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 OF m 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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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
ProQuest Number: 27558473
All rights reserved
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uestProQuest 27558473
Published by ProQuest LLO (2019). C opyrigh t of the Dissertation is held by the Author.
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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.
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
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-
3.3. Cluster-Based Differential Energy Detection 38
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
3.3. Cluster-Based Differential Energy Detection 39
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
3.3. Cluster-Based Differential Energy Detection 40
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)
3.3. Cluster-Based Differential Energy Detection 41
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
3.3. Cluster-Based Differential Energy Detection 42
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
3.4. Performance Analysis 44
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.
3.4. Performance Analysis 45
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. Performance Analysis 46
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
3.4. Performance Analysis 47
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)
3.4. Performance Analysis 48
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. Performance Analysis 49
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-
3.4. Performance Analysis 50
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:
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.
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. 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.
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 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
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. 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
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)
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.
4.3. Optimal and Sub-Optimal Solutions 62
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
4.3. Optimal and Sub-Optimal Solutions 63
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-
....... ................................... 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
in the magnitude. This point will allow us to distinguish components from different
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