Optimal GLRT-based Robust Spectrum Sensing for ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2015.2500183, IEEE Transactions on Signal Processing

Optimal GLRT-based Robust Spectrum Sensing for MIMO Cognitive Radio Networks with CSI Uncertainty Adarsh Patel, Student Member, IEEE, Sinchan Biswas, Student Member, IEEE, and Aditya K. Jagannatham, Member, IEEE

Abstract—In this work, we develop generalized likelihood ratio test (GLRT)-based detectors for robust spectrum sensing in multiple-input multiple-output (MIMO) cognitive radio networks considering uncertainty in the available channel state information (CSI). Initially, for a scenario with known CSI uncertainty statistics, 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 estimates of the channel coefficients. Subsequently, for a scenario with unknown CSI uncertainty statistics, we develop a generalized likelihood ratio test (GLRT) based composite hypothesis robust detector (CHRD) for spectrum sensing. Closed form expressions are presented for the probability of detection (PD ) and the probability of false alarm (PF A ) to characterize the detection performance of the proposed robust spectrum sensing schemes. Further, a deflection coefficient based optimization framework is also developed and solved to derive closed form expressions for the optimal beacon sequences. Simulation results are presented to demonstrate the performance improvement achieved by the proposed robust spectrum sensing schemes and to verify the analytical results derived. Index Terms—Cognitive Radio, Spectrum Sensing, MultipleInput Multiple-Output (MIMO), Channel State Information (CSI) uncertainty.

I. I NTRODUCTION

T

HE proliferation of wireless multimedia applications has lead to a natural increase in the demand for a higher data rate in current and upcoming 3G/ 4G wireless communication systems. The radio frequency (RF) spectrum is a valuable resource for wireless communication applications and its usage is regulated by agencies such as the Federal Communications Commission (FCC) of the United States etc. However, recent reports such as [1] on the temporal and geographic RF spectrum utilization patterns point to an extremely low efficiency of current spectrum utilization. Thus, to cope with the tremendous increase in the demand for RF spectrum, coupled with the motivation for efficient spectrum utilization, the FCC has recently proposed the cognitive radio paradigm [2]–[4] which allows a set of unlicensed/ secondary users to opportunistically access unused spectrum bands licensed to Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. A. Patel, S. Biswas and A. K. Jagannatham are with the Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016. Email: {adarsh, sinchan, adityaj}@iitk.ac.in. This work is supported in part by the TCS Research Scholarship Program, DST-SERB and DST-IUATC Project Grants.

primary users. This in turn leads to an increase in the efficiency of spectrum utilization. This strategic reuse of the licensed spectrum by the unlicensed secondary users in cognitive radio networks necessitates the need to reliably detect the presence of vacant spectral bands, termed as spectral holes or white spaces, without causing a significant interference to the licensed primary users. This key task of spectral occupancy detection in cognitive radio networks is termed as spectrum sensing. Several spectrum sensing techniques have been proposed in literature [5] for the detection of primary user signals in cognitive radio networks. Among these, the energy detector [6] has a simple structure and does not require any prior knowledge of the primary user signal. However, the noncoherent energy detector has a poor performance compared to other techniques for spectrum sensing [7]–[9]. Other popular spectrum sensing techniques exploit statistical properties of the primary user signal for the detection of spectral holes. These techniques can be extended to wideband spectrum sensing, with Nyquist and sub-Nyquist sampling, as described in [10], [11] and to a distributed wideband sensing scenario with finite feedback in [12]. For instance, the cyclostationarity based detection scheme [5] requires the existence and knowledge of the cyclic frequency of the primary user signal. Similarly, the matched-filter based detection scheme [5] requires perfect channel state information (CSI) of the primary user, in the absence of which its performance degrades drastically [13]. Further, the work in [14] demonstrates that soft combination based schemes, such as equal gain combining (EGC) and maximal ratio combining (MRC), exhibit a significant improvement in the detection performance over conventional hard decision fusion schemes. However, it is shown in [15], that the improvement in the detection performance obtained from such soft-decision based spectrum sensing schemes depends significantly on the accuracy of the CSI at the secondary user. An adaptive spectrum sensing scheme based on a finite state Markov channel (FSMC) model is presented in [16]. However, the FSMC with finite states for the channel coefficient does 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 the signal of the licensed user. One major advantage of the GLRT based approach is that it achieves joint primary user detection and unknown parameter estimation. In literature, several spectrum sensing algorithms have been proposed for cogni-

1053-587X (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


tive radio scenarios, which compute the maximum likelihood estimates (MLE) of the unknown system parameters towards employing GLRT based detection schemes. For example, [17]– [20] compute the MLE of the noise variance and the signal correlation matrix, whereas [21]–[23] compute the MLE only for the noise variance in order to employ the GLRT framework for primary user signal detection. However, it is shown [7], [8] that the noise power uncertainty agnostic detectors, such as the energy detector, are limited by a signal-to-noise ratio (SNR) wall below which the detector fails to detect the presence of the 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 the detector can not be guaranteed at a given SNR, regardless of the number of samples of the primary signal acquired by the secondary user. Further, [17], [18], [25] employ the GLRT framework with the unknown parameter being the CSI between the primary transmitter and the secondary receiver. The various contributions of these works are listed below. •

•

•

The work in [17] presents a GLRT based framework for spectrum sensing with unknown channel gains and considering a single antenna at the primary user and multiple antennas at the secondary user The work in [18] presents various partial GLRT schemes with each of the channel gain, noise variance and signal variance components unknown considering block as well as fast fading channel scenarios The work in [25] presents a framework for GLRT based spectrum sensing with Bernoulli nonuniform sampling (BNS) considering unknown signal power.

However, a significant shortcoming of all these works above is that they do not consider CSI uncertainty, which is of significance in communication scenarios where frequently nominal CSI is available at the receiver. The work in [26] considers the Wilks’ detector based spectrum sensing for a scenario with unknown noise covariance. However, the authors therein consider a simplistic scenario with a deterministic channel matrix and analyze the instantaneous detection performance without averaging over the channel statistics. Obtaining the true channel coefficients in a cognitive radio network is challenging, owing to estimation/ quantization errors, limited feedback, Doppler shift etc, in practical wireless scenarios. These uncertainties in the system parameters can potentially lead to a performance degradation of uncertainty agnostic spectrum sensing schemes. Therefore, this paper considers a GLRT based approach for spectrum sensing, unlike conventional techniques such as matched filtering, energy detection and soft combining etc described in [5], [6], [14] respectively. Works such as [10]–[12], exploit only second order statistical information. Further, the motivation of this work is to develop detection schemes which are robust against the performance degradation caused by the uncertainty in the available CSI estimates, unlike the conventional GLRT schemes [17]–[23], [25] which consider completely unknown CSI, noise variance or other parameters. We propose novel primary user detection schemes, namely the robust estimatorcorrelator detector (RECD) and the generalized likelihood

ratio test (GLRT) based robust generalized likelihood detector (RGLD), which are robust against the uncertainty in the nominal estimates of the channel coefficients for a scenario with known uncertainty statistics. An analytical framework is developed to characterize the theoretical performance of the proposed RECD scheme and derive the expressions for the probability of detection (PD ) and probability of false alarm (PF A ). Also, the work in this paper considers a general scenario with multiple antennas, while [10]–[12], [14]– [16], [23] are restricted to single antenna scenarios. Next we present the composite hypothesis based robust detector (CHRD), for primary user detection in scenarios with unknown CSI uncertainty statistics. Further, we also derive closed form expressions to determine the probability of detection (PD ) and probability of false alarm (PF A ) performance of the proposed CHRD scheme. Also, the analysis for the PF A , PD in our work is carried out by averaging with respect to the distributions of both the CSI uncertainty and also the available nominal channel estimate, unlike the work in [26]. The subsequent section presents a novel deflection coefficient based optimization framework to derive the optimal beacon matrices for the proposed robust detection schemes, with known/ unknown uncertainty covariance statistics, which further enhance the detection performance. Simulation results demonstrate the improvement in the detection performance achieved by the proposed spectrum sensing schemes and also validate the analytical results. This paper is organized as follows. Section II describes the multiuser MIMO system model for cognitive radio networks. Section III presents the robust estimator-correlator detector (RECD) and the robust generalized likelihood detector (RGLD) with known CSI uncertainty statistics and the associated detection and false alarm probabilities. Next, in section IV we present the composite hypothesis based robust detector (CHRD) with unknown CSI uncertainty statistics. Closed form expressions for the PD and PF A to analytically characterize the detection performance of the proposed scheme are also derived therein. Section V describes the optimization framework to derive the optimal beacon sequence with both known and unknown CSI uncertainty statistics. Simulation results are given in section VI followed by the conclusion in section VII. Throughout this paper we use boldface uppercase/ lowercase letters to denote matrices/ vectors, respectively. All the vectors are column vectors. The operations (·)∗ , (·)T , (·)H and E{·} denote the conjugate, transpose, conjugate transpose and expectation operators. The random vector x, when defined as x ∼ CN (µ, R) follows a complex Gaussian distribution with mean µ and covariance R. Similarly, x ∼ χ2n implies that x follows a central chi-squared distribution with n degrees of freedom. The L2 norm of a vector and the trace of a matrix are represented by k · k and Tr(·) respectively. The matrix D(a) denotes the diagonal matrix with elements of the vector a along its principal diagonal. The identity matrix of dimension N × N is denoted by IN and the various detector test statistics are denoted by T (·). Finally, the functions Γ(·), Γ(·, ·) and Q(·) denote the Gamma function, the incomplete Gamma function and the Gaussian Q-function respectively.



II. S YSTEM M ODEL Consider a spectrum sensing scenario with a primary user base-station and a secondary user. Further, we assume a multiple-input multiple-output (MIMO) cognitive radio network with Nt transmit antennas at the primary user basestation and Nr receive antennas at the secondary user. The baseband system model for the scenario described above at the nth time instant is given as, y(n) = Hx(n) + η(n),

III. ROBUST S PECTRUM S ENSING WITH K NOWN U NCERTAINTY S TATISTICS Let the beacon matrix Xi = [xi (1), xi (2), . . . , xi (L)]H ∈ L×Nt C where xi (n) ∈ CNt ×1 , 1 ≤ n ≤ L denote the beacon signals from the primary user base-station and i = 0, 1 correspond to the absence, presence of the primary users respectively. From (3) it follows that the signal yk at the kth receive antenna is given as, ˆ k + uk ) + η H0 : yk = X0 (h k

ˆ k + uk ) + η , H1 : yk = X1 (h k

Nr ×1

where y(n) ∈ C is the receive signal vector at the secondary user corresponding to the primary user base-station broadcast beacon signal x(n) ∈ CNr ×1 and the vector η(n) ∈ CNr ×1 is the additive spatio-temporally white Gaussian noise at the secondary user with covariance R = E{η(n)η H (n)} = σ 2 INr . Each H ∈ CNr ×Nt , is the MIMO channel matrix between the primary user base-station and the secondary user. The system model considered follows in the same spirit as those in works such as [17], [18], [22], [25]. From the Cognitive Radio system model described above, the signal yk∗ (n) for the kth receive antenna can be equivalently expressed as, yk∗ (n) = xH (n)hk + ηk∗ (n), where (·)∗ denotes the complex conjugate and the vectors 1×Nt , 1 ≤ k ≤ Nr constitute the rows of the hH k ∈ C channel matrix H = [h1 , h2 , . . . , hNr ]H . Let the L broadcast beacon vectors x(1), x(2), . . . , x(L) be concatenated to form a beacon matrix X = [x(1), x(2), . . . , x(L)]H ∈ CL×Nt . The concatenated signal yk corresponding to the L symbols can be equivalently represented as, yk = Xhk + η k ,

(1)

where yk = [yk (1), yk (2), . . . , yk (L)]H ∈ CL×1 is the signal at the kth receive antenna corresponding to the broadcast beacon matrix X. Similarly, the concatenated noise vector η k = [ηk (1), ηk (2), . . . , ηk (L)]H ∈ CL×1 has covariance 2 matrix Rη = E{ηk η H k } = ση IL . In practical wireless scenarios it is significantly challenging to obtain accurate CSI as described previously. Hence, we model the channel matrix H by incorporating the CSI uncertainty as, ˆ + U, H=H (2) H ˆ = h ˆ1, h ˆ 2, . . . , h ˆN where H is the available nomir nal CSI and the matrix U ∈ CNr ×Nt denotes uncertainty in H. The uncertainty matrix the channel matrix H U = u1 , u2 , . . . , uNr , where each row vector uH k ∈ C1×Nt , 1 ≤ k ≤ Nr follows a complex Gaussian distribution i.e., uk ∼ CN (0, Ru ) with the uncertainty covariance matrix Nt ×Nt Ru = E{uk uH . The concatenated system model k } ∈ C in (1), incorporating the uncertainty model described in (2), can be equivalently obtained as,

(4)

where the hypotheses H0 and H1 correspond to the absence and presence of the primary user respectively for the binary hypothesis testing problem towards primary user detection. For example, in a practical wireless scenario, the beacon matrix X0 = 0L×Nt denotes the absence of primary transmission i.e. when the spectral band is vacant. This non-antipodal signaling model is well suited to the context of a cognitive radio scenario where the non-zero beacon matrix X1 denotes the presence of the primary user signal while the all zero beacon matrix X0 denotes the absence of the primary user signal. We now present the RECD scheme for robust spectrum sensing in MIMO cognitive radio scenarios. A. Robust Estimator-Correlator Detection (RECD) The observation vectors yk in (4), corresponding to the hypotheses H0 , H1 are distributed as, H0 :

H1 :

yk ∼ CN (0, Rη ) ˆ k , Γ), yk ∼ CN (X1 h

L×L and Ru ∈ CNt ×Nt is where Γ = X1 Ru XH 1 + Rη ∈ C the uncertainty covariance matrix. Let the concatenated observation matrix Y corresponding to the Nr receive antennas be defined as Y = [y1 , y2 , . . . , yNr ] ∈ CL×Nr . The likelihood ratio L(Y) corresponding to the concatenated observation matrix Y can be expressed as,

L(Y) = log

Q Nr

k=1 p(yk ; H1 ) Q Nr k=1 p(yk ; H0 )

!

,

Nr X ˆk) + c ˆ H −1 (yk − X1 h = ykH R−1 η yk − (yk − X1 hk ) Γ

. =

k=1 Nr X

k=1

ˆk, ˆ k yk + 2yH Γ−1 X1 h ˆ H H −1 X1 h ykH R−1 k η − hk X1 Γ

(3)

(5) . |R | and the operator = where the constant c = Nr log |Γ|η represents an equivalence to a constant factor. Using the matrix inversion identity from [27] in (5), the likelihood ratio L(Y) can be simplified to obtain the robust estimator-correlator detector (RECD) test statistic,

In the next section we describe robust spectrum sensing schemes, which consider CSI uncertainty, for primary user detection in multiuser MIMO cognitive radio networks, with known uncertainty covariance statistics.

TRECD (Y) Nr X ˆ k + yH R−1 X1 Ru XH Γ−1 yk . (6) = 2ykH Γ−1 X1 h k η 1

ˆ k + uk ) + η . yk = X(h k

k=1



Therefore the optimal Neyman-Pearson criterion based LRT, which maximizes the probability of detection PD for a given probability of false alarm PF A can be obtained as, H1

TRECD(Y) ≷ γ,

(7)

H0

with two degrees of freedom, described as, 1 H0 : 2 TRECD(ykl ) ∼ χ22 ση 2 1 H1 : 2 TRECD(ykl ) ∼ χ′2 (θl ), σγ 2

where the presence/ absence of the primary user signal is determined depending on whether the test statistic TRECD (Y) exceeds or falls short of the detection threshold γ. The proposed RECD exploits the statistical properties of the CSI uncertainty, thereby leading to an improvement in the accuracy of primary user detection. Consider the specific case when the CSI uncertainty covariance matrix Ru is given as Ru = σu2 INt and the beacon matrix X1 corresponding to the alternative hypothesis is an orthogonal matrix with X1 XH 1 = LI. Under these assumptions, the matrix Γ in (5) is seen to be given as, Γ = σγ2 I with σγ2 = Lσu2 + ση2 . Similarly, let the matrix H 2 2 2 −2 V = R−1 η X1 Ru X1 . Therefore V = σv I with σv = Lσu ση . The Lemma below characterizes the asymptotic PD versus PF A performance of the RECD scheme described above for spectrum sensing. Lemma 1. The probabilities of detection and false alarm, denoted by PD and PF A respectively, for the RECD detector in (7) towards robust spectrum sensing for a large number of receive antennas Nr are given as,   ′ γ − (θ + 2LN ) 2 r σγ  (8) PD = Q  p 2(θ + 2LNr )   ′ γ ση2 − 2LNr , (9) PF A = Q  √ 4LNr where θ =

PNr PL k=1

l=1

ˆ 2 σγ−2 (1 + σv−2 )2 |xH 1 (l)hk | .

Proof. It can be seen that the test statistic TRECD (Y) in (5) can be equivalently written as, TRECD (Y) Nr X ˆ k + yH R−1 X1 Ru XH Γ−1 yk = 2ykH Γ−1 X1 h k η 1 =

k=1 Nr X

ˆ k + yH (σ 2 σ −2 )Iyk 2ykH (σγ−2 I)X1 h k v γ

k=1

v

. =

k=1 l=1

γ

1

ˆ ykl + σv−2 xH 1 (l)hk

|

ˆ ykl + σv−2 xH 1 (l)hk , {z }

N

H0 :

∗

TRECD (ykl )

(10)

where ykl is the lth component of the vector yk defined as yk = [yk1 , . . . , ykl , . . . , ykL ]T . The component test statistic TRECD (ykl ) defined above follows a chi-squared distribution

L

r X X 1 1 T TRECD (ykl ) ∼ χ22LNr (Y) = RECD ση2 ση2

1 TRECD (Y) = σγ2

k=1 l=1 Nr X L X

2 1 TRECD (ykl ) ∼ χ′2LNr (θ), 2 σ k=1 l=1 γ (11) PNr PL where the non-centrality parameter θ = k=1 l=1 θl . The chi-squared distribution corresponding to the two hypotheses in (11) has 2LNr degrees of freedom. Using the asymptotic property of the chi-squared random variable for a large number of receive antennas Nr [6], the moments of the scaled test statistic TRECD (Y) corresponding to the two hypotheses in (11) can be equivalently obtained as, o n1 = 2LNr E 2 TRECD (Y); H0 ση o n1 = 4LNr var 2 TRECD (Y); H0 ση o n 1 = θ + 2LNr E 2 TRECD (Y); H1 σγ n 1 o var 2 TRECD (Y); H1 = 2(θ + 2LNr ). (12) σγ

H1 :

Using the results in (12), the analytical expression for the detection probability PD of the proposed RECD can be equivalently obtained as, PD

Nr X L σv2 X ˆk ˆ ∗ ykl + σ −2 xH (l)h ykl + σv−2 xH = 2 v 1 1 (l)hk σγ k=1 l=1 ˆ k 2 − σ −2 σ 2 xH (l)h Nr X L X

respectively. The quantity χ′2 (θl ) denotes the non-central chisquared distribution with two degrees of freedom, and nonˆ 2 centrality parameter θl = σγ−2 (1 + σv−2 )2 |xH 1 (l)hk | . The distributions of the scaled test statistic TRECD (Y) for the Nr received vectors stacked as Y = [y1 , y2 , . . . , yNr ] corresponding to the two hypotheses H0 , H1 can be equivalently derived as,

= P r {TRECD (Y) > γ ′ ; H1 } γ′ 1 TRECD (Y) > 2 ; H1 = Pr σ2 σ  ′γ γ γ σγ2 − (θ + 2LNr ) , = Q p 2(θ + 2LNr )

where γ ′ is the detection threshold and Q(·) denotes the standard Gaussian Q-function [6], [28]. Similarly, the probability of false alarm PF A can be obtained as, PF A = P r TRECD (Y) > γ ′ ; H0 γ′ 1 T (Y) > ; H = Pr RECD 0 σ2 ση2  ′η  γ ση2 − 2LNr  . √ = Q 4LNr



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 radio networks with CSI uncertainty. Let the vector rk be defined as ˆ k corresponding to the alternative hypothesis rk = yk − X1 h H1 . Thus, the signal described in (3) corresponding to the L concatenated sensed samples at the kth receive antenna of the total Nr receive antennas can be equivalently written as,

nitive radio scenarios can be derived as, ! QNr ˆk , H1 ) k=1 p(yk ; z TRGLD (Y) = log QNr p(yk ; H0 )  Q k=1  Nr 1 H −1 k=1 π L+Nt |Rz | exp(−zk Rz zk )  = log  QNr 1 H −1 k=1 π L+Nt |Rη | exp(−yk Rη yk ) Nr

. X H −1 = −ˆzk Rz zˆk + ykH R−1 η yk , k=1

rk

= X1 uk + η k , uk X1 IL , = | {z } ηk {z } | A

(13)

zk

T where zk = uTk , η Tk ∈ C(L+Nt )×1 denotes the concatenated unknown random vector and the identity matrix IL is of dimension L × L. Lemma 2. The GLRT-based test statistic TGLRT (Y) corresponding to the uncertainty covariance matrix Ru , is given as, TRGLD (Y) =

Nr X

k=1

−1 −ˆ zH zk + ykH R−1 k Rz ˆ η yk ,

(14)

where zk = [uTk , ηTk ]T and Rz ∈ CL+Nt ×L+Nt is the block diagonal matrix with Ru , Rη along its principal diagonal. Proof. The likelihood of the observation vector yk parameterized by zk corresponding to the alternative hypothesis H1 can be derived as, p(yk ; zk , H1 ) =

1 −1 exp(−zH k Rz zk ), π L+Nt |Rz |

. 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 detection in spectrum sensing scenarios provides a performance edge over the conventional uncertainty agnostic matched filter detector. In the next section we further relax the CSI uncertainty model and consider the uncertainty covariance statistics to be unknown. We then derive the corresponding composite hypothesis based robust detector (CHRD) for spectrum sensing in MIMO cognitive radio scenarios. IV. ROBUST D ETECTION WITH U NKNOWN U NCERTAINTY C OVARIANCE S TATISTICS A. Composite Hypothesis based Robust Detector (CHRD) In this section we compute a generalized likelihood ratio test based composite hypothesis testing framework with unknown uncertainty statistics. The primary user detection problem from (1) can be equivalently recast as the hypothesis testing problem, H0 : INt hk = 0 H1 : INt hk 6= 0,

(15)

where Rz ∈ CL+Nt ×L+Nt denotes the covariance matrix of zk , given as, Ru 0Nt ×L H Rz = E{zk zk } = , 0L×Nt Rη and |Rz | denotes the determinant of the matrix Rz . The parameterized vector zk is obtained on maximizing the likelihood p(yk ; zk , H1 ) in (15) which can be formulated as the standard weighted minimum L2 norm optimization problem [27], −1 min. zH k Rz zk s.t. rk = Azk .

The solution of the above convex optimization problem yields the estimate of zˆk as, ˆzk = Rz AH (ARz AH )−1 rk . Employing the GLRT framework, the RGLD test statistic TRGLD (Y) for the primary user detection problem in cog-

where INt is the Nt × Nt identity matrix and the vector parameters hk , 1 ≤ k ≤ Nr form the rows of the channel matrix ˆk|0 , h ˆ k|1 denote the maximum likelihood H. Let the vectors h estimate (MLE) of the vector parameter hk corresponding to the null hypothesis H0 , alternative hypothesis H1 respectively, which can be obtained [6] as, ˆ k|1 h ˆ k|0 h

= = =

−1 H (XH X1 yk (16) 1 X1 ) H −1 H H −1 H −1 ˆ ˆ hk|1 − (X1 X1 ) I [I(X1 X1 ) I ] (INt hk|1 )

0.

The concatenated received vector yk , defined in (1), follows a complex Gaussian distribution given as, H0 : yk ∼ CN (0, Rη ) ˆ k|1 , Rη ), H1 : yk ∼ CN (X1 h corresponding to the hypotheses H0 , H1 . Let the matrix Y be obtained by stacking the received signal vectors yk , as Y = [y1 , y2 , . . . yNr ]. Hence, the test statistic TCHRD (Y) for the composite hypothesis testing based primary user detection problem can be computed by applying the generalized



as,

likelihood ratio test (GLRT) as, TCHRD (Y) ! Q Nr ˆ k=1 p(yk ; hk|1 , H1 ) (17) = log Q Nr k=1 p(yk ; H0 )   QN r H −1 ˆ ˆ k=1 exp −(yk − X1 hk|1 ) Rη (yk − X1 hk|1 )  = log  QNr H −1 k=1 exp(−yk Rη yk ) =

. =

Nr X

k=1 Nr X

ˆ k|1 ˆ H XH R−1 X1 h h η k|1 1 ˆ ˆH h h k|1 k|1 ,

(18)

k=1

where the last equality holds due to the orthogonality property of the beacon matrix, i.e. XH 1 X1 = LINt . The test statistic TCHRD (Y) obtained above yields the primary user detection rule that is robust against the uncertainty in the estimate of the MIMO channel matrix. We now consider a general scenario with non-isotropic CSI uncertainty covariance, i.e. the uncertainty covariance is not necessarily of the form Ru = σu2 I. It can be noted that the non-isotropic CSI uncertainty covariance matrix considered in our formulation is more general and practical in comparison to the isotropic model considered in works such as [29], [30]. We now characterize the theoretical performance of the CHRD for composite hypothesis testing based primary user detection in MIMO cognitive radio scenarios. Using (3) and the orthogonality property of the beacon ˆ k|1 of the matrix in (16), the maximum likelihood estimate h column vectors hk , 1 ≤ k ≤ Nr of the channel matrix H, corresponding to the alternative hypothesis H1 , can be equivalently written as, ˆ k|1 h

−1 H = (XH X1 yk 1 X1 ) −1 H ˆ k + uk ) + XH η = (XH X1 X1 (h 1 X1 ) 1 k ˆ k + uk + nk , = h (19)

where as nk = XH 1 η k with covariance Rn = nHk is defined 2 ˆ k ∈ CNt ×1 follows E nk nk = Lση INt . The CSI estimate h ˆ a complex Gaussian distribution hk ∼ CN (0, INt ). Thus, this ˆ k +uk is implies that the true channel coefficient vector hk = h also complex Gaussian distributed, leading to a Rayleigh fading wireless channel, which is a standard assumption for fading wireless scenarios. It follows from (19) that the maximum ˆ k|1 has a complex Gaussian distribution, likelihood estimate h ˆ k|1 ∼ CN (0, Λ) where Λ = (1 + Lσ 2 )IN + Ru . with h t η Let the eigenvalue decomposition of Λ be VΣVH , with the 2 T eigenvalue matrix Σ = D([σ12 , σ22 , . . . , σN ] ), where D(a) t denotes a diagonal matrix with the elements of vector a along ˆ kl|1 of the principal diagonal. Therefore, the lth element h T T T T ˆ ˆ ˆ ˆ the vector hk|1 = [hk1|1 , . . . , hkl|1 , . . . , hkNt |1 ] ∈ CNt ×1 follows a Gaussian distribution. Let zkl be defined as zkl = 1 ˆ σl hkl|1 . The test statistic in (18) can be equivalently described

TCHRD (Y)

= =

Nr X

ˆ H (ΣΣ−1 )h ˆ k|1 h k|1

k=1 Nr X Nt X k=1 l=1

=

Nr X Nt X k=1 l=1

σl2

ˆ kl|1 |2 |h σl2

σl2 |zkl |2 ,

(20)

where each zkl is Gaussian distributed with zkl ∼ CN (0, 1). Therefore, |zkl |2 , 1 ≤ k ≤ Nr , 1 ≤ l ≤ Nt is distributed as a χ22 random variable. Employing the above results, we now characterize the performance of the composite hypothesis detector corresponding to the test statistic TCHRD (Y) in (20). Theorem 1. The probability of false alarm PF A and probability of detection PD of the CHRD detector based on the test statistic TCHRD (Y) in (20), for primary user detection in MIMO cognitive radio networks can be derived as, γ PF A = Qχ22N N , (21) r t ση2 Nr L X X (−2σl2 )k γ PD = κ Al,k (22) Γ k, 2 , (k − 1)! 2σl l=1 k=1

where the coefficients Al,k are the partial fraction constants obtained using the residue method, Γ(·, ·) is the incomplete Gamma function defined in (30) and the constant κ is defined QNt −1 Nr . as κ = m=1 2σ2 m

Proof. The probability of false alarm PF A of the GLRT test statistic TCHRD (Y) can be derived as, PF A

= P r {TCHRD (Y) > γ; H0 } 1 γ = Pr T ; H (Y) > CHRD 0 σ2 ση2 η γ = P r χ22Nr Nt > 2 ση γ = Qχ22N N . r t σ2 η

Next, we derive the probability of detection PD for the test statistic TCHRD (Y). This can be expressed as, Z ∞ PD = P r {TCHRD (Y) > γ; H1 } = pT (t)dt, (23) γ

where pT (t) denotes the probability density function of the test statistic TCHRD (Y). Let ΦT (ω) denote the characteristic function of the test statistic TCHRD (Y) corresponding to the alternative hypothesis H1 . The probability density function pT (t) can be expressed in terms of ΦT (ω) as, ( R∞ 1 ΦT (ω) exp−jωt dω t ≥ 0 (24) pT (t) = 2π −∞ 0 t < 0.



The characteristic function ΦT (ω) for the CHRD test statistic corresponding to the alternative hypothesis H1 is derived as, E {exp(jωTCHRD (Y))} !) ( Nt X Nr X 2 ⋆ E exp jω σl zkl zkl

ΦT (ω) = =

PD =

=

l=1 k=1 Nt Y

=

l=1

(25) =κ

E exp(jωσl2 |zkl |2 )

=κ

Z

, N (1 − 2jσl2 ω) r

(26)

Nt ∞ Y

1 exp(−jωt) dω 2 Nr (1 − 2jσ −∞ l=1 l ω) Y Nr Nt Nt Y −1 1 −1 = F −t 2 2σm (jω − 2σ1 2 )Nr m=1 l=1 | {z l } (N N t X r X

) Al,k =κ (jω − 2σ1 2 )k l=1 k=1 l ) ( Nt X Nr X 1 −1 =κ , Al,k F−t (jω − 2σ1 2 )k l=1 k=1

(27)

l

where the coefficients Al,k correspond to the partial fraction expansion of ψ(ω) and the constant κ is defined as κ = QNt −1 Nr . The coefficients Al,k can be obtained using 2 m=1 2σm the standard residue method for partial fraction expansion [31] and can be explicitly expressed as,    Nr  dNr −k  jω − 2σ1 2 l  Q Al,k = βk lim1 Nr  , N −k L 1 jω= 2  djω r jω − 2 2σ n=1 2σ l n (28) where βk = (Nr1−k)! . Computing the inverse Fourier transform of the expression in (27), the probability density function pT (t) of the test statistic TCHRD (Y) in (20), corresponding to the alternative hypothesis H1 , can be obtained as, t

pT (t) = κ

− 2 Nt X Nr X Al,k (−1)k tk−1 e 2σl u(t) l=1 k=1

(k − 1)!

κ

− 2 Nt X Nr X Al,k (−1)k tk−1 e 2σl u(t)

l=1 k=1 N N r t XX Al,k (−1)k

l=1 k=1 Nt X Nr X

(k − 1)!

Al,k

,

(k − 1)!

Z

∞

tk−1 e

−

t 2σ2 l

dt

u(t) dt

γ

γ (−2σl2 )k Γ k, 2 , (k − 1)! 2σl

where γ is the threshold for detection and Γ(., .) denotes the incomplete Gamma function [28], defined as, Z ∞ Γ(s, γ) = ts−1 e−t u(t)dt. (30) γ

Further, we now derive the probability of detection PD for the restrictive case of an isotropic CSI uncertainty covariance matrix, i.e Σ = σ 2 INt with σi2 = σ 2 , ∀1 ≤ i ≤ Nt . Lemma 3. For an isotropic covariance matrix i.e with Σ = σ 2 I, the probability of detection for the detector in (18) reduces to, γ PD = Qχ22N N . (31) r t σ2

ψ(ω)

−1 F−t

t

∞

l=1 k=1

1

where (25) follows from the simplification of the test statistic TCHRD (Y) described in (20). The characteristic function ΦT (ω) obtained in (26) follows from the fact that each |zkl |2 follows a chi-squared distribution with 2 degrees of freedom as described earlier. Employing (26) above, the probability density function pT (t) of the test statistic TCHRD (Y) defined in (24) can be equivalently expressed as, 1 pT (t) = 2π

Z

γ

l=1 k=1

Nt Y Nr Y

expression for pT (t) in (23) as,

(29)

where u(t) is the unit step function defined as, ( 1 t>0 u(t) = 0 t ≤ 0. Hence, the probability of detection PD for the composite hypothesis detector can be derived by substituting the above

Proof. The characteristic function ΦT (ω) in (26) and the corresponding probability density function pT (t) of the test statistic TCHRD (Y) in (20), for the covariance matrix Σ = σ 2 I, can be obtained as, Nr Nt 1 , ΦT (ω) = 1 − 2jσ 2 ω N r Nt X 1 −1 pT (t) = κ0 Ak F−t (jω − 2σ1 2 )k k=1 = κ0

N r Nt X k=1

t

Ak (−1)k tk−1 e− 2σ2 u(t) , (k − 1)!

(32)

respectively, where the constant κ0 is defined as κ0 = −1 Nr Nt . The partial fraction constants Ak in (32) can now 2σ2 be equivalently obtained as, ( !) dNr Nt −k 1 Ak = βk lim1 Nr Nt jω= 2σ2 djω Nr Nt −k jω − 2σ1 2 ( 1, k = Nr Nt (33) = 0, k 6= Nr Nt , where the constant βk is defined as βk = (Nr N1t −k)! . Using the expression for pT (t) from (32), the corresponding probability of detection PD for this scenario with a diagonal covariance matrix Ru can be derived as, PD = κ0

N r Nt X k=1

Ak

γ (−2σ)k Γ k, 2 . (k − 1)! 2σ



Substituting the expression for the coefficient Ak from (33), the above expression for PD can be simplified as, γ 1 Γ Nr Nt , PD = (Nr Nt − 1)! 2σ NrX Nt −1 γ 1 γ k = e− 2σ2 (34) k! 2σ 2 k=0 γ = Qχ22N N , r t σ2 where (34) follows from the following property of the Gamma function [28], Γ(q, s) = (q − 1)! e−s

q−1 m X s . m! m=0

In the next section we develop the framework to obtain the optimal beacon sequence which can further enhance the detection performance of the proposed spectrum sensing schemes. V. O PTIMAL B EACON F ORMULATION This section presents a deflection coefficient based optimization framework to derive the optimal beacon sequence X1 , which can further improve the primary user detection performance for scenarios with known/ unknown CSI uncertainty statistics. A. Known Uncertainty Covariance Ru Let the stacked vector ˜r ∈ CLNr ×1 be defined as ˜r = for 1 ≤ k ≤ Nr corresponding to the Nr receive antennas. The equivalent system model considering this stacked observation vector ˜r can be derived as, [rT1 , . . . rTk , . . . , rTNr ]T

˜, ˜r = (INr ⊗ X1 ) vec(UH ) + η {z } |

(35)

˜ X

˜ = (IN ⊗ X1 ) ∈ CLNr ×Nt Nr , the identity matrix where X r INr has dimensions Nr × Nr and ⊗ denotes the matrix Kronecker product. The vector vec(UH ) ∈ CNr Nt ×1 is the column vector obtained by stacking the columns of the uncertainty matrix UH and has the covariance matrix RU = E{vec(UH )(vec(UH ))H } = INr ⊗ Ru ∈ CNr Nt ×Nr Nt . ˜ is obtained as Similarly, the concatenated noise vector η ˜ = [ηT1 , η T2 , . . . , η TNr ]T with the noise covariance matrix η ˜ H } = INr ⊗ Rη ∈ CNr L×Nr L . The result below Rη˜ = E{˜ ηη derives the optimal beacon matrix X1 for this scenario. Theorem 2. The optimal beacon matrix X1 for primary user detection towards spectrum sensing, for a known uncertainty covariance Ru , can be obtained as a solution of the optimization problem, max. s.t.

Tr(X1 WXH 1 ) Tr(X1 XH 1 ) ≤ P0 ,

(36) PNr ˆ ˆ H where W = k=1 (hk hk ) − µNr Ru ∈ CNt ×Nt , the quantity µ is an appropriate non-negative constant and P0 denotes the total transmit beacon power.

Proof. To obtain the optimal beacon sequence, we employ the deflection coefficient d2 (X1 ) for the binary hypothesis testing based primary user detection problem [6], defined as, 2

d2 (X1 ) ,

=

=

kE{Y; H1 } − E{Y; H0 }k2 , Tr(cov{Y; H1 })

2

˜ ˆ H )

X vec(H

2 , ˜ H + Rη˜ ˜ UX Tr XR

PNr

ˆ 2 h

X k 1 k=1 1 2 , H Nr Tr(X1 Ru X1 + Rη )

(37)

where E{Y; H0 }, E{Y; H1 } denote the expected values of the observation vector Y under the null hypothesis H0 , the alternative hypothesis H1 respectively and cov{Y; H1 } denotes the covariance of Y under the alternative hypothesis H1 . However, the direct optimization of the deflection coefficient above is intractable since it is non-convex. Therefore, we consider a simplified convex problem of maximizing the weighted difference of the distance between the hypothesis means and the trace of the covariance. This yields a tractable problem which can be solved to yield the closed form expressions for the optimal beacon matrices as shown below. Hence, the deflection coefficient based optimization framework to obtain the optimal beacon matrix X1 for a scenario with known uncertainty covariance, towards maximization of the primary user detection performance, can be formulated as, max.

Nr

X

ˆ 2

X1 hk − µ(Nr ) Tr(X1 Ru XH 1 + Rη ) 2

k=1

s.t.

Tr(X1 XH 1 )

≤ P0 .

(38)

It can be observed that the optimization framework above is a bi-criterion optimization problem [32] where the choice of µ allows for a tradeoff between the uncertainty variance and the separation between the vectors corresponding to the two hypotheses. The optimization framework in (38) above can be equivalently reduced to, Tr(X1 WXH 1 ) H Tr(X1 X1 ) ≤ P0 ,

max. s.t.

PNr ˆ ˆ H where W = k=1 (hk hk ) − µNr Ru . The solution to the above optimization problem is given q by the beacon vectors x1 (i), 1 ≤ i ≤ L, defined as x1 (i) = PL0 νmax (W), where νmax (W) denotes the unit-norm principal eigenvector of the matrix W. B. Unknown Uncertainty Covariance Ru ˆ k|1 where h ˆ k|1 Let qk ∈ CL×1 be defined as qk = yk −X1 h is the MLE of h corresponding to the alternative hypothesis H1 derived in (16). Hence, the equivalent system model for the ˜ = spectrum sensing scenario with the concatenated vector q [qT1 , qT2 , . . . , qTNr ]T ∈ CLNr ×1 for the Nr receive antennas can be equivalently written as, ˜, ˜ = (INr ⊗ X1 ) vec(UH ) + η q {z } | ˜ X


1

1

0.9

0.9

0.8 0.7 MF(Geine) RGLD RECD CHRD MF(nominal) ED

0.6 0.5 0.4 0

0.1 0.2 0.3 0.4 0.5 Probability of False Alarm (PFA)

0.6

(a) Case I

Probability of Detection (PD)



0.8 0.7 MF(Geine) RGLD RECD CHRD MF(nominal) ED

0.6 0.5 0.4 0


0.6

(b) Case II

Fig. 1. Receiver operating characteristic (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 2 = 1 for (a) L = 2, (b) L = 4. matched filter detector (MF) in (42) with SNR = −3 dB and σu

˜ and the concatenated noise vector η ˜ are where the matrix X as defined in (35). Lemma 4. The optimal beacon matrix X1 for a CHRD based robust spectrum sensing scenario with unknown CSI statistics can be obtained as the solution of the optimization problem, max. s.t. where Z =

Tr(X1 ZXH 1 ) H Tr(X1 X1 ) ≤ P0 ,

VI. S IMULATION R ESULTS (39)

PNr ˆ ˆ H Nt ×Nt . k=1 (hk|1 hk|1 ) ∈ C

Proof. Similar to the procedure in (37), the deflection coefficient d2MLE (X1 ) for the composite hypothesis based primary user detection problem with an unknown uncertainty covariance matrix, can be determined as,

2

˜ ˆ H )

X vec(H MLE 2 d2MLE (X1 ) = , Tr(Rη˜)

PNr ˆ 2 k=1 X1 hk|1 2 , (40) = N Nr Tr(Rη )

ˆ MLE ∈ CNr ×Nt is defined as H ˆ MLE = where H ˆ 1|1 , . . . , h ˆ k|1 , . . . , h ˆ N |1 ]H . The optimization problem to [h r obtain the optimal beacon matrix X1 that maximizes the performance of the proposed detection scheme can be readily derived as, max. s.t.

Tr(X1 ZXH 1 ) H Tr(X1 X1 ) ≤ P0 ,

q x1 (i) = PL0 νmax (Z), 1 ≤ i ≤ L, along the principal unitnorm eigenvector νmax (Z) of the matrix Z corresponding to the largest eigenvalue. This yields the optimal beacon matrix X1 for the MIMO cognitive radio spectrum sensing scenario with unknown CSI statistics.

PNr ˆ ˆ H where Z = k=1 (hk|1 hk|1 ). The above optimization problem is a quadratic constrained quadratic program (QCQP) [32] which can be solved by aligning each beacon vector as

This section presents simulation results to illustrate the performance of the spectrum sensing schemes proposed above for MIMO cognitive radio scenarios. We consider a system where the secondary user has Nr = 2 receive antennas and the primary user base-station has Nt = 2 transmit antennas. In Figs. 1-7, we consider a non-antipodal signaling system with the beacon matrix X0 ∈ CL×2 corresponding to the null hypothesis set as X0 = 0 and the beacon matrix X1 ∈ C2×L corresponding to the alternative hypothesis set as an orthogonal beacon matrix, i.e. satisfying XH 1 X1 = LI2×2 . In our simulations we consider different levels of CSI uncertainty with σu2 ∈ {1, 0.8, 0.6} in the CSI uncertainty covariance matrix Ru = σu2 D([1, 0.9]T ). We present the probability of detection PD versus the probability of false alarm PF A for the proposed RECD, RGLD and CHRD robust MIMO spectrum sensing schemes towards primary user detection. The performance of the uncertainty agnostic matched filter detector corresponding to the nominal CSI estimate is also given in the figures, similar to the comparison between the robust and the non-robust techniques in [30], [33]. The proposed schemes are also compared with the energy detector (ED). Additionally, in our simulations we present comparisons of the proposed robust detection schemes with the genie aided matched filter detector (MF genie) with perfect knowledge of the CSI uncertainty, i.e. with knowledge of the true MIMO


1

1

0.9

0.9




0.8 0.7 MF(Genie) RGLD RECD CHRD ED MF(nominal)

0.6 0.5 0.4 0

0.1

0.2 0.3 0.4 0.5 Probability of False Alarm (PFA)

0.6

0.8 0.7 MF(Geine) RGLD RECD CHRD ED MF(nominal)

0.6 0.5 0.4 0

(a) Case I


0.6

(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) 2 = 1 for (a) SNR = −4 dB, (b) SNR = −2 dB. with L = 4, σu

channel matrix H. This serves as an upper bound for the proposed detection techniques illustrating the best detection performance achievable for the corresponding scenario. We now describe the genie aided matched filter and the nominal CSI based matched filter detector employed to benchmark the performance of the proposed schemes.

A. Genie-aided Matched Filter Detector (MF genie)

B. Matched Filter Detector (MF) The joint log-likelihood ratio test for the matched filter detector ignoring CSI uncertainty can be derived as, LMF (Y) QNr k=1 p(yk ; H1 ) = log QNr k=1 p(yk ; H0 ) N r . X H −1 ˆk) . ˆ k )H R−1 (yk − Xk h = yk Rη yk − (yk − Xh η k=1

The optimal detector with perfect CSI for an additive white Gaussian noise scenario is given by the standard matched filter detector [6]. This can be derived employing the likelihood ratio test as,

Therefore, the test statistic TMF (Y) for the uncertainty agnostic matched filter detector can be equivalently obtained as, TMF (Y) =

Nr X

ˆ ykH R−1 η Xhk .

(42)

k=1

Lgenie (Y) QNr k=1 p(yk ; H1 ) = log QNr k=1 p(yk ; H0 ) Nr X . H −1 H −1 = yk Rη yk − (yk − Xhk ) Rη (yk − Xhk ) . k=1

From the likelihood ratio Lgenie (Y) above, the NeymanPearson (NP) based matched filter detector and the test statistic TMF (genie) (Y) are given as,

TMF (genie) (Y) =

Nr X

k=1

H1

ykH R−1 η Xhk ≷ γ. H0

(41)

C. Energy Detector (ED) The test statistic TED for the energy detector [5]–[7] for the MIMO cognitive radio system model described in (1), is given as, Nr X TED (Y) = ykH yk . (43) k=1

Consider the test statistic obtained for the CHRD in (18). Let the beacon matrix X1 ∈ CL×Nt corresponding to the alternative hypothesis be of dimension 2 × 2, i.e. the number of transmit antennas Nt = 2 and the number of beacon vectors L = 2. This makes the beacon matrix an orthogonal H square matrix with XH 1 X1 = X1 X1 = 2I2 and the resultant ˆ MLE hk|1 of the vector hk corresponding to the alternative


1

1

0.9

0.9




0.8 0.7 MF(Geine) RGLD, RECD CHRD MF(nominal) ED

0.6 0.5 0.4 0


0.8 0.7 MF(Geine) RGLD, RECD CHRD MF(nominal) ED

0.6 0.5 0.4 0

0.6


(a) Case I

0.6

(b) Case II

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) 2 = 1, (b) σ 2 = 0.6. for SNR = −6 dB, L = 4 and (a) σu u

1

1

0.8 0.7

RGLD RECD CHRD ED MF(nominal)

0.8

0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0

−6

10

RGLD RECD CHRD ED MF(nominal)

0.9

Probability of detection (PD)

D

Probability of detection (P )

0.9

−4

−2

10 10 Probability of false alarm (PFA) (a) Case I

0

10

0

−8

10

−6

−4

−2

10 10 10 Probability of false alarm (PFA)

0

10

(b) Case II

Fig. 4. Receiver operating characteristic (ROC) curves in logarithmic scale for the 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) 2 = 1, (a) L = 2 and (b) L = 4. in (42) with SNR = −3 dB, σu


1

1

0.9

0.9

0.8

0.7

4x4 MIMO, RGLD 4x4 MIMO, RECD 4x4 MIMO, CHRD 4x4 MIMO, MF(nominal) 2x2 MIMO, RGLD 2x2 MIMO, RECD 2x2 MIMO, CHRD 2x2 MIMO, MF(nominal)

0.6

0.5

0.4 0

0.1

0.2 0.3 0.4 0.5 Probability of false alarm (PFA)

0.6

Fig. 5. Receiver operating characteristic (ROC) curves in logarithmic scale for the robust generalized likelihood detector (RGLD), robust estimator-correlator detector (RECD), composite hypothesis based robust detector (CHRD) and nominal estimate based matched filter detector (MF) in (42) with L = 2, SNR = −1 dB for 2 × 2 MIMO with Ru = D([1, 0.9]T ) and for 4 × 4 MIMO with Ru = D([1, 0.9, 0.6, 0.4]T ).


Probability of detection (PD)


0.8 0.7 0.6

MF(Geine) CHRD(Theory) CHRD(Simulation) MF(nominal)

0.5 0.4 0

0.1


Fig. 7. ROC curves for the simulation based performance of the composite hypothesis based robust detector (CHRD simulation) and the corresponding plots from analytical expressions (CHRD theory) for SNR = −2 dB, L = 2, and Ru = D(1, 0.9).

TCHRD (Y)


1

= . =

Nr X

k=1 Nr X

ˆ ˆH h h k|1 k|1 ykH yk ,

(44)

k=1

0.9 0.8 0.7 0.6

MF(Geine) RGLD(Theory) RGLD(Simulation) MF(nominal)

0.5 0.4 0

0.6

0.1


0.6

Fig. 6. ROC curves for the simulation based performance of the robust estimator-correlator detector (RECD simulation) and the corresponding plots from analytical expressions (RECD theory) for SNR = −3 dB, L = 2, and Ru = D(1, 1).

ˆ k|1 = 1 XH yk . Hence the test hypothesis H1 reduces to h 2 1 statistic TCHRD (Y) for the CHRD in (18) can be equivalently written as,

which is identical to that of the conventional energy detector given in (43). In Figs. 1a, 1b we compare the primary user detection performance of the proposed robust estimator-correlator detector (RECD) in (6), the robust generalized likelihood detector (RGLD) in (14), the composite hypothesis based robust detector (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 aided matched filter detector (MF genie) in (41) for L ∈ {2, 4}. The proposed robust detection schemes can be seen to lead to an improved detection performance in comparison to the uncertainty agnostic matched filter detector. Further, RGLD demonstrates a performance edge over the other schemes. Across the figures, the primary user detection performance of the proposed schemes improves with an increase in the number of beacon vectors L, thereby decreasing the performance gap with respect to the genie aided matched filter detector. Simulation results also demonstrate that the detection performance of the CHRD and the ED is identical for Nt = L = 2. This is due to the fact that ED is a special case of the proposed CHRD for Nt = L = 2 as shown in section VI-C. Figs. 2a, 2b present a performance comparison of the competing spectrum sensing schemes for various SNR values in the set {−4, −2} dB and L = 4 beacon symbols. Similarly, Figs. 3a, 3b present the PD versus PF A performance with


1

1

0.9

0.9




0.8 0.7 Opt−RECD Opt−MF Opt−CHRD RECD CHRD MF

0.6 0.5 0.4 0

0.1


0.6

(a) Case I

0.8 0.7 Opt−RECD Opt−MF Opt−CHRD RECD CHRD MF

0.6 0.5 0.4 0

0.1


0.6

(b) Case II

Fig. 8. ROC curves for the optimal beacon matrix versus the orthogonal beacon matrix for the composite hypothesis based robust detector (CHRD) and matched filter (MF) in (42) with SNR = −5 dB, and Ru = D(1, 0.9) for (a) L = 2, (b) L = 4.

different levels of CSI uncertainty considering the uncertainty covariance matrices Ru = σu2 D([1, 0.9]T ) with σu2 ∈ {1, 0.6}. The simulation results demonstrate a similar trend in the detection performance of the proposed robust detection schemes. It can also be observed across the figures that the performance gap between the proposed robust schemes and the uncertainty agnostic matched filter detector along with the energy detector (ED) widens with increasing CSI uncertainty. Further, Fig. 4a and 4b present a PF A versus PD performance comparison of the proposed robust detection schemes with the uncertainty agnostic matched filter (MF) detector and the energy detector (ED) on a logarithmic scale for improved resolution 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 receive and transmit antennas. We consider 2×2 MIMO scenario with CSI uncertainty covariance matrix Ru = D([1, 0.9]T ) and 4 × 4 MIMO scenario with CSI uncertainty covariance matrix Ru = D([1, 0.9, 0.6, 0.4]T ). The detection performance of the proposed robust detection schemes significantly improves with the increase in the number of receive and transmit antennas. In Fig. 6 we plot the probability of detection (PD ) versus the probability of false alarm (PF A ) and compare the simulated detection performance with the analytical performance curves generated from the derived expressions for PD in (8) and PF A in (9) for the RECD with L = 2, SNR = −3 dB and Ru = D([1, 1]T ). It is evident from the figure that the detection performance of the proposed RECD technique obtained through simulations is in close agreement with the results obtained via theory. Similarly, in Fig. 7 we compare the detection performance from the expressions for PD in (22) and PF A in (21) corresponding to the CHRD with the simulation results for a scenario with L = 2, SNR = −1 dB and σu2 = 1.

The simulated detection performance of the CHRD scheme coincides with the analytical results. Figs. 8a, 8b present the PD versus PF A performance considering the optimal beacon matrix derived in section V for the scenarios with a known uncertainty covariance and unknown uncertainty covariance. The detection performance for the suboptimal orthogonal beacon sequence is also given therein. Figs. 8 clearly demonstrate a performance improvement for the derived optimal beacon sequence based spectrum sensing in comparison to the orthogonal beacon sequence. VII. C ONCLUSION This 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 robust estimator-correlator detector (RECD) and the robust generalized likelihood detector (RGLD), which are robust against CSI uncertainty, have been proposed for scenarios with known uncertainty statistics. Further, for the scenario with unknown CSI uncertainty statistics, we developed a GLRT based composite hypothesis robust detector (CHRD) for spectrum sensing in MIMO cognitive radio networks. Closed form analytical expressions have been derived to characterize the theoretical detection performance of the proposed RECD and CHRD schemes. Subsequently, an optimization framework has also been presented to obtain the optimal beacon sequences which further enhance the performance of the proposed detectors. Simulation results were presented to illustrate the improved detection performance of the proposed robust detection schemes which consider CSI uncertainty in MIMO cognitive radio networks. It has also been shown that the optimal beacon matrix significantly boosts the detection performance towards MIMO spectrum sensing. The proposed framework can be further



extended considering other additional challenging aspects such as noise uncertainty in future works. R EFERENCES [1] M. A. McHenry, P. A. Tenhula, D. McCloskey, D. A. Roberson, and C. S. Hood, “Chicago spectrum occupancy measurements & analysis and a long-term studies proposal,” in Proceedings of the first international workshop on Technology and policy for accessing spectrum, ser. TAPAS ’06. New York, NY, USA: ACM, 2006. [2] J. Mitola and J. Maguire, G.Q., “Cognitive radio: making software radios more personal,” Personal Communications, IEEE, vol. 6, no. 4, pp. 13– 18, Aug. 1999. [3] J. Mitola, “Cognitive radio: An integrated agent architecture for software defined radio,” Doctor of Technology, Royal Inst. Technol.(KTH), Stockholm, Sweden, pp. 271–350, 2000. [4] S. Haykin, “Cognitive Radio: Brain-Empowered Wireless Communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [5] E. Axell, G. Leus, E. Larsson, and H. Poor, “Spectrum sensing for cognitive radio : State-of-the-art and recent advances,” IEEE Signal Process. Mag., vol. 29, no. 3, pp. 101–116, May 2012. [6] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory. NJ, USA: Prentice Hall PTR, Jan. 1998. [7] A. Sonnenschein and P. Fishman, “Radiometric detection of spreadspectrum signals in noise of uncertain power,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 3, pp. 654–660, 1992. [8] A. Sahai, N. Hoven, and R. Tandra, “Some fundamental limits on cognitive radio,” pp. 1–11, 2004. [Online]. Available: http: //www.eecs.berkeley.edu/∼ sahai/Papers/cognitive radio preliminary.pdf [9] E. Axell and E. Larsson, “Optimal and sub-optimal spectrum sensing of OFDM signals in known and unknown noise variance,” IEEE J. Sel. Areas Commun., vol. 29, no. 2, pp. 290–304, 2011. [10] H. Sun, W.-Y. Chiu, J. Jiang, A. Nallanathan, and H. Poor, “Wideband spectrum sensing with sub-nyquist sampling in cognitive radios,” Signal Processing, IEEE Transactions on, vol. 60, no. 11, pp. 6068–6073, Nov 2012. [11] H. Sun, A. Nallanathan, C.-X. Wang, and Y. Chen, “Wideband spectrum sensing for cognitive radio networks: a survey,” Wireless Communications, IEEE, vol. 20, no. 2, pp. 74–81, April 2013. [12] O. Mehanna and N. Sidiropoulos, “Maximum likelihood passive and active sensing of wideband power spectra from few bits,” Signal Processing, IEEE Transactions on, vol. 63, no. 6, pp. 1391–1403, March 2015. [13] D. Cabric, “Addressing feasibility of cognitive radios,” IEEE Signal Process. Mag., vol. 25, no. 6, pp. 85–93, 2008. [14] J. Ma and Y. Li, “Soft Combination and Detection for Cooperative Spectrum Sensing in Cognitive Radio Networks,” in Global Telecommunications Conference. GLOBECOM’07.IEEE, Nov. 2007, pp. 3139– 3143. [15] S. Qing-hua, W. Xuan-li, S. Xue-jun, and Z. Nai-tong, “Cooperative spectrum sensing under imperfect channel estimation,” in Communications and Mobile Computing (CMC), 2010 International Conference on, vol. 2, April 2010, pp. 174–178. [16] H. He, G. Li, and S. Li, “Adaptive spectrum sensing for time-varying channels in cognitive radios,” Wireless Communications Letters, IEEE, vol. 2, no. 2, pp. 1–4, April 2013. [17] A. Taherpour, M. Nasiri-Kenari, and S. Gazor, “Multiple antenna spectrum sensing in cognitive radios,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 814–823, February 2010. [18] J. Font-Segura and X. Wang, “GLRT-based spectrum sensing for cognitive radio with prior information,” IEEE Trans. Commun., vol. 58, no. 7, pp. 2137–2146, July 2010. [19] R. Zhang, T. J. Lim, Y.-C. Liang, and Y. Zeng, “Multi-antenna based spectrum sensing for cognitive radios: A GLRT approach,” IEEE Trans. Commun., vol. 58, no. 1, pp. 84–88, January 2010. [20] T. J. Lim, R. Zhang, Y. C. Liang, and Y. Zeng, “GLRT-based spectrum sensing for cognitive radio,” in Global Telecommunications Conference, 2008. IEEE GLOBECOM 2008. IEEE, Nov 2008, pp. 1–5. [21] D. Ramirez, G. Vazquez-Vilar, R. Lopez-Valcarce, J. Via, and I. Santamaria, “Detection of rank-p signals in cognitive radio networks with uncalibrated multiple antennas,” IEEE Trans. Signal Process., vol. 59, no. 8, pp. 3764–3774, Aug 2011. [22] A. Taherpour, S. Gazor, and M. Nasiri-Kenari, “Invariant wideband spectrum sensing under unknown variances,” IEEE Trans. Wireless Commun., vol. 8, no. 5, pp. 2182–2186, May 2009.

[23] ——, “Wideband spectrum sensing in unknown white Gaussian noise,” Communications, IET, vol. 2, no. 6, pp. 763–771, July 2008. [24] J. Font-Segura, G. Vazquez, and J. Riba, “Nonuniform sampling walls in wideband signal detection,” IEEE Trans. Signal Process., vol. 62, no. 1, pp. 44–55, Jan 2014. [25] G. Vazquez-Vilar, R. Lopez-Valcarce, and J. Sala, “Multiantenna spectrum sensing exploiting spectral a priori information,” IEEE Trans. Wireless Commun., vol. 10, no. 12, pp. 4345–4355, December 2011. [26] L. Wei, O. Tirkkonen, and Y.-C. Liang, “Multi-source signal detection with arbitrary noise covariance,” Signal Processing, IEEE Transactions on, vol. 62, no. 22, pp. 5907–5918, Nov 2014. [27] T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing. Prentice Hall, Aug. 1999. [28] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed. Elsevier/Academic Press, Amsterdam, 2007. [29] S. Vorobyov, A. Gershman, and Z.-Q. Luo, “Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 313–324, Feb 2003. [30] J. Li, P. Stoica, and Z. Wang, “Doubly constrained robust capon beamformer,” IEEE Trans. Signal Process., vol. 52, no. 9, pp. 2407– 2423, Sept 2004. [31] D. Eustice and M. S. Klamkin, “On the coefficients of a partial fraction decomposition,” The American Mathematical Monthly, vol. 86, no. 6, pp. 478–480, 1979. [32] S. Boyd and L. Vandenberghe, Convex Optimization. NY, USA: Cambridge University Press, Mar. 2004. [33] R. Lorenz and S. Boyd, “Robust Minimum Variance Beamforming,” IEEE Trans. Signal Process., vol. 53, no. 5, pp. 1684–1696, May 2005.

Adarsh Patel received his B.Tech degree in electronics and communication engineering from Integral University, Lucknow, India, in 2008 and M.Tech degree in computer science and engineering (digital communication) from ABV-Indian Institute of Information Technology and management, Gwalior, India, in 2010. He is currently working toward the Ph.D. degree in electrical engineering at the Indian Institute of Technology, Kanpur, India. He is a recipient of the TCS Research Fellowship. His main research interests include detection and estimation theory within the areas of wireless communication and specially spectrum sensing aspect of cognitive radio.

Sinchan Biswas was born in India in 1988. He received the M.Tech degree from Department of Electrical Engineering, Indian Institute of Technology Kanpur in 2013. He is currently pursuing the Ph.D degree in Department of Engineering Sciences at the Uppsala University, Sweden. From 2013 to 2014, he was a research assistant at Indian Institute of Technology Kanpur. His research interest includes signal processing, information theory and stochastic control.



Aditya K. Jagannatham received his Bachelors degree from the Indian Institute of Technology, Bombay and M.S. and Ph.D. degrees from the University of California, San Diego, U.S.A.. From April ’07 to May ’09 he was employed as a senior wireless systems engineer at Qualcomm Inc., San Diego, California, where he worked on developing 3G UMTS/WCDMA/HSDPA mobile chipsets as part of the Qualcomm CDMA technologies division. His research interests are in the area of next-generation wireless communications and networking, sensor and ad-hoc networks, digital video processing for wireless systems, wireless 3G/4G cellular standards and CDMA/OFDM/MIMO wireless technologies. He has contributed to the 802.11n high throughput wireless LAN standard. He was awarded the CAL(IT)2 fellowship for pursuing graduate studies

at the University of California San Diego and in 2009 he received the Upendra Patel Achievement Award for his efforts towards developing HSDPA/HSUPA/HSPA+ WCDMA technologies at Qualcomm. Since 2009 he has been a faculty member in the Electrical Engineering department at IIT Kanpur, where he is currently an Associate Professor, and is also associated with the BSNL-IITK Telecom Center of Excellence (BITCOE). At IIT Kanpur he has been awarded the P.K. Kelkar Young Faculty Research Fellowship (June 2012 to May 2015) for excellence in research and the Gopal Das Bhandari Memorial Distinguished Teacher Award for the year 2012-13 for excellence in teaching. He has also delivered a set of video lectures on Advanced 3G and 4G Wireless Mobile Communications for the Ministry of Human Resource Development funded initiative NPTEL (National Programme on Technology Enhanced Learning).
