Decentralized Adaptive Eigenvalue-Based Spectrum ...

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Decentralized Adaptive Eigenvalue-Based Spectrum Sensing for Multi-antenna Cognitive Radio Systems Christos G. Tsinos, Member, IEEE, and Kostas Berberidis, Senior Member, IEEE

Abstract—Eigenvalue-Based Spectrum Sensing (EBSS) techniques may operate in a totally blind manner while they offer remarkably improved performance for specific types of signals compared to energy-based methods. In the literature so far, only batch and centralized cooperative EBSS techniques have been considered which, however, suffer from limitations that render them impractical in several cases. Thus, the aim of the present paper is to develop practical cooperative adaptive versions of typical EBSS techniques which could be applied in a completely decentralized manner. To this end, at first, novel adaptive EBSS techniques are developed for the Maximum Eigenvalue Detector (MED), the Maximum-Minimum Eigenvalue Detector (MMED), and the Generalized Likelihood Ratio Test (GLRT) scheme, respectively, for a single-user (no cooperation) case. Then, a novel distributed subspace tracking method is proposed which enables the cooperating nodes to track the joint subspace of their received signals. Based on this method, cooperative decentralized versions of the adaptive EBSS techniques are subsequently developed that overcome the limitations of the existing batch centralized approaches. The performance of the proposed methods is verified via indicative simulations. Index Terms—Eigenvalue-Based Spectrum Sensing, Cognitive Radio, Distributed Subspace Tracking, MED, MMED, GLRT.

I. I NTRODUCTION Cognitive Radio (CR) provides an efficient way to utilize the available bandwidth in wireless communication systems. The main idea of the CR concept is to allow non-licensed users (Secondary Users - SUs) to transmit their data in a spectrum area that is licensed to the so-called Primary Users (PUs) of the system. To this end, Opportunistic Spectrum Sharing techniques may be employed to optimize the SUs transmissions without degrading the PUs’ performance [1]. A common approach is that the SUs establish their communication links only in the so-called “spectrum holes” of the PU system, i.e., to spectrum areas that are temporarily unused by the corresponding PUs. Key role in those approaches play the spectrum sensing techniques which are employed by the SUs in order to detect the spectrum holes. Due to the fact that the performance of both the PUs and SUs depends highly on the successful detection of the corresponding spectrum holes, a major part of the recent CR literature has focused on the design of efficient spectrum sensing techniques. Spectrum sensing may be implemented either by a single radio architecture or by a dual radio one. In the single radio case, which is the most common one, a fixed portion The authors are with the Department of Computer Engineering and Informatics University of Patras & RACTI/RU8. Emails: [email protected] & [email protected] This work was supported in part by EU and national funds via the National Strategic Reference Framework (NSRF) - Research Funding Programs: Heracleitus II and Thales (ENDECON), and in part by the University of Patras.

of each time slot is dedicated to spectrum sensing and the remaining part is used for data transmission. In the dual radio architecture, which has been recently proposed (see [2] and references therein), one radio chain is dedicated to data transmission and reception while a second radio chain is dedicated to continuous spectrum monitoring. The differences between the two approaches are discussed in [2]. In general, the double radio approach provides higher spectrum efficiency and better sensing accuracy, however it should employ very low-complexity spectrum sensing techniques due to its “continuous” nature. Several spectrum sensing schemes have been proposed over the last few years including energy-based detectors (EBD) [3][5], eigenvalue - based spectrum sensing techniques (EBSS) [6]-[11], matched-filter based detection [12]-[13] covariance based spectrum sensing techniques [14], and cyclostationary based spectrum sensing techniques [15]. The EBD methods are usually simple to implement, however, they require knowledge of the involved noise variance. The EBSS methods (particularly, the MMED and GLRT) may operate without the knowledge of the noise variance and they may also offer remarkably improved performance for specific signal categories. However, this is done at the expense of high complexity since they require the EigenValue Decomposition (EVD) of the received signal’s sample covariance matrix. Moreover, the computation of the involved decision thresholds is generally based on the asymptotic (limiting) distributions of the corresponding test statistics and requires a large number of samples in order to attain a satisfactory performance. Recently, close approximations for the distribution functions of the test statistics have been proposed in the literature, [8]-[9], though they have quite a complex form. Thus, the computation of the decision thresholds is actually performed by numerical methods which increase even further the computational complexity. The high complexity required for implementing the batch EBSS techniques makes them impractical for cases where continuous monitoring of a specific spectrum band is required (dual radio approach). Thus, the first goal of the present paper is to develop low complexity adaptive EBSS techniques that track the involved test statistics and therefore are applicable to the dual radio spectrum sensing approach. The spectrum sensing techniques that were mentioned above refer to single-user systems. The performance of a single-user spectrum sensing technique is limited to environments where the fading and shadowing effects of the wireless channels degrade the quality of the received signals. Cooperation among multiple secondary users has been proposed in literature in order to improve the sensing performance. Note however, that the existing approaches (see [16]-[24] and references therein),

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are cooperative variations of the generic energy detector. To the best of our knowledge, cooperative techniques of the EBSS type have not been considered in literature so far, apart from the case where a fusion center collects the sensed data and applies the EBSS in a centralized batch manner [8]. A centralized approach comes with a number of limitations concerning the high power costs in transmitting local information to the fusion center and conveying global decisions back to the SUs. Moreover, a centralized SU network is quite sensitive to node and link failures. Contrariwise, a decentralized approach exhibits low communication overhead and it is robust to node and link failures. Each SU node communicates only with its adjacent SU nodes via one-hop transmissions resulting in transmissions with reduced power consumption during the sensing period. The SU nodes exchange information for several rounds so as to reach global convergence. Upon convergence, each node can reach the same decision concerning the PU existence without the need of a fusion center node. Motivated by the above, the second and main goal of the present paper is to develop cooperative adaptive EBSS techniques that function in a completely decentralized manner so as to overcome the limitations of the centralized batch approaches. It is noteworthy to mention, that a constituent part of the new cooperative EBSS technique is a new distributed subspace tracking algorithm which can be considered as a byproduct of the present work. Note that little work has been published so far for the problem of distributed subspace tracking [25]-[26]. The existing approaches are suitable for large sensor networks rather than small scale topologies of adjacent SU nodes as it is typically assumed in cooperative spectrum sensing scenarios. Moreover, the existing approaches present slow convergence speed and bad performance since they do not guarantee the orthonormality of the tracked eigenvectors (see also Subsection V.A). As it will be shown later, the distributed subspace tracking algorithm does not suffer from these limitations. More specifically, the contributions of the present paper are the following ones. At first, novel single SU (non-cooperative) adaptive EBSS techniques are derived based on well-known subspace tracking methods. Note that, to the best of our knowledge, this is the first time that adaptive versions of the EBSS techniques are derived, even for the non-cooperative case. Then, the distributions of the adaptive test-statistics are derived in order to compute the required decisions thresholds. It turns out that accurate approximations of the test statistics correspond to well-known tabulated functions improving further the practicality of the new adaptive schemes over the batch ones. Next, a novel cooperative adaptive EBSS method is proposed by first developing a distributed adaptive subspace tracking algorithm. The proposed adaptive EBSS and cooperative adaptive EBSS methods are compared to the corresponding batch EBSS approaches, in terms of performance for single (fixed spectrum sensing time) and double (continuous spectrum monitoring) radio architectures. The rest of the paper is organized as follows. In Section II the system description is provided. In Section III a brief description is given for the EBSS techniques considered in the present paper. In Section IV, adaptive versions of the

EBSS (AD-EBSS) techniques are derived. In Section V, the test statistics’ distribution functions of the proposed ADEBSS techniques and the corresponding decision thresholds are derived. In Section VI, the cooperative EBSS scheme is presented. Section VII presents the simulations, and Section VIII concludes the work. II. S YSTEM D ESCRIPTION Let us assume that a single-antenna PU node and K SU nodes of Rx antennas each one, are operating in the same frequency band in a typical interweave CR scheme [27]. The system description is given for the single radio approach, though it can be appropriately modified so as to be applied to the dual radio one. Thus, the time axis is assumed to be divided into transmit time intervals (time-slots). The time-slot is considered as the basic unit of time scheduling. At the beginning of each SU time-slot, the SUs sense the frequency band so as to determine if it is occupied by a PU transmission. Let us also assume that during each sensing period, each one of the K SUs collects N sample vectors yni , 1 ≤ n ≤ N and 1 ≤ i ≤ K, of dimensions 1 × Rx . In a centralized approach, a node that plays the role of the fusion center receives from the SUs the collected vector samples. Then, the fusion center node processes the samples jointly, decides if active PU transmissions exist in the environment and notifies accordingly the SUs. More specifically, in the associated cooperative spectrum sensing problem the following hypothesis test is considered at the fusion center, H 0 : y n = zn

(1)

H1 : yn = hxn + zn , (2) 1 T where yn = yn , . . . , ynK , zn is an KR x × 1 additive noise random variable modeled as CN 0, σz2 , h is a KRx ×1 vector that contains all complex flat channel gains that correspond to the links between the PU and the antennas of all SUs, and xn is the transmitted PU symbol. That is, under the hypothesis H0 , the PU is idle and the received signals yni at the SUs contain only noise. On the other hand, under the hypothesis H1 , the PU transmits data and the received SUs’ signals are a superposition of these data (scaled by the channel gains) and noise. An EBSS technique decides between the two hypotheses by employing test statistics that are functions of the eigenvalues of the received signals’ sample covariance matrix. In the following section a brief description of the existing EBSS techniques is given. III. C ENTRALIZED BATCH C OOPERATIVE E IGENVALUE -BASED S PECTRUM S ENSING In literature so far, only batch EBSS techniques have been considered, in the sense that the derivation of the test statistics is based on the EVD of the sample covariance matrix formed by a number of N received vectors. Moreover, only centralized cooperative extensions of the EBSS techniques have been proposed so far. As long as a fusion center exists to collect the SU vector samples yni in the cooperative case, the application of a batch EBSS technique is almost identical in both the noncooperative (single) SU and the cooperative cases.



In the rest of this section, we provide a brief description of the existing approaches for the cooperative case, though they are directly applicable to the non-cooperative case by setting the number of SU nodes equal to K = 1. We consider that the fusion center receives the SUs’ sample vectors yni through noise-free links and forms the aggregate vector communication T yn = yn1 , . . . , ynK as discussed in the previous section. The covariance matrix R of the received signal yn , under the two hypotheses, is given by, σz2 IKRx H0 H R = E{yn yn } = , (3) 2 σx hhH + σz2 IKRx H1 where E{·} denotes the expected value operator, σx2 is the PU’s transmitted signal’s variance and (·)H denotes the Hermitian of a matrix. In practice, the fusion center (or a single SU) computes the sample covariance matrix of the received signals, which is given by, N 1 X ˆ yn ynH . R= N i=1

(4)

Let us denote with λ1 ≥ · · · ≥ λRx the ordered eigenvalues ˆ As it is known from the relevant literature, the of matrix R. eigenvalues of the sample covariance matrix can be used to form a sufficient test statistics for the spectrum sensing problem. In the following equations three different test statistics are defined. λ1 (5) T M ED (λ1 ) = 2 , σz λ1 T GLRT (λ1 , . . . , λRx ) = 1 PRx , (6) m=2 λm N −1 λ1 . (7) T M M ED (λ1 , λRx ) = λ Rx The noise eigenvalues of the sample covariance matrix are random variables due to the finite number of samples that are used for its computation. The distribution of these noise eigenvalues is used to compute the decision threshold in a Neyman-Pearson sense for a pre-defined probability of false alarm. Detailed information on how to compute the decision thresholds is given in [6], [7] and [28]. We can now proceed with the development of the cooperative adaptive EBSS methods. For clarity purposes, we first deal with the non-cooperative case in which only a single SU is involved. Then the extension of the proposed adaptive approaches to the cooperative case is derived. IV. A DAPTIVE E IGENVALUE -BASED S PECTRUM S ENSING FOR THE S INGLE SU C ASE As already mentioned in the Introduction, the existing batch EBSS techniques are suitable in cases where single radio architectures are assumed. In the case of a dual radio approach, the aim is to continuously monitor the spectrum area of interest so as to detect as soon as possible an abrupt change (i.e. the PU starts or stops its transmission at a random time). To this end a number of different solutions are proposed in literature so far based on the quickest detection of change / sequential detection approaches [29]-[32]. According to the

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aforementioned approaches, the SU updates sequentially its corresponding tests statistics upon the reception of a new data sample and decides if a change have occurred in the spectrum area of interest. The decision is based on thresholds which are computed in order to optimize a metric i.e., mean detection delay which is usually defined as the the mean time which is required by the SU to detect a change subject to the targeted probability of false alarms. To the best of our knowledge sequential / quickest detection approaches that involve the EBSS techniques have not considered so far in literature. A technique that applies the existing EBSS approaches in a sequential manner will exhibit very high computational complexity due to the periodic computation of the EVD of the sample covariance matrix of eq.(4). Thus we were motivated to develop on-line implementations of the EBSS techniques based on low-complexity subspace tracking (ST) techniques [33]-[36] (and references therein). The aim of a ST technique is to update in an adaptive manner, the estimation of the subspace of an input signal provided that a new data sample of the signal is available at each time index. In other words the EVD of the sample covariance matrix of eq.(4) is updated at each time index when a new data sample is available. The main advantage of a ST technique is that the latter update can be done with complexity of O(Rx L) operations, where L is the rank of the desired subspace. Contrariwise in the batch approach recall that the subspace of the signal is updated by computing the EVD of the sample covariance matrix at each time index which exhibits the prohibited computational complexity of O(Rx3 ) operations. A. Single Radio Approach Let us now proceed to the description of the adaptive EBSS technique for the single radio approach. In this paper the complex version of the Fast Data Projection Method (FDPM) is employed [36]. The FDPM steps are summarized in the following equations. rn = UH n−1 yn Un = Un−1 ±

(8) µyn rH n

(9)

Un = orth{Un }

(10) 2

λn = αλn−1 + (1 − α)|rn | ,

(11)

where Un and λn are the Rx ×L matrix and L×1 vector that contain the L principal (minor) eigenvectors and eigenvalues, respectively, |rn |2 is the vector of the squared absolute values of the elements of vector rn and α, µ are step size parameters. Note that in equation (9) the sign is a (+) or (−) depending on whether the signal or the noise subspace, respectively, is updated. Eq.(11) tracks the corresponding eigenvalues. An orthonormalization procedure is applied in Eq.(10) based on a low complexity Householder transformation [36] given by the following equations H

an = rn − krn kej(angle(e1 yn )) e1 2 (Un an ) aH Un = Un − n kan k2 Un = norm{Un }


(12) (13) (14)



In eq.(12) e1 = [1, 0 . . . 0, 0] and in eq.(13), norm{·} denotes the operator that normalizes the columns of a matrix. For the MED test statistic, only the first (maximum) eigenvalue is required, so Un is a Rx × 1 matrix and λn is a scalar that contains the current estimate of the maximum eigenvalue. The orthonormalization step is also reduced to a simple normalization one, i.e. Un = Un /kUn k. In a similar way, the MMED test statistic is estimated. At first, the signal subspace version of the FDPM is employed (with (+) in (9)) to track the maximum eigenvalue and then the noise subspace version follows (with (−) in (9)) to track the minimum one. For the GLRT test statistic the complete FDPM (eqs.(8)-(11)) is used ˆ n are required in eq.(6). since all the Rx eigenvalues of R It is evident now that, by employing the FDPM method, the SU is capable of tracking, with low complexity, the value of any of the test-statistics under consideration at every time index within the sensing period. Therefore, once a new signal vector is received, the SU updates the employed test-statistic and decides if a change in the state has been occurred (from H0 to H1 and vice-versa). Thus, the adaptive EBSS techniques can be applied in both the single and in the dual radio approach. In the single radio approach the adaptive techniques can be applied in fixed sensing periods in a similar way to the batch ones. That is for each time slot within the sensing period, the corresponding test statistics are updated and at the end of the sensing period a decision is taken according to a threshold whose computation is given in Section V. B. Complexity Analysis for the Single Radio Approach Let us now derive the complexity of the adaptive and the batch EBSS techniques for the single radio case. The complexity is given in terms of the total required complex mathematical operations (additions/subtractions, multiplications/divisions and square roots) at a sensing period of N timeslots. According to [36] the FDPM algorithm requires at each timeslot 12Rx L+5L+2 operations, to update L principal components of the covariance matrix R at the SU receiver, provided that a new Rx × 1 sample yn is available (see [36] for a detailed analysis). In the aforementioned complexity we must add 3L operations since eq.(11) is not included in the original version of the FDPM presented in [36]. Note that eq.(11) requires only 3L operations since the entries of the vector |rn | are already computed in eq.(8) and in eq.(12). Thus, the overall complexity is 12Rx L + 8L + 2 per update/timeslot. Since the sensing period is associated with N signals, a total of N (12Rx L + 8L + 2) operations are required. Now the complexity of each one of the adaptive EBSS techniques can be derived by properly setting the value of the parameter L and then adding the number of operations that are required for the computation of the specific test statistic according to eqs.(5)-(7). That is, for the MED test statistic L = 1 and the complexity is N (12Rx + 10) + 1 operations. In a similar way the MMED test statistic requires two applications of the FDPM with L = 1 (Subsection IV.A) resulting in complexity of 2N (12Rx + 10) + 1 operations. Finally L = Rx for the GLRT test statistic and the complexity is N (12Rx2 + 8Rx + 2) + Rx operations.

Let us now derive the complexity of the batch approaches. ˆ is updated at each timeslot. The latter In this case matrix R 2 requires 2N Rx complex operations. Then, the SVD of the R is computed requiring approximately 13Rx3 operations according to [37]. Thus, the overall computational cost of each one of the batch EBSS techniques is equal to 13Rx3 + 2N Rx2 . Now, in order to derive the explicit complexity of each one of the test statistics, we should count for the operations of eqs.(5)(7), similarly to the adaptive case. Therefore, the MED and MMED test statistics require 13Rx3 + 2N Rx2 + 1 operations and the GLRT one requires 13Rx3 + 2N Rx2 + Rx operations. It is evident, the adaptive methods exhibit significantly reduced complexity compared to the batch ones. As it is shown in the following section, the complexity savings can be more significant for the case of the dual radio approach. C. Dual Radio Approach Let us now develop the adaptive EBSS techniques for the dual-radio architecture. It is reminded that in this approach the SU constantly monitors the spectrum area of interest in order to detect as fast as possible a change in the PU activity. There are two standard formulations in the area of change detection: Bayesian and minimax. The Bayesian formulation was developed by Shiryaev [38]-[40]. In his approach the change point is assumed to be a random variable with a certain (known) prior distribution and the objective is to minimize the mean detection delay subject to an upper bound on the false alarm probability. In the min-max approach introduced by Lorden [41], the change point is assumed to be an unknown deterministic parameter and the objective is to minimize the worst-case conditional detection delay subject to an inequality constraint on the average run false alarm length. Each one of the approaches is suited for different regimes and both of them were applied in the cognitive radio literature during the past years [32], [42]-[43]. The existing approaches are mainly based on the energy detector whereas there is lack of approaches that are based on the EBSS techniques, as it was already mentioned above. An approach that is based on the batch EBSS techniques will require high computational complexity in order to achieve small mean detection delay since the values of the corresponding test statistics must be updated periodically so as to detect as quickly as possible an abrupt change in the PU’s behaviour. Let us now assume that the SU employs the FDPM method so as to track continuously the test statistics during consecutive periods of N samples. We follow the approach of [32] where it is assumed that the PU changes its behaviour at a random sample number within the sensing period. The change point T0 is assumed to follow a known geometric distribution parametrized by p, that is P {T0 = n} = p(1 − p)n−1 (1 − λ0 ), 0 < n ≤ N,

(15)

where λ0 = P {T0 = 0} denotes the probability that a change occurs before the sensing period. According to the Bayesian formulation, the problem of quickest detection of change is to design a detection rule that gives the stopping time Td so that the mean detection delay E{(Td − T0 )+ } is minimized



Table I: Adaptive EBSS for the single SU case Initialization: The SU detects the present state (H0 or H1 ) by applying the adaptive EBSS technique for a sufficient number of received signals, until an initial decision is taken. For each block of N symbols for n = 1 → N do Update the employed Test-Statistic T via the FDPM (8)-(11) if T ≥ ηd under H0 || T < ηd under H1 then Raise an alarm, change in the state has been detected; end if end for

subject to a constraint on the probability of false alarm, that is P {Td < T0 } ≤ ζ. Note that the definition of the probability of false alarm here differs from the single radio approach, since here is denoted as the probability that a change is detected prior the actual time of change (Td < T0 ). The solution to that problem requires the knowledge of the exact distribution of the samples under both hypothesis which is not possible to derive in the case of the EBSS techniques. Moreover a closed form is in general intractable even in case of the simple energy detector. Thus, in order to develop an efficient and simple technique the following approach is adopted which is summed up in the Table I. The SU for each block of N samples updates sequentially the EBSS test statistics under consideration. At each time index n of the sensing block the updated test statistic is compared to a global threshold. The global threshold is computed so as the probability of false alarm is set to a specific value P {Td < T0 } = ζ. The threshold computation in this approach is more complex than the simple radio one and it is described in Section V. Remark 1: One may argue that the batch EBSS approaches can be extended easily to the dual radio case. Indeed, let us assume that within the sensing period of N symbols the EVD of the sample covariance matrix is periodically computed every Ns < N samples so as to update the values of the test statistics. This could increase significantly the complexity due to a number of b NNs c SVD that are required. Clearly a small number of Ns reduces the detection delay though it results in high complexity and vice versa. A detailed analysis of the complexity is given in the following subsection. D. Complexity Analysis for the Dual Radio Approach Let us assume that a dual radio sensing approach is applied for a period of N symbols. For the subspace update which is given in Subsection IV.B, the adaptive techniques require the same complexity with the single radio case. The difference in the dual radio case is that the computation of the test statistics is done at every timeslot n ∈ [1, N ] and thus the operations of eqs.(5)-(7) must be added N times to the overall complexity. Thus, it is easy to see that in the dual radio approach the complexity for the MED technique is N (12Rx + 10) + N , for the MMED one it is 2N (12Rx + 10) + N and for the GLRT one it is N (12Rx2 + 8Rx + 2) + N Rx operations respectively. According to Remark 1, the dual radio version of the batch approaches requires the computation of b NNs c SVDs for a sensing period of N symbols. According to Subsection IV.B, the latter costs 13b NNs cRx3 operations. In order to derive the

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overall complexity, one should also add b NNs c times the operations of eqs.(5)-(7) similarly to the adaptive case. Therefore N 3 the MED and MMED 2 testN techniques require 13b N s cRx + N 2N + b N s c − 1 Rx + b N s c operations and the GLRT one requires 13b NNs cRx3 + 2N + b NNs c − 1 Rx2 + b NNs cRx operations. Clearly, the complexity gains for employing the adaptive EBSS techniques are even greater in the dual radio sensing approach especially for small values of the parameter Ns . In the simulations section (Figure 8) the complexity of the adaptive EBSS techniques is compared to the one of the batch techniques for different values of the involved parameters. V. T EST S TATISTICS D ISTRIBUTIONS AND D ECISION T HRESHOLDS In this section, the Cumulative Distribution Functions (CDF) of the three test statistics of Eqs.(5)-(7) are derived for the adaptive case, under the hypothesis H0 , i.e., when no information signal is present in the signal received by the SU. Based on eq.(11) of the FDPM, the distribution functions of the involved test statistics can be tracked at every time index n. The following lemma provide expressions for the distribution functions of the corresponding test statistics. Lemma 1: The distribution functions of the adaptive teststatistics updated by eq.(11) for a SU of Rx antennas under the hypothesis H0 can be approximated by the functions, γ(ξx, ρ) , Γ(ρ) FT GLRT (x; ρ, Rx ) ≈ I (Rx −1)x (ρ, (Rx − 1)ρ), FT M ED (x; ξ, ρ) ≈

(16) (17)

x+1

x (ρ, ρ), FT M M ED (x; ρ) ≈ I x+1 n

(18) n 2

(1−α )(1+α) (1−α ) (1+α) respectively, where ξ = ((1−α)(1−α 2n )) , ρ = ((1−α)(1−α2n )) , R x ρ−1 −t γ(x, ρ) = 0 t R e dt is the lower incomplete gamma ∞ function, Γ(ρ) = 0 tρ−1 e−t dt R xdenotes the ordinary gamma function and Ix (ρ1 , ρ2 ) = 0 tρ1 −1 (1 − t)ρ2 −1 dt is the incomplete beta function.

Proof: Observe that, under hypothesis H0 , the signal vector yn received by the SU consists of i.i.d. complex Gaussian noise samples CN (0, σz2 ). Since matrix Un−1 is orthonormal, the (i) 2 entries rn of vector rn = UH n−1 yn are also i.i.d. CN (0, σz ). From eq.(11), the l-th eigenvalue of the covariance matrix is estimated as, n X 2 λl (n) = αi (1 − α)|r(i) (19) n | . i=0

According to the previous equation, the MED test statistic can be expressed as a weighted sum of chi-squared variables with each one derived by squaring the absolute value of a random variable φi ∼ CN (0, 1). A close approximation to the previous distribution can be derived by properly applying the results of [44] to our case. Let us consider the following RV T ,

m X

wi ζi2 ,

(20)

i=1

where ζi ∼ N (0, 1) and wi ∈ R. Welch, in a 1938 paper [44], proposed an approximation of the distribution of variable T by a scaled chi-squared distribution of ρ degrees of freedom.



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w

i i=1 That is, T ∼ 1ξ χ2ρ , where ξ = Pm 2, ρ = i=1 wi the corresponding CDF is given by

γ(ξ/2x, ρ/2) FT (x; ξ, ρ) = . Γ(ρ/2)

(

Pm wi )2 Pi=1 m 2 i=1 wi

and

(21)

Now, by defining wi = αi (1 − α)/2 and using eq.(21) in the case of complex normal variables φi , we can apply the previous results to the MED test statistic (5). Moreover, observe that the weights wi are actually terms of a geometric sequence enabling as to compute closed forms for the parameters ρ 2(1−αn )(1+α) and ξ. Thus, it can be verified that ξ = ((1−α)(1−α 2n )) and n 2

2(1−α ) (1+α) ρ = ((1−α)(1−α 2n )) . Finally, by combining these expressions with eq.(21) the proof for MED is completed. In the case of the MMED test statistic, we seek for the ˆ n , that distribution of the ratio of two eigenvalues of matrix R are estimated via Eq.(11). The required distribution is equivM ED alent to the distribution of the ratio TTM ED (λ(λR1 )) . Therefore, x the corresponding CDF is equal to the one of the ratio of two independent T M ED random variables. It turns out that the distribution of the MMED statistic can be approximated by the beta prime distribution with the corresponding CDF given by eq.(18). The detailed proof is given at Appendix A. Finally, the CDF of the GLRT test statistic canP be computed Rx by firstly observing that the random variable m=2 λm ∼ 1 2 χ , where the parameters ξ and ρ are defined similar (R −1)ρ ξ x to the case of the MED distribution. Then the CDF of the ratio of the two independent random variables PRxλ1 λ is m=2 m computed in a similar way to the MMED case. As long as the CDF of the ratio under consideration is computed, it is easy then to compute the CDF of the scaled random variable λ1 T GLRT = 1 P which is given by eq.(17). Rx λ Rx −1

m=2

m

A. Threshold Computation for the Single Radio Approach As it can be seen from the results of Lemma 1, the CDFs of the adaptive test statistics involve the computation of wellknown tabulated functions and the same comment is true for their corresponding inverse functions. The latter observation enables the SU to easily compute the decision thresholds in a Neyman-Pearson sense, for a pre-defined probability of false alarm Pf , as opposed to the case of the batch test statistics where usually numerical methods are required. To proceed further, first, the decision threshold η M ED for the M ED test statistic is computed. We equivalently have Pf = P T M ED > η M ED |H0 = 1 − FT M ED (x; ξ, ρ) ⇒ η M ED = FT−1 M ED (1 − Pf ; ξ, ρ) .

(22)

The decision thresholds for the adaptive MMED and GLRT test statistics can be computed in a similar way, and they are given by the following equations, I −1 (1 − Pf ; ρ, ρ) , (23) 1 − I −1 (1 − Pf ; ρ, ρ) I −1 (1 − Pf ; ρ, (Rx − 1)ρ) = . (24) (Rx − 1) − I −1 (1 − Pf ; ρ, (Rx − 1)ρ)

η M M ED = η GLRT

B. Threshold Computation for the Dual Radio Approach According to Section IV in the dual radio approach, the threshold is computed so as to set the probability of false alarm P {Td < T0 } to a specific value ζ. In order to compute the previous probability, we require the probability that a PU is detected at least one time before the time index of change T0 under H0 . Given the fact that the test statistics are updated by adding each time index a new positive random variable and that the parameter α is set to values close to 1, the latter probability can be approximated by the probability the test statistic is greater than the threshold at time index T0 − 1, that is P {TT0 −1 > ηd |H0 } = 1 − FTT0 −1 (ηd ), where FTT0 −1 (ηd ) denotes the CDF of the corresponding test statistic given by eqs.(16)-(18) for each one of the cases considered here. Now by assuming that the time of change is independent to the values of the test statistics and taking the summation over all the 0 < n ≤ N , it can be seen that the required false alarm probability can be approximated by the following equation P {Td < T0 } ≈

N X

(1−FTn−1 (ηd ))p(1−p)n−1 (1−λ0 ) (25)

n=0

Finally, by setting P {Td < T0 } = ζ and solving numerically eq.(25), the desired threshold is computed. VI. C OOPERATIVE D ECENTRALIZED A DAPTIVE E IGENVALUE -BASED S PECTRUM S ENSING In this section, the previously derived adaptive EBSS techniques are extended to the cooperative case, in which multiple SUs sense the spectrum in a collaborative manner. The new distributed technique is designed so as, a) to improve the sensing performance by forming a virtual multiple antenna system that performs a joint EBSS technique, b) to distribute the computation overhead among the SU nodes, and c) to enable each user to track the joint test statistics at each time index within the sensing period so as to reach a common decision in a decentralized manner. Apart from the benefits of a decentralized approach discussed in the Introduction, the latter feature is crucial in scenarios where the multiple nodes employ a cooperative transmission scheme (e.g. cooperative beamforming [45]), and thus all the involved nodes should reach a common decision concerning the PU existence. Clearly, the proposed decentralized adaptive EBSS methods require the development of a distributed adaptive subspace tracking method in order to track the corresponding test statistics of eqs.(5) - (7). Therefore, a novel distributed subspace tracking scheme is first developed by extending the FDPM of [36]. The present section is divided into three subsections. Subsection A provides a brief discussion on distributed subspace tracking and the related work that has published in literature. In Subsection B the proposed distributed subspace tracking method is described. Finally, in Subsection C, the cooperative EBSS methods are described. A. Related Work To the best of our knowledge, little work concerning the problem of distributed subspace tracking has been published



in the literature. The authors in [25] develop a consensus based technique of OJA rule in which several subspace tracking methods are based. In [26] a distributed version of the wellknown PAST technique is presented. The previous methods are more suitable for ad hoc wireless sensor networks with a large number of nodes since they are based on consensus strategies. In the scenario studied in this paper the number of nodes that are involved in the network topology is relative small, and thus we seek for suitable distributed strategies that minimize the required communication overhead. Moreover, the approaches of [25] and [26] provide estimations for the eigenvectors only and not for the corresponding eigenvalues. Also, they converge rather slowly since they are gradient flow based approaches [36], and finally they do not guarantee the orthogonality of the estimated eigenvectors resulting in severe performance degradation. The new approach is based on an incremental strategy that requires no fusion center, it exhibits minimum communication overhead [46] and distributes evenly the computation cost among the SU nodes, thus enabling each one of them to perform the spectrum sensing test at every time index within the sensing period. Moreover, it converges faster since the Data Projection Method (DPM) is based on the power method [47] and guarantees the orthogonality of the tracked eigenvectors. In the following subsection the new distributed subspace tracking scheme is derived.

Table II: The DDPM Algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21:

Initialization: U0 ← IRx ×L & λ0 ← 0L×1 Previous Instant Data: Node i has Uin−1 and λin−1 New Data: Node i has a new vector sample yni Apply: 1st Spatial Iteration: r0loc ← 0L×1 TK+1 ← 0L×1 loc for i = 1 → K do ∗ i−1 riloc ← rloc + Uin−1 yn (i) % Incremental Comp. of rn end for 2nd Spatial Iteration: for i = K → 1 do λin = αλin−1 + (1 − α)|rn |2 , % Update of λin Uin ← Uin−1 ± µyni rH % Update of Uin n an ← rn − krn ke1 % Distributed Orthogonalization of Un Uin ← Uin − kan2 k2 Uin an aH n PL j 2 Tiloc ← Ti−1 % Incr. Comp. of Tn loc + l=1 |Un (:, l)| end for 3rd Spatial Iteration: for i = 1 → K do Uin ← diag (Tn )1/2 Uin % Distr. Normalization of Un end for

Now, in order to execute the local update step of eq.(26) each node must know the value of rn . Observe that the computation of rn requires a spatial iteration in which the i-th node performs the local computation of eq.(27) and sends the quantity riloc to the (i + 1)-th node. That is, i

∗ i riloc = ri−1 loc + Un−1 yn

(27)

Thus, starting from the 1st node the variable riloc is updated in an incremental manner and at the end of this first spatial iteration the K-th node has the exact value of rn given by

B. Distributed Data Projection Method (DDPM) Let us assume that K SU nodes form a network where the i-th node is connected via direct links only to the (i − 1)th and the (i + 1)-th nodes. The communication links are assumed to be established via ideal (noise free) channels. Note that the latter assumption is typical in both the distributed adaptive signal processing [46] and cooperative spectrum sensing literature [16] - [20]. For simplicity, we assume that each SU node has the same number of Rx antennas, though the results are directly applicable in cases where the nodes may have different numbers of antennas. At each time index of the sensing period, the i-th SU obtains a Rx ×1 vector of samples yni . Let us now assume that we are interested in tracking the L first principal (or minor) components of the correlation matrix 1 K T R of the aggregate vector yn , . . . , yn . The task of tracking each one of the L eigenvectors is properly distributed to the SU nodes in a way that the i-th node tracks the elements of the sub-matrix Un ((i − 1)Rx + 1 : iRx , 1 : L) of the aggregate eigenvectors matrix Un . For simplicity, the later sub-matrix will be denoted by Uin . Thus, each node needs to update only a specific part of the subspace. Moreover, each node is capable of tracking the corresponding L principal (minor) eigenvalues of the aggregate system, denoted by λin . As explained below, at each time index n, the DDPM requires three spatial iterations in order to update the eigenvectors and eigenvalues matrices. By observing eq.(9), the subspace update can be done in a local computation step at node i given by Uin = Uin−1 ± µyni rH n

7

(26)

rn =

K X

riloc

(28)

i=1

The second spatial iteration starts from the last node K of the network. As it was mentioned the K-th node has already the exact value of rn eq.(28) and therefore it is able to update the corresponding sub-matrix of eigenvectors as determined by eq.(26), then update the eigenvalues by using eq.(11) locally and apply the orthogonalization transformation to the updated eigenvectors sub-matrix, that is 2 Uin = Uin − Uin an aH (29) n, 2 kan k where the quantity an is computed by applying eq.(12) locally. Clearly, in order to complete the eigenvectors update, the normalization step of eq.(13) is required. Therefore, the last local computation of the i−th node at the 2nd spatial iteration is given by i−1 Tiloc = Tloc +

L X

|Ujn (:, l)|2

(30)

l=1

Now observe that, by incrementally applying eq.(30) at the end of the 2nd spatial iteration the 1st node has the vector Tn = kUn (:, l)k2 , . . . , kUn (:, L)k2 whose i-th element is the squared normed of the i-th eigenvector. Note that as already mentioned, the incremental strategy requires the transmission of the quantities rn and Tiloc from the i-th node to




the (i+1)-th one. The aim of the third iteration is to normalize the updated eigenvalues matrix Un . It is easy to see that the latter can be done by performing sequentially the following steps at the i-th node Uin = diag (Tn )

1/2

Uin

(31)

where diag (·) denotes the operator that transforms the vector operand to a diagonal matrix. The complete DDPM algorithm is summarized in Table II. C. Cooperative EBSS techniques In this subsection the extension of the adaptive EBSS technique to the case of multiple SUs is described. Let us consider that the nodes of SUs network apply the DDPM algorithm of Table II so as to jointly track the subspace of the received data covariance matrix of the aggregate virtual multiantenna SU system. As it is evident from line 13 of the Table II, each node, at each time index, updates the local estimates of the eigenvalues of the sample covariance matrix. Therefore, each node is also able to compute the test statistics of eqs.(5)(7) of the aggregate system and can perform the detection tests independently. Moreover, the test statistics distributions, under the hypothesis H0 , are again given by Lemma 1, by replacing Rx with K ∗ Rx . Observe also that in a dual radio approach, each SU node can employ independently the algorithm of Table I so as to reach a common decision concerning a detection of a change in the PU activity. Let us now comment on the specificities of each one of the cooperative EBSS techniques. If the MED method is employed, then it suffices to track only the maximum eigenvalue and thus in the DDPM algorithm the cooperating SUs may set L = 1. Moreover, the distributed orthogonalization steps (lines 15-16 on Table II) are not performed at all, in a way similar to the single SU case. In the case of the MMED method, again in a similar way to the single SU one, each of the SUs tracks the maximum and the minimum eigenvalues of the covariance matrix by employing twice the DDPM for L = 1. Specifically, first the signal subspace version (with (+) in (9)) is employed so as to track the maximum eigenvalue and then the noise subspace version follows (with (−) in (9)) in order to track the minimum one. Recall, that this approach has been adopted in order to avoid the complexity of tracking the complete subspace of the covariance matrix. Of course the steps of each one of the two versions of the DDPM can be applied simultaneously at each time index. Finally, for the adaptive GLRT EBSS method the SU nodes should apply the complete DDPM algorithm with L = K ∗ Rx . D. Computational and Network Complexity of the Decentralised Adaptive EBSS Techniques Let us first examine the computational complexity. From the description of the DDPM technique it is evident that its computational complexity equals to the one of a centralized FDPM for a system of K × Rx receiver antennas, though eq.(11) is executed K times, that is one time for each one of the participating SUs. Therefore according to [36] and the discussion of Subsection IV.B the complexity for subspace estimation for a

sensing period of N timeslots is N (12KRx L+5L+3KL+2) operations. Now by noting that each one of the SUs employs eqs.(22)-(23) and following the same procedure with the single SU case the complexity at both the single and the dual radio case can be derived easily. Thus, we have for the single radio case N (12KRx + 3K + 7) + K operations for the MED test statistic, 2N (12KRx + 3K + 7) + K operations for the MMED one and N (12K 2 Rx2 + 11KRx + 2) + KRx for the GLRT one, respectively. Accordingly we have for the dual radio case N (12KRx + 3K + 7) + N K operations for the MED test statistic, 2N (12KRx + 3K + 7) + KN operations for the MMED one and N (12K 2 Rx2 + 11KRx + 2) + N KRx operations for the GLRT one respectively. Note that in a similar way, one may also derive the complexity of the batch EBSS techniques by simply replacing Rx with KRx in the expressions derived in Subsections IV.B and IV.D (omitted here due to space limitations). The network complexity is derived in terms of the complex scalar quantities that must be transmitted during a sensing period of N data samples. From the description of the DDPM technique (Table II) the MED test statistic requires 3N K scalar quantities to be transmitted and the MMED and the GLRT require 6N K and 3N KRx , respectively. A centralized approach requires N KRx scalar transmissions in order for the SUs to transmit the data samples to the fusion center and K transmissions to notify the users for its decision. Clearly for SUs with number of antennas greater than 3 the decentralized MED test statistic exhibits lower complexity than the corresponding centralized approach. In a similar way, the MMED test statistic exhibits less communication burden when the number of antennas of each node is greater than 6. Finally the GLRT exhibit always greater communication complexity than the centralized. The network complexity is depicted also in Figure 8 in the simulations section. VII. S IMULATION R ESULTS In this section, some indicative simulation results are presented in order to evaluate the performance of the adaptive EBSS methods. We assume that the PU transmits a BPSK modulated signal. The step of the FDPM algorithm is set to µ = 0.8/kyn k2 . First, the methods for the single SU case are tested. In Fig. 1, the theoretical CDFs of the MED, MMED and GLRT adaptive test statistics under the H0 (Lemma 1) are compared to the empirical ones when a block of N = 20 received signal vectors at the SU is used to estimate them. The results of 10000 simulations are averaged so as to compute the empirical CDFs for a SU receiver with Rx = 4. As it is shown, the derived theoretical CDFs are very close to the empirical ones even for this small number of received signals. In Fig. 2, the performance of the AD-EBSS techniques is compared to the one of the batch techniques in terms of the achieved probability of detection Pd under different SNR values for probability of false alarm Pf = 0.1 for the single radio case. The parameter a of the FDPM algorithm (11) is set to a = 0.98. The performance is examined considering constant channels within each timeslot of duration



1

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Fig. 1: CDFs of the test statistics under H0 (Eq.(16)-(18))

Probability of Detection (P )

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Fig. 4: Pd for non i.i.d. noise samples and uncorrelated fading

N = 100 symbols. The taps of all the involved channels are derived as CN (0, 1) and the results of 10000 realizations are averaged. Note that, even in single radio approach, the ADEBSS techniques exhibit reduced complexity compared to the batch ones, as they do not require the computation of the sample covariance matrix given by Eq.(4). As it is shown, the adaptive versions of the test statistics achieve, in general, performance close to the batch ones (or even better, in some cases). This is due to the fact that the CDFs of the adaptive test statistics Eq.(16)-(18) are close approximations to the exact ones, whereas for the batch case asymptotic expressions are used. This can be also verified by the results depicted in Fig. 3 where the simulations of Fig. 2 were repeated but now the decision thresholds were computed numerically. The batch techniques exhibit improved performance due to accurate values of the employed thresholds. In Fig. 4, the same experiment of Fig. 2 is repeated, though now, we consider that the noise samples are no longer i.i.d. due to calibration errors at the SU. According to the relevant literature [48]-[49], the correlation matrix of the noise samples can be considered diagonal. In

the simulations of the present figure the noise covariance matrix is given by Σz = diag{1.1, 0.9, 0.8, 1.2}. As it was expected both the adaptive and the batch techniques exhibit worse performance compared to the i.i.d. case. Nevertheless for high SNR regimes, the EBSS techniques achieve high detection rates. Similar conclusions can be derived in the case of correlated fading coefficients at the SU antennas. The results are depicted in Figure 5, where the fading coefficients are correlated such that each entry of their covariance matrix is |m−n| given by cm,n = ρc [50], where ρc ∈ R is the correlation coefficient that satisfies |ρc | ≤ 1. Note that the noise samples are assumed to be i.i.d. in order to evaluate the effect of correlated fading to the EBSS techniques’ performance. Again for low SNR regimes performance degradation is observed for both the adaptive and the batch EBSS methods. In Fig. 6, the mean detection delay in an abrupt change is compared for each one of the batch and the adaptive EBSS techniques for different SNR values in order to evaluate the performance of the method in the dual radio case. We consider timeslots of N = 1000 symbols in which an abrupt change in




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Fig. 8: Computational Complexity vs. Number of SUs

the channel state occurred at a random symbol time. The time at which a change occurs is assumed to follow the geometric distribution of eq.(15) with parameters p = 0.1, λ0 = 0.01 and a targeted probability of false alarm Pf = 0.01. The decision threshold of the adaptive methods is computed by eq.(25), as it is described in Subsection V.B. The threshold of the batch EBSS techniques is computed by simulations since a similar expression of the probability of false alarm was not possible to be found for this case. The adaptive EBSS techniques employ the algorithm of Table I. The batch EBSS techniques are employed periodically at every block of Ns = 50, 100, 200 symbols as it is described in Remark 1. It is evident that the adaptive EBSS techniques detect very fast the change even in low SNR regimes, since the detection test is applied at every sample time. The batch techniques detection delay is related to the number Ns that dictates the frequency of their application. Let us now study the multiple SUs case. In Fig. 7, the experiments of Fig. 2 are repeated, though now 7 SUs of 4 antennas each one are cooperating. The performance of the cooperative adaptive EBSS methods that employ the DDPM

of Table II is compared to the one of the single user adaptive EBSS methods. The performance improvement offered by the cooperative methods is evident and it is due to the fact that they process jointly the samples of the aggregate system of the K ∗ Rx receive antennas. In the same figure the performance of a fusion based method, that collects the samples for all the users and performs the batch EBSS techniques, is shown for comparison purposes. As it is shown, the cooperative adaptive EBSS schemes, apart from their benefits due to their completely decentralized nature and their low complexity, exhibit also satisfactory performance compared to the centralized batch ones, as it was also shown in the single SU case. Finally in Figs. 8 and 9 the computational and network complexity of the decentralized dual radio approach is examined for a different number of SUs K. Each one of the SUs is considered to have Rx = 7 antennas. In the same figures we present also the corresponding complexities for the batch centralized EBSS techniques. As it is evident, the adaptive EBSS techniques can provide significant reduction in the required complexities, especially for large network sizes.



where τi ∈ R+ . The inverse functions of the ones of eq.(33) are given by

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Fig. 9: Network Complexity vs. Number of SUs VIII. C ONCLUSION In this work, cooperative decentralized adaptive versions of the well-known EBSS techniques were developed for multiantenna cognitive receivers. The proposed techniques offer low complexity and improved performance especially in cases when continuous spectrum monitoring is applied. Moreover due to their decentralized nature the proposed approaches exhibit reduced power consumption at the transmissions during the sensing period and provide robustness against node and link failures. In order to compute the decision thresholds for each one of the adaptive test statistics, close approximations for the associated distribution functions were derived. A novel distributed subspace tracking method was also derived as a constituent part of the proposed decentralized EBSS techniques. The new distributed ST method enable the SUs to track jointly the subspace of their received signals in a completely decentralized manner. The performance of the proposed EBSS techniques was verified via indicative simulations and compared to those of the corresponding batch centralized approaches. A PPENDIX A D ERIVATION OF THE DISTRIBUTION OF THE MMED TEST STATISTIC UNDER THE H0 H YPOTHESIS Let us assume that two RVs T1 and T2 follow the same distribution with that of the MED test statistic. The CDF of the later distribution is given by eq.(16). The corresponding Probability Density Function (PDF) is computed by taking the derivative of eq.(16). That is, fTi (τi ) =

ξ ρ ρ−1 −ρτi τ e . Γ(ρ) i

(32)

We are interested in the distribution of the random variable Y1 = T1 /T2 . Let us also define the auxiliary random variable Y2 = T2 . Thus we have, t1 f1 (t1 , t2 ) = t2 f2 (t1 , t2 ) = t2 , (33)

(34)

The joint PDF of variables Y1 and Y2 is given by fY1 ,Y2 (y1 , y2 ) = fT1 ,T2 f1−1 (y1 , y2 ), f2−1 (y1 , y2 ) |J(y1 , y2 )|, (35) ∂(t1 ,t2 ) is the Jacobian matrix of the where J(y1 , y2 ) = ∂(y ,y ) 1 2 transformation and |J(y1 , y2 )| = y2 is its determinant. Observe now that, since the eigenvalues are estimated via eq.(11), there are statistically independent. That is, the joint PDF of the variables under consideration T1 and T2 can be computed as the product of the corresponding marginal ones (eq.(16)). Therefore, from Eq.(35) the joint PDF of Y1 and Y2 is given by fY1 ,Y2 (y1 , y2 ) = fT1 (y1 y2 )fT2 (y2 )y2 ξ 2ρ = 2 y1ρ−1 y22ρ−1 e−ξy2 (y1 +1) Γ (ρ)

(36)

In order to compute the marginal PDF of RV Y1 , we integrate the joint one of eq.(36) with respect to y2 . That is Z +∞ y1ρ−1 fY1 (y1 ) = 2ρ 2 y22ρ−1 e−ξy2 (y1 +1) dy2 ξ Γ (ρ) 0 y1ρ−1 y1ρ−1 ξ 2ρ Γ(2ρ) = , (37) = (1 + y1 )2ρ ξ 2ρ Γ2 (ρ) B(ρ, ρ)(1 + y1 )2ρ where the following property of the beta function [51] was used Z 1 Γ(ρ1 + ρ2 ) B(ρ1 , ρ2 ) = xρ1 (1 − x)ρ2 −1 dx = . (38) Γ(ρ1 )Γ(ρ2 ) 0 By integrating eq.(37) we derive the corresponding CDF of the beta prime distribution given by Eq.(18) of Lemma 1 and the proof is completed. R EFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201 – 220, Feb. 2005. [2] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms for cognitive radio applications,” IEEE Communications Surveys Tutorials, vol. 11, no. 1, pp. 116–130, 2009. [3] A. Ghasemi and E. Sousa, “Collaborative spectrum sensing for opportunistic access in fading environments,” in First IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks, DySPAN 2005, Baltimore, Maryland USA, Nov. 2005, pp. 131 –136. [4] V. Kostylev, “Energy detection of a signal with random amplitude,” in IEEE International Conference on Communications- ICC 2002, vol. 3, 2002, pp. 1606–1610 vol.3. [5] F. Digham, M.-S. Alouini, and M. K. Simon, “On the energy detection of unknown signals over fading channels,” in IEEE International Conference on Communications - ICC 2003., vol. 5, May 2003, pp. 3575–3579 vol.5. [6] Y. Zeng, C. Koh, and Y.-C. Liang, “Maximum eigenvalue detection: Theory and application,” in Proc, of the 2008 IEEE International Conference on Communications, ICC 2008, Beijing, China, May 2008, pp. 4160 –4164. [7] Y. Zeng and Y.-C. Liang, “Eigenvalue-based spectrum sensing algorithms for cognitive radio,” IEEE Trans. on Communications, vol. 57, no. 6, pp. 1784 –1793, June 2009.




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