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2 EURASIP Journal on Advances in Signal Processing

2. System Model

In hypothesis RI> s;(n) is the received source signal atantenna/receiver i, which may include the channel multipathand fading effects. In general, si(n) can be expressed as

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: Jeo,signal absent, and Jet, signal present. The received signal atantenna/receiver i is given by

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 [45-52].The rest of this paper is organized as follows. Thesystem 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 primary signals, is reviewed in Section 7. Cooperative sensingis discussed in Section 8. The impacts of noise uncertaintyand noise power estimation to the sensing performanceare analyzed in Section 9. The test statistic distribution andthreshold setting for sensing are reviewed in Section 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 ofLRT and matched filteringdetection methods. Practical challenges and future researchdirections for spectrum sensing are discussed in Section 12.Finally, Section 13 concludes the paper.

(2)

(1)

K qiksi(n) = l: l:hik(l)sk(n - I),k=I/=O

Jeo : xi(n) = 'lien),Jel :Xi(n) = Si(n) + 'lien), i = 1, ... ,M.

200 KHz. If a secondary user is several hundred metersaway from the microphone device, the received SNR maybe well below - 20 dB. Secondly, multi path fading and time...dispersion of the wireless channels co~ the sensing

( problem. Multipath fading may cause the sfgMrpower tofluctu . as much as 30 dB. On the other hand, unknowntf~ Gpersion in wireless channels may turn the ~ .-detection unjel~ble. C Thirdly, the noise/interferenc"t1ev~ v"may changl wit . ('ti~e"and ~, which yields the noisepower uncertainty ~ for detection [9-12]. S,'-~ n, .Eadng 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) detection [10, 13, 15], cyclostationary detection (CSD) [16-19],and some newly emerging methods such as eigenvalue-basedsensing [6, 20-25], wavelet-based sensing [26], covariancebased 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 Band 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 distributed at different locations, it is possible for them tocooperate to achieve higher sensing reliability. There arevarious sensing cooperation schemes in the current literature[34-44]. 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


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EURASIP Journal on Advances in Signal Processing

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, and qik denotes the channel order for hik.It is assumed that the noise samples l1i(n)'s are independentand identically distributed (LLd) over both nand 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 (choose Jio or Jil) based onthe received signal. If the decision is Ji] , 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 Jil, the probability of the algorithm correctlydetecting the presence of the primary signal; and probabilityof false alarm, Pfa, which defines, at the hypothesis Jio,the probability of the algorithm mistakenly declaring thepresence of the primary signal. A sensing algorithm is called"optimal" ifit achieves the highest Pd for a given Pfa with afixed number of samples, though there could be other criteriato evaluate the performance of a sensing algorithm.

Stacking the signals from the Mantennaslreceivers yieldsthe following M X 1 vectors:

x(n) = [XI (n) xM(n) r,s(n) = [SI (n)

TsM(n)] , (3)l1M(n)] T.

The hypothesis testing problem based on N signal samples isthen obtained as

source signal distribution, the wireless channels, and thnoise distribution, while the distribution of x under Jiorelated to the noise distribution. In order to use the LRT,wneed to obtain the knowledge of the channels as well as thsignal and noise distributions, which is practically difficult trealize.

If we assume that the channels are flat-fading, and threceived source signal sample si(n)'s are independent overthe PDFs in LRT are decoupled as

N-(p(x I Jit> = np(x(n) I Jid,

11=0 (6N-Ip(x I Jio) = n p(x(n) I Jio).

11=0

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

N-lTEdx) = L xT(n)Rs(Rs + a;Irlx(n). (11=0

From (4), we see that Rs(Rs + 2a~I)-lx(n) is actually thminimum-mean-squared-error (MMSE) estimation 'Of thsource signal s(n). Thus, TEdx) in (7) can be seen as thcorrelation of the observed signal x(n) with the MMSestimation of s(n).

The EC detector needs to know the source signcovariance matrix Rs and noise power a~.When the signpresence is unknown yet, it is unrealistic to require the sourcsignal covariance matrix (related to unknown channels) fodetection. Thus, if we further assume that Rs = a;I, the Edetector in (7) reduces to the well-known energy detecto(ED) [9, 14] for which the test statistic is given as follows (bdiscarding irrelevant constant terms):

Jio : x(n) = ,,(n),Jil : x(n) = s(n) + ,,(n), n = 0, ... ,N - 1. (4)

N-]Tw(x) = I xT(n)x(n).

11=0(8

3. Neyman-Pearson TheoremThe 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

(5)

Note that for the multi-antennalreceiver case, TED is actuallthe summation of signals from all antennas, which isstraightforward cooperative sensing scheme [41,56,57]. Igeneral, the ED is not optimal ifRs is non-diagonal.If we assume that noise is Gaussian distributed ansource signal s(n) is deterministic and known to the receivewhich is the case for radar signal processing [32,33,58], iteasy to show that the LRT in this case becomes the matchedfiltering-based detector, for which the test statistic is

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

where p(.) denotes the probability density function (PDF),and x denotes the received signal vector that is the aggregation of x(n), n = O,I, ... ,N - 1. Such a likelihood ratiotest decides Jil when TLRT(X) exceeds a threshold y, and Jiootherwise.

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

N-]TMP(X) = L sT(n)x(n).

11=0

4. Bayesian Method and the GeneralizedLikelihood Ratio Test

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uncertainty about one or more parameters in these functions. For instance, we may not know the noise power cr~and/or source signal covariance Rs Hypothesis testing in thepresence of uncertain parameters is known as "composite"hypothesis testing. In classic detection theory, there are twomain approaches to tackle this problem: the Bayesian methodand the generalized likelihood ratio test (GLRT).In the Bayesian method [13], the objective is to evaluate the likelihood functions needed in the LRT throughmarginalization, that is,

It is not guaranteed that the GLRT is optimal orapproaches to be optimal when the sample size goes toinfinity. Since the unknown parameters in 80 and 81 arehighly dependent on the noise and signal statistical models,the estimations of them could be vulnerable to the modelingerrors. Under the assumption of Gaussian distributed sourcesignals and noises, and flat-fading channels, some efficientspectrum sensing methods based on the GLRT can be foundin [60].

p(x I Ho) = f p(x I Ho,8o)p(8o I Ho)d8o, (l0) 5. Exploiting Spatial Correlation ofMultiple Received Signals

Finally, the GLRT decides HI if TGLRT(X) > y, where y is athreshold, and Ho otherwise.

eo = argmax p(x I Ho,80),Elo

(16)

(14)

(15)

1 N-I 2TOCED(X) = N 2: 11 z(n) 11 n=O

r(B) = Tr(BTRsB),crFr(BTB)where Tr() denotes the trace of a matrix. Let Amax be themaximum eigenvalue of Rs and let PI be the correspondingeigenvector. It can be proved that the optimal combiningmatrix degrades to the vector PI [29].Upon substituting PI into (13), the test statistic for theenergy detection becomes

The received signal samples at different antennas/receiversare usually correlated, because all s;(n)'s are generated fromthe same source signal sk(n)'s. As mentioned previously, theenergy detection defined in (8) is not optimal for this case.Furthermore, it is difficult to realize the LRT in practice.Hence, we consider suboptimal sensing methods as follows.

If M > 1, K = I, and assuming that the propagationchannels are flat-fading (qik = 0, Vi,k) and known to thereceiver, the energy at different antennas can be coherentlycombined to obtain a nearly optimal detection [41, 43,57]. This is also called maximum ratio combining (MRC).However, in practice, the channel coefficients are unknownat the receiver. As a result, the coherent combining may notbe applicable and the equal gain combining (EGC) is used inpractice [41,57], which is the same as the energy detectiondefined in (8).

In general, we can choose a matrix B with M rows tocombine the signals from all antennas as

zen) = BTx(n), n = O,I, ... ,N - 1. (13)The combining matrix should be chosen such that theresultant signal has the largest SNR. It is obvious that theSNR after combining is

where E() denotes the mathematical expectation. Hence,the optimal combining matrix should maximize the valueof function f(B). Let Rs = E[s(n)sT(n)] be the statisticalcovariance matrix of the primary signals. It can be verifiedthat

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(11)

To make the LRT applicable, we may estimate theunknown parameters first and then use the estimatedparameters in the LRT. Known estimation techniques couldbe used for this purpose [59]. However, there is one majordifference from the conventional estimation problem wherewe know that signal is present, while in the case of spectrumsensing we are not sure whether there is source signal or not(the first priority here is the detection of signal presence). Atdifferent hypothesis (Ho or Hd, the unknown parametersare also different.

The GLRT is one efficient method [13, 55] to resolve theabove problem, which has been used in many applications,for example, radar and sonar signal processing. For thismethod, the maximum likelihood (ML) estimation of theunknown parameters under Ho and HI is first obtained as

where 80 and 81 are the set of unknown parameters underHo and HI, respectively. Then, the GLRT statistic is formedas

where 80 represents all the unknowns when Ho is true. Notethat the integration operation in (l0) should be replacedwith a summation if the elements in 80 are drawn from adiscrete sample space. Critically, we have to assign a priordistribution p(o I Ho) to the unknown parameters. Inother words, we need to treat these unknowns as randomvariables and use their known distributions to express ourbelief in their values. Similarly, p(x I HI) can be defined.The main drawbacks of the Bayesian approach are listed asfollows.

(1) The marginalization operation in (10) is often nottractable except for very simple cases.

(2) The choice of prior distributions affects the detectionperformance dramatically and thus it is not a trivialtask to choose them.


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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 than Tw(x) in terms of SNR.The OCED needs an eigenvector of the received source

signal covariance matrix, which is usually unknown. Toovercome this difficulty, we provide a method to estimatethe eigenvector using the received signal samples only.Considering the statistical covariance matrix of the signaldefined as

5

(1) The received signal is oversampled. Let .60 be theNyquist sampling period of continuous-time signal seCt) andlet se(n.6o) be the sampled signal based on the Nyquistsampling rate. Thanks to the Nyquist theorem, the signalseCt) can be expressed as

n= -00

we can verify that

Since Rx and Rs have the same eigenvectors, the vector f31is also the eigenvector of Rx corresponding to its maximumeigenvalue. However, in practice, we do not know thestatistical covariance matrix Rx either, and therefore wecannot obtain the exact vector f31'An approximation of thestatistical covariance matrix is the sample covariance matrixdefined as

(17)

(18)

where get) is an interpolation function. Hence, the signalsamples s(n) = se(n.6s) are only related to se(n.6o), where.6s is the actual sampling period. If the sampling rate atthe receiver is Rs = 1I.6s > 11.60, that is, .6s < .60, thens(n) = se(n.6s) 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 resultingsignal samples are highly correlated in time.(2) The propagation channel is time-dispersive. In thiscase, the received signal can be expressed asN-l

Rx(N) = ~ L x(n)xT(n)."=0 (19) seCt) = f~oo h(r)so(t - r)dr, (24)~ ~ 2Let f31(normalized to 1If3!1 = 1) be the eigenvector of thesample covariance matrix corresponding to its maximum

eigenvalue. We can replace the combining vector f31by Ppthat is,

where so(t) is the transmitted signal and het) is the responseof the time-dispersive channel. Since the sampling period .6sis usually very small, the integration (24) can be approximated as

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

Tzen) = PI x(n).

N-ITBCED(X) = ~ L Izen) 12"=0

It can be verified that

(20)

(21)Hence,

00

seCt) ::::.6s L h(k.6s)so(t - k.6s).k=-ooIt

se(n.6s) :::: .6sLh(k.6s)son - k).6s),k=/()

(25)

(26)

6. Combining Space and Time CorrelationIn addition to being spatially correlated, the received signalsamples are usually correlated in time due to the followingreasons.

where Xmax(N) is the maximum eigenvalue of Rx(N). Thus,TBCEf)(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) [20-25].

N-J1 ~ ~T T ~TBCED(X) = N L. f31x(n)x (n)f3111=0

rvT A rv= f31RxCN)f31= Amax(N),

(22)

where (Jo.6s,!J.6s] is the support of the channel responsehet), with het) = 0 for t ~ (J0.6s,/1 .6s]. For time-dispersivechannels, h > 10 and thus even if the original signal samplesso(n.6s)'s are LLd., the received signal samples se(n.6s)'s arecorrelated.(3) The transmitted signal is correlated in time. In this

case, 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 of

independent (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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ennas and over L sampling periods all together and definecorresponding ML X 1 signal/noise vectors: Especially, when we have some prior information on thsource signal correlation, we may choose a corresponding

subset of the elements in the sample covariance matrixform a more efficient test.Another effective usage of the covariance matrix f

sensing is the eigenvalue based detection (EBD) [2G-25which uses the eigenvalues of the covariance matrix as testatistics.

XM(n - L+ l)f(27)

xl(n-L+I)

= [51 (n)51 (n - L + 1) 7. Cyclostationary Detection

ed on the sample covariance matrix, we could develop theariance absolute value (CAV) test [27,28] defined as1 ML ML

TCAV(X) = ML L L Irnm(N)I, (33)n=lm=1rnm(N) denotes the (n, m)th element of the sample

riance matrix RL.x(N).There are other ways to utilize the elements in theple covariance matrix, for example, the maximum value

(28)= ['11 (n) '" '1M(n) '11 (n - 1) ... '1M(n - 1)

'11(n-L+1) '1M(n-L+1)]T.(29)

(a) Amplitude-Shift Keying: x(t) = [2:;=-00 anP- nt"s - to)] cos(2rrlet + 0)' It has cyclfrequencies at k/ t"s, k l' 0 and 2 le + k/ t"s, k0, 1, 2, ....

(b) Phase-Shift Keying: x(t) cos[2rrlct2:;=-00 anP(t- nt"s - to)]. For BPSK, it has cyclfrequencies at k/ t"s, k l' 0, and 2 le +k/ t"s, k0, 1, 2,. , .. For QPSK, it has cycle frequencieat k/t"s, k l' O.

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

Practical communication signals may have special statistical features. For example, digital modulated signals havnonrandom components such as double sidedness duesinewave carrier and keying rate due to symbol period. Sucsignals have a special statistical feature called cyclostationarity, that is, their statistical parameters vary periodicallin time. This cyclostationarity can be extracted by thspectral-correlation density (SCD) function [16-18]. Forcyclostationary signal, its SCD function takes nonzero valueat some nonzero cyclic frequencies. On the other hand, noisdoes not have any cyclostationarity at all; that is, its SCfunction has zero values at all non-zero cyclic frequenciesHence, we can distinguish signal from noise by analyzing thSCD function. Furthermore, it is possible to distinguish thsignal type because different signals may have different nonzero cyclic frequencies.In the following, we list cyclic frequencies for som

signals of practical interest [17, 18].

(l) Analog TV signal: it has cyclic frequencies at mutiples of the TV-signal horizontal line-scan ra(15.75 KHz in USA, 15.625 KHz in Europe).

(2) AM signal: xCt) = aCt) cos(2rr let + 0)' It has cyclfrequencies at 2!c.

(3) PM and FM signal: x(t) = cos(2rr /ct+(t)). It usuallhas cyclic frequencies at 2/c. The characteristicsthe SCD function at cyclic frequency 2~ depend o(t).

(4) Digital-modulated signals are as follows

(32)

(30)

(31)

RL~= E[xL(n)x[(n)],RL,s= E[sLCn)s[(n)],

N-IA 1,,- TRL~(N) = N L. xLCn)xL (n).n=O

ectively, we can verify thatRl.,x = RL,s+ C1~It.

en, by replacing x(n) by xI.(n), we can directly extend theviously introduced OCED and BCED methods to incorte joint space-time processing. Similarly, the eigenvalueed detection methods [21-24] can also be modified tork for correlated signals in both time and space. Anotherroach to make use of space-time signal correlation iscovariance based detection [27, 28, 61] briefly describedfollows. Defining the space-time statistical covariancetrices for the signal and noise as

he signal is not present, RL,s= 0, and thus the off-diagonalents in RL~are all zeros. If there is a signal and the signalles are correlated, RI..s is not a diagonal matrix. Hence,nonzero off-diagonal elements of RL~ can be used fornal detection.In practice, the statistical covariance matrix can only beputed using a limited number of signal samples, wherecan be approximated by the sample covariance matrixned as


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(3x (I-P .)K+i.1 ,

(37=I, ... ,M,

M-K( M ) M K .fa = ~ K + i (1 - Pfa.i) - -,

M-K( M )d = I .(1 - Pd.i)M-K-;;=0 K+I

8.2. Decision Fusion. In decision fusion, each user sends ione-bit or multiple-bit decision to a central processor, whicdeploys a fusion rule to make the final decision. Specifically,each user only sends one-bit decision ("I" for signal presenand "0" for signal absent) and no other informationavailable at the central processor, some commonly adoptedecision fusion rules are described as follows (42).(1) "Logical-OR (LO)" Rule: If one of the decisions is "1the final decision is "1:' Assuming that all decisionare independent, then the probability of detectioand probability of false alarm of the final decision aPd = 1-n~I(1-Pd.i) andPfa = 1-n~I(1-Pfa.;),respectively, where Pd.; and P fa.i are the probabilitof detection and probability of false alarm for userrespectively.

(2) "Logical-AND (LA)" Rule: If and only if all decisionare "1:' the final decision is "1." The probabilitydetection and probability of false alarm of the findecision are Pd = n~IPd.i and Pfa = n~IPfa,i,respectively.

(3) "K out of M" Rule: If and only if K decisionor more are "1" S, the final decision is "1." Thincludes "Logical-OR (LO)" (K = 1), "Logical-AND(LA)" (K = M), and "Majority" (K 7' rM/21)special cases (34). The probability of detection anprobability of false alarm of the final decision are

where u1 is the received source signal (excluding the noisepower of user i.A fusion scheme based on the CAV is given in [53

which has the capability to mitigate interference and noisuncertainty.

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

Data Fusion. If the raw data from all receivers are senta central processor, the previously discussed methodsmulti-antenna sensing can be directly applied. However,mmunication of raw data may be very expensive forctical applications. Hence, in many cases, users only sendessed/compressed data to the central processor.A simple cooperative sensing scheme based on the energyection is the combined energy detection. For this scheme,h user computes its received source signal (including theise) energy as TED,i = (l/N) I.~:01IXi(n)12 and sends it tocentral processor, which sums the collected energy valuesng a linear combination (LC) to obtain the following test

ere ~ denotes the convolution, and hCt) denotes thennel response. It can be shown that the SCD function ofis

* denotes the conjugate, a denotes the cyclic frency for x(t), H(f) is the Fourier transform of theh(t), and Sx(f) is the SCD function of x(t). Thus,

unknown channel could have major impacts on thength of SCD at certain cyclic frequencies.Although cyclostationary detection has certain advanes (e.g., robustness to uncertainty in noise power andpagation channel), it also has some disadvantages: (1) itds a very high sampling rate; (2) the computation of SCDction requires large number of samples and thus highputational complexity; (3) the strength of SCD couldaffected by the unknown channel; (4) the sampling timeor and frequency offset could affect the cyclic frequencies.

Cooperative Sensingen there are multiple users/receivers distributed in differlocations, it is possible for them to cooperate to achieveher sensing reliability, thus resulting in various cooperve sensing schemes [34-44, 53, 62). Generally speaking,each user sends its observed data or processed data to acific user, which jointly processes the collected data andkes a final decision, this cooperative sensing scheme ised data fusion. Alternatively, if multiple receivers processir observed data independently and send their decisions topecific user, which then makes a final decision, it is calledcision fusion.

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

wer received by each user, the EGC can be used, that is,

MTrdx) = Igi TED.;,i=1 (36) ( )K+i1 - Pfa.i ,respectively.

Alternatively, each user can send multiple-bit decisiosuch that the central processor gets more informationmake a more reliable decision. A fusion scheme based omultiple-bit decisions is shown in (41). In general, there i


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

(47)

(48)

(46)

(44)

under Ro.

under Jfo.

N-lYl = Q-l (Pfa) 2:: /ls(n)/l2.

n=O

1 ( 2 2CT~)M TED(x) -.N CT", NM

1 f+ooQ(t) = -- e-u'/2du.J2ii tFor the matched-filtering detection defined in (9), for a

sufficiently large N, we have

Thereby, for given Pfa and N, it can be shown that

where

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

Accordingly, for given Pfa and N, the corresponding YI canbe found as

For the GLRT-based detection, it can be shown that theasymptotic (as N - 00) log-likelihood ratio is central chisquare distributed [13]. More precisely,

2ln TGLRT(X) - X; under Ro, (49)where r is the number of independent scalar unknownsunder Ro and RI' For instance, if CT~s known while Rs isnot, r will be equal to the number of independent real-valuedscalar variables in Rs However, there is no explicit expressionfor Yl in this case.

Random matrix theory (RMT) is useful for determiningthe test statistic distribution and the parameter Yl forthe class of eigenvalue-based detection methods. In thefollowing, we provide an example for the BCED detectionmethod with known noise power, that is, To(x) = CT~. Forthis method, we actually compare the ratio of the maximumeigenvalue of the sample covariance matrix RAN) to thenoise power CT~with a threshold Yl . To set the value for YI , weneed to know the distribution orXmax(N)/CT~for any finite N.With a finite N, RAN) may be very different from the actualcovariance matrix Rx due to the noise. In fact, characterizingthe eigenvalue distributions for Rx(N) is a very complicatedproblem [66-69], which also makes the choice of Yldifficultin general.

When there is no signal, RxCN) reduces to R,,(N), whichis the sample covariance matrix of the noise only. It is knownthat R,/(N) is a Wish art random matrix [66]. The studyof the eigenvalue distributions for random matrices is a

(40)

(41)

(42)

(43)

To(x) = CT;.

1 MLTo(x) = ML 2:: Irnn(N)/.n=l

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

independent taps of equal power) is assumed. For Figure 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, whileMME 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 Figures 1and 2, 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 Test

Statistic DistributionTo make a decision on whether signal is present, we need toset a threshold y for each proposed test statistic, such thatcertain Pd and/or Pfa can be achieved. For a fixed samplesize N, we cannot set the threshold to meet the targets forarbitrarily high Pd and low Pfa 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 on Pfa under hypothesis Jfo.Without loss of generality, the test threshold can bedecomposed into the following form: y = Yl To(x), where Ylis related to the sample size N and the target Pfa, and To(x)is a statistic related to the noise distribution under Jfo. Forexample, for the energy detection with known noise power,we have

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

In practice, the parameter YI can 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 Jfo. In general, suchdistributions are difficult to find, while some known resultsare given as follows.


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hot research topic over recent years in mathematics,munications engineering, and physics. The joint PDF ofordered eigenvalues of a Wishart random matrix has beenn for many years [66]. However, since the expressione joint PDF is very complicated, no simple closed-formessions have been found for the marginal PDFs of theed eigenvalues, although some computable expressionsbeen found in [70]. Recently, Johnstone and Johanssonfound the distribution of the largest eigenvalue [67,68]Wishart random matrix as described in the foIlowing

~ 2orem 1. Let A(N) = (N/(J,~)R'1(N), fl = (v'N -1 + /M) ,v = (,jN - 1+ /M)(II,jN - 1+ 1I.JM)1/3. Assume that_ co (M/N) = Y (0 < Y < 1). Then, (Amax(A(N converges (with probability one) to the Tracy-Widombution of order 1 [71, 72).

The Tracy-Widom distribution provides the limiting lawthe largest eigenvalue of certain random matrices [71,Let FI be the cumulative distribution function (CDF)e Tracy-Widom distribution of order 1.We have

0.90.8

e 0.7~ 0.61;l;:a 0.5'0~ 0.4~o 0.3D:: 0.2

0.1

0.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945 0.95l/threshold

-e- Theoretical Pia-"iI- Actual PIa

FIGURE 3: Comparison oftheoretical and actual PI'"

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

From the definitions of fl and v in Theorem 1, we finallyobtain the value for Yl as

(55)

( (.jN + .JM)-213 _I )x 1+ 1/6 FI (I-Pfa).(NM)(51)"(u) = uq(u) + 2q3(U).

ordingly, numerical solutions can be found for function) at different values of t.Also, there have been tables fores of FI (t) [67] and Matlab codes to compute them [73].Based on the above results, the probability of false alarmthe BCED detection can be obtained as

Note that YI depends only on Nand Pfa. A similar approachlike the above can be used for the case of MME detection, asshown in [21,22].

Figure 3 shows 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.

= P(Amax(A(N > y,N)

= pCmax(A~N - fl > ylNv - fl);::,1 - FI (YI Nv- fl),

h leads to

ivalently,

(52)

(53)

(54)

11. Robust Spectrum SensingIn many detection applications, the knowledge of signaland/or noise is limited, incomplete, or imprecise. This isespecially 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 interference 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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where q(u) is the solution of the nonlinear Painleve IIdifferential equation given by (55)

( (IN + JM) -213 _I )X 1 + 1/6 FI (1 - Pfa) .(NM)

From the definitions of fI and v in Theorem 1, we finallyobtain the value for YI as

-&- Theoretical Pia-"'iI- Actual PIa

0.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945 0.95l/threshold

0.9

FIGURE 3: Comparison of theoretical and actual PI'"

0.8

e 0.7~ 0.61;l~ 0.5'Cl~ 0.4l0.3P< 0.2

0.1

Note that YI depends only on Nand Pfa. A similar approachlike the above can be used for the case of MME detection, asshown in [21,22].

Figure 3 shows 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.

(51)"(u) = uq(u) + 2q3(U).

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 theordered 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.

Accordingly, numerical solutions can be found for functionFI (t) at different values of t. Also, there have been tables forvalues of FI (t) [67] and Matlab codes to compute them [73].

Based on the above results, the probability of false alarmfor the BCED detection can be obtained as

A 2Theorem 1. Let A(N) = (N/CT,~)R'1(N), fI = (IN -1+JM) ,and v = (IN - 1+ JM)( 1/IN - 1+ 1/JM) 1/3. Assume thatlimN_ 00 (M/N) = Y (0 < Y < 1). Then, (Amax(A(N fI)/v converges (with probability one) to the Tracy-Widomdistribution of order 1 [71, 72].

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

= P(Amax(A(N > yIN)

= pCmax(A~N - fI > Y1Nv - fI)"" 1 - FI (YI Nv- fI),

which leads to

or equivalently,

(52)

(53)

11. Robust Spectrum SensingIn many detection applications, the knowledge of signaland/or noise is limited, incomplete, or imprecise. This isespecially 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 interference 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,


11/12

12

noise samples are usually correlated. This will cause manysensing methods unworkable, because they usually assumethat the noise samples are i.i.d. For some methods, a noiseprewhitening process can be used to make the noise samples.i.i.d. prior to the signal detection. For example, this methodhas been deployed in [22] to enable the eigenvalue-baseddetection methods. The similar method can be used forcovariance-based detection methods, for example, the CAY.

(2) Spurious signal and interference. The received signalmay contain not only the desired signal and white noise butalso some spurious signal and interference. The spurioussignal may be generated by Analog-to-Digital Converters (ADC) due to its nonlinearity [79] or other intentional/unintentional transmitters. If the sensing antenna isnear some electronic devices, the spurious signal generatedby the devices can be strong in the received signal. For somesensing methods, such unwanted signals will be detected assignals rather than noise. This will increase the probabilityof false alarm. There are methods to mitigate the spurioussignal at the device level [79]. Alternatively, signal processingtechniques can be used to eliminate the impact of spurioussignal/interference [53]. It is very difficult, if possible, toestimate the interference waveform or distribution becauseof its variation with time and location. Depending onsituations, the interference power could be lower or higherthan the noise power. If the interference power is muchhigher than the noise power, it is possible to estimate theinterference first and subtract it from the received signal.However, since we usually intend to detect signal at verylow SNR, the error of the interference estimation could belarge enough (say, larger than the primary signal) such thatthe detection with the residue signal after the interferencesubtraction is still unreliable. If the interference power islow, it is hard to estimate it anyway. Hence, in general wecannot rely on the interference estimation and subtraction,especially for very low-power signal detection.

(3) Fixed point realization. Many hardware realizationsuse fixed point rather than floating point computation. Thiswill limit the accuracy of detection methods due to the signaltruncation when it is saturated. A detection method shouldbe robust to such unpredictable errors.

(4) Wideband sensing. A cognitive radio device mayneed to monitor a very large contiguous or noncontiguousfrequency range to find the best available band(s) fortransmission. The aggregate bandwidth could be as largeas several GHz. Such wideband sensing requires ultrawide band RP frontend and very fast signal processingdevices. To sense a very large frequency range, typicallya corresponding large sampling rate is required, which isvery challenging for practical implementation. Fortunately,if a large part of the frequency range is vacant, that is, thesignal is frequency-domain sparse, we can use the recentlydeveloped compressed sampling (also called compressedsensing) to reduce the sampling rate by a large margin[80-82]. Although there have been studies in widebandsensing algorithms [26,83-87], more researches are neededespecially when the center frequencies and bandwidths of theprimary signals are unknown within the frequency range ofinterest.


(5) Complexity. This is of course one of the major factorsaffecting the implementation of a sensing method. Simplebut effective methods are always preferable.

To detect a desired signal at very low SNR and in a harshenvironment is by no means a simple task. In this paper,major attention is paid to the statistical detection methods.The major advantage of such methods is their little dependency on signal/channel knowledge aswell as relative ease forealization. However, their disadvantage is also obvious: theyare in general vulnerable to undesired interferences. How wecan effectively combine the statistical detection with knownsignal features is not yet well understood. This might bea promising research direction. Furthermore, most exitingspectrum sensing methods are passive in the sense that theyhave neglected the interactions between the primary andsecondary networks via their mutual interferenceS. If thereaction of the primary user (e.g., power control) uponreceiving the secondary interference is exploited, some activespectrum sensing methods can be designed, which couldsignificantly outperform the conventional passive sensingmethods [88,89]. At last, detecting the presence of signal ionly the basic task of sensing. For a radio with high leveof cognition, further information such as signal waveformand modulation schemes may be exploited. Therefore, signaidentification turns to be an advanced task of sensing. If wcould find an effective method for this advanced fusk, it inturn can help the basic sensing task.13. ConclusionIn this paper, various spectrum sensing techniques have beenreviewed. Special attention has been paid to blind sensingmethods that do not need information of the source signalsand the propagation channels. It has been shown that spacetime joint signal processing not only improves the sensingperformance but also solves the noise uncertainty problem tosome extent. Theoretical analysis on test statistic distributionand threshold setting has also been investigated ..References[1] FCC, "Spectrum policy task force report;' in Proceedings of th

Federal Communications Commission (FCC '02),WashingtonDC,USA,November 2002.12) M. H. Islanl, C. L. Koh, S. W. Oh, et al.) USpectrum surveyin Singapore: occupancy measurements and analysis;' i

Proceedings of the 3rd International Conference on Cognitive Radio Oriented Wireless Networks and Communications(CROWNCOM '08), Singapor,May2008.[3] J. Mitola and G. Q. Maguire, "Cognitive radio: making sofware radios more personal;' IEEE Personal Communications,vo\. 6, no. 4, pp. 13-18, 1999.

[4] S.Haykin, "Cognitive radio: brain-empowered wireless communications;' IEEE Transactions on Communications, vo\. 23no. 2, pp. 201-220,2005.[5] N. Devroye, P. Mitran, and V. Tarokh, "Achieveablerates icognitive radio channels;' TREE Transactions on InformationTheory, vo!. 52, no. 5, pp. 1813-11\27,2006. .[6] 802.22 Working Group, "IEEE 802.22 D1: draft standard for wireless regional area networks;' March 200http://grouper.ieee.org/ groups/802/22/.


12/12


(5) Complexity. This is of course one of the major factorsaffecting the implementation of a sensing method. Simplebut effective methods are always preferable.

To detect a desired signal at very low SNR and in a harshenvironment is by no means a simple task. In this paper,major attention is paid to the statistical detection methods.The major advantage of such methods is their little dependency on signal/channel knowledge as well as relative ease forrealization. However, their disadvantage is also obvious: theyare in general vulnerable to undesired interferences. How wecan effectively combine the statistical detection with knownsignal features is not yet well understood. This might bea promising research direction. Furthermore, most exitingspectrum sensing methods are passive in the sense that theyhave neglected the interactions between the primary andsecondary networks via their mutual interferences. If thereaction of the primary user (e.g., power control) uponreceivin the secondar interference is Joite some:ildi

() t basic task of sensing. 'For a radio with high levelof cognition, further information such as signal waveformand modulation schemes may be exploited. Therefore, signalidentification turns to be an advanced task of sensing. If wecould find an effective method for this advahced tas~, it inturn can help the basic sensing task.

noise samples are usually correlated. This will cause manysensing methods unworkable, because they usually assumethat the noise samples are LLd. For some methods, a noiseprewhitening process can be used to make the noise samples.i.i.d. prior to the signal detection. For example, this methodhas been deployed in [22] to enable the eigenvalue-baseddetection methods. The similar method can be used forcovariance-based detection methods, for example, the CAY.

(2) Spurious signal and interference. The received signalmay contain not only the desired signal and white noise butalso some spurious signal and interference. The spurioussignal may be generated by Analog-to-Digital Converters (ADC) due to its nonlinearity [79] or other intentional/unintentional transmitters. If the sensing antenna isnear some electronic devices, the spurious signal generatedby the devices can be strong in the received signal. For somesensing methods, such unwanted signals will be detected as~.~ \baa ~ . increase the robabilit~~ ..

~ ,estimate the interference waveform or distribution becauseof its variation with time and location. Depending onsituations, the interference power could be lower or higherthan the noise power. If the interference power is muchhigher than the noise power, it is possible to estimate theinterference first and subtract it from the received signal.HDwL'v.!; iw:'e W"' .w.ualI? ..in.t.et:tQ J;p .ae..tN".t ~ 4lt Mtrf

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