Abstract—In this paper, we develop generalized likelihoodratio test (GLRT)-based detectors for robust spectrum sensing inmultiple-input multiple-output (MIMO) cognitive radio networksconsidering uncertainty in the available channel state information(CSI). Initially, for a scenario with known CSI uncertainty sta-tistics, we derive the novel robust estimator-correlator detector(RECD) and the robust generalized likelihood detector (RGLD),which are robust against the uncertainty in the available estimatesof the channel coefficients. Subsequently, for a scenario withunknown CSI uncertainty statistics, we develop a generalizedlikelihood ratio test (GLRT) based composite hypothesis robustdetector (CHRD) for spectrum sensing. Closed form expressionsare presented for the probability of detection and theprobability of false alarm to characterize the detectionperformance of the proposed robust spectrum sensing schemes.Further, a deflection coefficient based optimization framework isalso developed and solved to derive closed form expressions forthe optimal beacon sequences. Simulation results are presentedto demonstrate the performance improvement achieved by theproposed robust spectrum sensing schemes and to verify theanalytical results derived.Index Terms—Cognitive radio, spectrum sensing, multiple-input
multiple-output (MIMO), channel state information (CSI) un-certainty.
I. INTRODUCTION
T HE proliferation of wireless multimedia applications haslead to a natural increase in the demand for a higher data
rate in current and upcoming 3G/4G wireless communicationsystems. The radio frequency (RF) spectrum is a valuable re-source for wireless communication applications and its usageis regulated by agencies such as the Federal CommunicationsCommission (FCC) of the United States etc. However, recentreports such as [1] on the temporal and geographic RF spec-trum utilization patterns point to an extremely low efficiencyof current spectrum utilization. Thus, to cope with the tremen-dous increase in the demand for RF spectrum, coupled with the
Manuscript received November 20, 2014; revised April 15, 2015 and July15, 2015; accepted October 18, 2015. Date of publication November 13, 2015;date of current version February 16, 2016. The associate editor coordinatingthe review of this manuscript and approving it for publication was Dr. JosephCavallaro. This work was supported in part by the TCS Research ScholarshipProgram, DST-SERB and DST-IUATC Project Grants.The authors are with the Department of Electrical Engineering, Indian Insti-
at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2015.2500183
motivation for efficient spectrum utilization, the FCC has re-cently proposed the cognitive radio paradigm [2]–[4] which al-lows a set of unlicensed/secondary users to opportunistically ac-cess unused spectrum bands licensed to primary users. This inturn leads to an increase in the efficiency of spectrum utiliza-tion. This strategic reuse of the licensed spectrum by the unli-censed secondary users in cognitive radio networks necessitatesthe need to reliably detect the presence of vacant spectral bands,termed as spectral holes or white spaces, without causing a sig-nificant interference to the licensed primary users. This key taskof spectral occupancy detection in cognitive radio networks istermed as spectrum sensing.Several spectrum sensing techniques have been proposed in
literature [5] for the detection of primary user signals in cogni-tive radio networks. Among these, the energy detector [6] hasa simple structure and does not require any prior knowledge ofthe primary user signal. However, the non-coherent energy de-tector has a poor performance compared to other techniques forspectrum sensing [7]–[9]. Other popular spectrum sensing tech-niques exploit statistical properties of the primary user signalfor the detection of spectral holes. These techniques can be ex-tended to wideband spectrum sensing, with Nyquist and sub-Nyquist sampling, as described in [10], [11] and to a distributedwideband sensing scenario with finite feedback in [12]. For in-stance, the cyclostationarity based detection scheme [5] requiresthe existence and knowledge of the cyclic frequency of the pri-mary user signal. Similarly, the matched-filter based detectionscheme [5] requires perfect channel state information (CSI) ofthe primary user, in the absence of which its performance de-grades drastically [13]. Further, the work in [14] demonstratesthat soft combination based schemes, such as equal gain com-bining (EGC) and maximal ratio combining (MRC), exhibit asignificant improvement in the detection performance over con-ventional hard decision fusion schemes. However, it is shownin [15], that the improvement in the detection performance ob-tained from such soft-decision based spectrum sensing schemesdepends significantly on the accuracy of the CSI at the sec-ondary user. An adaptive spectrum sensing scheme based on afinite state Markov channel (FSMC) model is presented in [16].However, the FSMCwith finite states for the channel coefficientdoes not capture the continuous variation of the channel.On the other hand, the generalized likelihood ratio test
(GLRT) based approach exploits the prior information of thesignal of the licensed user. One major advantage of the GLRTbased approach is that it achieves joint primary user detectionand unknown parameter estimation. In literature, several spec-
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trum sensing algorithms have been proposed for cognitive radioscenarios, which compute the maximum likelihood estimates(MLE) of the unknown system parameters towards employingGLRT based detection schemes. For example, [17]–[20] com-pute the MLE of the noise variance and the signal correlationmatrix, whereas [21]–[23] compute the MLE only for thenoise variance in order to employ the GLRT framework forprimary user signal detection. However, it is shown [7], [8]that the noise power uncertainty agnostic detectors, such as theenergy detector, are limited by a signal-to-noise ratio (SNR)wall below which the detector fails to detect the presence ofthe primary user signal, irrespective of the sensing duration.Similarly, [24] shows the existence of the sampling wall, i.e.,the sampling density below which the performance of thedetector can not be guaranteed at a given SNR, regardless ofthe number of samples of the primary signal acquired by thesecondary user. Further, [17], [18], [25] employ the GLRTframework with the unknown parameter being the CSI betweenthe primary transmitter and the secondary receiver. The variouscontributions of these works are listed below.• The work in [17] presents a GLRT based framework forspectrum sensing with unknown channel gains and con-sidering a single antenna at the primary user and multipleantennas at the secondary user
• The work in [18] presents various partial GLRT schemeswith each of the channel gain, noise variance and signalvariance components unknown considering block as wellas fast fading channel scenarios
• The work in [25] presents a framework for GLRT basedspectrum sensing with Bernoulli nonuniform sampling(BNS) considering unknown signal power.
However, a significant shortcoming of all these works aboveis that they do not consider CSI uncertainty, which is of signif-icance in communication scenarios where frequently nominalCSI is available at the receiver. The work in [26] considers theWilks’ detector based spectrum sensing for a scenario with un-known noise covariance. However, the authors therein considera simplistic scenario with a deterministic channel matrix andanalyze the instantaneous detection performance without aver-aging over the channel statistics.Obtaining the true channel coefficients in a cognitive radio
network is challenging, owing to estimation/quantization er-rors, limited feedback, Doppler shift etc, in practical wirelessscenarios. These uncertainties in the system parameters canpotentially lead to a performance degradation of uncertaintyagnostic spectrum sensing schemes. Therefore, this paperconsiders a GLRT based approach for spectrum sensing, unlikeconventional techniques such as matched filtering, energydetection and soft combining etc described in [5], [6], [14] re-spectively. Works such as [10]–[12], exploit only second orderstatistical information. Further, the motivation of this workis to develop detection schemes which are robust against theperformance degradation caused by the uncertainty in the avail-able CSI estimates, unlike the conventional GLRT schemes[17]–[23], [25] which consider completely unknown CSI, noisevariance or other parameters. We propose novel primary userdetection schemes, namely the robust estimator-correlator de-tector (RECD) and the generalized likelihood ratio test (GLRT)based robust generalized likelihood detector (RGLD), whichare robust against the uncertainty in the nominal estimates of
the channel coefficients for a scenario with known uncertaintystatistics. An analytical framework is developed to characterizethe theoretical performance of the proposed RECD schemeand derive the expressions for the probability of detection
and probability of false alarm . Also, the workin this paper considers a general scenario with multiple an-tennas, while [10]–[12], [14]–[16], [23] are restricted to singleantenna scenarios. Next we present the composite hypothesisbased robust detector (CHRD), for primary user detection inscenarios with unknown CSI uncertainty statistics. Further, wealso derive closed form expressions to determine the proba-bility of detection and probability of false alarmperformance of the proposed CHRD scheme. Also, the analysisfor the in our work is carried out by averaging withrespect to the distributions of both the CSI uncertainty andalso the available nominal channel estimate, unlike the workin [26]. The subsequent section presents a novel deflectioncoefficient based optimization framework to derive the optimalbeacon matrices for the proposed robust detection schemes,with known/unknown uncertainty covariance statistics, whichfurther enhance the detection performance. Simulation resultsdemonstrate the improvement in the detection performanceachieved by the proposed spectrum sensing schemes and alsovalidate the analytical results.This paper is organized as follows. Section II describes
the multiuser MIMO system model for cognitive radio net-works. Section III presents the robust estimator-correlatordetector (RECD) and the robust generalized likelihood de-tector (RGLD) with known CSI uncertainty statistics and theassociated detection and false alarm probabilities. Next, inSection IV we present the composite hypothesis based robustdetector (CHRD) with unknown CSI uncertainty statistics.Closed form expressions for the and to analyticallycharacterize the detection performance of the proposed schemeare also derived therein. Section V describes the optimizationframework to derive the optimal beacon sequence with bothknown and unknown CSI uncertainty statistics. Simulationresults are given in Section VI followed by the conclusion inSection VII.Throughout this paper we use boldface uppercase/lowercase
letters to denote matrices/vectors, respectively. All the vectorsare column vectors. The operations anddenote the conjugate, transpose, conjugate transpose and expec-tation operators. The random vector , when defined as
follows a complex Gaussian distribution with meanand covariance . Similarly, implies that follows a
central chi-squared distribution with degrees of freedom. Thenorm of a vector and the trace of a matrix are represented
by and respectively. The matrix denotes thediagonal matrix with elements of the vector along its prin-cipal diagonal. The identity matrix of dimension is de-noted by and the various detector test statistics are denotedby . Finally, the functions and denotethe Gamma function, the incomplete Gamma function and theGaussian -function respectively.
II. SYSTEM MODEL
Consider a spectrum sensing scenario with a primary userbase-station and a secondary user. Further, we assume a mul-tiple-input multiple-output (MIMO) cognitive radio network
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with transmit antennas at the primary user base-stationand receive antennas at the secondary user. The basebandsystem model for the scenario described above at the th timeinstant is given as,
where is the receive signal vector at thesecondary user corresponding to the primary user base-sta-tion broadcast beacon signal and thevector is the additive spatio-temporallywhite Gaussian noise at the secondary user with covariance
. Each , is theMIMO channel matrix between the primary user base-stationand the secondary user. The system model considered followsin the same spirit as those in works such as [17], [18], [22],[25]. From the Cognitive Radio system model described above,the signal for the th receive antenna can be equivalentlyexpressed as,
where denotes the complex conjugate and the vectorsconstitute the rows of the channel
matrix . Let the broadcast beaconvectors be concatenated to form a beaconmatrix . The con-catenated signal corresponding to the symbols can beequivalently represented as,
(1)
where is the signalat the th receive antenna corresponding to the broadcastbeacon matrix . Similarly, the concatenated noise vector
has covariancematrix . In practical wireless sce-narios it is significantly challenging to obtain accurate CSI asdescribed previously. Hence, we model the channel matrixby incorporating the CSI uncertainty as,
(2)
where is the available nominal CSIand the matrix denotes uncertainty in the channelmatrix . The uncertainty matrix ,where each row vector follows acomplex Gaussian distribution i.e., with theuncertainty covariance matrix .The concatenated system model in (1), incorporating the uncer-tainty model described in (2), can be equivalently obtained as,
(3)
In the next section we describe robust spectrum sensingschemes, which consider CSI uncertainty, for primary userdetection in multiuser MIMO cognitive radio networks, withknown uncertainty covariance statistics.
III. ROBUST SPECTRUM SENSING WITH KNOWNUNCERTAINTY STATISTICS
Let the beacon matrixwhere denote the beacon
signals from the primary user base-station and corre-spond to the absence, presence of the primary users respectively.From (3) it follows that the signal at the th receive antennais given as,
(4)
where the hypotheses and correspond to the absenceand presence of the primary user respectively for the binary hy-pothesis testing problem towards primary user detection. Forexample, in a practical wireless scenario, the beacon matrix
denotes the absence of primary transmission i.e.,when the spectral band is vacant. This non-antipodal signalingmodel is well suited to the context of a cognitive radio scenariowhere the non-zero beacon matrix denotes the presence ofthe primary user signal while the all zero beacon matrix de-notes the absence of the primary user signal.We now present theRECD scheme for robust spectrum sensing in MIMO cognitiveradio scenarios.
A. Robust Estimator-Correlator Detection (RECD)The observation vectors in (4), corresponding to the hy-
potheses are distributed as,
where and is theuncertainty covariance matrix. Let the concatenated observationmatrix corresponding to the receive antennas be definedas . The likelihood ratiocorresponding to the concatenated observation matrix can beexpressed as,
(5)
where the constant and the operator repre-sents an equivalence to a constant factor. Using the matrix in-version identity from [27] in (5), the likelihood ratio canbe simplified to obtain the robust estimator-correlator detector(RECD) test statistic,
(6)
Therefore the optimal Neyman-Pearson criterion based LRT,which maximizes the probability of detection for a givenprobability of false alarm can be obtained as,
(7)
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where the presence/absence of the primary user signal is deter-mined depending on whether the test statistic ex-ceeds or falls short of the detection threshold . The proposedRECD exploits the statistical properties of the CSI uncertainty,thereby leading to an improvement in the accuracy of primaryuser detection. Consider the specific case when the CSI uncer-tainty covariance matrix is given as and thebeacon matrix corresponding to the alternative hypothesis isan orthogonal matrix with . Under these assump-tions, the matrix in (5) is seen to be given as, with
. Similarly, let the matrix .Therefore with . The Lemma belowcharacterizes the asymptotic versus performance of theRECD scheme described above for spectrum sensing.Lemma 1: The probabilities of detection and false alarm, de-
noted by and respectively, for the RECD detector in(7) towards robust spectrum sensing for a large number of re-ceive antennas are given as,
(8)
(9)
where .Proof: It can be seen that the test statistic in
(5) can be equivalently written as,
(10)
where is the th component of the vector defined as. The component test statistic
defined above follows a chi-squared distributionwith two degrees of freedom, described as,
respectively. The quantity denotes the non-central chi-squared distribution with two degrees of freedom, and non-cen-trality parameter . The distribu-tions of the scaled test statistic for the received
vectors stacked as corresponding to thetwo hypotheses can be equivalently derived as,
(11)
where the non-centrality parameter . Thechi-squared distribution corresponding to the two hypotheses in(11) has degrees of freedom. Using the asymptotic prop-erty of the chi-squared random variable for a large number ofreceive antennas [6], the moments of the scaled test statistic
corresponding to the two hypotheses in (11) can beequivalently obtained as,
(12)
Using the results in (12), the analytical expression for the detec-tion probability of the proposed RECD can be equivalentlyobtained as,
where is the detection threshold and denotes the stan-dard Gaussian -function [6], [28]. Similarly, the probability offalse alarm can be obtained as,
B. Robust Generalized Likelihood Detector (RGLD)
We now employ the generalized likelihood ratio test (GLRT)paradigm to develop the robust generalized likelihood detector(RGLD) for primary user detection in MIMO cognitive radionetworks with CSI uncertainty. Let the vector be defined as
corresponding to the alternative hypothesis. Thus, the signal described in (3) corresponding to the
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concatenated sensed samples at the th receive antenna of thetotal receive antennas can be equivalently written as,
(13)
where denotes the concatenatedunknown random vector and the identity matrix is of dimen-sion .Lemma 2: The GLRT-based test statistic corre-
sponding to the uncertainty covariance matrix , is given as,
(14)
where and is the blockdiagonal matrix with along its principal diagonal.
Proof: The likelihood of the observation vector param-eterized by corresponding to the alternative hypothesiscan be derived as,
(15)
where denotes the covariance matrix of, given as,
and denotes the determinant of the matrix . The pa-rameterized vector is obtained on maximizing the likelihood
in (15) which can be formulated as the standardweighted minimum norm optimization problem [27],
The solution of the above convex optimization problem yieldsthe estimate of as,
Employing the GLRT framework, the RGLD test statisticfor the primary user detection problem in cognitive
radio scenarios can be derived as,
where the operator denotes equivalence to a constant factor.This completes the proof.The inherent ability of the proposed RECD and RGLD
schemes to exploit the CSI uncertainty for primary user de-tection in spectrum sensing scenarios provides a performanceedge over the conventional uncertainty agnostic matched filterdetector. In the next section we further relax the CSI uncer-tainty model and consider the uncertainty covariance statisticsto be unknown. We then derive the corresponding composite
hypothesis based robust detector (CHRD) for spectrum sensingin MIMO cognitive radio scenarios.
IV. ROBUST DETECTION WITH UNKNOWN UNCERTAINTYCOVARIANCE STATISTICS
A. Composite Hypothesis Based Robust Detector (CHRD)In this section we compute a generalized likelihood ratio test
based composite hypothesis testing framework with unknownuncertainty statistics. The primary user detection problem from(1) can be equivalently recast as the hypothesis testing problem,
where is the identity matrix and the vector parame-ters form the rows of the channel matrix . Letthe vectors denote the maximum likelihood estimate(MLE) of the vector parameter corresponding to the null hy-pothesis , alternative hypothesis respectively, which canbe obtained [6] as,
(16)
The concatenated received vector , defined in (1), follows acomplex Gaussian distribution given as,
corresponding to the hypotheses . Let the matrixbe obtained by stacking the received signal vectors , as
. Hence, the test statistic forthe composite hypothesis testing based primary user detectionproblem can be computed by applying the generalized likeli-hood ratio test (GLRT) as,
(17)
(18)
where the last equality holds due to the orthogonality propertyof the beacon matrix, i.e., . The test statistic
obtained above yields the primary user detectionrule that is robust against the uncertainty in the estimate of theMIMO channel matrix. We now consider a general scenariowith non-isotropic CSI uncertainty covariance, i.e., the uncer-tainty covariance is not necessarily of the form . Itcan be noted that the non-isotropic CSI uncertainty covariance
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matrix considered in our formulation is more general and prac-tical in comparison to the isotropic model considered in workssuch as [29], [30]. We now characterize the theoretical perfor-mance of the CHRD for composite hypothesis testing based pri-mary user detection in MIMO cognitive radio scenarios. Using(3) and the orthogonality property of the beacon matrix in (16),the maximum likelihood estimate of the column vectors
of the channel matrix , corresponding to thealternative hypothesis , can be equivalently written as,
(19)
where is defined as with covariance. The CSI estimate follows
a complex Gaussian distribution . Thus, thisimplies that the true channel coefficient vector isalso complex Gaussian distributed, leading to a Rayleigh fadingwireless channel, which is a standard assumption for fadingwireless scenarios. It follows from (19) that the maximum like-lihood estimate has a complex Gaussian distribution, with
where . Let theeigenvalue decomposition of be , with the eigen-value matrix , where denotesa diagonal matrix with the elements of vector along the prin-cipal diagonal. Therefore, the th element of the vector
follows aGaussian distribution. Let be defined as . Thetest statistic in (18) can be equivalently described as,
(20)
where each is Gaussian distributed with .Therefore, is distributed as a
random variable. Employing the above results, we now char-acterize the performance of the composite hypothesis detectorcorresponding to the test statistic in (20).Theorem 1: The probability of false alarm and proba-
bility of detection of the CHRD detector based on the teststatistic in (20), for primary user detection inMIMOcognitive radio networks can be derived as,
(21)
(22)
where the coefficients are the partial fraction constantsobtained using the residue method, is the incompleteGamma function defined in (30) and the constant is definedas .
Proof: The probability of false alarm of the GLRTtest statistic can be derived as,
Next, we derive the probability of detection for the teststatistic . This can be expressed as,
(23)
where denotes the probability density function of the teststatistic . Let denote the characteristic func-tion of the test statistic corresponding to the alterna-tive hypothesis . The probability density function canbe expressed in terms of as,
(24)
The characteristic function for the CHRD test statisticcorresponding to the alternative hypothesis is derived as,
(25)
(26)
where (25) follows from the simplification of the test statisticdescribed in (20). The characteristic function
obtained in (26) follows from the fact that eachfollows a chi-squared distribution with 2 degrees of freedomas described earlier. Employing (26) above, the probabilitydensity function of the test statistic definedin (24) can be equivalently expressed as,
(27)
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where the coefficients correspond to the partial frac-tion expansion of and the constant is defined as
. The coefficients can be obtainedusing the standard residue method for partial fraction expansion[31] and can be explicitly expressed as,
(28)
where . Computing the inverse Fourier transformof the expression in (27), the probability density functionof the test statistic in (20), corresponding to the al-ternative hypothesis , can be obtained as,
(29)
where is the unit step function defined as,
Hence, the probability of detection for the composite hy-pothesis detector can be derived by substituting the above ex-pression for in (23) as,
where is the threshold for detection and denotes theincomplete Gamma function [28], defined as,
(30)
Further, we now derive the probability of detection forthe restrictive case of an isotropic CSI uncertainty covariancematrix, i.e., with .Lemma 3: For an isotropic covariance matrix i.e., with, the probability of detection for the detector in (18) reduces
to,(31)
Proof: The characteristic function in (26) andthe corresponding probability density function of thetest statistic in (20), for the covariance matrix
, can be obtained as,
(32)
respectively, where the constant is defined as. The partial fraction constants in (32) can now
be equivalently obtained as,
(33)
where the constant is defined as . Using theexpression for from (32), the corresponding probability ofdetection for this scenario with a diagonal covariance matrix
can be derived as,
Substituting the expression for the coefficient from (33), theabove expression for can be simplified as,
(34)
where (34) follows from the following property of the Gammafunction [28],
In the next section we develop the framework to obtain theoptimal beacon sequence which can further enhance the detec-tion performance of the proposed spectrum sensing schemes.
V. OPTIMAL BEACON FORMULATION
This section presents a deflection coefficient based opti-mization framework to derive the optimal beacon sequence
, which can further improve the primary user detection per-formance for scenarios with known/unknown CSI uncertaintystatistics.
A. Known Uncertainty CovarianceLet the stacked vector be defined as
for correspondingto the receive antennas. The equivalent system modelconsidering this stacked observation vector can be derived as,
(35)
where , the identity ma-trix has dimensions and denotes the matrixKronecker product. The vector is thecolumn vector obtained by stacking the columns of the un-certainty matrix and has the covariance matrix
.Similarly, the concatenated noise vector is obtained as
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with the noise covariance matrix. The result below
derives the optimal beacon matrix for this scenario.Theorem 2: The optimal beacon matrix for primary user
detection towards spectrum sensing, for a known uncertaintycovariance , can be obtained as a solution of the optimizationproblem,
(36)
where , the quantityis an appropriate non-negative constant and denotes the
total transmit beacon power.Proof: To obtain the optimal beacon sequence, we em-
ploy the deflection coefficient for the binary hypothesistesting based primary user detection problem [6], defined as,
(37)
where denote the expected values of theobservation vector under the null hypothesis , the alterna-tive hypothesis respectively and denotes thecovariance of under the alternative hypothesis . How-ever, the direct optimization of the deflection coefficient aboveis intractable since it is non-convex. Therefore, we considera simplified convex problem of maximizing the weighted dif-ference of the distance between the hypothesis means and thetrace of the covariance. This yields a tractable problem whichcan be solved to yield the closed form expressions for the op-timal beacon matrices as shown below. Hence, the deflectioncoefficient based optimization framework to obtain the optimalbeacon matrix for a scenario with known uncertainty co-variance, towards maximization of the primary user detectionperformance, can be formulated as,
(38)
It can be observed that the optimization framework above isa bi-criterion optimization problem [32] where the choice ofallows for a tradeoff between the uncertainty variance and
the separation between the vectors corresponding to the twohypotheses. The optimization framework in (38) above can beequivalently reduced to,
where . The solution to theabove optimization problem is given by the beacon vectors
defined as where
denotes the unit-norm principal eigenvector of thematrix .
B. Unknown Uncertainty CovarianceLet be defined as whereis theMLE of corresponding to the alternative hypothesisderived in (16). Hence, the equivalent system model for the
spectrum sensing scenario with the concatenated vectorfor the receive antennas can
be equivalently written as,
where the matrix and the concatenated noise vector are asdefined in (35).Lemma 4: The optimal beacon matrix for a CHRD based
robust spectrum sensing scenario with unknown CSI statisticscan be obtained as the solution of the optimization problem,
(39)
where .Proof: Similar to the procedure in (37), the deflection coef-
ficient for the composite hypothesis based primaryuser detection problemwith an unknown uncertainty covariancematrix, can be determined as,
(40)
where is defined as. The optimization problem to obtain the
optimal beacon matrix that maximizes the performance ofthe proposed detection scheme can be readily derived as,
where . The above optimization problemis a quadratic constrained quadratic program (QCQP) [32]which can be solved by aligning each beacon vector as
along the principalunit-norm eigenvector of the matrix correspondingto the largest eigenvalue. This yields the optimal beacon matrix
for the MIMO cognitive radio spectrum sensing scenariowith unknown CSI statistics.
VI. SIMULATION RESULTSThis section presents simulation results to illustrate the per-
formance of the spectrum sensing schemes proposed above forMIMO cognitive radio scenarios. We consider a system wherethe secondary user has receive antennas and the primaryuser base-station has transmit antennas. In Figs. 1–7,we consider a non-antipodal signaling system with the beaconmatrix corresponding to the null hypothesis setas and the beacon matrix corresponding
PATEL et al.: OPTIMAL GLRT-BASED ROBUST SPECTRUM SENSING 1629
Fig. 1. Receiver operating characteristic (ROC) curves for the genie aided matched filter detector (MF genie), robust generalized likelihood detector (RGLD), ro-bust estimator-correlator detector (RECD), composite hypothesis based robust detector (CHRD), energy detector (ED) in (43) and nominal estimate based matchedfilter detector (MF) in (42) with SNR dB and for (a) , (b) . (a) Case I, (b) Case II.
Fig. 2. ROC curves for the genie aided matched filter detector (MF genie), robust generalized likelihood detector (RGLD), robust estimator-correlator detector(RECD), composite hypothesis based robust detector (CHRD), energy detector (ED) in (43) and nominal estimate based matched filter detector (MF) in (42) with
for (a) dB, (b) dB. (a) Case I, (b) Case II.
to the alternative hypothesis set as an orthogonal beacon matrix,i.e., satisfying . In our simulations we considerdifferent levels of CSI uncertainty with in theCSI uncertainty covariance matrix . Wepresent the probability of detection versus the probabilityof false alarm for the proposed RECD, RGLD and CHRDrobust MIMO spectrum sensing schemes towards primary userdetection. The performance of the uncertainty agnostic matchedfilter detector corresponding to the nominal CSI estimate is alsogiven in the figures, similar to the comparison between the ro-bust and the non-robust techniques in [30], [33]. The proposedschemes are also comparedwith the energy detector (ED). Addi-tionally, in our simulations we present comparisons of the pro-posed robust detection schemes with the genie aided matchedfilter detector (MF genie) with perfect knowledge of the CSI un-certainty, i.e., with knowledge of the true MIMO channel matrix. This serves as an upper bound for the proposed detection
techniques illustrating the best detection performance achiev-able for the corresponding scenario. We now describe the genieaided matched filter and the nominal CSI based matched filterdetector employed to benchmark the performance of the pro-posed schemes.
A. Genie-Aided Matched Filter Detector (MF Genie)The optimal detector with perfect CSI for an additive white
Gaussian noise scenario is given by the standard matched filterdetector [6]. This can be derived employing the likelihood ratiotest as,
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Fig. 3. ROC curves for the genie aided matched filter detector (MF genie), robust generalized likelihood detector (RGLD), robust estimator-correlator detector(RECD), composite hypothesis based robust detector (CHRD), nominal estimate based matched filter detector (MF) in (42) and energy detector (ED) in (43) for
dB, and (a) , (b) . (a) Case I, (b) Case II.
Fig. 4. Receiver operating characteristic (ROC) curves in logarithmic scale for the robust generalized likelihood detector (RGLD), robust estimator-correlatordetector (RECD), composite hypothesis based robust detector (CHRD), energy detector (ED) in (43) and nominal estimate based matched filter detector (MF) in(42) with dB, , (a) and (b) . (a) Case I, (b) Case II.
From the likelihood ratio above, the Neyman-Pearson (NP) based matched filter detector and the test statistic
are given as,
(41)
B. Matched Filter Detector (MF)The joint log-likelihood ratio test for the matched filter de-
tector ignoring CSI uncertainty can be derived as,
Therefore, the test statistic for the uncertainty agnosticmatched filter detector can be equivalently obtained as,
(42)
C. Energy Detector (ED)The test statistic for the energy detector [5]–[7] for the
MIMO cognitive radio system model described in (1), is givenas,
(43)
Consider the test statistic obtained for the CHRD in (18). Let thebeacon matrix corresponding to the alternativehypothesis be of dimension 2 2, i.e., the number of transmit
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Fig. 5. Receiver operating characteristic (ROC) curves in logarithmic scalefor the robust generalized likelihood detector (RGLD), robust estimator-cor-relator detector (RECD), composite hypothesis based robust detector (CHRD)and nominal estimate based matched filter detector (MF) in (42) with
dB for 2 2 MIMO with and for 4 4MIMO with .
Fig. 6. ROC curves for the simulation based performance of the robust esti-mator-correlator detector (RECD simulation) and the corresponding plots fromanalytical expressions (RECD theory) for dB, , and
.
antennas and the number of beacon vectors .This makes the beacon matrix an orthogonal square matrix with
and the resultant MLE of thevector corresponding to the alternative hypothesis re-duces to . Hence the test statisticfor the CHRD in (18) can be equivalently written as,
(44)
which is identical to that of the conventional energy detectorgiven in (43).In Figs. 1(a),(b), we compare the primary user detection
performance of the proposed robust estimator-correlator de-
Fig. 7. ROC curves for the simulation based performance of the composite hy-pothesis based robust detector (CHRD simulation) and the corresponding plotsfrom analytical expressions (CHRD theory) for dB, , and
.
tector (RECD) in (6), the robust generalized likelihood detector(RGLD) in (14), the composite hypothesis based robust de-tector (CHRD) in (18), with the energy detector (ED) in (43),the nominal channel estimate based matched filter detector(MF) in (42) and the true channel coefficient based genie aidedmatched filter detector (MF genie) in (41) for .The proposed robust detection schemes can be seen to leadto an improved detection performance in comparison to theuncertainty agnostic matched filter detector. Further, RGLDdemonstrates a performance edge over the other schemes.Across the figures, the primary user detection performance ofthe proposed schemes improves with an increase in the numberof beacon vectors , thereby decreasing the performance gapwith respect to the genie aided matched filter detector. Simula-tion results also demonstrate that the detection performance ofthe CHRD and the ED is identical for . This is dueto the fact that ED is a special case of the proposed CHRD for
as shown in Section VI-C.Figs. 2(a), (b) present a performance comparison of the
competing spectrum sensing schemes for various SNR valuesin the set dB and beacon symbols. Similarly,Figs. 3(a), (b) present the versus performance withdifferent levels of CSI uncertainty considering the uncertaintycovariance matrices with .The simulation results demonstrate a similar trend in the detec-tion performance of the proposed robust detection schemes. Itcan also be observed across the figures that the performancegap between the proposed robust schemes and the uncer-tainty agnostic matched filter detector along with the energydetector (ED) widens with increasing CSI uncertainty. Fur-ther, Fig. 4(a) and (b) present a versus performancecomparison of the proposed robust detection schemes withthe uncertainty agnostic matched filter (MF) detector and theenergy detector (ED) on a logarithmic scale for improvedresolution and show a trend similar to Fig. 1.Fig. 5 presents a performance comparison of the proposed
detection schemes for various values of the number of receiveand transmit antennas. We consider 2 2 MIMO scenario withCSI uncertainty covariance matrix and 44 MIMO scenario with CSI uncertainty covariance matrix
1632 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 64, NO. 6, MARCH 15, 2016
Fig. 8. ROC curves for the optimal beacon matrix versus the orthogonal beacon matrix for the composite hypothesis based robust detector (CHRD) and matchedfilter (MF) in (42) with dB, and for (a) , (b) . (a) Case I, (b) Case II.
. The detection performance of theproposed robust detection schemes significantly improves withthe increase in the number of receive and transmit antennas.In Fig. 6 we plot the probability of detection versus
the probability of false alarm and compare the simulateddetection performance with the analytical performance curvesgenerated from the derived expressions for in (8) andin (9) for the RECD with dB and
. It is evident from the figure that the detection per-formance of the proposed RECD technique obtained throughsimulations is in close agreement with the results obtained viatheory. Similarly, in Fig. 7 we compare the detection perfor-mance from the expressions for in (22) and in (21)corresponding to the CHRD with the simulation results for ascenario with dB and . The sim-ulated detection performance of the CHRD scheme coincideswith the analytical results.Fig. 8(a), (b) present the versus performance con-
sidering the optimal beacon matrix derived in Section V for thescenarios with a known uncertainty covariance and unknownuncertainty covariance. The detection performance for thesub-optimal orthogonal beacon sequence is also given therein.Fig. 8 clearly demonstrate a performance improvement for thederived optimal beacon sequence based spectrum sensing incomparison to the orthogonal beacon sequence.
VII. CONCLUSIONThis paper considers the problem of primary user detection
for MIMO cognitive radio scenarios with CSI uncertainty.In this context, novel detection schemes such as the robustestimator-correlator detector (RECD) and the robust gener-alized likelihood detector (RGLD), which are robust againstCSI uncertainty, have been proposed for scenarios with knownuncertainty statistics. Further, for the scenario with unknownCSI uncertainty statistics, we developed a GLRT based com-posite hypothesis robust detector (CHRD) for spectrum sensingin MIMO cognitive radio networks. Closed form analyticalexpressions have been derived to characterize the theoreticaldetection performance of the proposed RECD and CHRD
schemes. Subsequently, an optimization framework has alsobeen presented to obtain the optimal beacon sequences whichfurther enhance the performance of the proposed detectors.Simulation results were presented to illustrate the improved de-tection performance of the proposed robust detection schemeswhich consider CSI uncertainty in MIMO cognitive radio net-works. It has also been shown that the optimal beacon matrixsignificantly boosts the detection performance towards MIMOspectrum sensing. The proposed framework can be furtherextended considering other additional challenging aspects suchas noise uncertainty in future works.
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Adarsh Patel (S’12) received the B.Tech. degreein electronics and communication engineering fromIntegral University, Lucknow, India, in 2008 and theM.Tech. degree in computer science and engineering(digital communication) from ABV-Indian Insti-tute of Information Technology and management,Gwalior, in 2010.He is currently working toward the Ph.D. degree
in electrical engineering at the Indian Institute ofTechnology, Kanpur. His main research interestsinclude detection and estimation theory within the
areas of wireless communication and specially spectrum sensing aspect ofcognitive radio.Mr. Patel is a recipient of the TCS Research Fellowship.
Sinchan Biswas (S’14) was born in India in 1988.He received the M.Tech. degree from Departmentof Electrical Engineering, Indian Institute of Tech-nology (IIT), Kanpur, in 2013.He is currently pursuing the Ph.D. degree with the
Department of Engineering Sciences, Uppsala Uni-versity, Sweden. From 2013 to 2014, he was a Re-search Assistant with the IIT Kanpur. His research in-terest includes signal processing, information theory,and stochastic control.
Aditya K. Jagannatham (M’09) received the Bach-elors degree from the Indian Institute of Technology(IIT), Bombay, and the M.S. and Ph.D. degrees fromthe University of California, San Diego, USA.From April 2007 to May 2009, he was a Senior
Wireless Systems Engineer at Qualcomm, Inc.,San Diego, where he worked on developing 3GUMTS/WCDMA/HSDPA mobile chipsets as partof the Qualcomm CDMA technologies division. Hisresearch interests are in the area of next-generationwireless communications and networking, sensor
and ad-hoc networks, digital video processing for wireless systems, wireless3G/4G cellular standards and CDMA/OFDM/MIMO wireless technologies.He has contributed to the 802.11n high throughput wireless LAN standard.Since 2009, he has been with Electrical Engineering Department, IIT Kanpur,where he is currently an Associate Professor, and is also associated with theBSNL-IITK Telecom Center of Excellence (BITCOE).Dr. Jagannatham was awarded the CAL(IT)2 fellowship for pursuing grad-
uate studies at the University of California San Diego and, in 2009, received theUpendra Patel Achievement Award for his efforts towards developing HSDPA/HSUPA/HSPA+ WCDMA technologies at Qualcomm. At IIT Kanpur, he hasbeen awarded the P.K. Kelkar Young Faculty Research Fellowship (June 2012to May 2015) for excellence in research and the Gopal Das Bhandari MemorialDistinguished Teacher Award for the year 2012–2013 for excellence in teaching.He has also delivered a set of video lectures on Advanced 3G and 4G Wire-less Mobile Communications for the Ministry of Human Resource Develop-ment funded initiative National Programme on Technology Enhanced Learning(NPTEL).
