ACMAFRESHWhiteningFilterforBlindInterferenceRejection · 2 Chapter1.Introduction Thisthesisbuildsonthisbodyofworkandpresentsamethodforco-channelinterference...

A CMA FRESH Whitening Filter for Blind Interference Rejection

Ahmad S. Jauhar

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Electrical Engineering

Jeffrey H. Reed, Chair

R. Michael Buehrer

Vuk Marojevic

September 17, 2018

Blacksburg, Virginia

Keywords: Cyclostationarity, FRESH filter, Constant Modulus Algorithm, Interference

Rejection, Signal Detection, Hidden Node Problem, Spectrum Sharing

Copyright 2018, Ahmad S. Jauhar


Ahmad S. Jauhar

(ABSTRACT)

The advent of spectrum sharing has increased the need for robust interference rejection

methods. The Citizens Broadband Radio Service (CBRS) band is soon to be occupied by

LTE waveforms and License Assisted Access (LAA) will have LTE signals coexisting with

other signals in the 5 GHz band. In anticipation of this need, we present a method for

interference rejection of cyclostationary signals, which can also help avoid interference through

better detection of low power co-channel signals.

The method proposed in this thesis consists of a frequency-shift (FRESH) filter which acts

as a whitening filter, canceling the interference by exploiting its cyclostationarity. It learns

the cyclostationary characteristics of the interferer blindly, through a property restoration

algorithm which aims to drive the spectrum to white noise. The property restoration

algorithm, inspired by the constant modulus algorithm (CMA), is applied to each frequency

bin to determine the optimal coefficients for the proposed CMA FRESH whitening filter

(CFW).

The performance of the CFW in interference rejection is compared to a time-invariant version,

and proposed use cases are analyzed. The use cases consist of the rejection of a high powered,

wider bandwidth interferer which is masking the signal-of-interest (SOI). The interferer is

rejected blindly, with no knowledge of its characteristics. We analyzed signal detection

performance in the case that the SOI is another user with much lower power, for multiple

types of SOIs ranging from BPSK to OFDM. We also deal with the case that the SOI is to

be received and demodulated; we recover it and compare resulting bit error rates to state of

the art FRESH filters. The results show significantly better signal detection and recovery.


Ahmad S. Jauhar

(GENERAL AUDIENCE ABSTRACT)

Wireless communication is complicated by the fact that multiple radios may be attempting to

transmit at the same frequency, time and location concurrently. This scenario may be a due

to malicious intent by certain radios (jamming), or mere confusion due to a lack of knowledge

that another radio is transmitting in the same channel. The latter scenario is more common

due to congested wireless spectrum, as the number of devices increases exponentially. In

either case, interference results.

We present a novel interference rejection method in this work, one that is blind to the

properties of the interferer and adapts to cancel it. It follows the philosophy of property

restoration as extolled by the constant modulus algorithm (CMA) and is a frequency shift

(FRESH) filter, hence the name. The process of restoring the wireless spectrum to white

noise is what makes it a whitening filter, and is also how it adapts to cancel interference.

Such a filter has myriad possible uses, and we examine the use case of rejecting interference

to detect or recover the signal-of-interest (SOI) that we are attempting to receive. We present

performance results in both cases and compare with conventional time-invariant filters and

state of the art FRESH filters.

Dedication

To my family.

iv

Acknowledgments

I am grateful to a lot of people for helping me advance my career in wireless communications.

First of all, I would like to thank my mentors Dr. Jeffrey H. Reed and Dr. Vuk Marojevic for

their supervision and support through my Master’s degree program. The research directions,

advice, and resources provided by them were crucial to the progress of my research and the

completion of this work. I am honored to work with researchers of your caliber and am

grateful for the freedom that you gave me to pursue my research goals.

I would also like to thank Dr. R. Michael Buehrer for serving on my defense committee,

and for his helpful feedback which developed this thesis. Additionally, I would like to thank

Drs. Vuk, Reed, and Buehrer again for their leadership and guidance during the DARPA

Spectrum Collaboration Challenge, which formed the majority of my research experience at

Virginia Tech, as well as for other research projects which helped me develop my curiosity

into the design and practical implementation of ideas.

I owe a debt of gratitude to Dr. Harpreet Dhillon, for his course Stochastic Signals and

Systems. This course proved to be the solid foundation of my wireless background, and I

could not have asked for a better teacher. I would also like to thank my undergraduate

advisors, Dr. Shabbir Merchant, and Dr. Nitin Sharma, for sparking and nurturing my

interest in wireless communications.

I would like to thank Defense Advanced Research Projects Agency (DARPA), Army Research

Lab (ARL), and Wireless Systems Solutions (WSS), since their cooperation and funding

helped me advance my research.

I would like to thank all my friends at VT. To Tarun: I’m glad we got to hang out through

my time at grad school, it would have been very dull without your black humor! To Komal:

v

I’m happy I met you in grad school and got you to listen to all my rants about it (and about

everything else).

I would like to thank my colleagues Siddharth, Shem, Natalie, Nistha, Taiwo, Chris, Dan,

Hanif, Tahsin, Jet and Kaleb for their collaboration and camaraderie throughout our time

working together. Also, to Raghu, Tad, Brian, Xavier, Aaron and Aditya: you guys were

fantastic and very helpful when it comes to bouncing off ideas and providing criticism. I would

especially like to thank Matt, for helping me with the theoretical basis and jump-starting my

practical implementation of cyclostationary signal processing.

I would also like to thank my past and present roommates, Vihan, Faizan, Rounak, Oscar,

and Carlos for their friendship and companionship through our time in Blacksburg. To Vihan:

I’m glad we made it through grad school together!

To Hilda and Nancy: Thanks for making the navigation of administrative logistics easy for

the rest of us!

Last but not the least, I would like to thank my parents for their constant support through a

difficult period in my life. Finishing my education would not have been possible without their

continuous support and encouragement, and my interest in science and engineering would

have been much harder to develop if I did not live in a home of three scientists! I would like

to thank my sister Roshni for her advice and encouragement through all stages of my career.

I could not have made it this far without her pushing me to be better, throughout my life.

vi

Contents

List of Figures x

List of Tables xiv

List of Abbreviations xvii

1 Introduction 1

2 Background 4

2.1 Adaptive Filtering and the Constant Modulus Algorithm . . . . . . . . . . . 4

2.1.1 Wiener Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 RLS Adaptive filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 LMS Adaptive filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 Blind Equalization and the Constant Modulus Algorithm (CMA) . . 13

2.2 Whitening Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Cyclic Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Basic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 FRESH filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

vii

2.4.1 Energy Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Matched Filter Detection . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 Cyclostationary Feature Detection . . . . . . . . . . . . . . . . . . . 31

2.4.4 The Hidden Node Problem . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.5 Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Methodology 36

3.1 CMA FRESH Whitening Filter (CFW) . . . . . . . . . . . . . . . . . . . . . 36

3.1.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.2 Selecting Cycle Frequencies . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.3 Convergence and Initialization . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 SOI recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Results 49

4.1 Interference Rejection Performance of the CFW . . . . . . . . . . . . . . . . 50

4.2 Performance of the CFW in the detection of single-carrier signals . . . . . . 55

4.2.1 Detection of one single-carrier SOI . . . . . . . . . . . . . . . . . . . 55

4.2.2 Detection of multiple single carrier SOIs . . . . . . . . . . . . . . . . 66

4.2.3 Performance of the CFW with bandwidth limited SNOI . . . . . . . 72

viii

4.3 Performance of the CFW in detection of multicarrier signals . . . . . . . . . 76

4.4 Performance of the CFW in SOI data recovery . . . . . . . . . . . . . . . . . 83

4.5 OFDM SNOI Cancellation with Paramorphic Multicarrier Waveforms . . . . 86

4.5.1 OFDM SOI Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5.2 Narrow-Band SOI Detection . . . . . . . . . . . . . . . . . . . . . . . 91

5 Conclusion and Future Work 94

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.1 Interference Rejection Performance of CFW and TIW . . . . . . . . 95

5.1.2 CFW Detection Performance in Hidden Node Scenario . . . . . . . . 95

5.1.3 SOI data recovery performance . . . . . . . . . . . . . . . . . . . . . 97

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 101

Appendices 107

Appendix A Histograms and pdfs of deflection 108

ix

List of Figures

2.1 An adaptive filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 A FIR Wiener filter. The filter coefficients are derived from Eq. 2.12 . . . . 7

2.3 Mean Squared Error surface for a 2-tap FIR filter . . . . . . . . . . . . . . . 8

2.4 SCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 A FRESH filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 A typical ROC curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Hidden Node Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 SCF of AWGN, with a noise power of -90 dB . . . . . . . . . . . . . . . . . 37

3.2 SCF of BPSK (oversampled 4 times) . . . . . . . . . . . . . . . . . . . . . . 37

3.3 SCF of CFW output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 CFW filter structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 CFW filter coefficients after convergence. Each color denotes filter coefficients

for a different CF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Scenario for testing the CFW filter . . . . . . . . . . . . . . . . . . . . . . . 44

3.7 Deflection for different scenarios. c shows the deflection when only the SNOI

is passed into the CFW filter. d shows the deflection when both SOI and

SNOI are passed into the filter. . . . . . . . . . . . . . . . . . . . . . . . . . 46

x

3.8 Scenario for recovering the SOI . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 SCF for the output of TIW and CFW. a SCF of the SNOI. b SCF of the TIW

output. c SCF of the CFW output. d SCF of the noise. . . . . . . . . . . . . 52

4.2 SCF for the output CFW with different CFs. a shows the SCF of CFW output

without CFs of the SNOI . b shows the SCF of CFW output with CFs of the

SNOI. Note: b is the same as c, but on a different scale. . . . . . . . . . . . 52

4.3 Spectral correlation for the output CFW with different CFs. a shows the

spectral correlation of TIW output. b shows the spectral correlation of CFW

output with CFs of the SNOI. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 PSD for single carrier SOI detection, case 1 . . . . . . . . . . . . . . . . . . 57

4.5 Deflection plots for TIW and CFW. The scales are different for visibility. . . 59

4.6 ROC curve for detection of SOI in the time domain, for a 30 dB SNOI . . . 60

4.7 ROC for frequency domain localization, for a 30 dB SNOI . . . . . . . . . . 60

4.8 PD vs decreasing SIR for frequency domain localization, for a 30 dB SNOI . 61

4.9 PSD for single carrier SOI detection, case 2 . . . . . . . . . . . . . . . . . . 63

4.10 ROC curve for time-domain detection, for a 10 dB SNOI . . . . . . . . . . . 64

4.11 ROC curve for frequency domain localization, for a 10 dB SNOI . . . . . . . 65

4.12 PD vs decreasing SIR for frequency domain localization, for a 10 dB SNOI . 66

4.13 PSD for multiple single-carrier SOI detection . . . . . . . . . . . . . . . . . . 67

4.14 Deflection for two SOIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.15 ROC curve for time-domain detection . . . . . . . . . . . . . . . . . . . . . . 70

xi

4.16 ROC curve for frequency domain localization . . . . . . . . . . . . . . . . . 71

4.17 PD vs decreasing SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.18 Deflection for an SRRC SNOI . . . . . . . . . . . . . . . . . . . . . . . . . . 73




4.22 PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.23 Deflection for an OFDM SNOI . . . . . . . . . . . . . . . . . . . . . . . . . 79


4.25 ROC curve for frequency domain lozalization . . . . . . . . . . . . . . . . . . 81


4.27 PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.28 BER vs SIR for 10 dB SNOI . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.29 BER vs SIR for 30 dB SNOI . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.30 SCFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.31 CFW filter coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.32 Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90




xii


A.1 pdf and Histogram plots for CFW deflection, SNOI SNR = 30 dB . . . . . . 108

A.2 pdf and Histogram plots for TIW deflection, SNOI SNR = 30 dB . . . . . . 109

A.3 pdf and Histogram plots for CFW deflection, SNOI SNR = 10 dB . . . . . . 110

A.4 pdf and Histogram plots for TIW deflection, SNOI SNR = 10 dB . . . . . . 111

A.5 pdf and Histogram plots for CFW deflection for two SOIs . . . . . . . . . . 112

A.6 pdf and Histogram plots for TIW deflection for two SOIs . . . . . . . . . . . 113

A.7 pdf and Histogram plots CFW deflection for OFDM SOI . . . . . . . . . . . 114

A.8 pdf and Histogram plots TIW deflection for OFDM SOI . . . . . . . . . . . 115

xiii

List of Tables

4.1 TIW vs CFW simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Single carrier 30 dB SOI simulation parameters . . . . . . . . . . . . . . . . 56

4.3 Single carrier 30 dB SOI modulation types . . . . . . . . . . . . . . . . . . . 56



4.6 Multiple single carrier SNOI simulation parameters . . . . . . . . . . . . . . 68

4.7 Multiple single carrier SOI modulation types . . . . . . . . . . . . . . . . . . 68



4.10 OFDM SOI simulation parameters . . . . . . . . . . . . . . . . . . . . . . . 76

4.11 OFDM SOI parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.12 SOI recovery: SNOI and filter parameters . . . . . . . . . . . . . . . . . . . 84

4.13 SOI recovery: SOI parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.14 OFDM SOI simulation parameters . . . . . . . . . . . . . . . . . . . . . . . 87

4.15 OFDM SOI parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.16 Narrowband SOI parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

xiv

List of Abbreviations

CMA Constant Modulus Algorithm

FRESH FREquency SHift (filter)

RF Radio Frequency

CDMA Code Division Multiple Access

DSSS Direct Sequence Spread Spectrum

NBI Narrow-Band Interference

DS-CDMA Direct Sequence - Code Division Multiple Access

OFDM Orthogonal Frequency Division Multiplexing

SAIC Single Antenna Interference Cancellation

AWGN Additive White Gaussian Noise

LMS Least Mean Squares

SOI Signal Of Interest

SNOI Signal Not Of Interest

CFW CMA FRESH Whitening filter

TIW Time-Invariant Whitening filter

LTI Linear Time-Invariant (filter)

FIR Finite Impulse Response (filter)

LMSE Least Mean Square Error

MMSE Minimum Mean Square Error

RLS Recursive Least Squares

ZF Zero Forcing

MAP maximum a posteriori

ISI Inter-Symbol Interference

xv

FM Frequency Modulation

FSK Frequency Shift Keying

PSK Phase Shift Keying

BPSK Binary Phase Shift Keying

QPSK Quadrature Phase Shift Keying

QAM Quadrature Amplitude Modulation

SGD Stochastic Gradient Descent

DFE Decision Feedback Equalizer

SCF Spectral Correlation Function

CAF Cyclic Autocorrelation Function

FSM Frequency Smoothing Method

TSM Time Smoothing Method

CF Cycle Frequency

LCL Linear Conjugate Linear

SINR Signal to Interference and Noise Ratio

SIR Signal to Interference Ratio

SNR Signal to Noise Ratio

SPS Samples per Symbol

LTE Long Term Evolution

CBRS Citizens Broadband Radio Service

ROC Receiver Operating Characteristic

PSD Power Spectral Density

GSM Global system for Mobile Communication

CR Cognitive Radio

POMDP Partially Observable Markov Decision Process

xvi

NP Neyman-Pearson

FFT Fast Fourier Transform

BER Bit Error Rate

DSA Dynamic Spectrum Access

3GPP Third Generation Public Partnership

5G Fifth Generation (wireless networks)

PHY Physical (layer)

xvii

Chapter 1

Introduction

Interference mitigation has been the focus of much research since the advent of wireless

communications. Over the last century, the scientific community has developed various

methods to work around or reject interference in wireless systems. The development of spread

spectrum techniques, CDMA, and smart antennas are salient examples of this effort.

A key feature present in most man-made communication signals is their cyclostationarity;

the statistical properties of these signals being periodic rather than constant over time.

Cyclostationarity manifests itself as spectral correlation in the frequency domain. It is this

property that can be used to reject cyclostationary signals when they are present as interferers.

Filtering structures that perform this function are known as frequency-shift (FRESH) filters,

and the investigation and development of these filters is the essence of this work.

The literature already contains work on the application of cyclostationarity to interference

mitigation. For example, [1] applied FRESH filters to reject narrowband interference (NBI)

in DSSS systems, [2] exploited cyclostationarity with neural networks to reject the same

in DS-CDMA systems, [3] used FRESH filters to reject pulsed radar interferers to OFDM

systems, and [4] used a widely-linear FRESH filter to improve on previous methods of single

antenna interference cancellation (SAIC). In [5], the authors present a novel OFDM waveform

which contains induced cyclostationarity and exploits it with a FRESH filter at the receiver

to suppress interference.

1

2 Chapter 1. Introduction

This thesis builds on this body of work and presents a method for co-channel interference

cancellation, predicated on the cyclostationary nature of said interference.

The principal contribution made in this thesis is the development of a novel FRESH filter

inspired by the Constant Modulus Algorithm (CMA), which exploits the cyclostationarity

present in wireless communication signals to reject them. Said filter requires little knowledge

of the interference, only needing time-samples which include the complete bandwidth of the

signal. It adapts to this interference in time-steps on the order of 103, making it a robust and

adaptive blind interference rejection method. By canceling interference, the FRESH filter can

extract other signals present in the spectrum and can do so for signals-of-interest (SOI) that

are well below the interference power (so long as the SOI is above the AWGN noise floor).

The filter developed in this work follows the principle of a whitening filter, as defined

in [6]; ”The basic idea in employing a whitening filter is to flatten the spectrum of the

signal and interference”. Previous works on this concept utilized algorithms such as Least

Mean Squares (LMS) to adapt a whitening filter to its goal of rejecting specific signals

by performing extraction of other signals. For example, DSSS spectrum is typically flat,

so the spikes of NBI can be removed by a whitening filter while leaving the DSSS signal

intact. Such methods require that some signal property be known so that it can be used as

the target for the adaptation algorithm. The new concept presented here uses a property

restoration approach to reject interference by restoring the received signal to the spectrally

flat, temporally-uncorrelated white Gaussian noise that is present in all wireless spectra. Thus,

it requires no known or training signals to achieve its goal and works as a genuinely blind

interference rejection method for cyclostationary signals. It relies on the interference having

different cyclostationary properties from the SOI, as the filter works to cancel interference

by canceling all signals that have cycle frequencies of the interfering signal. It whitens the

spectrum by attempting to remove the interference (signal-not-of-interest, or SNOI).

3

We refer to this filter as a CMA FRESH whitening filter (CFW) in the remainder of this

work. A time-invariant version of the CFW dubbed the time-invariant whitening filter (TIW)

is also presented for comparison. The rest of this work addresses the development and

analysis of these methods. Chapter 2 presents the technical background involved, while

Chapter 3 presents the mathematical formulation and design of the CFW. Chapter 4 analyzes

performance of the CFW, based on proposed use cases. Chapter 5 comprises of conclusions

and future research directions for this work.

Chapter 2

Background

This chapter covers the background material in signal detection, whitening filters, cyclic

spectral analysis and the principle of the constant modulus algorithm, as well as its

modification for adaptive whitening filters. An overview of ‘deflection’, a metric used

for signal detection, is also presented.

2.1 Adaptive Filtering and the Constant Modulus Algorithm

Adaptive filters are a class of filters in which the tap weights are updated based on an

optimization problem, directed towards attaining the desired filter output. A schematic of

such a filter is shown in Fig. 2.1.

Figure 2.1: An adaptive filter

4

2.1. Adaptive Filtering and the Constant Modulus Algorithm 5

yn is the output signal, xn is the filter input vector, wn is the filter vector, and dn is the

desired signal. en = yn − dn is the error signal that is used to update the filter weights [7].

Hence,

yn = wnx′n (2.1)

wn+1 = wn + F(en) (2.2)

where F is a function of the error term; the exact function itself depends on the algorithm

being used to update the weights. This is known as the loss function or cost function.

Adaptive filtering is commonly applied to these fields[8]:

• Equalization: Channel distortion can be removed by adapting the equalizer output to

some known pilot or training sequence [7]. This section deals with the adaptive filtering

aspect of equalization, as well as blind equalization.

• Estimation and system identification: Adaptive estimation of the parameters of a

non-stationary unknown signal can be done, in the same vein as Wiener filtering. This

work presents an algorithm for blind estimation and rejection of interference, and this

will be covered most widely in upcoming sections.

We will only deal with FIR filters in this work, though they will not generally be Linear

Time-Invariant (LTI) structures. In this section and section 2.2, we cover the optimization

algorithms typically used in adaptive filtering. The following section provides an overview of

Wiener filtering.

6 Chapter 2. Background

2.1.1 Wiener Filtering

A Wiener filter is the optimal filter for the estimation of a desired stationary signal from a

noisy input process [8]. At the nth time step, given a desired signal dn, and a noisy input

process xn, denoting the Wiener filter coefficients as w, we may write the equations for

Wiener filtering as

dn =L−1∑k=0

w[k]xn−k (2.3)

= wTx (2.4)

n is the discrete time index, dn is the filter output, x = [x(n), x(n− 1), · · · , x(n− P )] is the

input vector, and L is the filter length. Wiener filtering utilizes a Least Mean Square Error

(LMSE) criterion E [e2(n)]. The filter error may be expressed as

en = dn − dn (2.5)

= xn − wTx (2.6)

e = d − Xw (2.7)

e(n) is the sample error, e is the error vector over n time steps, and X is the N × L input

matrix, created over n discrete time steps. A diagram of the Wiener filter is shown in Fig.

2.2.


Figure 2.2: A FIR Wiener filter. The filter coefficients are derived from Eq. 2.12

The solution of Eq. 2.7 depends on N , the number of samples of xn, and L, the filter length.

N is also the number of linear equations which make up the matrix equation 2.7, so the case

of N = L should theoretically yield a unique solution w with zero error. If N ≤ L, then the

equation is under-determined and we cannot obtain a unique solution.

Practically, we would have more samples of xn than the filter length, which means that

N ≥ L. This leads to an over-determined equation for which we can obtain a unique solution

with typically non-zero error. The filter coefficients are calculated to minimize the LMSE


metric (E [e2(n)]) w.r.t. w.

E [e2n] = E [(dn − wTx)2] (2.8)

= E [d2n]− 2wTE [xdn] + wTE [xxT ]w (2.9)

= rdd(0) + 2wT rxd + wTRddw (2.10)

∂E [e2n]∂w = −2rxd + 2wTRdd (2.11)

As displayed in Fig. 2.3, the cost function is convex and has a global minimum. The minimum

is naturally where the gradient equals zero. So, the minimum mean squared error (MMSE)

FIR Wiener filter is obtained as the linear and closed form solution:

rxd = wTRdd (2.12)

Figure 2.3: Mean Squared Error surface for a 2-tap FIR filter


The problem of Wiener filtering now becomes one of calculating the correct autocorrelation

and correlation matrices. This calculation is generally done by assuming ergodicity and

calculating these values for blocks of samples [8]. It can be shown that the MMSE criterion

makes the Wiener filter the optimal filter for generally complex stationary signals in Gaussian

noise.

Since most of the content in this work is presented in the frequency domain, we will also

mention the formulation of Wiener filtering in that domain.

Given the input signal X(f), filter output D(f), and filter frequency response W (f),

D(f) = W (f)X(f) (2.13)

E(f) = D(f)−W (f)X(f) (2.14)

E [|E(f)|2] = E [(D(f)−W (f)X(f)∗)(D(f)−W (f)X(f))] (2.15)

∂E [|E(f)|2]∂W (f)

= 2W (f)SXX(f)− 2SDX(f) = 0 (2.16)

where SXX and SDX are the power spectrum and cross power spectrum of X and D respectively.

This gives the filter coefficients W (f) as

W (f) =SDX(f)

SDD(f)(2.17)

Eq. 2.17 is the frequency domain equivalent of Eq. 2.12 [8]. We will explore the extension of

Wiener filtering to non-stationary signals in section 2.3.

Until now, we have discussed filters that use state space models or block updates based on the

calculation of signal statistics. We will now discuss methods which update the filter weights

at each sample, producing lower processing delay and faster adaptation for non-stationary

signals. Adaptive filters follow the sample update equation 2.2, and the structure is typically


that of 2.1.

For this work, the most relevant methods of filter adaptation are Recursive Least Squares

(RLS) and Least Mean Squares (LMS).

2.1.2 RLS Adaptive filters

RLS is a sample adaptive version of the Wiener filter, which updates the tap gain values by

recalculating the inverse correlation matrix Φxx(n) = R−1xx (n) at each time step n [8].

Given an input xn and a desired signal dn (as in the last section), the equations, as derived

in [8], are

Φxx(n) = R−1xx (n) (2.18)

wn = Φxx(n)rxd(n) (2.19)

(2.20)

Defining a tap update vector kn, and initializing the inverse correlation matrix as Φxx(n) = δI

and the weight vector as w0, we can write the tap update vector as

kn =λ−1Φxx(n− 1)xn

1 + λ−1xTnΦxx(n− 1)xn

(2.21)

the error as

en = dn − wTn−1xn (2.22)


the inverse correlation matrix update as

Φxx(n) = λ−1Φxx(n− 1)− λ−1knxTnΦxx(n− 1) (2.23)

and finally, the tap update equation as

wn = wn−1 + knen (2.24)

Eq. 2.21 follows the matrix inversion lemma 2.1 as proven in Section 7.6.1 of [8].

Lemma 2.1. Let A and B be two positive-definite P × P matrices related by

A = B−1 + CD−1CT (2.25)

where D is a positive-definite N ×N matrix and C is a P ×N matrix. The matrix inversion

lemma states that the inverse of matrix A can be expressed as

A−1 = B − BC(D + CTBC)−1CB (2.26)

2.1.3 LMS Adaptive filters

Let us move on to LMS adaptive filters, which utilize a somewhat simpler approach, producing

gains in memory requirement and computational complexity. The loss function of an adaptive

FIR filter is generally convex, with a global minima. Therefore, we can use steepest descent

along the gradient of the mean squared error. LMS takes this one step further by using the


gradient of the instantaneous squared error. The tap update equation simply becomes

wn+1 = wn + µ(∂e2n∂wn

) (2.27)

where µ is the step-size and

en = dn − wTnxn (2.28)

∂e2n∂wn

= −2xnen (2.29)

This gives the update equation

wn+1 = wn + µ[xnen] (2.30)

There are slight modifications which can be made to the LMS algorithm, such as

• Leaky LMS Algorithm: The update equation is modified to

wn+1 = αwn + µ[xnen] (2.31)

where α is called the leakage factor. This reduces increases stability and adaptability to

changes in the input signal characteristics, improving tracking of non-stationary signals.

• Normalized LMS Algorithm: The update equation is modified to

wn+1 = wn +µ

a+∑L−1

k=0 y2n−k

[ynen] (2.32)

This normalizes the step size with signal energy, which is useful when dealing with

multiple SNRs and typically results in a faster convergence time.


While efficacious sample adaptive filtering algorithms in their own right, LMS and RLS

have one principal disadvantage: they require knowledge of the desired signal which we are

attempting to extract out of the noisy input process. A class of algorithms to alleviate this

need, known as blind adaptation algorithms, have been a field of active research for the last

half-century. We will explore one popular blind adaptation algorithm, known as the constant

modulus algorithm (CMA), next.

2.1.4 Blind Equalization and the Constant Modulus Algorithm

(CMA)

In the latter half of the 20th century, much work was done on blind equalization through

adaptive algorithms. This effort spawned a class of property restoration algorithms, the most

prominent of which were the Sato algorithm and its generalizations, such as the Bussgang

and Godard algorithms. Of our interest in this work is the constant modulus algorithm

(CMA); a classical blind equalization algorithm that is popular for its simple and efficient

implementation.

Generally, blind equalization algorithms attempt to adjust the filter weights w such that

yn = gxn[k − ν] (2.33)

where g is a positive scalar gain and ν is a constant time delay. To this end, they define a

cost function as

D(w)∆= E[Ψ(yn)] (2.34)

where Ψ(yn) is some error function based on the filter output yn. The cost function can then


be used to update the filter weights as

wn+1 = wn − µ∂D(w)

∂wn

(2.35)

For blind equalization algorithms, while the exact inverse of a non-minimum phase channel

H(z) is unstable, we can use a truncated non-casual or anti-causal solution, delayed by ν, to

approximate a causal FIR zero-forcing (ZF) equalizer. However, the existence of zeros of the

channel H(z) on the unit circle will prevent convergence, making any FIR approximation

impossible [9].

The first blind equalizer was the Sato algorithm, developed for M-level PAM signals[10]. It

uses the cost function

Ψ(yn)∆= yn −R1sgn(yn) (2.36)

where

R1 =E{|xn|2}E{|xn|}

Bussgang algorithms, on the other hand, utilize the maximum a posteriori (MAP) estimate

of xn[k − ν] to update the filter weights [9]. Since this work focuses on CMA, an in-depth

derivation of the MAP estimate will be omitted, and merely an overview provided.

Given knowledge of the probability distribution of the inter-symbol interference (ISI) of a

channel, the MAP estimate of xn[k − ν] is

xn[k − ν]MAP = arg maxx

pyn|xn[k−ν](yn|xn) (2.37)

where pyn|xn−ν is the conditional distribution of yn given xn−ν . By assuming a proper model

for the distribution of the ISI, we can obtain this estimate, and then use it to update the


weights in two ways. The first is to recursively solve the Least Squares equation

wn+1 = (E[xnx′n])

−1E[xnxn[k − ν]MAP ] (2.38)

for every input xn. Alternatively, we can use gradient descent to update the weights as

wn+1 = wn − µ(xn[k − ν]MAP − yn)xn (2.39)

Eq. 2.38 and Eq. 2.39 are sub-optimum MAP algorithms, known as the Bussgang algorithms.

Moving closer to our algorithm of interest here, the direct precursor of CMA is the Godard

algorithm for equalization [7, 11]. The algorithm used a cost function of

D(p)(w)∆= E(|yn|p −Rp)

2 (2.40)

yn is the filter output signal at time step n, while p is a parameter. For p = 2, this is called

the constant modulus algorithm. In the case of equalization,

yn = wnx′n (2.41)

D(p)(w)∆= E(|yn|2 −Rp)

2 (2.42)

wn+1 = wn − µ∇D(p)(wn) (2.43)

where yn is the equalizer output signal, xn is the tap-gain output vector, and wn is the tap

gain vector [11]. Rp is a positive real constant

Rp =E{|xn|2p}E{|xn|p}

(2.44)


Taking the instantaneous value of the gradient, by differentiating D(p)(w) w.r.t. w and

removing the expectation from the cost function, we can obtain a weight update equation as

wn+1 = wn + µx∗n(x′

nyn|yn|

(|yn| −R1) (2.45)

CMA has also seen implementation in fields such as blind beamforming [12], and in this

work we adapt it to spectral whitening and interference cancellation. Variations of the CMA

algorithm are [13–16]

• 1-2 CMA: Setting p = 1 gives the alternative cost function and update equation

D(p) ∆= E(|yn|p −R1)

2 (2.46)

wn+1 = wn + µx∗n(yn −R1

yn|yn|

) (2.47)

The cost function used in this work is similar to the 1-2 CMA cost function.

• Normalized CMA: µ is made scaling independent, like so

wn+1 = wn +µ

||xn||2xn(yn −R1

yn|yn|

) (2.48)

• Orthogonal CMA: The data covariance matrix C is used to whiten the input.

wn+1 = wn + µCn−1xn(yn −Rp

yn|yn|

) (2.49)

• Least Squares CMA: Instead of updating the filter coefficients at every step, we

update it after a block of inputs of size N (block update). We get the best estimate s


of the complete source vector, like so

yi = wHn xi i = 1, 2, ..., N (2.50)

sn := [y1|y1|

,y2|y2|

, · · · , yN|yN |

] (2.51)

wn+1 = (snX†)H (2.52)

We may then solve

minw

||sn − wHX|| (2.53)

The defining feature of the CMA algorithm, which sets it apart from other techniques such

as the least mean squares (LMS), is the error function. The error function does not target

a specific signal of interest, but rather a constant modulus value which the algorithm tries

to reach. This approach follows the philosophy of recovery, with the algorithm penalizing

samples that deviate from the constant modulus property. It works well for tracking sources

of communication signals which have a constant modulus/envelope property (such as FM,

FSK, PSK), and can be useful in beginning the adaptation of non-constant envelope signals.

The solution is attained by minimizing the cost function with stochastic gradient descent

(SGD). Thus, the convergence of CMA is dependent on the geometry of Dp. This dependence

makes it particularly challenging to do a global analysis of the convergence of CMA or

similar blind equalization algorithms. In particular, the convergence analysis in [17] showed

that for CMA, local minima on the cost surface exist even in noiseless channels with no

inter-symbol-interference (ISI). For this reason, the literature presents simulation results and

not much rigorous mathematical analysis of the convergence of such algorithms [9]. Thus,

initialization of equalizer parameters is particularly crucial so that the algorithm can avoid


local minima.

One technique worth discussing here is that of tap-centering[18]. The tap-centering strategy

avoids having large coefficient values near the edges of the weight vector, by tap-shifting

periodically, or re-initializing [19]. A closely related initialization scheme is center-spike

initialization [20], in which all but the center tap are initialized as zero. Both these schemes

have seen widespread usage in most equalization or beamforming applications.

The advantages of the CMA algorithm lie, as mentioned above, in the simplicity of implementation

and its adaptive tracking of sources. Like any blind equalization algorithm, it can also find

non-causal solutions with a delay. The downsides are the limitation of converging to a single

source, as well as its tendency to get stuck on local minima. We will examine how these

characteristics transfer over to our implementation in Chapter 3.

2.2 Whitening Filters

A white signal is a signal whose frequency spectrum is flat, without any significant peaks or

humps. Such signal are known as white because they have the same property as white light;

they contain equal amounts of power from all frequencies.

Whitening filters are a class of filters which can be used to cancel interference by predicting

and then subtracting it out, thus leaving only white noise. The primary contribution of

this work is, in fact, a sample adaptive filter designed to reject interference by whitening.

Though a version of the whitening filter can be derived from Kalman filters using a state

space equation, we will focus on sample adaptive whitening filters in this work.

Whitening filters have been used in the literature to suppress noise in mobile communication

system [21] and as feed-forward filters in a Decision Feedback Equalizer (DFE) in [22]. The

2.2. Whitening Filters 19

authors in [6] present a structure for using whitening filters driven by a Least Mean Square

(LMS) estimation algorithm to reject interference. Since this method is a sample adaptive

whitening filter, it is close to the interference suppression method presented in this work, and

it is instructive to lay it out in detail.

yk = wHk xk (2.54)

εk = dk − yk (2.55)

wk+1 = wk + µxkε∗k (2.56)

where k is the time-step, xk is the input vector, wk is the filter tap vector, yk is the filter

output, H and ∗ denote conjugate transposition and conjugation operators respectively, and

µ is the step-size parameter.

The authors of [6] lay this out as a method of canceling narrow-band interference (NBI) from

a Direct Sequence Spread Spectrum (DSSS) signal, which allowed the use of a delayed version

of the input as dk. Since DSSS signals are uncorrelated in time, this allowed the adaptive

filter to estimate and subtract out the NBI, leaving the DSSS signal as the estimation error.

If dk was set to a known signal-not-of-interest (SNOI), the LMS estimation technique could

theoretically reject this signal by using the estimation error as the signal-of-interest (SOI).

Furthermore, in the case where the SNOI is the only signal present, the estimation error

would be white noise. Such a filter, when applied to a scenario with SOIs present, would work

as the optimal interference rejection method. However, since we generally do not possess

perfect knowledge of SNOIs, this is an unrealistic scenario. This work, therefore, attempts to

design an interference rejection filter which is agnostic 1 of the characteristics of the SNOI,1Agnostic here means that the system does not know the modulation, coding or bandwidth of the SNOI.


but can be trained to reject it through captured samples. We will explore this in Chapter 3.

2.3 Cyclic Spectral Analysis

When attempting to do analysis or separation of signals in the Fourier domain, we often

use distinguishing features. For the larger part, these are statistical features such as

mean, auto-correlation, or higher-order moments. For stationary signals, these features are

time-invariant. Unfortunately, most signals that are of interest are non-stationary. Man-made

modulated signals, however, often possess the characteristic of being cyclostationary, which

means that their statistical features themselves are periodic. This cyclostationarity is

introduced through the periodicity of components such as codes, chip rates, modulation and

such.

Cyclostationary signals have been well studied in the last half-century [23, 24]. They have

found applications in interference rejection [5], signal detection [25], and beamforming [26],

among others. In this work, the cyclostationary properties of signals are applied to interference

rejection through linear-conjugate-linear (LCL) filtering, which is explained in this section.

2.3.1 Basic Functions

Let us lay down an initial mathematical framework for cyclic spectral analysis in this section.

Given a generally complex signal x(t), we may say x(t) is Nth-order cyclostationary in the

strict sense [27] if it’s Nth-order moment is periodic with some period T0. Of more relevance

The only knowledge required is that the SNOI lies entirely within the frequency range the filter is beingapplied to, as well as knowledge of the thermal noise floor.

2.3. Cyclic Spectral Analysis 21

is wide-sense cyclostationarity, in which the mean and autocorrelation

R§∆= E{x(t)x(t+ τ)} (2.57)

are periodic with a period T0.

We may write the Fourier series expansion of R§(t, τ) as

Rx =∞∑

n=−∞

Rn/T0x (τ)e

−j2π nT0

t (2.58)

where

Rn/T0x

∆=

1

T0

∫ T0)2

−T02

Rx(t, τ)e−i2π n

T0tdt (2.59)

Rn/T0x is known as a cyclic autocorrelation function at a cycle frequency n

T0 n∈Z.

A more general definition is that of almost cyclostationary signals. x(t) is Nth-order almost

cyclostationary [27] if

Rx =∑α

Rαx(τ)e

−j2παt (2.60)

where α is in the set of cycle frequencies. Rαx(τ) is defined as

Rαx(τ) = lim

T→∞

1

T

∫ T2

−T2

Rx(t, τ)e−i2παtdt (2.61)

For a cyclostationary signal, a set of given αs will have Rαx(τ) 6= 0. α = 0 will yield the

standard autocorrelation function of x(t). Simply put, if a function has non-zero cyclic


autocorrelation at any non-zero cycle frequencies, it is cyclostationary.

Cyclostationarity manifests itself as spectral correlation in the frequency domain. A simple

proof is provided in [28], and outlined here.

Rαx(τ) can also be written as the cross-correlation of two frequency shifted versions of x(t),

u(t) = x(t)eiπαt and u(t) = x(t)e−iπαt. Therefore, it is also the inverse Fourier transform of

the spectral correlation function (SCF), Sαx , defined as the cross spectral density of u(t) and

v(t).

Sαx (f) =

∫ ∞

−∞Rα

x(τ)e−j2πfτdτ (2.62)

Sαx (f)

∆= Suv∗(f) (2.63)

Since this is valid for all signals which exhibit second-order periodicity, we can say that a

signal is wide sense cyclostationary if and only if it exhibits spectral correlation with frequency

shifts of its cycle frequencies.

Most modulated signals can thus be shown to be cyclostationary. This is because most

modulation types can be written as a product modulation:

x(t) = m(t)w(t) (2.64)

If w(t) is a linear periodically time varying function (such as cos(2πf0t+φ0)), y(t) will exhibit

cyclostationarity. This is proven in [28].


For the case of phase shift keying, it can be written as [29]

x(t) = sin(2 pif0t+n=∞∑n=−∞

q(t− nTs)M∑

m=1

δm(n)θm) (2.65)

where Ts is the symbol rate, and q(t) is the rectangular pulse shaping function. δ(n) is the

data, and M is the modulation order. This can also be written as the digitally modulated

pulse train

x(t) =n=∞∑n=−∞

M∑m=1

δm(n)qm(t− nTs)

qm(t) = sin(2πf0t+ θm)

Given that the data is stationary, [29] derives the cyclic spectrum to be

Qm(f) =sin[π(f − f0)Ts]e

iθm

2πi(f − f0)+

sin[π(f + f0)Ts]e−iθm

2πi(f + f0)(2.66)

It can be seen in this equation that the cyclic spectrum is essentially the sum of two phase

shifted sinc functions. This results in spectral correlation, which can intuitively be seen as

a manifestation of the cyclostationarity of phase-shift keyed signals. If the pulse shaping

function q(t) were changed, these cyclic spectra would change along with it.

As an intuitive explanation of why communication signals exhibit cyclostationarity, the

regularity of the the modulated signal structure causes periodicity in the statistics of the

stochastic signal itself. This is also why oversampling allows us to see cyclostationarity. If

we took one sample per signal, we give the resulting digital signal no chance to exhibit the

regularity introduces by the pulses themselves, since it is sampling the same point in the

symbol pulse at periodic intervals. If we take multiple samples-per-symbol, we are allowing


the regularity present in the pulse in the analog domain to manifest itself in the digital signal.

The peaks of the SCF for a cyclostationary signal are nonzero for the same cycle frequencies

as the CAF. The SCF can be computed for a single known α from the cyclic periodogram IαT

at α, by the frequency smoothing method(FSM) [30]. FSM uses a pulselike window function

to smooth the periodogram, through convolution in the frequency domain.

IαT (t, f) =1

TXT (t, f +

α

2)XT

∗(t, f − α

2) (2.67)

Sαx (f) = h(f)⊗ IαT (t, f) (2.68)

An alternative method is the time smoothing method (TSM). This is similar to the Bartlett

spectrum estimation method, but with non-zero and conjugate CFs. It simply takes the time

average of multiple cyclic periodograms to generate the cyclic power spectrum [31, 32]. An

example SCF for an oversampled BPSK signal, generated with TSM, is shown in Fig. 2.4.

Figure 2.4: SCF


2.3.2 FRESH filters

A Wiener filter is the optimum LTI filter that can produce an estimate of a desired stationary

signal by applying it to a noisy input process. Cyclic Wiener filtering, also known as Frequency

Shift (FRESH) filtering, is the optimal linear filter for cyclostationary signals [33]. It exploits

the spectral redundancy in the input signal due to cyclostationarity to estimate the desired

signal. The FRESH filter itself is periodically time variant, with (ideally) as many periods as

the cyclostationary signal has cycle frequencies.

The input signal and its conjugate are passed through parallel frequency shifts, and each

branch is passed through an LTI FIR filter. This is known as linear-conjugate-linear FRESH

filtering. The output of this is 2.69 [33].

dn =M∑

m=1

am(t)⊗ xαm(t) +N∑

n=1

bn(t)⊗ x∗−βn

(t) (2.69)

where M and N are the number of cycle frequencies and conjugate cycle frequencies respectively.

A representation of the FRESH filter is shown in Fig. 2.5.

The problem of finding the optimal cyclic Wiener filter becomes equivalent to the multivariate

(dimension = M+N) Wiener filtering problem [33] and can be solved for a minimum mean

square error (MMSE) metric. The author formulates the problem as

h(t) = [a1(t), ..., am(t), b1(t), ..., bm(t)] (2.70)

x(t) = [xα1 , ..., xαm , xβ1 , ..., xβn ] (2.71)

d(t) = h′(t)x(t) (2.72)

S ′xx(f)H(f) = Sdx(f) (2.73)


Figure 2.5: A FRESH filter

where Sxx(f) and Sdx are auto-correlation and cross-correlation of x(t) and d(t), respectively.

Defining the spectrum of error e(t)∆= d(t)− d(t) as Se,

Se = Sd(f)− SHdx(f)H(f)

For an infinite set of cycle frequencies, this is called the cyclic Wiener filter. For a constrained

set, it is called a constrained optimum FRESH filter.

This work utilizes a similar LCL-FRESH structure for the whitening filter, though it is

dissimilar to a cyclic Wiener filter in other respects. The solution is not attained by solving

the system of linear equations. Instead, we converge to a solution through gradient descent

2.4. Signal Detection 27

over a cost function. We will explore the methodology in Chapter 3, and an understanding of

the operation of an LCL-FRESH filter will be useful then.

2.4 Signal Detection

Signal detection has been a field of active interest for many decades, and the advent of

cognitive radio [34] has only helped intensify inquiry into the field. Spectrum sensing, as a

method of identifying available spectrum in an inefficiently used frequency domain [35], is an

essential part of cognitive radio systems.

Generally, a cognitive radio is a user of secondary priority (unlicensed/secondary user), using

the spectrum only when it is unused by a licensed ’primary user’. So, a cognitive radio must

periodically sense the radio frequency spectrum, to identify ’spectrum holes’ which it can use

[36]. In the majority of cases, spectrum holes are ’white space’; spectrum that is completely

unused by a primary user for a period in a given location. ’Gray space’ has been used to

refer to spectrum that is used by primary users but can still be used by a secondary user [37].

Forms of this include cooperative spectrum sharing between cellular carriers and secondary

users [38], or the primary user tolerating a degradation in SINR that does not affect its

quality of service [39]. Coexistence of LTE with RADAR waveforms in the 3.5 GHz CBRS

band is also a relevant example of gray space spectrum sharing [40].

A mathematical framework to model the signal should be formed to explain better and

compare methods of detection.

The observed signal can be modeled as [36]

y(t) = x(t) + z(t) (2.74)


x(t) is the signal that is of interest, while n(t) is additive noise and/or interference. The

detection problem can be modeled as a hypothesis test

H0 : y(t) = z(t) (2.75)

H1 : y(t) = x(t) + z(t) (2.76)

In practice, y(t) is sampled, say into N samples, giving us

yn = xn + zn (2.77)

The detection problem then becomes a discrete time binary hypothesis testing problem,

DN : CN → {H0, H1} (2.78)

Performance of a detector can be measured by comparing Hi with Hi, allowing us to define

probability of false alarm (PFA) and probability of detection (PD) as

PFA ≡ P[Dn = H1|H0] (2.79)

PD ≡ P[Dn = H1|H1] (2.80)

The graph comparing PD and PFA is known as the receiver operating characteristic (ROC)

curve, and is commonly used to analyze the performance of detectors. A typical ROC curve

is shown in Fig. 2.6.

Now, we will move on to the various methods of detection. Initial forays into spectrum

sensing involved the use of techniques such as energy detection spectrum sensing, matched


Figure 2.6: A typical ROC curve

filter detection, and cyclostationary feature detection. Much work has been done on the

enhancement of these basic techniques, and we will briefly describe each one in this section.

2.4.1 Energy Detection

Energy Detection Spectrum Sensing is a method that decides whether a signal is present in

an AWGN channel using a simple threshold on the energy of the signal. This is done by

using the power spectral density (PSD) of the signal.

If the PSD of the signal being searched for is known, a simple correlation of the measured

periodogram with the known PSD is the ideal detection statistic, forming the optimal energy

detector [30]

Yoed =

∫S0x(f)I

0xT (t, f)df (2.81)


where the periodogram I0xT is

I0xT (f) =1

T|XT (t, f)|2 (2.82)

XT (t, f) =

∫ t+T2

t−T2

x(u)e−i2πfudu (2.83)

XT (t, f) is the Fourier transform of data x(t) over T time.

If the PSD of the signal is not known, the sub-optimal energy detector [30] is

Ysed =

∫|I0xT (t, f)|2df (2.84)

which simply integrates the squared periodogram over a given bandwidth.

Since we usually work with discrete frequency samples, the practical decision metric is Y

constructed from N signal samples [36],

Y =

∑Nn=1 |yn|2

N(2.85)

A simple threshold η is applied to Y to determine the presence of the signal of interest. While

computationally efficient, this method suffers from degradation in performance from noise

uncertainty. It can also not distinguish between multiple primary/secondary users. It can be

augmented with feature detection to improve its performance, which can identify features

such as autocorrelation or cyclostationarity.

2.4.2 Matched Filter Detection

In some cases, we possess knowledge about pilot or training samples transmitted in the signal.

We can test for the presence of these signals, using the technique known as matched filter


detection or coherent detection.

A matched filter is a filter matched to the time-reversed and conjugated impulse response

of a signal x(t). It provides the maximum SNR when the signal is present in the spectrum,

making it the optimal detector for known signals in AWGN channels.

Examples of cases where matched filter detection may be used are the 127-bit pilot synchronization

sequence of 802.11a/g or the 26-bit midamble in the 156-bit traffic time slots in GSM [30].

Matched filtering does not work well in non-AWGN channels, and the performance of the

filter degrades severely with lack of frequency synchronization. It also requires knowledge of

the signal being searched for, making it unsuitable for blind signal detection. Its upsides are

computational simplicity and the fact that it is the optimal detector for AWGN channels.

2.4.3 Cyclostationary Feature Detection

Communication signals of most modulation types can be distinguished from noise by their

lack of stationarity. While some signals are wide-sense stationary, many lack even that

characteristic. Fortunately, the majority of these signals are cyclostationary, a property

discussed in section 2.3. Gaussian noise is wide-sense stationary and has no cyclostationary

properties.

A signal detector based on cyclostationarity will attempt to distinguish signals based on their

cycle frequencies. The simplest implementation of such a detector is the optimal single cycle

detector [30]

Yosd =

∫Sαx (f)I

αT (t, f)df (2.86)

which is the correlation of the cyclic periodogram with the known SCF a CF α of the SOI. A


simplex extension is the optimal multicycle detector, which takes a summation of the previous

statistic over L known CFs[30].

Yomd =L∑i

∫Sαix (f)Iαi

T (t, f)df (2.87)

Similar to the energy detection case, if we do not exactly known the SCF of the received SOI,

we may use the magnitude of the cyclic periodogram at known CFs. So, the sub-optimal

single cycle detector becomes [30, 36]

Yosd =

∫|IαT (t, f)|2df (2.88)

Cyclostationary detectors show good performance in low-SNR regions or with high noise

uncertainty and do not require comprehensive knowledge of the signal of interest. Their

downside is the relative complexity of calculating the test statistics, and a priori knowledge

of the CFs.

If the CFs are not known, a search or extraction must be conducted to gain this knowledge.

The most straightforward approach is an exhaustive search of candidate α to procure the

ones with nonzero Rαx(τ). The authors in [41] presented an estimator to this end.

2.4.4 The Hidden Node Problem

In most communication systems which employ spectrum sensing, the aim is to avoid

transmitting when a licensed user is using the concerned RF channel. Generally, the sensing

entity is a cognitive radio (CR), which is looking for spectrum holes to transmit in. However,

conflicts may occur when the CR incorrectly infers that a channel is unoccupied. The hidden


node problem refers to the case when this happens due to the propagation environment

reducing the primary user’s signal power to the extent that the CR does not detect it. This

scenario is depicted in Fig. 2.7, where an obstacle causes attenuation.

Figure 2.7: Hidden Node Problem

The literature proposes cooperative sensing as the most common solution to the hidden

node problem. In cooperative sensing, multiple RF spectrum sensors are deployed over a

geographical area, with the CR(s) making the decision based on information from all of

them. Improvements on this simple technique are proposed in the literature, such as using a

Partially Observable Markov Decision Process (POMDP) to model the distributed sensing

[42], mining the historical sensing data of multiple user [43], and using compressed sensing to

reduce the data transfer between multiple users [44].

2.4.5 Deflection

In this work, we have used a metric for signal detection known as deflection, or generalized

SNR [45, 46]. Deflection is the metric used to achieve optimal detection performance when

the distribution of the noise is unknown. This proves useful; a detailed analysis of the effects

of the several transforms the data in this work goes through would be tedious.


To put down a brief mathematical framework, we may define our hypotheses as

H1 : x = s + e1 (2.89)

H0 : x = e0 (2.90)

(2.91)

where s is the transmitted signal and ei is the noise under each hypothesis i.

If the pdf under both hypotheses is known, the the optimal detector is the Neyman-Pearson

(NP) detector [47]. It uses the following likelihood ratio test as the optimal test statistic:

L(x) = p(x;H1)

p(x;H0)≶ ν (2.92)

where p(x;H1) and p(x;H0) are the pdfs under their respective hypotheses.

For defining deflection, the received length-N vector x can be said to have the following

statistics under the two hypotheses H0 (noise) and H1 (signal and noise):

E0(x) = 0 E0(xxT ) = C0 (2.93)

E1(x) = s E0(xxT ) = C1 + ssT (2.94)

(2.95)

where C0 and C1 are covariance matrices of x under H0 and H1 and s is the transmitted

signal.


The linear quadratic detector for this system is then derived in [45] to be

S(x) ∆= hTx + xTMx − Tr(CM) (2.96)

where M is an N ×N matrix and h is a real vector. The deflection is then defined by

D(S) =[E0(S)

E1(S2)(2.97)

The optimal detector can then be obtained to maximize deflection. While we will not solve

linear-quadratic equations to determine the optimal detector in this work, we will use a

similar metric of deflection. We will see that we do not know what the pdfs of the received

vector are, which makes the Neyman-Pearson (NP) detector non-optimal [45, 46].

Chapter 3

Methodology

This chapter describes the methodology of the adaptive filter. Conceptually, it is based on

CMA, with a similar loss function and optimization technique. There are also sections on the

use cases tested in this work, describing the signal detection and recovery methodology.

3.1 CMA FRESH Whitening Filter (CFW)

The method proposed in this work uses an estimation algorithm inspired by CMA, applied in

the frequency domain to discrete frequency bins. The idea is to drive the power in each bin

to a constant value across the entire band of interest, thus giving it the statistical properties

of white noise. If applied merely to the zeroth CF spectrum of a signal, this method would

flatten the spectrum but not remove the spectral correlation that makes it distinct from white

noise. However, when applied in the form of an LCL-FRESH filter, it cancels out the signal

by exploiting its second-order cyclostationarity to create a white signal. Fig. 3.1 shows an

example of the SCF of AWGN (which we are trying to achieve through the whitening filter),

while Fig. 3.2 shows the SCF of a BPSK signal, which may be input to the CMA-FRESH

whitening filter (CFW) filter.

Whitening the SCF at the CFs of the signal removes the spectral correlation reduces it to

white noise, effectively subtracting out the SNOI. An example output of the CFW filter

applied to the above BPSK signal is shown in Fig. 3.3. The cyclostationary features of the

36

3.1. CMA FRESH Whitening Filter (CFW) 37

Figure 3.1: SCF of AWGN, with a noise power of -90 dB

Figure 3.2: SCF of BPSK (oversampled 4 times)

input BPSK signal are removed and the spectrum is flattened.

The development of this filtering structure is thus the primary contribution of this work. The

38 Chapter 3. Methodology

Figure 3.3: SCF of CFW output

remainder of this section will lay out the mathematical framework of the CFW filter, as well

as discuss initialization strategies and convergence.

3.1.1 Mathematical Framework

Given a signal s(t), we may combine it with additive white Gaussian noise (AWGN) n(t),

with noise power being Pn dB, to obtain x(t).

Taking the NFFT -point Fast Fourier Transform (FFT) of the given signal to move into the

Fourier domain, we obtain x[k] as the frequency bins, k ∈ [1, NFFT ].

We use a FRESH filter with cycle frequencies αk, k ∈ [1, N ], and conjugate cycle frequencies

βk, k ∈ [1,M ]. In all parts of this work, N = M and we will use only N for both

sets of CFs. The set of cycle frequencies has α1 = 0 and β1 = 0. In practice, the cycle

frequencies are used as bin shifts in the discrete frequency domain. Since we are operating on


the data in the frequency domain following a DFT, this will be a circular convolution filter

in the time domain.

Consequently, for each frequency bin x[k], we may write the output of the FRESH filter as

y[k] = x[k] ∗ wα1 [k] + x[k − α2] ∗ wα2 [k] + · · ·+ x[k − αN ] ∗ wαN[k]

+ x∗[k] ∗ wβ1 [k] + x∗[k − β2] ∗ wβ2 [k] + · · ·+ x∗[k − βN ] ∗ wβN[k]

(3.1)

The cost function for the algorithm may now be defined as

D(p)k

∆= E{(|y[k]|p −Rp)

2} (3.2)

Fig. 3.4 is a diagram of the CFW filter.

Figure 3.4: CFW filter structure


Looking at Fig. 3.4, it is evident that each bin has its own set of relevant FRESH filter

coefficients that will need to be adapted. We will denote this wk, and the set of input bins

corresponding to these coefficients as xk. The tap update equation for bin k at time step n

may be written as

wk(n+ 1) = wk(n) + µp[∂D

(p)k

∂wk

] (3.3)

y[k] = x′kwk (3.4)

Take the derivative of D(p)k w.r.t. wk,

∂D(p)k

∂wk

= E{2p∂|x′kwk|

∂wk

|x′kwk|p−1

(|x′kwk|p −Rp)} (3.5)

Using

∂|x′kwk|

∂wk

= x∗k(x′

kwk)|x′kwk|−1 (3.6)

This gives us the equation

∂D(p)k

∂wk

= E{2px∗k(x′

kwk)|x′kwk|p−2

(|x′kwk|p −Rp)} (3.7)

If we set p = 1, and take the instantaneous value of the gradient estimate by removing the


expectation, we get the tap update equation

wk(n+ 1) = wk(n) + λ1x∗k(|y[k]| −R1

y[k]

|y[k]|) (3.8)

where λ1 is a constant multiple of µ. The constant R1 is selected to drive |y[k]|2 to the noise

power. In this work, we used√Pn as R1. Therefore, the implicit assumption in all parts of

this work is that we known the thermal noise floor, and are trying to achieve this at the filter

output by canceling out interference. Since we will generally know the thermal noise floor

of the device and location where we set up a wireless system, this is not an unreasonable

assumption.

3.1.2 Selecting Cycle Frequencies

Given knowledge of cycle frequencies, an LCL-FRESH filter is an optimal filter for estimating

a cyclostationary signal. The method proposed in this work designs an LCL-FRESH filter that

minimizes an error function to estimate the signal-not-of-interest(SNOI), without actually

knowing the SNOI.

Given the cycle frequencies of the SNOI, we could design a CFW filter with just those

frequency shifts. This design would be the ideal case since training time would be reduced to

only that which is essential. Chapter 4 compares the performance with and without CFs of

the SNOI included.

Generally speaking, a CFW filter may be designed with an arbitrary set of cycle frequencies

for blind interference cancellation. The basic idea is that with enough cycle frequencies, the

filter will include some if not all the cycle frequencies of the SNOI. Even without the inclusion

of all the cycle frequencies, interference cancellation can be remarkable, as we will show later.


Knowledge about the signal type or its cyclic properties can optimize the filtering process in

terms of performance and complexity. However, the salient feature of this approach is that it

is a blind interference rejection algorithm, merely needing enough training samples of the

SNOI, while being agnostic to its characteristics, to learn how to reject it. Once the algorithm

converges to an optimal solution, the LCL-FRESH filter weights can be ”frozen”, and the

resulting filter directly applied to an input signal to cancel out the SNOI. Any other signals

in the spectrum will also be frequency shifted, which makes this method suitable for signal

detection, but makes signal recovery more involved. That is to say, since the signal-of-interest

(SOI) has been frequency shifted and summed several times, we must reverse these frequency

shifts to recover the data within the SOI, if that is the intention.

3.1.3 Convergence and Initialization

We found that the CFW filter did not converge to a global minimum without proper

initialization. The cost function for each tap is multimodal, and a mathematical analysis

of the surface is made difficult by the large number of cycle frequencies which are typically

needed to yield good results. An initialization strategy which produced excellent results in a

reasonable time was selected arbitrarily through experimentation.

It was observed that large values of filter coefficients for cycle frequencies other than the

zeroth CF (including conjugate CFs) resulted in solutions with high spectral correlation. A

good solution has the FIR filter coefficient values for non-zero cycle frequencies approaching

zero, except the cycle frequencies characteristic to the SNOI. As shown in Fig. 3.1, a white

Gaussian noise SCF has no energy at non-zero CF and a white spectrum at the zeroth CF.

Fig. 3.5 shows an example diagram of the CFW filter coefficients after convergence. The

x-axis is frequency bins, and the y-axis is coefficient magnitude on a logarithmic scale. Each


differently colored curve is the filter coefficients for a different CF/bin-shift. As can be seen,

a few CFs corresponding to the CFs of the SNOI have coefficients converged to a significant

magnitude, while the rest die off towards zero. These are the 6 CFs ([0, 0.25,−0.25]) of BPSK

oversampled 4 times (including conjugate CFs). A total of 34 CFs was used to generate this

diagram.

Figure 3.5: CFW filter coefficients after convergence. Each color denotes filter coefficientsfor a different CF.

The error function typically has initial values in the order of signal magnitude. Since the

signal magnitude is generally far less than unity on a linear scale, the loss function may be

unable to reduce the coefficients to the global minima if they start at unity. It converges

to a solution that has a ’white’, or rather ’flat’, spectrum, but has high non-zero CF peaks.

These peaks are due to spectral correlation in the filter output, present due to the incomplete

removal of the SNOI. A desirable solution would be one in which all spectral correlation

stemming from the SNOI is removed, and the output spectrum is just white noise.

Since this solution is undesirable, an initialization strategy was arbitrarily selected which


makes it easier for the algorithm to converge to a desirable solution but is still general enough

not to have to be modified at all for different SNOIs.

The initial filter coefficients for the zeroth CF were set to unity, and to zero for all other CFs

(including conjugate CFs). This initialization strategy resulted in convergence to interference

cancellation in a reasonable number of steps on the order of 103, depending on the SNR of

the SNOI. Fig. 3.5 is an example of such a solution.

3.2 Scenario

Figure 3.6: Scenario for testing the CFW filter

To demonstrate its efficacy, the algorithm developed in this work is tested on a scenario

where the presence of the SNOI makes it difficult to detect the SOI by conventional methods

such as energy detection or cyclostationary sensing. This is especially the case if the SNOI

has cyclostationary characteristics that are similar to the SOI.


This scenario is where the CFW proves useful. This filter can learn to cancel out the SNOI,

given enough samples. Then, when applied to a spectrum in which both SOI and SNOI are

present, it will extract the SOI, allowing us to detect its presence reliably. As shown in Fig.

3.6, the output spectrum contains the SOI and several artifacts. These artifacts are images

of the SOI, generated due to the frequency shifts administered by the CFW filter. Since the

filter structure is wholly known, these shifts are trivial to determine, and recovering the SOI

is addressed in a later section.

3.3 Signal Detection

The goal of signal detection in this scenario is not to merely detect the presence of an SOI

in the spectrum, but also to localize it in frequency. We attempt to do so by localizing the

frequency bins where the SOI is present.

To this end, we calculate the deflection in the discrete frequency domain for the CFW output.

The deflection is calculated by using Eq. 2.97. The expected values of the data in the

frequency domain are calculated, over time. That is to say, frequency domain data vectors

are taken, and the deflection is calculated for each frequency bin across multiple frequency

domain data vectors.

This output spectrum contains the SOI, and several of its frequency shifted versions. The

deflection is significantly different based on the presence of the SOI. Fig. 3.7 is an example

image of what deflection may look like for different inputs to the CFW filter.

To detect whether an SOI is present at all in the spectrum, we may use a simple threshold on

the deflection. The threshold can be applied independently to each frequency bin, and if any

bin is above the threshold, we may declare an SOI present and follow up with the methods


(a) SNOI (b) SOI

(c) Deflection for only SNOI (d) Deflection for SOI+SNOI

Figure 3.7: Deflection for different scenarios. c shows the deflection when only the SNOI ispassed into the CFW filter. d shows the deflection when both SOI and SNOI are passed intothe filter.

described above to localize the SOI in frequency. In summary, we find the detection decision

in all the bins separately and then use an AND function on all the decisions.

The thresholding problem for each bin can be modeled as a hypothesis test, such that H0

denotes the hypothesis of the SOI being absent, and H1 is the hypothesis for the SOI being

present. If the pdfs of the deflection in each bin are found to be separable, then a threshold

on the deflection can be used to detect the presence of the SOI in the spectrum. In chapter

4, Sections 4.2 and 4.3 use a frequentist approach to find the pdf of a given frequency bin

under different hypotheses and show that the pdfs are separable by a threshold. A set of

thresholds ranging from the minimum to the maximum deflection values is used to generate

3.4. SOI recovery 47

ROC curves.

The multiple humps in the deflection plots for the SOI input are due to the artifacts created

by the CFW filter. These artifacts occur because the frequency shifting operations of the

FRESH filter introduce cyclostationarity in the SOI. The location of these artifacts is thus

dependent on the CFs used by the filter. However, for a well-designed filter, these artifacts

will always have a far lower magnitude (at least an order of magnitude lower) than the SOI.

This is so that the SOI may be reliably localized in frequency using detection.

Given this metric, we may use a predetermined threshold to detect the location in frequency

of an SOI. For this method, we have presented ROC curves in the next chapter. An

alternative detection method used in this work was the MATLAB findpeaks1 function, which

can conveniently localize the peaks of the deflection. The largest peak is the SOI, with

other peaks being its frequency shifted images. For this method, graphs of the probability of

detection (PD) against signal to interference ratio 2(SIR) are presented.

3.4 SOI recovery

As shown in Section 3.3, the SOI is distorted coming out of the CFW. The multiple frequency

shifts and phase distortion due to the CFW must be reversed before the SOI can be

demodulated. Ideally, this would be done by knowledge of the CFW, in a manner that is

blind to the SOI itself. However, designing such a blind filter is beyond the scope of this

work, and is to be addressed as a future research direction. In lieu of a blind FRESH filter to

correct the distortion in the SOI, we have utilized the FRESH filter presented in [5], which

exploits spectral redundancy to improve the SNR of a known signal. Since the CFW, with its1The findpeaks function returns a vector of the local maxima (peaks) of the input data. It returns the

prominences, locations and widths of each maximum.2The SOI is the signal, and the SNOI is the interference.


multiple frequency shifting operations, introduced cyclostationarity and spectral redundancy

into the SOI (in addition to any cyclostationarity the SOI may have possessed already), we

can exploit this to reconstruct the SOI.

The process of reconstruction merely requires knowledge of the CFs of the SNOI, since these

are the frequency shifts distorting the SOI. Since we know which CFs acquired non-trivial

coefficient values in the adaptation process, we essentially know the CFs of the SNOI. We

may use this knowledge to design a linear minimum mean squared error (MMSE) FRESH

filter or cyclic Wiener filter as described in [5, 33]. The reconstruction process is depicted in

Fig. 3.8.

Figure 3.8: Scenario for recovering the SOI

This method forms a powerful interference rejection process, one which can learn to reject the

SNOI in a blind manner, then use the knowledge gained through adaptation to reconstruct

the SOI and recover it. In section 4.4, it is shown to produce excellent bit error rates (BER)

at SIRs below zero, achieving the performance of the optimal FRESH filter developed in

[33].

Chapter 4

Results

This chapter presents simulation results and analysis for the CFW. We used the following

computer system for running simulations:

CPU Intel® Core™ i7-6920HQ CPU @ 2.90GHz × 8

RAM 32 GB

OS Linux, Ubuntu 16.04 LTS

Simulation Software MATLAB r2015b

We use two types of filters in this chapter: the CMA-FRESH whitening filter (CFW) and

the time-invariant whitening filter (TIW). Both types of filters are FIR and are as defined in

Chapter 3. The TIW is the variant of the CFW in which only a cycle frequency of zero is

used 1.

Section 4.1 presents a comparison of the CFW and TIW, as well as an analysis of the effect

of CFs on performance. Sections 4.2 and 4.3 present the performance of the CFW and TIW

in the scenario from Section 3.2. Section 4.4 presents performance results for the CFW in the

scenario defined in Section 3.4. Section 4.5 presents detection results for an OFDM SNOI,

using the paramorphic multicarrier waveforms of [5]. Overall, the purpose of this chapter is

to highlight a possible use case and discuss the CFW’s performance characteristics.1While an adaptive filter is not time-invariant, the use of this term is justified since the filter weights are

frozen after adaptation has converged. Once this has happened, the resulting TIW is an LTI filter, while theCFW is a periodically time-varying filter.

49

50 Chapter 4. Results

A list of the assumptions made in these simulations:

•AWGN was added to signals after oversampling in MATLAB. So, the noise is not oversampled.

•In cases where the signals have different oversampling rates, with or without pulse shaping,

samples per symbol (SPS) were different. Therefore, different numbers of symbols were

generated to ensure different symbol rates/bandwidths, and then oversampled to reach the

same resulting ’sample rate’ at the ’receiver’ input. That is, if one signal has an SPS of 4, and

another has an SPS of 8, the first one would have 1M symbols generated and oversampled,

while the second one had 500K symbols generated and oversampled, to a total of 4M samples

for both signals.

•Bandwidth is defined as frequency spectrum containing ≥ 90% of the signal power. So, upon

applying a pulse shaping filter, it reduces significantly. Bandwidth as defined for OFDM does

not include its sidelobes, but bandwidth for single-carrier signals (BPSK, QPSK, 16QAM,

64QAM) with square pulses does.

•It was assumed in all cases that the power of the noise floor is known, as a value for the

CFW to target during adaptation.

•SIR in all graphs is the ratio of absolute power of two signals, unadjusted for bandwidth.

So, SIR = PSOI

PSNOIin the case of comparing a SOI and SNOI.

4.1 Interference Rejection Performance of the CFW

This section presents an analysis of the effects of the cycle frequencies of the CFW on

interference rejection performance. The performances of a TIW and a CFW are compared for

the same SNOI, to present the advantages of using a FRESH filter, rather than an FIR LTI

whitening filter. The effects of the cyclostationary properties of the SNOI are also discussed.

4.1. Interference Rejection Performance of the CFW 51

In the first simulation, a BPSK signal, oversampled four times to introduce cyclostationarity,

is used as SNOI. The signal is passed through the TIW and CFW, and the output signals are

presented. The CFW had cycle frequencies in normalized frequency of [−0.5 : 0.0625 : 0.5].

The SCF and spectral correlation are shown in Fig. 4.1, and the simulation parameters are

shown in Table 4.12.

SNOI Modulation BPSKSPS 4No. of symbols 1MSNR 10 dBAWGN power -90 dB

(a) SNOI Parameters

No. of freq. bins 1024No. of steps(CFW)

300

No. of steps(TIW)

3000

µ 10000CFs (normalizedfreq.)

[−0.5 : 0.0625 : 0.5]

(b) Filter parameters

Table 4.1: TIW vs CFW simulation

It is clear from Fig. 4.1 that although the TIW does attenuate the SNOI, it does not eliminate

it. This failure to cancel the SNOI results in relatively high spectral correlation due to the

remaining BPSK signal. That is, it attenuates the signal, but retains the statistical properties

of BPSK. It will be shown in section 4.2 that the TIW cannot reject high powered interference

as well when the interference has SNR of 30 dB or higher, but will leave artifacts at the bins

corresponding to the CFs in normalized frequency of the SNOI.

The CFW produces a solution much closer to white noise and is thus a far better choice

for rejecting interference. More precisely, it eliminates its cyclostationary features and thus

changes its statistical properties. Another line of inquiry into its functioning would be the

effect of the CFs used by the filter. The CFs of the SNOI, −0.25 and 0.25 in normalized

frequency, are removed from the CFW. The CFs are present in the SNOI before filtering due

to the four times oversampling. Fig. 4.2 shows the results of this analysis.2No. of steps is the number of time steps required to for the filter to converge.


(a) SNOI (SNR = 10 dB) (b) TIW output

(c) CFW output (d) AWGN

Figure 4.1: SCF for the output of TIW and CFW. a SCF of the SNOI. b SCF of the TIWoutput. c SCF of the CFW output. d SCF of the noise.

(a) CFW output without CFs of SNOI (b) CFW output with CFs of SNOI

Figure 4.2: SCF for the output CFW with different CFs. a shows the SCF of CFW outputwithout CFs of the SNOI . b shows the SCF of CFW output with CFs of the SNOI. Note: bis the same as c, but on a different scale.

4.1. Interference Rejection Performance of the CFW 53

Clear peaks are present at offsets of -0.25 and 0.25 from the center in the CFW output when

the CFs of the SNOI are not used. So, it is evident that the CFW provides an output much

closer to white noise when all the CFs of the SNOI are included in its set of bin-shifts. This

makes sense, since the aim of the CFW is to cancel interference, and it can do so correctly

only by acting on the CFs of the interference.

We can also look at the spectral correlation remaining in the filter output to determine the

degree of interference rejection. Fig. 4.3 shows the spectral correlation for the TIW and the

CFW with SNOI CFs.

In Fig. 4.3, the spectral correlation is noticeably high in the TIW output when compared

to the CFW. There is not a significant difference in spectral correlation when the CFs of

the SNOI are not included in the CFW, but the output spectrum is not white noise; it is

distorted as seen in Fig. 4.2.

Based on these results, it is clear that the CFW has better interference cancellation

performance than a TIW, and does so correctly only when it uses the CFs of the SNOI.

An effect of the properties of the CFW, as described in the previous section, is that it is

ineffective in the cancellation of signals that are not cyclostationary. Though a relatively

’white’ spectrum might be obtained, the filter is merely flattening the power spectrum and not

interacting in any meaningful way with the non-zero cycle frequencies of the SNOI (since there

are no non-zero CFs.). So, it is not canceling the interference but merely reducing its power,

while in the case of cyclostationary interference it actually cancels it. One example of signals

lacking cyclostationarity is an OFDM signal, which does possess some weak cyclostationarity,

but only if a cyclic prefix is appended. Cyclostationarity can be induced in OFDM by

duplicating subcarriers, thus explicitly introducing spectral correlation.

This inability to reject non-cyclostationary signals is a limitation of the CFW proposed in


(a) Spectral correlation of TIW output

(b) Spectral correlation of CFW output

Figure 4.3: Spectral correlation for the output CFW with different CFs. a shows the spectralcorrelation of TIW output. b shows the spectral correlation of CFW output with CFs of theSNOI.

4.2. Performance of the CFW in the detection of single-carrier signals 55

this work, and one that is to be addressed in future work.

In the following sections, we will examine the performance of the CFW in the test scenario

proposed in Section 3.2.

4.2 Performance of the CFW in the detection of single-carrier

signals

This section presents results for detection of a single-carrier SOI, of less bandwidth than

the SNOI. The SOI has lower SNR than the SNOI and is thus being hidden by it since it is

co-channel interference. This scenario is possible in Dynamic Spectrum Access systems and

the industrial, scientific, and medical radio band (ISM band), due to a large number of radios

competing for spectrum. The hidden node problem for DSA presented in Section 2.4.4 is a

fundamental problem addressed by this work; the CFW provides the ability to look through

nearby interferers and detect SOIs which are attenuated by the path loss of the environment.

This scenario was presented in Section 3.2, and here we will analyze the performance of the

CFW and TIW with different SOIs.

4.2.1 Detection of one single-carrier SOI

In this scenario, only one SOI was present, co-channel to the SNOI. The center frequency of

the SOI is slightly offset from the SNOI, to demonstrate resilience to frequency offsets. The

scenario is run for various SNRs of SOI, and two SNRs of SNOI.

For the first set of simulation results, the simulation parameters are shown in Table 4.2. The

SNOI SNR, in this case, is deliberately set to a high value, to underline the difference in


CFW and TIW performance for high powered interferers.


(a) SNOI Parameters


300

No. of steps(TIW)

3000


[−0.5 : 0.0625 : 0.5]


Table 4.2: Single carrier 30 dB SOI simulation parameters

SOI Modulation BPSK, QPSK, 16QAM,64QAM

Pulse Shaping SRRC, roll-off 0.5No. of symbols 500KBandwidth 1

8th of SOI

SPS 8SIR 0 to -28 dBFreq. offset (Normalized) 0.125

Table 4.3: Single carrier 30 dB SOI modulation types

The PSD of all signals involved is shown in Fig. 4.4. The SIRs 3 in this image are 0 and -20

dB. SOIs of modulation types BPSK, QPSK, 16QAM, and 64QAM were used, with a square

root raised cosine filter with a roll-off of 0.5.

Fig. 4.4a shows the PSD of the SNOI and the SOI, as well as their combined spectra for

SIRs of 0 and -20 dB. Fig. 4.4b shows the PSD of the CFW and TIW outputs for SIRs of

0 and -20 dB. As can be seen, the TIW output has many artifacts, and the SOI is heavily

attenuated. The CFW output only has artifacts resulting from the frequency shifted images

of the SOI, and extracts the SOI with much higher SNR.

3SIR = PSOI

PSNOI. That is, the SOI is the signal, and everything else is the interference


(a) Input PSD

(b) Output PSD

Figure 4.4: PSD for single carrier SOI detection, case 1


The deflection plots for a single carrier SOI, with various SIRs and modulation types, are

shown in Fig. 4.5. Deflection reduces in magnitude with the SIR of the SOI but maintains

the characteristic that the signal position has a higher peak than the spectrally redundant

images. As can be seen by looking at Fig. 4.4, the SOI is present on frequency bins [576,704]

(this is the ground truth). While the CFW output deflection shows a clear, very prominent

peak at this location, the TIW output contains artifacts which overshadow the actual SOI.

These artifacts arise from the imperfect cancellation of high powered interferers by the TIW.

The CFW adapts well to different SNOI SNRs and does not face this issue.

To detect whether the SOI is present in the channel, we may use a simple threshold. To show

that the pdfs are separable by this method, we use a frequentist approach to show the pdfs

under different hypotheses as discussed in Section 3.3. Fig. A.1 in Appendix A shows the

histograms and pdfs for this case. It can be seen that the pdfs are easily separable at SIR =

0 dB and even at SIR = -20 dB for the CFW, but not so for the TIW.

The ROC curve for detecting the presence of the SNOI in the time domain is shown in Fig.

4.6.

It can be seen that the ROC curve for the CFW is almost rectangular, due to the high

separability of the pdfs. Thus, the deflection metric if robust enough that time domain

detection is not an issue for the CFW. The TIW does not manage to detect the SOI as

effectively. Localizing the SOI in the frequency domain is an even tougher problem and one

which we tackle next.

Two methods were tried to detect the frequency domain location of the SOI. The first is a

simple threshold on the value of the deflection. A set of thresholds ranging from the minimum

to maximum deflection values were used to generate a ROC curve. A threshold with a desired

false alarm rate can then be selected through this curve. The generated ROC curve is shown


(a) Deflection without SOI

(b) Deflection with SOI

Figure 4.5: Deflection plots for TIW and CFW. The scales are different for visibility.


Figure 4.6: ROC curve for detection of SOI in the time domain, for a 30 dB SNOI

Figure 4.7: ROC for frequency domain localization, for a 30 dB SNOI


Figure 4.8: PD vs decreasing SIR for frequency domain localization, for a 30 dB SNOI

in Fig. 4.7. It does not show any variation over modulation types, since the modulation

types in this simulation do not have many variations in their cyclostationary features, and

their PSD is typically quite similar. The use of an SRRC also suppresses any cyclostationary

features the SOI may have.

The other method was the MATLAB findpeaks function. As the name implies, it finds peaks

in the deflection, and then the peak with maximum prominence can be selected. Since a well

designed CFW will always have the SOI peak that is far more prominent than any other

artifacts in the output deflection, this method works well over a range of SIRs. The variation

of PD over SIR for this method is shown in Fig. 4.8. Similar to the ROC curve, the PD curve


does not show any variation over modulation types.

We will discuss these results in conjunction with the next set of results. Therefore, first, we

shall present a second simulation which used a much lower SNR for the SNOI. The simulation

parameters are shown in Table. 4.4.


(a) SNOI Parameters


300

No. of steps(TIW)

3000


[−0.5 : 0.0625 : 0.5]




SPS 8Pulse Shaping SRRC, roll-off 0.5No. of symbols 500KBandwidth 1

8th of SOI

SIR 0 to -28 dBFreq. offset (Normalized) 0.125


The PSD of all signals involved is shown in Fig. 4.9. The SIRs in this image are 0 and -20

dB. SOIs of modulation types BPSK, QPSK, 16QAM, and 64 QAM were used, with a square

root raised cosine filter with a roll-off of 0.5.

As before, Fig. 4.9a shows the PSD of the SNOI and the SOI, as well as their combined

spectra for SIRs of 0 and -20 dB. Fig. 4.9b shows the PSD of the CFW and TIW outputs

for SIRs of 0 and -20 dB. Once again, the TIW output has many artifacts, and the SOI is


(a) Input PSD

(b) Output PSD

Figure 4.9: PSD for single carrier SOI detection, case 2


heavily attenuated and distorted. The CFW output extracts the SOI much more cleanly,

though it also contains frequency shifted images of the SOI.

As before, the pdfs were found to be quite separable for time domain detection. The pdfs are

shown in Appendix A Fig. A.3.

The ROC curve for time domain detection is shown in Fig. 4.10. The CFW again has better

performance than the TIW.

Figure 4.10: ROC curve for time-domain detection, for a 10 dB SNOI

As above, the first method used to test localization capability is a threshold on the deflection.

The generated ROC curve is shown in Fig. 4.11. As before, the ROC curve does not show


any variation over modulation types

The second method was the MATLAB findpeaks function. The variation of PD over SIR

for this method is shown in Fig. 4.12. The PD curve does not show any variation over

modulation types.

Figure 4.11: ROC curve for frequency domain localization, for a 10 dB SNOI

It can be seen that the detection capability of the CFW decreases with the decreasing SNR

of the SOI, which is expected as the SOI moves farther below the noise floor. For the CFW

filter, the SNR of the SNOI does not matter, since it adapts well to differently powered

SNOIs (given enough time-samples for convergence). However, the TIW is unable to reject

the SNOI at higher SNRs, and the resulting spectral artifacts make its use in SOI detection


Figure 4.12: PD vs decreasing SIR for frequency domain localization, for a 10 dB SNOI

untenable. Indeed, comparing the performance of the TIW between different SNRs of the

SNOI, detection performance is practically nil for the high-SNR SNOI simulation.

4.2.2 Detection of multiple single carrier SOIs

We will now numerically analyze the performance of the CFW in detecting multiple single

carrier SOIs, co-channel with the SNOI but offset in frequency. We will explore the detection

of multicarrier signals such as OFDM in the next section, so the results produced here are

only for two single-carrier SOIs. The simulation parameters are shown in Table 4.6.


(a) Input PSD

(b) Output PSD

Figure 4.13: PSD for multiple single-carrier SOI detection



(a) SNOI Parameters


300

No. of steps(TIW)

3000


[−0.5 : 0.0625 : 0.5]

(b) CFW parameters

Table 4.6: Multiple single carrier SNOI simulation parameters

SOI Modulation BPSK, QPSK,16QAM, 64QAM

SPS 8Pulse Shaping SRRC, roll-off

0.5No. of symbols 500KBandwidth 1

8th of SOI

SIR 0 to -24 dBFreq. offset(Normalized)

0.125

(a) SOI-1 Parameters

SOI Modulation BPSK, QPSK,16QAM, 64QAM

SPS 8Pulse Shaping SRRC, roll-off

0.5No. of symbols 500KBandwidth 1

8th of SOI

SIR 0 to -24 dBFreq. offset(Normalized)

-0.125

(b) SOI-2 parameters

Table 4.7: Multiple single carrier SOI modulation types

The PSD of all signals involved is shown in Fig. 4.13. As in the single SOI case, the SIRs in

this image are 0 and -20 dB. SOIs of modulation types BPSK, QPSK, 16QAM, and 64 QAM

were used, with a square root raised cosine filter with a roll-off of 0.5.

Fig. 4.13a shows the PSD of the SNOI and the SOI, as well as their combined spectra for

SIRs of 0 and -20 dB. Fig. 4.13b shows the PSD of the CFW and TIW outputs for SIRs of 0

and -20 dB. Once again, the TIW output has the SOI being attenuated and distorted. The

CFW output extracts the SOI even when it is below the noise floor, though it does contain

artifacts due to its FRESH filtering nature. These artifacts are clearer in the deflection plot in

Fig. 4.14. The deflection shows clear peaks for the CFW, along with peaks at the locations of


the frequency shifts. The TIW deflection is quite distorted and results in a poorer detection

capability, as shown in latter graphs.

Figure 4.14: Deflection for two SOIs

The pdfs of the deflection were calculated by a frequentist approach. The pdfs are shown in

Appendix A in Fig. A.5.

A threshold was used to detect the SOIs in the time domain. The ROC curve for this method

is shown in Fig. 4.15. The CFW has much better performance than the TIW, due to the

separation in pdfs. These can be examined in Appendix A.

As in previous simulations, the first method used to test localization capability is a threshold

on the deflection. The generated ROC curve is shown in Fig. 4.16. The ROC curve does not

show any variation over modulation types.

The second method was the MATLAB findpeaks function. For this scenario, the findpeaks

function was tasked to search for the two most prominent peaks and these peaks were

compared to the ground truth, generating a PD metric. The variation of PD over SIR is


Figure 4.15: ROC curve for time-domain detection

shown in Fig. 4.17. The PD curve does not show any variation over modulation types.

It seems counter-intuitive that the TIW has better frequency localization performance with

multiple single carrier SOIs. This is, in fact, a flaw in the methodology. The TIW, due

to its poor interference rejection, leaves many artifacts in its output spectrum, and these

are reflected in the deflection plot in Fig. 4.14. The peak-finding method identifies these

artifacts as the SOI, and this lifts up the detection performance. This phenomenon is more

pronounced with multiple SOIs because the number of bins of the ground truth corresponding

to the artifacts is greater. This is not to say that none of the detected peaks correspond to

the SOI, but to point out the limitations of this method of analysis. It does underline the


Figure 4.16: ROC curve for frequency domain localization

superiority of the CFW in rejecting the SNOI without leaving residual artifacts.


Figure 4.17: PD vs decreasing SIR

4.2.3 Performance of the CFW with bandwidth limited SNOI

This section describes performance results for detection of an SOI when the SNOI is also

bandwidth limited by the usage of a square root raised cosine (SRRC) filter. The introduction

of an SRRC filter reduces the spectral redundancy and thus suppresses cyclic properties,

slightly degrading interference cancellation performance.The simulation parameters are shown

in Table. 4.8.

The deflection plots for this case are shown in Fig. 4.18.

The presence of the SOI in the spectrum was detected using a threshold as in previous cases.


SNOI Modulation BPSK (SRRC)SPS 4No. of symbols 1MSNR 10 dBAWGN power -90 dB

(a) SNOI Parameters


800

No. of steps(TIW)

3000


[−0.5 : 0.0625 : 0.5, 0.1,−0.1]




SPS 8Pulse Shaping SRRC, roll-off 0.5No. of symbols 500KBandwidth 1

2th of SOI

SIR 0 to -25 dBFreq. offset (Normalized) 0.125


Figure 4.18: Deflection for an SRRC SNOI


The ROC curve for this method is shown in Fig. 4.19. The CFW has much better performance

than the TIW, due to the separation in pdfs. These can be examined in Appendix A.


As in previous simulations, the first method used to test localization capability is a threshold

on the deflection. The generated ROC curve is shown in Fig. 4.20. The ROC curve does not

show any variation over modulation types.

It can be seen that the ROC curve for the CFW typically has a slightly higher false alarm

rate than that for the TIW. This can be understood by examining the deflection plots in Fig.

4.18. Due to the frequency shifting nature of the CFW, it creates peaks at location other

than thatbof the SNOI. A thresholding method detects these peaks, and thus has a higher

false alarm rate for the CFW. However, the CFW achieves better detection performance than

the TIW simply due to the clarity of the peaks generated through a FRESH filter which

properly cancels the SNOI.

The second method was the MATLAB findpeaks function. The variation of PD over SIR is



shown in Fig. 4.21. The PD curve does not show any variation over modulation types.


It can be seen that detection performance degrades moderately when the SNOI has less cyclic


spectral content.Thus, more cyclostationarity, as in a larger set of CFs, aids in cancellation of

the SNOI. However, the CFW is able to successfully reject the SNOI even in cases of pulse

shaping reducing cyclic spectral content.

4.3 Performance of the CFW in detection of multicarrier

signals

This section presents the detection performance when the SOI is a multicarrier signal, such as

OFDM. A generic OFDM waveform is used here, with its subcarriers being QPSK modulated.

The SNOI, in this case, is a BPSK signal. The simulation parameters are shown in Table

4.10. The modulation format of the OFDM subcarriers is irrelevant, since the cyclic prefix

determines the cyclostationarity of an OFDM signal.


(a) SNOI Parameters


300

No. of steps(TIW)

3000


[−0.5 : 0.0625 : 0.5]

(b) CFW parameters

Table 4.10: OFDM SOI simulation parameters

4.3. Performance of the CFW in detection of multicarrier signals 77

SOI Modulation OFDM (128 subcarriers,QPSK)

Oversampling 2Bandwidth 1

2of SOI

No. of OFDM symbols 7812IFFT size 256SIR 0 to -18 dB

Table 4.11: OFDM SOI parameters

Two broad cases for detection of multicarrier signals have been found to exist in the context

of the CFW: the width of each frequency bin is larger than the bandwidth of each sub-carrier,

or it is smaller. In the latter case, the method proposed in this work performs very poorly.

While the thresholding method of detection does not change its performance, the peak-finding

method fails since it cannot detect the peaks of the subcarriers that make up the OFDM

signal. This is because the DFT resolution is too small and does not show the subcarriers of

the OFDM.

However, it is possible to detect multicarrier signals reliably with the method proposed in

this work. When the width of each frequency bin is larger than that of each subcarrier,

the deflection reflects the multicarrier nature of the SOI. This allows each subcarrier to be

detected independently, and the union of these results in detection of the entire multicarrier

signal. This will be articulated better in the deflection plots shown below.

The PSD of all signals involved is shown in Fig. 4.22. As before, the TIW plots are quite

heavily distorted. The CFW plot extracts the SOI well at higher SIRs up -10 dB.

The deflection plots for this case with various SIRs are shown in Fig. 4.23. The deflection

shows every individual subcarrier of the OFDM signal, making it easy for the peak finding

method to detect it.


(a) Input PSD

(b) Output PSD

Figure 4.22: PSD


Figure 4.23: Deflection for an OFDM SNOI

A frequentist approach calculated the pdfs of the deflection. The pdfs are shown in Fig. A.7.

A threshold was used to detect the SOIs in the time domain. The ROC curve for this method

is shown in Fig. 4.24.

The TIW has better time-domain performance than in previous cases, though the CFW still

outperforms it.

As in previous cases, the first method used to test localization capability is a threshold on

the deflection. The generated ROC curve is shown in Fig. 4.25.

The second method was the MATLAB findpeaks function. The variation of PD over SIR is

shown in Fig. 4.26. The function was set to search for 128 peaks since the OFDM signal has

128 subcarriers.

It can be seen the TIW has the same artifacts in its output spectrum as in previous cases.

These artifacts are reflected in its processing of the OFDM signal. The deflection plot



produced in Fig. 4.23 for the TIW is not a good representation of the OFDM SOI. Regardless,

due to the limitations of the method of analysis, it presents a PD curve that is unexpectedly

high in magnitude in comparison to the CFW. Essentially, since the ground truth covers

half the spectrum, false alarms are highly probable because of the many artifacts in the

TIW output spectrum. The CFW manages to eliminate these artifacts far better through its

superior interference rejection properties.


Figure 4.25: ROC curve for frequency domain lozalization



4.4. Performance of the CFW in SOI data recovery 83

4.4 Performance of the CFW in SOI data recovery

This section describes performance results for recovering the SOI by reversing the frequency

shifts of the CFW, by using a standard FRESH filter. Once the CFW has completed operation,

the CFs which have non-trivial filter coefficients are selected as the CFs of the SNOI. These

are used in a FRESH filter which is designed to extract the SOI through a linear MMSE

formulation, as described in [5, 33].

Once the second FRESH filter has completed operation, the BER of the SOI is measured

as a metric for the performance of the CFW in rejecting interference, in comparison to the

standard FRESH filter. To aid in this comparison, the MMSE FRESH filter is also run on

the un-whitened spectrum, i.e., the spectrum of (SNOI+SOI). For this baseline simulation,

we assumed the MMSE FRESH filter knew the CFs of the SNOI, so that it could remove the

SNOI from the SOI. This was essential to make it a fair comparison. This is because the

MMSE FRESH filter could not possibly function without knowing the CFs of the signal it is

attempting to recover. Since the SOI is not required to be cyclostationary in out simulation,

we gave the MMSE FRESH filter the CFs of the SNOI, so that it can cancel it, to maintain

consistency in our assumptions. Simulation parameters are shown in Table 4.124.

The PSD of all signals involved is shown in Fig. 4.27 for SIR = 0 dB. The CFW+FRESH

method manages to eliminate the distortion due to the SNOI better than just the FRESH

filter.

The BER resulting from both techniques is shown for a range of SIRs from [20 : −30] dB.

The CFW shows performance that is comparable to the optimal MMSE FRESH filter. It

can be seen that after 3 dB SIR, the FRESH filter perform just as well as a system with

no filtering, demarcating this as the boundary between the noise and interference limited4AWGN power had to be increased due to numerical errors in MATLAB with the MMSE FRESH filter.

Low signal magnitudes caused underflows



(a) SNOI Parameters


300

µ 10000CFW CFs(normalizedfreq.)

[−0.5 : 0.0625 : 0.5]

Baseline FRESHCFs (normalizedfreq.)

[−0.25, 0, 0.25]


Table 4.12: SOI recovery: SNOI and filter parameters

SOI Modulation 16QAMPulse Shaping SRRC, roll-off 0.5No. of symbols 500KBandwidth 1

8th of SOI


Table 4.13: SOI recovery: SOI parameters

regions.

We repeated the experiment for an SNOI with an SNR of 30 dB. The SIR range is adjusted

to [0 : −50] dB to maintain the same SNR for the SOI. All other simulation parameters are

the same. A BER curve is shown in Fig. 4.29. As before, the CFW produces an output that

is the same as that of an optimal cyclic Wiener filter which had complete knowledge of the

SNOI CFs. Thus, our two-stage interference rejection method performs as well as the optimal

FRESH filter. It provides 13 dB gain in BER over the case without any sort of FRESH

filtering for 10 dB SNR SNOI, and 33 dB gain in BER for a 30 dB SNR SNOI.

This simulation proves that reconstruction of the SOI following application of the CFW

is possible with a resulting low BER. As a future research direction for this work, this

4.4. Performance of the CFW in SOI data recovery 85

Figure 4.27: PSD

Figure 4.28: BER vs SIR for 10 dB SNOI


Figure 4.29: BER vs SIR for 30 dB SNOI

reconstruction could be done in a blind manner, without knowledge of the SOI modulation

or coding characteristics. This would result in a novel interference rejection algorithm that

has many advantages over the standard FRESH filter, not the least of which is not requiring

any knowledge of the SOI.

4.5 OFDM SNOI Cancellation with Paramorphic Multicarrier

Waveforms

In this section, results are presented for the cancellation of an OFDM SNOI which has

been generated using the work of [5]. This paramorphic multicarrier waveform [5] uses

redundant symbols scattered in time and frequency to introduce cyclostationarity in the

OFDM waveform. This cyclostationarity can then be exploited by the FRESH filter to

effectively reject the OFDM SNOI. Performance is compared in an SOI detection scenario to

4.5. OFDM SNOI Cancellation with Paramorphic Multicarrier Waveforms 87

a generic OFDM waveform with no symbol repetition. In this simulation, 64 QPSK symbols

were repeated accross 256 subcarriers in frequency 4 times, i.e [1 : 64; 1 : 64; 1 : 64; 1 : 64].

This introduces an explicit spectral redundancy which can be exploited by the CFW.

The motivation for this particular simulation was to showcase the compatibility of the method

proposed in this work with modern communications which utilize OFDM, such as LTE.

While an OFDM signal does not possess cyclostationarity (save for the cyclic prefix), LTE

can easily repeat symbols to introduce this property, and thus make the resulting waveform

compatible with the CFW. Thus, the CFW can be utilized in coexistence scenarios to solve

the hidden node problem for OFDM waveforms. It can look through a designated set of

OFDM subcarriers to detect interference behind them.

It is assumed that the SNOI was available for training the CFW.

SNOIModulation

OFDM (256subcarriers,QPSK)

No. of OFDMsymbols

10000

SPS 2IFFT size 512SNR 10 dBAWGN power -90 dB

(a) OFDM SNOI Parameters


3000


[−0.5 : 0.0625 : 0.5,]

(b) CFW parameters

Table 4.14: OFDM SOI simulation parameters

4.5.1 OFDM SOI Detection

The first test case has an OFDM SOI, similar to the SNOI in bandwidth but different in

other parameters. It has no symbol repetition. Table 4.15 shows the SOI parameters.


SOI Modulation OFDM (64 subcarriers,16QAM)

Bandwidth Same as SOINo. of OFDM symbols 40000SPS 2IFFT size 128SIR 0 to -25 dB

Table 4.15: OFDM SOI parameters

(a) SCF for OFDM SNOI with symbol repetition of 4

(b) SCF for OFDM SNOI with no symbol repetition

Figure 4.30: SCFs

The SCFs of the SNOI with and without repetition is shown in Fig. 4.30.

The CFW filter coefficients are shown in Fig. 4.31. It can be seen that without symbol

repetition, the CFW is no better than an LTI filter, since no non-zero or non-conjugate CFs

have any significant magnitude.


Figure 4.31: CFW filter coefficients

The deflection with and without repetition is shown in Fig. 4.32. It can be seen the the SOI

is clearly detected with symbol repetition in the SNOI, but suppressed and distorted when

there is no repetition.

As in previous simulations, to detect the presence of the SOI in the spectrum, a frequentist

approach was used to calculated the pdfs of the delection, and separated by a threshold to

detect the SOI. The generated ROC curve is shown in Fig. 4.33.

A threshold on deflection over frequency bins was used to localize the SOI over frequency

bins. The ROC curve for this is shown in Fig.

It can be seen that the CFW performs better at SOI recovery when the SNOI possesses

cyclostationarity, as expected. The findpeaks function was not used as the OFDM SNOI of

similar bandwidth has caused enough distortion to not reflect the subcarriers of the SOI in

the deflection.


Figure 4.32: Deflection




4.5.2 Narrow-Band SOI Detection

The second simulation tested detection of a narrow-band SOI, which lies within the SNOI,

interfering with a subset of subcarriers. The SNOI was the same as the previous case for

consistency, and the SOI parameters are shown in Table 4.16.

SOI Modulation BPSKPulse Shaping SRRC, roll-off 0.5No. of symbols 500KBandwidth 1

4th of SOI


Table 4.16: Narrowband SOI parameters

So, the SOI is interfering with subcarriers [128-192], and we will try to detect its presence in

time and frequency in this simulation. To detect the presence of the SOI in the spectrum, a


frequentist approach was used to calculated the pdfs of the deflection, and the deflection was

separated by a threshold to detect the SOI. The generated ROC curve is shown in Fig. 4.35.


A threshold on deflection over frequency bins was used to localize the SOI over frequency

bins. The ROC curve for this is shown in Fig.

As before, the CFW performs better when the SNOI has cyclostationarity. It is able to exploit

the cyclostationarity to perform interference cancellation rather than mere attenuation.



Chapter 5

Conclusion and Future Work

5.1 Conclusion

In this work, we have developed a blind adaptive interference cancellation algorithm and

discussed possible use cases. After providing a review of the technical background in relevant

topics in Chapter 2, the mathematical model of the CFW was presented in Chapter 3. The

development of the CFW is the primary contribution of this work. Chapter 3 also discussed

the convergence of the adaptive filter and suggested a use case for the CFW. While the uses of

a blind interference cancellation algorithm are myriad, we introduced the case of detecting an

SOI hidden behind a more powerful wideband SNOI. As noted previously, this is commonly

observed in DSA systems as the hidden node problem and the also in the ISM band.

Chapter 4 presented the analyses of the CFW in two areas:

•The interference rejection performance of the CFW

•The performance of the CFW for signal detection

•The performance of the CFW in recovering the data in the SOI, compared to the optimal

MMSE FRESH filter

94

5.1. Conclusion 95

5.1.1 Interference Rejection Performance of CFW and TIW

The interference rejection performance of the CFW was analyzed by examining the SCF of its

output when fed different SNOIs to cancel. It was compared to the performance of the TIW,

a time-invariant version of the CFW, which does not take advantage of the cyclostationary

nature of SNOIs. It was found that the CFW has superior interference rejection properties

compared to the TIW, producing a filter output signal with the statistical properties of

white noise. The output of the TIW, on the other hand, had remnants of the SNOI, evident

through the spectral correlation present.

A corollary of this result is that the CFW is ineffective at canceling interference that is not

cyclostationary. Since there are no cyclic properties to interact with, the filter develops

non-trivial coefficients only on the zeroth CF. This scenario essentially makes it a TIW,

which is ineffective at canceling interference and merely attenuates it instead.

The TIW was also unable to reject high powered interferers (SNR > 20 dB), while the CFW

can adapt to any SNR, given enough time for the algorithm to converge. With a suitable µ,

convergence occurred in time-steps on the order of 103 points for both filters.

5.1.2 CFW Detection Performance in Hidden Node Scenario

The CFW and TIW were tested for the use case of detecting an SOI of lower power than a

co-channel SNOI, a situation that prevents detection of the SOI. A range of SIRs typically

from 0 dB to -20 dB were tested. A deflection metric of the frequency domain CFW output

was utilized for the detection method. A predetermined threshold was applied to the deflection

in each bin to detect if a signal was present in the spectrum at any bin. ROC curves were

presented in Chapter 4 for this simulation.

96 Chapter 5. Conclusion and Future Work

Following this, two techniques were used for localizing the SOI in the frequency domain

through the deflection:

•A predetermined threshold on the deflection. The deflection was highest at the location of

the SOI and thus made it possible to localize it. ROC curves were presented in Chapter 4 for

this method. The CFW typically had better performance than the TIW across all test cases.

•Searching for peaks in the deflection by the MATLAB findpeaks function. The SOI had the

most prominent peak and could be easily differentiated from any other artifacts, allowing it

to be detected easily. A set of curves of PD against SIR were presented in Chapter 4 for this

method. The CFW typically has better performance than the TIW.

It was found that the CFW has better performance than the TIW in all scenarios tested.

The scenarios were:

•One single-carrier SOI co-channel with a wideband single-carrier SNOI

•Two single-carrier SOIs, offset from each other in frequency, co-channel with a wideband

single-carrier SNOI.

•One generic OFDM SOI, co-channel with a wideband single-carrier SNOI.

•One single carrier SOI, co-channel with a wideband single carrier SNOI with SRRC pulse

shaping.

For the OFDM SOI, the peak detector was found to function only when the frequency

resolution was higher than the subcarrier bandwidth, allowing the peak detector to detect

each subcarrier independently.

Since the CFW is ineffective at canceling SNOIs which are not cyclostationary, an alternative

in the form of paramorphic multicarrier waveforms [5] was used for testing. These waveforms

use symbol repetiion on an OFDM waveform to induce cyclostationarity. The detection

5.1. Conclusion 97

performance of the CFW was compared for the paramorphic multicarrier waveform against a

generic OFDM waveform with no repetition for the following scenarios.

•An OFDM SOI of the same bandwidth as the SNOI.

•A narrow-band SOI, interfering with a subset of the SNOI’s subcarriers.

The CFW showed better detection performance when the OFDM waveform had symbol

repetition. This simulation demonstrated compatibility of the CFW with modern communication

systems that utilise OFDM waveforms, since LTE for example can easily introduce cyclostationarity

in its physical layer by repeating data across subcarriers. Such a waveform could be used in

conjunction with the CFW to solve the hidden node problem in spectrum sharing scenarios,

in the CBRS band or for LAA in the 5 GHz band, among others.

As a downside, the frequency shifting nature of the CFW distorts the SOI, so it cannot be

demodulated right away. However, since we know the frequency shifts that have occurred, the

removal of this distortion and complete recovery and demodulation of the SOI are possible

and are covered in the next section.

5.1.3 SOI data recovery performance

The SOI was passed through a standard cyclic Wiener/ MMSE FRESH filter following

removal of the SNOI by the CFW. The frequency shifts of this FRESH filter were determined

by using filter weights associated with the CFs which converged to non-trivial values during

SNOI rejection. These CFs are the CFs of the SNOI and also characterize the cyclostationarity

introduced into the SOI by the CFW.

It was found that the CFW performed as well as an optimal MMSE FRESH filter which is

omniscient of the CFs of the SNOI. Tests were done over an SIR range of 20 to -30 dB for a

98 Chapter 5. Conclusion and Future Work

10 dB SNR SNOI, and over an SIR range of 0 to -50 for a 30 dB SNR SNOI. The difference

in SIR in the two cases is accounted for by the fact that the SNR of the SOI remained the

same in both cases. So, given a constant SNR for the SOI, performance was maintained for

interferers with different but high SNRs.

The primary caveat of this method is that we need to know the SOI to use the MMSE

FRESH filter. This is not an unreasonable requirement since training sequences are usually

used to receive and demodulate the SOI. It should be noted that the cyclic Wiener filter

alone will only work to recover the SOI if the SOI is cyclostationary. The CFW introduces

cyclostationarity artificially and thus our two-stage approach of CFW followed by a cyclic

Wiener filter also works for a much wider class of SOIs. For the purposes of performance

comparison in our work, we assumed the cyclic Wiener filter knew the cyclic properties of

the SNOI and could thus remove it from the SOI.

This method is well suited for mitigating jammers and coexisting with other users in a DSA

environment. The speed of the CFW in adapting to interferers makes the system agile and

flexible for canceling incoming interference, to continue its transmission and reception.

5.2 Future Work

The next steps in the development of this algorithm are to develop a blind method to

reverse the frequency shifts on the SOI and recover a clean version of it which can then

be demodulated. This would result in a new and powerful interference rejection and data

recovery algorithm. Some potential approaches could be using the constant envelope property

of an SOI to develop a blind algorithm or using its known alphabet for a decision directed

technique. Thus, potential candidate algorithms could be CMA, decision directed algorithms,

maximum SNR and Bussgang techniques. Alternatively, predistortion in the SOI could be

5.2. Future Work 99

explored as a transmitter side method to resist the effect of the CFW.

Such an algorithm could be used in conjunction with communication systems such as LTE to

increase resilience to jamming. It could, therefore, be used to reject pulsed radar interference

in scenarios of spectrum sharing in the CBRS band. This method also has the advantage that

one does not have to know the SOI, removing signaling overhead necessary to transmit this

information. It can also work well at very high interference signal powers, making it robust.

A natural extension of the signal detection functionality is to look through wideband jammers,

detecting signals hidden behind the jammer. If the user is conducting the jamming, it does

not need to be turned off to check if the targeted signal is still present. Thus, the CFW

lends itself well to network security and wireless forensics. Its adaptive nature would also

allow it to handle different jamming scenarios on the fly, with functionality achieved over a

reasonable number of time samples (on the order of 107).

The ability to reliably detect OFDM SOIs makes the CFW relevant to modern communication

systems since OFDM is at the core of the PHY layer in the 3GPP standard for LTE and 5G.

Also, as shown in Section 4.5, the work of [5] can be used to identify interference behind a

given set of subcarriers, but introducing redundant symbols to aid detection. This can find

application in various spectrum sharing scenarios such as LAA or the CBRS band.

Since the CFW filter weights are non-trivial only for those CFs which are present in the

SNOI, it could be developed as a method to find the cyclic properties of interference. This

knowledge can be used to identify signal modulation parameters. The performance could be

compared to alternative state of the art methods for cyclic spectral analysis such as the strip

spectral correlation analyzer [30].

As an application of the FRESH filter, it could be used to combat the non-linearity of power

amplifiers. Such amplifiers often restore sidelobe levels due to their non-linearity, which then

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have to be filtered out. Instead, a blind FRESH filter which detected the location of the

sidelobe levels could then be used to shift these sidelobes and utilise them to increase the

SNR of the received signal.

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Appendices

107

Appendix A

Histograms and pdfs of deflection

The histograms and pdfs used to in Sections 4.2 and 4.3 are placed here for referral.

(a) pdf and Histogram at SIR = 0 dB

(b) pdf and Histogram at SIR = -20 dB

Figure A.1: pdf and Histogram plots for CFW deflection, SNOI SNR = 30 dB

108

Bibliography 109



Figure A.2: pdf and Histogram plots for TIW deflection, SNOI SNR = 30 dB

Bibliography Appendix A. Histograms and pdfs of deflection



Figure A.3: pdf and Histogram plots for CFW deflection, SNOI SNR = 10 dB

Bibliography 111



Figure A.4: pdf and Histogram plots for TIW deflection, SNOI SNR = 10 dB




Figure A.5: pdf and Histogram plots for CFW deflection for two SOIs

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Figure A.6: pdf and Histogram plots for TIW deflection for two SOIs




Figure A.7: pdf and Histogram plots CFW deflection for OFDM SOI

Bibliography 115



Figure A.8: pdf and Histogram plots TIW deflection for OFDM SOI

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