performance analysis of volume-based spectrum...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 1, JANUARY 2015 317

Performance Analysis of Volume-Based SpectrumSensing for Cognitive Radio

Lei Huang, Member, IEEE, Cheng Qian, Student Member, IEEE, Yuhang Xiao, and Q. T. Zhang, Fellow, IEEE

Abstract—In this work, the volume-based method for spectrumsensing is analyzed, which is able to provide the desirable prop-erties of constant false-alarm rate, robustness against deviationfrom independent and identically distributed (IID) noise and be-ing free of noise uncertainty. By computing the first and secondmoments for the signal-absence and signal-presence hypothesestogether with using the Gamma distribution approximation, wederive accurate analytic formulae for the false-alarm and detectionprobabilities for IID noise situations. This enables us to developtheoretical decision threshold as well as receiver operating charac-teristic. Numerical results are presented to validate our theoreticalfindings.

Index Terms—Cognitive radio, spectrum sensing, volume,Gamma distribution, multiple antenna.

I. INTRODUCTION

A S a fundamental element in cognitive radio (CR) [1], [2],spectrum sensing has received much attention in the liter-

ature. Basically, a secondary (unlicensed) user (SU) is allowedto borrow the frequency channels from the primary (licensed)users (PUs) in the same CR network provided that it does notcause intolerable interference to the latter. To maximize thespectral utilization and minimize the harmful interference tothe PUs, the SU usually needs to employ multiple antennas toreliably detect the PUs particularly at low signal-to-noise ratio(SNR) and/or small samples [3], [4]. Nevertheless, the multi-antenna receiver is typically uncalibrated or contains calibrationerror in practice, thereby calling for robust methodologies forspectrum sensing.

Numerous methodologies have been devised for spectrumsensing in the literature, varying from non-blind approach toblind approach. Basically, the non-blind algorithms need toemploy some a priori knowledge of the noise, signal or channelto construct their test statistics, such as the energy detection(ED) method [5]–[7] and feature detection approaches [8]–[11].

Manuscript received March 4, 2014; revised June 6, 2014; accepted July 29,2014. Date of publication August 7, 2014; date of current version January 7,2015. The work described in this paper was supported by the National NaturalScience Foundation of China (No. 61222106) and by the Shenzhen Kongqietalent program under Grant KQC201109020061A. The associate editor co-ordinating the review of this paper and approving it for publication wasG. V. V. Sharma.

L. Huang, C. Qian, and Y. Xiao are with Department of Electronic andInformation Engineering, Harbin Institute of Technology Shenzhen GraduateSchool, Shenzhen 518055, China (e-mail: [email protected]).

Q. T. Zhang is an Emeritus Professor with Department of Electronic En-gineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2014.2345660

With the known noise variance, it is proved that the ED methodis optimal for independent and identically distributed (IID)observations. Nevertheless, its optimality cannot be guaran-teed for the situation of unknown noise variance because itis rather sensitive to the noise uncertainty in the estimatednoise variance, particularly for the non-IID noise. It is shownin [10], [11] that the feature detector is robust against thenoise uncertainty and provide superior detection performance.However, it usually suffers from synchronization errors andfrequency offsets in practical situations, thereby limiting itsapplications. Indeed, the presence of primary signals not onlychanges the energy in the observation data but the correlationstructure as well. The correlation structure inherent in the ob-servation covariance matrix leads to the most spread-out eigen-spectrum, providing a good indication for the primary signals.In addition, unlike the ED and feature detection schemes, theeigenvalue-based approach is free of the noise variance andsignal features, thereby being a blind detector. As a result, theeigenvalue-based spectrum sensing approaches have receivedmuch attention [3], [12]–[16]. As a generalized likelihood ratiotest (GLRT) variant, the spherical test (ST) detector [17] isable to reliably identify the correlated signals embedded inadditive IID noise. In fact, the ST detector is equivalent tothe eigenvalue arithmetic-to-geometric mean (AGM) algorithm[18]. Nevertheless, it is indicated in [19] that, as the locallymost powerful invariant test for sphericity, John’s detector [20]is superior to the ST detector when the numbers of antennas andsamples tend to infinity at the same rate. As a matter of fact, thespectrum sensing algorithms above are developed upon the IIDnoise assumption and thereby not robust against the deviationfrom the IID noise, which is quite relevant in the real-worldapplications since the SU receiver is typically uncalibrated.Even though the receiver can be calibrated, the calibration errormakes the thermal noise to be non-ideal IID to some extent,which poses a big challenge for practical spectrum sensing.Indeed, besides the non-IID noise, the radio frequency (RF) IQimbalance [21], [22] can also lead to performance degradationfor the spectrum sensing methods, which, however, is beyondthe scope of this paper.

Some approaches have been suggested for robust spectrumsensing in the literature, such as the GLRT test [23], indepen-dence test [24], Hadamard ratio test [25], [26], Gerschgorindisk test [27], locally most powerful invariant test (LMPIT)[28] and volume-based test approaches [29]. Being robustagainst the non-IID noise and derived in the GLRT paradigm,the Hadamard ratio approach [30]–[32] has received muchattention in the community of spectrum sensing, such as [25],[26]. In this approach, spectrum sensing is cast as the issue of

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318 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 1, JANUARY 2015

distinguishing between a diagonal matrix and an arbitraryHermitian matrix. A variant of the Hadamard ratio approach forspectrum sensing has been proposed in [25], where the numberof PUs needs to be known a priori to the receiver, which, how-ever, is not rational in the practical spectrum sensing situations.On the other hand, the performance of the Hadamard ratioalgorithm for spectrum sensing has been analyzed in [23], [24].However, as pointed out in [29], although the Hadamard ratiotest is robust against the non-IID noise, its detection perfor-mance needs to be further enhanced. Unlike the Hadamard ratioapproach, the volume-based detector [29] is able to employ thecorrelation among the primary signals to further increase thedenominator of the test statistic, leading to significant reduc-tion in the test variable under the signal-presence hypothesis.This eventually leads to considerable improvement in detectionperformance for the volume-based approach.

It is worth pointing out that the volume-based detector isoriginally developed in [29] for real-valued observation and itsdetection performance has not yet been analyzed, which is themajor contribution of this work. In particular, by computingthe first and second moments for the signal-absence andsignal-presence hypotheses together with using the Gammadistribution approximation, we derive accurate analyticformulae for the false-alarm and detection probabilities for thescenario of IID noise. This enables us to develop the theoreticaldecision threshold for practical primary signal detection as wellas receiver operating characteristic (ROC) for performanceevaluation.

The remainder of the paper is organized as follows. Section IIpresents the problem formulation, including the signal modelas well as relevant sensing solution. Performance analysis ofthe volume-based method is provided in Section III. Simulationresults are presented in Section IV. Finally, conclusions aredrawn in Section V.

Throughout this paper, we use boldface uppercase letter todenote matrix, boldface lowercase letter for column vector,and lowercase letter for scalar quantity. Superscripts (·)T and(·)H represent transpose and conjugate transpose, respectively.The E[a] and a denote the expected value and estimate ofa, respectively. The |A| is the determinant of A, diag(·)stands for a diagonal matrix and IM represents the M ×Midentity matrix. The x ∼ N (μ,Σ) means x follows a complexGaussian distribution with mean μ and covariance matrix Σ,and ∼ signifies “distributed as.” The R ∼ WM (N,Σ) denotesa complex Wishart distribution with N degrees of freedom andassociated covariance matrix Σ. The aij stands for the (i, j)element of A and A � B means that A−B is a positivedefinite matrix.

II. PROBLEM FORMULATION

A. Signal Model

Consider the standard signal model where a SU with Mantennas tries to detect the signals emitted by d PUs with singleantenna:

xt = Hst + nt. (1)

Here, H ∈ CM×d denotes the channel matrix between the PUs

and SU, which is unknown deterministic during the sensingperiod, and the

xt = [x1(t), . . . , xM (t)]T (2)

st = [s1(t), . . . , sd(t)]T (3)

nt = [n1(t), . . . , nM (t)]T (4)

are the observation, signal and noise vectors, respectively.Suppose that si(t) ∼ N (0, σ2

si)(i = 1, . . . , d) with σ2

sibeing

the ith unknown signal variance, and that ni(t) ∼ N (0, τi)(i =1, . . . ,M) with τi being the unknown noise variance. Note thatτi is not necessarily equal to τj for i �= j in practice, whichcorresponds to the case of uncalibrated receiver. Furthermore,we assume that the noises are statistically independent of eachother and also independent of the signals. In the signal-absencehypothesis H0, the population covariance matrix of the obser-vation xt is Σ = E[xtx

Ht ] = diag(τ1, . . . , τM ). However, the

presence of primary signals destroys this diagonal structure,leading to

Σ = HΣsHH + diag(τ1, . . . , τM ) (5)

where Σs = E[stsHt ]. Therefore, the spectrum sensing issue is

cast as the binary hypothesis test:

H0 : Σ =diag(τ1, . . . , τM ) (6)

H1 : Σ � diag(τ1, . . . , τM ) (7)

where H1 denotes the signal-presence hypothesis. The problemat hand is to test Σ = diag(τ1, . . . , τM ) for a null hypothesisagainst all any other possible alternatives of Σ by using thenoisy observations X = [x1, . . . ,xN ]. Here, N denotes thenumber of samples which, without loss of generality, is as-sumed to be not less than the number of antennas, i.e., N ≥ M .

B. Sensing Solution

The determinant of Σ is the hyper-volume of the geometrydetermined by the row vectors of Σ. For example, considerthe situation of three receiving antennas where the observeddata with zero mean and unity variance can be independent,correlated or coherent. The corresponding covariance matricesturn out to be the 3 × 3 identity matrix, full-rank non-identitymatrix and rank-one arbitrary matrix. Fig. 1 plots these ge-ometries, namely, cube, parallelepiped and line, formed by therow vectors of the matrices. We assume that all edges of thegeometries are unity such that ‖Σ(i, :)‖ = 1 where Σ(i, :) isthe i-th row of Σ and ‖ · ‖ is the Euclidean norm. Moreover,the volumes of the cube, parallelepiped and line, are denoted byv1, v2 and v3, respectively. The cube is referring to the situationof signal-absence while the other two geometries correspondto the scenario of signal-presence. For the situation of signalabsence, the covariance matrix is a 3 × 3 identity matrix,meaning that v1 = 1. For the scenario of signal presence, never-theless, the structure of diagonal matrix is destroyed, resultingin significant volume reduction. Therefore, the volume is able

HUANG et al.: VOLUME-BASED SPECTRUM SENSING FOR COGNITIVE RADIO 319

Fig. 1. Volume comparison for uncorrelated, correlated and coherent observations. (a) v1 = 1; (b) v2 < v1; (c) v1 = 0.

to differentiate the primary signals from the background noise,providing a new approach for accurate spectrum sensing.

In the signal-absence situation, the elements of x(k), i.e.,xi(t), t = 1, . . . , N , i = 1, . . . ,M , are statistically indepen-dent. To utilize the correlation structure for practical spectrumsensing, we need to calculate the sample covariance matrix(SCM) rather than the population covariance matrix Σ, whichis given as

S =1

N

N∑t=1

xtxTt . (8)

Moreover, the edge lengths of the row vectors of the SCMare computed as ρi = ‖S(i, :)‖(i = 1, . . . ,M). Setting D =diag(ρ1, . . . , ρM ), the volume of the geometry with unity edgeis |D−(1/2)SD−(1/2)|. Consequently, we have

ξVOLΔ=

|S||D| . (9)

For the scenario of signal-absence, D−(1/2)SD−(1/2) asymp-totically approaches the identity matrix as the number of sam-ples tends to infinity, leading to the volume of one. For thesituation of signal-presence, however, the correlation structureleads to notable reduction in |S| but considerable increase in|D|, eventually leading to significant reduction of the totalvolume and thereby providing a good indication for the primarysignals. Therefore, compared with a predetermined thresholdγVOL, the statistic is able to yield the detection of the PUs.That is

ξVOL

H0

≷H1

γVOL. (10)

It is worth pointing out that the volume-based detector offersthe same expression as the Hadamard ratio test but with adifferent diagonal matrix in the denominator. The Hadamardratio rule is given as

ξHDMΔ=

|S||G|

H0

≷H1

γHDM (11)

where G = diag(r11, . . . , rMM ) with rii(i = 1, . . . ,M)being the main diagonal elements of S. Noting thatD−(1/2)SD−(1/2) and G−(1/2)SG−(1/2) asymptoticallyare an identity matrix under H0, the decision threshold γVOL

is approximately equal to γHDM. On the other hand, ρi ismuch larger than rii due to the correlation structure, making

Fig. 2. PDF versus threshold. M = 4, d = 3 and SNR = [−2,−3,−4] dB.(a) N = 5000. (b) N = 50.

ξVOL to be much smaller than ξHDM under H1. Therefore,the volume-based approach is able to employ the correlationstructure to further enhance the detection performance.

To illustrate the different behaviors of the volume-based andHadamard ratio approaches, the empirical probability densityfunctions (PDFs) of the test statistics for these two methodsare plotted in Fig. 2, where the number of antennas is 4, threeprimary signals with powers of [−2,−3,−4] dB are assumedto exist in the sensed channel and the noise is IID with unitypower. In Fig. 2(a), the number of samples is sufficiently large,i.e., N = 5000, so that the SCM is the very accurate estimate


of the population covariance matrix. For the signal-absencesituation, the covariance matrix is an identity matrix, which inturn means that the PDF of ξVOL is equal to that of ξHDM,as validated in Fig. 2(a). Moreover, it is seen that the PDFunder H1, denoted as p1, is much farther away from the PDFunder H0, denoted as p0, of the volume-based detector than thatof the Hadamard ratio scheme. Indeed, the Kullback-Leibler(KL) distance between p0 and p1 is calculated as D(p1‖p0) =∑Q

i=1 p1(i) log(p1(i)/p0(i)) with Q = 200, which is DVOL =7.8659 for the volume-based algorithm and DHDM = 4.3039for the Hadamard ratio approach. The larger KL distance resultsin considerable enhancement in detection performance for thevolume-based algorithm. When the number of samples be-comes small, say, N = 50, the SCM is the inaccurate estimateof the population one. The corresponding numerical resultsare depicted in Fig. 2(b). Accordingly, the KL distance of thevolume-based approach is DVOL = 84.0485 whereas the KLdistance of the Hadamard ratio method is DHDM = 59.0912.Here, we set Q = 50. It is seen that the volume-based approachis still superior to the Hadamard ratio testing because it providesa larger KL distance than the latter. This thereby implies that thevolume-based algorithm is able to correctly detect the PUs withhigher probability than the Hadamard ratio approach especiallyin small sample and low SNR situations.

To easy the theoretical analysis, the volume-based test statis-tic is modified as

ξΔ= − log

|S||D|

H0

≷H1

γ. (12)

where γ is the decision threshold of the volume-based de-tector. Note that the volume-based test statistic in (12) hasbeen derived in [29] for accurate and robust spectrum sensing.Nevertheless, its theoretical performance analysis has not yetbeen investigated. In the next section, exploiting the Gammadistribution approximation, accurate analytic formulae are de-rived for the false-alarm probability, detection probability, the-oretical decision threshold as well as ROC. This enables usto accurately determine the theoretical decision threshold forpractical spectrum sensing. Moreover, the analytic expressionfor the probability of detection can be employed to evaluate thedetection performance of the volume-based approach.

III. PERFORMANCE ANALYSIS

In this section, analytic formulae for the false-alarm proba-bility, detection probability, decision threshold as well as ROCare derived by assuming the additive noise is IID. Note that|S|/|D| ∈ [0, 1] while −log(|S|/|D|) ∈ [0,∞). Therefore, weemploy the Gamma approximation with the same support bymeans of moment matching to determine the distributions ofξVOL under H0 and H1. Since the computation of the first twomoments under the signal-absence hypothesis is the simplifiedcase of that under the signal-presence hypothesis, we will firstaddress the detection probability of the volumed-based method.

A. Detection Probability

Let p1(y) be the PDF of the statistic variable ξ under H1,which is determined by the following proposition.

Proposition 1: For any antenna number M and sample num-ber N with N ≥ M , the two-first-moment Gamma approxima-tion to the PDF of ξ under H1 is

p1(y) ≈yα1−1β−α1

1 e−yβ1

Γ(α1), y ∈ [0,∞) (13)

where Γ(·) is the complete Gamma function and

α1 =μ21

ν21(14a)

β1 =ν21μ1

(14b)

with μ1 and ν1 being the second-order approximation to themean of ξ and the first-order approximations to the variance ofξ, whose computations are provided in (A.4).

Proof: The proof is seen in Appendix A. �It follows from Proposition 1 that the detection probability is

computed as

Pd(γ)Δ= Prob(ξ < γ|H1) = F (γ;α1, β1) (15)

provided that the population covariance matrix Σ is given.Here, F (γ;α1, β1) is the cumulative distribution function(CDF) of Gamma distribution. From the constant false-alarmrate (CFAR) perspective, the decision threshold γ usually isdetermined under the signal-absence hypothesis, which must bedeterministic and independent of the noise variance.

B. False Alarm Probability

The false-alarm probability can be easily determined by thefollowing proposition.

Proposition 2: For any antenna number M and sample num-ber N with N ≥ M , the two-first-moment Gamma approxima-tion to the PDF of ξ under H0 is

p0(y) ≈yα0−1β−α0

0 e−yβ0

Γ(α0), y ∈ [0,∞) (16)

where

α0 =μ20

ν20(17a)

β0 =ν20μ0

(17b)

with μ0 and ν0 being the second order approximation to themean of ξ and the first order approximations to the variance ofξ, whose calculation are given in (B.1).

Proof: The proof is seen in Appendix B. �It follows from Proposition 2 that the false-alarm probability

is determined as

Pfa(γ)Δ= Prob(ξ < γ|H0) = F (γ;α0, β0). (18)

Inversely, given a false-alarm probability Pfa, the deci-sion threshold can be obtained by numerically inverting


F (γ;α0, β0). That is,

γ = F−1(Pfa;α0, β0) (19)

where F−1(·) represents the inverse function of F (·). On theother hand, with the so-obtained threshold, the correspond-ing detection probability is computed by (15). The mappingbetween the false-alarm probability and detection probabilityyields the ROC. Hence, the analytic ROC formula for thevolume-based test is

Pd = F[F−1(Pfa;α0, β0);α1, β1

]. (20)

Remark: Note that the IID noise assumption is not requiredfor the volume-based detector but necessary for theoreticallyderiving the decision threshold, which under the non-IID noisescenario can alternatively be determined by the Monte-Carlosimulation. Nevertheless, it is quite hard to calculate the the-oretical threshold of the volume-based approach for non-IIDnoise, which will be studied in our future work. On the otherhand, recall that the volume-based method involves the compu-tations of SCM and determinant, requiring about O(M2N +M3) flops. Some computationally simpler schemes, such as[33]–[36], might be adopted to alleviate the computationalintensity of the volume-based approach, which will be ourfuture work.

IV. NUMERICAL RESULTS

Simulation results are presented in this section to validateour derived analytic false-alarm and detection probabilities.Moreover, the superiority of the volume-based approach overthe representative approaches for spectrum sensing under theIID and non-IID noise conditions are demonstrated.

A. Analytic False Alarm and Detection Probabilities

In this subsection, the accuracy of the analytic formulaefor the false-alarm and detection probabilities is numericallyevaluated. For the purpose of comparison, the exact false-alarm and detection probabilities, which are empirically deter-mined by 105 Monte Carlo simulation trials, are presented aswell. Fig. 3(a) plots the false-alarm probability versus decisionthreshold in the presence of IID noise, where the numberof antennas is M = 4 whereas the number of samples is setas N = [100, 200, 400]. It is indicated in Fig. 3(a) that theproposed Gamma approximate false-alarm probability is veryaccurate in terms of fitting the empirical false-alarm probabilityfor IID noise. The simulation results for the derived Gamma ap-proximate and empirical false-alarm probabilities are depictedin Fig. 3(b) for M = 6 and N = [200, 400, 600]. It is observedthat the Gamma approximate Pfa is very close to the exact one.This in turn implies that the derived false-alarm probabilityprovides accurate theoretical threshold calculation for practicalspectrum sensing.

Now let us examine the accuracy of the detection probabilityfor the proposed Gamma approximation. Similarly, the exactdetection probability empirically determined by 105 Monte-Carlo simulations are also presented for comparison. In the

Fig. 3. False alarm probability versus threshold in IID noise. (a) M = 4 andN varies from 100, 200 to 400. (b) M = 6 and N varies from 200, 400 to 600.

presence of primary signals, the Rayleigh-fading channel isadopted to evaluate the accuracy of the derived Gamma ap-proximate Pd, which has been well studied in [37]–[40]. Itis shown in [38] that, in the Rayleigh-fading situation, thecolumns of the channel matrix H follow a complex Gaussiandistribution with mean zero and covariance matrix Ψ due to thecorrelation among the signals at the receiving antennas whichcannot be sufficiently spaced for physical size constraints. Asis indicated in [37], [38], [40], the correlated Rayleigh-fadingchannel model is able to precisely characterize the behaviorof the practical channels. The (k, ) entry of Ψ is determinedas [38, Eq.(27)]

φk� =I0(√

κ2 − 4π2d2k� + j4πκ sin(ψ)dk�

)I0(κ)

,

(k, = 1, . . . ,M) (21)

where I0(·) denotes the zero-order modified Bessel function,κ controls the width of the angles-of-arrival (AOAs) of theprimary signals impinging upon the receiving antennas of the


Fig. 4. Detection probability versus threshold for three primary signals in IIDnoise. σ2

s1= −5 dB. (a) M = 4 and N varies from 100, 200 to 300. (b) M =

6 and N varies from 200, 300 to 400.

SU, which can vary from 0 (isotropic scattering) up to ∞(extremely non-isotropic scattering), ψ ∈ [−π, π) representsthe mean direction of the AOAs, and dk� stands for the distance,which is normalized with respect to the wavelength λ, betweenthe k-th and -th antennas of the SU. In the simulation, weset κ = 80 and ψ = π/2, and suppose that the antennas ofthe SU are of the linear uniform array structure with theinter-element distance being λ/2. As a result, the normalizeddistance between the adjacent antennas equals 0.5. Without lossof generality, the channel is normalized as gi = hi/‖hi‖ andthe noise variance is set to one for IID noise.

Fig. 4(a) demonstrates the numerical results for a singleprimary signal in the Rayleigh fading channel and under thesituation of IID noise. Here, the number of antennas is 4, thenumber of samples increases from 100, 200 to 300, andthe power of the primary signal is set as −5 dB. It is implied thatthe derived detection probability is able to accurately predict thedetection performance for the volume-based approach. Fig. 4(b)plots the Gamma approximate and exact detection probabilityof the volume-based detection method for IID noise, where the

Fig. 5. Detection probability versus threshold for three primary signals in IIDnoise. [σ2

s1, σ2

s2, σ2

s3] = [−2,−3,−4] dB. (a) M = 4 and N varies from 100,

200 to 400. (b) M = 6 and N varies from 200, 400 to 800.

number of antennas is 6, the number of samples increases from200, 300 to 400, and the signal power is the same as that inFig. 4(a). The numerical results again validate the efficiencyof the derived Gamma approximate Pd. The numerical resultsfor three primary signals with powers of [−2,−3,−4] dB areplotted in Fig. 5. The number of antennas is 4 and the numberof samples is set as [100, 200, 400] in Fig. 5(a). We can observethat the proposed Gamma approximate detection probability isquite precise in terms of fitting the exact one. In Fig. 5(b), thenumber of antennas and number of samples are set as M = 6and N = [200, 400, 800], respectively. It is observed that thederived Pd is very accurate in terms of predicting the detectionperformance for the volume-based algorithm.

B. Detection Performance

In this subsection, let us evaluate the robustness as well asaccuracy of the volume-based detection algorithm for complex-valued observed data by comparing its empirical ROC withthose of other representative methods. In particular, the decision


threshold is varied to calculate the false-alarm probability andits corresponding detection probability, leading to the ROCcurve. For the purpose of comparison, the numerical resultsof the John’s, AGM (or ST), Hadamard ratio as well as EDdetectors are provided. Recalling that the true noise variance isunknown a priori to the receiver in practice, the ED approach isthereby used herein as the benchmark. In addition, the Rayleighfading channel model addressed above is employed. All thenumerical results are obtained from 105 Monte Carlo trials.

The ROCs of the volume-based, AGM, John’s, Hadamardratio and ED algorithms in Rayleigh fading channel and IIDnoise are demonstrated in Fig. 6, where the number of antennasequals 4. It is indicated from Fig. 6(a), where the numberof samples is 10, the number of primary signals is one andits power equals 8 dB, that the volume-based approach issuperior to the AGM and Hadamard ratio methods but inferiorto John’s detector which is known to be the locally mostpowerful invariant test for sphericity. Moreover, all of themare inferior to the true noise variance based ED approach. Thesimilar results are observed in Fig. 6(b), where the numbersof antennas and samples remain unchanged while the numberof primary signals increases to three and their powers are setas [σ2

s1, σ2

s2, σ2

s3] = [5, 2, 0] dB. This indicates that the volume-

based detector surpasses the AGM and Hadamard ratio methodsbut is not as accurate as John’s approach in the scenarioof IID noise. However, as the sample number increases to50, the volume-based algorithm is able to provide the sameaccuracy as John’s algorithm which has been proved to bethe most powerful for the small sample case. The numericalresults for the same signal parameters and channel model butdifferent noise model, namely, the non-IID noise, are plottedin Fig. 7, where the noise variances are [1, 1.7,−0.7,−2] dBand the averaged noise variance is equal to one. Under such anon-IID noise condition, the volume-based detector is capableof offering the best detection performance among the blindmethods. This in turn implies that the volume-based approachis superior to the Hadamard ratio method in accuracy and to theAGM and John’s schemes in robustness against the deviationof IID noise. As pointed out in Section II-B, the Hadamardratio algorithm cannot utilize the correlation among the primarysignals to improve the detection accuracy although it is robustagainst the non-IID noise. Unlike the Hadamard ratio approach,the denominator in the volume-based test expression consid-erably increases while the numerator, i.e., the determinant ofcovariance matrix, significantly decreases due to the signalcorrelation, finally resulting in notable reduction in the totalvolume. This is why the volume-based test scheme is able toprovide the superiority over the Hadamard ratio method. Asthe AGM and John’s detectors are derived from the assumptionof IID noise, they cannot provide the robustness against thenon-IID noise.

The ROCs of various algorithms for M = 6 and IID noiseare depicted in Fig. 8. Similar to the numerical results inFig. 6, the volume-based method is superior to the AGM andHadamard ratio schemes but inferior to John’s scheme in de-tection performance. The numerical results of various methodsfor M = 6 and non-IID noise are shown in Fig. 9, where thenoise powers are set as [−1.2,−0.3, 2.6,−0.8, 2.4,−2.7] dB.

Fig. 6. ROCs of various detectors for Rayleigh fading channel in IIDnoise. (a) M = 4, N = 10, d = 1, and σ2

s1= 8 dB. (b) M = 4, N = 10,

d = 3, and [σ2s1, σ2

s2, σ2

s3] = [5, 2, 0] dB. (c) M = 4, N = 50, d = 3, and

[σ2s1, σ2

s2, σ2

s3] = [0,−3,−5] dB.

Again, it is seen in Fig. 9 that the volume-based detector out-performs the robust Hadamard ratio test and non-robust AGMas well as John’s approaches in detection accuracy. Moreover,


Fig. 7. ROCs of various detectors for Rayleigh fading channel in non-IIDnoise. (a) M = 4, N = 10, d = 1, and σ2

s1= 8 dB. (b) M = 4, N = 10,

d = 3, and [σ2s1, σ2

s2, σ2

s3] = [5, 2, 0] dB. (c) M = 4, N = 30, d = 3, and

[σ2s1, σ2

s2, σ2

s3] = [0,−3,−5] dB.

for M = 6, N = 30, and [σ2s1, σ2

s2, σ3

s3] = [0,−3,−5] dB, the

volume-based algorithm is even superior to the ED scheme, asillustrated in Fig. 9(c).

Fig. 8. ROCs of various detectors for Rayleigh fading channel in IIDnoise. (a) M = 6, N = 10, d = 1, and σ2

s1= 8 dB. (b) M = 6, N = 10,

d = 3, and [σ2s1, σ2

s2, σ2

s3] = [5, 2, 0] dB. (c) M = 6, N = 50, d = 3, and

[σ2s1, σ2

s2, σ2

s3] = [0,−3,−5] dB.

V. CONCLUSION

The analytic formulae for the false-alarm and detectionprobabilities of the volume-based approach have been derived


Fig. 9. ROCs of various detectors for Rayleigh fading channel in non-IIDnoise. (a) M = 6, N = 10, d = 1, and σ2

s1= 8 dB. (b) M = 6, N = 10,

d = 3, and [σ2s1, σ2

s2, σ2

s3] = [5, 2, 0] dB. (c) M = 6, N = 50, d = 3, and

[σ2s1, σ2

s2, σ2

s3] = [0,−3,−5] dB.

for the IID noise, which provide accurate theoretical thresholdcalculation and efficient approach to evaluate its sensing per-formance. By means of calculating the accurate first and second

moments of the test statistic under the signal-absence hypoth-esis, we are able to employ the moment-matching method toaccurately derive the Gamma distribution. Accordingly, thetheoretical decision threshold can be accurately determinedas well. On the other hand, by computing the accurate first-and second-order moments of the test statistic under thesignal-presence hypothesis, we are capable of approximatingthe Gamma distribution for the volume-based test statistic,ending up with accurate analytic expression for the detectionprobability. Extensive numerical results validate our theoreticalanalysis.

As has been pointed out in Sections I and IV-A, it is quitedifficult to derive the false-alarm probability of the volume-based detector for the non-IID noise situations because thenoise covariance matrix is unknown in practice. More specif-ically, the noise covariance matrix can be diagonal for non-uniform spatially white noise but arbitrary for spatially colornoise. Derivation of the analytic false-alarm probability of thevolume-based method in these non-IID noise scenarios will beour future work.

APPENDIX APROOF OF PROPOSITION 1

Let R = NS =∑N

t=1 xtxHt , which under H1 follows the

complex Wishart distribution, i.e., R ∼ WM (N,Σ). Setting�k =

∑Mi=1 |rki|2 with rki being the (k, i) entry of R, it follows

from (12) that the volume-based test statistic becomes

ξ =1

2

M∑k=1

log �k − log |R|. (A.1)

We utilize the Delta method to approximately calculate the firsttwo moments of ξ. To proceed, we need to the following resultsof complex Bartlett decomposition thanks to Goodman [41].

Lemma 1: Let R′ ∼ WM (N, IM ) and R′ = TTH whereT = (tij) is a complex lower triangular matrix with real andpositive diagonal elements, i.e., tii > 0, and non-diagonal el-ements tij = tRij + jtIij (j > i) with R

ij and Iij being the real

and imaginary parts of ij , respectively. Then, the elements ofT are all independent, 2tii ∼ χ2

2(N−i+1) (1 ≤ i ≤ M) with

χ2 being chi-square distribution and tRij ∼ N (0, 1/2), tIij ∼N (0, 1/2)(1 ≤ j < i ≤ M).

According to Lemma 1, R can be represented in the formof R = ATTHAH with A = Σ1/2. Then, the Taylor seriesexpansion of

ξΔ= f(T ) =

1

2

M∑k=1

log �k − 2 log |T | − 2 log |A| (A.2)

in the neighborhood of the means of tii, tRij and tIij for 1 ≤ j <i ≤ M is given in (A.3), shown at the bottom of the next page.In the sequel, the second-order approximation to the mean of ξ,

denoted as μ1Δ= E[ξ], is

μ1 ≈ f(T )|T=E[T ]

+1

2

⎡⎣ M∑

j<i

∂2f

∂(tRij)2 |T=E[T ]Var

(tRij)


+

M∑j<i

∂2f

∂(tIij)2 |T=E[T ]Var

(tIij)

+M∑i=1

∂2f

∂ (tii)2 |T=E[T ]Var (tii)

]. (A.4a)

Additionally, the first-order approximation to the variance of ξ,

denoted as ν1Δ= Var[ξ], is

ν1 ≈M∑j<i

∂f

∂tRij

∣∣∣∣∣∣T=E[T ]

Var(tRij)+

M∑j<i

∂f

∂tIij

∣∣∣∣∣∣T=E[T ]

Var(tIij)

+

M∑i=1

∂f

∂tii

∣∣∣∣∣T=E[T ]

Var (tii) . (A.4b)

We are now at a position to determine the first two momentsof the elements of T , and the first and second-order partialderivatives of f(T ) with respect to the elements of T . It followsfrom Lemma 1 that

E[tRij]=E

[tIij]= 0 (1 ≤ j < i ≤ M) (A.5a)

Var[tRij]=Var

[tIij]=

1

2(1 ≤ j < i ≤ M) (A.5b)

E [tii] =Γ(n− i+ 3

2 )

Γ(n− i+ 12 )

(1 ≤ i ≤ M) (A.5c)

Var[tii] =n− i+ 1−(Γ(n− i+ 3

2 )

Γ(n− i+ 12 )

)2

(1 ≤ i ≤ M).

(A.5d)

On the other hand, the first-order partial derivatives of f(T )with respect to tRij , tIij , and tii are computed in (A.6), shownat the bottom of the page. In addition, the second-order partialderivatives of f(T ) with respect to tRij , tIij , and tii are calculatedby (A.7), shown at the bottom of the next page. Here, the first-order partial derivatives of rRij and rIij with respect to tRij , tIij ,and tii are given, respectively, as

∂rRk�∂tRij

=(aRika

Rj� + aIika

Ij�

)E[tii] (A.8a)

∂rRk�∂tIij

=(aIika

Rj� − aRika

Ij�

)E[tii] (A.8b)

f(T )≈f(T )|T=E[T ]+

M∑j<i

∂f

∂tRij

∣∣∣∣∣T=E[T ]

(tRij − E

[tRij])+

M∑j<i

∂f

∂tIij

∣∣∣∣∣T=E[T ]

(tIij−E

[tIij])+

M∑i=1

∂f

∂tij

∣∣∣∣T=E[T ]

(tij−E [tij ])

+1

2

⎡⎢⎣ M∑

p<q

M∑j<i

∂2f

∂tRij∂tRpq

∣∣∣∣∣∣T=E[T ]

(tRij−E

[tRij]) (

tRpq−E[tRpq])

+M∑p<q

M∑j<i

∂2f

∂tIij∂tIpq

∣∣∣∣∣∣T=E[T ]

(tIij − E

[tIij]) (

tIpq − E[tIpq])

+M∑p=1

M∑i=1

∂2f

∂tpp∂tii

∣∣∣∣∣T=E[T ]

(tpp − E [tpp]) (tii − E [tii]) + 2M∑p<q

M∑j<i

∂2f

∂tRij∂tIpq

∣∣∣∣∣∣T=E[T ]

(tRij − E

[tRij]) (

tIpq − E[tIpq])

+ 2

M∑p=1

M∑j<i

∂2f

∂tRij∂tpq

∣∣∣∣∣∣T=E[T ]

(tRij−E

[tRij])

(tpp−E [tpp]) + 2

M∑p=1

M∑j<i

∂2f

∂tIij∂tpq

∣∣∣∣∣∣T=E[T ]

(tIij − E

[tIij])

(tpp − E [tpp])

⎤⎥⎦

(A.3)

∂f

∂tRij

∣∣∣∣∣T=E[T ]

=1

2

M∑k=1

M∑�=1,k �=�

[2rRk��k

∂rRk�∂tRij

+2rIk��k

∂rIk�∂tRij

]+

1

2

M∑k=1

2rkk�k

∂rkk∂tRij

(A.6a)

∂f

∂tIij

∣∣∣∣∣T=E[T ]

=1

2

M∑k=1

M∑�=1,k �=�

[2rRk��k

∂rRk�∂tIij

+2rIk��k

∂rIk�∂tIij

]+

1

2

M∑k=1

2rkk�k

∂rkk∂tIij

(A.6b)

∂f

∂tii

∣∣∣∣T=E[T ]

=1

2

M∑k=1

M∑�=1,k �=�

[2rRk��k

∂rRk�∂tii

+2rIk��k

∂rIk�∂tii

]+

1

2

M∑k=1

2rkk�k

∂rkk∂tii

− 2

E[tii](A.6c)


∂rRk�∂tii

=2(aRika

Ri� + aIika

Ii�

)E[tii] (A.8c)

and

∂rIk�∂tRij

=(aRika

Ij� − aIika

Rj�

)E[tii] (A.8d)

∂rIk�∂tIij

=(aRika

Rj� + aIika

Ij�

)E[tii] (A.8e)

∂rIk�∂tii

=2(aIika

Rj� − aRika

Ij�

)E[tii]. (A.8f)

Moreover, the second-order partial derivatives of rRij and rIijwith respect to tRij , tIij and tii are given, respectively, as

∂2rRk�

∂(tRij)2 =2

(aRika

Ri� + aIika

Ii�

)(A.8g)

∂2rRk�∂(tIij)2

=2(aRika

Ri� + aIika

Ii�

)(A.8h)

∂2rRk�∂t2ii

=2(aRika

Ri� − aIika

Ii�

)(A.8i)

∂2f

∂(tRij)2∣∣∣∣∣T=E[T ]

=1

2

M∑k=1

M∑�=1,k �=�

⎧⎨⎩

M∑p=1,p �=k,p �=�

[4rRk�r

Rkp

�2k

∂rRk�∂tRij

∂rRkp∂tRij

+4rRk�r

Ikp

�2k

∂rRk�∂tRij

∂rIkp∂tRij

+4rIk�r

Rkp

�2k

∂rIk�∂tRij

∂rRkp∂tRij

+4rIk�r

Ikp

�2k

∂rIk�∂tRij

∂rIkp∂tRij

]

+4rRk�rkk

�2k

∂rRk�∂tRij

∂rkk∂tRij

+4rIk�rkk

�2k

∂rIk�∂tRij

∂rkk∂tRij

+2�k − 4

(rRk�)2

�2k

(∂rRk�∂tRij

)2

+2rRk��k

∂2rRk�

∂(tRij)2

+2�k − 4

(rIk�)2

�2k

(∂rIk�∂tRij

)2

+2rIk��k

∂2rIk�

∂(tRij)2 +1

2

M∑k=1

⎡⎣2�k − 4r2kk

�2k

(∂rkk∂tRij

)2

+2rkk�k

∂2rkk

∂(tRij)2⎤⎦⎫⎬⎭

(A.7a)∂2f

∂(tIij)2∣∣∣∣∣T=E[T ]

=1

2

M∑k=1

M∑�=1,k �=�

⎧⎨⎩

M∑p=1,p �=k,p �=�

[4rRk�r

Rkp

�2k

∂rRk�∂tIij

∂rRkp∂tIij

+4rRk�r

Ikp

�2k

∂rRk�∂tIij

∂rIkp∂tIij

+4rIk�r

Rkp

�2k

∂rIk�∂tIij

∂rRkp∂tIij

+4rIk�r

Ikp

�2k

∂rIk�∂tIij

∂rIkp∂tIij

]

+4rRk�rkk

�2k

∂rRk�∂tIij

∂rkk∂tIij

+4rIk�rkk

�2k

∂rIk�∂tIij

∂rkk∂tIij

+2�k − 4

(rRk�)2

�2k

(∂rRk�∂tIij

)2

+2rRk��k

∂2rRk�

∂(tIij)2

+2�k − 4

(rIk�)2

�2k

(∂rIk�∂tIij

)2

+2rIk��k

∂2rIk�

∂(tIij)2

+1

2

M∑k=1

⎡⎣2�k − 4r2kk

�2k

(∂rkk∂tIij

)2

+2rkk�k

∂2rkk∂(tIij)2⎤⎦⎫⎬⎭

(A.7b)∂2f

∂t2ii

∣∣∣∣T=E[T ]

=1

2

M∑k=1

M∑�=1,k �=�

⎧⎨⎩

M∑p=1,p �=k,p �=�

[4rRk�r

Rkp

�2k

∂rRk�∂tii

∂rRkp∂tii

+4rRk�r

Ikp

�2k

∂rRk�∂tii

∂rIkp∂tii

+4rIk�r

Rkp

�2k

∂rIk�∂tii

∂rRkp∂tii +4rIk�r

Ikp

�2k

∂rIk�∂tii

∂rIkp∂tii

]

+4rRk�rkk

�2k

∂rRk�∂tii

∂rkk∂tii

+4rIk�rkk

�2k

∂rIk�∂tii

∂rkk∂tii

+2�k − 4

(rRk�)2

�2k

(∂rRk�∂tii

)2

+2rRk��k

∂2rRk�∂t2ii

+2�k − 4

(rIk�)2

�2k

(∂rIk�∂tii

)2

+2rIk��k

∂2rIk�∂t2ii

+1

2

M∑k=1

[2�k − 4r2kk

�2k

(∂rkk∂tii

)2

+2rkk�k

∂2rkk∂t2ii

]}+

2

E2[tii]

(A.7c)


and

∂2rIk�

∂(tRij)2 =2

(aIjka

Rj� − aRjka

Ij�

)(A.8j)

∂2rIk�

∂(tIij)2 =2

(aIjka

Rj� − aRjka

Ij�

)(A.8k)

∂2rIk�∂t2ii

=2(aIika

Ri� − aRika

Ij�

). (A.8l)

Furthermore, we easily obtain

∂rkk∂tRij

=(aRika

Rjk + aIika

Ijk

)E[tii] (A.8m)

∂rkk∂tIij

=(aIika

Rjk − aRika

Ijk

)E[tii] (A.8n)

∂rkk∂tii

=2((

aRik)2

+(aIik)2)

E[tii] (A.8o)

and

∂2rkk

∂(tRij)2 =2

((aRjk)2

+(aIjk)2)

(A.8p)

∂2rkk

∂(tIij)2 =2

((aRjk)2

+(aIjk)2)

(A.8q)

∂2rkk∂t2ii

=2((

aRik)2

+(aIik)2)

. (A.8r)

Substituting (A.8) into (A.6) and (A.7), and then inserting theso-obtained results along with (A.5) into (A.4), we eventuallyattain the mean μ1 and variance ν1 for calculating α1 and β1 in(14). This completes the proof of Proposition 1.

APPENDIX BPROOF OF PROPOSITION 2

For the signal-absence hypothesis, we have A = IM . Itfollows from (A.4) and (A.5) that the mean and variance of ξunder H0, denoted by μ0 and ν0, respectively, can be given as

μ0 ≈ f(T )|T=E[T ] +1

4

M∑j<i

(∂2f

∂(tRij)2 +

∂2f

∂(tIij)2)∣∣∣∣∣∣

T=E[T ]

+1

2

M∑i=1

(n− i+ 1−

(Γ(n− i+ 3

2 )

Γ(n− i+ 12 )

)2)

∂2f

∂ (tii)2

∣∣∣∣∣T=E[T ]

(B.1a)

and

ν0 ≈ 1

2

M∑j<i

(∂f

∂tRij+

∂f

∂tIij

)∣∣∣∣∣∣T=E[T ]

+

M∑i=1

(n− i+ 1−

(Γ(n− i+ 3

2 )

Γ(n− i+ 12 )

)2)

∂f

∂tii

∣∣∣∣∣T=E[T ]

.

(B.1b)

where f(T ) = (1/2)∑M

k=1 log �k − 2 log |T |. Moreover, itfollows from (A.6), (A.7), and (A.8) that the first and second-order partial derivatives of f(T ) with respect to the elements ofT can be simplified, respectively, as

∂f

∂tRij

∣∣∣∣∣T=E[T ]

=

(rRij�j

+rRij�i

)∂rRij∂tRij

(B.2a)

∂f

∂tIij

∣∣∣∣∣T=E[T ]

=

(rIij�j

+rIij�i

)∂rIij∂tIij

(B.2b)

∂f

∂tii

∣∣∣∣T=E[T ]

=rii�i

∂rii∂tii

− 2

E[tii](B.2c)

and

∂2f

∂(tRij)2∣∣∣∣∣T=E[T ]

=

(�j−2

(rRij)2

�2j+�i−2

(rRij)2

�2i

)(∂rRij∂tRij

)2

+rjj�j

∂2rjj

∂(tRij)2 (B.3a)

∂2f

∂(tIij)2∣∣∣∣∣T=E[T ]

=

(�j−2

(rIij)2

�2j+�i−2

(rIij)2

�2i

)(∂rIij∂tIij

)2

+rjj�j

∂2rjj

∂(tIij)2 (B.3b)

∂2f

∂ (tii)2

∣∣∣∣T=E[T ]

=�i−2r2ii

�2i

(∂rii∂tii

)2

+rii�i

∂2rii

∂ (tii)2 +

2

E2[tii]

(B.3c)

where

∂rii∂tii

=2∂rRij∂tRij

= 2∂rIij∂tIij

= 2E[tii] (B.4a)

∂2rjj

∂(tRij)2 =

∂2rjj

∂(tIij)2 =

∂2rii

∂ (tii)2 = 2. (B.4b)

Thus, inserting (B.4) into (B.2) and (B.3), and then substitutingthe so-obtained results into (B.1) yield μ0 and ν0 for computingα0 and β0 in (17). The proof of Proposition 2 is finished.

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Lei Huang (M’07) was born in Guangdong, China.He received the B.Sc., M.Sc., and Ph.D. degreesin electronic engineering from Xidian University,Xi’an, China, in 2000, 2003, and 2005, respectively.

From 2005 to 2006, he was a Research Associatewith the Department of Electrical and ComputerEngineering, Duke University, Durham, NC. From2009 to 2010, he was a Research Fellow with theDepartment of Electronic Engineering, City Univer-sity of Hong Kong and a Research Associate with theDepartment of Electronic Engineering, The Chinese

University of Hong Kong. Since 2011, he has joined the Department of Elec-tronic and Information Engineering, Harbin Institute of Technology ShenzhenGraduate School, where he is currently a Professor. His research interests in-clude spectral estimation, array signal processing, statistical signal processing,and their applications in radar and wireless communication systems.

He currently is an editorial board member of Digital Signal Processing.

Cheng Qian was born in Zhejiang on November 27,1988. He received the B.E. degree in communica-tion engineering from Hangzhou Dianzi University,Hangzhou, China, in 2011, and M.E. degree in infor-mation and communication engineering from HarbinInstitute of Technology (HIT), Shenzhen, China, in2013. He is currently pursuing the Ph.D. degree inthe field of information and communication engi-neering at HIT. His research interests are in arraysignal processing and MIMO radar.


Yuhang Xiao was born in Anhui on January 20,1992. He received the B.E. degree from HarbinEngineering University, Harbin, China, in 2012. Heis currently pursuing the Ph.D. degree in the fieldof communication and information engineering atHIT. His research interests are in statistical signalprocessing and spectrum sensing.

Q. T. Zhang (S’84–M’85–SM’95–F’09) receivedthe B.Eng. degree from Tsinghua University,Beijing, China and M.Eng. degree from South ChinaUniversity of Technology, Guangzhou, China, bothin wireless communications, and the Ph.D. degreein electrical engineering from McMaster University,Hamilton, ON, Canada.

After graduation from McMaster in 1986, he helda research position and Adjunct Assistant Profes-sorship at the same institution. In January 1992, hejoined the Spar Aerospace Ltd., Satellite and Com-

munication Systems Division, Montreal, as a Senior Member of Technical Staff.At Spar Aerospace, he participated in the development and manufacturing ofthe Radar Satellite (Radarsat). He joined Ryerson University, Toronto in 1993and became a Professor in 1999. In 1999, he took one-year sabbatical leave atthe National University of Singapore, and was a Chair Professor of InformationEngineering at the City University of Hong Kong. His research interest is onwireless communications with current focus on wireless MIMO, cooperativesystems, and cognitive radio. He served as an Associate Editor for the IEEECommunications Letters from 2000 to 2007.

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