Cyclostationarity-based cooperative spectrum sensing over imperfect reporting channels

Int. J. Electron. Commun. (AEÜ) 66 (2012) 833–840

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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue

Cyclostationarity-based cooperative spectrum sensing over imperfect reporting channels Hamed Sadeghi, Paeiz Azmi ∗ , Hamid Arezumand Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 26 July 2011 Accepted 27 February 2012 Keywords: Cooperative spectrum sensing Cyclostationary Cognitive radio Primary user detection

a b s t r a c t Reliable detection of weak primary user signals is a crucial problem for cognitive radio networks. To address the above issue, cooperative spectrum sensing (CSS) methods based on cyclostationary detection (CD) have been introduced in the literature. In this paper, a soft decision-based CSS method based on the second-order CD at secondary users (SUs) is proposed. The proposed scheme aims to maximize the deflection criterion at the fusion center (FC), while the reporting channels are characterized by Rayleigh fading. To this end, a fusion rule which does not require to know the noise variances of sensing channels is developed. Since the fusion rule assumes the perfect knowledge of channel state information (CSI) of reporting links, it has theoretical significance and provides an upper bound for the performance of cyclostationarity-based CSS. We have also proposed a more practical suboptimum fusion rule and studied its detection performance in the presence of uncertainties in noise variance and channel power gain estimations. Furthermore, in order to be able to evaluate the performance of the CSS, an analytic threshold estimation method has been proposed. Extensive simulation results have been illustrated the robustness of the proposed method compared to the existing cyclostationary detectors. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction Cognitive radio (CR), first introduced by Mitola in 1999 [1], provides a way to use the valuable radio spectrum in an efficient manner. These radios are actually unlicensed wireless devices that temporarily utilize the unused primary spectral bands [2,3]. However, the first step in opportunistic access to the licensed spectrum is the detection of unused spectral bands [4]. In addition, CR should vacate the primary band as soon as a primary user (PU) starts transmitting. Briefly speaking, reliable PU detection in low SNR regime is one of the main challenges to cognitive radio networks. Although cognitive radios should reliably detect the presence of PUs, it has been shown that the local spectrum sensing may not provide enough reliability in fading environments [5]. To address this problem, cooperative spectrum sensing (CSS) methods were proposed in the literature to improve the detection performance of secondary networks [6–8]. However, most existing scenarios assume that all the cooperating users use the commonly adopted energy detection (ED) technique for their local sensing. Nevertheless, despite the simplicity, quickness and no requirements of pre-knowledge about the PU’s signal, the ED method has some challenging issues. For example, it cannot differentiate PUs from secondary users (SUs), and requires to know the noise variance

∗ Corresponding author. Tel.: +98 21 82883303. E-mail address: [email protected] (P. Azmi). 1434-8411/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2012.02.004

to ensure proper detection performance [9]. Even though the cyclostationary detection (CD) method can solve these problems [10–12,8,13–16], there are rather limited developments in CSS strategies based on CD. Recently, soft decision-based CSS methods based on ED at SUs have been introduced in the literature [6,17–20]. These methods assume that the perfect knowledge about the noise variances of sensing channels are available at the fusion center (FC). However, in practice, this assumption may be unrealistic. Therefore, the CSS methods based on local ED may be very susceptible to noise uncertainties and thus their performance can be dictated by the accuracy of the noise power estimate. Furthermore, in these studies, it is assumed that the perfect knowledge of noise variances and channel states of reporting links are available at the FC. To overcome the above issues, CSS based on CD has been proposed in the literature [8,21,22]. Cyclostationarity-based detection methods exploit inherent cyclostationary properties of digitally modulated signals and have acceptable performance in very low SNRs [11,8]. Since CD can distinguish PUs from SUs and also is robust against noise uncertainty, the CD-based CSS can achieve an acceptable detection performance in low-SNR conditions, while the performance is not degraded due to the noise uncertainty problem at cognitive radios. Despite the advantages of CD over ED-based spectrum sensing, there are rather limited researches on CD-based CSS, mostly because of its complicated analytical expressions and also complexities that may arise in implementing the CD algorithms.

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Fig. 1. System model for the cyclostationarity-based cooperative spectrum sensing over imperfect reporting channels.

The main objective of this paper is to develop a cyclostationarity-based CSS framework for reliable detection of PUs when the reporting and observation channels are under fading. To this end, we first provide an overview of the Dandawaté–Giannakis’s algorithm [11,8] as the local sensing method. This statistical test does not require any specific assumption about the distribution of PU signals. Based on this local sensing method, we then propose a weighted soft combination method based on deflection criterion maximization at the FC and show that our approach achieves better detection performance compared to the conventional cooperative detectors. Note that our proposed method does not require any prior knowledge about the noise variances of sensing channels or their fading statistics. Since the proposed weighted soft combination method requires the knowledge of some parameters that may be unknown at the FC, we further simplify it heuristically, while showing a comparable performance to the previous one. Because there is not a closed-form expression for the probability density function (PDF) of the proposed decision statistic, we first derive its characteristic function and then propose a numerical inversion method for evaluating the corresponding distribution function. Simulation results show that the proposed analytical threshold setting procedure provides adequate accuracy for performance analysis purposes. The remainder of this paper is organized as follows. System model is presented in Section 2. The local sensing strategy and the proposed cooperative spectrum sensing method are described in Section 3. Performance of proposed scheme is investigated in Section 4. Finally, conclusions are drawn in Section 5.

2. System model It is assumed that the base-band discrete-time received signal for ith SU, xi (n), i = 1, 2, . . ., L, at a time instance n is given by:

After the decision statistic Ti at the ith SU is computed, it is transmitted to the FC through an independent reporting channel that experiences flat fading. Hence, yi = hi Ti + zi ,

i = 1, . . . , L,

(2)

zi ∼N(0, i2 )

and hi is a real-valued fading envelope with where hi > 0, which is assumed to be constant during the sensing period. Without loss of generality, we assume that the fading channels have unit powers (i.e. E{h2i } = 1). In many cognitive radio scenarios, the envelope of the fading channel and the noise variance can be estimated in advance. Thus, at the first step, the quantities {hi }Li=1 and {i2 }Li=1 are assumed to be perfectly known to the FC. Then, the impact of presence of uncertainties in the estimations of the above quantities is studied. The whole system model is depicted in Fig. 1. 3. The proposed cyclostationarity-based cooperative sensing method In this section, we consider the proposed CSS method which is based on the second-order cyclostationary detection at each SU. 3.1. Overview of non-cooperative cyclostationary detection method In this section, we only present the key steps of Dandawaté–Giannakis’s algorithm and omit details. For more detailed background on this algorithm and its implementation, the readers are referred to [11,12,8]. Assume that we want to test for the presence of the cyclostationarity at a candidate cycle frequency ˛ (known prior or can be estimated [11,23]) in the received signal x(n). For a given set of time lags {i }N , the estimated cyclic autocorrelation vector rˆ xx∗ is i=1 defined to be [11] ˛ ˛ ˛ ˆ xx ˆ xx rˆ xx∗  [Re{Rˆ xx ∗ (1 )}, . . . , Re{R ∗ (N )}, Im{R ∗ (1 )}, . . . , ˛ Im{Rˆ xx ∗ (N )}],

xi (n) = gi x˜ (n) + wi (n),

n = 1, . . . , M,

(1)

where L refers to the number of SUs existing in the network, gi denotes the channel fading coefficient between PU and ith SU, and wi (n)∼CN(0, ı2i ) with ı2i as the variance of the complex additive Gaussian noise. Note that ı2i and gi are generally unknown. Moreover,  = 0 and  = 1 correspond to null (inactive PU) and alternative (active PU) hypotheses, respectively. We assume that the PU is either active or inactive during the sensing duration. The signal transmitted by PU is denoted by x˜ (n). Without loss of generality, x˜ (n), gi and wi (n) are assumed to be independent of each other. Furthermore, conditional independence of spatially distributed SUs is assumed [8].

in which

˛ Rˆ xx ∗ (i ) 

⎧ M ⎪ ⎨ 1 M

(3)

x(n)x∗ (n + i )e−j2n˛ ,

⎪ ⎩ ˆ ˛n=1 (Rxx∗ (−i ))∗ ,

i ≥ 0

(4)

i < 0

denotes the estimated cyclic autocorrelation function (CAF) at the time lag  i and the cycle frequency√˛. It has been proven that subject to certain mixing conditions, Mˆ rxx∗ is asymptotically (i.e. as D def√ ˆ ∗ M→ ∞) distributed as u = M rxx ∼N(, xx∗ ), where =0 under √ D H0 , and = Mrxx∗ under H1 [8]. Here, ∼ denotes the convergence in distribution, and N(, V ) denotes the multivariate normal distribution with mean vector  and covariance matrix V . The

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asymptotic covariance matrix xx∗ can be computed using the formulae in Appendix A. In addition, rxx∗ is the true value of rˆ xx∗ , which is unknown to SUs but is non-random [10,11]. Exploiting the above-noted asymptotic distribution for rˆ xx∗ , one can obtain the following generalized log-likelihood ratio (GLLR) [11,8]: a



22N , 2 (), 2N

H0 , H1

(5) λ

−1 T

ˆ ∗ rˆ ∗ ∼ TG = Mˆrxx∗  xx xx a

where ∼ represents an asymptotic probability density function (PDF), 22N denotes a chi-squared PDF with 2N degrees of freedom, and 2 () denotes a noncentral chi-squared PDF with r degrees of 2N ˆ xx∗ freedom and noncentrality parameter . In the above equation,  is the estimated covariance matrix. The noncentrality parameter is −1 ˆ ∗ rT ∗ , where the superscript (·)T represents given by  = Mrxx∗  xx

xx

the transpose operator. It is worth noting that the true value of noncentrality parameter is unknown to the SU. The false alarm probability can be obtained as Pfa = Q2 (), where  is the decision threshold, and Q2 tail probability for distribution probability can be derived as:



N−1  k (/2)

1 Pfa = exp −  2

k!

22N

denotes the right-

2N

[24]. Therefore, the false alarm

.

τ

2N

(6)

k=0

Fig. 2. Comparison between the exact and approximated ML estimation of noncentrality parameter of 2 (). 2N

likelihood estimation of the noncentrality parameter at the ith SU as ˆ i,ML = arg maxf (Ti |H1 ), 

In this section, we propose a CSS scheme where soft cyclostationary decisions transmitted by several SUs are combined at the FC. Following Section 3.1, we propose to use the Dandawaté–Giannakis’s algorithm at all the SUs, aiming to simultaneously test for a candidate cycle frequency ˛. We denote the GLLR test statistics computed at SUs by Tj , j = 1, . . ., L. We propose the weighted combination decision rule [17,6] for fusing the soft decisions received from SUs: y˜ =

L  i=1

T

where f (Ti |H1 ) =

H1

 (N−1)/2

1 (−(T + ))/2 Ti i i e 2 i



IN−1 (

i Ti ),

Ti > 0.

(11)

In the above equation, I (·) is a modified Bessel function of the first kind and order [25]: I (x) =

wi yi = w y   ˜

(10)

i

3.2. The proposed test for cooperative detection

(7)

∞ 2r+  ((1/2)x) r=0

r! ( + r + 1)

,

(12)

H0

where w = [w1 , w2 , . . . , wL ]T ,

wi ≥ 0

(8)

is the weight vector used to build the weighted fusion rule, and y = [y1 , y2 , . . . , yL ]T

(9)

denotes the vector of observations. 3.2.1. Weight vector computation The problem that we are encountering is how we should calculate the weight vector in (7), thereby the resultant detector achieves the best performance. To this end, we choose the deflection coefficient [7,24,20] as a performance metric. In what follow, we derive the optimal weight vector for one sensing slot, as defined in (1). In each sensing slot, the weight vector should be updated according to the new PU activity and/or channel conditions. We supposed that the PU activity does not change in a sensing slot. First- and second-order moments of y˜ are required for derivation of the deflection of the cooperative detector. However, computation of these moments under H1 requires the knowledge of values of noncentrality parameters at the SUs. To overcome this problem, we propose that each SU directly estimates the noncentrality parameter from the received signal. To this end, let us define the maximum

where function (·) is the Gamma function. However, Eq. (10) can not be solved analytically and therefore numerical methods should be used. A simple alternative solution is provided in [25, Eq. (29.45h)]. Considering this method, we can approximately compute the noncentrality parameter under H1 as follows:

 ˆi = 

Ti − 2N, (N + 1)−1 Ti ,

if Ti ≥ 2N + 2 if Ti < 2N + 2

(13)

Fig. 2 shows the accuracy of the approximation method proposed in (13). It is evident that the approximation (13) follows the ML estimation very closely. Note that a common approach to the approximation of the noncentrality parameter is based on the method of [11]. The authors in [11] stated that the alternative distribution in (5) can be approximated through the substitution of rxx∗ by rˆ xx∗ . In other words, −1

ˆ ∗ rˆ T ∗ , ˆ i Mˆrxx∗   xx xx

(14)

ˆ i Ti . As it is shown in Fig. 2, this approximation or equivalently,  is just a simplification of (13) and is valid only for small values of N. Furthermore, we will show in Section 4 that the approximation (13) provides better accuracy than (14) for analytical performance evaluations.

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After estimating the noncentrality parameters, we can find the required statistics as:

⎧ L  ⎪ ⎪ ⎪ E{˜y|H0 } = 2N wi hi ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ L ⎨  E{˜y|H1 }

ˆ i) wi hi (2N + 

⎪ ⎪ i=1 ⎪ ⎪ L ⎪  ⎪ ⎪ 2 ⎪ var{˜ y |H } = wi2 (4Nhi + i2 ) = w T mw, ⎪ 0 ⎩

(15)

i=1

where



m = 4Nh2 + diag() h = diag([h1 , h2 , . . . , hL ]).

(16)

Here, diag(·) denotes a square diagonal matrix with the elements of the given vector on its diagonal. In the same equation,   [12 , 22 , . . . , L2 ]T represents the vector of reporting channel variances. After some manipulation, we can obtain the deflection of cooperative detector as:

We call this heuristic proposition the weight vector with the minimal pre-knowledge. Furthermore, the suboptimal equal gain com√ biner (EGC) is obtained by substituting w EGC = [1, 1, . . . , 1]T / L into (7). Note that the suboptimum rule (21) does not require the knowledge of reporting channel fading envelopes and noncentrality parameters. The only required information are the noise variances {i2 }Li=1 and the power gains {h2i }Li=1 of reporting links for computing m in (16). These power values can be estimated in practice if the SUs periodically transmit pilot tones or training sequences at known transmit powers in the secondary network [6,18,20]. Furthermore, we have assumed that the channel coherence time is much larger than the channel estimation period. Intuitively, the expression (21) reduces the contribution of those received signals that have low sensing channel SNR (since the noncentrality parameter would have a small value in low SNR conditions, this impacts on the value of yi ) and/or reporting channel SNR (because of the term m−T/2 m−1/2 ), in the final decision made at the FC. As it will be shown later, our simulation results support the heuristic proposition (21) and show a significant performance gain over EGC. In addition, it has a relatively close performance to ˆ vector. that of the CSS with the known ␭

2

D(w) 

ˆ (E{˜y|H1 } − E{˜y|H0 })2 (w T h␭) = , T var{˜y|H0 } w mw

(17)

ˆ = [ ˆ 1,  ˆ 2, . . . ,  ˆ L ]T is the vector of estimated noncentralwhere ␭ ity parameters at different SUs. We rewrite (17) and define the optimal weight vector w opt as the one that meets the following optimization problem: w opt = arg

max

w≥0, w 2 =1

˜ w T mw , w T mw

(18)

T

ˆˆ T ˜  h␭ where m L ␭2 h1/2 and · 2 denotes the Euclidian norm (i.e.

w 2 = ( i=1 wi ) ). In order to achieve a unique solution for the optimization problem, we have confined the weight vector to have a unit norm. If m1/2 be the square root obtained from the Cholesky decomposition of m, using w = m−T/2 v we get:

3.3. Threshold selection at the FC In this section, we introduce a method to estimate the decision threshold in (7) for a given probability of false alarm. Since an analytical closed-form expression for the null distribution of the test statistic does not exist, we propose to approximate the distribution function by the numerical inversion of corresponding characteristic function (CF). Under null hypothesis, the CF of a random variable X is X|H0 (t)  E{ejtX |H0 }. From (2), the CF of received statistic can be obtained as



yi |H0 (t) = (1 − 2jhi t)−N exp

Applying the fact that y˜ |H0 (t) =



T

vT (m−1/2 h␭ˆ ␭ˆ hT m−T/2 )v D(v) = , vT v

(19)

tor as: ˆ m−T/2 m−1/2 h␭ . ˆ

m−T/2 m−1/2 h␭

(20)

Note that the above fusion rule has theoretical significance since it provides the best achievable detection performance to which any suboptimal approach requires to be compared. Computation of the proposed weight vector (20) requires the ˆ and fading envelopes at the FC. To address a more knowledge of ␭ ˆ practical fusion rule, we alternatively propose to substitute the ␭ T vector with  = [y1 /h1 , y2 /h2 , . . ., yL /hL ] . In fact, the linear minimum mean-square estimation of Ti from the received signal yi can ˆ i Ti [11], be expressed as yi /hi . Then, using the approximation  ˆ Note that the only we can achieve  as an approximation for ␭. information available in practice at the FC are the received signals {yi }Li=1 . Doing so, the resulting suboptimum weight vector can be rewritten as m−T/2 m−1/2 y

m−T/2 m−1/2 y 2

.

.

2

L k=1

−N

L 

(1 − 2jwk hk t)

yk |H0 (wk t), we can write:

 exp

(22)

−t 2  2 2 wk k 2 L

(21)

 (23)

k=1

A usual approach for obtaining the cumulative distribution function (CDF) is to evaluate the Gil-Pelaez inversion integral [27]. Doing so, we can compute the following expression for the false alarm probability at the FC:



˜ 0 Pfa = Pr y˜ > |H

2

˘M = w

y˜ |H0 (t) =

i2 t 2

k=1

which is in the Rayleigh’s quotient form [26]. Hence, vopt = ˆ and normalizing the result gives the optimal weight vecm−1/2 h␭

w opt =

−



where

Fy˜ |H0 ( ) =

1 − 2



= 1 − Fy˜ |H0 (), ˜



∞

−∞

L

exp (−t 2 /2)



(24)

k=1

wk2 k2

N

L (1 − 2jwk hk t) k=1

e−jt dt. 2jt

(25)

Since a closed-form result for this integral can not be obtained, we numerically estimate (25) by employing an algorithm introduced in [28]. This technique approximately calculates the distribution function of a standardized (i.e. zero-mean unit-variance) random variable, when its characteristic function is known. Following the strategy discussed in [28], we first define the standardized test statistic Z

y˜ − E{˜y|H0 } , var{˜y|H0 }

(26)

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so the approximate distribution function of Z can be obtained as: ˆ Z|H (z) F 0

H−1 

1

z + − 2 2

Z|H0 ( ) 2j

=1−H = / 0

e−j z ,

(27)

where is a constant variable which ensures that the full range of FZ|H0 (z) is considered (i.e. it includes 0 and 1) and H defines the number of points used in the approximation of CDF. The values for z may be chosen as the Fourier frequencies, that is, zk = (2(k − H)/2 (H − 1)) for k = 1, 2, . . ., 2H − 1. The sum in (27) may be efficiently calculated using the fast Fourier transform, if the  = / 0 term is subtracted from the result. Furthermore, the characteristic function Z|H0 ( · ) in (27) can be calculated as follows: L  ((−j2N

Z|H0 (t) = e

wi hi )/(w T mw)1/2 )t

y˜ |H0 ((w T mw)−1/2 t)

i=1

(28) Fig. 3. Local detection performance curves for different SNR values over frequencyflat Rayleigh fading channels.

Finally, the intended distribution function can be obtained as





(w T mw)−1/2

ˆ y˜ |H ( ) = F ˆ Z|H F 0 0

−

L 



wi hi

.

(29)

i=1

Therefore, the global decision threshold in (7) can be analytically determined as −1

ˆ y˜ |H (1 − Pfa ).  ˜ F 0

(30)

4. Simulation results and discussions In this section, we evaluate the performance of the proposed CSS method. The simulated primary signal is a DVB-T ((Digital video broadcasting, Terrestrial television) signal with 64-QAM subcarrier modulation. Following the settings in DVB standard, we set the values of FFT length, number of occupied channels, and the length of guard interval as Nfft = 8192, Nocc = 6817, and Ng = 1024, respectively. The transmission mode is selected to be 8 K mode and carrier frequency is set to 750 MHz. It is assumed that Pfa =0.01, L=10 and the sensing duration is 3 OFDM symbols. It is well-known that the peaks of the cyclic autocorrelation function of an OFDM signal occurs at  = ± Nfft and ˛ = k/(Nfft + Ng ) for integer k [10]. In all simulations, each SU uses the cycle frequency ˛ = 1/(Nfft + Ng ), the time-lag  = Nfft . Furthermore, the cyclostationary detectors employ a Kaiser window with length P = 2048 and ˇ = 10 (see Appendix A). The average signal-to-noise ratio (SNR) in the ith observation channel is defined as SNRi = 10 log10 (ı˜ 2x /ı2i ), where ı˜ 2x is the variance of the PU’s signal. In all CSS scenarios, the vector of reporting channel variances is set to  = [1.9, 1.2, 4.6, 2.2, 3.7, 1.5, 1.0, 1.7, 1.8, 2.8]T . In each simulation run, fading coefficients of reporting channels are estimated by the FC for use in weight vector computations. 4.1. Performance of local sensing For N=1, the decision threshold introduced in (6) can be reduced to  = − 2 ln (Pfa ). Therefore, the probability of missed detection can be obtained as: Pm = Pr{TG < |H1 } = F () (−2 ln Pfa ), 2

where



x

F () (x) = 2

−∞



(31)



1 t+ I0 ( t) dt. exp − 2 2

(32)

In the above equation, I0 (·) is the modified Bessel function of the first-kind and order-zero [29]. Fig. 3 presents the complementary receiver operating characteristics (CROC) curves for the non-cooperative spectrum sensing over frequency-flat Rayleigh fading channel, where the SU is assumed to be mobile with speed 5 km/h. Analytical curves are obtained from (31) and substitution of  by the approximations (13) and (14), respectively. It is evident that the analytical curves obtained using (13) leads to better reliability as compared with the curves obtained using (13). 4.2. Performance of cooperative sensing Since in practical situations the SUs do not know the true noncentrality parameter, the first step in the proposed CSS method is that each SU should estimate the unknown noncentrality parameter based on its local observations. The accuracy of the ML estimation of the noncentrality parameter as well as the proposed approximation (13) is illustrated in Fig. 4. In Fig. 4(a), the SU estimates the noncentrality parameter from the received signal affected by AWGN. In Fig. 4(b), the noncentrality parameter is estimated from the received signal through a frequency-flat Rayleigh fading channel. As we can see, the approximation (13) follows the ML estimation very closely, especially in low-SNR conditions that are of our interest. The true noncentrality parameter is computed −1 ˆ ∗ rT ∗ . The ML estimate is obtained from numerical evalas Mrxx∗  xx

xx

uations. All curves are averages over 4000 independent trials. It is obvious that the proposed approximation achieves better reliability as compared to the method of [11]. Fig. 5 represents the CROC curve for the proposed method. Each SU has a different average channel SNR for its observation channel. We assume that  = [ − 16, − 12, − 8, − 4, , , + 4, + 8, + 12, + 16], where  = [SNR1 , . . . , SNRL ] is the vector of average channel SNRs and is the arithmetic mean value of all the channel SNRs in decibels. As it is evident, the proposed method significantly outperforms the EGC as well as the non-cooperative detection methods. We observe that the cooperative CD with minimal pre-knowledge follows the optimal fusion. Furthermore, the EGC does not necessarily result in performance improvement as compared to the non-cooperative sensing. The accuracy of the employed threshold estimation method is investigated in Fig. 6. As we can see, the thresholds obtained from numerical inversion of characteristic functions are almost accurate. In addition, to justify the reason of the close performance of the proposed

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a

b

Fig. 5. Complementary ROC performances of proposed cyclostationarity-based cooperative sensing as compared with non-cooperative and equal gain combining methods. Sensing channels are under frequency-flat Rayleigh fading. All cooperative sensing curves are simulated with  = [ − 16, − 12, − 8, − 4, , , + 4, + 8, + 12, + 16].

test statistic T with a uniform quantizer and sends the quantized decision Tq, to the FC. Here, we do not deal with the optimization of the local quantization levels. We can observe that the 4-bit quantization is sufficient, while the 5-bit quantization is practically indistinguishable from the analog transmission. 4.4. Impact of imperfect CSI

Fig. 4. Comparison between the true value, ML estimation, and proposed approximations given in (13) and (14) for the noncentrality parameter over (a) AWGN channel and (b) frequency-flat Rayleigh fading channel.

Sensitivity of our proposed method to the lack of precise knowledge of channel gain powers and noise variances is investigated in Fig. 10. We assume h2i = h · h˜ 2i and i2 =  · ˜ i2 , where h˜ 2i and ˜ i2 denote the nominal values (i = 1, 2, . . ., L). In addition,  h and   denote the uncertainty factors of channel power gains and noise variances, respectively. Results exhibit good robustness against the presence of uncertainty in the noise variance and the channel power gain estimations. Furthermore, the CSS with 2 dB uncertainties provides good detection performance compared to the local

suboptimum curve to the optimal one, the mean-squared error (MSE) between the optimal and suboptimal weights of each SU is the same simulation. For each SU, the MSE is computed measured in It as E = (1/It ) i=1 |ω ˘ M,i − ωopt,i |2 , where It denotes the number of iterations. As we can see in Fig. 7, the difference between the computed optimal and suboptimal weights is negligible. For this reason, the detection curve of cooperative CD with minimal pre-knowledge follows relatively closely the optimal fusion curve in Fig. 5. However, it should be noted that the difference between the optimal and suboptimal weights is a function of the noise powers of the reporting links. In other words, considering higher noise powers for the reporting channels may lead to more considerable MSEs in Fig. 7. Performance of the proposed schemes over Rician fading channels is investigated in Fig. 8. It is assumed that sensing channels seen by different SUs are independent, and each one has a Rician factor K = 10 dB. As we can see, the proposed weighted scheme offers significant performance gain over local detectors and EGC. 4.3. Impact of quantization Sensitivity of our proposed method to quantization process is investigated in Fig. 9. It is assumed that the th SU quantizes its GLLR

Fig. 6. Comparison between the proposed analytical threshold estimation method and the exact threshold values obtained from numerical simulation. The parameters are the same as Fig. 5.

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Fig. 7. Mean-squared error between the computed optimal and suboptimal weights at the FC, associated to different SUs in the network.

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Fig. 10. Impact of imprecise knowledge of power estimations. Sensing channels undergo frequency-flat Rayleigh fading with fd =150 Hz ( = −22 dB for cooperative cases).

sensing with the highest sensing channel SNR. Note that 1 dB uncertainty is a typical value in the spectrum sensing scenarios [8]. 5. Conclusions We proposed a weighted cyclostationarity-based cooperative spectrum sensing scheme which has adequate robustness to the variations introduced by fading impairments of both the reporting and sensing channels. We further derived a cooperative fusion rule that only needs to know the noise variances and fading power gains of the reporting channels. In addition, an efficient method for threshold setting and performance evaluation is proposed. The impact of quantization of the decision statistics was investigated. Simulation results showed that the detection performance is not significantly affected if a reasonable number of quantization bits is used. Furthermore, numerical simulations showed that the proposed cooperative method is relatively robust against the presence of uncertainties in fading and noise power estimations. Fig. 8. Complementary ROC performances of the proposed cyclostationarity-based cooperative sensing methods over Rician fading channels with K = 10 dB.

Acknowledgments The authors acknowledge the partial research support provided by the Research Institute for ICT (ITRC) under project number of T8755-500 and would also like to thank the anonymous reviewers for their valuable comments and helpful suggestions which greatly improved the manuscript. Appendix A. Computation of covariance matrix of rˆxx∗ The (2N × 2N)-dimensional asymptotic covariance matrix xx∗ is defined as below [11]: xx∗

1 = 2



Re{Q + S} Im{Q + S}

Im{Q − S} Re{S − Q }



,

(A.1)

where the (m, n)th entries of the two covariance matrices Q and S are given by (1 ≤ m, n ≤ N): ˛ ˛ ˆ xx Q mn = lim M cov{Rˆ xx ∗ (m ), R ∗ (n )}

Fig. 9. Comparison of analog forwarding with quantized decision transmission for the cooperative CD with minimal pre-knowledge. Sensing channels are under frequency-flat Rayleigh fading ( = −22 dB).

(A.2)

M→∞

∗

˛ ˛ ˆ xx S mn = lim M cov{Rˆ xx ∗ (m ), (R ∗ (n )) }. M→∞

(A.3)

840

H. Sadeghi et al. / Int. J. Electron. Commun. (AEÜ) 66 (2012) 833–840

In practice, these elements can be estimated respectively by:





(P−1)/2

1 Qˆ mn = MP

WP (s)Fn

ϑ−

s M

ϑ+

s M



 Fm

ϑ+

s M

ϑ+

s M

,

(A.4)

,

(A.5)

s=−(P−1)/2



(P−1)/2

1 Sˆ mn = MP

WP (s)F∗n





 Fm



s=−(P−1)/2

where F (ϑ) =

M 

x(t)x∗ (t + )e−j2ϑt ,

(A.6)

t=1

and WP is a normalized smoothing window with an odd length P. References [1] Mitola J. Cognitive radio: an integrated agent architecture for software defined radio. PhD thesis. Sweden: KTH Royal Institute of Technology; 2000. [2] Cabric D, Mishra SM, Brodersen RW. Implementation issues in spectrum sensing for cognitive radios. In: Proc 38th Asilomar conf signals, systems and computers, vol. 1. 2004. p. 772–6. [3] Sridhara K, Chandra A, Tripathi PSM. Spectrum challenges and solutions by cognitive radio: an overview. Wireless Pers Commun 2008;45(3):281–91. [4] Arslan H. Cognitive radio, software defined radio, and adaptive wireless systems. Springer; 2007. [5] Letaief KB, Zhang W. Cooperative communications for cognitive radio networks. Proc IEEE 2009;97(5):878–93. [6] Quan Z, Cui S, Sayed AH. Optimal linear cooperation for spectrum sensing in cognitive radio networks. IEEE J Sel Top Signal Process 2008;2(1):28–40. [7] Unnikrishnan J, Veeravalli VV. Cooperative sensing for primary detection in cognitive radio. IEEE J Sel Top Signal Process 2008;2(1):18–27. [8] Lunden J, Koivunen V, Huttunen A, Poor HV. Collaborative cyclostationary spectrum sensing for cognitive radio systems. IEEE Trans Signal Process 2009;57(11):4182–95. [9] Tandra R, Sahai A. Fundamental limits on detection in low SNR under noise uncertainty. In: Proc int conf wireless networks, commun and mobile computing, vol. 1. 2005. p. 464–9. [10] Oner M, Jondral F. Air interface identification for software radio systems. Int J Electron Commun (AEU) 2007;61(2):104–17. [11] Dandawaté AV, Giannakis GB. Statistical tests for presence of cyclostationarity. IEEE Trans Signal Process 1994;42(9):2355–69. [12] Dandawaté AV, Giannakis GB. Asymptotic theory of mixed time averages and kth-order cyclic-moment and cumulant statistics. IEEE Trans Inf Theory 1995;41(1):216–32. [13] Lunden J, Koivunen V, Huttunen A, Poor HV. Spectrum sensing in cognitive radios based on multiple cyclic frequencies. In: Proc 2nd int conf cognitive radio oriented wireless network commun (CrownCom). 2007. [14] Sutton PD, Nolan KE, Doyle LE. Cyclostationary signatures in practical cognitive radio applications. IEEE J Sel Top Signal Process 2008;26(1):13–24. [15] Arezumand H, Azmi P, Sadeghi H. A robust reduced-complexity spectrum sensing scheme based on second-order cyclostationarity for OFDM-based primary users. In: Proc 19th Iranian Conf Elect Eng (ICEE). 2011. [16] Sadeghi H, Azmi P. A novel primary user detection method for multiple-antenna cognitive radio. In: Proc Int Symp Telecommun (IST 2008). 2008. p. 188–92. [17] Ma J, Zhao G, Li YG. Soft combination and detection for cooperative spectrum sensing in cognitive radio networks. IEEE Trans Wireless Commun 2008;7(11):4502–7. [18] Taricco G. Optimization of linear cooperative spectrum sensing for cognitive radio networks. IEEE J Sel Top Signal Process 2011;5(1):77–86.

[19] Taricco G. On the accuracy of the Gaussian approximation with linear cooperative spectrum sensing over Rician fading channels. IEEE Signal Process Lett 2010;17(7):651–4. [20] Shen B, Ullah S, Kwak K. Deflection coefficient maximization criterion based optimal cooperative spectrum sensing. Int J Electron Commun (AEU) 2010;64(9):819–27. [21] Derakhshani M, Le-Ngoc T, Nasiri-Kenari M. Efficient cooperative cyclostationary spectrum sensing in cognitive radios at low SNR regimes. IEEE Trans Wirelss Commun 2011;10(11):3754–64. [22] Sadeghi H, Azmi P. Cyclostationarity-based cooperative spectrum sensing for cognitive radio networks. In: Proc Int Symp Telecommun (IST 2008). 2008. p. 429–34. [23] Mosquera C, Scalise S, Lopez-Valcarce R. Non-data-aided symbol rate estimation of linearly modulated signals. IEEE Trans Signal Process 2008;56(2):664–75. [24] Kay SM. Fundamentals of statistical signal processing, vol. II: Detection theory. Prentice Hall; 1998. [25] Johnson NL, Kotz S, Balakrishnan N. Continuous univariate distributions, vol. 2. 2nd ed. New Jersey: John Wiley and Sons; 1995. [26] Seber GAF. A matrix handbook for statisticians. New Jersey: John Wiley and Sons; 2008. [27] Gil-Pelaez J. Note on the inversion theorem. Biometrika 1951;38(3/4): 481–2. [28] Waller LA, Turnbull BW, Hardin JM. Obtaining distribution functions by numerical inversion of characteristic functions with applications. Am Stat 1995;49(4):346–50. [29] Proakis JG, Salehi M. Digital communications. 5th ed. New York: McGraw-Hill; 2008. Hamed Sadeghi was born in Tehran, Iran, in 1983. He received the M.Sc. degree in communication systems engineering from Tarbiat Modares University, Tehran, Iran, in 2008. He is currently pursuing the Ph.D. degree in communication systems engineering at Tarbiat Modares University. He is a student member of IEEE. His research interests are in the areas of statistical signal processing and its applications, including detection, estimation, data fusion, and spectrum sensing. Paeiz Azmi was born in Tehran-Iran, on April 17, 1974. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Sharif University of Technology (SUT), Tehran-Iran, in 1996, 1998, and 2002, respectively. Since September 2002, he has been with the Electrical and Computer Engineering Department of Tarbiat Modares University, Tehran-Iran, where he became a professor on December 2011. From 1999 to 2001, he was with the Advanced Communication Science Research Laboratory, Iran Telecommunication Research Center (ITRC), Tehran, Iran. From 2002 to 2005, he was with the Signal Processing Research Group at ITRC. Hamid Arezumand was born in Tehran, Iran, on September 18, 1985. He is currently working toward the M.Sc. degree in Tarbiat Modares University, Tehran, Iran. His research focus is on spectrum sensing in cognitive radio networks.