Multiple Antenna Spectrum Sensing in Cognitive Radios-DPd

814 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010

Multiple Antenna Spectrum Sensing inCognitive Radios

Abbas Taherpour, Member, IEEE, Masoumeh Nasiri-Kenari, Member, IEEE,and Saeed Gazor, Senior Member, IEEE

Abstract—In this paper, we consider the problem of spectrumsensing by using multiple antenna in cognitive radios when thenoise and the primary user signal are assumed as independentcomplex zero-mean Gaussian random signals. The optimal multi-ple antenna spectrum sensing detector needs to know the channelgains, noise variance, and primary user signal variance. In

practice some or all of these parameters may be unknown, so wederive the Generalized Likelihood Ratio (GLR) detectors underthese circumstances. The proposed GLR detector, in which all theparameters are unknown, is a blind and invariant detector with alow computational complexity. We also analytically compute themissed detection and false alarm probabilities for the proposedGLR detectors. The simulation results provide the availabletraded-off in using multiple antenna techniques for spectrumsensing and illustrates the robustness of the proposed GLRdetectors compared to the traditional energy detector when thereis some uncertainty in the given noise variance.

Index Terms—Cognitive radio, spectrum sensing, multipleantenna, eigenvalue decomposition, opportunity detection, GLRdetector, noise variance mismatch.

I. INTRODUCTION

RECENT measurements reveal that many portions ofthe licensed spectrum are not used during significant

time periods [1]. Since the number of users and their datarates steadily increase, the traditional fixed spectrum policyis inefficient and is no longer a feasible approach. Oneproposal for alleviating the spectrum scarcity is allowinglicence-exempted Secondary Users (SU) to exploit the unusedspectrum holes over some frequency ranges by using CognitiveRadio (CR) technology [2]. One of the major challenges ofimplementing this technology is that the CRs must accuratelymonitor and be aware of the presence of the Primary Users(PUs) over a particular spectrum. To address this challenge,several efficient methods have been proposed [3]–[7]. In [7]the spectrum sensing in a wideband Orthogonal FrequencyDivision Multiplexing (OFDM) scenario, when the receivedpower of PU is unknown and there are different amount ofthe priori knowledge about PU signal, has been investigated.The Energy Detector (ED) (a.k.a. radiometer) is a commonmethod to detect an unknown signal in additive noise [8].

Manuscript received March 16, 2009; revised August 20, 2009; acceptedNovember 13, 2009. The associate editor coordinating the review of this paperand approving it for publication was S. Wei.

A. Taherpour and M. Nasiri-Kenari are with the Department of ElectricalEngineering, Sharif University of Technology, Tehran, Iran (e-mail: [email protected], [email protected]).

S. Gazor is with the Department of Electrical and Computer Engineering,Queen’s University, Kingston, ON, Canada (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2009.02.090385

This method is optimal for white Gaussian noise if the noisevariance is known. Unfortunately, the performance of the EDis susceptible to errors in the noise variance [9]. It has beenshown that to achieve a desired probability of detection underuncertain noise variance, the Signal-to-Noise Ratio (SNR) hasto be above a certain threshold [10]. For the case of unknownnoise variance, the cyclostationarity property of communi-cation signals is exploited in [5], [11], [12]. In contrast tonoise, which is a wide-sense stationary random signal withimpulse autocorrelation function, in general, the modulatedsignals have the periodical mean and autocorrelation function.These features can be used to distinguish the noise from themodulated signal. The drawbacks of this method are that themethod requires a significantly long observation time and ishighly computationally complex for practical implementation.

While there has been an intensive work on the spectrumsensing problem in the case of known noise variance, notenough attention has been made to the spectrum sensing underunknown noise variance except [10], [13]–[15].

Multiple antenna techniques currently are used in commu-nications and their effectiveness have been shown in differentaspects [16]. In the context of dynamic spectrum sharing,multiple antenna SU can be used for a reliable signal trans-mission and also spectrum sensing. In fact, using multipleantenna techniques in CRs is one of possible approaches forthe spectrum sensing by exploiting available spatial domainobservations and has been proposed in [17]–[19]. In [17],the authors have shown the efficiency of multiple antennaspectrum sensing in terms of required sensing time andhardware by using a two-stage sensing method. In [18],the ED has been proposed for spectrum sensing by usingmultiple antennas. The PU signal has been treated as anunknown deterministic signal and based on this model theperformance of the energy detector has been evaluated inRayleigh fading channels. In [19], it has been shown that amultiple antenna OFDM based CR scheme, when using thesquare law combing energy detector, has better performancethan the single antenna scheme, even at low SNRs. In [20], ablind energy detector based on SNR maximization has beenproposed and its performance has been evaluated in differencecases.

In this paper, we investigate the spectrum sensing problemby using multiple antennas when the PU signal can be wellmodeled as a complex Gaussian random signal in the presenceof an Additive White Gaussian Noise (AWGN). We derive theoptimum detector structure for spectrum sensing and investi-

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TAHERPOUR et al.: MULTIPLE ANTENNA SPECTRUM SENSING IN COGNITIVE RADIOS 815

Fig. 1. SU with multiple antennas.

gate its performance. The optimal detector needs to know thenoise and PU signal variances, and also the channel gains. Inpractice one or more of these parameters may be unknown, soin what follows we derive the Generalized Likelihood Ratio(GLR) detector when some or all of these parameters areunknown.

The remaining of the paper is organized as follows. InSection II, we describe the system model and the basic as-sumptions about the PU signal, noise, and channel gain vector.Conditioned on knowing the noise and PU variances andchannel gain vector, we derive the optimal multiple antennadetection rule and evaluate its performance analytically. InSection III, we derive the GLR detectors for three cases,namely, unknown channel gain, unknown channel gain andthe PU signal variance, and finally unknown all the afore-mentioned parameters. In this Section, we also show that theGLR detector when all of the parameters are unknown, is aninvariant detector. In Section IV, we evaluate the performanceof the proposed GLR detectors analytically. In Section V, wepresent some numerical results to evaluate the performanceof the GLR detectors based on both analytical derivationsand simulations, and investigate the available tarde-offs inthe GLR detector performance. Also in order to evaluate theperformance of the proposed GLR detectors in a practicalscenario, we compare the performance of the GLR detectorswith the cyclostionarity based detector and ED, when the PUsignal is considered as a DTV signal in IEEE 802.22 standard.Finally, Section VI concludes the paper.

Throughout this paper, we use boldface letters for columnvectors and boldface capital letters for matrices. We also de-note x� = [x]�, X�,� = [X]�,�, and X� = [X]� , respectively,as the elements of a vector x, the elements of a matrix X, andthe �th row of a matrix X. We use the notation � to denotethe estimation of unknown parameter �, which may be scalar,vector or matrix.

II. BASIC ASSUMPTIONS AND OPTIMAL DETECTOR

Suppose that the SU has � receiving antennas and eachantenna receives � samples as shown in Figure 1. We assumethat the PU signal samples are independent zero-mean randomvariables with complex Gaussian distribution. This assump-tion, for instance, is valid for an OFDM signal in which eachcarrier is modulated by independent data streams. We denotethe hypothesis of the PU signal being active and inactive(within the range of the CR) by ℋ0, and ℋ1, respectively. We

assume that the additive noise samples at different antennas areindependent zero-mean Gaussian random variables. Under ℋ1,we assume that the PU signal and noise are independent. LetY = [y1, ⋅ ⋅ ⋅ ,y�] ∈ C�×� be a complex matrix containingthe observed signals at � antennas. The multiple antenna PUdetection problem can be expressed as the following binaryhypothesis test:{ℋ0: Y ∼ �� (0, �2�I� ) if the PU is inactive,ℋ1: Y ∼ �� (0, �2�hh

� + �2�I� ) if the PU is active.(1)

where h ∈ C�×1 denotes the channel gain vector betweenthe PU and � antennas, and �2� and �2� are the variances ofnoise and PU signal, respectively. We assume that the channelgain vector, i.e., h, is a constant parameter at each sensingtime. For the optimal detection, the channel gain vector isassumed to be known, and for the other practical detections,we assume that the channel gain is unknown and we estimatethis unknown parameter, as will be discussed in the followingsubsections.

A. Optimal Detector

For the optimal detector, the SU knows the channel gains,noise and PU signal variances. In this case, from (1), under ℋ0

the Probability Density Function (PDF) of observation matrix,Y, is as follows [21]:

�(Y;ℋ0, �2�) =

�∏

�=1

1

(��2�)�

exp

{− 1

�2�y�� y�

}

=1

(��2�)��

exp

{− 1

�2�

�∑

�=1

y�� y�

}

=1

(��2�)��

exp

{−tr(YY�)

�2�

}, (2)

where tr(.) denotes the trace of the matrix. By taking log-arithm of PDF of observations under hypothesis ℋ0, i.e.,ℒ0(Y) = ln �(Y;ℋ0, �

2�), we will have:

ℒ0(Y) = − tr(YY�)

�2�−�� ln� −�� ln�2�. (3)

Similarly, from (1) under the hypothesis ℋ1, the PDF can bewritten as [21]:

�(Y;ℋ1,h, �2�, �

2� ) =

�∏

�=1

exp{−y�� R

−1y�}

��det(R)

=exp{−∑��=1 y

�� R

−1y�

}

��det(R)�

=exp{−tr(R−1YY�)

}

��det(R)�, (4)

where R ≜ �[YY� ∣ℋ1] = �2�hh

� + �2�I is the covariancematrix. We can easily show that det(R) = (�2�∥h∥2 +�2�)(�

2�)

(�−1) and also by using the matrix inversion lemma[22], we obtain:

R−1 = �−2� I− �−2

�

hh�

�2��2�

+ ∥h∥2. (5)



By taking the logarithm from (4) and denoting ℒ1(Y) =ln �(Y;ℋ1,h, �

2�, �

2� ), we obtain:

ℒ1(Y) = −tr(R−1YY�)−�� ln� (6)

−� ln det(�2�∥h∥2 + �2�)− �(� − 1) ln�2�,

which substituting (5) in (6) results in:

ℒ1(Y) = − tr(YY�)

�2�+

∥h�Y∥2

(�2��2�

+ ∥h∥2)�2�(7)

− �� ln� − � ln(�2��2�

∥h∥2 + 1)− �� ln�2�.

For optimal detector in Neyman-Pearson sense, we need tocompare the Likelihood Ratio (LR) function or Logarithm ofLikelihood Ratio (LLR) function with a threshold. From (6)and (3), the LLR function is equal to:

LLR = ln�(Y;ℋ1,h, �

2� , �

2�)

�(Y;ℋ0, �2�)

= ℒ1(Y)− ℒ0(Y)

=∥h�Y∥2

(�2��2�

+ ∥h∥2)�2�− � ln(

�2��2�

∥h∥2 + 1). (8)

Comparing the LLR function with a threshold results in thefollowing optimal decision rule:

∥h�Y∥2

(�2��2�

+ ∥h∥2)�2�− � ln(

�2��2�

∥h∥2 + 1)ℋ1

≷ℋ0

�1. (9)

Where �1 is the decision threshold. Considering that in optimaldetector, the channel gains and noise and PU variance areknown, with some straightforward simplifications, the optimaldecision rule can be rewritten as follows:

�opt = ∥h�Y∥2ℋ1

≷ℋ0

�, (10)

where � ≜�1+� ln(

�2�

�2�∥h∥2+1)

(�2�

�2�+∥h∥2)�2�

denotes the detection threshold.

In general, the detection threshold � is obtained by solving� (�) = 1−�fa, where �fa denotes the false alarm probabilityand � (�) is the Cumulative Distribution Function (CDF) ofthe decision statistic. Also, it is possible to find this thresholdby Monte-Carlo simulation method. From (10), the optimaldetector is a maximum ratio combiner, which gives weightsto the observations at the different antennas according to theircorresponding channel gains, where the antenna with betterreception has more contribution in the summation in (10).

In the context of dynamic spectrum sharing, the false alarmprobability �fa indicates the probability that a spectrum hole(a vacant band) is falsely detected as an occupied band, i.e.,�fa represents the percentage of the spectrum holes whichare not used. Therefore, the SUs must reduce the false alarmprobability �fa as much as possible. On the other hand, themissed detection probability, i.e., �� = 1 − ��, determinesthe probability that an occupied band is mistakenly detectedas a spectrum hole. Such a missed detection induces harmfulinterference for PU. Thus, the missed detection probabilitymust be small enough to avoid perceptible performance lossfor the PU.

B. Performance of Optimal Detector

To evaluate the performance of the optimal detector, wefirst compute the Complementary Cumulative DistributionFunction (CCDF) of the decision statistic under ℋ0 and ℋ1.Under hypothesis ℋ0, we notice that the elements of theobserved matrix are i.i.d. Gaussian variables with zero meanand variance of �2�, i.e. Y ∼ �� (0, �2�I). As a result,conditioned on channel coefficients, the random vector h�Yhas a Gaussian distribution,i.e., h�Y ∼ �� (0, ∥h∥2�2��).Then, from (10), the decision statistic under hypothesis ℋ0

has the following distribution:

�opt(Y)

∥h∥2�2�=

∥h�Y∥2∥h∥2�2�

∼ �22�. (11)

Therefore, the false alarm probability �fa is easily obtainedusing CCDF of �opt(Y) as follows [14]:

�fa = � [��(Y) > �∣ℋ0]

=Γ(�, �

∥h∥2�2�)

Γ(�), (12)

where Γ(�, �) =∫∞� ��−1�−�� and Γ(�) =

∫∞0 ��−1�−��

are the upper incomplete and complete gamma functions,respectively.

Similarly under hypothesis ℋ1, we have Y ∼�� (0, �2�hh

� + �2�I). Then, again conditioned on thechannel coefficients, we easily conclude that

h�Y ∼ ��(0, ∥h∥2(∥h∥2�2� + �2�)I�

), (13)

and hence from (10)

�opt(Y)

∥h∥2(∥h∥2�2� + �2�)=

∥h�Y∥2∥h∥2(∥h∥2�2� + �2�)

∼ �22�. (14)

Therefore, the detection probability �� is easily evaluated asfollows

�� = � [��(Y) > �∣ℋ1]

=Γ(�, �

∥h∥2(∥h∥2�2�+�2�)

)

Γ(�), (15)

which if we define the received SNR at the SU as � ≜ �2�∥h∥2

�2�,

the detection probability can be written as:

�� =Γ(�, �

∥h∥2�2�(1+�)

)

Γ(�). (16)

III. GLR DETECTORS

The optimal detector needs to know the values of channelgains, noise and PU variances. In practice, we may have noknowledge about the values of some or all of these parameters.In these cases, we can use the GLR test to decide about thepresence or absence of the PU. In this section, we derivethe GLR test in the different cases, in which some or allof the parameters are unknown. Then, we investigate theinvariancy of the derived GLR tests under several groups oftransformations.



A. Case 1: Unknown Channel Gains (GLRD1)

In this part, we assume that the SU has knowledge about thenoise and PU signal variances, but the channel gain vector his unknown. In this case, since the variance of noise is known,the logarithm of PDF of observations Y under hypothesis ℋ0

is computed as (3). Under hypothesis ℋ1 the channel gainsare unknown and in order to derive the GLR test, we firstmaximize (4) with respect to h to find the ML estimation ofthe channel gains. From (7), by setting ∂

∂hℒ1(Y) = 0, wehave:

(�I+ �hh�

)YY�h = �h, (17)

where � = 1

(�2�

�2�+∥h∥2)�2�

, � = �

(�2�

�2�+∥h∥2)�2�

and � =

1

(�2�

�2�+∥h∥2)�2�

. If we define A ≜(�I+ �hh�

), then since �

and � are real positive numbers, the matrix A = �I+ �hh�

is a full rank matrix and therefore it has an inverse. Usingmatrix inversion lemma and multiplying both sides of (17) byA−1 = 1

� (I− hh�

��+∥h∥2 ), we obtain

1

�YY�h =

�

�A−1h

=�

��(I− hh�

�� + ∥h∥2 )h. (18)

This obviously means that h is a eigenvector of R ≜ 1�YY� ,

i.e.,

Rh = �h (19)

and � = ��(

��+�∥h∥2 ) is a real number and the corresponding

eigenvalue of sample covariance matrix R = 1�YY� . From

(19), it is obvious that the ML estimation of the vector h is aneigenvector of R. Now we must determine which eigenvectorsof R maximizes the likelihood function. We first normalizethe eigenvectors such that ∥h∥2 = 1. From Rh = �h andthe definition of R, we obtain ∥h�Y∥2 = ��∥h∥2 = ��.Replacing this equation in (6) we obtain

ℒ1(Y) = −� tr(R)

�2�+

��

(�2��2�

+ 1)�2�(20)

− �� ln� − � ln(�2��2�

+ 1)− �� ln �2�.

Since the above function is an increasing function with respectto �, the ML estimation h is the eigenvector correspondingto the maximum eigenvalue of the matrix R which we denotethis vector by h, i.e.

Rh = �maxh, (21)

where, by assumption, we have ∥h∥ = 1. By replacing theabove estimation in (20), we obtain:

ℒ1(Y) = −� tr(R)

�2�+

��max

(�2��2�

+ 1)�2�(22)

− �� ln� − � ln(�2��2�

+ 1)− �� ln �2�.

From (3) and (22), the LLR function is equal to:

LLR = ℒ1(Y)− ℒ0(Y)

=��max

(�2��2�

+ 1)�2�− � ln(

�2��2�

+ 1). (23)

For decision making, we must compare the LLR in (23) to athreshold which results in

��max

(�2��2�

+ 1)�2�− � ln(

�2��2�

+ 1)ℋ1

≷ℋ0

�1 ⇒

�GLRD1(Y) =�max

�2�

ℋ1

≷ℋ0

�, (24)

where � = 1� [�1 + � ln(

�2��2�

+ 1)](�2��2�

+ 1).

B. Case 2: Unknown Channel Gains and PU Variance

(GLRD2)

In this part, in addition to unknown channel gains, weassume that the PU variance is also unknown. We maximize(22) with respect to �2� to find the ML estimation of PUvariance �2� . By setting ∂

∂�2�ℒ1(Y) = 0, we obtain

�2� = �max − �2�. (25)

We now replace (25) in (22), as

ℒ1(Y) = −� tr(R)

�2�+��max

�2�− �−�� ln�

−� ln(�max

�2�)− �� ln�2�. (26)

From (3) and (26), the LLR function is equal to:

LLR = ℒ1(Y)− ℒ0(Y)

=��max

�2�− � ln(

�max

�2�)− �. (27)

For decision making, the LLR function must be compared bya threshold. We easily obtain:

�max

�2�− ln(

�max

�2�)ℋ1

≷ℋ0

�1. (28)

We notice that �(�) = �− ln(�) is an increasing function for� ≥ 1. Under both ℋ0 and ℋ1 hypotheses, the ratio of �max

�2�is greater than one with high probability [23] and hence wecan simplify the GLR detector as the following form:

�GLRD2(Y) =�max

�2�

ℋ1

≷ℋ0

�. (29)

Interestingly, the derived GLR detector (GLRD1) has the sameform as the detector given in (24), in which only the channelgains are unknown (GLRD1). So we can conclude that theknowledge about the PU variance, when the channel gains areunknown, does not lead to a better GLR detector. This is alsointuitively expected as this case is equivalent to the case thatthe PU variance is one and the channel gain is ��h.



C. Case 3: Unknown Channel Gains and PU and Noise

Variance (GLRD3 or blind GLRD)

In the following, we derive the GLR detector when all ofthe mentioned parameters are unknown. Up to know, we haveobtained the ML estimations of the channel gain vector h

and �2� . In order to derive the GLR test, we maximize thePDFs in (26) and (3) with respect to �2� and then form theLR function. By maximizing (3) with respect to �2�, we obtainthe ML estimation of noise variance under hypothesis ℋ0 asfollows:

ℋ0 : �2� =tr(YY�)

��=

tr(R)

�, (30)

and hence from (3) and (30), for PDF under hypothesis ℋ0,we get:

sup�2�

�(Y;ℋ0, �2�) =

(��)��

(��)��(tr(R)

)�� . (31)

Similarly under hypothesis ℋ1, from (26) by setting∂∂�2�

ℒ1(Y) = 0, we obtain

�2� =1

� − 1

(tr(R)− �max

). (32)

From (21), (25) and (32), the ML estimations of unknownparameters under ℋ1 are summarized as follows

ℋ1 :

⎧⎨⎩

�2� = 1�−1

(tr(R)− �max

),

�2� =1�−1

(��max − tr(R)

),

h = vmax

∥vmax∥

(33)

where vmax denotes the eigenvector corresponding to thelargest eigenvalue. Using the ML estimations in (33) leadsto the following PDF under hypothesis ℋ1:

suph,�2�,�

2�

�(Y;ℋ1,h, �2�, �

2�) = (34)

(�(� − 1))�(�−1)��

(��)��(�max)�(tr(R)− �max)�(�−1).

From (31) and (34), the LR function is derived as:

LR(Y) =

sup(h,�2�,�

2�)

�(Y;ℋ1,h, �2�, �

2�)

sup�2�

�(Y;ℋ0)(35)

=

(�(�−1))�(�−1)��

(��)��(�max)�(tr(R)−�max)�(�−1)

(��)��

(��)��(tr(R))��

=(� − 1)(�−1)�

��(tr(R))��

��max(tr(R)− �max)�(�−1).

If we define �def= �max

tr(R)then above LR function can be written

as:

LR(Y) =(� − 1)(�−1)�

��

(1

�(1 − �)�−1

)�. (36)

For decision making, (36) must be compared with a threshold�1, i.e., we must compare

1

�(1− �)�−1

ℋ1

≷ℋ0

�2, (37)

where �2 =(

��

(�−1)(�−1)� �1

) 1�

.Let �1 = �max ≥ �2 ≥ ⋅ ⋅ ⋅ ≥ �� denote the eigenvalues

of matrix R in descending order. It can be easily observed thatthe supremum and infinimum of the expression � = �max

tr(R)=

�1∑��=1 ��

are respectively 1 and 1� , i.e., 1

� < � < 1. Now sincethe above LR is an increasing function of � in the interval( 1� , 1), the GLR test in (37) can be simplified as:

� =�1∑��=1 ��

ℋ1

≷ℋ0

�, (38)

where we have used the fact that tr(R) =∑��=1 ��. As

another form, by manipulation of the above detector, we canexpress the GLR detector as the following equivalent form:

�GLRD3(Y) =�1∑��=2 ��

ℋ1

≷ℋ0

�, (39)

where � = �1−� .

D. Computational Complexity

The computational complexity of the proposed detectorscomes from two major operations: computation of samplecovariance matrix and eigenvalue decomposition of the covari-ance matrix. Here we consider the computational complexityof a trivial approach (which may be numerically non-efficient)in which the largest root of the sample covariance matrix iscalculated. For the first part, since the covariance matrix is ablock Toeplitz and Hermitian, we need only to compute itsfirst block row. Hence �2� multiplications and �2(� − 1)additions are required. For the second part, in general at most�(�3) multiplications and additions are needed. Thus thetotal computational complexity is as follows :

�2(2�− 1) +�(�3) (40)

Note that the considered approaches for the calculations of theabove parameters are not necessarily the best ones, from com-putational complexity aspect. For instance, the computationalcomplexity can be reduced by determining the dominant sin-gular value of the observation matrix Y, instead of calculatingthe the largest root of the sample covariance matrix. In practicethe number of temporal samples � is usually much larger thanthe number of antennas � and the dominant term is the firstterm. On the other hand, the ED needs �� multiplicationsand�(�−1) additions and thus the computational complexityof the proposed detectors is about � times that of the ED.

E. Invariancy of the GLR Detectors

In the following, we investigate the invariancy of the GLRdetectors derived in (39), (29) and (24) under two groups oftransformations namely orthogonal transformation and scale:

�� ={��∣��(Y) = QY, ∀Q ∈ C

�×� ,Q�Q = I�}

�� = {��∣��(Y) = �Y, ∀� > 0} . (41)

The above transformations are groups, since they are closed,associative, and contain the identity and inverse elements.

The invariancy of the GLR detector under orthogonal trans-formation indicates that the proposed GLR detector has the



same form if we use the frequency samples instead of temporalsamples. The reason is that taking FFT from the temporalsamples is equal to applying a unitary matrix transformationto the temporal samples which is an orthogonal transformation.Also invariancy under scale implies that the received samplesat the antennas can be amplified or attenuated during sensingprocess.

In the following, we prove that for blind GLRD, the dis-tribution of the observations and the parameter spaces remaininvariant under any compositions of the transformation groupsin (41), and the GLRD1 is invariant only under orthogonaltransformation.

1) Orthogonal Transformation: under ℋ0, from Y ∼�� (0, �2�I� ), we get

��(Y) = QY ∼ �� (0,QQ��2�I� )

= �� (0, �2�I� ). (42)

under ℋ1, Y ∼ �� (0, �2�hh� + �2�I� ) and thus we

get

��(Y) = QY ∼ �� (0, �2��hh�Q� +QQ��2�I� )

= �� (0, �2�h′h′� + �2�I� ). (43)

where h′ = Qh. Since the channel gain h is unknown,the transformed channel gain h′ is also unknown withthe same unity norm ∥h′∥2 = ∥Qh∥2 = h�Q�Qh =h�h = ∥h∥2. Hence the distribution family of thetransformed signal is unchanged. The above discussionis valid for both GLRD1 and blind GLRD. As an-other approach, we know from linear algebra theorythat the orthogonal transformation does not change theeigenvalues of a given matrix. Thus, under orthogonaltransformation, the decision statistics of both GLR de-tectors remain unchanged and they are invariant underorthogonal transformation.

2) Scale: For blind GLR under ℋ1, from Y ∼�� (0, �2�hh

� + �2�I� ) we obtain

��(Y) = �Y ∼ �� (0, �2�2�hh� + �2�2�I� ). (44)

Since �2� and �2� are unknown, the distribution familyof the transformed signal is not changed. Under thenull hypothesis, the proof is similar. It is obvious thatthe GLRD1 is not invariant under scale transformation.In fact, by scaling the observation matrix, all of theeigenvalues are scaled and hence their ratio will beconstant for blind GLRD.

IV. ANALYTICAL PERFORMANCE EVALUATION

In this section we evaluate the performance of the GLRdetectors in terms of detection probability, ��, and false alarmprobability, �fa. For computing the detection probability, weneed to compute the statistics of the principal componentswhen the PU signal is present. Also for the false alarmprobability, we need to determine the behavior of eigenvaluesunder null hypothesis, i.e., ℋ0. We evaluate these probabilitiesin the following parts.

A. False Alarm Probability

In this part, we first introduce a statistical result for theeigenvalues of the sample correlation matrix , i.e., R, underhypothesis ℋ0. We then use the result for the calculation of�fa.

Lemma 1: The normalized largest eigenvalue of a complexcorrelation matrix, i.e., �1/�2�, in null case is distributed asTracy-Widom distribution of order 2 [24], i.e,

�1/�2� − ��

→�2 ∼ ��2, (45)

where the limit is in distribution, and

�� =

(1 +

√�

�

)2

(46)

�� =1√�

(1 +

√�

�

)(1√�

+1√�

)1/3

. (47)

For the analytical formula of ��2 refer to [24], and for thetables of its CDF refer to [25].

In the following parts, we assume that the number ofantennas, i.e., � , and received sample, i.e., �, are largeenough and using (45), we compute the false alarm probability�fa for different GLR detectors derived in parts IV-B1 andIV-B2.

We note that the detector derived in (29) is the same as (24)and thus they have the same performance.

1) GLRD1 (or GLRD2): In this case from (24) and (45),we have:

�fa = � [�GLRD1(Y) > �∣ℋ0]

= � [�1/�2� > �∣ℋ0]

= � [�1/�

2� − ��

>� − ��

∣ℋ0]

= 1− ��2(� − ��

), (48)

where ��2(�) is the CDF of Tracy-Widom distribution oforder 2. Now for a given �fa, the threshold can be easilyobtained as:

� =

(1 +

√�

�

)2

+1

�

(√�+

√�)4/3

(√��)1/3

�−1��2(1 − �fa), (49)

where �−1��2(.) denotes the inverse CDF of Tracy-Widom

distribution of order 2. We note that, unlike the optimumdetector, the threshold does not depend on noise variance andchannel gains, and can be pre-computed based only on � , �and �fa.

2) GLRD3 (Blind GLRD): In this case, since the distri-bution of largest eigenvalue is known, from (39) we mustcompute the distribution of the summation of other eigenvaluesunder hypothesis ℋ0. Under ℋ0, since we have assumedthat the number of antennas and samples are large enough,the summation 1

�−1

∑��=2 �� is approximately constant and

equals to the variance of noise[26], i.e.,

ℋ0 :1

� − 1

�∑

�=2

�� ≈ �2�, (50)



and we conclude that∑��=2 �� ≈ (�−1)�2�. Therefore, under

hypothesis ℋ0 the decision statistics can be written as thefollowing form:

ℋ0 :�1∑��=2 ��

≈ 1

� − 1

�1�2�. (51)

Thus from (45), the false alarm probability can be easilycalculated as:

�fa = 1− ��2

(� − ��

�−1��

�−1

), (52)

and for a given ��, the threshold can be easily obtained as:

� =

(1 +√��

)2

� − 1(53)

+1

�(� − 1)

(√�+

√�)4/3

(√��)1/3

�−1��2(1− �fa).

Again, we note that, the threshold does not depends onnoise variance and channel gains and can be pre-computedbased only on � , � and �fa.

B. Detection Probability

Let �1 ≥ �2 ≥ ⋅ ⋅ ⋅ ≥ �� and �1 ≥ �2 ≥ ⋅ ⋅ ⋅ ≥ ��denote the eigenvalues, in the descending order, of the actualcovariance matrix R and the sample covariance matrix R

defined in (4) and (18), respectively. Under hypothesis ℋ1, wehave the following lemma for the distribution of the largesteigenvalue:

Lemma 2: Under hypothesis ℋ1, the largest eigenvalue, �1,of a sample matrix has the normal distribution as follows [23]:

�1 ∼ �(�1 +

(� − 1)�1�2�

�(�1 − �2�),�21�

). (54)

Under hypothesis ℋ1, from (4), we have R = �2�hh� +

�2�I and det(R) = (�2�∥h∥2 + �2�)(�2�)(�−1). Thus, we canconclude that �1 = �2�∥h∥2+�2� and �2 = �3 = ⋅ ⋅ ⋅ = �� = �2�.By using the received SNR definition, i.e., �, in (16), (54) canbe rewritten as:

�1�2�

∼ �((1 + �)

(1 +

� − 1

��

),(1 + �)2

�

). (55)

In the following, by using (55), we compute the detectionprobability, i.e., �� for different scenarios.

1) GLRD1 (or GLRD2): In this case, from (24) and (55),the probability of detection can be computed as follows:

�� = � [�GLRD1(Y) > �∣ℋ1]

= �

⎛⎝� − (1 + �)

(1 + �−1

��

)

1+�√�

⎞⎠

= �

(√��

1 + �− � − 1√

��−√�

). (56)

The above detection probability is conditional on instan-taneous SNR, i.e., �. In the fading channel, the averageprobability of detection can be computed by averaging over �distribution, i.e., ��(�), as follows:

�� =

∫ ∞

0

��(�)��(�)�� (57)

=

∫ ∞

0

�

(√��

1 + �− � − 1√

��−√�

)��(�)��,

For instance, in Rayleigh fading channel, ��(�) = 1� �

− ��

and the average probability can be computed by numericalintegration.

2) GLRD3 (Blind GLRD): Having the largest eigenvaluedistribution, from (39), we must compute the distribution ofthe summation of the other eigenvalues. We have the followingresult on the summation of the other eigenvalues:

Lemma 3: Under hypothesis ℋ1, for the averaged summa-tion of eigenvalues except the largest one, we have [26]:

ℋ1 :1

� − 1

�∑

�=2

�� ≈ �2� −�2��1

�(�1 − �2�). (58)

Then, we obtain:

�∑

�=2

�� ≈ (� − 1)(1− 1 + �

��)�2�, (59)

where � is defined in (16). From (55) and (59), the decisionstatistic will have a Gaussian distribution as follows:

ℋ1 :�1∑��=2 ��

∼ (60)

�((1 + �)(1 + �−1

�� )

(� − 1)(1− 1+�� )

,(1 + �)2

�(� − 1)2(1− 1+�� )

2

).

So, the detection probability can be calculated as:

�� = �

(�√�(� − 1)(1− 1+�

�� )

1 + �−

√�

1 + �−1��

), (61)

and in fading channel, the average detection probability canbe computed by averaging the above probability over SNR,i.e., �.

�� = (62)∫ ∞

0

�

(�√�(� − 1)(1− 1+�

�� )

1 + �−

√�

1 + �−1��

)��(�)��.

V. SIMULATION RESULTS AND DISCUSSION

In this section, we present some numerical results to evalu-ate the performance of the proposed detectors. Figure 2 depictsthe probability of missed detection �� of the optimal detector,the proposed GLR detectors and the energy detector versusSNR at a false alarm rate of �fa = 10−2, � = 16 and� = 4 . In order to determine the threshold for a givenfalse alarm probability, we have generated the decision statisticrandomly according to its distributions for 106 independenttrials (in absence of PU signal) and chosen the detectionthreshold as 100�fa percentile of the generated data, i.e., for�fa = 10−3, 100 × 10−3 = 0.1% of the generated decisionstatistic (out of 106) are above the determined threshold. Ascan be observed, by increasing the SNR the performance of



Ŧ10 Ŧ5 0 5 10 1510

Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

SNRdB

Pro

bability

ofm

isse

ddete

ction,P

m

Optimal detectorGLRD1EDBlind GLRD

Fig. 2. The probability of missed detection of the GLR detectors, ED andoptimal detector versus SNR for �fa = 10

−2, � = 4 and � = 16.

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

10Ŧ5

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

Probability of false alarm, Pfa

Pro

bability

ofm

isse

ddete

ction,P

m


Fig. 3. The complementary ROC (�� vs. �fa) of different detectors inAWGN channel, for �� = 5 dB, � = 2 and � = 8.

the detectors improves and the ED which knows the exactvalue of noise variance performs only better than the blindGLRD. Also, the performance of the GLRD1 is better thanthe ED and blind GLRD. Note that the GLRD1 and GLRD2derived in Sections III-A and III-B are identical.

Figure 3 show the complementary ROC (Receiver Operatingcharacteristics) or the probability of missed detection, i.e. ��,versus probability of the false alarm, i.e., �fa, for differentdetectors in AWGN channel for SNR = 5 dB, � = 2,and � = 8. Also, Figures 4, 5, and 6 show these curvesfor average SNR, � = 5 dB and � = 8, respectively, for� = 2, � = 4 and � = 6 in Rayleigh fading channel. Ascan be observed, the performance of all detectors degradesslightly in fading channel compared with AWGN channel.One approach to improve the performance in the fadingchannels, is to use collaborative spectrum sensing [3], [4],[27]. By collaboration among the SUs, the deleterious effect offading can be mitigated and a more reliable spectrum sensingcan be achieved. In fact, in collaborative spectrum sensing,the SUs use the available spatial diversity to improve theirperformance.

As can be seen from Figures 4,5,6 by using more antennas,like in a collaborative spectrum sensing, the performanceimproves due to the spatial diversity provided.

Our further simulation results, which have not providedhere, indicates that by increasing the number of samples, i.e.,

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

10Ŧ5

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100


Pro

bability

ofm

isse

ddete

ction,P

m

Optimal detectorGLRD1EDBlind detector

Fig. 4. The complementary ROC of different detectors in Rayleigh fadingchannel, for average �� = 5 dB, � = 2 and � = 8.

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

Probability of fale alarm, Pfa

Pro

bability

ofm

isse

ddeete

ction,P

m



�, the performance improves. Note that, we can not increase �arbitrarily since � determines the acquisition time (the waitingtime-lag before a decision can be made). Thus in practice, wehave to make a trade-off between �fa (the spectrum usageefficiency), �m (PU interference protection level) and � (theacquisition time). However as expected, the simulation resultsindicate that increasing the number of antennas, i.e., � ,compared to increasing the number of samples, i.e., �, hasmore substantial effect on the performance improvement ofthe different detectors in fading channels.

In Figure 7, we compare the proposed GLR detectorswith the optimal detector and ED under noise variancemismatch of 0.5 dB. For noise mismatch, it is assumedthat ∣10 log10( �

2�

�2�)∣ = �dB, where �2� is the actual noise

variance, and ��, defined as noise uncertaincy factor, isconsidered as a uniform distribution variable in the interval�� ∼ � [−0.5, 0.5]. In practice, the noise uncertaincy factor,i.e., �dB, in receiver is normally 1-2 dB which due to theexistence of interference can be much higher [10], [15]. As canbe realized, the GLR detectors are more robust to the noise un-certaincy than the optimal detector and ED, and in fact undernoise variance mismatch, the optimal detector performs similarto the GLRD1. Also, the blind GLRD performs slightly betterthan the GLRD1 in which only the variance of noise is known.Under a greater noise uncertaincy factor, the performance ofED and optimal detectors degrades more substantially and the



10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100


Probabilit

yofm

issed

detectio

n,P

m

Optimal detectorGLRD1EDBlind detector


10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100


Pro

bability

ofm

isse

ddete

ction,P

m

EDBlind GLRDOptimal detectorGLRD1

Fig. 7. The effect of noise variance mismatch on the performance of theED and GLR and optimal detectors, for �dB=0.5, �� = 3dB, � = 16

and � = 4.

GLR detectors present a better performance. From this figureand our further simulations, we conclude that the optimaldetector can outperform the GLR detectors provided that theoptimal detector knows the noise variance accurately enough.However in practice, there are uncertainty about the noisevariance which under these unavoidable circumstances, theblind GLRD outperforms the optimal detector and ED. In Fig-ure 8, we evaluate the performance of the proposed detectorsin a typical practical applications. In IEEE 802.22 standard(WRAN), the CRs need to detect the presence or absence ofwireless microphone, and digital and analog TV signals whichin north America these signals are respectively FM, NTSC,and ATSC signals[28]. This figure illustrates the performanceof blind GLR, GLRD1, cyclostationarity based detecor, andED, when the PU signals are considered as captured DTVsignals and the parameters are set as �dB = 0.5, �� =−10��, � = 4096, and � = 4. For simulation, the capturedDTV signal samples have been taken from [29]. As can beseen, in this case even thought the PU signal is not a Gaussiansignal, the performance of the proposed detectors, i.e., blinddetector and GLRD1, are acceptable, and the blind detectorperforms like and even slightly better that the cyclostationaritybased detector. We note that the cyclostionarity based detectoruses the available information of DTV signal such as timeduration and waveform and cyclic frequencies, and is highly

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100


Pro

bability

ofm

isse

ddete

ction,P

m

Cyclostationary detector

GLRD1

Blind detector

ED

Fig. 8. The Performance of Blind detector, GLRD1, Energy detector (ED)and cyclostationarity based detector, for �dB=0.5, �� = −10dB, � =

4096, and � = 4.

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100

10Ŧ4

10Ŧ3

10Ŧ2

10Ŧ1

100


Pro

bability

ofm

isse

ddete

ction,P

m

Blind GLRD, simulationBlind GLRD, analyticalGLRD1, analyticalGLRD1, simulation

Fig. 9. Comparison between simulation and analytical performance of GLRdetectors, for SNR= 3 dB, � = 8 and � = 32.

computationally complex for practical implementation. On theother hand, as mentioned before, the blind detector can beimplemented easily and does not use any information of PUsignal. Our further simulations indicate similar behaviors forthe other considered IEEE 802.22 potential signals, i.e., forFM wireless microphone and analog TV signals. In Figure9, we have presented the performance evaluation of the GLRdetectors based on both analytical and simulations, for SNR= 3 dB, � = 8, and � = 32. As can be observed, thesimulation results well confirm the analytical derivations. Itis notable that because of asymptotic approximations used inderiving analytical results, the simulation and analytical resultswill well coincide provided that the number of samples andantennas are large enough. In fact, the available gap betweenthe analytical and the simulation results will decrease byincreasing the number of samples or the number of antennas.

VI. CONCLUSION

In this paper, we considered the spectrum sensing for theCRs equipped with multiple antenna receivers. We derivedthe optimal detector which needs to know the variances of thePU signal and noise as well as the channel gains. We alsopresented the GLR detectors in which some or all of theseparameters are unknown. We evaluated the performance ofthe proposed detectors in terms of false alarm and detection



probabilities. The simulation results revealed that the proposedGLR detectors perform better than the ED and almost identicalto the optimal detector under noise variance mismatch.

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Abbas Taherpour (S’2007-M’2009) received theB.Sc. degree in Electrical Engineering in 2002 fromSharif University of Technology and the M.Sc.degree in Communication Systems Engineering in2004, from Tehran University. He also received hisPh.D. in Communication Systems Engineering fromSharif University of Technology in 2009, where hewas a member of Wireless Research Lab. He hasbeen also a visiting scholar at Queen’s University,ON, Canada form June 2007 until June 2008. His re-search interests are cognitive radio, statistical signal

processing and information theory.

Saeed Gazor (S’94-M’95-SM’98) received theB.Sc. degree in Electronics Engineering in 1987and the M.Sc. degree in Communication SystemsEngineering in 1989, from Isfahan University ofTechnology, both with Summa Cum Laude, thehighest honors. He received his Ph.D. in Signal andImage Processing from Telecom Paris, DépartementSignal (École Nationale Supérieure des Télécommu-nications/ENST, Paris), France in 1994. Since 1999,he has been on the Faculty at Queen’s Universityat Kingston, Ontario, Canada and currently holds

the position of Professor of the Department of Electrical and ComputerEngineering. He is also cross-appointed to the Department of Mathematicsand Statistics at Queen’s University. Prior joining Queen’s University, he wasan Assistant Professor in Isfahan University of Technology, Department ofElectrical and Computer Engineering from 1995 to 1998. He was a researchassociate at University of Toronto from January 1999 to July 1999.

His main research interests are statistical and adaptive signal processing,cognitive radio and signal processing, array signal processing, speech process-ing, image processing, detection and estimation theory, MIMO communicationsystems, collaborative networks, channel modeling, and information theory.His research has earned him a number of awards including; a ProvincialPremier’s Research Excellence Award, a Canadian Foundation of InnovationAward and an Ontario Innovation Trust Award. Dr. Gazor is a memberProfessional Engineers Ontario and is currently serving as Associate Editorfor the IEEE SIGNAL PROCESSING LETTERS.

Masoumeh Nasiri-Kenari (S’90-M’1994) received the B.S and M.S. degreesin electrical engineering from Isfahan University of Technology, Isfahan, Iran,in 1986 and 1987, respectively, and the Ph.D. degree in electrical engineeringfrom the University of Utah, Salt Lake City, in 1993. From 1987 to 1988,she was a Technical Instructor and Research Assistant at Isfahan Universityof Technology. Since 1994, she has been with the Department of ElectricalEngineering, Sharif University of Technology, Tehran, Iran, where she isnow a Professor. Dr. Nasiri-Kenari also serves as the head of WirelessLab. of Advanced Communications Research Institute, Sharif Universityof Technology. From 1999-2001, She was a Co-Director of the AdvancedCommunication Science Research Laboratory, Iran Telecommunication Re-search Center, Tehran, Iran. Her current research interests are in wirelesscommunication systems, error correcting codes, and optical communicationsystems.


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Multiple Antenna Spectrum Sensing in Cognitive Radios-DPd