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2 EURASIP Journal on Advances in Signal Processing

200 KHz. If a secondary user is several hundred metersaway from the microphone device, the received SNR maybe well below20 dB. Secondly, multipath fading and timedispersion of the wireless channels complicate the sensingproblem. Multipath fading may cause the signal power tofluctuate as much as 30 dB. On the other hand, unknown

time dispersion in wireless channels may turn the coherentdetection unreliable. Thirdly, the noise/interference levelmay change with time and location, which yields the noisepower uncertainty issue for detection [912].

Facing these challenges, spectrum sensing has reborn asa very active research area over recent years despite its longhistory. Quite a few sensing methods have been proposed,including the classic likelihood ratio test (LRT) [13], energydetection (ED) [9,10,13,14], matched filtering (MF) detec-tion [10,13, 15], cyclostationary detection (CSD) [1619],and some newly emerging methods such as eigenvalue-basedsensing [6, 2025], wavelet-based sensing [26], covariance-based sensing [6, 27, 28], and blindly combined energydetection [29]. These methods have different requirementsfor implementation and accordingly can be classified intothree general categories: (a) methods requiring both sourcesignal and noise power information, (b) methods requiringonly noise power information (semiblind detection), and(c) methods requiring no information on source signal ornoise power (totally blind detection). For example, LRT,MF, and CSD belong to category A; ED and wavelet-basedsensing methods belong to category B; eigenvalue-basedsensing, covariance-based sensing, and blindly combinedenergy detection belong to category C. In this paper, wefocus on methods in categories B and C, although someother methods in category A are also discussed for the sakeof completeness. Multiantenna/receiver systems have beenwidely deployed to increase the channel capacity or improvethe transmission reliability in wireless communications. Inaddition, multiple antennas/receivers are commonly usedto form an array radar [30, 31] or a multiple-inputmultiple-output (MIMO) radar [32, 33] to enhance theperformance of range, direction, and/or velocity estimations.Consequently, MIMO techniques can also be applied toimprove the performance of spectrum sensing. Therefore,in this paper we assume a multi-antenna system model ingeneral, while the single-antenna system is treated as a specialcase.

When there are multiple secondary users/receivers dis-tributed at different locations, it is possible for them to

cooperate to achieve higher sensing reliability. There arevarious sensing cooperation schemes in the current literature[3444]. In general, these schemes can be classified into twocategories: (A) data fusion: each user sends its raw data orprocessed data to a specific user, which processes the datacollected and then makes the final decision; (B) decisionfusion: multiple users process their data independently andsend their decisions to a specific user, which then makes thefinal decision.

In this paper, we will review various spectrum sensingmethods from the optimal LRT to practical joint space-timesensing, robust sensing, and cooperative sensing and discusstheir advantages and disadvantages. We will pay special

attention to sensing methods with practical applicationpotentials. The focus of this paper is on practical sensingalgorithm designs; for other aspects of spectrum sensing incognitive radio, the interested readers may refer to otherresources like [4552].

The rest of this paper is organized as follows. The

system model for the general setup with multiple receiversfor sensing is given in Section 2. The optimal LRT-basedsensing due to the Neyman-Pearson theorem is reviewedin Section 3. Under some special conditions, it is shownthat the LRT becomes equivalent to the estimator-correlatordetection, energy detection, or matched filtering detection.The Bayesian method and the generalized LRT for sensingare discussed in Section 4. Detection methods based onthe spatial correlations among multiple received signals arediscussed in Section 5, where optimally combined energydetection and blindly combined energy detection are shownto be optimal under certain conditions. Detection methodscombining both spatial and time correlations are reviewed inSection 6, where the eigenvalue-based and covariance-baseddetections are discussed in particular. The cyclostationarydetection, which exploits the statistical features of the pri-mary signals, is reviewed inSection 7. Cooperative sensingis discussed inSection 8. The impacts of noise uncertaintyand noise power estimation to the sensing performanceare analyzed inSection 9. The test statistic distribution andthreshold setting for sensing are reviewed inSection 10,where it is shown that the random matrix theory is veryuseful for the related study. The robust spectrum sensingto deal with uncertainties in source signal and/or noisepower knowledge is reviewed in Section 11, with specialemphasis on the robust versions of LRT and matched filteringdetection methods. Practical challenges and future researchdirections for spectrum sensing are discussed inSection 12.Finally,Section 13concludes the paper.

2. System Model

We assume that there are M 1 antennas at the receiver.These antennas can be sufficiently close to each other toform an antenna array or well separated from each other.We assume that a centralized unit is available to process thesignals from all the antennas. The model under considerationis also applicable to the multinode cooperative sensing [3444,53], if all nodes are able to send their observed signals toa central node for processing. There are two hypotheses: H0,

signal absent, and H1, signal present. The received signal atantenna/receiveriis given by

H0 : xi(n) = i(n),

H1: xi(n) = si(n)+i(n), i = 1, . . . ,M.(1)

In hypothesis H1, si(n) is the received source signal atantenna/receiveri, which may include the channel multipathand fading effects. In general,si(n) can be expressed as

si(n) =K

k=1

qikl=0

hik (l)

sk(n l), (2)


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EURASIP Journal on Advances in Signal Processing 3

where K denotes the number of primary user/antennasignals,sk(n) denotes the transmitted signal from primaryuser/antenna k, hik(l) denotes the propagation channelcoefficient from the kth primary user/antenna to the ithreceiver antenna, andq ik denotes the channel order forhik.It is assumed that the noise samples i(n)s are independent

and identically distributed (i.i.d) over both n and i. Forsimplicity, we assume that the signal, noise, and channelcoefficients are all real numbers.

The objective of spectrum sensing is to make a decisionon the binary hypothesis testing (chooseH0or H1) based onthe received signal. If the decision is H1, further informationsuch as signal waveform and modulation schemes may beclassified for some applications. However, in this paper, wefocus on the basic binary hypothesis testing problem. Theperformance of a sensing algorithm is generally indicated bytwo metrics: probability of detection, Pd, which defines, atthe hypothesis H1, the probability of the algorithm correctlydetecting the presence of the primary signal; and probabilityof false alarm,

Pf a, which defines, at the hypothesis H

0,

the probability of the algorithm mistakenly declaring thepresence of the primary signal. A sensing algorithm is calledoptimal if it achieves the highest Pdfor a givenPf a with afixed number of samples, though there could be other criteriato evaluate the performance of a sensing algorithm.

Stacking the signals from theMantennas/receivers yieldsthe followingM 1 vectors:

x(n) =

x1(n) xM(n)T

,

s(n) =

s1(n) sM(n)T

,

(n) = 1(n) M(n)T.(3)

The hypothesis testing problem based onNsignal samples isthen obtained as

H0 : x(n) = (n),

H1: x(n) = s(n)+ (n), n = 0, . . . , N 1.(4)

3. Neyman-Pearson Theorem

The Neyman-Pearson (NP) theorem [13,54,55] states that,for a given probability of false alarm, the test statistic thatmaximizes the probability of detection is the likelihood ratiotest (LRT) defined as

TLRT(x) = p(x| H1)p(x| H0) , (5)

where p() denotes the probability density function (PDF),and xdenotes the received signal vector that is the aggre-gation ofx(n), n= 0,1, . . . , N 1.Such a likelihood ratiotest decides H1whenTLRT(x) exceeds a threshold, and H0otherwise.

The major difficulty in using the LRT is its requirementson the exact distributions given in (5). Obviously, thedistribution of random vector xunder H1 is related to the

source signal distribution, the wireless channels, and thenoise distribution, while the distribution ofxunder H0 isrelated to the noise distribution. In order to use the LRT, weneed to obtain the knowledge of the channels as well as thesignal and noise distributions, which is practically difficult torealize.

If we assume that the channels are flat-fading, and thereceived source signal samplesi(n)s are independent overn,the PDFs in LRT are decoupled as

p(x| H1) =N1n=0

p(x(n) | H1),

p(x| H0) =N1n=0

p(x(n) | H0).(6)

If we further assume that noise and signal samples are bothGaussian distributed, that is, (n) N(0, 2 I) and s(n)N(0, Rs), the LRT becomes the estimator-correlator (EC)[13] detector for which the test statistic is given by

TEC(x) =N1n=0

xT(n)Rs

Rs+2 I1

x(n). (7)

From (4), we see that Rs(Rs+ 22 I)1

x(n) is actually theminimum-mean-squared-error (MMSE) estimation of thesource signal s(n). Thus, TEC(x) in (7) can be seen as thecorrelation of the observed signal x(n) with the MMSEestimation ofs(n).

The EC detector needs to know the source signalcovariance matrixRs and noise power 2 . When the signalpresence is unknown yet, it is unrealistic to require the sourcesignal covariance matrix (related to unknown channels) for

detection. Thus, if we further assume thatRs= 2sI, the ECdetector in (7) reduces to the well-known energy detector(ED) [9,14] for which the test statistic is given as follows (bydiscarding irrelevant constant terms):

TED(x) =N1n=0

xT(n)x(n). (8)

Note that for the multi-antenna/receiver case,TEDis actuallythe summation of signals from all antennas, which is astraightforward cooperative sensing scheme [41,56,57]. Ingeneral, the ED is not optimal ifRsis non-diagonal.

If we assume that noise is Gaussian distributed and

source signals(n) is deterministic and known to the receiver,which is the case for radar signal processing [32,33,58], it iseasy to show that the LRT in this case becomes the matchedfiltering-based detector, for which the test statistic is

TMF(x) =N1n=0

sT(n)x(n). (9)

4. Bayesian Method and the GeneralizedLikelihood Ratio Test

In most practical scenarios, it is impossible to know thelikelihood functions exactly, because of the existence of


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The resulting detection method is called optimally combinedenergy detection (OCED) [29]. It is easy to show that this teststatistic is better thanTED(x) in terms of SNR.

The OCED needs an eigenvector of the received sourcesignal covariance matrix, which is usually unknown. Toovercome this difficulty, we provide a method to estimate

the eigenvector using the received signal samples only.Considering the statistical covariance matrix of the signaldefined as

Rx= E

x(n)xT(n)

, (17)

we can verify that

Rx= Rs+2 IM. (18)

Since Rx and Rs have the same eigenvectors, the vector 1is also the eigenvector ofRxcorresponding to its maximumeigenvalue. However, in practice, we do not know thestatistical covariance matrix Rx either, and therefore we

cannot obtain the exact vector 1. An approximation of thestatistical covariance matrix is the sample covariance matrixdefined as

Rx(N) = 1N

N1n=0

x(n)xT(n). (19)

Let1 (normalized to12 = 1) be the eigenvector of thesample covariance matrix corresponding to its maximum

eigenvalue. We can replace the combining vector 1 by1,

that is,

z(n) =T

1 x(n). (20)

Then, the test statistics for the resulting blindly combinedenergy detection (BCED) [29] becomes

TBCED(x) = 1

N

N1n=0

z(n)2. (21)It can be verified that

TBCED(x) = 1N

N1n=0T1 x(n)xT(n)1

=

T

1Rx(N)

1

= max(N),(22)

wheremax(N) is the maximum eigenvalue ofRx(N). Thus,TBCED(x) can be taken as the maximum eigenvalue of thesample covariance matrix. Note that this test is a special caseof the eigenvalue-based detection (EBD) [2025].

6. Combining Space and Time Correlation

In addition to being spatially correlated, the received signalsamples are usually correlated in time due to the followingreasons.

(1) The received signal is oversampled. Let 0 be theNyquist sampling period of continuous-time signalsc(t) andlet sc(n0) be the sampled signal based on the Nyquistsampling rate. Thanks to the Nyquist theorem, the signalsc(t) can be expressed as

sc(t) = n=

sc(n0)g(t n0), (23)

where g(t) is an interpolation function. Hence, the signalsamples s(n) = sc(ns) are only related to sc(n0), wheres is the actual sampling period. If the sampling rate atthe receiver is Rs = 1/s > 1/0, that is, s < 0, thens(n) = sc(ns) must be correlated over n. An example ofthis is the wireless microphone signal specified in the IEEE802.22 standard [6,7], which occupies about 200 KHz in a6-MHz TV band. In this example, if we sample the receivedsignal with sampling rate no lower than 6 MHz, the wirelessmicrophone signal is actually oversampled and the resulting

signal samples are highly correlated in time.(2) The propagation channel is time-dispersive. In this

case, the received signal can be expressed as

sc(t) =

h()s0(t )d, (24)

wheres0(t) is the transmitted signal andh(t) is the responseof the time-dispersive channel. Since the sampling period sis usually very small, the integration (24) can be approxi-mated as

sc(t)

s

k=h(ks)s0(t ks). (25)Hence,

sc(ns) sJ1

k=J0h(ks)s0((n k)s), (26)

where [J0s,J1s] is the support of the channel responseh(t), with h(t)= 0 for t / [J0s,J1s]. For time-dispersivechannels,J1 > J0 and thus even if the original signal sampless0(ns)s are i.i.d., the received signal samples sc(ns)s arecorrelated.

(3) The transmitted signal is correlated in time. In thiscase, even if the channel is flat-fading and there is nooversampling at the receiver, the received signal samples arecorrelated.

The above discussions suggest that the assumption ofindependent (in time) received signal samples may be invalidin practice, such that the detection methods relying on thisassumption may not perform optimally. However, additionalcorrelation in time may not be harmful for signal detection,while the problem is how we can exploit this property. Forthe multi-antenna/receiver case, the received signal samplesare also correlated in space. Thus, to use both the spaceand time correlations, we may stack the signals from the M


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antennas and overLsampling periods all together and definethe correspondingML 1 signal/noise vectors:

xL(n) =[x1(n) xM(n) x1(n 1) xM(n 1)

x1(n L+ 1) xM(n L+ 1)]T(27)

sL(n) =[s1(n) sM(n) s1(n 1) sM(n 1)

s1(n L+ 1) sM(n L+ 1)]T(28)

L(n) =

1(n) M(n) 1(n 1) M(n 1)

1(n L+ 1) M(n L+ 1)T

.(29)

Then, by replacingx(n) byxL(n), we can directly extend thepreviously introduced OCED and BCED methods to incor-

porate joint space-time processing. Similarly, the eigenvalue-based detection methods [2124] can also be modified towork for correlated signals in both time and space. Anotherapproach to make use of space-time signal correlation isthe covariance based detection [27,28,61] briefly describedas follows. Defining the space-time statistical covariancematrices for the signal and noise as

RL,x= E

xL(n)xTL (n)

,

RL,s= E

sL(n)sTL (n)

,

(30)

respectively, we can verify that

RL,x= RL,s+2 IL. (31)

If the signal is not present,RL,s= 0, and thus the off-diagonalelements inRL,xare all zeros. If there is a signal and the signalsamples are correlated,RL,s is not a diagonal matrix. Hence,the nonzero off-diagonal elements ofRL,xcan be used forsignal detection.

In practice, the statistical covariance matrix can only becomputed using a limited number of signal samples, whereRL,xcan be approximated by the sample covariance matrixdefined as

RL,x(N) = 1

N

N1

n=0 xL(n)xTL (n). (32)Based on the sample covariance matrix, we could develop thecovariance absolute value (CAV) test [27,28] defined as

TCAV(x) = 1ML

MLn=1

MLm=1

|rnm(N)|, (33)

where rnm(N) denotes the (n, m)th element of the sample

covariance matrixRL,x(N).There are other ways to utilize the elements in the

sample covariance matrix, for example, the maximum valueof the nondiagonal elements, to form different test statistics.

Especially, when we have some prior information on thesource signal correlation, we may choose a correspondingsubset of the elements in the sample covariance matrix toform a more efficient test.

Another effective usage of the covariance matrix forsensing is the eigenvalue based detection (EBD) [2025],

which uses the eigenvalues of the covariance matrix as teststatistics.

7. Cyclostationary Detection

Practical communication signals may have special statisti-cal features. For example, digital modulated signals havenonrandom components such as double sidedness due tosinewave carrier and keying rate due to symbol period. Suchsignals have a special statistical feature called cyclostation-arity, that is, their statistical parameters vary periodicallyin time. This cyclostationarity can be extracted by thespectral-correlation density (SCD) function [1618]. For a

cyclostationary signal, its SCD function takes nonzero valuesat some nonzero cyclic frequencies. On the other hand, noisedoes not have any cyclostationarity at all; that is, its SCDfunction has zero values at all non-zero cyclic frequencies.Hence, we can distinguish signal from noise by analyzing theSCD function. Furthermore, it is possible to distinguish thesignal type because different signals may have different non-zero cyclic frequencies.

In the following, we list cyclic frequencies for somesignals of practical interest [17,18].

(1) Analog TV signal: it has cyclic frequencies at mul-tiples of the TV-signal horizontal line-scan rate(15.75 KHz in USA, 15.625 KHz in Europe).

(2) AM signal:x(t)= a(t) cos(2 fct+ 0). It has cyclicfrequencies at 2fc.

(3) PM and FM signal: x(t) = cos(2 fct+(t)). It usuallyhas cyclic frequencies at2fc. The characteristics ofthe SCD function at cyclic frequency2fcdepend on(t).

(4) Digital-modulated signals are as follows

(a) Amplitude-Shift Keying: x(t)= [n=anp(t ns t0)] cos(2 fct + 0). It has cyclicfrequencies atk/s, k/= 0 and 2fc+ k/s, k =

0, 1, 2, . . . .(b) Phase-Shift Keying: x(t) = cos[2 fct +

n=anp(tnst0)]. For BPSK, it has cyclicfrequencies at k/s, k/= 0, and 2fc +k/s, k =0, 1, 2, . . . . For QPSK, it has cycle frequenciesatk/s, k/= 0.

When source signal x(t) passes through a wirelesschannel, the received signal is impaired by the unknownpropagation channel. In general, the received signal can bewritten as

y(t) = x(t) h(t), (34)


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where denotes the convolution, and h(t) denotes thechannel response. It can be shown that the SCD function ofy(t) is

Sy

f

= H

f +

2

H

f 2

Sx

f

, (35)

where denotes the conjugate, denotes the cyclic fre-quency for x(t), H(f) is the Fourier transform of thechannelh(t), and Sx(f) is the SCD function ofx(t). Thus,the unknown channel could have major impacts on thestrength of SCD at certain cyclic frequencies.

Although cyclostationary detection has certain advan-tages (e.g., robustness to uncertainty in noise power andpropagation channel), it also has some disadvantages: (1) itneeds a very high sampling rate; (2) the computation of SCDfunction requires large number of samples and thus highcomputational complexity; (3) the strength of SCD couldbe affected by the unknown channel; (4) the sampling time

error and frequency off

set could aff

ect the cyclic frequencies.

8. Cooperative Sensing

When there are multiple users/receivers distributed in differ-ent locations, it is possible for them to cooperate to achievehigher sensing reliability, thus resulting in various cooper-ative sensing schemes [3444,53, 62]. Generally speaking,if each user sends its observed data or processed data to aspecific user, which jointly processes the collected data andmakes a final decision, this cooperative sensing scheme iscalled data fusion. Alternatively, if multiple receivers processtheir observed data independently and send their decisions to

a specific user, which then makes a final decision, it is calleddecision fusion.

8.1. Data Fusion. If the raw data from all receivers are sentto a central processor, the previously discussed methodsfor multi-antenna sensing can be directly applied. However,communication of raw data may be very expensive forpractical applications. Hence, in many cases, users only sendprocessed/compressed data to the central processor.

A simple cooperative sensing scheme based on the energydetection is the combined energy detection. For this scheme,each user computes its received source signal (including the

noise) energy asTED,i=

(1/N)N1n=

0

|xi(n)

|2 and sends it to

the central processor, which sums the collected energy valuesusing a linear combination (LC) to obtain the following teststatistic:

TLC(x) =Mi=1

giTED,i, (36)

where gi is the combining coefficient, with gi 0 andMi=1gi= 1. If there is no information on the source signal

power received by each user, the EGC can be used, that is,gi = 1/M for all i. If the source signal power received byeach user is known, the optimal combining coefficients can

befound[38, 43]. For the low-SNR case, it can beshown [43]that the optimal combining coefficients are given by

gi= 2iM

k=12k

, i = 1, . . . ,M, (37)

where2

i is the received source signal (excluding the noise)power of useri.A fusion scheme based on the CAV is given in [53],

which has the capability to mitigate interference and noiseuncertainty.

8.2. Decision Fusion. In decision fusion, each user sends itsone-bit or multiple-bit decision to a central processor, whichdeploys a fusion ruleto make the final decision. Specifically, ifeach user only sends one-bit decision (1 for signal presentand 0 for signal absent) and no other information isavailable at the central processor, some commonly adopteddecision fusion rules are described as follows [42].

(1) Logical-OR (LO) Rule: If one of the decisions is 1,the final decision is 1. Assuming that all decisionsare independent, then the probability of detectionand probability of false alarm of the final decision are

Pd= 1M

i=1(1Pd,i) andPf a= 1M

i=1(1Pf a,i),respectively, wherePd,i and Pf a,i are the probabilityof detection and probability of false alarm for user i,respectively.

(2) Logical-AND (LA) Rule: If and only if all decisionsare 1, the final decision is 1. The probability ofdetection and probability of false alarm of the final

decision are Pd =

Mi=1Pd,i and Pf a =

Mi=1Pf a,i,

respectively.(3) K out of M Rule: If and only if K decisions

or more are 1s, the final decision is 1. Thisincludes Logical-OR (LO) (K= 1), Logical-AND(LA) (K = M), and Majority (K = M/2) asspecial cases [34]. The probability of detection andprobability of false alarm of the final decision are

Pd=MK

i=0

MK+ i

1 Pd,iMKi

1 Pd,i

K+i

,

Pf a=MK

i=0

MK+ i

1 Pf a,iMKi

1 Pf a,iK+i

,

(38)

respectively.

Alternatively, each user can send multiple-bit decisionsuch that the central processor gets more information tomake a more reliable decision. A fusion scheme based onmultiple-bit decisions is shown in [41]. In general, there is atradeoffbetween the number of decision bits and the fusion


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reliability. There are also other fusion rules that may requireadditional information [34,63].

Although cooperative sensing can achieve better perfor-mance, there are some issues associated with it. First, reliableinformation exchanges among the cooperating users mustbe guaranteed. In an ad hoc network, this is by no means

a simple task. Second, most data fusion methods in literatureare based on the simple energy detection and flat-fadingchannel model, while more advanced data fusion algorithmssuch as cyclostationary detection, space-time combining,and eigenvalue-based detection, over more practical prop-agation channels need to be further investigated. Third,existing decision fusions have mostly assumed that decisionsof different users are independent, which may not be truebecause all users actually receive signals from some commonsources. At last, practical fusion algorithms should be robustto data errors due to channel impairment, interference, andnoise.

9. Noise Power Uncertainty and EstimationFor many detection methods, the receiver noise power isassumed to be known a priori, in order to form the teststatistic and/or set the test threshold. However, the noisepower level may change over time, thus yielding the so-called noise uncertainty problem. There are two types ofnoise uncertainty: receiver device noise uncertainty andenvironment noise uncertainty. The receiver device noiseuncertainty comes from [911]: (a) nonlinearity of receivercomponents and (b) time-varying thermal noise in thesecomponents. The environment noise uncertainty is causedby transmissions of other users, either unintentionally orintentionally. Because of the noise uncertainty, in practice,it is very difficult to obtain the accurate noise power.

Let the estimated noise power be2 = 2 , where iscalled the noise uncertainty factor. The upper bound on (in dB scale) is then defined as

B = sup

10log10

, (39)

whereB is called the noise uncertainty bound. It is usuallyassumed that in dB scale, that is, 10 log10, is uniformlydistributed in the interval [B, B] [10]. In practice, thenoise uncertainty bound of a receiving device is normallybelow 2dB [10, 64], while the environment/interferencenoise uncertainty can be much larger [10]. When there is

noise uncertainty, it is known that the energy detection is noteffective [911,64].

To resolve the noise uncertainty problem, we need toestimate the noise power in real time. For the multi-antennacase, if we know that the number of active primary signals,K, is smaller thanM, the minimum eigenvalue of the samplecovariance matrix can be a reasonable estimate of the noisepower. If we further assume to know the difference MK, the average of the M K smallest eigenvalues can beused as a better estimate of the noise power. Accordingly,instead of comparing the test statistics with an assumed noisepower, we can compare them with the estimated noise powerfrom the sample covariance matrix. For example, we can

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probabilityofdetection

102 101 100

Probability of false alarm

BCED

MME

EME

ED

ED-0.5 dB

ED-1dB

ED-1.5 dB

ED-2dB

Figure1: ROC curve: i.i.d source signal.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probabilityofdetection

102 101 100

Probability of false alarm

BCED

MME

EME

ED

ED-0.5 dB

ED-1dB

ED-1.5 dB

ED-2dB

Figure2: ROC curve: wireless microphone source signal.

compareTBCED and TED with the minimum eigenvalue ofthe sample covariance matrix, resulting in the maximumto minimum eigenvalue (MME) detection and energy tominimum eigenvalue (EME) detection, respectively [21,22].These methods can also be used for the single-antenna caseif signal samples are time-correlated [22].

Figures 1 and 2 show the Receiver Operating Charac-teristics (ROC) curves (Pd versus Pf a) at SNR= 15 dB,N = 5000, M= 4, and K = 1. In Figure 1, the sourcesignal is i.i.d and the flat-fading channel is assumed, whilein Figure 2, the source signal is the wireless microphonesignal [61,65] and the multipath fading channel (with eight


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independent taps of equal power) is assumed. ForFigure 2,in order to exploit the correlation of signal samples in bothspace and time, the received signal samples are stacked as in(27). In both figures, ED-x dB means the energy detectionwith x-dB noise uncertainty. Note that both BCED and EDuse the true noise power to set the test threshold, while

MME and EME only use the estimated noise power as theminimum eigenvalue of the sample covariance matrix. It isobserved that for both cases of i.i.d source (Figure 1) andcorrelated source (Figure 2), BCED performs better than ED,and so does MME than EME. Comparing Figures1and2, wesee that BCED and MME work better for correlated sourcesignals, while the reverse is true for ED and EME. It is alsoobserved that the performance of ED degrades dramaticallywhen there is noise power uncertainty.

10. Detection Threshold and TestStatistic Distribution

To make a decision on whether signal is present, we need toset a threshold for each proposed test statistic, such thatcertain Pd and/or Pf a can be achieved. For a fixed samplesizeN, we cannot set the threshold to meet the targets forarbitrarily highPd and low Pf a at the same time, as theyare conflicting to each other. Since we have little or no priorinformation on the signal (actually we even do not knowwhether there is a signal or not), it is difficult to set thethreshold based on Pd. Hence, a common practice is tochoose the threshold based onPf aunder hypothesis H0.

Without loss of generality, the test threshold can bedecomposed into the following form:=1T0(x), where1is related to the sample sizeNand the targetPf a, andT0(x)

is a statistic related to the noise distribution underH

0. Forexample, for the energy detection with known noise power,we have

T0(x) = 2 . (40)

For the matched-filtering detection with known noise power,we have

T0(x) = . (41)

For the EME/MME detection with no knowledge on thenoise power, we have

T0(x)= min(N), (42)

wheremin(N) is the minimum eigenvalue of the samplecovariance matrix. For the CAV detection, we can set

T0(x) = 1ML

MLn=1

|rnn(N)|. (43)

In practice, the parameter1can be set either empiricallybased on the observations over a period of time when thesignal is known to be absent, or analytically based on thedistribution of the test statistic under H0. In general, suchdistributions are difficult to find, while some known resultsare given as follows.

For energy detection defined in (8), it can be shown thatfor a sufficiently large values ofN, its test statistic can be wellapproximated by the Gaussian distribution, that is,

1

NMTED(x) N

2 ,

24NM

under H0. (44)

Accordingly, for givenPf a andN, the corresponding1 canbe found as

1= NM 2

NMQ1

Pf a

+ 1

, (45)where

Q(t) = 12

+t

eu2/2du. (46)

For the matched-filtering detection defined in (9), for asufficiently largeN, we have

1N1n=0s(n)2

TMF(x) N

0, 2

under H0. (47)

Thereby, for givenPf aandN, it can be shown that

1= Q1

Pf aN1

n=0s(n)2. (48)

For the GLRT-based detection, it can be shown that theasymptotic (asN ) log-likelihood ratio is central chi-square distributed [13]. More precisely,

2 ln TGLRT(x) 2r under H0, (49)

where r is the number of independent scalar unknownsunder H0 and H1. For instance, if2 is known while Rs isnot, rwill be equal to the number of independent real-valuedscalar variables in Rs. However, there is no explicit expressionfor1in this case.

Random matrix theory (RMT) is useful for determiningthe test statistic distribution and the parameter 1 forthe class of eigenvalue-based detection methods. In thefollowing, we provide an example for the BCED detectionmethod with known noise power, that is, T0(x)= 2 . Forthis method, we actually compare the ratio of the maximum

eigenvalue of the sample covariance matrixRx(N) to thenoise power2with a threshold1. To set the value for1, we

need to know the distribution ofmax(N)/2for any finiteN.With a finiteN,Rx(N) may be very different from the actualcovariance matrixRxdue to the noise. In fact, characterizing

the eigenvalue distributions forRx(N) is a very complicatedproblem [6669], which also makes the choice of1difficultin general.

When there is no signal,Rx(N) reduces toR(N), whichis the sample covariance matrix of the noise only. It is known

that

R(N) is a Wishart random matrix [66]. The study

of the eigenvalue distributions for random matrices is a


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very hot research topic over recent years in mathematics,communications engineering, and physics. The joint PDF ofthe ordered eigenvalues of a Wishart random matrix has beenknown for many years [66]. However, since the expressionof the joint PDF is very complicated, no simple closed-formexpressions have been found for the marginal PDFs of the

ordered eigenvalues, although some computable expressionshave been found in [70]. Recently, Johnstone and Johanssonhave found the distribution of the largest eigenvalue [67,68]of a Wishart random matrix as described in the followingtheorem.

Theorem 1. LetA(N) = (N/2 )R(N), = (N1+M)2,and = (N 1 + M)(1/N 1 + 1/M)1/3. Assume thatlimN(M/N) = y(0 < y < 1). Then, (max(A(N)))/ converges (with probability one) to the Tracy-Widomdistribution of order 1 [71,72].

The Tracy-Widom distribution provides the limiting law

for the largest eigenvalue of certain random matrices [71,72]. Let F1 be the cumulative distribution function (CDF)of the Tracy-Widom distribution of order 1. We have

F1(t) = exp 1

2

t

q(u)+(u t)q2(u)du, (50)

where q(u) is the solution of the nonlinear Painleve IIdifferential equation given by

q(u) = uq(u)+ 2q3(u). (51)

Accordingly, numerical solutions can be found for function

F1(t) at diff

erent values oft. Also, there have been tables forvalues ofF1(t) [67] and Matlab codes to compute them [73].Based on the above results, the probability of false alarm

for the BCED detection can be obtained as

Pf a= Pmax(N)> 12

= P

2N

max(A(N))> 12

= Pmax(A(N))> 1N= P

max(A(N))

>1N

1 F1

1N

,

(52)

which leads to

F1

1N

1 Pf a (53)

or equivalently,

1N

F11

1 Pf a

. (54)

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

Probabilityof

falsealarm

0.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945 0.95

1/threshold

TheoreticalPf aActualPf a

Figure

3: Comparison of theoretical and actual Pf a.

From the definitions of and in Theorem 1, we finallyobtain the value for1as

1

N+

M2

N

1 +

N+

M2/3

(NM)1/6 F11

1 Pf a

.(55)

Note that1depends only onNand Pf a. A similar approach

like the above can be used for the case of MME detection, asshown in [21,22].

Figure 3shows the expected (theoretical) and actual (bysimulation) probability of false alarm values based on thetheoretical threshold in (55) for N = 5000, M= 8, andK= 1. It is observed that the differences between these twosets of values are reasonably small, suggesting that the choiceof the theoretical threshold is quite accurate.

11. Robust Spectrum Sensing

In many detection applications, the knowledge of signaland/or noise is limited, incomplete, or imprecise. This is

especially true in cognitive radio systems, where the primaryusers usually do not cooperate with the secondary usersand as a result the wireless propagation channels betweenthe primary and secondary users are hard to be predictedor estimated. Moreover, intentional or unintentional inter-ference is very common in wireless communications suchthat the resulting noise distribution becomes unpredictable.Suppose that a detector is designed for specific signal andnoise distributions. A pertinent question is then as follows:how sensitive is the performance of the detector to the errorsin signal and/or noise distributions? In many situations,the designed detector based on the nominal assumptionsmay suffer a drastic degradation in performance even with


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small deviations from the assumptions. Consequently, thesearching for robust detection methods has been of greatinterest in the field of signal processing and many others [7477]. A very useful paradigm to design robust detectors is themaxmin approach, which maximizes the worst case detectionperformance. Among others, two techniques are very useful

for robust cognitive radio spectrum sensing: the robusthypothesis testing [75] and the robust matched filtering[76, 77]. In the following, we will give a brief overviewon them, while for other robust detection techniques, theinterested readers may refer to the excellent survey paper [78]and references therein.

11.1. Robust Hypothesis Testing. Let the PDF of a receivedsignal sample be f1 at hypothesis H1 and f0 at hypothesisH0. If we know these two functions, the LRT-based detectiondescribed inSection 2is optimal. However, in practice, dueto channel impairment, noise uncertainty, and interference,it is very hard, if possible, to obtain these two functions

exactly. One possible situation is when we only know that f1and f0 belong to certain classes. One such class is called the-contamination class given by

H0 : f0 F0, F0=

(1 0)f00 + 0g0

,

H1 : f1 F1, F1=

(1 1)f01 + 1g1

,(56)

wheref0j (j= 0,1) is the nominal PDF under hypothesisHj ,j in [0, 1] is the maximum degree of contamination, andgjis an arbitrary density function. Assume that we only knowf0j and j(an upper bound for contamination), j= 1, 2. Theproblem is then to design a detection scheme to minimizethe worst-case probability of error (e.g., probability of false

alarm plus probability of mis-detection), that is, finding adetector such that=argmin

max(f0 ,f1)F0F1

Pf af0,f1,

+ 1Pd

f0,f1,

.

(57)

Hubber [75] proved that the optimal test statistic is acensored version of the LRT given by

= TCLRT(x) = N1n=0

r(x(n)), (58)

where

r(t) =

c1, c1 f01(t)

f00(t),

f01(t)

f00(t), c0

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