solution for Ay = z, i.e. v0 = A†z.

(b) Choose a decreasing sequence of σ = [σ1, σ2, · · · , σJ ], where σ1 can be chosen

as two to four times of max v0.

2. For j = 1, 2, · · · , J

(a) Set σ = σj.

(b) Maximize the Fσ on feasible set Y using U iterations as follows:

i. Set y = vj−.

ii. For u = 1, 2, · · · , U

A. Calculate δ = [y0e−

|y0|2

2σ2 , y1e−

|y1|2

2σ2 , · · · , yn−1e−

|yn−1|2

2σ2 ]T .

B. Calculate y← y − µδ.

C. Project y back onto feasible set

Y: y← y −A†(Ay − z).

End for

(c) Set vj = y.

End for

3. Output: y0 = vJ .

Now consider the noisy case z = Ay + n, where n is the AWGN. The minimisation

problem can be formulated as follows

yp = argminy

‖y‖0 subject to ‖Ay − z‖2 ≤ ǫ. (6.12)


Note that the previous (noiseless) algorithm can also be applied to this noisy case.

However, in this case the accuracy of the estimated y is bounded by noise power [65].

For impulsive noise detection, our problem formulation of (6.3) is similar to (6.12)

by simply replacing the parameters as follows: A,y, z → F1, i,S.

6.1.3 Postprocessing

The postprocessing to the amplitude of raw estimated impulsive noise is conducted to

get ip. The procedure is as follows [25, 33]:

1. Solve (6.3) to get the raw estimated values ip.

2. Estimate the support Ip = j : |ip(j)| > th, where th = k ×√

2σ2w and k is the

multiplication constant given by k =√

2 ln ((1− p)/p · σg/σw) [66].

3. Recalculate the amplitude of ip by using least-square (LS) or MMSE as follows.

An N ×m selection matrix Sm, where m is the cardinality of estimated impulsive

noise samples, m = |Ip|, is constructed. The elements of the matrix is Sm(i, j)=1

for i ∈ I, j = 1, 2, · · · ,m and 0 otherwise. Finally, the calculation can be done as

follows

(a) LS

ip = B−SHmFH1 S, (6.13)

or

(b) MMSE

ip = [(σw/σg)I+B]−SHmFH1 S, (6.14)

where B = SHmFH1 F1Sm.


6.2 Simulation Results

An OFDM system with QPSK modulation is simulated, N = 256 and N − K = 128

(19% lower than PRIME) to show the ability of (P0) algorithm1 to recover the impulsive

noise samples. Impulsive noise occurrence is rare (at most a few impulsive samples in

every OFDM block) in practical systems [25]. In this paper, without loss of generality, it

is assumed that every OFDM symbol is contaminated with three or five impulsive noise

samples with random positions, i.e. p ≈ 0.01 or p ≈ 0.02, respectively. The channel

impulse response is as in Fig. 5.5 and the SNR and INR are set to be SNR = 20 dB

and INR = 30 dB, respectively. Moreover, the parameters for (P0) algorithm are as

follows: µ = 2.5, σJ = 0.3×√

2σ2w, σ decreasing factor 0.5, and U = 3.

As a baseline, the comparison between (P0) and (P1) is done. The conventional

ℓ1-norm minimisation is shown as follows

(P1) ip = argmini‖i‖1 subject to ‖F1i− S‖2 ≤ ǫ, (6.15)

where ‖ · ‖1 is the ℓ1-norm and ‖ · ‖2 is the ℓ2-norm. Following [31, 33], a threshold, ǫ,

is set such that ‖w‖2 ≤ ǫ with probability ξ. To be feasible ξ = 0.95 is chosen. Note

that ‖w‖22 is a chi-squared distribution with 2(N −K) degrees of freedom, χ22(N−K) so

that ǫ2 = χ22(N−K)(0.95)2σ

2w , where χ

22(N−K)(0.95) is the 95th percentile of χ2

2(N−K).

The postprocessing steps are similar to those in Section 6.1.3, except that k = 1 is

used in the second step. ℓ1-magic with a log-barrier algorithm is used for (6.15) [67].

From now on, the terms (P1) and ℓ1-magic (with a log-barrier algorithm) will be used

interchangeably.

Fig. 6.2 shows the amplitude parts of the impulsive noise samples for original and

estimated values using (P0) and (P1) algorithm before the postprocessing for one random

OFDM symbol. It can be seen that the impulsive noise by using (P0) is recovered better

than (P1). The amplitude values are then refined by using LS as discussed in Section

6.1.3.

To analyse the capability of impulsive noise detection, the residual interference-plus-

1The publicly available program code for ℓ0-norm minimisation from

http://ee.sharif.edu/~SLzero is used.

http://ee.sharif.edu/~SLzero


50 100 150 200 2500

0.5

1

1.5

2

2.5

3

3.5

Samples

Am

plitu

de v

alue

estimated (P0)estimated (P1)original

Figure 6.2: Recovery of impulsive noise using (P0) and (P1) algorithms.

(background)-noise signal is introduced as follows [25, 33]

ρ = i− ip +w (6.16)

The normalised variance of (6.16) is θ = var(ρ)/var(w). Furthermore, the mean-error

square (MSE) of the estimated samples is MSE = ‖i− ip‖2 and the ”signal-to-noise”

ratio is η(dB) = 20 log ‖i‖2/‖i− ip‖2. Next, the experiments are performed and re-

peated 100 times. The average statistical performance comparison of those parameters

is shown in Table. 6.1. Note that if the estimated samples are the same as the original

samples, the θavg = is θavg = 1. In addition, the number of experiments that yielded η

larger than 20 dB was 91 (m = 3) and 95 (m = 5) among 100 runs for (P0). However, it

was just 76 (m = 3) and 73 (m = 5) for (P1). It is clear that (P0) algorithm outperforms

(P1) algorithm, by having lower MSE and higher η.

The average F-measure, precision of recovery, and recall of support recovery for the

estimated samples after postprocessing are also presented as shown in Table. 6.2. F-

measure is F = (2PR)/(R+P ), the precision of recovery is defined as P = |Ip ∩ I|/|Ip|,and the recall of support recovery is R = |Ip ∩ I|/|I| [68].

Next, the complexity in terms of the CPU processing time is compared. The average

processing time for one run was 2.56 milliseconds for (P0) and 1.15 seconds for (P1). It

shows that the estimated value using (P0) minimisation can represent the original value

./Chapter5/new12a.eps

6.3. Conclusions 61

Table 6.1: Statistical performance comparison of (P0) compared to (P1) minimisation.

mθavg MSEavg(×10−3) ηavg (dB)

(P0) (P1) (P0) (P1) (P0) (P1)

3 1.06 1.08 0.837 1.09 25.8 24.1

5 1.06 1.14 1.09 1.81 25.9 23.6

Table 6.2: F-measure F , precision of recovery P , and recall of support R performance

comparison of (P0) compared to (P1) minimization.

mFavg Pavg Ravg

(P0) (P1) (P0) (P1) (P0) (P1)

3 0.971 0.924 0.970 0.975 0.980 0.897

5 0.970 0.940 0.965 0.988 0.980 0.906

better while having three orders of magnitude faster in running time than ℓ1-magic.

Fig. 6.3 depicts the ηavg performance vs. the number of null subcarriers for (P0)

and (P1) algorithms with m = 3 and m = 5. Consistent with the nature of compressive

sensing, the larger the number of null subcarriers the better the detection accuracy.

Furthermore, for a large number of null subcarriers, the ηavg values for both m = 3 and

m = 5 are almost the same.

6.3 Conclusions

The impulsive noise can be considered as a sparse signal that is to be recovered by using

compressive sensing approach by utilising the property of null subcarriers in OFDM

systems. The recovery of sparse signal is commonly performed by using the conventional

ℓ1-norm minimisation algorithm. However, the method has relatively high complexity.

In this chapter, the use of smoothed ℓ0-norm minimisation algorithm for detecting

impulsive noise in OFDM-based PLC has been proposed and analysed. The comparison


60 80 100 120 140 1600

5

10

15

20

25

30

Number of null subcarriers

η avg(d

B)

m=3

m=5 (P0)

(P1)

Figure 6.3: Performance of ηavg vs. the number of null subcarriers.

between the performance of the smoothed ℓ0-norm minimisation algorithm and that of

the ℓ1-norm minimisation conventional recovery algorithm (using the ℓ1-magic tool with

a log-barrier algorithm) is performed. Simulation results have shown that the proposed

method yields lower complexity in terms of CPU processing time and provides a good

estimate.

./Chapter5/noz_vs_snr.eps

Chapter 7

Conclusions and Future Works

7.1 Conclusions

This thesis has proposed several impulsive noise mitigation techniques in BB-PLC sys-

tems. In particular, in Chapter 3 deep clipping technique has been proposed. Deep clip-

ping is a new variant of nonlinear preprocessor which has two parameters, i.e. threshold

and depth factor. The output SNR expression of the deep clipping has been derived. In

addition, using the deep clipping output SNR expression, the output SNR expressions

for the conventional preprocessors such as conventional clipping, blanking, and joint

blanking/clipping can be derived easily. It has been shown that the performance of

deep clipping in terms of output SNR outperforms the conventional preprocessors, such

as clipping, blanking, and joint blanking/clipping.

In Chapter 4 linear combination of two nonlinear preprocessors has been proposed.

This technique is motivated by the use of the diversity technique to overcome the fad-

ing problem in the wireless communications. The generic output SNR expression has

been derived. The expression depends on the nonlinear preprocessor employed. Two

examples have been presented and analysed. The results have shown that the proposed

method yields better performance than the separately used nonlinear preprocessors. In

addition, the proposed result has slightly better (or the same) performance than the

deep clipping (with optimum threshold and depth factor) while having lower complexity.

In Chapter 5 it has been shown that in OFDM-based PLC, high PAPR results in false

64 Chapter 7. Conclusions and Future Works

detection for nonlinear preprocessors. Therefore, a joint PAPR reduction and impulsive

noise mitigation method has been proposed. Guel’s method has been chosen as the

PAPR reduction technique as this method does not reduce the BER performance. The

clipping nonlinear preprocessor has been employed at the receiver as the impulsive noise

mitigation technique. The simulation results have shown that this method improves the

BER performance of OFDM-based PLC.

In Chapter 6 it has been shown that the impulsive noise is sparse in nature and

the OFDM-based PLC systems commonly use null subcarriers. Therefore, the principle

of compressive sensing can be used to detect the impulsive noise. Smoothed ℓ0-norm

sparse recovery algorithm has been proposed to detect the impulsive noise samples. The

reason is that the smoothed ℓ0-norm algorithm has a lower complexity but has a better

(or the same) performance compared with the conventional ℓ1-norm algorithm. The

simulation results have shown that the proposed method outperforms the conventional

one.

7.2 Future Works

Following this thesis there are several works that can be done as future works. In

this thesis, we employ Bernoulli-Gaussian impulsive noise. In some literature, aperi-

odic impulsive noise can be modelled as a Middleton Class A, Markov-Middleton, and

Markov-Gaussian [20]. The analysis using those models can be useful.

As mentioned in the Chapter I, this thesis focuses on the impulsive noise detection

and mitigation for BB-PLC. The impulsive noise detection and mitigation in the NB-

PLC is also of interest. Different from the BB-PLC, the periodic impulsive noise in

NB-PLC is more dominant [9]. The periodic impulsive noise can be modeled by using

a linear periodically time varying (LPTV) system model [69]. In [69], a non-parametric

mitigation technique based on sparse Bayesian learning was proposed. The improvement

of the non-parametric methods to mitigate the periodic impulsive noise can be done as

a future work.

Appendix A

Derivation of

E[(T + µT )ejϕks∗k

∣∣C2, I

]and

E[µ|rk|ejϕks∗k

∣∣C2, I

]

Let T + µT = T ′ and ζ = arg(rk)− arg(sk), then

E[(T + µT )ejϕs∗k

∣∣C2, I

]= E

[T ′|sk| cos ζ

∣∣C2, I

](A.1)

As wk = rk − sk we have |wk|2 = |rk|2 + |sk|2 − 2|rk||sk| cos ζ. Multiplying both sides

with T ′/|rk| and substituting it into (A.1) yields

E[T ′ejϕs∗k

∣∣C2, I

]=T ′

2

(E[|rk|∣∣C2, I

]+ E

[z1∣∣C2, I

]− E

[z2∣∣C2, I

])(A.2)

where z1 = |sk|2/|rk| and z2 = |wk|2/|rk|.It is more convenient to calculate E[T ′ejϕs∗k|C2, I ]P (C2|I) than E[T ′ejϕs∗k|C2, I ]

alone. Therefore, we need to calculate E[|rk|∣∣C2, I

]P (C2|I), E

[z1∣∣C2, I

]P (C2|I), and

E[z2∣∣C2, I

]P (C2|I) as shown in (A.3), (A.4), and (A.5), respectively. In (A.4) and

(A.5), I0(x) is the modified Bessel function of the first kind zeroth order and the inner

integrals can be obtained by using the formula found in [70, p. 4].

E[|rk|∣∣C2, I

]P (C2|I) =

√2π√

σ2I

Q

T√

σ2I

−Q

βT√

σ2I

+ T

[

e− T2

2σ2I − βe

−β2T2

2σ2I

]

(A.3)

66 Chapter A. Derivation of E[(T + µT )ejϕks∗k

∣∣C2, I

]and E

[µ|rk|ejϕks∗k

∣∣C2, I

]

E[z1|C2, I]P (C2|I) =

∫ βT

T

∫ ∞

0

|sk|2|rk||sk|σ2s

e−

|sk|2

2σ2s|rk|σ2w

e−

(|rk|2+|sk|2)

2σ2w I0

( |sk||rk|σ2w

)

d|sk|d|rk|

=

√2πσ2s(σ

4s + 3σ2sσ

2w + 2σ4w)

σ5I

Q

T√

σ2I

−Q

βT√

σ2I

+σ4sT

σ4I

[

e− T2

2σ2I − βe

−β2T2

2σ2I

]

(A.4)

E[z2|C2, I]P (C2|I) =

∫ βT

T

∫ ∞

0

|wk|2|rk|

|wk|σ2w

e−

|wk|2

2σ2w|rk|σ2s

e−

(|rk|2+|wk|2)

2σ2s I0

( |rk||wk|σ2s

)

d|wk|d|rk|

=

√2πσ2w(σ

4w + 3σ2sσ

2w + 2σ4s )

(σI)5

Q

T√

σ2I

−Q

βT√

σ2I

+σ4wT

σ4I

[

e− T2

2σ2I − βe

−β2T2

2σ2I

]

(A.5)

Substituting (A.3), (A.4), and (A.5) into (A.2), applying Bayes’ rule (noting that

P (C1, I) = P (C1|I)P (I)), and after simplification we have (A.6).

E[T ′ejϕks∗k|C2, I ]P (C2, I) =

√2πσ2s(T + µT )

√

σ2I

Q

T√

σ2I

−Q

βT√

σ2I

+σ2sT (T + µT )

σ2I

[

e− T2

2σ2I − βe

−β2T2

2σ2I

]

(1− p) (A.6)

Now, to calculate E[µ|rk|ejϕks∗k

∣∣C2, I

], we consider

E[µ|rk|ejϕks∗k

∣∣C2, I

]=µ

2

(E[|rk|2

∣∣C2, I

]+ E

[|sk|∣∣C2, I

]− E

[|wk|2

∣∣C2, I

])(A.7)

where,

E[|rk|2

∣∣C2, I

]= 2σ2I +

T 2

e− T2

2σ2I − e

−β2T2

2σ2I

×(

e− T2

2σ2I − β2e

−β2T2

2σ2I

)

(A.8)

E[|sk|2

∣∣C2, I

]= 2σ2s +

σ4sT2

(

e− T2

2σ2I − e

−β2T2

2σ2I

)

σ4I

(

e− T2

2σ2I − β2e

−β2T2

2σ2I

) (A.9)

67

and

E[|wk|2

∣∣C2, I

]= 2σ2w +

σ4wT2

(

e− T2

2σ2I − β2e

−β2T2

2σ2I

)

σ4I

(

e− T2

2σ2I − e

−β2T2

2σ2I

) (A.10)

As a result, we have

E[µ|rk|ejϕks∗k

∣∣C2, I

]= 2σ2sµ+

σ2sµT2

(

e− T2

2σ2I − β2e

−β2T2

2σ2I

)

σ2I

(

e− T2

2σ2I − e

−β2T2

2σ2I

) (A.11)

Appendix B

Derivation of E[|yk|2

∣∣C2, I

]

The term E[|yk|2

∣∣C2, I

]can be calculated as follows

E[|yk|2

∣∣C2, I

]= E

[(T + µT )2

∣∣C2, I

]− E

[2(T + µT )µ|rk|

∣∣C2, I

]+ E

[µ2|rk|2

∣∣C2, I

]

= E[(T + µT )2

∣∣C2, I

]− 2(T + µT )µE

[|rk|∣∣C2, I

]+ µ2E

[|rk|2

∣∣C2, I

]

(B.1)

where E[|rk|2

∣∣C2, I

]is given by (A.8),

E[(T + µT )2|C2, I] = (T + µT )2, (B.2)

and

E[|rk|∣∣C2, I

]=

√2π√

σ2I

e− T2

2σ2I − e

−β2T2

2σ2I

Q

T√

σ2I

−Q

βT√

σ2I

+T

e− T2

2σ2I − e

−β2T2

2σ2I

[

e− T2

2σ2I − βe

−β2T2

2σ2I

]

(B.3)

Bibliography

[1] Sources of greenhouse gas emissions. online, available at

http://www.epa.gov/climatechange/ghgemissions/sources.html.

[2] Quarterly update of australia’s national greenhouse

gas inventory: June 2015. online, available at

https://www.environment.gov.au/system/files/resources/cb14abbb-3a4b-406f-

a22d-86f565674c3e/files/nggi-quarterly-update-jun-2015.pdf.

[3] A. A. Amarsingh, H. A. Latchman, and D. Yang, “Narrowband power line com-

munications: Enabling the smart grid,” IEEE Potentials, vol. 33, no. 1, pp. 16–21,

Jan.-Feb. 2014.

[4] S. Kim, H.-K. Kim, and H. J. Kim, “Climate change and ICTs,” in Proc. Telecom-

munications Energy Conf., Incheon, Korea, 2009, pp. 1–4.

[5] S. Galli, A. Scaglione, and Z. Wang, “For the grid and through the grid: The role

of power line communications in the smart grid,” Proc. IEEE, vol. 99, no. 6, pp.

998–1027, Jun. 2011.

[6] S.-G. Yoon, S. Jang, Y.-H. Kim, and S. Bahk, “Opportunistic routing for smart

grid with power line communication access network,” IEEE Trans. Smart Grid,

vol. 5, no. 1, pp. 303–311, Jan. 2014.

[7] M. Schwartz, “Carrier-wave telephony over power lines: Early history,” IEEE Com-

mun. Mag., vol. 47, no. 1, pp. 14–18, Jan. 2009.

72 BIBLIOGRAPHY

[8] W. Liu, H. Widmer, and P. Raffin, “Broadband PLC access systems and field

deployment in European power line networks,” IEEE Commun. Mag., vol. 41,

no. 5, pp. 114–118, May 2003.

[9] M. Nassar, J. Lin, Y. Mortavi, A. Dabak, I. H. Kim, and B. L. Evans, “Local utility

power line communications in the 3-500 kHz band: Channel impairments, noise,

and standards,” IEEE Signal Process. Mag., vol. 29, no. 5, pp. 116–127, Sep. 2012.

[10] M. Spahiu and H. P. Partal, “High frequency modeling and impedance matching of

power transformers for PLC applications,” in Proc. IEEE PES Innovative Smart

Grid Technol. Conf., Washington, DC, USA, 2014, pp. 1–5.

[11] H. Meng, Y. L. Guan, and S. Chen, “Modeling and Analysis of Noise Effects on

Broadband Power-Line Communications,” IEEE Trans. Power Del., vol. 20, no. 2,

pp. 630–637, Apr. 2005.

[12] M. Zimmermann and K. Dostert, “A multi-path signal propagation model for the

power line channel in the high frequency range,” in Proc. IEEE Int. Symp. Power

Line Commun. Appl., Lancaster, U.K., 1999.

[13] N. Andreadou and F.-N. Pavlidou, “Modelling the noise on the OFDM power-line

communications systems,” IEEE Trans. Power Del., vol. 25, no. 1, pp. 150–157,

Jan. 2010.

[14] G. Ndo, P. Siohan, M.-H. Hamon, and J. Horard, “Optimization of turbo decoding

performance in the presence of impulsive noise using soft limitation at the receiver

side,” in Proc. IEEE Global Telecom. Conf., New Orleans, LA, USA, 2008, pp. 1–5.

[15] G. Ndo, P. Siohan, and M.-H. Hamon, “Adaptive noise mitigation in impulsive

environtment: Application to power-line communications,” IEEE Trans. Power

Del., vol. 25, no. 2, pp. 647–656, Apr. 2010.

[16] J. Lin, M. Nassar, and B. L. Evans, “Impulsive noise mitigation in powerline com-

munications using sparse Bayesian learning,” IEEE J. Sel. Areas Commun., vol. 31,

no. 7, pp. 1172–1183, Jul. 2013.

BIBLIOGRAPHY 73

[17] M. Zimmermann and K. Dostert, “Analysis and modeling of impulsive noise

in broad-band powerline communications,” IEEE Trans. Electromagn. Compat.,

vol. 44, no. 1, pp. 249–258, Feb. 2002.

[18] K. M. Rabie and E. Alsusa, “Quantized peak-based impulsive noise blanking in

power-line communications,” IEEE Trans. Power Del., vol. 29, no. 4, pp. 1630–

1638, Aug. 2014.

[19] K. M. Rabie and E. Alsusa, “Preprocessing-based impulsive noise reduction for

power-line communications,” IEEE Trans. Power Del., vol. 29, no. 4, pp. 1648–

1658, Aug. 2014.

[20] T. Shongwe, A. J. H. Vinck, and H. C. Ferreira, “On impulsive noise and its

models,” in Proc. IEEE Int. Symp. Power Line Commun. Appl., Glasgow, SCT,

U.K., 2014, pp. 12–17.

[21] E. Biglieri and P. Torino, “Coding and modulation for a horrible channel,” IEEE

Commun. Mag., vol. 41, no. 5, pp. 92–98, May 2003.

[22] M. Gotz, M. Rapp, and K. Dostert, “Power line channel characteristics and their

effect on communication system design,” IEEE Commun. Mag, vol. 42, no. 4, pp.

78–86, Apr. 2004.

[23] H. Hranisca, A. Haidine, and R. Lehnert, Broadband Powerline Communications

Networks: Network Design. New York: John Wiley & Sons, 2004.

[24] D.-F. Tseng, R.-B. Yang, T.-R. Tsai, Y. S. Han, andW. H. Mow, “Efficient Clipping

for Broadband Power Line Systems in Impulsive Noise Environment,” in Proc.

IEEE Int. Symp. Power Line Communications and Its Applications, Beijing, China,

2012.

[25] G. Caire, T. Y. Al-Naffouri, and A. K. Narayanan, “Impulse noise cancellation

in OFDM: An application of compressed sensing,” in Proc. IEEE Int. Symp. Inf.

Theory, Toronto, ON, Canada, 2008, pp. 1293–1297.

74 BIBLIOGRAPHY

[26] S. V. Zhidkov, “Analysis and comparison of several simple impulsive noise mitiga-

tion schemes for OFDM receivers,” IEEE Trans. Commun., vol. 56, no. 1, pp. 5–9,

Jan. 2008.

[27] S. V. Zhidkov, “Performance analysis and optimization of OFDM receiver with

blanking nonlinearity in impulsive noise environment,” IEEE Trans. Veh. Technol.,

vol. 55, no. 1, pp. 234–242, Jan. 2006.

[28] H. A. Suraweera, C. Chai, J. Shentu, and J. Armstrong, “Analysis of impulse noise

mitigation techniques for digital television systems,” in 8th Int. OFDM Workshop,

Hamburg, Germany, 2003, pp. 172–176.

[29] C.-H. Yih, “Iterative interference cancellation for OFDM signals with blanking

nonlinearity in impulsive noise channels,” IEEE Signal Process. Lett., vol. 19, no. 3,

pp. 147–150, Mar. 2012.

[30] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact

signal reconstruction from highly incomplete frequency information,” IEEE Trans.

Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006.

[31] E. J. Candes and P. A. Randall, “Highly robust error correction by convex pro-

gramming,” IEEE Trans. Inf. Theory.

[32] T. Y. Al-Naffouri, A. A. Quadeer, and G. Caire, “Impulse noise estimation and

removal for OFDM systems,” IEEE Trans. Commun., vol. 62, no. 3, pp. 976–989,

Mar. 2014.

[33] L. Lampe, “Bursty impulse noise detection by compressed sensing,” in Proc. IEEE

Int. Symp. Power Line Commun. Appl., Udine, Italy, 2011, pp. 29–34.

[34] Y. G. Li and G. L. Stuber, Eds., Orthogonal Frequency Division Multiplexing for

Wireless Communications. New York: Springer, 2006.

[35] H. Phillipps, “Modelling of powerline communication channels,” in Proc. Int. Symp.

Power Line Commun. Appl., Lancaster, U.K., 1999, pp. 14–21.

BIBLIOGRAPHY 75

[36] R. Pighi, M. Franceschini, G. Ferrari, and R. Raheli, “Fundamental performance

limits of communications systems impaired by impulse noise,” IEEE Trans. Com-

mun., vol. 57, no. 1, pp. 171–182, Jan. 2009.

[37] L. D. Bert, P. Caldera, D. Schwingshack, and A. M. Tonello, “On noise modeling

for power line communications,” in Proc. IEEE Int. Symp. Power Line Commun.

Appl., Udine, Italy, 2011, pp. 283–288.

[38] M. Ghosh, “Analysis of the effect of impulse noise on multicarrier and single carrier

QAM systems,” IEEE Trans. Commun., vol. 44, no. 2, pp. 145–147, Feb. 1996.

[39] P. G. Georgiou, P. Tsakalides, and C. Kyriakakis, “Alpha-stable modeling of noise

and robust time-delay estimation in the presence of impulsive noise,” IEEE Trans.

Multimedia, vol. 1, no. 3, pp. 291–301, Sep 1999.

[40] A. Zhang and L. Wu, “Performance of PAPR in the clipped OFDM system,” in

IEEE Int. Conf. Multimedia Signal Process., Guilin, China, 2011.

[41] Y.-C. Wang and Z.-Q. Luo, “Optimized iterative clipping and filtering for PAPR

reduction of OFDM signals,” IEEE Trans. Commun., vol. 59, no. 1, pp. 33–37,

Jan. 2011.

[42] L. Wang and C. Tellambura, “A simplified clipping and filtering technique for PAR

reduction in OFDM system,” IEEE Signal Process. Lett., vol. 12, no. 6, pp. 453–

456, Jun. 2005.

[43] S. Kimura, T. Nakamura, M. Saito, and M. Okada, “PAR reduction for OFDM

signals based on deep clipping,” in IEEE Int. Symp. Commun., Control, Signal

Process., 2008, pp. 911–916.

[44] A. Goldsmith, Wireless Communications. New York: Cambridge University Press,

2005.

[45] H. E. Rowe, “Memoryless nonlinearities with Gaussian inputs: Elementary results,”

Bell Syst. Tech. J., vol. 61, no. 7, pp. 1519–1525, Sep. 1982.

76 BIBLIOGRAPHY

[46] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw–Hill, 2001.

[47] A. L. Garcia, Probability, Statistics, and Random Processes for Electrical Engineer-

ing, 3rd ed. Upper Saddle River, New Jersey: McGraw-Hill, 1991.

[48] D. Guel and J. Palicot, “Clipping formulated as an adding signal technique for

OFDM peak power reduction,” in Proc. IEEE Veh. Technol. Conf., Barcelona,

Spain, 2009.

[49] S. S. Rao, Engineering Optimization: Theory and Practice, 4th ed. Hoboken, New

Jersey: John Wiley & Sons, inc., 2009.

[50] F. H. Juwono, Q. Guo, D. Huang, and K. P. Wong, “Deep clipping for impulsive

noise mitigation in OFDM-based power-line communications,” IEEE Trans. Power

Del., vol. 29, no. 3, pp. 1335–1343, Jun. 2014.

[51] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terres-

trial TV broadcasting,” IEEE Commun. Mag., vol. 33, no. 2, pp. 100–109, Feb.

1995.

[52] A. Mengi and A. J. H. Vinck, “Successive impulsive noise suppression in OFDM,”

in Proc. IEEE Int. Symp. Power Line Commun. Appl., Rio de Janeiro, Brazil,

2010, pp. 33–37.

[53] M. Korki, N. Hosseinzadeh, H. L. Vu, T. Moazzeni, and C. H. Foh, “Impulsive

noise reduction of a narrowband power line communication using optimal nonlin-

earity technique,” in Proc. Aust. Telecomun. Netw. Appl. Conf., Melbourne, VIC,

Australia, 2011, pp. 1–4.

[54] S. V. Zhidkov, “Impulsive noise suppression in OFDM based communication sys-

tems,” IEEE Trans. Consum. Electron., vol. 49, no. 4, pp. 944–948, Nov. 2003.

[55] F. H. Juwono, Q. Guo, D. Huang, and K. P. Wong, “Joint peak amplitude and

impulsive noise clippings in OFDM-based power line communications,” in Proc.

Asia Pacific Conf. Commun., Bali, Indonesia, 2013, pp. 567–571.

BIBLIOGRAPHY 77

[56] D. Guel and J. Palicot, “OFDM PAPR reduction based on nonlinear functions

without BER degradation and out-of-band emission,” in Proc. IEEE Int. Conf.

Signal Process. Syst., Singapore, 2009.

[57] S.-K. Deng and M.-C. Lin, “OFDM PAPR reduction using clipping with distortion

control,” in Proc. IEEE Int. Conf. Commun., 2005.

[58] H. Ochiai and H. Imai, “On clipping for peak power reduction of OFDM signals,” in

Proc. IEEE Global Telecommun. Conf. (GLOBECOM), San Fransisco, CA, USA,

2000.

[59] S. Haykin, Communication Systems, 4th ed. New York: John Wiley & Sons, 2001.

[60] J. Anatory and N. Theethayi, Broadband Power-Line Communication Systems:

Theory and Applications. Southampton, UK: WIT Press, 2010.

[61] K. M. Rabie and E. Alsusa, “Improving blanking/clipping based impulsive noise

mitigation over powerline channels,” in Proc. IEEE Int. Symp. Personal Indoor

Mobile Radio Commun., London, U.K., 2013, pp. 3413–3417.

[62] J. Lin, M. Nassar, and B. L. Evans, “Non-parametric impulsive noise mitigation in

OFDM systems using sparse bayesian learning,” in Proc. IEEE Global Commun.,

2011, pp. 289–301.

[63] G. H. Mohimani, M. Babaie-Zadeh, and C. Jutten, “Fast sparse representation

based on smoothed l0 norm,” in Proc. 7th Int. Conf. Independent Component Anal-

ysis Signal Separation (ICA), London, U.K., 2007, pp. 389–396.

[64] G. H. Mohimani, M. Babaie-Zadeh, and C. Jutten, “Complex-valued sparse repre-

sentation based on smoothed ℓ0 norm,” in Proc. IEEE Int. Conf. Acoustics, Speech,

Signal Process. (ICASSP), Las Vegas, NV, USA, 2008, pp. 3881–3884.

[65] H. Mohimani, M. Babaie-Zadeh, and C. Jutten, “A fast approach for overcomplete

sparse decomposition based on ℓ0 norm,” IEEE Trans. Signal Process., vol. 57,

no. 1, pp. 289–301, Jan. 2009.

78 BIBLIOGRAPHY

[66] H. Zayyani and M. Babaie-Zadeh, “Thresholded smoothed-ℓ0 (sl0) dictionary learn-

ing for sparse representation,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal

Process., Taipei, Taiwan, 2009, pp. 1825–1828.

[67] E. Candes and J. Romberg. (2005) ℓ1-magic: Recovery of

sparse signals via convex programming. online, available at

www.acm.caltech.edu/l1magic/downloads/l1magic.pdf.

[68] P. Ghosh, M. E. Sargin, and B. Manjunath, “Robust dynamical model for simul-

taneous registration and segmentation in a variational framework: A Bayesian ap-

proach,” in Proc. The 12th IEEE Int. Conf. Comput. Vision, Kyoto, Japan, 2011,

pp. 709–716.

[69] J. Lin and B. L. Evans, “Non-parametric mitigation of periodic impulsive noise in

narrowband powerline communications,” in Proc. IEEE Global Commun. (Globe-

com), Atlanta, GA, USA, 2013, pp. 2981–2986.

[70] A. H. Nuttall, “Some Integrals Involving the Q-Function,” Naval Underwater Sys-

tems Center, New London, Connecticut, Tech. Rep. TR 4297, 1972.

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