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Fast and Accurate Spectrum Sensing in Low Signal to Noise Ratio Environment

Parisa Cheraghi

Submitted for the Degree of Doctor of Philosophy

from the University of Surrey

4 UNIVERSITY OFm SURREY

Centre for Communication Systems Research Faculty of Engineering and Physical Sciences

University of Surrey Guildford, Surrey GU2 7XH, U.K.

September 2012

@ Parisa Cheraghi 2012

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AbstractOpportunistic Spectrum Access (OSA) [1] promises tremendous gain in improving spectral efficiency. The main objective of OSA is to offer the ability of identifying and exploiting the under-utilised spectrum in an instantaneous manner in a wireless device, without any user intrusion. Hence, the initial requirement of any OSA device is the ability to perform spectrum sensing. Local narrow-band spectrum sensing has been quite well investigated in the literature. However, it is realised that existing schemes can hardly meet the requirements of a fast and accurate spectrum sensing particularly in very low signal-to-noise-ratio (SNR) range without introducing high complexity to the system. Furthermore, increase in the spectrum utilisation calls for spectrum sensing techniques that adopt an architecture to simultaneously search over multiple frequency sub-bands at a time. However, the literature of sub-band spectrum sensing is rather limited at this time. The main contributions of this thesis is two-fold:

• First a clusterd-based differential energy detection for local sensing of multicarrier based system is proposed. The proposed approach can form fast and reliable decision of spectrum availability even in very low SNR environment. The underlying initiative of the proposed scheme is applying order statistics on the clustered differential Energy Spectral Density (BSD) in order to exploit the channel frequency diversity inherent in high data-rate communications.

• Second contribution is three-fold : 1) re-defining the objective of the subband level spectrum sensing device to a model estimator, 2) deriving the optimal model selection estimator for sub-band level spectrum sensing for fixed and variable number of users along with a sub-optimal solution based on Bayesian statistical modelling and 3) proposing a practical model selection estimator with relaxed sample size constraint and limited system knowledge for sub-band spectrum sensing applications in Orthogonal Frequency-Division Multiple Access (OFDMA) systems.

The result obtained showed that through exploitation of the channel frequency selectivity the performance of the stat-of-the-art spectrum sensing techniques can be significantly improved. Furthermore, by modelling the sub-band level spectrum sensing through model estimation allows for new spectrum sensing approach. It was proved both analytically and through simulations that the proposed approach have significantly extended to state-of-the-art spectrum sensing.

K ey words: Differential, energy detection, low signal-to- noise ratio (SNR), multicarrier, opportunistic spectrum access, spectrum sensing.

Acknowledgements

Any attempt to list the people and opportunities with which my life has been richly blessed would be like trying to count the stars in the sky. Yet among these stands individuals whose profound impact deserves special acknowledgment and to whom I would like to dedicate this thesis.

First and foremost, I would like to express my gratitude to my principle supervisor, Dr. Yi Ma for constant and generous support and guidance, whom his mind provoking discussions, careful comments and criticism have greatly influenced my research. The care and time he put into all his students set an example I hope to follow.

Secondly, I would like to express my most sincere appreciations to my co-supervisor. Professor Rahim Tafazolli for his endless support throughout my Ph.D. His patient, thought-provoking guidance and instruction provided a foundation that will continue to influence my research.

I wish to express my deepest regards to my parents for their endless love and encouragement and without the support of whom, I could not have been able to reach this stage.

I would like to thank all my friends and colleagues in CCSR for their support during my Ph.D.

Last but no means least, I would like to thank all my teachers and lecturers from the first day of school up until now for their undeniable contributions towards my academic achievements.

Contents

IX

1 Introduction 1

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

1.2 Motivation and O b jec tiv e ....................................................................... 3

1.3 Major Contribution ................................................................................ 3

1.4 Thesis O rgan isa tion ................................................................................ 5

1.5 Publication L is t .......................................................................................... 5

1.5.1 Journal Publications..................................................................... 5

1.5.2 Conference P roceed ings............................................................... 5

1.5.3 P a t e n t s ........................................................................................... 6

2 State-of-the-art Spectrum Sensing Techniques 7

2.1 Problem Formulation for Narrow-Band Spectrum S en sin g ................. 7

2.2 Exploiting Energy D ifference................................................................. 9

2.2.1 Energy Detection........................................................................... 9

2.2.2 Energy Detection Under Noise U n ce rta in ty ............................. 10

2.3 Exploiting Stationarity Difference.......................................................... 12

2.3.1 Second Order Cyclostationarity D etection ................................ 12

2.3.2 Covariance Based D etection......................................................... 14

2.3.2.1 Eigenvalue Based D etection........................................ 16

2.3.3 Matched F i l te r in g ........................................................................ 17

2.3.3.1 Pilot Based Match Filtering D etection....................... 17

2.4 Exploiting The Distribution Difference................................................. 18

2.4.1 Entropy Based D e te c tio n ........................................................... 19

2.4.2 Kurtosis Based D e tec tio n ........................................................... 20

2.5 Summary of The Narrow-Band Spectrum Sensing Approaches . . . 21

2.6 Sub-Band Level Spectrum Spectrum S e n s in g .................................... 23

2.6.1 Sub-Band Level Spectrum Sensing System M o d e l .................. 23

ii

Contents iii

2.6.2 Filter-Bank Based Spectrum Sensing....................................... 23

2.6.3 Joint Multi-Band D etection ....................................................... 24

2.6.4 Sequential Multi-Band D etection .............................................. 25

2.6.5 Wavelet Based Detection .......................................................... 26

2.6.6 Wigner-Ville Based D etection.................................................... 27

2.7 Summary of The Sub-Band Spectrum Sensing Approaches................ 28

2.8 S u m m a ry ................................................................................................... 29

3 Cluster-Based Differential Energy D etection for Spectrum Sensingin Multi-Carrier System s 31

3.1 Introduction................................................................................................ 31

3.2 System Model and Problem Formulation.............................................. 32

3.2.1 Multi-Carrier S ystem s................................................................. 32

3.2.2 Effect of Second Order M om ent................................................. 34

3.2.3 Statement of The Spectrum Sensing P ro b le m ......................... 34

3.3 Cluster-Based Differential Energy D etection ........................................ 35

3.3.1 Sensing of CP-Based Multi-Carrier Signals ............................ 35

3.3.2 Overcoming Timing Offset........................................................... 41

3.3.3 Extension to the ZP-Based S ystem ........................................... 42

3.3.4 Knowledge of Key Parameters ................................................. 42

3.4 Performance Analysis ............................................................................. 43

3.4.1 Probability of False A larm .......................................................... 43

3.4.2 Probability of D etection ............................................................. 46

3.4.3 Numerical Results and Discussions........................................... 49

3.4.4 Computational C om plexity ....................................................... 52

3.5 Simulation Results and Discussions....................................................... 52

3.6 S u m m a ry .................................................................................................. 57

4 A Bayesian M odel Based Approach for Joint Sub-Band Level Spectrum Sensing 62

4.1 Introduction................................................................................................ 62

4.2 Problem Formulation................................................................................ 63

4.3 Optimal and Sub-Optimal S o lu tio n s .................................................... 64

4.3.1 Optimal Decision R u le ................................................................ 64

4.3.2 Optimal Solution For Fixed â .................................................... 65

4.3.3 Sub-optimal Solution................................................................... 67

4.4 Link to OFDMA based Systems............................................................. 69

Contents iv

4.4.1 Transmitted S ignal....................................................................... 70

4.4.2 Received Signal............................................................................. 70

4.4.3 Join Sub-Band Level Spectrum Sensing in OFDMA Systems 70

4.5 Performance Analysis for The Practical S o lu tio n .............................. 73

4.5.1 Asymptotic Probability of False Alarm and Threshold . . . . 73

4.5.2 Asymptotic Probability of “knee-point” D e te c tio n .................. 74

4.5.3 Optimal Asymptotic Threshold S e t t i n g .................................... 76

4.5.4 Computational complexity........................................................... 77

4.6 Further Discussions.................................................................................. 78

4.6.1 Overcoming Timing Offset........................................................... 78

4.6.2 Overcoming the Energy L ea k ag e ............................................... 78

4.6.3 Extension to the ZP-based s y s te m ........................................... 79

4.7 Simulation R e s u lts .................................................................................. 80

4.8 S u m m a ry ..................................................... ............................................ 84

5 Conclusion and Future Work 86

5.1 Conclusion .............................................................................................. 86

5.2 Future W o r k ................ 90

Appendix A 91

Appendix B 92

Appendix C 94

Appendix D 95

Appendix E 96

Bibliography 98

List of Figures

2.1 Flow chart of conventional energy detection........................................................... 9

2.2 Flow chart of frequency-domain energy detection................................................. 9

2.3 The SNR wall phenomenon.......................................................................................... 11

2.4 Flow chart of second order cyclostaionary based detection technique. . . . 12

2.5 Flow chart of covariance based detection................................................................ 14

2.6 Block diagram of Eigenvalue based detection technique..................................... 16

2.7 Block diagram of the pilot based detection technique......................................... 18

2.8 Flow chart of entropy based detection...................................................................... 19

2.9 Flow chart of kurtosis based detection..................................................................... 20

2.10 Block diagram of the wavelet based detection technique.................................... 26

3.1 Block diagram of the cluster-based differential energy detection algorithm .................................................................................................................................. 36

3.2 Effect of the sort function on the output, for iV = 50 and r = 7 on various distributions. This figure illustrates how the sort function focuses on a particular point of a distribution based on the value of r. Furthermore the shape difference for various distribution all having a mean value of 0.42 isalso shown in this figure............................................................................................... 39

3.3 The relationship between the PD and the observation length for M = 64and £ = 6 .......................................................................................................................... 49

3.4 The relationship between the PD and the coherence bandwidth, £ , andthe observation length A = 1 0 ................................................................................... 50

3.5 Complementary ROC curves of the Test I and it’s comparison with energy detection for various uncertainty factors (U), the optimal detector based on Neyman-Pearson criteria. 7 = —IQdB, £ = 8 and M = 64 based onthe analytical results in Section 3.4.......................................................................... 50

3.6 Complementary ROC curves of the Test II and it’s comparison with energy detection for various uncertainty factors (U). 7 = —IQdB, £ = 5 andM = 64 based on the analytical results in Section 3.4........................................ 51

3.7 Comparison of the simulation results and its equivalent analytical results (in Section 3.4) for Test I. Furthermore the effect of the sorting operationis shown............................................................................................................................. 53

3.8 Comparison of the simulation results of the proposed technique with andwithout the sorting operation and its equivalent analytical results (in Section 3.4) for Test II........................................................................................................ 54

V

List of Figures vi

3.9 The relationship between the PD and the observation length for M = 64and £ = 6 .......................................................................................................................... 59

3.10 The relationship between the PD and the coherence bandwidth, £ , andthe observation length K = 1 0 ................................................................................... 59

3.11 Complementary ROC curves of the Test I and it’s comparison with energy detection for various uncertainty factors (U), the optimal detector based on Neyman-Pearson criteria. 7 = —IQdB, £ = 8 and M = 64 based onthe analytical results in Section 3.4.......................................................................... 60

3.12 Complementary ROC curves of the Test II and it’s comparison with energydetection for various uncertainty factors (U). 7 = —lOdB, £ = 5 andM = 64 based on the analytical results in Section 3.4........................................ 60

3.13 The performance comparison of the proposed technique, frequency-domain energy detection, second order cyclostationarity, pilot based detection and differential energy detection for K = 7.................................................................... 61

3.14 The effect of the differential and clustering stages on the performance ofthe proposed spectrum sensing technique............................................................... 61

4.1 The performance evaluation of the optimal solution introduced in Section4.3 for fixed â and K = 100..................................................................................... 67

4.2 The performance comparison of the proposed optimal and sub-optimalsolutions derived in Section 4.3......................................... 69

4.3 Flow chart of the proposed OFDMA sub-band level spectrum sensing. . . 71

4.4 ROC Curve of the proposed algorithm in Section 4.4.3 for p = 2 dB. . . . 76

4.5 Step by step process of the proposed algorithm in Section 4.4.3 in order todetermine the vacancy of unused sub-bands.......................................................... 81

4.6 ROC curve comparison of the proposed algorithm and energy detectionbased filter-bank approach........................................................................................... 82

4.7 Performance comparison of the the proposed algorithm and energy detection based filter bank in terms of probability of detection for PFA = 0.01. 83

List of Tables

2.1 Summary of the state-of-the-art local narrow-band spectrum sensingapproaches................................................................................................... 22

2.2 Summary of the state-of-the-art local sub-band spectrum sensing approaches........................................................................................................ 29

3.1 Comparison of the state-of-the-art local narrow-band spectrum sensing approaches and the cluster-based energy detection......................... 56

4.1 Extended Pedestrian A m odel................................................................. 80

4.2 Comparison of the state-of-the-art local sub-band spectrum sensingapproaches and proposed Bayesian approach......................................... 85

VII

List of Abbreviation

OSA Opportunistic Spectrum AccessFCC Federal Communication CommitteeOfcom Office of CommunicationCSMA Carrier Sense Multiple AccessPFA Probability of False AlarmPD Probability of DetectionSNR Signal-to-Noise RatioBSD Energy Spectral DensityOFDMA Orthogonal Frequency-Division Multiple AccessOFDM Orthogonal Frequency Division MultiplexingFFT Fast Fourier TransformIFFT Inverse Fast Fourier TransformIDFT Inverse Discrete Fourier TransformAWGN Additive White Gaussian NoiseDFT Discrete Fourier TransformCP Cyclic PrefixIBI Inter-Block InterferenceCFO Carrier Frequency Offseti.i.d. Independent and Identically DistributedMGF Moment Generating Functionp.d.f. Probability Distribution FunctionCDF Cumulative Distribution FunctionMC-CDMA Multi-Carrier Code Division Multiple AccessSC-FDMA Single-Carrier Frequency Division Multiple AccessMIMO Multiple-input multiple-outputROC Receiver Operating Characteristic3GPP 3rd Generation Partnership Project (Telecommunication)LTE Long Term EvaluationPDP Power Delay ProfileZP Zero PaddingPSD Power Spectral Density

Vlll

List of Symbols

s{t) Transmitted Signalv{t) Additive White Gaussian Noisey{t) Received signal at the spectrum sensing device'Ho Hypothesis condition that the spectrum band of interest is vacant7^1 Hypothesis condition that the spectrum band of interest is occupiedA Threshold value^ Test statistic used for spectrum sensing^ s c Test statists for second order cyclostationarity based detection^CD Test statistics for covariance based detection^EDt Threshold value to time domain energy detectionN Observation lengthn Discrete time sample indext Continuous time sample index^ N x N N X N Discrete Fourier Transform Matrix<7 Additive white Gaussian noise varianceCTg Transmitted signal varianceXe d Threshold value used for energy detectionU Noise uncertainty factorSNRwaii Signal to noise ratio walli F i ( . ; .) Hypergeometric functionF(.) Gamma Function'P Signal periodRy Autocorrelation of variable yX Cyclic frequencyL n ( - ) Laguerre polynomial functionIf, L x L Identity matrixCy Covariance matrix of the received signalCs Covariance matrix of the transmitted signalRy Frequency representation of cyclic autocorrelation function of variable yQmax Maximum eigenvalue of the covariance matrixQmin Minimum eigenvalue of the covariance matrix^EV Test statistic for eigenvalue based detectionXe v Threshold value for eigenvalue based detectionâ Average eigenvalueSp{ t ) Pilot signal^PM Test statistics used for pilot based match filtering detectionApM Threshold value used for pilot based match filtering detection^ e b Test statistics used for pilot based entropy based detectionXe b Threshold value used for pilot based entropy based detectionlog Logarithm operationy,y Average value of the received signal

ix

^ k b Test statistics used for Kurtosis based detectionXk b Threshold value used for Kurtosis based detectionK Total number of sub bands^WB Test statistics used for Wavelet based detectionXwB threshold value used for Wavelet based detectionI m M X M identity matrix$ Pre-coding MatrixJ Total number of sub-carriersM Number of data Sub-carriersh Frequency selective channele Frequency offset normalised by the sub carrier spacinge Timing offsetTb Block durationTs Sampling periodAfo Noise PowerX OFDM modulated transmitted signalB Estimate of the channel coherence bandwidthC Upper bound of channel bandwidthL Cluster size[.J Floor functionIf Estimated power spectral densityà Estimated spectrum availability indicator functionQ Normalised covariance matrixdet(.) Determinant functionerfc(.) Complementary error functionA(.) Upper triangle channel matrixV(.) Lower triangle channel matrix

Central Chi squared distribution with N degrees of freedom X^(A) Chi squared distribution with N degrees of freedom with non-centrality parameter At q Values of q sorted in increasing order0(a, b) Guassian distribution with mean and variance a and b respectivelyI j J X J identity matrixC m M X M circulant channel matrixp Signal-to-noise ratio^ Matrix transpose operation* Complex conjugate operation

Hermitian function E(.) Expectation operationmax Maximum function, return the maximum value in a setmin Minimum function, returns the minimum value in a set|.| Absolute Value functionI.II Frobenius norm

Chapter 1Introduction

1.1 Background

Opportunistic spectrum access (OSA) [1], first coined by Mitola et al. [2] under

the term “spectrum pooling” in cognitive radio terminology, promises tremendous

gain in improving spectral efficiency. The main objective of OSA is to offer the

ability of identifying and exploiting the under-utilized spectrum in an instantaneous

manner in a wireless device, without any user intrusion. This allows the wireless

devices to rapidly change their modulation scheme and communication protocol

so as to better and more efficient communication. The initial requirement of any

OSA device is to determine the spectrum availability. There are three possible

solutions for monitoring the spectrum availability proposed in the literature: 1)

through an ubiquitous connection to the database, 2) a dedicated standardised

channel to broadcast a beacon signal, 3) spectrum sensing [3]. Recently, Federal

Communication Committee (FOG) [4] considered database connection for inclusion

in the IEEE 802.22 standard [5]. However, it has been shown in [6] that the geo

location database solution might incur additional costs, e.g., signalling overhead,

scheduling complexity and database maintenance costs. Nevertheless, spectrum

sensing, thanks to its relatively low infrastructure cost, still receives more and more

attention. Therefore, proposed by the Office of Communications (Ofcom) [7] a

complementary application of both spectrum sensing and database connection can

provide a practical solution for enabling spectrum availability monitoring.

Spectrum sensing is a traditional topic in the scope of signal processing for mobile

communications. It is quite mature for carrier sense multiple access (CSMA) [8]

1.1. Background

based random access networks, i.e., “listen before talk” communication protocols.

In such communication protocols the transmitter uses feedback from the receiver

in order to evaluate the availability of a channel. Hence, the transmitter has to

detect the presence of an encoded signal from another station before attempting to

transmit. If a carrier is sensed, the station waits for the transmission in progress

to finish before initiating its own transmission. New challenges and problems arise

for spectrum sensing in flexible networks, e.g., cognitive radio, where it is required

to meet the following three requirements.

• Fast sp ec tru m sensing. Since it is not possible to transmit and sense

simultaneously at a particular frequency band, sensing has to be interleaved

with data transmission. Hence, the required observation time (or window)

should be as short as possible in order to maximise the overall throughput.

• H igh accuracy. The spectrum sensing device identifies vacant spectrum

bands by detecting presence of primary signal, i.e., by performing a binary

hypothesis test. With noise and fading available in any communication sys

tem, sensing errors are inevitable. For example, false alarm occurs when an

idle channel is detected as busy, and miss detection occurs when an occupied

channel is declared as idle. In the occasion of a false alarm a transmission

opportunity is overlooked, resulting in waste of the spectrum, while miss detec

tion leads to collision with primary users and hence, interference. According

to FCC the performance of spectrum sensing should reach the probability

of detection (PD) of more than 90% (or equivalently the probability of mis-

detection smaller than 10%) whilst the probability of false alarm (PFA) does

not exceed 10%.

• Low com plexity. The computational complexity of the sensing device should

be kept as low as possible in order to reduce the signal processing time, device

energy consumption as well as the infrastructure cost.

In addition, it is becoming increasingly demanding for delivering reliable spectrum

sensing in very low signal-to-noise ratio (SNR) range, such that heavily shadowed

signals can be identified.

1.2. Motivation and Objective

1.2 M otivation and Objective

A lot of research has been carried out in the area of local spectrum sensing, how

ever, it is realised that existing spectrum sensing schemes can hardly meet the

requirements of a fast and accurate spectrum sensing particularly in very low SNR

range, (considering that the target SNR for a reliable spectrum sensing sensitivity

is about —20 dB [5]) without introducing high complexity to the system. This ob

servation motivates us to develop a new local spectrum sensing scheme, which can

significantly improve the state-of- the-art and provides a practical solution, while

showing robustness towards physical impairments which exists in communication

systems, e.g., timing and frequency offsets and noise uncertainty issue.

Furthermore, with increase in the spectrum utilisation, spectrum scarcity increases.

This would call for spectrum sensing techniques that adopt an architecture to si

multaneously search over multiple frequency sub-bands at a time. However, the

literature of sub-band spectrum sensing is rather limited. This observation moti

vated us to develop a local wide-band spectrum sensing spectrum, specifically for

multi-carrier based environments, which can offer reliable performance in low SNR

environments without introducing high complexity to the systems.

1.3 Major Contribution

The major contributions of this thesis are considered to be two-fold.

I. Firstly, a novel spectrum sensing scheme namely, cluster-based differential

energy detection is proposed. It has several distinctive features including low

latency, high accuracy, reasonable computational complexity, as well as robust

ness to very low SNR. Since most of the current and future mobile networks

are multi-carrier based systems, thus the proposed approach has a wide impli

cation for practical applications. The key idea of the proposed scheme is to

exploit the channel frequency diversity inherent in high data-rate communi

cations using the clustered differential ordered energy spectral density (BSD).

Specifically, after the BSD computation, the clustering operation is utilised

to group uncorrelated subcarriers based on the coherence bandwidth to enjoy

a good frequency diversity. The knowledge of coherence bandwidth does not

need to be very accurate. Furthermore, making use of order statistics of the

1.3. Major Contribution

estimated ESD, we further increase the reliability of the sensing algorithm. In

order to exploit the second order moment diversity of the observed signal, a

differential operation is performed on the rank ordered ESD. When the chan

nel is frequency selective and the noise is white, the differential process can

effectively remove the noise floor resulting in ehmination of the noise uncer

tainty impact which is the main factor making energy detection reluctant [9].

At the final stage of the proposed scheme, the differential rank ordered ESD

within different clusters are linearly combined in order to further reduce the

effect of impulse/spike noise. Binary hypothesis testing is then applied on

either the maximum or the extremal quotient (maximum-to-minimum ratio)

depending on the wireless channel characteristics of the sensed environment.

More importantly, the proposed spectrum sensing scheme is designed to allow

robustness in terms of both time and frequency offset. In order to analytically

evaluate the proposed scheme, both PD and PFA are derived for Rayleigh

fading channels. The closed-form expression shows a clear relationship be

tween the sensing performance and the cluster size, i.e., channel coherence

bandwidth, which is an indicator of the diversity gain. Computer simulations

are carried out in order to evaluate the effectiveness of the proposed approach

and to compare the performance of the proposed scheme with state-of-the-

art spectrum sensing schemes where up to 10 dB gain in performance can be

observed.

II. The second contribution, is three-fold; 1) re-defining the objective of the

sub-band level spectrum sensing device to a model estimator, 2) deriving the

optimal model selection estimator for sub-band level spectrum sensing for

fixed and variable number of users along with a sub-optimal solution based

on Bayesian statistical modelling and 3) proposing a practical model selection

estimator with relaxed sample size constraint and limited system knowledge

for sub-band spectrum sensing applications in Orthogonal Frequency-Division

Multiple Access (OFDMA) systems. The proposed technique takes advantage

of the second order moment channel frequency diversity. More interestingly,

it does not require a priori knowledge of noise power and the propagation

channel gain, and is designed in such a way to show robustness towards energy

leakage. The proposed model selection based sub-band level spectrum sensing

approach is analytically evaluated through PFA and PD along with closed

form expression for the threshold value. Furthermore, computer simulations

are carried out in order to evaluate the effectiveness of the proposed scheme.

1.4. Thesis Organisation

1.4 Thesis Organisation

The rest of thesis is organised as follows, in chapter 2 the existing approaches

for local spectrum sensing are reviewed, a novel cluster-based differential energy

detection for spectrum sensing in multi-carrier systems is proposed in Chapter 3.

Furthermore, in Chapter 4 a Bayesian model based approach for joint sub-band

level spectrum sensing is presented. Finally, conclusions are drawn in Chapter 5

along with the set of possible future work.

1.5 Publication List

1.5.1 Journal Publications

P. Cheraghi, Y. Ma, and R. Tafazolli, and Z. Lu, “Cluster-based Differential Energy

Detection for Sensing Multi-carrier Sources in Very Low SNR”, IEEE Transaction

on Signal Processing, To appear, December, 2012.

P. Cheraghi, Y. Ma, and R. Tafazolli, and Z. Lu, “A Bayesian Model Based Ap

proach for Joint Sub-Band Level Spectrum Sensing”, IEEE Transaction on Signal

Processing, (Submitted, July 2012)

Z. Lu, Y. Ma, P. Cheraghi, and R. Tafazolli, Novel Pilot Assisted Spectrum Sensing

for OFDM Systems by Exploiting Statistical Difference between Subcarriers,” IEEE

Transaction on Communications, (First revision submitted, September, 2012)

P. Cheraghi, Y. Ma, R. Tafazolli and Z. Lu, “Spectrum Sensing in OFDM environ

ments: Advances and Challenges” , IEEE Journal on Selected Areas in Communi

cations, (To be submitted, October 2012)

1.5.2 Conference Proceedings

P. Cheraghi, Y. Ma, and R. Tafazolli, “A Novel Blind Spectrum Sensing Approach

for Cognitive Radios,” PCNET 2010, 21-22, Jun., 2010

P. Cheraghi, Yi Ma, and Tafazolli, R. , “Frequency-Domain Differential Energy

Detection Based on Extreme Statistics for OFDM Source Sensing,” 2011 IEEE

73rd Vehicular Technology Conference (VTC Spring), pp.1-5, 15-18 May 2011

1.5. Publication List

P. Cheraghi, Y. Ma, Z. Lu, and R. Tafazolli, “A Novel Low Complexity Differential

Energy Detection For Sensing OFDM Sources In Low SNR Environment,” 2011

IEEE GLOBECOM Workshops (GO Wkshps), pp. 378-382, 5-9 Dec. 2011

Z. Lu, P. Cheraghi, Y. Ma, and R. Tafazolli, “Extreme Statistics Based Spectrum

Sensing for OFDM Systems by Exploiting Frequency-Domain Pilot Polarity,” 2011

IEEE GLOBEGOM, 5-9 Dec. 2011

1.5.3 Patents

International Patent No: PCT/CB2012/050764 , “Spectrum Sensing of OFDM”,

filed 4 April 2012.

Chapter 2

State-of-the-art Spectrum Sensing

Techniques

The basic idea behind development of opportunistic spectrum access technologies

such as cognitive radio is to increase the spectral utilisation. This goal can be

achieved by identifying and utilising the spectrum holes; given the, the conventional

definition of spectrum hole: “a band of frequencies that are not being used by the

licensed user of that band at a particular time in a particular geographical area” [4].

Hence, based on this definition, identifying spectrum holes requires exploitation of

three dimensions of frequency, time and space by the spectrum sensing device. In

other words, the main task of any spectrum sensing device is to determine if the

frequency band of interest is occupied by the licensed user during a time slot within

a certain geographical area [10].

In what follows in this chapter, firstly, we will consider the original problem of

local narrow-band spectrum sensing and provide a comprehensive study of existing

solutions and secondly, the problem of local sub-band sensing is considered along

with the state-of-the-art sub-band spectrum sensing algorithms.

2.1 Problem Formulation for Narrow-Band Spec

trum Sensing

The general problem of spectrum sensing can be modelled as the binary hypothesis

testing with hypothesizes: when the frequency band of interest is vacant and

2.1. Problem Formulation for Nairow-Band Spectrum Sensing 8

no other users is utilising this frequency band; and 77i, when the frequency band

of interest is occupied by other users and not available for opportunistic usage.

Therefore, this problem mathematically reads as

(2.1)y s{t) + v{t), Hi

where y{t) denotes the received signal at the spectrum sensing device, v{t) is the

Additive White Gaussian Noise (AWGN) with zero mean, and s{t) represents the

signal transmitted by the existing active users. Hence, based on (2.1) spectrum sens

ing techniques can be evaluated through two classical metrics, namely probability

of detection (PD) and probability of the false alarm (PFA).

Mathematically, the probability of false alarm is defined by [11]

P F A A P r ( . f > A | % ) , (2.2)

where ^ denotes the test statistics and A is the detection threshold. As it can be

observed from (2.2), PFA refiects the probability of an event where the spectrum

sensing device reports an alarm when the signal is actually not being transmitted.

While, probability of detection is defined by

PD A P r ( . ^ > A I % ) . (2.3)

From (2.3) it can be concluded that PD reflects the probability of an event where

the spectrum sensing device reports an alarm when the signal is indeed there.

Spectrum sensing problem explained in (2.1) has been quite well investigated in the

literature, for which there are many approaches reported. In general the existing

spectrum sensing approaches can be divided into three main categories:

• Exploiting the energy difference

• Exploiting the stationarity difference

• Exploiting the distribution difference.

In the following sections each of the above categories is explained and compared in

detail.

2.2. Exploiting Energy Difference

2.2 Exploiting Energy Difference

The spectrum sensing approaches which fall into this category make a decision based

on the estimated energy of the received signal at the spectrum sensing device. The

most well-known spectrum sensing approach developed under this category is the

energy detection [11].

2.2.1 Energy Detection

Average DecisionMakingA/DFilter

Figure 2.1: Flow chart of conventional energy detection.

y(n)FFTA/D Average Decision

Making

Figure 2.2: Flow chart of frequency-domain energy detection.

Energy detection (radiometer) is the simplest spectrum sensing approach introduced

in the literature [11]- [15]. Due to its low computational complexity it is also the

most common technique used in practice. This approach can be thought of as a

blind spectrum sensing approach since it does not require a priori knowledge of

the signal being detected. Therefore, it is robust to variation of the signal being

detected and thus it is known as the optimal detector in the case where we assume

absolutely no deterministic knowledge about the signals being detected. Figure 2.1

and Figure 2.2 show block diagrams of the energy detection algorithm. As it can be

observed this technique can be implemented in either frequency or time domain. It

is worth mentioning that the conventional energy detection is implemented in time

domain, while the frequency domain version was later introduced for sub-band [16]

purposes. Frequency domain energy detection is usually used to simultaneously

detect the vacancy of several sub-bands, i.e. filter bank based energy detection [16].

The test statistics ^ ed for time domain energy detection is given by

(2.4)n = 0

2.2. Exploiting Energy Difference 10

where y(n) and N represents the sampled received signal and the observation length

respectively. While the test statistics in frequency domain energy detection is also

calculated as (2.4), with a minor difference that it is a function of ÿ = ^ n x n Y

given that ^ n x N denotes an N x N Discrete Fourier Transform (DFT) matrix and

y = [?/(0),2/(1), y{N - 1)]^.

Using central limit theorem [17] it can be observed that the alternative hypothesis

testing in (2.1) for energy detection can be expressed as [9]

(“ )

where and denote the noise and signal variance respectively. As it can be

observed from (2.5) the energy difference which exists between the two hypotheses

can be used as a metric for performing spectrum sensing. Thus, the availability of

the frequency band of interest using energy detection approach can be determined

using

^ E D >^ED (2 .6 )

where X e d denotes the energy detection threshold. It is clear from (2.5) and also

from the derivations in [11] and [9] that the threshold value is directly proportional

to the noise power. Thus, it can be concluded that the energy detection approach

requires accurate knowledge of the noise power, specifically in low SNR values, in

order to deliver a reliable performance.

However, noise power estimation error is unavoidable in practical systems, hence,

this will give rise to a phenomenon called noise uncertainty [11] [9].

2.2.2 Energy Detection Under Noise Uncertainty

In most communication systems noise is an aggregation of various independent

sources, i.e. thermal noise, interference due to nearby unintended emissions, etc.

Thanks to the central limit theorem [17], one can assume that noise at the receiver

is a Gaussian random variable. We should bear in mind that the error due to this

assumption will tend to zero as where N is the number of independent random

variables being summed up. In a practical scenario N is usually moderate therefore

this error can not be neglected, especially in low SNR environments. For some

2.2. Exploiting Energy Difference 11

Test Statistics (Estimated Energy)

III

Target Signal

. Noise power ^ uncertainty zone

Figure 2.3: The SNR wall phenomenon.

constant K the error due to this assumption can be modelled as [17]:

K\F v ix ) - A T { x ) \< - ^ (2,7)

where Fv{x) denotes the actual noise distribution and M{x) denotes a zero mean

Gaussian distribution. Nevertheless, most detectors operate under the assumption

of the received noise being Gaussian. The other main factors causing noise uncer

tainty is the temperature variations at the receiver which leads to inaccurate noise

power measurements. Hence, it can be concluded that if the SNR at the spectrum

sensing device is sufficiently low, there would be enough uncertainty in the noise

to render the energy detection useless. Consider the case with noise uncertainty

factor C/, since the energy detector only sees the the energy, distribution for the

noise uncertainty can be summarised in a single interval = [(l/f/)(j„2 , U a^.

It is further proved in [9] that as the observation length is increased, i.e., N oo,

the minimum operating SNR in which the energy detection can operate desirably

will converge to S N R I (U — (^))- This introduces a phenomenon called “«SiVR

w a ir . SNR wall of a spectrum sensing device is defined as the maximal SNR such

that for any SNR < SNRwaii detection is impossible for that particular detector.

The SNR wall for energy detector can be expressed as:

SNRwaii =C /2-1

U(2.8)

In the last four decades since the publication of [11], many solutions have been

2.3. Exploiting Stationarity Difference 12

developed to make energy detection more robust in terms of SNR wall (e.g. [18]-

[20]), yet the noise uncertainty problem in spectrum sensing approaches based on

the energy difference still exists. Hence, small noise power estimation error can

result in significant performance loss [21]. Consequently the noise power has to be

estimated dynamically. This is done by separating the noise and signal subspaces

using multiple signal classification algorithm [22]. Noise variance estimated by using

the value of the smallest eigenvalue of the incoming signals autocorrelation. In

[23] an iterative algorithm is proposed to find the near optimum threshold value.

The performance of energy detection technique over various fading channel models

has been investigated in [14], where it is shown that fading channels can have

a detrimental effect on the performance of the energy detection based spectrum

sensing.

2.3 Exploiting Stationarity Difference

Stationarity is defined as a quality of a process in which the statistical parameters

of the process do not change with time [24]. The spectrum sensing approaches

which fall into this category exploit the various stationarity difference which exists

between the noise and signal. In what follows the state-of-the-art spectrum sensing

approaches which fall into this category are explained in detail.

2.3.1 Second Order Cyclostationarity Detection

y(n)A/D Decision

MakingAuto

correlation FFT

Figure 2.4: Flow chart of second order cyclostaionary based detection technique.

The initial works of spectrum sensing through stationarity difference can be traced

back to work of Dandawate et al. in [25], where second order cyclostionarity is

employed. Cyclostationarity detection is based on exploiting the cyclostationarity

feature of the received signal [26]- [27]. This feature is caused by periodicity in the

statistics of the transmitted signal which could be a result of modulation, coding


or intentionally to assist spectrum sensing [28]- [30]. Process y{n) is considered sub

sense cyclostationary process if

Py = E [y{n)] = E [y{n -f V)] (2.9)

and

Ry (rj) = E [y{n)y* (n + 7])]

- E [y{n + V)y* {n + V + rj)] (2.10)

Vn, rj E Z . The smallest value of V for which (2.9) and (2.10) hold is called the

period. Being periodic, Ry{n) follows Fourier Series expansions over cyclic frequen

cies with the set of cycles A := [>c = A; = 0 ,1 , . . . , "P — 1]. Hence, the Fourier

coefficients also called cyclic auto correlations are related to Ry{n) using

1(^) Gxp { - p n x n ) . (2.11)

n = 0

As it can be observed from (2.11) the cyclic autocorrelation function at a given

cyclic frequency determines the correlation between spectral components of the sig

nal separated in frequency by an amount of hence, given sufficient observation

length the cyclic autocorrelation function of cyclostationary signals is nonzero only

for set of cycles which fall in the set A. Thus, one can determine the vacancy of

the frequency band of interest by analysing the cyclic autocorrelation function of

the received signal [25], i.e. using the second order cyclostationary feature. More

interestingly, cyclostionarity based detection can be employed to differentiate differ

ent types of signals [31]. Figure 2.4 illustrates the block diagram of a second order

cyclostationarity based detection technique.

In the above procedures it has been assumed that the cyclic frequency is known

at the receiver. However, this assumption may not be reasonable when the spec

trum sensing device is required to perform sensing in a sub spectrum band. Since,

the sensing device may not have knowledge of the period of all the users operating

and storing these information covering the whole geographical area in which the

sensing device (specifically for non stationary users) requires connection to an accu

rate location aided database. Thus, an exhaustive search is required to determine

the operating cyclic frequencies. This will increase the complexity of the detector

significantly and furthermore the detector will loss the ability of distinguishing be


tween the signal and the interference which also have cyclic characteristics. It is

shown in [29]- [25] that the cyclostationarity based scheme can trade latency with

high sensing reliability. Furthermore, It is less sensitive to the noise uncertainty

provided the knowledge of signals cyclic frequency [32].

2.3.2 Covariance Based Detection

yi(t)

DecisionMaking

Qmax

Qmin

Figure 2.5: Flow chart of covariance based detection.

To overcome the requirement of cyclic frequency information in cyclostationary

based detection while not suffering from the noise uncertainty problem which exists

with energy detection, the covariance based detection was introduced [33]. This

spectrum sensing method, as the name implies, is based on the estimated covari

ance matrix of the received signal, and utilises the correlation which exists in the

transmitted signal to determine the vacancy/occupancy of the frequency band of

interest. Hence, it is proved to be very effective when the transmitted signal is

highly correlated [33]- [34].

The correlation of the received signal samples can be due to many factors, e.g.,

modulation, multi-path fading, multiple receivers or can be intentionally introduced

by oversampling at the spectrum sensing device [35], while the noise samples are

independent. Moreover, since the covariance matrix of noise is determined by the

receiving filter at the receiver its structure is known to the spectrum sensing device,

allowing us to differentiate the two hypothesis in (2.1) [36].

Consider L to be the number of consecutive samples used for estimation the covari

ance matrix, i.e..

y{n) = [y{n) y { n - l ) . . . y { n - L + l)y (2.12)


hence, the estimated covariance matrices can be expressed as

\ \ ^ C y = C s + a % (2.13)

where

C , =E[s(n)s*(n)] (2.14)

Cy =E[y(n)y*(n)] , (2.15)

1l denotes an L x L Identity matrix and s(n) = [s{n) s{n — 1) . . . s(n — L + 1)]^.

Given that the noise samples are uncorrelated based on (2.13), it can be concluded

that considering the 'Ho scenario in (2.1), if the signal s(t) is not present, i.e. when

Cg = 0, the off-diagonal elements of Cy are all zero. On the other hand, if the

signal s(n) is present, i.e. 'Hi hypothesis in (2.1), Cy will no longer be a diagonal

matrix due to the correlation between the s(n) samples, resulting in off diagonal

elements. Hence, the vacancy of the frequency band of interest can be determined

based on the sum value of the off-diagonal elements. Based on [33], an effective test

statistic for this purpose would be ^ where

Ï Ê É (2.16)n = 0 m = 0

1Î2 = T E K " l' (2.17)

n = 0

given that Cm,n denote the element of the matrix Cy at the mth row and nth

column.

Considering the 'Ho scenario, = 1 while given 'Hi hypothesis 3 cd > 1 •

However, this is based on the assumption that L ^ oo, which is not a practical

assumption, hence in order to make a reliable decision 3 cd should be compared

to a threshold value which is function of observation length and the required PFA.

Details of the threshold setting can be found in [33]. It should be noted that the

performance of the covariance based detection is highly dependent on the correlation

of the received signal hence, in the extreme case where the received sample are

Independent and Identically Distributed (i.i.d.) this spectrum sensing approach

will fail.


2.3.2.1 Eigenvalue Based D etection

yi(t)

p ( t )

yM(t)

Iyi(n)

yzM

yM(n) u

Eigenvalue Qmax DecisionDecomposition Qm in Making

Figure 2.6: Block diagram of Eigenvalue based detection technique.

Following the development of the covariance based spectrum sensing approach the

eigenvalue based detection was introduced in [35]- [38]. The eigenvalue based detec

tion scheme exploits orthogonality between the signal subspace and noise subspace

using covariance matrix, i.e., second order stationarity features, to offer highly reli

able spectrum sensing [35].

Hence, based on this approach the vacancy of the frequency band of interest is

determined based on the fluctuation of the eigenvalues [24] of the covariance ma

trix, and hence many test statistics have been proposed to efficiently utilise this

fluctuation [35]- [38].

Amongst the well known test statistics for eigenvalue based detection is the ratio of

the maximum to minimum eigenvalues [35]. Let Pmin and Pmax denote the minimum

and the maximum eigenvalues of the covariance matrix Cy, respectively, hence, the

availability of the spectrum can be determined using

^E v = Aay. (2.18)

In [35], the asymptotic statistical characteristics of gmax and gmin under Hq have

been investigated and furthermore, closed from expression of the decision threshold

X e v for a given probability of false alarm has been provided.

An other popular test statistic used in eigenvalue based detection is the ratio of the

maximum eigenvalue to the average eigenvalue, i.e, ^ e v = where

(2.19)1=0


The asymptotic threshold value of the above test statistic is provided in [35]. How

ever, like covariance based detection, the eigenvalue based detection will fail if and

only if C5 = (TsIl i.e., the received signal samples are i.i.d. . However, the correla

tion between the received signal can be forced with employment of multiple receive

antennas or oversampling.

2.3.3 Matched Filtering

Matched filtering is known to be the optimum method for detecting signals when the

transmitted signals air interface is completely known to the sensing device [39]. The

main advantage of matched filtering is low latency and computational complexity.

Match filtering requires a very short time to achieve the desired probability of false

alarm or probability of detection as compared to all the existing detection methods

while introducing a linear complexity even in a very low SNR environment since

it maximise the received SNR at the sensing device [39]. On the other hand the

shortcoming of the match filtering technique is that the spectrum sensing device

needs to demodulate the received signal prior to determine the vacancy of the

frequency band of interest. Hence it will introduce to main concerns 1) requiring

a perfect knowledge of the transmitted signals signalling feature, 2) security issue,

since this allows the spectrum sensing device to have access to the transmitted

message.

Since the spectrum sensing device should be able to detect all the possible signals

transmitted within the bandwidth of interest, the implementation complexity of

such detection technique is impractically large, due to perfect knowledge of all the

available signalling information of users [40]. One of the other disadvantages of

match filtering is large power consumption as all the possible receiver algorithms

needs to be executed for spectrum availability decision making.

2.3.3.1 Pilot Based M atch Filtering D etection

In practical communication systems, pilots are usually transmitted periodically for

time or frequency synchronisation applications, channel estimation, etc. These pi

lots, if known to the spectrum sensing device, can be utilised for coherent detection

2.4. Exploiting The Distribution Difference 18

y(n)CorrelationA /D Decision

Making

Figure 2.7: Block diagram of the pilot based detection technique.

of the transmitted signal with the aid of match filtering. Therefore, it works even un

der a very low SNR region. Furthermore, it has lower complexity and latency than

statistics based cyclostationary and covariance based detection while overcoming

the noise uncertainty problem. Moreover, it does not require demodulation of the

transmitted signal as the conventional match filtering since orthogonal to the data

and can be considered independently. Therefore, pilot based coherent detection is

always one of the preferred spectrum sensing schemes in practice.

In the scenarios where the pilot structure is known to the spectrum sensing device

the optimum detection technique would be match filtering. Therefore, the test

metric can be expressed as:

S^PM = ~^{y{n)sl{n)} (2 .20)

where Sp{ t ) denotes the known pilot signal. Hence, a decision can be made using,

^PM ^PMi where \p m denotes the threshold value to satisfy the required

probability of false alarm.

In (2.20) perfect synchronisation between the sensing device and the transmitter is

assumed, while this condition is not feasible in practice. Hence, the sensing device

has to perform an exhaustive search to find the timing offset which maximises the

^PM value. Recently, various robust pilot-based coherent detection schemes have

been proposed for spectrum sensing applications [41]- [43] .

2.4 Exploiting The Distribution Difference

Given that in almost all communication system models, noise is assumed to be ad

ditive white and Gaussian, one can determine the vacancy of a particular frequency

band by observing the difference of the received signals distribution and that of the

AWGN. Based on this feature a number of well known spectrum sensing algorithms

have been proposed, which are explained in this section.


2.4.1 Entropy Based Detection

y(n)Partition Decision

MakingFFTA/D EntropyEstimation

Figure 2.8: Flow chart of entropy based detection.

In information theory, entropy is a measure of the uncertainty associated with a

discrete random variable. The term by itself usually refers to the Shannon entropy,

which quantifies the information conveyed in a message [44]. Recently, entropy based

detection approaches have been employed for spectrum sensing applications [45] [46].

Entropy based spectrum sensing can be thought of an approach which exploits the

distribution difference in order to determine the vacancy of the frequency band of

interest.

In order to allow robustness to noise uncertainty issue, the entropy based detection

makes a decision based on the estimated entropy of the measured signal in the

frequency domain with the probabihty space partitioned into fixed dimensions. This

is due the fact that the entropy of the received signal in the time domain is related to

the signal power and is sensitive to noise uncertainty [45]. Hence, the test statistic

for this spectrum sensing approach can be expressed as [46]

L-l= Xe b ,

i=0(2.21)

where L denotes the dimension of probability space, N is the number of Discrete

Fourier Transform (DFT) points, k{ is the total number of occurrences at the

probability state, and Xe b is the threshold value used for decision making. As

suming that the estimated noise entropy follows a Gaussian distribution, the value

of Xe b can be easily calculated based on the desired PFA and value of L [46].

However, the entropy based detection will fail to deliver accurate results, when the

transmitted signals also follow a Gaussian distribution and since the convergence to

normality could be extremely slow, this approach will require relatively high obser

vation length. However, this would is in contradiction to achieving channel capacity,

which benefit from Gaussian noise like transmit signals.


2.4.2 Kurtosis Based Detection

y MA/D Decision

MakingK urtosis

Estim ation

Figure 2.9: Flow chart of kurtosis based detection.

In statistics, kurtosis is a descriptor of the shape of a probability distribution, i.e.,

it is a measure of the “peakedness” of the probability. Hence, the kurtosis based

detection was introduced to exploit the non-Gaussianity of communication signals

in order to determine the availability of the frequency band of interest [47]- [48]. For

example for randomly occurring signals that produce non Gaussian distributions,

the kurtosis estimate can be less than 3 or it can have a value much greater.

This scheme features excellent accuracy at the price of large latency due to higher-

order statistics. A critical point is that the sensing performance degrades signifi

cantly when signals are approximately Gaussian. Kurtosis is defined by the ratio

of the expected value of the fourth-order central moment and the square of the

expected value of the second-order central moment. Hence, the test statistic of the

kurtosis based detection can be formulated as

E \{y{n) - f l y ) ] ^KB = — -------------- ^

E [(y(n) - fly) J(2 .22)

This scheme features excellent accuracy at the price of high computational complex

ity due to higher-order statistics. Furthermore, the convergence to normality could

be extremely slow, and the sample estimate of the kurtosis can deviate substantially

from its true value even with a large number of observations. Thus, for moderate

sample sizes which is a prerequisite for any spectrum sensing device, the kurtosis

test cannot be expected to be accurate.

2.5. Summary of The Narrow-Band Spectrum Sensing Approaches 21

2.5 Summary of The Narrow-Band Spectrum Sens

ing Approaches

In the above discussion, we have introduced various state-of-the-art narrow-band

spectrum sensing techniques. As explained, various spectrum sensing techniques

have different advantages/ disadvantages and hence are applicable in different sens

ing scenarios. In general, the existing spectrum sensing approaches can be divided

into three main categories:

• E xplo iting energy difference. The most well-known spectrum sensing

approach developed under this category is the energy detection [11]. The

energy detection is recognized as a blind sensing scheme with advantages such

as low complexity and low latency. However, it is very sensitive to the noise

uncertainty such that its performance is limited by the SNR wall [9]. In the last

four decades since the publication of [11], many solutions have been developed

to make energy detection more robust in terms of SNR wall (e.g. [18]- [20]),

yet the noise uncertainty problem in spectrum sensing approaches based on

the energy difference still exists.

• Explo iting s ta tio n a rity difference. The initial works of spectrum sensing

through stationarity difference can be traced back to work of Dandawate et

al. in [25], where second order cyclostionarity is employed. The cyclosta

tionarity based scheme can trade latency with high sensing reliability. It is

less sensitive to the noise uncertainty provided the knowledge of signals cyclic

frequency [32]. To overcome the requirement of cyclic frequency in cyclosta

tionary based detection while not suffering from the noise uncertainty problem

which exists with energy detection, the covariance based detection was intro

duced [33]. This spectrum sensing method, utilises the correlation which exists

in the transmitted signal to determine the vacancy/occupancy of the frequency

band of interest. However, the performance of this approach degrades dramat

ically as the correlation of the transmitted signal decreases. Matched filtering

is known to be the optimum method for detecting signals when the transmit

ted signals air interface is completely known to the sensing device [39]. The

main advantage of matched filtering is low latency and computational com

plexity. However, this approach requires perfect synchronisations between the

transmitter and the spectrum sensing device. Matched-filtering pilot based

2.5. Summary of The Narrow-Band Spectrum Sensing Approaches 22

Approach Air-InterfaceInformation Synchronisation Computational

Complexity Latency

Energy Detection X X Low Low

Cyclostationarity Detection / / High High

Covariance Based Detection X X Medium Low

Eigenvalue Based Detection X X High Medium

Matched Filtering / / Low Low

Pilot Based Matched Filtering / / Low Low

Entropy Based Detection / X Medium Medium

Kurtosis Based Detection / X High High

Table 2.1: Summary of the state-of-the-art local narrow-band spectrum sensing approaches.

detection, given the knowledge of pilot symbols and reasonably good timing

and frequency synchronizations, exploits the cyclostationary property of the

pilot symbols, to deliver fast and reliable sensing. The eigenvalue-based de

tection scheme exploits orthogonality between the signal subspace and noise

subspace using second order stationarity features to offer highly reliable spec

trum sensing [35]. However, it often needs the support of multiple antennas,

and the subspace decomposition costs cubic complexity.

• Explo iting th e d istrib u tio n difference. Given that in almost all commu

nication system models, noise is assumed to be additive white and Gaussian,

one can determine the vacancy of a particular frequency band by observing

the difference of the received signals distribution and that of the AWGN. An

example of such approaches would be the kurtosis-type scheme, which exploits

the non-Gaussianity of communication signals [47]- [48]. This scheme features

excellent accuracy at the price of large latency due to higher-order statistics.

A critical point is that the sensing performance degrades significantly when

signals are approximately Gaussian. Entropy based spectrum sensing can be

thought of an approach which also benefits from this property [45], where

the probability space is partitioned into fixed dimensions and the Shannon

entropy is employed as the information measure of the received signal as the

test statistic.

2.6. Sub-Band Level Spectrum Spectrum Sensing 23

2.6 Sub-Band Level Spectrum Spectrum Sensing

With increase in the spectrum utilisation, spectrum scarcity increases. This would

call for spectrum sensing techniques that adopt an architecture to simultaneously

search over multiple frequency sub-bands at a time, while meeting the mandatory

requirements of spectrum sensing, i.e.,1) low latency, 2) high reliability and 3) low

complexity.

However, the literature of sub-band spectrum sensing is rather limited at this time.

In this section we will provide a system model for sub-band level spectrum sensing

and further explain the state-of-the-art sub-band level spectrum sensing techniques

in detail.

2.6.1 Sub-Band Level Spectrum Sensing System Model

Consider a communication system operating over a sub-band channel that is di

vided into K non-overlapping sub-bands, e.g., multi-carrier systems. However, in

a particular geographical region within a certain time frame only I number of the

sub-bands are utilised by the users, where I < K . Thus, {K — I) sub-bands are

available for opportunistic access. The essential task of the spectrum sensing device

is to determine the availability of these {K — I) sub-bands.

Let y{n) denote the received sub-band signal at the spectrum sensing device. Hence:

K - l

2/W = ^ AfcSfc(n) 4- v{n) (2.23)fe=o

where Aq is the indicator function which denotes the presence of the transmitted

signal q. The opportunistic user needs to determine which of the spectrum bands

are unoccupied, in order to utilise them efficiently. Based on (2.23) a number of

solutions were proposed in the literature which are fully explained in the following

sections.

2.6.2 Filter-Bank Based Spectrum Sensing

The Filter-bank architecture allows sub-band sensing with the aid of multiple narrow

band, band-pass filters [16] [49]. Filter banks are often implemented based on a


prototype Iter. The prototype filter is a lowpass Iter that is also used to realise the

first sub-band of the filter bank. Other bands are realised through repetition of the

prototype filter. Hence, all the N sub-bands of interest share the same structure.

The implementation of an spectral estimator that uses a filter bank for signal anal

ysis is as follows: 1) the input process is passed through a bank of filters and 2) the

output power of each filter is measured as an estimate of the spectral power over the

associated sub-bands and finally the vacancy of each sub-band is determined based

on the estimated power of that particular sub-band. However, like conventional

energy detection this approach will face the noise uncertainty problem.

It is shown in [49] that the filter bank based spectrum sensing performs significantly

better in filter bank-based multi-carrier communication systems, since the same

filters can be utilised for sensing purposes. Hence, in such systems, channel sensing

is done at virtually no cost. This is only possible given that all the users within

the geographical area of interest share the same air-interface and furthermore the

opportunistic user also employs the same air interface. However, in a more general

case where users may have different signalling format the filter-bank approach will

result in increased number of components and energy consumption.

2.6.3 Joint Multi-Band Detection

In order to improve the performance of the filter bank detection the joint multi

band detection, was proposed in [50]. This approach jointly optimises a bank of

multiple narrow band detectors to improve the aggregate opportunistic throughput

of the opportunistic users while limiting the interference to the existing users. In

particular, the joint multi-band detection reformulates the original problem of sub

band spectrum sensing into a class of optimisation problems, where the objective

is to maximise the aggregate opportunistic throughput in an interference-limited

network given the opportunistic rate and interference penalty on each sub-band are

known to the spectrum sensing device. Hence, the optimisation problem can be

summarised by [50]


[1 — PFA (7 )]

K - l

s.t. : ^ 2 ^ [1 ~ FD j(7i)] < €j , j = 1 ,2 ,. . . Ji=o (2.24;

PFA (7 ) < a

[ 1 - P D (7 )]< /S

where r = [ r i ,r2, . . . ,tk]'^ is a vector with the throughput achievable over all K

sub-bands, 7 = [71,72, - - - , 7k]^ is the vector denoting the threshold value for all

sub-bands, 1 —a = [ l - a i , l — o:2,---,l — ock]^ and f3 = [Pi, /32, . . . , Pk ]" are the

minimum limit for opportunistic spectral utilisation required from the spectrum

sensing device and the upper limit for the interference introduced by the oppor

tunistic users, respectively.

Hence, the threshold setting in this approach is in a such a way to firstly assure

that the sub-band with a higher opportunistic rate has a higher threshold . In

other words reduce PFA for the corresponding sub-band to ensure best possible use

by the opportunistic users. Secondly, the higher priority sub-band, i.e., sub-bands

carrying important messages, have a lower threshold resulting in smaller PD in order

to prevent opportunistic users interference. Finally, a little compromise on the sub

bands carrying less important information which might boost the opportunistic rate

considerably. Thus, in the determination of the optimal threshold for each sub-band,

it is necessary to balance the channel conditions, the opportunistic throughput,

and the relative priority of each sub-band. It has been shown in [50] that the

joint multi-band detection can improve the performance of the filter-bank spectrum

sensing significantly, and that the performance of this approach can be improved

considerably by further exploiting the spatial diversity, i.e., cooperation between

the spectrum sensing devices.

However, this technique requires the knowledge of noise power and the squared

values of the channel frequency responses, which makes this approach only practical

in fixed wireless networks, i.e., TV broadcast bands.

2.6.4 Sequential Multi-Band Detection

In, [51] a sequential detection scheme has been developed for multi-band spectrum

sensing. This approach employs a bank of sequential probability ratio tests [52],

i.e., one per sub-band. The sequential probability ratio test has a very simple


structure where the likelihood ratio of the observed samples is tested against two

thresholds. The sequential probability is known to minimise the average sample

number amongst all detectors given the PD and PFA requirements of the system.

Hence, this algorithm can be particularly useful in delay sensitive applications. The

hypothesis testing for this approach for the sub-band can be expressed as

logB^ < < log take the next sample

(2.25)

where the test statistic ^SM is the likelihood ratio estimated from the received

signal and the threshold values are related to the false alarm probability and the

miss detection probability, i.e., [52]

PD*PFA^

1 - P D '1 - PFAk' (2.26)

However, the key challenge associated with this detector is that the parallel sequen

tial probability tests do not yield the same sample sizes. This is due to the fact

that the observation length is variable which depends on the random received sig

nal. Thus, the overall sensing delay will be considered as the largest detection delay

among those of the parallel detectors, until the set of bands that can support the

requested rate is discovered.

2.6.5 Wavelet Based Detection

y(n) WaveletTransfrom

PSDEstimation

DecisionMaking

EdgeDetectionA/D

Figure 2.10: Block diagram of the wavelet based detection technique.

The wavelet based spectrum sensing is able to perform sub-band sensing with the

aid of edge detection [53] [54]. Assuming that the power spectral characteristic is

smooth within each sub-band but exhibits a discontinuous change between adjacent

sub-bands, wavelet based detection has been proposed to identify and locate the

spectrum holes by exploiting the irregularities within the estimated Power Spectral


Density (PSD) [24] with the aid of the wavelet transform, an attractive mathemati

cal tool for analysing singularities and irregular structures of signals. Wavelet based

detection has proved useful for fast coarse spectrum sensing based on a number of

non-stationary samples, by making use of the signals non-stationarity features.

The wavelet based detection has been developed under four main assumptions: 1)

The total bandwidth for detection is known to the spectrum sensing device , 2)

The number of licensed users are unknown to the spectrum sensing device, 3) The

PSD of all occupied sub-band is smooth and almost flat, 4) The noise is AWGN, i.e.

noise process has a flat PSD within the whole observed bandwidth.

Hence, once the region of support is determined, wavelet-based approach will firstly

estimate the PSD of the received signal and determine the number of sub-bands and

the corresponding frequency boundaries. Later, the PSD for each sub-band will be

employed to determine the vacancy of the estimated sub-bands. As a result, the

wavelet based detection is also known as the wavelet based edge detection. Hence,

the availability of the sub-band can be determined using the

^WB — y - V - t d / ^WB (2.27)Jk — Jk-1 '//k-l

where fk — f k - i denotes the estimated frequency boundaries of the k*^ sub-band

obtained using the wavelet transform [54] [53]. Based on the above, one of the

main advantages of the wavelet based detection is that it does not require any prior

knowledge about the signals features. However, the most important limitation

of this spectrum sensing approach is determining the correct smoothing function

(mother wavelet) for the wavelet transformation. Even though some common fea

tures are shared by most mother functions, some can perform better than others

in a given environment. Hence, in order to obtain the best possible results in the

wavelet based detection, the specific wavelet family should be designed based on

the characteristics of the transmitted signal.

2.6.6 W igner-Ville Based Detection

The Wigner-Ville based spectrum sensing [55] derives a greyscale image of the time-

frequency description of the received signal through the Wigner-Ville transform,

and similar to wavelet based detection with the aid edge detection is able to detect

occupied frequency bands.

2.7. Summary of The Sub-Band Spectrum Sensing Approaches 28

With the aid of the Wigner-Ville transform, it is possible to show the spectral

components of a signal with respect to the time variable and therefore have a bi-

dimensional description of the perceived signal [56]. The resulting image from the

Wigner-Ville based detection shows the spectrum occupancy in both time and fre

quency, marking the occupied zones with higher brightness. Hence, such zones are

to be avoided by the opportunistic user, who, thanks to an edge detection, is able

to detect the vacant sub-bands.

Such two dimensional strategies such as Wigner-Ville and wavelet based detection,

tend to improve the performance of the spectrum sensing device with respect to sin

gle dimensional approaches due to the phenomenon known as “uncertainty relation

ship” which describes the trade off between the spectral and temporal resolution.

At the final stage, the measured energy level is employed as the slot availability

criterion, entailing that slots are considered occupied even when they present high-

energy even in a narrow spectral and time components. However, these approaches

may suffer from noise uncertainty problem, due to use of energy detection.

2.7 Summary of The Sub-Band Spectrum Sensing

Approaches

As explained in this section, different spectrum sensing scenarios demand for differ

ent sub-band spectrum sensing approaches based on their requirements. Amongst

the existing practical solutions, there are filter-bank [49] and wavelet [54] based

spectrum sensing techniques. The Filter-bank architecture allows sub-band sens

ing with the aid of multiple narrow bands, which results in increased number of

components and energy consumption. However, the filter-bank approach is one

of the preferred solutions when the spectrum sensing device does not have any

a-priori knowledge about the signalling information. Wavelet based spectrum sens

ing exploits the multi-resolution features of the wavelet transform to estimate the

power spectral density. With the aid of edge detection, the spectrum band of in

terest is divided into a number of sub-bands. This technique is particularly useful,

when there are limited number of non-stationary samples. Fine spectrum sensing

is further required, in order to determine the vacancy of specified frequency sub

bands. The Wigner-Ville based spectrum sensing [55] derives a greyscale image

of the time-frequency description of the received signal through the Wigner-Ville

transform. Similar to the wavelet based detection, with the aid of edge detection.

2.8. Summary 29

Approach Air-interfaceInformation Synchronisation Computational

Complexity Latency

Filter Bank X X Low Low

Joint M ulti-Band Detection / / High High

Sequential M ulti-Band Detection / X High Low

Wavelet Based Detection X X Medium Low

W igner-Ville Based Detection X / High High

Table 2.2: Summary of the state-of-the-art local sub-band spectrum sensing approaches.

it is able to detect occupied frequency bands. Recently, a multi-band joint detec

tion for spectrum sensing has been introduced in [50], where spectrum sensing is

performed through a class of optimisation problem with the objective of improv

ing the aggregate opportunistic throughput of the opportunistic spectrum access

user while limiting the interference to the other users in the system. However, this

technique requires the knowledge of noise power and the squared values of the chan

nel frequency responses, which makes this approach only practical in fixed wireless

networks, i.e., TV broadcast bands. While [51] investigates multi-band spectrum

sensing algorithm, which supports quality-of-service traffic. In particular, [51] pro

poses a sequential sensing, where a bank of sequential probability ratio tests are

run in parallel to detect the availability of sub-bands, while ensuring a fixed min

imum rate for the opportunistic user. This approach is based on the assumption

that the propagation channel between the transmitter and the spectrum sensing

device is fixed and deterministically known to the opportunistic users, making this

multi-band spectrum sensing approach also only suitable for fixed networks.

2.8 Summary

Spectrum sensing device needs to continuously monitor the spectrum for possible

presence of the vacant frequency bands. In this chapter, we have discussed various

local spectrum sensing techniques that exploit null, minimal, or full knowledge of

the transmitted signal characteristics. We have also addressed the state-of-the-

art local sub-band spectrum sensing approaches, which can effectively improve the

overall spectrum utilisation by simultaneously search over multiple frequency sub

bands at a time. Furthermore, it is realised that existing schemes can hardly meet

the requirements of a fast and accurate spectrum sensing, particularly, in low SNR

range, (considering that the target SNR for a reliable spectrum sensing sensitivity

is about -20 dB [5]) without introducing high complexity to the system. This

2.8. Summary ' 30

observation motivates us to develop a new local spectrum sensing scheme, which

can significantly improve the state-of- the-arts and provides a practical solution.

Chapter 3Cluster-Based Differential Energy

Detection for Spectrum Sensing in

Multi-Carrier Systems

3.1 Introduction

In this chapter a novel spectrum sensing scheme namely, cluster-based differential

energy detection is presented. The proposed spectrum technique has several dis

tinctive features, including low latency, high accuracy, reasonable computational

complexity, as well as robustness to low SNR. The proposed scheme is specially de

signed for sensing multi-carrier sources since most of the current and future mobile

networks are multi-carrier based systems. Hence, it has a wide range of practical

applications.

The key idea of the proposed scheme is to exploit the channel frequency diversity

inherent in high data-rate communications using the clustered differential ordered

energy spectral density (ESD). Specifically, after the ESD computation, the cluster

ing operation is utilised to group uncorrelated subcarriers based on the coherence

bandwidth to enjoy a good frequency diversity. The knowledge of coherence band

width does not need to be very accurate (here we employ the reciprocal of the

maximal channel delay). Furthermore, making use of order statistics of the esti

mated ESD, we further increase the reliability of the sensing algorithm.

31

3.2. System Model and Problem Formulation 32

In order to exploit the second order moment diversity of the observed signal, a

differential operation is performed on the rank ordered ESD. When the channel

is frequency selective and the noise is white, the differential process can effectively

remove the noise floor resulting in elimination of the noise uncertainty impact which

is the main factor making energy detection reluctant [9]. At the final stage of the

proposed scheme, the differential rank ordered ESD within different clusters are

linearly combined in order to further reduce the effect of impulse/spike noise. Binary

hypothesis testing is then applied on either the maximum or the extremal quotient

(maximum-to-minimum ratio) depending on the wireless channel characteristics of

the sensed environment. More importantly, the proposed spectrum sensing scheme

is designed to allow robustness in terms of both time and frequency offset.

In order to analytically evaluate the proposed scheme, both PD and PFA are derived

for Rayleigh fading channels. The closed-form expression shows a clear relationship

between the sensing performance and the cluster size, i.e., channel coherence band

width, which is an indicator of the diversity gain. Computer simulations are carried

out in order to evaluate the effectiveness of the proposed approach and to compare

the performance of the proposed scheme with state-of-the-art spectrum sensing

schemes where up to 10 dB gain in performance can be observed.

3.2 System M odel and Problem Formulation

3.2.1 Multi-Carrier Systems

Transmitted Signal

A general framework of multi-carrier systems has been presented in [57]. The trans

mitted signal can be expressed in the matrix form, Xfc = ^Sk , where is an J x 1

transmitted signal block, Sfc is an M x 1 information-bearing symbol block with the

covariance o-^Im (M stands for the number of subcarriers, and Im for the identity

matrix of size M ), Ÿ is an J x M (J > M) tall pre-coding matrix with full column

rank, and subscript k is the block index. There are two conventional approaches

for implementing the pre-coding matrix Ÿ, i.e.,

C P : ^ (3.1)

ZP: ^ A 0^"^$ (3.2)


where !F is the M x M normilized discrete Fourier transform (DFT) matrix [24],

in (3.2) is formed by collecting the last ( J — M) columns of JF, $ is an M x M full

rank matrix. This work is focused on the cyclic prefix (CP) based system since it has

been widely deployed in practical networks due to its advantages, e.g., eliminating

inter-symbol interference and handling multi-path channels [58]. Nevertheless, it

is shown in Section 4.6.3 how the proposed spectrum sensing scheme can be easily

extended to the zero-padding (ZP) based system in (3.2).

Signal Analysis at the Sensing Device

Consider a wireless device sensing a particular frequency band, in the absence of

the multi-carrier signal, the device can only receive noise, otherwise, it receives a

signal distorted by the frequency-selective channel (denoted by h), timing offset

(denoted by e), frequency offset normalized by the subcarrier spacing (denoted by

e), and additive white Caussian noise (denoted by v). Indeed, there are many other

distortions such as phase noise and non-linear distortions due to imperfect electro

components [58]. However, we will focus on those major physical distortions, (i.e.,

frequency selective channel, noise, timing and frequency offsets) in order not to

diverge the presentation of the key concept.

Given that the spectrum sensing device knows some key parameters of the operating

air-interface such as the block length J , the number of subcarriers M, and the block

duration Tb, the received continuous-time signal is sampled at the sampling period

of Ts = (Tb)/(J). Hence, the timing offset can be expressed into two parts: the

integer timing offset = [(e/7^)J and the fractional timing offset (e — rie), where

[•J denotes the floor operator. It is understood that the fractional timing offset can

be incorporated into the channel impact. Hence, the discrete-time equivalent form

of the received signal is [59]

cYn — ^ ^ y ^t^kJ-\-n—t—Til "bî/n, (3.3)

1=0

where C denotes the upper bound of channel order {C < J — M), O" = exp ,

and the block index k = [(n)/(J)J. Consider an J x 1 vector

Yk = [ykJ+i,YkJ+2,- ,YkJ+j]'^, where (-)^ stands for the matrix transpose, then


(3.3) can be expressed as the matrix form

Yk = flk(A(ne)xk + V(ne)xfc-i) + v&, (3.4)'------V------ '

IBI

where flk — diag{SH(*’ +^), Vk is the corresponding noise vec

tor, A(ne) is a lower triangle channel matrix, and V(ne) is a upper triangle channel

matrix. The detailed layout of both channel matrices depends on the timing offset

He, and the term 'V(ri£)xk-i is the inter-block interference (IBI).

3.2.2 Effect of Second Order Moment

The second-order moments of yk in (3.4) can be computed as below

E (y ty f )=

+ < 7 jV (% )* * " V " (n ,)+ V o Ij , (3.5)

where Afo is the noise power. It is observed that the above result is constant with

respect to the block index k, and the carrier frequency offset (CFO) impact has been

completely removed. This means that the second-order moments of y„ has a period

of J. Furthermore, highly likely the diagonal entries of E(yfcyj^), for n = 1,2,..., J ,

are not constant with respect to the index n due to the frequency selectivity nature

of the communication channels in high data rates.

Remark: In practice, the processing (3.5), i.e., ensemble average, is replaced by the

time average

H y k V k ) (y&yf)' (3-6)fc=0

where K is the number of observation windows and C”' is an J x J matrix. The

above substitution is due to limited processing time available. This would result

in fluctuation of the ESD of AWCN (generation of impulse or heavily tailed noise).

Hence, affecting the performance of any spectrum sensing algorithms regardless of

what scheme is being employed.

3.2.3 Statement of The Spectrum Sensing Problem

The general problem of local spectrum sensing is modelled as the binary hypothesis

testing with hypothesises: Tlo, when the signal is absent; and Hi, when the signal

3.3. Cluster-Based Differential Energy Detection 35

is present.

The specific problem of interest in this chapter is: given the noise to be white

Gaussian, and independent of the multi-carrier signal which is second-order

cyclostationary with the period of N, what is the efficient way to determine the

presence of the signal formulated in (3.3) specifically, in low SNR range?

It is understood that the random sequence x^ due to it cyclostaionairty character

istics has the property E(xfcX^) = E{xk+N^k+N) [24], and the random sequence

Vk satisfies E(vjkVj^) = Afo. In other words, in the absence of signal, i.e.. Ho, the

random sequence of observation {y^} is a white process, otherwise, i.e.. Hi, a second-

order cyclostationary process. Furthermore, as mentioned in Section 3.2.2, making

use of second-order moment of {y^} results in overcoming the CFO phenomenon.

Hence employing second order moment yields the hypothesises

and

where and (a) denote a central Chi squared distribution with 2K degrees

of freedom and a non-central Chi squared with non-centrality factor a, respectively.

Thus, if the SNR of the received signal, i.e., was fairly high, the hypothesis test

in (3.7) will be trivial. The problem of interest in this chapter is to consider spectrum

sensing in very low SNR which, given (3.7), is a rather challenging problem.

3.3 Cluster-Based Differential Energy D etection

3.3.1 Sensing of CP-Based Multi-Carrier Signals

Form an M X M matrix C”' by collecting the last M columns and rows of C”'

defined in (3.6). Consider the special case where the timing offset ric = 0, i.e., Cy.

Due to the effect of CP, the second term at the right hand of (3.5) vanishes, i.e.,

IBI is removed, and the residual term can be written as

cl = + JVoIm, (3.9)


Receive^Signal

Order

Order

Order Differentiation

Differentiation

Differentiation

Figure 3.1: Block diagram of the cluster-based differential energy detection algorithm.

where Cm is an M x M circulant channel matrix defined in [57]. Then, an M-point

DFT operation is performed on Cy leading to

c j â :FCy = + JVoI m , (3.10)

where T>m — ^ C m ^ ^ is an M x M diagonal matrix, whose diagonal entries are

in fact the channel frequency response (denoted by hm)- Let 0 ^ be the mth row

vector of 0 . Hence, the mth diagonal entry of Cy reads as

(3.11)

where || • |p denotes the Frobenius norm. Note that, the above expression is equiva

lent to the ESD computation. In many multi-carrier systems, such as OFDM, multi

carrier code division multiple access (MC-CDMA), and single-carrier frequency di

vision multiple access (SC-FDMA), the term ||0 ^ |p is normalized [58]. Therefore,

the ||0 ^ |P term in (3.11) can be ignored leading to

[Cyj = + Vo, m = 1, 2, • • • , M. (3.12)

Based on (3.12), we propose a cluster-based differential energy detection technique

with the following steps. An overview of the proposed technique is illustrated in

Figure 4.1.

81) Group [Cy]m, for m = 1,2, ...,M , into B, where B — M /L , clusters with


each cluster having L elements. The mathematical form of each cluster can

be expressed by

- [py]%, Py]t+B ,- , [Cy]i+(L-1)BV', « = 1,2,..., B. (3.13)

The grouping criteria are: cl) elements within each cluster are statistically un

correlated or weakly correlated; c2) all clusters are almost identical or strongly

correlated in the noiseless case, i.e., q i = q2 = ... = qg. The criterion cl) is

to assure that the channel gain within each cluster is sufficiently selective since

the proposed differential energy detection technique aims to take advantage of

the spectrum fluctuation induced by channel frequency selectivity. The crite

rion c2) is mainly for the purpose of de-noising through linear combination of

all clusters on the step S3). Here, the noise is mainly referred to the residual

noise after the second-order statistics (3.6).

In order to fulfil the criteria cl) and c2), we first divide the whole frequency

band into L sub-bands with each accommodating B subcarriers. The mathe

matical form of the Ith sub-band is expressible as:

Pi - [PJ](i-i)B+i>Py](i-i)B+2v I = 1,2,... ,L. When the band

width of each sub-band is smaller than the channel coherence bandwidth, all

elements in p; are highly correlated or approximately identical. Moreover,

we can configure the parameter B such that the bandwidth of the group

[p^, [êy];g+i]^ is larger than the coherence bandwidth such that any two

adjacent sub-bands are weakly correlated or even statistically independent.

With the above configuration to be satisfied, the cluster q can be generated

through block wise interleaving of pz, / = 1,2,..., L.

The above statement implicitly indicates that the clustering process requires

the knowledge of the coherence bandwidth which can be computed assuming

the availability of accurate channel models. In case the accurate channel

models are not available at the sensing device, we can use the upper bound

of channel order C to approximately estimate the coherence bandwidth (for

instance we can let B = [M /£J since the coherence bandwidth is generally

inversely proportional to the channel order). Although, there is no optimal

approach proposed to configure the parameter B, our simulation results in

Section 3.5 demonstrate excellent performance when using the configuration

B = [ M / jC\.

It might also be worth mentioning that the idea of subcarrier clustering has re-


cently received a lot of interests particularly for improving the communication

quality and spectral efficiency in cognitive communications [60]- [63]. How

ever, in our work, the subcarrier clustering is for improving the performance

of spectrum sensing.

S2) Sort Qi in an ascending manner, and apply differentiation on each cluster

respectively. This can be viewed as a rank conditioned rank selection pro

cess [64], where the order can change in an adaptive manner from zero to L.

Advantages of such filtering process would be the insensitivity towards heavy

tailed noise and impulsive noise while preserving the edge information [64]-

[66]. The sorting operation allows smoothing of the input without affecting

the statistics of the overall input. Furthermore, the differential operation

allows us to observe the available second order moment diversity.

As it can be observed from (3.12), the sorting function will not have an effect

in 7^0 scenario given that qij'Ho = A/q. When considering a more practical

scenario, i.e., limited number of samples, (ensemble average E(.) replaced by

the time average (3.6)) we will experience noise power fluctuations. Thus,

qil'Ho will no longer be constant and will follow the distribution described

in (3.7). Given that the input signal at this stage, q^, is independent (due

to the clustering operation performed in the previous stage) and identically

distributed, with cumulative density function Fg (q), the probability density

function of the output of the sorting operation is given by [67]

/,.,.{q) = ’- ( ^ ) j ’r * ( q ) ( i - - F i ( q ) ) ' '‘ ’'/ ,(q ) , (3.i4)

where r (1 < r < L) is the rth value returned after the sorting operation, and

/g(q) is the input probability density function. It can be observed from (3.14)

that fq^.^ (q) is the product of the density function of the input, i.e., /g(q),

and the function

»nL(q) = r ( 4 f - ' ( q ) (1 - • (3.15)

It can be concluded that (3.15) is equivalent to beta probability density func

tion [24]. Hence, the sorting operation is equivalent to multiplication of the

input distribution function with a beta function, with shape parameters equal

to r and L — r + l. Replacing u = Fg{q), the expression of the expected value


of the value of the output can be calculated using

< r \ r o o

lE(gr:Z<)—

Lr j J-oo

1

q F : - X q ) ( i - ^ g ( q ) r V , ( q )

(3.16)

Wr:I,(u)

where F~^{u) = q (since Fq is increasing in addition to being continuous) and

Wr:L{u) is the sorting function corresponding to rth highest value from set

containing L elements. The above equation reveals that the expected value

after sorting operation is the integral of the product between the sort function,

Wrxiiu), and the inverse distribution function. Figure 3.2 shows the sorting

function and the input distribution superimposed and further demonstrates

how sorting operations allows focusing on a particular region. Thus, the

sorting operation will reduce the effect of noise power fluctuation through

smoothing the sudden changes by focusing on a specific region of the input

density function out one time, this can be particularly useful when dealing

with impulse/spike noise hence, having a direct effect on the error probability.

1.4S o rt Function for r = 7 G um be! Distribution

Rayieigh Distribution Exponentiai Distribution

1.2

0.8

0.6

0 .4

0.2

Figure 3.2: Effect of the sort function on the output, for AT = 50 and r = 7 on various distributions. This figure illustrates how the sort function focuses on a particular point of a distribution based on the value of r. Furthermore the shape difference for various distribution all having a mean value of 0.42 is also shown in this figure.

The sorting problem has attracted a great deal of research and since early


1950s many sorting algorithm have been introduced in the literature, e.g.,

bucket sort, counting sort, spread sort. A comprehensive description of various

search algorithms can be found in [68]. Hence, sorting operation in this step

can be implemented using one of many developed sorting algorithm based on

the memory/efficiency trade-off the spectrum sensing device requires. There

fore, the device does not need to perform the operations explained in (3.14)-

(3.16) to sort the data.

The main objective of the differential operation, which is further performed

in this stage, is to remove the constant noise floor, i.e., A/q, contained in all

elements. The output of differentiation is denoted as with its Zth element

given by

[q<]i = I (3.17)

It is clear that [q ][ is zero for all I in the absence of the signal, and under

goes a fluctuation in the presence of the signal due to the channel frequency

selectivity. This distinctive feature motivates the test statistics presented in

S3) and allows us to overcome the noise uncertainty problem inherent in the

conventional energy detection.

Furthermore, this stage is intended to exploit the second order moment diver

sity of the input signal distribution. Figure 3.2 illustrates the shape/feature

difference [69] (in terms of inverse CDF) which exists between various dis

tributions. All three distributions in this flgure have equal mean value, yet

regions exist where the distributions are very distinct from one to an other.

In the case of no shape/feature difference, the performance of the proposed

technique will degrade. Since today’s high data rate communications always

leads to frequency selective channel, we will experience shape difference and

consequently second order moment diversity.

S3) Perform linear combination of q for i = 1 , 2 , . . . , # for the purpose of de-

noising, and then the following test

Test I : max

maxTest II : -----

B

' (3 18)I

mm

t=i

i E L o ' i1 A2, (3.19)


where the threshold Ai, A2 should be carefully configured to manage the PD

and PFA, which will be discussed in the performance analysis (see Section IV).

The test metrics presented in (3.18) and (3.19) represent the maximum and

the maximum to minimum ratio of the clustered ESD respectively, which have

been widely used for sub-optimum decision with low computational cost [70].

It is shown in Section IV-C that the proposed differential energy detection tech

nique can offer comparable performance to the optimal detector in Neyman-

Pearson sense [71], however, the latter requires the knowledge of channel gain,

noise power and signal power, which are often not available in practice for the

spectrum sensing application.

3.3.2 Overcoming Timing Offset

As mentioned in Section 3.2.2 the effect of CFO has been already solved through

employment of second-order statistics. Now, our main concern is to overcome the

timing offset. In fact, the special case of rie = 0 can be hardly captured due to the

lack of timing synchronization mechanism before the spectrum sensing component.

In order to handle the problem of unknown timing offset effectively, we propose

an “one ballot veto" policy to reject the hypothesis H q. The policy is stated as

follows:

51) Form J x 1 vectors, = [ykJ+i+ô,ykJ+2+ô,- ,ykJ+J+ô]'^, k=o,i,...,K, where

Ô denotes the offset in time,

52) Compute A E{yk,57^,5) according to (3.5), for

(J = 0 , (J - M), 2 (J - M ) , . . . , M;

53) Apply the cluster-based differential energy detection explained in Section 4.4.3

on V . If for any value of 6 the test statistic satisfies Hi criterion

it is understood that the signal is present and the cluster-based differential

energy detection algorithm would not be applied on the input after detecting

the first value of meeting the Ho condition.

The underlying idea is, in the presence of a signal, there exists such a S fulfilling the

condition | ne— < J —M, and under this condition, the proposed spectrum sensing

scheme can successfully reject the IBI. In the absence of signal, is approx

imately constant with respect to 5, due to constant energy of AWGN throughout


the spectrum. Most certainly, this stage will add to the overall complexity of the

algorithm which would be shown in Section 3.4.4. However, in order to increase the

reliability of the sensing device, implementation of this stage is necessary.

3.3.3 Extension to the ZP-Based System

Let us start from the special case of 71 = 0. Using the result in [57], we can

easily justify that the second term at the right hand of (3.5) vanishes due to the

implementation of ZP, i.e., (3.2). Therefore, (3.5) can be expressed by

E (yityf )= < 72A (n ,)**"A '^(n ,) + M o h (3,20)

= + V olj. (3.21)

Performing J-point DPT on (3.21) yields

:^./E(yfcyf ) : f " = + jVoIj, (3.22)

where $ = J~j is an J x J DPT matrix normalized by the factor (1)/(V J),

C j is an J X J circulant channel matrix with "Dj formed by the corresponding

channel frequency response. It is easy to observe that (3.22) has the same form as

(4.23). Therefore, the three step spectrum sensing algorithm proposed in Section

4.4.3 for the CP-based system can, be straightforwardly, applied on (3.21).

Furthermore, the “one ballot veto” policy can be applied on the ZP-based system

to handle the problem of unknown timing offset.

3.3.4 Knowledge of Key Parameters

The proposed spectrum sensing technique requires the knowledge of several key pa

rameters about the operating air-interface as well as channel models (i.e.,, the block

length J , the number of subcarriers M, the sampling rate Tg, as well as the upper

bound of channel order C). Those knowledge of parameters are very commonly

assumed in almost all estimation and detection techniques including spectrum sens

ing, e.g., in [11] [25] [50] [49]. Lack of the knowledge of these parameters would

result in performance degradation for all spectrum sensing techniques. Practically,

it is possible to obtain the mentioned parameters through accessing a geo-location

database. For example, the new Ofcom regulations [7] allow for sensing devices to ac

cess location-aided databases for obtaining key parameters about local air-interfaces

3.4. Performance Analysis 43

and channel power delay profiles (PDPs). Design and maintenance of location-aided

databases is an ongoing research activity in both Europe and US [4], [72]. Surely, the

impact of imperfect knowledge of air-interface parameters on the spectrum sensing

performance is of interest to telecommunication engineers.

3.4 Performance Analysis

Conventionally, the metrics of interests for performance evaluation of spectrum

sensing are mainly the PFA, PD, and computational complexity. The PFA is often

formulated for the AWGN case since it would not be affected by the channel fad

ing. However, the PD is related to the channel fading behaviour, and here we are

interested in the Rayleigh fading scenario. In addition to the PFA and PD analysis,

we will present numerical results as well as the computational complexity of the

proposed approach.

3.4.1 Probability of False Alarm

Let’s consider the special case of Ue = 0. It is understood that elements of q%

(see (3.13)) under the hypothesis Ho follow independent and identical central Chi

squared distributions with 2K degrees of freedom [34], i.e.,

exp (-a /2 ) (3.23)

where F(.) represents the Gamma function [24]. Hence, after the differentiation (ig

noring the effect of the sorting operation), the I th element of qj based on Appendix

A follows the p.d.f.

(3.24)

Remark: In the derivation of (3.24), we ignored the effect of the sorting operation.

This is mainly because the exact probability density function of the r th order statis

tic from any continuous population is rather difficult to deal with (see (3.14)) and

in most cases requires numerical evaluation of a nontrivial integral [67]. Since the

earliest known bounds for the expected value of highest order statistic with was

derived by Gumbel, Hartley and David, much work has been done on statistical

properties of order statistics, the summary of which can be found in [67]. Despite


all the work carried out in the area of the order statistics still the only effective way

for determining the distribution of /r:L(q) would be evaluating them numerically.

However, using the probability-integral transformation we are able to approximate

the variance of the r th order statistic, of any continuous distribution as

4 ^ (9 ) - (x + i]2’|j;+2)^ (E^hr.1.]))-", (3.25)

where E[qr-.L]i or in other words the expected value of rth order statistics, can be

approximated by:

(3-26)

where denotes the inverse cumulative distribution of the input signal. Please

note that the above approximations will converge as L ^ oo (see [67, Chapter 3]

for proof). The above approximations indicate that the sorting operation will have

a direct effect on the performance of the proposed algorithm since it will reduce

the variance of the data significantly. Thus, it can be concluded that the sorting

operation will reduce the effect of noise power fluctuation resulting from the limited

observation length. Hence, having a direct effect on the error probability as the test

statistic is subject to less variation. Since it is not mathematically feasible to derive

the performance incorporating the sorting operation we have shown the effect of

the sorting operation in Section V through simulations.

The linear combination q[ = l]^i[q^]z employed in (3.18)-(3.19) will result in the

following moment generating function (MCF) [24]

Ai([q'];|Mo)= n ^ ( |q ; i l l % ) = ((1 - ‘i t y " ) ^ ■ (3.27)

It can be observed that the random variable [q']i|'#o has an Erlang distribution [24]

with the shape and rate parameter equal to w = K B and rj = 0.5. Hence, its p.d.f

is given by

/[q'ldWoH = ex p (-7/0') (3.28)

Accordingly for Test II (see Appendix B), we can derive the p.d.f. of [q']f/[q%|'#o,

V 1 < Z, j < # and j ^ I, bearing in mind that the values of q are non-negative.


as [24]

roo rq[z

/[q']j/[q']il«o(^)= / / /[q']i,[q'bl«o dg'-Jo Jo

rOO

= //[q'Ii.Iq'Jil o g!) dgjyW —' r ( 2ro)

Finally, we can obtain the PFA as

(3.29)

Test I : PFA = 1 - (3.30)

Test II :PFA =, A AfT(2ro)2Fi([tî7,2ro],B + l , - A 2) V - V -------------------

where if) = (g), Q{.,.) is the lower Gamma incomplete function, and 2F i([a ,6],c,d)

is the Gauss hypergeometric function [73].

The PFA formulas above indicate the probability where the second order moment

diversity observed from the noise only input is higher than the test statistic. It can

be observed from (4.5) that Test II can only be applied and is meaningful if the

channel order is, L > 3. Hence, given the maximum channel order, one can choose

which test to employ. Furthermore, it can be concluded from (3.30) and (3.31) that

the PFA of proposed schemes is a function of the cluster size L, the number of

clusters, B, and sample complexity K, as well as the thresholds Ai, A2. Specifically,

it is exponentially related to the inverse of the channel delay, i.e., L, implying

that the performance is exponentially eflFected by the frequency selectivity of the

environment. This was expected as the key idea behind the proposed spectrum

sensing approach is to make a decision based on the observed second order moment

diversity resulting from the frequency selective channel. Furthermore, PFA will be

reduced dramatically as AT —>• 00. Given that for practical applications, the PFA is

often given a fixed value, such as 10% as per the FCCs requirement [4], (3.30) and

(4.5) can be employed to determine the appropriate thresholds Ai, A2 for a given air

interface, channel order and the required observation length, i.e., F (A) = 1 — PFA.

The exact effect of threshold value on the performance of the proposed approach is

shown in Section 3.4.3.


3.4.2 Probability of Detection

It has been proved that the random variable q,i\Hi follows non-central Chi squared

distribution with the p.d.f. [11]

K -1

W — ^ ^ ^ (3.32)

where X{.) denotes the modified Bessel’s function of the first kind, and j i the SNR

affecting the value.

Furthermore, we consider an interesting case when the SNR, 7 , follows an indepen

dent and identical exponential distribution

/y(o:) = - exp ^ , (3.33)

where 7 denotes the SNR mean.

Remark: In fact, modelling the SNR as an i.i.d. exponential distribution implies

that the communication channel is a Rayleigh fading channel. Rayleigh fading is

considered as one of the most practical models for tropospheric and ionospheric

signal propagation as well as for the effect of heavily built-up urban environments

on radio signals. Rayleigh fading is mostly applicable when there is no dominant

propagation along a line of sight between the transmitter and receiver [8]. Since,

based on FOG regulations [4] there is no guarantee that there would exist a line

of sight between the sensing device and the transmitter, it would be a reasonable

assumption to model the fading channel as Rayleigh fading.

The distribution of A7; = 7; — 7f- i, whose MGF is given by

Af(A7() = — p—. (3.34)1 + W

Hence, it can be concluded that A7f follows a Laplace distribution [24]. Considering

that q'il'Hi follows a non-central Chi square distribution with 2K degrees of freedom

and the non-centrality factor of 2A7/, and also the fact that A7 is non-negative,

the term in Appendix B is computed using the following

PDJ^ = = / Qk (^ , \ A i ) exp dA'yi (3.35)7Ja7!=o ' V 7 /

with (f = 2A7/, where Qk {cl, 6) denotes the generalised Marcum Q-function defined


by1 ro o / 2 I -2 \

QK{a,b) = J lK -i{ax)dx . (3.36)

The PD for Test II can be evaluated using Appendix C, where the p.d.f. of

A7i/A 7d^i|?/i given by

pOO

yA 7 ; /A 7 d ^ ; |% i (( ) /J 3 = 0

(l + a )2 -(3.37)

Hence, the term P D ^ in Appendix B given Rayleigh fading is given by

Once more considering the special case of = 0, after the differentiation under

the hypothesis Hi, the differential SNR A ji corresponding to (see (3.13))

follows the Laplace distribution with the p.d.f. based on the derivation in (3.27).

Furthermore, the average of differential SNR A7 ; can be computed by

Then, the term PDJ^ can be evaluated by

X exp dA7 f.

A ji7

xu—l

Based on the analysis in Appendix C, we can further write (3.40) into

PDT = _ _ exp ( _ I X2 + 7 \ 2 + 7

1 + 3 2(24-7)7 V 7

2 4-7Ai 7

2(2 + 7)

+ 2 + 7K -l _

exp ( - y ) X

(3.39)

(3.40)


where iF i(.;.;.) denotes the hypergeometric function [73], and Ln{.) the Laguerre

polynomial function defined by

r=l(3.41)

We can obtain the PD for Test I by applying (3.41) into Appendix B.

Evaluating the PD of Test II requires the p.d.f of the ratio of A'y^/A'yjl'Hi. Based

on the derivation in (3.29), we have

-^T{2w)/A7i/A7, |7£i (^) (1 + ^) (3.42)

Then, the term PDf^ can be computed by

PJJT2 _ r ( t o ) (y , V ^ ) dA7 ,. (3.43)

Considering considerably Low SNR such that 1 A7 , the integration in (3.43) can

be computed by using Appendix C and the analysis in [74, Appendix A]. Hence,

P D ^ can be expressed by:

P D P = ®exp ( ^,fc=0

+$exp ( - y ) E iF i (ro; & + l ; y ) (3.44)

where 0 = (the full proof can be obtained by using [75, Eqn. (25)]). Finally,

we can obtain the PD for Test II by applying (3.44) into Appendix C.

It can be observed from (3.41) and (3.44) that the performance of the proposed

spectrum sensing technique, in terms of PD, is affected by the average SNR value 7 ,

sample complexity K and the threshold value Ai and Ag and further exponentially

effected by the channel order L. Moreover, it can be observed that the performance

of Test II improves much faster with the increase in channel order, L. The effect of

various parameters on the PD of the proposed approach will be discussed in detail

and illustrated pictorially in Section 3.4.3.


3.4.3 Numerical Results and Discussions

In this section, numerical results based on the PFA and PD expressions found

the Sections 3.4.1-3.4.2, are provided to visually demonstrate the effect of various

factors. Figure 3.9 illustrates how PD is affected by the observation length (latency)

in Test I. The results are generated for the configuration where the number of sub

carriers M = 64, and the number of clusters B = 6. The threshold Ai was fixed

for achieving PFA = 10% with the noise uncertainty factor set to 2 dB (the noise

uncertainty factor in practical scenarios is typically between 1 to 2 dB [9]). The

main factor causing noise uncertainty is the temperature variations at the receiver

which leads to inaccurate noise power measurements. The uncertainty is created

by fixing assumed/ estimated noise power based on the SNR value mentioned, while

the real noise power varies with each realization by a certain degree according to

the uncertainty factor. It is observed that the proposed approach features fast

convergence rate. For example observing the point of PD = 90%, the PD improves

by 5 dB in the SNR when the number of multi-carrier symbols K varies from 3 to

5, while this improvement is as small as approximately 1 dB when K varies from

20 to 30.

Figure 3.10 shows how the channel length C would infiuence the PD when the

observation length is set to K = 10. Take the point PD = 90% as an example, 8 dB

gain in the SNR can be observed when C varies from 0 to 4. Furthermore, 10 dB

improvement when it varies from 4 to 12. It is an interesting result which clearly

indicates the channel frequency-diversity gain inherent in the proposed spectrum

sensing scheme.

The complementary receiver operating characteristic (ROC) curve for both Test

I and Test II (in Rayleigh fading channel) are shown in Figure 3.11 and Figure

3.12 respectively. These Figures refiect a fundamental tradeoff between PFA and

PD. Furthermore, the effect of the threshold value on both PFA and PD can be

also observed, since different threshold values were employed to produce the PFA-

PD tradeoff. In order to have a benchmark and also for performance comparison,

the ROC curve for conventional energy detection with various uncertainty factors

(U) are also illustrated. It is observed that the performance of the energy detection

severely degrades as the uncertainty factor is introduced (this phenomenon has been

fully investigated in [9]). While, due to differential stage of the proposed technique,

it is considerably robust to uncertainty factor. For the sake of comprehensive perfor-


mance comparison, Figure 3.11 also illustrates the ROC of the optimal detector in

Neyman-Pearson sense [71]. It should be noted that the optimal detector requires

channel gain, noise power and the transmitted signal power (which is not a feasible

solution in practical scenarios). Hence, as expected it delivers better performance.

3.4.4 Computational Complexity

The main complexity of the proposed scheme is due to the following stages:

1. The second-order time average: for the case of rig = 0, this stage requires

X J complex multiplications and additions.

2. Discrete Fourier Transform: M —point DFT is implemented which introduces

the complexity by 0 (Mlog(M)).

3. Sorting: there are B clusters consisting of L elements, hence, the complexity

of this stage is BLO{L).

4. Differentiation: this stage consists of subtracting every element of from its

previous one for each cluster, hence the computational complexity is given by

BO{L).

5. Linear combination: This would add a further complexity of 0{B).

6. Decision making: Finally the extreme value(s) is selected and compared to

the predetermined threshold value. Consequently adding a complexity factor

of 0{L).

Resulting in the overall computational complexity:

0 { K ‘ J) + 0{M\og{M)) + B{L + 1)0{L) + 0{B) + 0{L) (3.45)

Note that the above complexity is for the case = 0. When employing the “one bal

lot veto” scheme for arbitrary ng (see Section HI-A), the computational complexity

is increased by a factor of ((M )/(J — M)) (in the worst case scenario).

The last three terms are negligible in (4.39), hence, the overall complexity of the

proposed scheme is approximately M [O(A’ J) + 0(M log(M ))] / { J — M). This

reflects that the proposed scheme requires a relatively low computational complexity,

making it suitable for practical scenarios, where computational efficiency is a key

issue.

3.5. Simulation Results and Discussions 51

3.5 Simulation Results and Discussions

Computer simulations were performed to evaluate the proposed spectrum sensing

scheme. The system investigated in this section has M = 2,048 sub-carriers with

the sub-carrier spacing of 15 kHz (3GPP LTE-advanced system [76]), each frame

consists of 7 OFDM blocks with the CP length oi J — M = 160, the sampling

frequency is the same as the signal bandwidth of 30.72 MHz. The carrier frequency

is also set at 5 GHz. The communication channel is generated according to the

WINNER channel model under B2 outdoor scenario [77], and the sensing device

is moving at the speed of 3 km/h. The SNR is defined by the average received

symbol energy to noise ratio at the sensing device. The threshold for hypothesis

test is carefully chosen so that the PFA is fixed to 10%. All simulation results were

obtained by averaging over 2,000 Monte Carlo realizations.

Experiment 1

The objective of this experiment is to examine the analytical analysis obtained in

the previous sections by comparing them against the simulation results based on

the configuration explained above and further to show the effect of the sorting op

eration on the performance of the proposed scheme. Figure 3.7 and Figure 3.8

demonstrate the probability of detection for different observation lengths given var

ious average SNRs for Test I and Test II, respectively. We can observe a very small

difference between analytical results and simulation results when the observation

length is larger than two symbols duration. The difference becomes large when the

observation length is less than two symbols duration. This is mainly caused by the

insufficient statistics used in signal processing. Comparing Figure 3.7 and Figure 3.8

verifies that Test II outperforms Test I, particularly, when the observation length

is short. This difference is mitigated with the increase of observation length. We

have also shown the effect of the sorting operation through simulations in Figure

3.7 and Figure 3.8. As it can be observed, the sorting operation can improve the

performance as the observation length is increased. This was expected as previously

explained in Section 3.3.1 and Section 3.4.1.


Experiment 2

The objective of this experiment is to examine the proposed scheme with respect

to the state-of-the-art spectrum sensing approaches. Since, the proposed approach

is based on exploiting the second order moment frequency diversity, it is essen

tial to check how much gain is introduced due to this exploitation by having the

frequency-domain energy detection as a benchmark for performance comparison.

The threshold setting for energy detection can be found in [11]. The simulation

performed for energy detection are based on noise uncertainty factor, U = 0,1,3

dB and the threshold is based on the assumed/estimated noise power while the real

noise power varies with each Monte Carlo realization by a certain degree depending

on the uncertainty factor. Figure 3.13 shows the performance comparison when ob

servation length, K = 7 symbols. It can be observed that the performance of energy

detection is considerably dependent on the noise uncertainty factor. It is further

proved in [9] that increasing the observation length does not affect the performance

of the energy detection scheme when the the exact noise power is not known, i.e.,

U 0. Figure 3.13 also illustrates the performance of the second order cyclosta-

tionarity based detection. The proposed approach is able to outperform the second

order cyclostationarity by at least 8 dB when K = 7. Cyclostationarity based de

tection relies on the cyclic frequency of the received signal to determine existence of

a source. Hence, deep fading at cyclic frequency can have a detrimental effect on its

performance while the proposed technique takes advantage of this fading to exploit

the moment diversity. The performance of the proposed approach is also compared

to Wigner-Ville based spectrum sensing [55]. For this purpose, in order to have fair

comparison, we have modified the original work in [55] to accommodate a SISO en

vironment. As shown in Figure 3.13 the mentioned approach can deliver acceptable

performance up to SNR of —16 dB. However, as the SNR further decreases, the

performance of Wigner-Ville transform based approach also decreases. This was ex

pected since noise power fluctuation increases such that it makes the edge detection

used in this approach reluctant. The performance comparison is also carried out

for pilot based detection [78] and differential based energy detection [79]. In order

to carry out simulations for the pilot based detection, it is assumed that the pilot

symbols are embedded in each OFDM block, which are equally spaced for every

16 or 32 sub-carriers. Since, the mentioned pilot based detection is based on the

energy of the pilot symbols it is affected by the noise uncertainty factor. Therefore,

the performance was evaluated for Z7 = 0 dB and [7 = 2 dB. Figure 3.13 shows that


the proposed technique is able to outperform the mentioned technique by at least

10 dB. It should be also noted that pilot based detection requires synchronization

and pilot information while this problem can be overcome in the proposed technique

with the implementation of the “one ballot veto” policy.

Since the detection technique introduced in [79], also exploits the frequency diver

sity of the channel our main concern is to observe how much improvement can be

delivered by the clustered-based energy detection. Figure 3.13 further shows that

the proposed approach can outperform the differential energy detection by at least

5 dB in low SNR environments. This improvement is mainly due to the clustering,

the linear combination and the “one ballot veto” policy which are implemented. It

is noteworthy to mention that the clustering operation not only improves the per

formance of the proposed techniques but also reduces the complexity by a factor of

0(2M ).

Experiment 3

The objective of this experiment is to observe the performance improvement due

to the clustering and differential stages. As it was explained, the main purpose of

the differential stage is to remove the AWGN which is available in all the frequency

bands and to further exploit the frequency diversity, while the clustering operation

is to remove any possible correlation in the ESD due to the fading channel. In

order to observe how much improvement can be achieved when incorporating these

two stages (i.e., clustering and differential stages), we have set an experiment were

the proposed technique in Section 4.4.3 is compared to its equivalent without the

mentioned two stages. Without differential and clustering operations the proposed

technique can be thought of as a simplified eigenvalue detection [35] where instead of

making decision based on the ratio of the eigenvalues of the covariance matrix of the

received signal, the decision is based on the ratio of the maximum and the minimum

of the ESD of the received signal. This comparison is possible since, in multi-carrier

systems, parallel transmission is performed, hence, the DFT decomposition can

be considered as a special case of eigenvalue decomposition. The result of this

performance comparison for different observation lengths is shown in Figure 3.14.

As it can be observed we are able to achieve up to 10 dB gain in performance, this

gain is more apparent as the observation is increased.

3.6. Summary 54

Approach Air-interfaceInformation Synchronisation Computational

Complexity Latency

Energy Detection X X Low Low

Cyclostationarity Detection / / High High

Covariance Based Detection X X Medium Low

Eigenvalue Based Detection X X High Medium

Matched Filtering / / Low Low

Pilot Based Matched Filtering / / Low Low

Entropy Based Detection / X Medium Medium

Kurtosis Based Detection / X High High

Clustered-Based Energy Detection / X Medium Low

Table 3.1: Comparison of the state-of-the-art local narrow-band spectrum sensing approaches and the cluster-based energy detection.

3.6 Summary

In this chapter, a novel differential energy detection scheme for multi-carrier sys

tems, namely, cluster-based differential energy detection, which can form fast and

reliable decision of spectrum availability even in very low SNR environment, has

been proposed. It has several distinctive features including low latency, high accu

racy reasonable computational complexity, as well as robustness to very low SNR.

For example, the proposed scheme can reach 90% in probability of detection and

10% in probability of false alarm for an SNR as low as —21 dB, while the observation

window is equivalent to 2 multi-carrier symbol duration. The proposed scheme at

this stage is specially designed for sensing multi-carrier sources but we would argue

that since most of the current and future mobile networks are multi-carrier based

systems, that the proposed scheme has wider ranging practical applications. The

proposed approach can deliver desirable performance in high density communication

networks and urban environments due to its robustness in low SNR environments.

Furthermore, the clustered-based differential energy detection can be employed in

vehicular communication systems.



energy spectral density. Initially, the ESD of the received signal is estimated. Follow

ing the ESD computation, the clustering operation is utilized to group uncorrelated

subcarriers based on the coherence bandwidth to enjoy a good frequency diversity.

The knowledge of coherence bandwidth does not need to be very accurate (we em

ploy the reciprocal of the maximal channel delay). Furthermore, making use of

order statistics of the estimated ESD, we increase the reliability of the sensing al

3.6. Summary 55

gorithm. This will allow us to smooth the fluctuation in noise ESD, resulting from

limited observation length which affects the statistics of the received signal. Hence,

this will stage will have a direct effect on the PFA of the proposed approach.


differential operation is performed on the rank ordered ESD. When the channel is

frequency selective and the noise is white, the differential process can effectively re

move the noise floor resulting in elimination of the noise uncertainty impact which

is the main factor making energy detection reluctant [9]. Furthermore, the differ

ential stage will allow us to exploit the frequency selectivity which available in the

received signal. At the flnal stage of the proposed scheme, the differential rank

ordered ESD within different clusters are linearly combined in order to further re

duce the effect of impulse/spike noise. Binary hypothesis testing is then applied

on either the maximum or the extremal quotient (maximum-to-minimum ratio) de

pending on the wireless channel characteristics of the sensed environment. More

importantly, the proposed spectrum sensing scheme is designed to allow robustness

in terms of both, time and frequency offset, without compromising computational

complexity. Additionally, it is worth mentioning that given not a very frequency

selective environment, i.e., when the transmitted signal is experiencing flat fading,

the performance of the proposed scheme is degraded to the performance of the

frequency energy detection with approximately zero uncertainty factor.

To analytically evaluate the proposed scheme, both PD and PFA were derived for

Rayleigh fading channels. The closed-form expression showed a clear relationship





schemes where up to 10 dB gain in performance can be observed. This would imply

that employing the proposed approach in a communication system will make the

network of interest more robust to hidden node problem, i.e., mitigate interference

to heavily shadowed licensed users. More interestingly, it has been shown through

simulation that the proposed spectrum sensing algorithm has a long convergence

time, which allows it a suitable in delay sensitive systems.

3.6. Summary 56

0 .95

0 .9

0 .85

IS 0.8

0 .75K = 3 K = 5•9

S 0 .72Q.

0 .65 K = 20 K = 30

0 .5

SN R (dB)

Figure 3.3: The relationship between the PD and the observation length for M = 64 and £ = 6 .

A -0.9

0.8

S 0 .7

0.5

Q- 0 .4 — Fl at F ading - A - C h an n e l Length= 2

^ C han n e l Length = 4 ' - D - ' C h an n e l Length = 8 — O — C h an n e l Length = 12

0 .3

S N R (dB)

Figure 3.4: The relationship between the PD and the coherence bandwidth, C, and the observation length K =\6.

3.6. Summary 57

- A - P ro p o se d T e chn ique Ti K = 6 ' P ro p o se d T e chn ique TI K = 10

' " O ' " P ro p o se d T e chn ique TI K = 15 • -O" ' P ro p o se d T ech n iq u e TI K = 20 E nergy D etection U=2 dB , K= 10' - ' - ' E nergy D etection U=1 dB, K= 10 ' E nergy D etection U=0 dB, K= 10- - - O ptim al D etector, K =6 — — — O ptim al D etecttor, K=10

0.01 0 .02 0 .03 0 .04 0 .05 0 .06 0 .0 7 0 .08 0 .09 0.1Probability of F a ise Alarm

Figure 3.5: Complementary ROC curves of the Test I and it’s comparison with energy detection for various uncertainty factors (U), the optimal detector based on Neyman-Pearson criteria. 7 = —lOdB, C = 8 and M = 64 based on the analytical results in Section 3.4.

Q 0.8

= 0 .75 -A - P ro p o se d T ech n iq u e Til K P ro p o se d T ech n iq u e Til K P ro p o se d T ech n iq u e Tii K P ro p o se d T echn ique Til K E nergy D etection U=0 dB, E nergy D etection U=1 dB,

E nergy D etection U=2 dB,

0 .02 0 .0 3 0 .04 0 .05 0 .06 0 .07Probabiiity of F a ise Alarm

Figure 3.6: Complementary ROC curves of the Test II and it’s comparison with energy detection for various uncertainty factors (U). 7 = —lOdB, £ = 5 and M = 64 based on the analytical results in Section 3.4.

3.6. Summary 58

o 0 .7 -A - Cyclostionarity D etection P ro p o se d T ech n iq u e T e s t I P ro p o se d T echn ique T e s t li

■ - O * ■ E nergy D etection U = 3 dB■ - 0 - ' E nergy D etection U = 0 dB • ' E nergy D etection U = 1 dB- O - Pilot B a se d D etection U = 2 dB

Differentiai E nergy D etection W igner-V iie b a s e d D etection

- 2 5 - 2 3 -2 1 - 1 9 - 1 7 - 1 5 - 1 3 -1 1 - 9S N R (dB)

Figure 3.7: The performance comparison of the proposed technique, frequency-domain energy detection, second order cyclostationarity, pilot based detection and differential energy detection for K = 7.

A , À - - é T ^ '

- n - K= 2 OFDM S ym bois- O - K= 7 OFDM S ym bo is

A - K= 14 OFDM S ym boisK = 2 OFDM S ym boi (Differntiai)K = 7 OFDM S ym bo is (Differentiai) K = 14 OFDM S ym bo is (Differntiai)

-2 5 - 2 3 -2 1 - 1 9 - 1 7 - 1 5 - 1 3 -1 1 - 9 - 7 - 5SN R (dB)

-3 -1 1

Figure 3.8: The effect of the differential and clustering stages on the performance of the proposed spectrum sensing technique.

Chapter 4

A Bayesian Model Based Approach

for Joint Sub-Band Level Spectrum

Sensing

4.1 Introduction

The problem of model estimation/ selection for array processing has been well inves

tigated in the literature. For example, the approach developed by Lawley [80] and

Bartlett [81] using a sequence of hypothesis tests with subjective thresholds for each

test, the model selection developed by Akaike [82], Schwartz [83], which addresses

this problem by selecting the model which results in minimum information criteria

using the log-likelihood of the maximum likelihood estimator of the parameters in

the model. Model estimators based on the eigenvalue of the covariance matrix was

further proposed by Kailath and Wax [84], namely, the Kailath-Wax Akaike infor

mation criteria estimator and Kailath-Wax minimum descriptive length criterion

estimator. Unfortunately, due to high computational complexity and sample size

requirements of the available approaches, they are not able to fulfil the demand

ing requirements of practical spectrum sensing for opportunistic spectrum access

applications, e.g., cognitive radio.

The main contributions of this chapter are 1) re-defining the objective of the sub

band level spectrum sensing device to a model estimator, 2) deriving the optimal

model selection estimator for sub-band level spectrum sensing for fixed and variable

59

4.2. Problem Formulation 60

number of users along with a sub-optimal solution based on Bayesian statistical mod

elling and 3) proposing a practical model selection estimator with relaxed sample

size constraint and limited system knowledge for sub-band spectrum sensing appli

cations in OFDM A systems. The proposed technique takes advantage of the second

order moment channel frequency diversity. More interestingly, it does not require a

priori knowledge of noise power and the propagation channel gain, and is designed

in such a way to show robustness towards energy leakage. The proposed model

selection based sub-band level spectrum sensing approach is analytically evaluated

through probability of false alarm and probability of detection along with closed

form expression for the threshold value. Furthermore, computer simulations are

carried out in order to evaluate the effectiveness of the proposed scheme.

4.2 Problem Formulation

Consider a communication system operating over a wide-band channel that is di

vided into K non-overlapping sub-bands, e.g., multi-carrier systems. However, in

a particular geographical region within a certain time frame only I number of the

sub-bands is utilized by the users, where I < K . Thus, [K — I) sub-band are avail

able for opportunistic access. The essential task of the spectrum sensing device is

to determine the availability of these {K — Ï) sub-bands.

The estimated power spectral density (PSD) [24] of the received signal, sampled at

frequencies {fk}k=v cem be modelled as

(fk — oikSk + Vk + jk , l < k < K (4.1)

where Sk denotes the power of the signal transmitted at the sub-band, ak is

an indicator function taking a value of 0 if the k^^ sub-band is vacant or 1 if it

is occupied by a user, Vk is the additive white Gaussian noise (AWGN) power at

the k^^ sub-band, and 7fc denotes the PSD estimation error due to limited obser

vation length at the k^^ sub-band. Please note that the effects of various physical

impairments in a communication systems is further discussed in Section 4.4.2.

Hence, (4.1) leads us to consider inferring the vacant sub-bands in the bandwidth

of interest, converts the spectrum sensing problem to a model estimation problem

where the parameter of interest is a = [o;i,a;2, . . . ,o:/ï-]^. Based on the model

above the task of an ideal spectrum sensing device, given ot is the estimation of a .

4.3. Optimal and Sub-Optimal Solutions 61

is to select the binary sequence ex. f o r which the a posteriori probability

distribution Pr{a.\(p), where (p = - - -, is m a x im u m over all

the possible binary sequence o f o l .

4.3 Optimal and Sub-Optimal Solutions

4.3.1 Optimal Decision Rule

Based on the problem stated in Section 4.2, there are at most 2^ sequences which

can be associated to a . Assume that the transmitted signal in each sub-band has

a unit power, i.e., = 1. This assumption holds when the users adopt a

uniform transmission strategies given no channel knowledge at the transmitter side.

Considering that each sequence is equiprobable using Bayes rule [85] we have

Based on the above equation, in order to maximize P(â|<^), our objective is to

maximize P {ip\ôc), given that

P(V5|a) = y 'p ( y ) |â ,7 ) P ( 7 )d 7 . (4.3)

Moreover, based on the variable definitions in Section 4.2, we have

/ 1 \ ^ / 2 f I JL \P ( v |â ,7 ) = j exp ( - * - 7i) j , (4.4)

where denotes the noise variance. For the sake of reaching a conclusion it is now

necessary to specify the distribution which 7 follows, hence, we will assume jk is a

random variable following a Gaussian distribution with variance of cr and zero

mean. Hence, substituting P (7 ) into (4.4) and integrating would result in

/ 1 \ K/2f(vl«)= ( 2 ^ ) (det(Q))-'/"

x e x p ( - ^ { ¥ ; - â ) ^ Q “ 7 v - « ) ) . (4.5)

^In information theory, Gaussian distribution is a saddle point of many optimization problems. Therefore, we are safe to ignore the distributional uncertainties and considering the worst-case scenario, i.e., Gaussian case.


where det(-) denotes the déterminante operation and Q is the normalized covariance

matrix of the noise plus the estimation error, defined by

Q = ^ P { ( v + 7 ) (v + 7 )^}. (4.6)

Based on (4.5), it can be concluded that the optimal decision making procedure

should select from the set of the possible binary sequence 6 , which minimizes the

quadratic form

S { (p ,à )= { (p -à ) '^ (4.7)

For simplicity, assume that all the sub-bands experience equal estimation error,

hence, (4.7) can be expressed as

^ 2-2S (y, â) = \ \ c p - à f - i<p - â f , (4.8)

given that (f = ^ and â = ^ E&Li

Based on the above formulation, the optimal decision rule should compute S (y, a)

for all the possible 2^ possible binary sequences and further to search for the se

quence which returns the minimum value. However, this procedure is both compu

tational costly and timely, which makes this approach not a practical solution given

the demanding requirements for the spectrum sensing device, e.g., FCCs require

ments [4].

4.3.2 Optimal Solution For Fixed â

Let â to be fixed within a time frame, e.g, on average I sub-bands are occupied

within a time frame. Resulting in âi = l /K . Re-writing (4.8) based on the fixed âi

and minimizing leads to

S (y, Qi) = mm ||y - â ||" - - , (ÿ - S , f , (4.9)

where denotes the set of sequences with â = âj. It can be concluded from

(4.9) that the sequence ài which minimizes S {(p,oci) is the one consisting of I I ’s

corresponding to the largest I elements of (p. Therefore, the next task would be to

determine the I largest values of (p given that I is not known.

This can be done by ordering the elements of (p in terms of their magnitude. Let


fi be the K x K matrix which allows this transformation, hence we have u = flip,

where ui >U2, . . . , > u k - This would also result in /3 = f la . Now, the problem of

interest would be

S (u, ( 3 ) = min S (u. P i) . (4.10)

Using (4.8) and the definitions explained above we can re-write (4.10) as

I< 5 (u ,/3 i)= ^ (w fc - l)^

k=l^ 7 2 2 / I \ 2

It should be noted that ü = <p. More interestingly, (4.11) can be solved using the

recursion formula

5 (u, A ) = 5 (u, A _ i ) - 4 (ui + ' ~ ^ ^ | ) ~ 1 ) , (4.12)

with the initial condition

«5(u,/3o) = ^ 2 /^2 • (4-13)fe=l

Hence, the index Î which allows S < S { vl,P i) Wl ^ Î, determines P = Pi.

Consequently,

0, k > l

Finally, we have to re-store the original order of the indicator function, i.e., a =

f f^ p . As it can be observed by having a assuming a fixed â (which is acceptable

assumption in communication systems), we can reduce the search size from 2 ^

used in Section 4.3.1 to K , while maintaining the same performance. It should be

noted that f t f l ^ = f t ^ f l = I. Figure 4.1 shows the performance of the optimal

solution explained in this section for various SNR values and different I. As it can

be observed the proposed approach can easily detect the number of the occupied

sub-bands even in low SNR range. As S (u, /3) returns in a clear minimum value

at P = I. The effect of the SNR value of the performance can be easily observed,

i.e., as the SNR value decreases the S (u,/3) function will return a smoother curve

making the decision more prone to errors.

Even though the above procedure simplifies the ideal decision making process ex-


- - 1= 40, SNR = 0 dB — 1=40, SNR = 5 dB- ' 1= 40, SNR = 10 dB- -1= 70, SNR = 0 dB — 1 = 70, SNR = 5 dB ■-•1=70, SNR = 10dB

S ' 4 0

-10100

Figure 4.1: The performance evaluation of the optimal solution introduced in Section 4.3 for fixed â and K = 100.

plained in Section 4.3.1, it still needs to compute the recursive formula satisfying

(4.11). This would imply that the decision time would still be high, given sys

tems which employ high number of sub-bands, and motivates us to formulate a

sub-optimal solution to meet demanding spectrum sensing requirements.

4.3.3 Sub-optimal Solution

Consider the ideal case where the dimension of the observation sample approaches

infinity, i.e., limÜT —> oo, this results in

lim O' = £, and lim ip = 'tp,K ^ o o K -^ o o ^

(4.15)

where 'ip is the expected value of (p, which can be calculated based on the definition

in (4.1).

Furthermore, it should be noted that (4.8) can also be expressed as

K

s iv, 0 ) = ^ [<Pk - à k - { ( p - â ) Ÿ + ^2 '^ ^ 2 ( v - ô i f ■ (4.16)K a l


Hence, incorporating (4.15) into (4.16) will lead to

lim = . (4.17)K -> oo (TXj

Ignoring the constant term on the right hand side in (4.17) it can be concluded that

ÔC = 8 {p - {'ip - ^ ) ) , (4.18)

where £{ip) is an indicator function which returns 1 if ^ > 0 and 0 otherwise. It

can be observed from (4.18) that in order to obtain a satisfactory estimation of d,

an accurate knowledge of 'ip and should be provided or estimated. Therefore, the

performance of this solution is directly proportional to the number of sub-bands,

since the estimate of ip and ^ converges as K oo. In addition, employing ac

curate estimation approaches plays an important role in the performance of (4.18).

An example of an effective estimation method in this scenario would be minimum

variance linear estimation [86].

It would be important to evaluate the performance of the sub-optimal solution

explained above. The parameter which will reflect the accuracy of the proposed

approach would be the average probability of error, Pre- Assuming P(o; = 1) =

P (a = 0) = 0.5 by replacing (4.18) in (4.3)-(4.8) we have

/ 2 \ 0 .5

P r e = erfc ( ) , K = 1 (4.19)

where erfc(.) denotes the complementary error function and p represents the SNR

value.

It can be observed from above that the performance of the sub-optimal solution

is a function of two parameters, the number of sub-bands, K , and the SNR value.

More interestingly, the effect of (4.18), which can be modelled as high-pass filter,

will result in degradation of p approximately by a factor of {K — 1)/(AT -f p), i.e.

This degradation can be made arbitrarily small by choosing sufiiciently large K.

Figure 4.2 illustrates the effect of K on the performance of (4.18) for different SNR

4.4. Link to OFDMA based Systems 66

Optimal, SNR=0 dBOptimal, SNR=5 dB Optimal SNR=10 dB Sub-optimal, SNR=0 dB Sub-optimal, SNR=5 dB

- — ■ Sub-optimal, SNR= 10 dB

10 20 30 40 50 60 70 80 90 100

Figure 4.2: The performance comparison of the proposed optimal and sub-optimal solutions derived in Section 4.3.

values. It is shown in Figure 4.2 that as the SNR value decreases, the effect of K on

the performance decreases. Furthermore, the performance comparison of the sub-

optimal and optimal solution can be observed in Figure 4.2. Note that for increasing

values of SNR and K the performance gap of the sub-optimal and optimal solution

decreases, which was expected.

In the next section we will propose a novel joint sub-band level spectrum sensing

in OFDMA based environment, specifically for downlink scenarios. The physical

impairments existing in various wireless communication systems will be also consid

ered will be also considered in Section 4.4.

4.4 Link to OFDM A based System s

OFDMA is the most popular multi-carrier based transmission scheme for wide

band digital communication used in applications such as digital television and au

dio broadcasting and 40 mobile communications [87], due to its advantages, e.g.,

spectral and implementation efficiency and robustness towards propagation channel

fading. Hence, in this section, we will propose a novel joint sub-band level spectrum

sensing technique (inspired from the findings in Section 4.3).


4.4.1 Transmitted Signal

The transmitted downlink OFDMA signal can be expressed in the matrix form,

Xfc = ^ X {T>{ak) X Sk), where Xfc is an J x 1 transmitted signal block, T>{ak)

denotes an M x M diagonal matrix with diagonal entries, ak acting as a frequency

mapping unit, indicating which sub-carriers are used for transmission, Sfc is an

M X 1 information-bearing symbol block with the covariance Og Im {M stands for

the number of subcarriers, and 1m for the identity matrix with the size M ), ^ is an

J X M ( J > M) tall pre-coding matrix with full column rank, and subscript k is the

block index. The pre-coding matrix Ÿ can be expressed as ^ . This

chapter is also focused on the CP based OFDMA systems. However, a solution forr -]T

zero padding based OFDMA system, i.e., where ^ 0 , is also provided

in Section 4.6.3.

4.4.2 Received Signal

In the absence of the signal, the spectrum sensing device can only receive noise,

otherwise, it receives a signal distorted by frequency-selective channel, timing offset

(denoted by e), frequency offset normalised by the subcarrier spacing (denoted by

e), and additive white Gaussian noise.

Please refer to Chapter 3 Section 1.2 for full details of received signal given the

mentioned distortions. Consider an J x 1 vector = [rfcj+i,rfcj+2v

then (3.3) can be formulated as the matrix form

rjt = Afc(A(ne)xfc -f V(ne)xfc_i) -f v&, (4.22)

where Ak = diag{A(^"^+^),

4.4.3 Join Sub-Band Level Spectrum Sensing in OFDMA

Systems

Incorporating the findings from Sections 4.3, a practical algorithm for sub-band

level sensing specifically for OFDM signals can be implemented using the following

steps: (the block diagram in Figure 4.3 demonstrates an overview of the proposed

algorithm)

S tep 1: Compute the BSD of the received signal.


ReceivedSignal

Store Original

Figure 4.3: Flow chart of the proposed OFDMA sub-band level spectrum sensing.

This step consists of two operations, performing a DPT operation and a second

order average, respectively. Consider the signal model presented in (4.22), it can

be observed that due to the effect of CP the second term in the brackets, i.e.,

V(ne)xfc_i, is vanished. Hence, in order to obtain the BSD of the received signal

after the CP removal, = [rkM+i,^kM+2, - , an M-point DPT operation

is performed on r^, i.e., ffc =

Furthermore, at this stage, an M x M matrix, Cr, is computed by 5 " ' = ^k= oi^k^k)^

where N denotes the number of observation windows. In order not to converge the

presentation of the proposed technique consider the special case where, and e

are set to zero (Section 4.6 aims to overcome these offsets). Hence,

c ;= r (a) c l V M V h + WLiM, (4.23)

where P m — is an M x M diagonal matrix whose diagonal entries are

actually the channel frequency response, given that C m is an M x M circulant

channel matrix defined in [57].

Hence, the diagonal entry of Cr reads as

Qm — l^m| “b Afo 171 — 1, 2, • • • , M.L J m

(4.24)

Rem ark: A time domain window operation can be performed in order to reduce

the leakage between DPT bins. This leakage is due to the fact that DPT implicitly

assumes that the signal is periodic. Hence, when a discontinuity is observed in the

input signal it will spreads power all across the spectrum. One solution would be to

multiply the time series with a window function in the time domain before applying

the DPT [88].


S tep 2 ; Order the BSD in ascending manner in terms of magnitude.

The values of BSD are sorted in increasing order, i.e., t 91 = where Ft is the

M x M sorting transformation matrix. The detailed explanation of the sorting

function can be found in Chapter 3 Section 1.3. The sorting function acts as a

weighting function to emphasise a particular region of the distribution function and

for different values of r the sort function allows us to focus on different regions of

the input distributions. This will have a direct effect on the error probability as it

will reduce the variance of the rth output i.e., t Qr considerably, see Section 4.5.

Furthermore, given a composite signal, by sorting the BSD in order of magnitude,

there would exist a point, namely the “knee-point”, where there is a sudden change

in the magnitude. This point will allow us to distinguish components from different

distributions. Hence, the next objective would be to identify the knee-point.

S tep 3: Apply differentiation on the ordered BSD.

The purpose of this step is to remove the energy component contained in all frequen

cies of interest. From (4.24) it can be observed that this would result in removing

the AWGN energy, A/q, from the BSD calculated in Step 1 and consequently allow

ing us to detect the knee point, k. In addition, the differentiation performed at

this stage will also allow us to exploit the frequency selectivity inherited in high

data rate communications [89]. This frequency selectivity will considerably have a

direct effect on the performance where all the sub-bands within the bandwidth of

the interest are all occupied.

In order to increase the reliability of the knee-point determination, a pre-determined

threshold, zu, can be employed. This will allow the proposed technique robust

towards heavily tailed noise and also power fluctuation due to limited sample size.

More interestingly, employing a threshold at this stage will signiflcantly improve

the performance of the proposed algorithm given the scenarios where the whole

bandwidth of interested is completely vacant. Thus, the knee-point will be the first

point in differentiate BSD where the corresponding magnitude is > zu.

Given the distribution of t q% one can determine the threshold value based on the

interference/throughput trade-off requirement of the system, see Section 4.5.3.

S tep 4: Zero force all the elements having a value less than the knee-point and

restore the original order.

Following the determination of the knee-point, the next step is to force the value of

all the elements in f <i' below the knee point to zero, i.e., t Qm = 0, Vm < «. Finally,

4.5. Performance Analysis for The Practical Solution 70

restore the original order of q using the inverse sorting transformation matrix used

in Step 2, i.e., q = t q ' (see Section 4.3.2). Sub-bands corresponding to a zero

value for m = 1, 2, • • • , M are assumed to be vacant.

From the above algorithm it can be observed that the proposed approach can be

used to offer spectrum sensing with sub-carrier level precision. This will benefit

system where low data rate/ bandwidth is required, e.g., sensor networks [90]. Since,

in most communications systems, operating devices require higher data rate, i.e.,

larger bandwidth, they would require more than one sub-carrier in order to transmit.

Hence, it would be to our interest to deliver a sub-band level accuracy.

4.5 Performance Analysis for The Practical Solu

tion

4.5.1 Asymptotic Probability of False Alarm and Threshold

In order to complete the construction of the spectrum sensing proposed in Section

4.4.3, we must provide a procedure to set the threshold vj. We propose to define

w in such a manner to keep PFA under a desired value. Given that for practical

applications, the PFA is often given a fixed value, such as 10% as per the FCCs

requirement. Hence, it would be necessary to derive the expression for PFA. As it

can be concluded from algorithm in Section 4.4.3, false alarm will occur when the

fluctuation resulting from two adjacent vacant sub-bands (i.e., noise only compo

nents) is greater than the threshold value. The exact probability density function

of the r th order statistic from any continues population is rather difficult to deal

with and in most cases requires numerical evaluation of a nontrivial integral [67].

Hence, in this section, we study the asymptotic behaviour of the order statistics of

q in order to determine a suitable threshold. Consider the special case of rig = 0.

It is understood that elements of q,„ (see (4.24)) given am = 0 follows [14]

/q (qrn|am=o) — X% (4.25)

where Xn denotes a central Chi squared distribution with N degrees of freedom.

Hence, after the sorting operation, t q (see 4.4.3 Step 2) based on the derivation


in Appendix E can be approximated as

t q . ~ g ( v n ■ — j (4.26)

where Q (a, b) denotes a Gaussian distribution with mean and variance equal to a

and b respectively, = f and $q(o) = i.e., CDF of qm|a,„=o- After

differentiation, f qrlar=ar-i=o will follow a Gaussian distribution with mean

Mq'r ~ (4.27)

and variance

+ j^o.5 • (4.28j

Hence, given a desired value of PFA in system, i.e., PFA< 'ijj the threshold value

can be obtained by

PFAr = q ( — (4.29)

where Q(.) is the standard Gaussian complementary CDF. As it can be observed,

the PFA is mainly dependent on two parameters, N and M. Since M is fixed

in any system, the performance can be improved by choosing longer observation

length, i.e., larger N. Furthermore, given a system with requirement of PFAr< w ,

the minimum threshold value, i.e., Wmm,r can be formulated as

^min,r — Mtqr (^r) • (4.30)

Since in the proposed algorithm one global threshold is used, zumin is set to max (tamin.r)

in order to meet the minimum requirement of the system. Given that in most sys

tems each sub-band has an equal PFA tolerance based on (4.27)-(4.30), rumin =

iUmin,K. In the case where the distribution of k is not known Wmin,K can be replaced

with Wrain,M to allow immunity in the worst case scenario.

4.5.2 Asymptotic Probability of “knee-point” Detection

As mentioned in Section 4.4.3, given a composite signal, by sorting the BSD in

order of magnitude, there would exist a point, namely the “fcnee-pomf’, where


there is a sudden change in the magnitude. In this section, we we will derive the

asymptotic distribution of the knee point. Given that the knee-point, «, is defined

as t q« = t Qk” t Q.k-1 where = 1 and 0:^-1 = 0. Based on Appendix E we have

t q . - i ~ S ( » " : ) , — j (4.31)

given that /g|a=o is defined in (4.25). Furthermore,

r , . ~ c (5.- (f--), " "<1: *’ '•1”") M

where [14]

9q (qm|am=l) ~ X n i^ P ) (4.33)

where, Xn (p) denote a non-central Chi squared distribution with the non-centrality

factor 2 p and H q (.) denoted the CDF of qm|am=i- Hence, after differentiation t q«

will follow a Gaussian distribution with mean

( i f ' " ) - i) (4-34)

and variance

fq;."' (M-K)OG

Given the distribution of a user activity within a region, i.e., /«(«;), probability of

detecting the right knee-point, PD«, can be formulated as

P D „ = / ° ° Q ( ” ~ / ^ ‘‘'-)/(K )dK .. (4.36)JQ \ /

11 I I. I y , —.1 II ✓

r(tu)

Given a system with requirement of PDk> r , the maximum threshold value, i.e.,

Wmax can be formulated as Wmax = F"^ (r). From the above derivations it can be

observed that the PD of the proposed sub-band level spectrum sensing is directly

prepositional to the number of sub-bands in the system of interest, M, observation

length, iV, the signal-to-noise ratio p at the sensing device and the threshold value,

VO.


4.5.3 Optimal Asymptotic Threshold Setting

It should be noted that value of the threshold, should be set in such a manner that

meets both the PD and PFA requirements of system, i.e., zumin < ro < zumax- More

specifically the exact value of threshold will be set in order to maximise the overall

throughput of the system while minimising the aggregate interference to the other

users. Given that the aggregate opportunistic throughput of the r th sub-band is

defined by

r..(ro) = J ^ 6 i( l-P F A j( ro ) ) (4.37)*eSr

where denotes the set of sub-carriers contained in the rth sub-band, bi denotes

the achievable rate of the i*^sub-carriers. Furthermore, the aggregate interference

to the user occupying the rth sub-band, can be evaluated as

^ Q (1 - PDi (a;)) (4.38)

where Q denotes the cost incurred if the user using the sub-carrier is inferred

with. As it can be observed from (4.37)-(4.38), opportunistic throughput and ag

gregate interference are directly promotional to PFA and PD respectively. Hence,

optimal threshold can be evaluated through receiver operating curve, known as

0.6

0.5

0.05 0.15

= 2, n , = 0 .5

= 5, rij = 0.5

= 10, = 0.5

_____N = 2. = 0.75

= 5, = 0.75

= 10, ti| = 0.75

0.25 0.3 0.35

Figure 4.4: ROC Curve of the proposed algorithm in Section 4.4.3 for p = 2 dB.


ROC curve. Figure 4.5.3 illustrates the ROC curve of the proposed algorithm in

Section 4.4.3. The ROC is shown for p = 2 dB, for different value of N and rjj^.

As it can be observed the performance improves for lower values of k . This was

expected as the noise power fluctuation for lower k would be less as it can be ob

served from (4.27)-(4.28) which would reflect in higher value for minra which in

turn would decrease the resulting PD. Detailed explanation on optimal threshold

setting based on interference/ throughput trade-off can be found in [50].

4.5.4 Computational complexity

The main complexity of the proposed scheme is due to the following stages:

1. The second-order time average: this stage requires x J complex multipli

cations and additions.

2. Discrete Fourier Transform: M —point DFT is implemented which introduces

the complexity by (9(Mlog(M)).

3. Sorting: the complexity of this stage on average is OMlog(M) (depending on

which sorting technique is employed).

4. Differentiation: this stage consists of subtracting every element of q from its

previous one, hence the computational complexity is given by 0{M ).

5. Knee point detection: This would add a further complexity of 0{M ).

6. Restoring the original order: a further complexity of O M log(M) is added due

to this stage.

Hence the overall computational complexity is:

JAT2 -K 3 0(M log(M )) 4- 2 0{M ). (4.39)

It can be concluded that the proposed algorithm is not very demanding in terms of

computational complexity.

4.6. Further Discussions 75

4.6 Further Discussions

4.6.1 Overcoming Timing Offset

In fact, the special case of rig = 0 can be hardly captured due to the lack of timing

synchronizations mechanism before the spectrum sensing component. Hence, in

order to handle the problem of unknown timing offset effectively, a “one ballot

ueto" policy is proposed to reject the vacancy of sub-bands. The policy is stated

as follows:

S tep 1; Form J x 1 vectors, = [rA=j+i+j,rA;j+2+j,...

where 6 denotes the offset in time.

S tep 2 : Compute according to (4.23), for ^ = 0, ( J — M), 2(J — M ), ..., M .

Apply the sub-band level spectrum sensing proposed in Section 4.4.3 on VS.

If for any value of Ô the energy within a sub-band falls above the knee-point value

it is understood that this particular sub-band is being used by an other user and is

not available for opportunistic use.

The key idea for proposing the “one ballot veto^’’ is that in the presence of a

signal, there exists such a S fulfilling the condition \n^ — S\ < J — M , and under

this condition, the proposed spectrum sensing scheme can successfully reject the

inter-block interference. In the absence of signal, is approximately constant

with respect to 5, due to constant energy of AWCN throughout the spectrum. Most

certainly, this stage will add to the overall complexity of the algorithm, however, in

order to increase the reliability of the sensing device, implementation of this stage

is necessary.

4.6.2 Overcoming the Energy Leakage

In all communication systems the transmitted signals are shaped by digital pulse-

shaping and analog band-pass filters according to a predefined frequency mask,

which in fact are not a perfect rectangular filters. As a result, the tail of the

transmitted signal in one sub-band might introduce significant interfering power

in adjacent sub-band provided that the signal is very strong. Moreover, the re

ceived samples at the sensing device are channelized with a non-ideal filter, which

also causes spectral leakage. Furthermore, the special case of perfect frequency

synchronisation, i.e., £ = 0 , can be hardly captured due to the lack of frequency

4.6. Further Discussions 76

synchronisation mechanism before the spectrum-sensing component. As a result

the energy of frequency bands will disperse [91] creating a challenge for sub-band

level spectrum sensing device. In order to combat this energy leakage and allow

robustness towards timing and frequency offsets requires the algorithm introduced

in Section 4.4.3 to be modified.

A possible solution would be to modify the condition stated in Step 4. Based on this

stage all the sub-carriers of a sub-band should be set to zero in order for that sub

band to be used by the opportunist user. Our proposed solution is to change this

condition by determining the vacancy of a sub-band not based on all, but at on a

portion of the sub-carriers within a sub-band. It is shown in [92] the energy leakage

has a significant effect on the sub-carriers situated at the edges of a sub-band while

the central sub-carriers are less effected. Based on this fact it can be concluded

that it is of our interest to determine the vacancy of a sub-band according to the

magnitude of the central sub-carriers. It was found through experiment that using

25 — 50% of the central sub-carriers we are able to overcome the energy leakage

problem significantly.

4.6.3 Extension to the ZP-based system

Let us start from the special case of = 0 and £ = 0. Using the result in [57],

we can easily justify that due to the implementation of ZP the second term in the

brackets, i.e., V(ne)xfc_i, of (4.22) is vanished. Therefore, performing a J-point

DFT on (4.22) yields

ffc = .FjAfc A(rie)xfc + .FjVfc, (4.40)

where JFj is an J x J DFT matrix normalized by the factor ( l ) / ( \ / j ) . Hence,

applying second order average on (4.40) will result in

(4.41)

where ^ C j is an J x J circulant channel matrix with T>j formed by

the corresponding channel frequency response. It is easy to observe that (4.41)

has the same form as (4.23). Therefore, the spectrum-sensing algorithm proposed

in Section 4.4.3 for the CP-based system can be straightforwardly applied on ZP-

based OFDMA systems. Furthermore, overcoming energy leakage policy explained

in Section 4.6.2 and the “one ballot veto” can also be employed in downlink ZP-

4.7. Simulation Results 77

Excess tap delay [ns] Relative power [dB]0 0

30 -170 -290 -3110 -8190 -17.2410 -20.8

Table 4.1: Extended Pedestrian A model

based OFDMA systems in order to show robustness towards physical impairments.

4.7 Simulation Results

Computer simulations were performed to evaluate the proposed spectrum-sensing

scheme. The system investigated in this section has M = 1,024 sub-carriers with the

sub-carrier spacing of 15 KHz,each frame consists of 7 OFDM blocks with the CP

length of J —M = 80, the sampling frequency is 30.72 MHz and the bandwidth is set

to 20 MHz. The channel is divided into 32 equal sub-bands, allowing each sub-band

to have 32 sub-carriers. The carrier frequency is also set at 5 GHz. We consider a

power delay profile, as given in Table 4.7 i.e., extended pedestrian A model, which

is defined for LTE based systems. This channel specified the frequency selective

channel between the transmitter and the spectrum sensing device. The sensing

device is assumed to be moving at the speed of 3 km/h. The SNR is defined by the

average received symbol energy to noise ratio at the sensing device.

Experiment 1

In this experiment we are aiming to observe the input behaviour and properties

at different stages of the algorithm explained in Section 4.4.3. Figure 4.5 shows

the step-by-step procedure which is carried out in the proposed solution. For this

experiment the SNR at the sensing device, p, is set to 2 dB and the observation

length N is equivalent to 2 frames duration. As it can be observed due to limited

sample size the estimated ESD at the sensing device experiences a large amount of

fluctuation. This energy fluctuation, due to limited size, will have a detrimental ef

fect on the performance of the existing spectrum sensing solution. More specifically,

this implies the conventional edge detection based approaches would not be deliv

ering an acceptable performance in such environments. The ESD is then ranked


according to the magnitude, while the original ordering indexes are saved. As it can

be observed this stage will allow more robustness in terms of limited sample size as

it smoothens the fluctuation of the estimated ESD, while not effecting any statisti

cal properties of the input ESD. More interestingly, the existence of the knee-point

is very apparent, reflecting the point where the two components of the composite

signal, in this case noise and distorted transmitted signal, can be distinguished,

as mentioned in the Section 4.4.3. An other interesting observation which can be

made is that the differential values after the knee point, i.e., reflecting the occupied

sub-carriers, are higher comparing to the value denoting the noise only sub-carriers.

This is due to the frequency selective propagation channel and is explained in detail

in [79] [89]. This phenomenon will further improve the performance of the proposed

approach and reflects that we can use the frequency diversity of the received signal

to further verify the final results. An interesting point worth noting is the sudden

change which occurs at the tail of the differential ESD, this sudden taking place is

mainly due to the propagation channel. Hence, the tail of the differential ESD can

be employed for characterising and estimating the channel coefficients. This might

be an interesting factor when considering interference cancellation at the cognitive

device, however, it is out of the scope of this thesis.

n . n n n ,nn Estimated ESD of the Distorted Onginal Signal ESD

Ordered Estimated ESD

« 0.8 K nee Point

0.015

dioi0.005

0

0 0.2 0.4 0.6 0.8Index

1.2 1.4 1.6 1.8

x i o ’— Differentiated Ordered Estimated ESD| ! ' ' '

-K nee Point ---------- >

-

to L . !0 0.2 0.4 0.6 0.8

InJex1.2 1.4 1.6 1.8 2

xIO ^

Frequency

Figure 4.5: Step by step process of the proposed algorithm in Section 4.4.3 in order to determine the vacancy of unused sub-bands.


Finally the frequency bands which reflect a zero value energy after the zero forcing

are considered as white space [16], i.e., vacant sub-bands and can be used via the

opportunistic spectrum access devices. The solution explained in Section 4.6.2 was

further employed to overcome the possible energy leakage in estimated ESD. Hence,

the flnal decision of whether a sub-band is vacant or not, was based on 50% of

the central sub-carriers in each sub-band. It is advisable to use > 25% of the

central sub-carriers to allow robustness in terms of the deep fading of the central

sub-carriers.

Experiment 2

The objective of this experiment is to evaluate the performance of the proposed

algorithm and compare it with the state-of-the-art sub-band level spectrum sensing

techniques. Firstly, we would like to compare the performance with the energy

detection based filter-bank approach, which is the most practical existing sub-band

level spectrum sensing solution. The energy detection is recognised as a blind

sensing scheme with advantages such as low complexity and low latency. However,

it is very sensitive to the noise uncertainty such that its performance is limited by

the SNR wall [9].

l o . e *s

1 — Pr opos ed algorithm N = 5 F ram es — Pr opos ed algorithm N = 10 F ram es

Energy D etection U=2 dB, N = 10 F ram es Energy D etection U=1 dB, N = 10 F ram es

1 1 i i i i0.01 0 .02 0 .03 0 .04 0.05 0.06 0 .07 0 .08 0 .09 0.1

Probability of F alse Alarm

Figure 4.6: ROC curve comparison of the proposed algorithm and energy detection based filter-bank approach.

The simulation performed for energy detection are based on noise uncertainty factor,

Î7 = 1,2 dB and the threshold is based on the assumed/ estimated noise power, while


0 .8£

0.7£

0 .65

-6 - 4

) — () ^ * 6 > 0 6 --------

.........•] ........ ..................... / < : ..........

............

....... ................................... L '..: ............... .................

A / u 7 :/ ; f i / ' ;

—a — P roposed T echnique, N =2 F ram es — Pr opos ed Technique, N=5 F ram es —O — P roposed Technique, N =10 F ram es - O - Energy D etection, N= 10 F ram es, U= IdB “ O - Energy D etection, N= 10 F ram es, U= 2dB• ' K urtosis B ased D etection, N= lO F ram es• • E igenvalue b ase d D etectio, N=10 F ram es

i

f

i

0 2 4S N R (dB)

10

Figure 4.7: Performance comparison of the the proposed algorithm and energy detection based filter bank in terms of probability of detection for PFA = 0.01.

the real noise power varies with each Monte Carlo realisation by a certain degree.

Figure 4.6 illustrates the ROC curve of the proposed algorithm and the filter-bank

solution for SNR level of 3 dB for = 0.65. As it can be observed the proposed

algorithm can easily outperform the energy detection based filter-bank approach in

low SNR region. Furthermore, it can be observed that the performance of energy

detection is considerably dependent on the noise uncertainty factor. It is further

proved in [9] that increasing the observation length does not affect the performance

of the energy detection where the exact noise power is not known, i.e., U ^ 0 .

The performance of the proposed algorithm in terms of PD for various values of

observation length, N , is shown in Figure 4.7. The threshold is set in such a way to

ensure PFA< 0.01. As it can be observed the performance of the energy detection

based filter-bank drops dramatically as it hits the SNR wall while the proposed

algorithm degrades slowly for decreasing value of SNR. This can be particularly

useful in practical scenarios, reflecting the fact the proposed algorithm can detect

heavily shadowed signal. The performance comparison has been also carried out

for kurtosis based fllter-bank, where a decision is made based on the kurtosis test

which exploits the non-Gaussianity of communication signals [47] [48]. This scheme

features excellent accuracy at the price of large latency due to higher-order statistics.

However, as expected due to limited sample size, the proposed scheme can outper

form the kurtosis based detection by more than 10 dB. Filter-bank eigenvalue based

detection [35] was also used to carry out a more comprehensive performance com

4.8. Summary 81

parison of the proposed technique. The eigenvalue-based detection scheme exploits

orthogonality between the signal subspace and noise subspace using second order

stationarity features to offer highly reliable spectrum sensing [35]. For this perfor

mance comparison we used the maximum to minimum ratio of the eigenvalue as the

test statistic. It can be observed from Figure 4.7, the proposed technique outper

forms the eigenvalue based detection in low SNR environment. This was expected

since eigenvalue detection requires higher observation length.

4.8 Summary

In this chapter, a novel Bayesian model based approach for joint sub-band level

spectrum sensing has been proposed. This contributions is three-fold 1) re-defining

the objective of the sub-band level spectrum sensing device to a model estimator, 2)

deriving the optimal model selection estimator for sub-band level spectrum sensing

for fixed and variable number of users along with a sub-optimal solution based on

Bayesian statistical modelling and 3) proposing a practical model selection estimator

with relaxed sample size constraint and limited system knowledge for sub-band

spectrum sensing applications in Orthogonal Frequency-Division Multiple Access

and Non-Contiguous Orthogonal Frequency-Division Multiplexing systems. The

key idea behind the proposed approach is to exploit the second order frequency

diversity between signal and noise. Based on this approach after the ESD estimation

at the sensing device, they are ordered in terms of magnitude. The sorting operation

allows robust to noise power fluctuation due to limited observation length.

Furthermore, given a composite signal, by sorting the ESD in order of magnitude,

there would exist a point, namely the knee-point, where there is a sudden change


distributions. A differentiation stage is further employed. The objective of this

stage is two-fold, to remove the noise floor and to observe the knee-point. This

stage will also allow us to exploit the frequency selectivity inherited in high data rate

communications. This frequency selectivity will considerably have a direct effect on

the performance where the all the sub-bands within the bandwidth of the interest are

all occupied. In order to increase the reliability of the knee-point determination, a

pre-determined threshold is employed. This will allow the proposed technique robust

towards heavily tailed noise and also power fluctuation due to limited sample size.

More interestingly, employing a threshold at this stage will significantly improve

4.8. Summary 82

Approach Air-InterfaceInformation Synchronisation Computational

Complexity Latency

Filter Bank X X Low Low

Joint M ulti-Band Detection CompleteKnowledge / High High

Sequential M ulti-Band Detection CompleteKnowledge X High Low

W avelet Based Detection X X Medium Low

W igner-Ville Based Detection X / High High

Bayesian Based Appraoch PartialKnowledge X Medium Low

Table 4.2: Comparison of the state-of-the-art local sub-band spectrum sensing approaches and proposed Bayesian approach.

the performance of the proposed algorithm given the scenarios where the whole

bandwidth of interested is completely vacant. Thus, the knee-point will be the first

point in differentiate ESD where the corresponding magnitude is greater than the

predetermined threshold. Following the determination of the knee-point, the next

step is to force the value of all the elements in the ordered differentiated ESD below

the knee point to zero. Finally, the ordered differentiated ESD is re-sorted using

the inverse sorting transformation matrix used in the sorting stage. The sub-bands

corresponding to zero value ESD are considered to be white space and are available

for opportunistic spectrum access use.

More interestingly, the proposed sub-band level spectrum sensing does not require

a priori knowledge of noise power and the propagation channel gain, and is designed

in such a way to show robustness towards energy leakage, without introducing high

complexity to the overall system, i.e., less the cubic computational complexity.

The proposed sub-band level spectrum sensing approach is analytically evaluated

through asymptotic probability of false alarm and probability of detection. It was

proved that the performance of the proposed approach is directly promotional to

the observation length, the total number of the sub-carriers and the SNR of the

received signal at the sensing device. Computer simulations are carried out in

order to evaluate the effectiveness of the proposed scheme. Through appropriate

threshold adjustments, the proposed approach can be extended to state- of-the-art

OFDMA/NC-OFDM joint sub-band level spectrum sensing.

hapter 5Conclusion and Future Work

5.1 Conclusion

In this thesis the original problem of local spectrum sensing in cellular networks was

investigated. Firstly, a comprehensive review of the existing narrow-band spectrum

sensing algorithms was presented. The existing approaches were divided into three

main categories:

• Explo iting energy difference. The most well-known spectrum sensing

approach developed under this category is the energy detection [11]. The

energy detection is recognised as a blind sensing scheme with advantages such

as low complexity and low latency. However, it is very sensitive to the noise

uncertainty, resulting in that its performance that is limited by the SNR

wall [9].

• Explo iting s ta tio n a rity difference. The initial works of spectrum sensing

through stationarity difference can be traced back to work of Dandawate et al

in [25]. This category of spectrum sensing schemes can trade latency with high

sensing reliability. Furthermore, they are sensitive to the noise uncertainty.

• Explo iting th e d istr ib u tio n difference. Given that in almost all commu

nication system models, noise is assumed to be additive white and Gaussian,

one can determine the vacancy of a particular frequency band by observing

the difference of the received signals distribution and that of the AWGN.

These schemes feature excellent accuracy at the price of large latency due to

83

5.1. Conclusion 84

higher-order statistics. A critical point of such sensing approaches is that their

performance degrades significantly when signals are approximately Gaussian.

Furthermore, the problem of sub-band level spectrum sensing was explored and it

was found that the literature of sub-band spectrum sensing is rather limited at this

time. However, a full survey of the state-of-the-art sub-band level spectrum sensing

was also provided.

In terms of contribution firstly, a novel differential energy detection scheme for multi

carrier systems, namely, cluster-based differential energy detection, which can form

fast and reliable decisions on spectrum availability even in very low SNR environ

ment, has been proposed. It has several distinctive features including low latency,

high accuracy reasonable and computational complexity, as well as robustness to

low SNR. For example, the proposed scheme can reach 90% in probability of detec

tion and 10% in probability of false alarm for an SNR as low as —21 dB, while the

observation window is equivalent to 2 multi-carrier symbol duration. The proposed

scheme at this stage is specially designed for sensing multi-carrier sources but we

would argue since most of the current and future mobile networks are multi-carrier

based systems, the proposed scheme has potentially wide-ranging practical appfi-

cations. The proposed approach can deliver desirable performance in high density

communication networks and urban area due to its ability to allow robustness in low

SNR environments. Furthermore, the clustered-based differential energy detection

can be employed in vehicular communication systems.



energy spectral density. Initially, the ESD of the received signal is estimated. Specif

ically, after the ESD computation, the clustering operation is utilized to group un

correlated subcarriers based on the coherence bandwidth to enjoy a good frequency

diversity. The knowledge of coherence bandwidth does not need to be very accurate

(we employ the reciprocal of the maximal channel delay). Furthermore, making use

of order statistics of the estimated ESD, we further increase the reliability of the

sensing algorithm. This will allow us to smooth the fluctuation in noise ESD, re

sulted from limited observation length with effecting the statistics of the received

signal. Hence, this will stage will have a direct effect on the PFA of the proposed

approach.


differential operation is performed on the rank ordered ESD. When the channel is

5.1. Conclusion 85

frequency selective and the noise is white, the differential process can effectively re

move the noise floor resulting in elimination of the noise uncertainty impact which

is the main factor making energy detection reluctant [9]. Furthermore, the differ

ential stage will allow us to exploit the frequency selectivity which available in the

received signal. At the final stage of the proposed scheme, the differential rank or

dered ESD within different clusters are linearly combined in order to further reduce

the effect of impulse/spike noise. Binary hypothesis testing is then applied on either

the maximum or the extremal quotient (maximum-to-minimum ratio) depending on

the wireless channel characteristics of the sensed environment. More importantly,

the proposed spectrum sensing scheme is designed to allow robustness in terms of

both, time and frequency offset, without compromising computational complexity.

To analytically evaluate the proposed scheme, both PD and PFA were derived for

Rayleigh fading channels. The closed-form expression showed a clear relationship





schemes where up to 10 dB gain in performance can be observed. This would imply

that employing the proposed approach in a communication system will make the

network of interest more robust to hidden node problem, i.e., mitigate interference

to heavily shadowed licensed users. More interestingly, it has been shown through

simulation that the proposed spectrum sensing algorithm has a long convergence

time, which allows it a suitable in delay sensitive systems.

Secondly, a novel Bayesian model based approach for joint sub-band level spectrum

sensing has been proposed. This contributions is three-fold 1) re-defining the objec

tive of the sub-band level spectrum sensing device to a model estimator, 2) deriving

the optimal model selection estimator for sub-band level spectrum sensing for fixed

and variable number of users along with a sub-optimal solution based on Bayesian

statistical modelling and 3) proposing a practical model selection estimator with re

laxed sample size constraint and limited system knowledge for sub-band spectrum

sensing applications in Orthogonal Frequency-Division Multiple Access and Non-

Contiguous Orthogonal Frequency-Division Multiplexing based systems. The key

idea behind the proposed approach is to exploit the second order frequency diversity

between signal and noise. Based on this approach, after the ESD estimation at the

sensing device, the ESD is ordered in terms of magnitude. The sorting operation

5.1. Conclusion 86

allows robust to noise power fluctuation due to limited observation length.

Furthermore, given a composite signal, by sorting the ESD in order of magnitude,

there would exist a point, namely the knee-point, where there is a sudden change


distributions. A differentiation stage is further employed. The objective of this

stage is two-fold, to remove the noise floor and to observe the knee-point. This

stage will also allow us to exploit the frequency selectivity inherited in high data

rate communications. This frequency selectivity will considerably have a direct

effect on the performance where the all the sub-bands within the bandwidth of

the interest are all occupied. In order to increase the rehability of the knee-point

determination, a pre-determined threshold is employed. This will ensure robustness

of the proposed technique against heavily tailed noise and also power fluctuation due

to limited sample size. More interestingly, employing a threshold at this stage will

significantly improve the performance of the proposed algorithm given the scenarios

where the whole bandwidth of interested is completely vacant. Thus, the knee-point

will be the first point in differentiate ESD where the corresponding magnitude

is greater than the predetermined threshold. Following the determination of the

knee-point, the next step is to force the value of all the elements in the ordered

differentiated ESD below the knee point to zero. Finally, the ordered differentiated

ESD is re-sorted using the inverse sorting transformation matrix used in the sorting

stage. The sub-bands corresponding to zero value ESD are considered to be white

space and hence available for opportunistic spectrum access use.

More interestingly, the proposed sub-band level spectrum sensing does not require

a priori knowledge of noise power and the propagation channel gain, and is designed

in such a way to show robustness towards energy leakage, without introducing high

complexity to the overall system, i.e., less the cubic computational complexity.

The proposed sub-band level spectrum sensing approach is analytically evaluated

through asymptotic probability of false alarm and probability of detection. It was

proved that the performance of the proposed approach is directly proportional to

the observation length, the total number of the sub-carriers and the SNR of the

received signal at the sensing device. Computer simulations are carried out in

order to evaluate the effectiveness of the proposed scheme. Through appropriate

threshold adjustments, the proposed approach can be extended to state- of-the-art

OFDMA/NC-OFDM joint sub-band level spectrum sensing.

5.2. Future Work 87

5.2 Future Work

To this point, further work in the context of local spectrum sensing can be consid

ered. These works can either be extensions to the proposed schemes or they can be

new and are summarised as follows:

Firstly, the clustered differential energy detection based approach proposed in Chap

ter 3, assumes a priori knowledge about some of the air-interface parameters , e.g.

number of sub-carriers and channel propagation bandwidth. It would be interest

ing if this work can be extended to a blind spectrum sensing technique where no

information of the operating users are required.

Secondly, the proposed sub-band level spectrum sensing technique presented in

Chapter 4, is developed for downlink OFDMA scenarios. Hence, this calls for an

updated system model which also accommodates uplink scenarios. It should be

noted that the problem of uplink spectrum sensing is a very promising research

topic, since, to this date very limited research has been carried on in this context.

This topic can be very challenging since it has to account for multi-dimensional

synchronisations due to multiple users. Furthermore, it would be desirable to extend

the proposed sub-band level spectrum sensing technique so that it would operate

without a prior knowledge of the signalling information.

Thirdly, for making the most out of the opportunities, spectrum sensing device

should keep track of variations in spectrum availability and should make predictions.

Hence, the history of the spectrum usage information can be used for predicting the

future prole of the spectrum. It will very interesting to incorporate the knowledge

about currently active users or prediction of the spectrum usage in to the final

decision made by the spectrum sensing device.

Appendix A

Probability Distribution Function of

the Differential ESD

The moment generating function (MGF) of the output of the differential process in

step S2 in Section 4.4.3, 9^1% , (which is expressed as the product of the MGF of

two central Chi squared random variables shown in (3.23)) is given by [24]

A4(g |9Yo) = A4(gi+i|%) x Ai(%|%o)

= (1 - 2 t) - ^ X (1 + 2 t) - ^ = (1 - 4 f ) " ^ (A.l)

where A4 (a) denotes the MGF of random variable a, and t denotes the time-domain

index. Hence, (A.l) indicates that follows a summation of K i.i.d. Laplace

distributions (with location parameter fi — 0 and scaling parameter ^ = 2), whose

their MGF functions are given by [24]

M { ^ ) = _ ^2 2

where J f denotes a random variable following a Laplace distribution. Moreover, we

know for a fact that qhlHo is non-negative due to the sorting process. Therefore, it

can be concluded that distribution of can be further simplified to summation

of K exponential distributions with scale factor 0.5. Hence, the probability density

function (p.d.f.) of

(A.3)

Appendix B Distribution of the Test I statistics

Since q ' is i.i.d., for 1 < m < L, we would have

FAax(ql%o)(Ai) = Pr(max(q'|?4o) < Ai)

= Pr(9z, > A i,çi, > 9m Vm 7 L^Hq)

UPr(9l_i > A i,9 i > ^ Vm f L - 1 |% ) U . . .

UPr(9 i > A i , q[ > 9 ^ Vm f l|74o)

= P r % > A i , 9 ^ > q'rrym f L\Hq)

^ Ai, 9 > q '^ m ^ L — l \ H o ) + . . .

+Pr(9i > Ai,9l > ôLVm f 1|% )

= [FgilWo(Ai)]^- (B.l)

PFA for Test I can then be evaluated using the cumulative density function (CDF)

of max(q'|'Ho)j given qi\Ho variables are i.i.d.

Test I : PFA= Pr(max(q'|74o) > Ai)

= 1 - [Fmax(q'|Wo) (Ai)]V---------- '

âPr(m ax(q'|H o)<Ai)

= 1 - ( A i ) ] . ( B .2 )

89

90

Furthermore, the PD for Test I can be expressed as

Test I : PD = Pr(max(Ay|94i) > Ai)

= 1 - [FmaxCqOI?^! ( A i ) ]L

= i - n (B.3)1 = 1 -------1-PDTi

Appendix C Distribution of the Test II statistics

The Hypothesis for Test II is based on the ratio of maximum to minimum of q^|% ,

whose CDF can be computed using

P - / ' max(q'IHo) ^ ,

= P r ( ^ > A2,5i > r„.Vm Lkq'i < %'ig + l|Wo)U

• . . U P r ( g - > A 2 , î1 > ÿm Vm ^ L & ÿ i < % | % )

= P r ( ^ > A z .îi > %,Vm + < % !% )+

• •. + P r ( ^ > A2,ÿ i > fLVm f 2) & g i < (C.l)

Due to max / min(q'|?4o) > 0, we can further express (C.l) as

fCX)poo pA 2 P

■ F n iax /m in (q '|?^ o ) ( ^ 2 ) — I I { o c , P ) d ( XJo J (x=0

poo pX zP

+ / / fq'L-iQil'^o (^» ^) do: d;d + . . .Jo J q := 0

pOO pX20C+ / /g' g '170 (a,/3)d/3 do: (C.2)

Jo J b= o

Once iTiax/min(q'|7{o) (^2)18 obtained, the PFA for Test II can be easily computed

by

Test II : PFA = 1 - F ’m a x /m in (q '|W o )(A 2 )- (C.3)

Furthermore, for Text II we have

Test II : PD = Pr(m ax/m in(q'|'H i) > Ag)

~ t “ [Fmax(g'/9'l'^i) (A2 )]

(D= 1 - n F(A72/A7d^0i?ii (A2) (C.4)

1- p o p

91

Appendix D

Probability Distribution Function of

ESD Over Rayleigh Fading Channel

Using the recursion for (3.36), the following result is obtained

Q niax, h)

= g - ” Tj^/2-i{axy) + QK-i{ax,h)

K - l

= Y ] e " In{axy)Qi{ax,b). (D.l)n = 0

Applying (D.l) into (3.40), the integration part can be computed by

/>oo

= g ( ^ y rn = 0

rOO/ rr^^^~^^exp~^ ^^Qi(ax,P)dx (D.2)

J x = 0

where p = 2, and = y/X. Using combination of Bessel functions and

exponentials in [73], the above terms can be evaluated.

92

Appendix E

Asymptotic Distribution of The rth

Order Statistics

We will first approximate the rth order statistics of the uniform distribution and

using probability-integral transformation transformation we derive the r th order

statistics from any continues distribution. Given that Ur is the rth ordered value

from uniform distribution with a total of M elements, we have

= (E.1)

Replacing Zr = in above leads to

/Zr (z) = (E-2)

where

© = (r - 1) log ^1 -t- -h(M - r ) log ■ (E.3)

93

94

Using Taylor series expansion

0 =z

2

— (r — 1) — { M — r )

n 1-/LI

L/

/T 2 (T 2— (r — 1) — ------ (M — r)w I - u

cr 3(M — r)

(E.4)

Given M o o and r / M —)■ 77 we can make the following approximations

M — r + 1

r { M + 2)

0.5

M r ] J0.5

M(1 — 77)

0.5

(E.6)

(EG)1 -M [ { M - r + l ) { M + 2 ) _

Substituting the above in to (E.4) it can be found that the coefficient of z and z ^ / 3

tends to 0 as 71 0 0 while the coefiicient of —z^ / 2 converges to 1 for increasing

value of n . Using these findings and replacing them into (E.4) and ignoring terms

of order n ~ ^ ' ^ and higher we have, lim - cx) 0 = — Using Stirling’s formula, i.e.,

k \ = \/27rexp { — k ) and the approximation

^ (M + 1 )^ (M + 2)0-5y .r -0 .5 ( M - r + l ) ^ - r + 0 5

(E.7)

( M + 1 ) M + o.5

we obtain the following

(E.8)

Hence,

(E.9)

The probability-integral transformation allows us to conclude that the same asymp

totic distribution holds for any continuous distribution as long as the appropriate

] i and a are replaced. Furthermore it has been shown in [67] the asymptotic mean

and standard deviation of the rth value of the random variable x containing total

elements of M can be approximated as

/ = Fx (?7) &: (T =(77(1 - 77))°-^(/x (//))

MO-5 (E.IO)

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