# Generalized likelihood ratio test for cyclostationary multi-antenna spectrum sensing.

1. IntroductionRecently, cognitive radio (CR) has drawn significant attention from academic and industrial communities to meet the ever-growing needs for spectrum resources [1]-[4]. In CR systems, cognitive unlicensed users are allowed to identify and exploit the local and instantaneous spectrum white space where no licensed user is present. A CR user is required to perform spectrum sensing [5]-[8] periodically to avoid significant interference to the licensed systems. If an idle channel is detected, the CR user can transmit or receive data on the channel; whereas, if a licensed user is detected, the CR user avoids data transmission on the channel and tunes to another idle channel. Cyclostationarity-based signal detection is one of the widely considered spectrum sensing techniques for CR systems, due to its capability of differentiating noise from licensed user signals. The cyclostationary spectrum sensing methods exploit the cyclic statistics such as cyclic auto-correlation for the detection of licensed user, which are nonzero at particular cyclic frequencies for the licensed user signal [9], [10].

Many existing cyclostationarity-based spectrum sensing methods [11]-[21] have employed the Dandawate-Giannakis algorithm [22] due to its constant false alarm rate (CFAR) property and robustness in the low signal-to-noise ratio (SNR) regime. The Dandawate-Giannakis algorithm can be viewed as the generalized likelihood ratio test (GLRT) for the presence of cyclostationarity. It has assumed that the distributions of the cyclic auto-correlation estimations under the null hypothesis and the alternative hypothesis have the same asymptotic covariance and differ only in mean. Based on this, the asymptotic covariance estimation can be generalized regardless of the hypothesis, which leads to a final test statistic having a squared Mahalanobis distance form.

The performance of spectrum sensing can be seriously degraded in Rayleigh fading situations. To remedy this problem, multi-antenna spectrum sensing that exploits spatial diversity could be employed. Recently, how to improve the performance of cyclostationary spectrum sensing by using multiple antennas has also been studied by researchers. In [20], we have extended the Dandawate-Giannakis algorithm to allow multi-antenna spectrum sensing, which takes into account the cyclic auto-correlations obtained from all receiver antennas and the cyclic cross-correlations obtained from all pairs of receiver antennas. Maximum radio combining was proposed in [21], which requires the channel state information as a priori information. In [23], the spectral correlation function (SCF) is used to estimate the phase difference between the channel responses of different antennas, and the test statistic is formed by combining the SCFs of the received signals from all the antennas. Collaborative spectrum sensing [11] can be viewed as a special case of multi-antenna spectrum sensing, which drops the cyclic cross-correlations in detection.

In this paper, we propose a novel GLRT for cyclostationary multi-antenna spectrum sensing. The framework of the proposed algorithm is similar to that in [20], i.e., taking into account all the achievable cyclic auto-correlations and cyclic cross-correlations. However, it is different with [20] in hypotheses problem formulation and asymptotic covariance estimation. More specifically, the the asymptotic covariances among the cyclic cross-correlations and auto-correlations under the null hypothesis are assumed to different from those under the alternative hypothesis, while they are assumed to be the same in the Dandawate-Giannakis algorithm and its extension [20]. We use an example, where the additive noise is low-pass zero-mean Gaussian noise, to illustrate the validity and necessity of this difference and conduct the proposed GLRT. The distribution and the computational complexity of the proposed test statistic are also derived. The proposed algorithm maintains the CFAR property and has better detection performance and much lower computational complexity than those of [20]. For example, a performance gain of 1.7dB is achieved for a four-antenna system. The necessity of taking into account the cyclic cross-correlations is also demonstrated via simulation experiments.

The rest of this paper is organized as follows: In Section 2, the multi-antenna spectrum sensing model is presented. In Section 3, we propose the GLRT for cyclostationary multi-antenna spectrum sensing, and then demonstrate how to construct the proposed GLRT under the assumption that the additive noise is low-pass zero-mean Gaussian noise with uncertain power. An example is presented in Section 4 to further illustrate the proposed test. In section 5, asymptotic distribution and computational complexity of the proposed test statistic are given. Simulation results and discussions are presented in Section 6. Finally, conclusions are drawn in Section 7.

2. System Model of Multi-Antenna Spectrum Sensing

As shown in Fig. 1 we consider a CR receiver with N (N [greater than or equal to] 2) antennas. We assume an independent flat Rayleigh fading channel [24] for each pair of antennas between the transmitter and receiver in the following derivations.

At each sensing period, the CR tries to distinguish between the following two hypotheses:

[H.sub.0] : [x.sub.i](t) = [w.sub.i](t), i = 1,2, ..., N,

[H.sub.1] : [x.sub.i](t) = [h.sub.i]s(t) + [w.sub.i] (t), i = 1, 2,..., N, (1)

where s(t) is the signal transmitted by the licensed user; i denotes the index of antenna; [x.sub.i] (t) and [w.sub.i](t) denote the received signal and the low-pass zero-mean complex Gaussian noise at the i-th antenna, respectively; [h.sub.i] denotes the channel response for the i-th antenna, which can be further expressed as

[h.sub.i] = [[gamma].sub.i] exp {-j 2[pi][[theta].sub.i]}, (2)

where [[gamma].sub.i] is Rayleigh distributed with the probability density function (PDF) as

f ([[gamma].sub.i]) = [[gamma].sub.i]/[[sigma].sup.2]exp{[[gamma].sup.2.sub.i]/2[[sigma].sup.2]]}, [[gamma].sub.i]>0, (3)

and [[theta].sub.i] denotes the phase offset of the channel response, which is uniformly distributed with the PDF as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Throughout this paper, we assume that [h.sub.i] is invariant over the sensing period, and both [w.sub.i] (t) and [h.sub.i] are independent for different antennas.

3. Cyclostationarity-Based Multi-Antenna Spectrum Sensing

3.1 Cyclostationarity

A continuous-time random process x(t) is a wide sense second-order cyclostationary process, if its mean and auto-correlation are periodical with some periods [10]. Define the time varying auto-correlation of x(t) as [R.sub.xx](t, [tau]) [??] E{x(t)[x.sup.*](t + [tau])}, where [tau] denotes the lag. Then, due to its periodicity, [R.sub.xx] (t, [tau]) can be represented as a Fourier series as

[R.sub.xx](t, [tau]) = [summation over ([alpha])] [R.sup.[alpha].sub.xx] ([tau]) [e.sup.j2[pi][alpha]t] (5)

where [tau] is called the cyclic frequency and the Fourier coefficients are called the cyclic auto-correlation functions, which are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Another useful function that can characterize the second-order cyclostationary process x(t) is called the conjugated cyclic auto-correlation (CCA) function, which is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [R.sub.xx]. (t, [tau]) [??] E{x(t)x(t + [tau])}. In the rest of this paper, we focus on the case using the CCA functions, while the one concerning cyclic auto-correlation functions can be derived similarly. The discrete time version of the CCA estimation, [[??].sup.[alpha].sub.xx]. (v), can be calculated as

[[??].sup.[alpha].sub.xx] (v) = 1/M [M-1.summation over (m=0)] x [m] x [m + v][e.sup.-j2[pi][alpha]m], (8)

where M is the number of available signal samples, v is the discrete version of the lag parameter, and m is the discrete time index, i.e., x[m] = x(m[T.sub.s]), where [T.sub.s] denotes the sampling period.

As for the multi-antenna spectrum sensing problem where there are N antennas at the CR receiver, N CCA functions and N(N -1)/2 conjugated cyclic cross-correlation (CCC) functions can be obtained, which can be defined together as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that when i =j , [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the CCA functions, and when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the CCC functions. Also note that we only take into account [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], since it is easy to verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Similar to (8), a discrete time version estimation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with M samples is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

3.2 GLRT for the Multi-Antenna Cyclostationary Spectrum Sensing

Taking into account all CCA and CCC estimations, we can define a vector [[??].sub.mul] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

By using [[??].sub.mul], the hypotheses problem for multi-antenna spectrum sensing is now formulated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where [r.sub.mul] is the asymptotic value of [[??].sub.mul] under [H.sub.1], [[??].sub.mul,0] is the estimation error vector resulted from [W.sub.i] (t) and [[??].sub.mul,1] is the estimation error vector resulted from both [w.sub.i] (t) and s(t), which are asymptotically distributed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Different from the Dandawate-Giannakis algorithm and its extension proposed in [20], we do not assume that the asymptotic covariance matrix under [H.sub.0] and [H.sub.1] is the same, i.e., we assume [[summation].sub.mul,0] [not equal to] [[summation].sub.mul,1] in our hypotheses problem. Our hypotheses problem is more realistic, due to the fact that both s(t)and [w.sub.i](t) contributes to the asymptotic covariance matrix under [H.sub.1], whereas only [w.sub.i](t) contributes to [H.sub.0]. We will use an example, where the additive noise is the low-pass zero-mean complex Gaussian noise, to illustrate its validity and necessity in the next subsection.

According to the binary hypotheses given in (14), the generalized likelihood ratio (GLR) for the test problem is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By substituting [[??].sub.mul] for [[theta].sub.1] and 0 for [[theta].sub.0], we can obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Taking the logarithm transformation of the GLR gives the test statistic as follows:

[T.sub.mul] = M[[??].sub.mul] [[summation].sup.-1.sub.mul,0] [[??].sup.T.sub.mul] + 2ln det[([[summation].sub.mul,0]).sup.1/2]/det[([[summation].sub.mul,1]).sup.1/2]. (16)

Since the second term of this test statistic is of the same value under [H.sub.1] and [H.sub.0], it has no contribution to the detection and can be eliminated, which yields the final test statistic as

[T.sub.mul] = M[[??].sub.mul] [[SIGMA].sup.-1.sub.mul,0] [[??].sup.T.sub.mul]. (17)

Remark 1: The GLR test statistic only uses the asymptotic covariance matrix under [H.sub.0], [[SIGMA].sub.mul,0], and it is unrelated to the asymptotic covariance matrix under [H.sub.1], [[summation].sub.mul,1].

Remark 2: The GLR test statistic suggests that knowledge or partial knowledge of the statistics and distributions of the additive noise can be exploited to improve the detection performance. In the next section, it is shown in an example that if the additive noise is low-pass zero-mean complex Gaussian, and its bandwidth and power spectral density is known, [[summation].sub.mul,0] can be calculated.

Remark 3: If no assumption about the additive noise can be made in the detection, the estimated [[??].sub.mul] calculated from the most recent detection that was decided to be under [H.sub.0] can be employed as a good approximation of [[summation].sub.mul,0] of the current detection. Since the noise statistic does not change rapidly in practical situations, [[??].sub.mul] calculated from the recent detection is a good approximation of current [[summation].sub.mul,0]. The test statistic of Dandawate-Giannakis algorithm employs current [[??].sub.mul], whereas our proposed method suggests using the [[??].sub.mul] calculated from the most recent detection that was decided to be under [H.sub.0].

4. An Example of [[summation].sub.mul,0] Estimation and the Corresponding Test

In this section, we demonstrate how to estimate [[summation].sub.mul,0] and construct the GLRT, with the partial knowledge that the noise is a low-pass zero-mean complex Gaussian noise with uncertain power, which is a typical and practical scenario for wireless communication (it models the equivalent baseband signal of bandpass additive white Gaussian noise). Moreover, we demonstrate the necessity of assuming [[summation].sub.mul,0] [not equal to] [[summation].sub.mul,1] during the derivation of the covariance matrix estimation.

With the assumption that w(t) is a low-pass zero-mean complex Gaussian noise, the power spectral density (PSD) of w(t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where B is the bandwidth of the equivalent band-pass signal of w(t) and [N.sub.0] is assumed to be unknown due to noise power uncertainty. The auto-correlation function and the conjugated auto-correlation function of w(t) are defined as

[R.sub.ww] ([tau]) [??] E{w(t)[w.sup.*] (t + [tau])} = 2[N.sub.0]sin([pi]B[tau])/[pi][tau], (19)

and

[R.sub.ww],([tau]) [??] E{w(t)w(t + [tau])} = 0, (20)

respectively, whose discrete time versions are given by

[R.sub.ww](v) [??] E{w[m][w.sup.*][m + v]} = 2[N.sub.0] sin([pi]Bv[T.sub.s])/[pi]v[T.sub.s], (21)

and

[R.sub.ww].(v) [??] E{w[m]w[m + v]} = 0, (22)

where [T.sub.s] denotes the sampling period, v is the discrete time version of lag, and w[m] is the discrete time version of w(t), i.e., w[m] = w(m[T.sub.s]).

Theorem 1: Suppose the PSD of [w.sub.i](t) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, when i < j, the asymptotic covariance of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Proof: Following the procedure presented in the proof of Theorem 1 of [22\, and applying the PSD of [w.sub.i] (t), (23) and (24) are proved.

Theorem 2: When i = j, the asymptotic covariance of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Proof: Similar to that of Theorem 1.

It can be seen from (23) to (26) that only [N.sub.i] needs to be estimated to calculate the asymptotic covariance of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], since the cyclic frequency of interest and the lag parameters are given in prior, and the bandwidth B and the sampling period [T.sub.s] are available as for a specific CR receiver. Under [H.sub.0], when [x.sub.i] t (t) = [w.sub.i] (t), [N.sub.i] can be consistently estimated by

[[??].sub.i] = 1/M[M-1.summation over (m=0)] [x.sub.i][m][x.sup.*.sub.i] [m]/2B (27)

Under [H.sub.1], using (27) to estimate [N.sub.i] tends to yield a larger result. However, in the regime of low SNR, which is the critical situation in spectrum sensing, the error introduced is not significant. Since this approximation error only happens under [H.sub.1], it has no impact on the CFAR property of the proposed algorithm.

By substituting (27) into (24) and (26), the asymptotic covariance of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be estimated. Actually, this covariance estimation method can be viewed as a special case of that in the Dandawate-Giannakis algorithm [22]. No matter it is under [H.sub.0] or [H.sub.1], [N.sub.i] is estimated as the energy of [x.sub.i] (t). This approximation is essentially equivalent to the assumption in the Dandawate-Giannakis algorithm that the cyclic correlation estimation has the same asymptotic covariance under [H.sub.0] and [H.sub.1]. It cannot be avoided, since under [H.sub.1] it is impossible to consistently estimate [N.sub.i].

Theorem 3: The asymptotic covariance between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Proof: Similar to that of Theorem 1, omitted due to space limit.

Theorem 3 indicates that under H0 the asymptotic covariance between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be simply and correctly estimated as zero under the low-pass zero-mean complex Gaussian noise. Apparently, the asymptotic covariance between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can not be guaranteed to be equal to zero under [H.sub.1]. Therefore, it is necessary to assume that the asymptotic covariance is different under [H.sub.0] and [H.sub.1].

As for the estimation of [[summation].sub.mul,0], which contains all the asymptotic covariance among the CCA and CCC estimations, the covariance matrix estimator [20] with the Dandawate-Giannakis algorithm is not valid, since under [H.sub.1] it fails to give good approximations for the entries of [[summation].sub.mul,0] corresponding to the asymptotic covariance between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This also demonstrate the necessity to formulate the hypotheses of cyclostationary multi-antenna spectrum sensing as in (14).

Denote [[??].sub.mul,1] as the estimation of the asymptotic covariance matrix [[summation].sub.mul,0], which is of size (([N.sup.2] + N) P) x (([N.sup.2] + N)P), and can be divided into (([N.sup.2] + N) / 2) x (([N.sup.2] + N) / 2) blocks as follows, one block for each pair of ((i, j), (a,b)), where i [less than or equal to] j [less than or equal to] N and a [less than or equal to] b [less than or equal to] N:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the asymptotic covariance matrix between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; under [H.sub.0], which can be calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

where [Q.sub.ij,ab] is a P x P matrix with the (p, q) -th entry defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

According to Theorem 3, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]with (i, j) [not equal to] (a, b) is a 2P x 2P zero matrix, thus, we can rewrite [[summation].sub.mul,0] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Combining (24), (26) and the equations from (31) to (33), the estimator of [[summation].sub.mul,0], [[??].sub.mul,0], can be calculated.

In summary, the proposed algorithm for cyclostationary multi-antenna spectrum sensing, under low-pass zero-mean complex Gaussian noise with uncertain power, can be implemented using the following steps:

Step 1 Declare a cyclic frequency [alpha] and a set of lags.

Step 2 Compute the CCA and CCC estimations as in (11) and construct [[??].sub.mul] as in (12).

Step 3 Calculate [[??].sub.mul,0], by using (24), (26), and (31) to (33).

Step 4 Substitute [[??].sub.mul,0], for [[summation].sub.mul,0] in (17) and calculate the test statistic as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Step 5 Let [GAMMA] denote the threshold that satisfies the required detection performance. If [T.sub.mul,Prop] > [GAMMA], accept [H.sub.1]; if [T.sub.mul,Prop] [less than or equal to] [GAMMA], accept [H.sub.0].

5. Asymptotic Distribution and Computational Complexity of the Proposed Test Statistic

5.1 Asymptotic Distribution of [T.sub.mul,Prop]

To derive the asymptotic distribution of [T.sub.mul,Prop], we follow [11] and borrow the following theorem from [25]:

Theorem 4: Let x ~ N ([mu], V), where V is L x L nonsingular, suppose that the real L x L matrix A is symmetric, and let r (A) denote its rank. Then the quadratic form xA[x.sup.T] follows a chi-square distribution if and only if AV is idempotent, in which case xA[x.sup.T] has r(A) degrees of freedom and noncentrality parameter [mu]A[[mu].sup.T].

As for the proposed test statistic under [H.sub.0], let x = [square root of (M)][[??].sub.mul], [mu] = 0 , V = [[summation].sub.mul,0] and A = [[??].sup.-1.sub.mul,1]. Since [[??].sub.i] is a mean-square sense consistent estimation of under [H.sub.0] and [[??].sup.-1.sub.mul,0] is only determined by [[??].sub.i], [[??].sup.-1.sub.mul,0] is also mean-square sense consistent under [H.sub.0]. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the matrix product, AV , is asymptotically idempotent. The convergence in probability ([??] above) follows from application of a Cramer-Wold device [26] and from the fact that convergence in the mean-square implies convergence in probability. Hence, from Theorem 4, it follows that under [H.sub.0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

where [N.sub.mul] = ([N.sup.2] + N) / 2. Clearly, this distribution is not related to the noise power, so uncertainty of the noise power has no impact on the CFAR property of the proposed algorithm. Deriving the distribution of the proposed test statistic under [H.sub.1] needs to introduce the following definition (See p. 67-88 of [27]):

Definition 1: If x has a multivariate normal distribution, N ([mu], V), then the value of the following form (x + a)A[(x + a).sup.T] + [b.sup.T]x , where A is a square matrix, has a generalized chi-square distribution.

As for the proposed test statistic under [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Under [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], thus, the test statistic [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is asymptotically generalized chi-square distributed.

5.2 Computational Complexity

Denote the test static of the extended Dandawate-Giannakis algorithm proposed in [20] as [T.sub.mul,Dan]. The computational complexity of [T.sub.mul,Dan] focuses on the calculation of the estimated asymptotic covariance [[??].sub.mul], which is given as follows when only multiplication is considered:

[[OMEGA].sub.mul,Dan] = O ([N.sub.mul] [PM.sub.FFT] [log.sub.2] [M.sub.FFT]/2) =) ([N.sup.2]/2 [PM.sub.FFT] [log.sub.2] [M.sub.FFT]/2). (36)

where [M.sub.FFT] is the length of the Fast-Fourier transform (FFT), which is employed for the calculation of [[??].sub.mul] (Refer to [20] and [22] for details). Note that [M.sub.FFT] must not be less than the sample size, i.e., [M.sub.FFT] [greater than or equal to] M.

As for the proposed test statistic given in (34), the computational complexity is

[[OMEGA].sub.mul] = O ([N.sub.mul]PM) = O ([N.sub.2]/2PM), (37)

Apparently, the calculation of [T.sub.mul,prop] has a lower complexity compared with Tmul Dan.

6. Simulation Results

The licensed communication system considered in the following simulations is a simplified GSM system with GMSK modulated signal of symbol rate [f.sub.GSM] = 1/[T.sub.GSM] = 270.833kbit/s. The baseband GMSK modulated signal is given by

s(t) = exp {j 2[pi][h.sub.m] [[infinity].summation over (k=-[infinity])] [I.sub.k] [[integral].sup.t.sub.-[infinity]] g([tau] - k[T.sub.GSM])d[tau]}, (38)

where [I.sub.k] is the k th data symbol, and [I.sub.k] [member of] {-1,1}; [h.sub.m] = 0.5 is the modulation index; g(t) is the impulse function given by

g(t) = 1/[T.sub.GSM] rect(t/[T.sub.GSM]) * p (t) , (39)

where rect(t) is the rectangular pulse function of unit length, and p(t) is a Gaussian impulse function with the time bandwidth product [B.sub.Gaussian] [T.sub.GSM] = 0.3. We assume that all the time slots of the simplified GSM system are occupied. It has been derived in [13] that a GMSK signal exhibits cyclostationarity at the cyclic frequency [alpha] = [f.sub.GSM]/2 and the lag [tau] = 0, which are employed in the following simulations as the cyclic frequency and lag of interest, respectively.

We assume flat Rayleigh fading channels in the simulations of Fig. 3 to Fig. 8, and the average SNR in dB of the received signal is defined by

SNR(dB) = 10[log.sub.10] E[[[absolute value of(/hs(t)].sup.2])]/[P.sub.N], (40)

where [P.sub.N] is the average power of the low-pass zero-mean complex Gaussian noise. The bandwidth of the equivalent band-pass Gaussian noise is B = 223.437kHz. The sampling rate of the CR receiver is set to be 10 times the symbol rate of the GSM system, i.e., [f.sub.s] = 10 [f.sub.GSM] and [T.sub.s] = [T.sub.GSM] /10.

For the estimation of the asymptotic covariance matrix with the Dandawate-Giannakis algorithm, a length-2049 Kaiser window with [beta] parameter of 10 is used [11].

We mainly use the false alarm rate, [P.sub.f], and the detection probability, [P.sub.d], to measure the detector performance, which are defined respectively as

[P.sub.f] [??] Prob(T > [GAMMA]|[H.sub.0]), (41)

and

[P.sub.d] [??] Prob(T > [GAMMA]|[H.sub.1]), (42)

where T is the test statistic.

6.1 Distributions of the Proposed Test Statistics under [H.sub.0]

To verify the distributions of the proposed test statistics given in (35), we plot in Fig. 2 the simulated cumulative distribution functions (CDFs) of [T.sub.mul,Prop] for different numbers of antennas under [H.sub.0]. The theoretical CDFs of [chi square] distribution with corresponding degrees of freedom are also presented for comparison. The sample size is M = 4000 .

It is observed from Fig. 2 that the curves of the simulated CDFs and the theoretical CDFs nearly coincide, which confirms that the proposed test statistics given in (34) are [chi square] distributed as (35) under [H.sub.0]. Moreover, since these distributions are not related to the noise power, the CFAR property can be guaranteed for spectrum sensing based on the proposed test statistics.

6.2 Multi-Antenna Cyclostationary Spectrum Sensing

We investigate the detection performance of the proposed multi-antenna cyclostationary spectrum sensing with flat Rayleigh fading channel in this subsection. We first compare the proposed algorithm using the test statistic [T.sub.mul,Prop] with the Dandawate-Giannakis algorithm for single-antenna sensing, whose test statistic is denoted as [T.sub.Dan], and the one simply extending the Dandawate-Giannakis algorithm for multi-antenna sensing, which was given in [20] and whose test statistic is denoted as [T.sub.mul,Dan]. Next, we show the necessity of taking into account the CCCs by comparing the proposed algorithm with the one excluding all the CCCs.

Fig. 3 plots the curves of detection probability vs. false alarm rate for an average SNR of -5dB and different numbers of antennas, and Fig. 4 plots the curves of detection probability vs. average SNR with a false alarm rate of 0.01 and different numbers of antennas. The sample size of both figures is 4000. From Fig. 3 and Fig. 4, it observed that as the number of antennas increases, the detection performance increases significantly. These two figures also demonstrate that our proposed algorithm has better detection performance than the extended Dandawate-Giannakis algorithm, i.e., the detection probability using [T.sub.mul,Prop] is larger than the one using [T.sub.mul,Dan] under the same number of antennas. Most importantly, the performance gap becomes more significant as the number of antennas increases. For a four-antenna system, the performance gain is about 1.7dB, which is much more significant than that for a singleantenna system. The reason is that the proposed algorithm gives better estimation of the asymptotic covariance matrices of [[??].sub.mul] under [H.sub.0], i.e., [[summation].sub.mul,0], than the Dandawate- Giannakis algorithm does, and the improvement increases as the number of antennas and the dimension of the matrices increase, which are verified in Fig. 5.

To demonstrate the contribution of CCCs, we simulate the performance of the multi-antenna cyclostationary spectrum sensing algorithms excluding all CCCs, which uses [[??].sub.mul,CCA] as following instead of [[??].sub.mul] in the proposed algorithm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Fig. 6 plots the curves of detection probability vs. false alarm rate for the proposed algorithm without the CCCs with an average SNR of -5dB and different numbers of antennas (N = 2 and N = 4). It shows that the multi-antenna spectrum sensing algorithms that take into account all CCAs and CCCs outperform the ones excluding the CCCs, which reveals the contribution of CCCs. Fig. 7 plots the curves of detection probability vs. average SNR for the proposed algorithm without CCCs with a false alarm rate of 0.01 and different numbers of antennas (N = 2 and N = 4), and Fig. 8 shows a zoom of the important area illustrating the differences in performance more clearly. From Fig. 8, it is observed that the proposed algorithm obtains a performance gain of nearly 2.0dB over the one excluding the CCCs for four antennas. Moreover, it is observed that the performance gain provided by the CCCs is more significant for the proposed algorithm than that for the extended Dandawate-Giannakis algorithm. As the number of antennas increases, the performance gain provided by CCCs also increases for the proposed algorithm, but stays unchanged for the extended DandawateGiannakis algorithm. This demonstrates that the extended Dandawate-Giannakis algorithm fails to take full advantages of the CCCs due to the false assumption that the asymptotic covariance matrix of [[??].sub.mul] is the same under [H.sub.0] and [H.sub.1] for spectrum sensing.

7. Conclusion

In this paper, we proposed a novel GLRT algorithm for cyclostationary multi-antenna spectrum sensing. Using an example where the additive noise is low-pass zero-mean Gaussian noise, we verified that the proposed GLRT has a more reasonable hypotheses problem formulation and employs a more suitable asymptotic covariance estimator than those of the Dandawate-Giannakis algorithm and its extension. We have also derived the asymptotic distributions of the proposed test statistics, and proved the CFAR property of the proposed algorithm. Theoretical analysis and simulation results showed that the proposed algorithm has better performance of detection probability and lower computational complexity compared with the Dandawate-Giannakis algorithm and its extension. Moreover, simulation results also verified the necessity of taking into account the CCCs in multi-antenna cyclostationary spectrum sensing.

http://dx.doi.org/ 10.3837/tiis.2014.08.011

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Received December 19, 2013; revised May 12, 2014; accepted June 21, 2014; published August 29, 2014

Guohui Zhong, Jiaming Guo, Daiming Qu, Tao Jiang, and Jingchao Sun

Department of Electronics and Information Engineering Huazhong University of Science and Technology Wuhan, 430074, China

[e-mail: zhonggh@hust.edu.cn, qudaiming@hust.edu.cn]

* Corresponding author: Daiming Qu

Guohui Zhong received his B.S. degree and M.S. degree from Huazhong University of Science and Technology. He is currently a Lecturer with the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His current research interests include signal processing, and dynamic spectrum techniques for wireless communications.

Jiaming Guo received his B.E. degree in telecommunication engineering from Huazhong University of Science and Technology in 2010. He is now a PhD student at National University of Singapore in the department of ECE. His research interests include signal processing, video segmentation, pattern recognition and machine learning.

Daiming Qu received the Ph.D. degree in information and communication engineering from Huazhong University of Science and Technology, Wuhan, P. R. China, in 2003.

He is currently a Full Professor with the Department of Electronics and Information Engineering, Huazhong University of Science and Technology. His current research interests include signal processing, coding, and dynamic spectrum techniques for wireless communications.

Tao Jiang received the B.S. and M.S. degrees in applied geophysics from China University of Geosciences, Wuhan, in 1997 and 2000, respectively, and the Ph.D. degree in information and communication engineering from Huazhong University of Science and Technology,Wuhan, P. R. China, in April 2004.

He is currently a Full Professor at Wuhan National Laboratory for Optoelectronics, Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan. From August 2004 to December 2007, he worked in some universities, such as Brunel University, U.K., and the University of Michigan, respectively. He has authored or coauthored more than 100 technical papers in major journals and conferences and five books/chapters in the areas of communications. His current research interests include the areas of wireless communications and corresponding signal processing, especially for cognitive wireless access, vehicular technology, OFDM, UWB and MIMO, cooperative networks, smart grid and wireless sensor networks.

Dr. Jiang was invited to serve as TPC Symposium Chair for the IEEE GLOBECOM 2 013 and IEEE WCNC 2013, and as a General Co-Chair for the workshop of M2M Co mmunications and Networking in conjunction with IEEE INFOCOM 2011. He served or is serving as Symposium Technical Program committee membership of many major IEEE conferences, including INFOCOM, ICC, and GLOBECOM, etc. He served or is serving as Associate Editor of some technical journals in communications, including the IEEE COMMUNICATIONS SURVEYS AND TUTORIALS, IEEE TRAN SACTIONS ON VEHICULAR TECHNOLOGY, etc. He is a recipient of Best Paper Awards in IEEE CHINACOM09 and WCSP09. He is a Member of the IEEE Commu nication Society, IEEE Vehicular Technology Society, IEEE Broadcasting Society, I EEE Signal Processing Society, and IEEE Circuits and Systems Society.

Jingchao Sun received the B.E. in Electronics and Information Engineering and the M.E. in Communication and Information System from Huazhong University of Science and Technology, China, respectively. He is currently a Ph.D. student in School of Electrical, Computer, and Energy Engineering at Arizona State University. His primary research interests are network and distributed system security and privacy, wireless networking, and mobile computing.

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Author: | Zhong, Guohui; Guo, Jiaming; Qu, Daiming; Jiang, Tao; Sun, Jingchao |
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Publication: | KSII Transactions on Internet and Information Systems |

Article Type: | Report |

Date: | Aug 1, 2014 |

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