Cooperative Spectrum Sensing over Correlated Log-Normal Channels in Cognitive Radio Networks based on Clustering Nima Reisi, Vahid Jamali, Mahmoud Ahmadian, Soheil Salari Faculty of Electrical and Computer Engineering KN-Toosi University of Technology, Tehran, Iran
[email protected],
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Abstract—In this paper, the problem of cooperative spectrum sensing in cognitive radio networks based on linear combination of local observations is considered. In particular, log-normal shadow-fading is considered in both sensing and reporting channels. To reduce the effects of imperfect channel conditions, a clustering algorithm is suggested in which final decision about the primary user activity is obtained based on linear combination of clusters transmits. To calculate the combination weights, we encounter with the problem of the joint distribution approximation for sum of correlated log-normal variables. A joint MGF matching algorithm is proposed to estimate the sums by a single lognormal vector. Monte Carlo simulations confirm the accuracy of the proposed MGF-based approach and efficiency of cluster based spectrum sensing algorithm in terms of primary signal detection. Keywords-cognitive radio, cooperative spectrum sensing, clustering, log-normal shadow-fading, joint MGF estimation I.
INTRODUCTION
Cognitive Radios (CR) are considered as secondary users utilizing a frequency band licensed to a primary user (PU). To avoid harmful interference to PU operation, CRs should first sense the spectrum and only in the case of primary user absence, start to transmit on the band. Therefore, spectrum sensing is one of the key enabling functionalities of the CR technology. Among various spectrum sensing methods including energy detection [14], matched filter [3], feature detection [5], wavelet detection [6], and Eigen value-based detection [7], energy detection is a commonly used method which has the lowest computational complexity and does not require a priori information about the primary signal. Spectrum sensing is so challenging in low SNR regions, which can be caused by severe fading, shadowing or blocking in the CR sensing channel. To address these challenges, cooperative spectrum sensing (CSS) has been investigated in which different CRs share their observations or decisions about the frequency band of interest so that the final decision becomes more reliable [8].
11th International Conference on Telecommunications - ConTEL 2011 ISBN: 978-953-184-152-8, June 15-17, 2011, Graz, Austria
Although most of papers analyze the performance of CSS in i.i.d. Rayleigh channels, several studies demonstrate that shadow-fading is more important to be considered in design of CSS scenario. As a case in point, in [9], it is stated that shadowing is the major aspect to be considered for detection of Digital TV signals, compared to multipath fading. Since cooperating CR nodes are geographically proximate, they will experience similar environmental shadowing effects and thus have correlated shadowing [10]. To cooperate, CR nodes need a common reporting channel to transmit their local observations / decisions to each other or a common receiver known as Fusion Center (FC) which can be either implemented as a dedicated frequency channel or as an underlay UWB channel. In addition to sensing channel, reporting channel is also imperfect and subject to shadow-fading. However, most papers consider reporting channel ideal. It is shown in [11] that reporting channel imperfection will limit the performance of the system. Some clustering and diversity based approaches to mitigate the effects of error in the Rayleigh fading reporting channels are proposed in [11, 12] but effect of shadowing is not considered. Combining the local information of cooperating nodes is one of challenges in CSS scenario. In addition to optimal Likelihood Ratio Test (LRT) method [13], which involves a quadratic form and has high computational complexity, some linear combination methods are also proposed in the literature such as selection combining (SC), Equal Gain Combining (EGC), and Maximum Ratio Combining (MRC) [14]. Linear combination of local observations of cooperating nodes is considered in [15] where the reporting channel is simply assumed to be an AWGN channel. In [16], Linear Quadratic detector is designed for the case of correlated log-normal sensing channels but ideal reporting channels. Performance analysis of CSS under correlated log-normal sensing and reporting channels is considered in [17]. This analysis is, however, only for equal gain combing (EGC) of local statistics and no attempt is performed to calculate the optimum weights of linear combination.
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Another important challenge in energy detection based spectrum sensing is performance degradation due to uncertainty in noise power estimation [2]. Different models for noise uncertainty have been suggested among which the log-normal model with known variance for the noise power at the output of the energy detector is more applicable [18]. Motivated from all mentioned above, the goal of this paper is to consider the CSS problem based on linear combination of local observations under log-normal shadow-fading in both sensing and reporting channels and log-normal uncertainty in noise power estimation and propose a cluster-based CSS to mitigate the effects of reporting channels imperfection. The rest of this paper is organized as follows. Section II is dedicated to the estimation of joint statistics of sum of lognormal RVs in different sets. In section III, an overview of the proposed cluster-based CSS is considered. System model and notations are introduced in section IV. Detection performance of the cluster-based CSS is given in section V. Section VI considers the accuracy of the joint statistics approximation method of log-normal sum. Simulation results of the proposed cluster-based CSS are also considered in this section. Finally, the paper is concluded in section VII. II.
SUM OF LOGNORMAL RANDOM VARIABLES IN DIFFERENT SETS
Before considering the cluster-based CSS, we study a mathematical problem which has a direct application in our scenario. Since sensing and reporting channels are modeled as log-normal variables, to calculate the received power, we need to estimate the statistics of sum of some correlated log-normal RVs. Estimating sum of log-normal RVs by another log-normal RV has been widely considered in the literature [19-22]; however, only one set of RVs is considered in these methods. Herein, we want to approximate the joint distribution of log-normal sums in different sets by a log-normal vector. Let “ ”ܭbe the number of correlated log-normal RVs. We divide these variables into “ ”ܬdifferent sets and sum the elements of each set: ೕ
ݔ ൌ ݁
௬ೕೖ
ǡ ݆ ൌ ͳǡʹǡ ǥ ܬǡ ራ ܭ ൌ ܭ
ୀଵ
ୀଵ
(1)
Our goal is to estimate the sum of log-normal variables in different sets as a log-normal vector. To do so, similar to the works done in [22] we utilize the joint MGF between the logarithm of sum of log-normal distributions in each set. First, consider the logarithm of ݔ and call it ݕ i.e.: ݕ ൌ ሺݔ ሻ. Since ݕ is normally distributed, the joint distribution of vector ࢟ has the form: ࢌࢅ ሺ࢟ሻ ൌ
162
ͳ
ଵ ݁ ݔቆെ
ሺʹߨሻ ଶ ȁȁଶ
ሺࢅ െ ࣆሻࢀ ି ሺࢅ െ ࣆሻ ቇ ʹ
(2)
where ࣆ and denote the mean vector and covariance ்
matrix of ࢟ ൌ ቂݕଵǡଵ ǡ ݕଵǡଶ ǡ ǥ ǡ ݕଵǡభ ǡ ǥ ǡ ݕǡଵ ǡ ݕǡଶ ǡ ǥ ǡ ݕଵǡ ቃ , respectively. The joint log-MGF of sum of variables in different sets can be calculated according to (3) on the top of the following page. Let ௦ ൌ ࢁ ܂be the square root of covariance matrix ൌ ࢁࢁࢀ . By the use of de-correlating transformation ࢅ ൌ ξʹ࢙ ࢆ ࣆ and Gauss-Hermite expansion with respect to ܼ , the joint MGF will be estimated via (4) on the top of the next page ( ࢙ is divided into J blocks each has ܭ rows and ܭcolumns. ߤ is also divided into ܬblocks each has ܭ elements). In ƍ this equation, ܿǡ is elements of ݆ 'th block of ࢙ , ܪൌ ςୀଵ ܪ . ܪ , ܲ are the weights and zeros of the Np-order Hermite polynomial. First and second order joint moments of ݕ ሾ݊ ൌ ͳǡ ǥ ǡ ܬሿ can be calculated by derivations of ߮ࢅ ሺ࢙ሻ according to (5) and (6) on the top of the next page. Hence, the covariance between variables ݕ and ݕ can be easily calculated by (7) on the top of the following page. To conclude, via joint MGF matching, we can estimate ் the vector ࢞ ൌ ൣݔଵ ǡ ݔଶ ǡ ǥ ǡ ݔ ൧ as a log-normal random ்
vector with parameters ൌ ൣ݉ଵ ǡ ݉ଶ ǡ ǥ ǡ ݉ ൧ and મ where ݉ and ሺમሻǡ can be calculated from (5) and (7), respectively. III. OVERVIEW OF CLUSTER-BASED CSS We consider a primary user (PU) and a secondary network consists of some CR nodes and a common receiver managing the secondary network. To mitigate the imperfection of reporting channel and reduce the complexity in calculation of linear combination weights cooperating CR nodes are divided into some clusters. Here, we assume that cooperative nodes are clustered according to their correlation coefficients in a way that nodes with highly correlated received signals, will be lied in the same cluster. Since sensing channel is subject to correlated log-normal shadowing and the correlation is exponentially related to the distances among nodes, choosing the most correlated nodes for each cluster is equivalent to clustering cooperative nodes based on their distances from each other. Our model is assumed to have infrastructure; hence, BS, which can be the same as FC, has knowledge about the location of CR nodes. Thus, clustering procedure is performed under control of BS. After the clusters are formed, clusters heads should be selected regarding to the status of reporting channels. Again, since BS has, or can obtain, information about the statistics of reporting channels, it has the responsibility for choosing cluster heads. Owing to the fact that nodes in the same cluster are near each other, it can be assumed that their reporting channels have equal means. Therefore, the node with the smallest fluctuation in its reporting channel is selected as the head of the cluster. The responsibility of cluster heads is to relay their own and other nodes observations to FC. Finally, by exploiting energy detection, average power of received
ConTEL 2011, ISBN: 978-953-184-152-8
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߮௬భǡ௬మ ǡǥǡ௬ ൫ݏଵ ǡ ݏଶ ǡ ǥ ǡ ݏ ൯ ൌ
߮ࢅ ሺ࢙ሻ ؆ ቆ
ͳ ξߨ
ͳ
ିஶ ିஶ
ே
ቇ భ ୀଵ మ ୀଵ
ൌቆ
ξߨ
ୀଵ
಼ ୀଵ
ା
ቇ
ೕ
ሺࢅ െ ࣆሻ் ିଵ ሺࢅ െ ࣆሻ ቇ ݀ݕଵଵ ݀ݕଵଶ ǥ ݀ݕ ʹ
ି௦ೕ
ƍ ǥ ܪ൦ෑ ቐ ݁ ݔ൭ ξʹ ܿǡ ܲ ߤǡ ൱ቑ
߲߮࢟ ൫࢙࢟ ൯ ȁ ߲ݏ ߲ݏ ௦ ୀǡבǡ ͳ
ିஶ ୀଵ
ே
߲߮࢟ ൫࢙࢟ ൯ ͳ ܧሺݕ ሻ ൌ െ ȁ௦ ୀǡב ൌ ቆ ቇ ߲ݏ ξߨ
ܧሺݕ ݕ ሻ ൌ
ଵ න න ǥ න ෑ ݁ݔ൫െݏ ݕ ൯ ݁ ݔቆെ
ሺʹߨሻ ଶ ȁȁଶ
ே
ே
ୀଵ
ே
ே
ே
(4)
൪
ୀଵ
ே
ƍ ା
ƍ ǥ ܪൈ ݈݊ ቐ ݁ ݔቌ ξʹ ܿǡ ܲ ߤǡ ቍቑ భ ୀଵ మ ୀଵ
಼ ୀଵ
ೕ
ே
಼ ୀଵ
ୀǡ
ୀଵ
signals from cluster heads are calculated in the FC and combined linearly to make the final test statistic. Clustering has at least two advantages: first, in cluster based sensing, all observations of nodes in a cluster will be sent through the best reporting channel in that cluster. Therefore, compared to conventional cooperation, performance degradation due to imperfect reporting channel will be decreased. And second, in conventional linear combination, number of weights is equal to the number of cooperating nodes. In contrast, in cluster based sensing, number of required weights will be reduced to the number of clusters which would be much less than the number of cooperating nodes. IV. SYSTEM MODEL Our model consists of a PU, a secondary network with ܰ CR nodes and a common receiver. ܩis the number of clusters in the secondary network. The number of nodes in ݃ ’th cluster is assumed to be ܰ݃ . Without loss of generality, the received power from primary user is assumed to be normalized to the transmitted power and the effect of path loss will be included in the sensing channel coefficients. Sensing channel is assumed to be noisy and subject to correlated log-normal shadowing. Hence, ࢻ, the vector of sensing channel coefficients, has parameters ࣆ࢙ and ܛwhere ߤ௦ǡ is the path loss value in dB divided by a constant ߞ ൌ ͳͲȀ݈݊ͳͲbetween PU and ݅ ’th CR node ௦ ൌ ߪ௦ଶ ߩିௗǡೕ is the covariance value between ݅'th and ሺǡሻ and ݆ ’th nodes with distance equals to ݀ǡ . ߩ ൌ ݁ ଵȀ represents the correlation between two nodes separated by unit distance, ܦ is correlation distance [23]
(6)
א
ܿݒሺݕ ǡ ݕ ሻ ൌ ܧሺݕ ݕ ሻ െ ܧሺݕ ሻܧሺݕ ሻ
ConTEL 2011, ISBN: 978-953-184-152-8
(5)
ƍ ାଵ ୀ
ୀଵ
ƍ ǥ ܪᇱ ෑ ݈݊ ቐ ݁ ݔቌ ξʹ ܿǡ ܲ ߤǡ ቍቑ భ ୀଵ మ ୀଵ
(3)
(7) and ߪ௦ଶ is the network variance [16] in the presence of primary signal. Reporting channel is subjected to uncorrelated lognormal shadowing. According to the same mean values of reporting channels of a cluster, the node with the least variance value of reporting channel is selected as the head of that cluster. When the cluster heads are selected, there are only ܩnodes that transmit to the FC. We model the reporting channels of cluster heads as ࢼ ൌ ሾߚଵ ǡ ߚଶ ǡ ǥ ǡ ߚீ ሿ் which is log-normally distributed with parameters ଶ ଶ ଶ ࣆ࢘ ൌ ሾߤଵ ǡ ߤଶ ǡ ǥ ǡ ߤீ ሿ் and ܚൌ ݀݅ܽ݃ൣߪǡଵ ǡ ߪǡଶ ǡ ǥ ǡ ߪǡீ ൧. Noise components at the nodes, cluster heads, and FC are called sensing, intra-cluster, and reporting noises, respectively. Noise uncertainty is considered by modeling the noise power as a log-normal distribution with known variance [16]. Without loss of generality, the power of all noise components can be modeled with the same parameters ߤ௪ and ߪ௪ଶ ࡵ where and ࡵ are the all-one vector and unitary matrix of appropriate size, respectively. V.
PERFORMANCE ANALYSIS
A. Problem Formulation According to the previous section, the received signal in each node is defined as follows: ௦ ሺ݊ሻ ݔǡ ሺ݊ሻ ൌ ܫ௨ ߙǡ ݏሺ݊ሻ ݓǡ ݅ ൌ ͳǡʹǡ ǥ ܰ ǡ ݃ ൌ ͳǡʹǡ ǥ ǡ ܩ
(8)
where ݏሺ݊ሻ is the transmitted primary signal which is assumed to have unit power, ߙǡ is the sensing channel coefficient, ܫ௨ אሼͲǡͳሽ is an indicator which shows the ௦ ሺ݊ሻ is sensing status of primary user activity and ݓǡ
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noise with log-normal uncertainty in estimation of its power. In local spectrum sensing, the ݅’th node calculates the average power of the M-sample received signal according ଵ ݔሺ݊ሻଶ ሺ݊ሻ. Hence, ܲ distribution under ܪ toܲ ൌ σெ ெ ୀଵ (idle PU) and ܪଵ (active PU) hypotheses can be stated as: ݈ܰ݃ሺߤ௪ ǡ ߪ௪ଶ ሻǡ ܲ ̱ ቊ ݈ܰ݃൫ߤଵǡ ǡ ȭଵǡ ൯ ǡ
ܪ ܪଵ
(9)
where ൫ߤଵǡ ǡ ȭଵǡ ൯ are the parameters of the log-normal estimate of sum of two uncorrelated log-normal RVs with parameters ሺߤ௪ ǡ ߪ௪ଶ ሻ and ൫ߤ௦ǡ ǡ ߪ௦ଶ ൯ related to noise and primary signal components, respectively, calculated from [17, equations 8-11]. This value is then compared with a predefined threshold࣎, to determine the status of the desired spectrum band. The probabilities of making correct decision underܪଵ and making erroneous decision underܪ , which are denoted by probabilities of detection ሺܲௗ ሻ and false alarm൫ܲ ൯, respectively, can be calculated as: ߬ െ ߤ௦ǡ ቇ ܲௗǡ ൌ ܲݎሺ݈݊ሺܲ ሻ ߬ȁܪଵ ሻ ൌ ܳ ቆ ඥߪ௦ଶ (10) ߬ െ ߤ௪ ሻ ሻ ܲ ൌ ܲݎሺ݈݊ሺܲ ߬ȁܪ ൌ ܳ ቆ ቇ ඥߪ௪ଶ On the other hand, in cluster-based CSS, cooperating nodes in each cluster transmit their observations to the cluster heads which in turn combine them and their own observations and transmit the resultant signals to the FC in an orthogonal manner (TDM or FDM). It is assumed that in each sensing period, the nodes in clusters are ordered in a manner that the index 1 will be assigned to the cluster head. As a result, the received signal at the FC from ݃'th cluster head is: ே
ୀଵ
(11)
ሺ݊ሻ and ݓ ሺ݊ሻ are the reporting and intrawhere ݓǡ cluster noises with log-normal uncertainty and ݏ ሺ݊ሻ and ݓ ሺ݊ሻ consist of terms related to primary signal and noises, respectively. In the FC, the received signals powers for ܯsamples are calculated individually according to ܲ ൌ ଵ ெ σ ݕଶ ሺ݊ሻ , and combined linearly to make the test ெ ୀଵ statistic ܼ as in (12): ீ
ܼ ൌ ݓ ܲǤ ǡ ܲǤ ൌ ݈݊൫ܲ ൯ ୀଵ
(12)
The test statistic ܼ is then compared to a predefined threshold to decide about presence/absence of the primary signal: if ܼ is larger than threshold, FC decides
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B. Null and alternate hypotheses When the primary user is idle ሺܪ ሻ , ݏ ሺ݊ሻ in (11) equals to zero and ܲ is the sum of ܰ ʹ uncorrelated log-normal variables with known parameters. Hence, the sum can be estimated as another log-normal variable [17, equations 8-11]. Consequently, ࡼ ൌ ሾܲଵ ǡ ܲଶ ǡ ǥ ǡ ܲீ ሿ் is estimated as a log-normal vector with parameters ࣆ and . On the other hand, when the primary system is active ሺܪଵ ሻ , ܲ is sum of ܰଶ correlated and ܰ ʹ uncorrelated log-normal variables with known parameters. Therefore, ࡼ can be estimated as another lognormal vector with parameters ࣆ and obtained according to (5-7). Therefore, power of received signals in FC can be estimated as log-normal variables with known parameters under ܪ and ܪଵ hypotheses: ݈ܰ݃ሺࣆ ǡ ሻ ǡ ࡼ̱ ൜ ݈ܰ݃ሺࣆ ǡ ሻ ǡ
ܪ ܪଵ
(13)
C. Calculating the linear combining weigths Inasmuch as obtaining the optimum weights mathematically is complicated, we will consider the suboptimum solution based on maximizing deflection criterion under normalized weights constraint [15]: ሾܧሺܼȁܪଵ ሻ െ ܧሺܼȁܪ ሻሿଶ ݎܽݒሺܼȁܪଵ ሻ ሺࣆଵ் ࢝ െ ࣆ் ࢝ሻଶ ൌ ்࢝ ଵ ࢝ ݏǤ ݐǤ ԡ࢝ԡ ൌ ͳ ݉ܽ ݀ ࢝ݔଶ ൌ
(14)
Using Cholesky decompositionଵ ൌ ࡸࡸ் and applying linear transformation ࢝ᇱ ൌ ࡸ் ࢝ and, the solution is given by: [24] ࢝ᇱ ̱ݒ௫ ሺࡸିଵ ࣆઢ ࣆ்ઢ ࡸି் ሻ ൌ ߩࡸିଵ ࣆઢ
ݕ ሺ݊ሻ ൌ ߚ ቐൣݔǡ ሺ݊ሻ൧ݓ ሺ݊ሻቑ ݓ ሺ݊ሻ ൌ ݏ ሺ݊ሻ ݓ ሺ݊ሻ ǡ ݃ ൌ ͳǡʹǡ ǥ ǡ ܩ
in presence of primary signal otherwise, the final decision will be the absence of primary signal.
(15)
where ݒ௫ is the eigen-vector corresponding to the largest eigen-value of matrix ࡸିଵ ࣆઢ ࣆ்ઢ ࡸି் , ࣆઢ ൌ ሺࣆ െ ࣆ ሻ் and ߩ is the adjustment actor depends on the optimization constraint. Hence, deflection-based optimum weights can be calculated according to (16). ࢝ௗ ൌ
ଵିଵ ࣆઢ ԡଵିଵ ࣆઢ ԡଶ
(16)
Let ࣎ be the decision threshold. Then, probabilities of detectionand false alarm can be calculated as: ܲௗ ൌ ܲݎሺܼ ߬ȁܪଵ ሻ ൌ ܳ ቆ ܲ ൌ ܲݎሺܼ ߬ȁܪ ሻ ൌ ܳ ቆ
߬ െ ࣆ ࢝ࢀ
ඥ࢝ ࢝ࢀ ߬ െ ࣆ ࢝ࢀ ඥ࢝
࢝ࢀ
ቇ
ቇ
(17)
After the weights are determined, decision threshold
ConTEL 2011, ISBN: 978-953-184-152-8
COST IC0802 Workshop
sense channel
report channel noise
variance vector mean vector
ોଶୱ ૄ୰
variance vector
ોଶ୰
power vector uncertainty vector
ૄ୵ ોଶ୵
value ʹ ͵ dB ---ୱ ൈ ି܌బ
S=[ 3 3 2 ] , Np = 6
0.9 0.8 0.7
ʹ mentioned in each simulation set
pairwise correlation
Table I. Parameters used for simulation Parameter symbol path loss factor Į PU power ୱ nodes distances from PU ܌ mean vector ૄୱ
ρ 13
0.5
ρ 12
ρ 23
0.6
0.4 0.3 0.2 0.1 0
߬ൌ
ܳିଵ ሺܲி ሻඥ࢝ ࢝ࢀ
ࢀ
ࣆ ࢝
(18)
VI. SIMULATION RESULTS The aim of this section is twofold: first, to illustrate the accuracy of the proposed algorithm for approximating the vector of log-normal sums. Second, to analyze the performance of the proposed cluster-based CSS via Monte Carlo simulations. A. Sum of log-normal RVs in different sets Estimation of covariance matrix between log-normal sums in different sets is the main motivation of the proposed joint MGF-based algorithm. To peruse the precision of the algorithm in covariance matrix estimation, different sets of correlated log-normal RVs are formed. The process of constructing RVs, especially the correlation matrix, is the same as procedure explained in section III for sensing channel coefficients. We use this form so that the performance of algorithm can be analyzed in terms of one variable i.e. ܦ . Therefore, increasing ܦ is equal to increasing the correlation coefficients between log-normal RVs and vice versa. Figures 1 and 2 compare the resultant covariance values between log-normal sums in different sets of proposed algorithm to covariance values obtained from Monte Carlo simulations in terms of correlation distanceܦ . In these figures, 8 correlated log-normal RVs are divided into 3 sets. Number of RVs in sets are ܵ ൌ ሺ͵ǡ ͵ǡ ʹሻ and (4, 2, 2) in Fig. 1 and Fig. 2, respectively. Maximum distance between elements of each set is fixed to 10 and ܦ changes from zero (uncorrelated case) to 20 (highly correlated). Vectors ࣆ and ࣌ଶ are uniformly generated in intervals (0, 4) and (0, 2), respectively. Both figures confirm the accuracy of the proposed approximation method for estimation of correlation coefficients between log-normal sums in different sets. It is necessary to mention that the accuracy of the proposed approach is validated for different number of scenarios. However, for the sake of conciseness, just the results of two scenarios are depicted in this article. B. Clustering performance To evaluate the performance of proposed cluster-based
ConTEL 2011, ISBN: 978-953-184-152-8
0
2
4
6
8
10 Dc
12
14
16
18
20
Fig. 1. Correlation values of log-normal sums in 3 sets. Solid lines represent joint-MGF-based approximation and markers denote Monte Carlo simulation. ࣆ ൌ ሾ͵ǤͻͶͲǡ ʹǤͶʹͶǡ ͵Ǥͷ͵Ͷǡ ͵Ǥͷͷǡ ʹǤͲͶͳǡ ʹǤͲͺǡ ͵Ǥ͵Ͷͻǡ ʹǤ͵ͳͷሿ் ࣌ଶ ൌ ሾͳǤͻ͵ͺǡ ͳǤͻͷǡ ͳǤʹͶǡ ͳǤ͵ͻǡ ͳǤ͵ͷǡ ͳǤͷʹǡ ͳǤͶͷǡ ͳǤͷʹ͵ሿ் S=[ 4 4 2 ] , Np = 6
1
0.8
pairwise correlation
can be calculated according to tolerable probability of false alarmሺܲி ሻ:
-0.1
ρ 13
0.6
ρ 12
ρ 23
0.4
0.2
0
-0.2 0
2
4
6
8
10 Dc
12
14
16
18
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Fig. 2. Correlation values of log-normal sums in 3 sets. Solid lines represent joint-MGF-based approximation and markers denote Monte Carlo simulation. ࣆ ൌ ሾʹǤ͵ͺǡ ʹǤʹͳͷǡ ʹǤͳͺͲǡ ʹǤͻͶǡ ʹǤͻͻǡ ʹǤ͵ͳͶǡ ʹǤͺʹͺǡ ʹǤͳͳሿ் ࣌ଶ ൌ ሾͳǤͺͷͷǡ ͳǤͶͶͷǡ ͳǤͻͳ͵ǡ ͳǤ͵͵ǡ ͳǤͻǡ ͳǤͷͻǡ ͳǤͺͷǡ ͳǤ͵ʹͶሿ்
CSS, different sets of simulations are performed. In our detection problem, the performance metric is the Receiver Operating Characteristic (ROC) curve which is the probability of detection in terms of probability of false alarm൫ܲ ൯. The nodes are assumed to be in squares with length ܮൌ ͳͲ in each cluster. Distances between PU and centers of clusters (at most 4 clusters in following simulations) are also assumed to be ൣࡸ௫ ǡ ࡸ௬ ൧ with ȁࡸ௫ ȁ ൌ หࡸ௬ ห ൌ ͷͲ. Correlation distance is also set to be ܦ ൌ ͷͲ . Other simulation parameters are shown in Table I. Fig. 3. compares the theoretical and simulation results of cluster-based CSS in terms of ROCs for 10 cooperating nodes. ROC curves are depicted for 2, 3, and 4 clusters. Simulation results are averaged on different conditions of reporting channels and different topologies. Variance values of reporting channels are generated from
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uniform distribution in the interval ofሺͲǡ ʹሻ. As it is obvious from the curves, differences between MonteCarlo simulations and analytical results are negligible which results the reliability of given formulas for performance analysis of cluster-based CSS. The comparison between cluster-based and conventional, i.e. no clustering, CSS methods is also depicted in Fig. 4. for ܰ ൌ ͳͲ cooperating nodes divided into clusters of sizes 4 and 6. To evaluate the performance of cluster-based CSS, we keep the fluctuations of cluster heads reporting channels fixed while increasing those of other nodes. As the curves show, conventional CSS performance in terms of PU detection decreases by increase in variance values of cooperating nodes. On the other hand, performance of cluster-based CSS relies only on the fluctuations of cluster heads reporting channels hence, does not change by the variations of other nodes reporting channels. Effect of increasing the number of cooperating nodes while number of clusters is kept constant, is realized from Fig. 5. where ROC curves of conventional and clusterbased CSS are depicted for 7, 10, and 15 nodes divided into two clusters, averaged on different conditions of reporting channels and different topologies. As it is apparent from the curves, increasing the number of nodes, the differences between ROC curves of noncluster and cluster-based schemes increase. Hence, it can be concluded that cluster-based CSS is more applicable for larger number of nodes. I.
CONCLUSION
In this paper, we proposed a cluster-based CSS in realistic environment i.e. imperfect sensing and reporting channels subjected to log-normal shadow-fading. The uncertainty in noise power estimation is modeled lognormally. It is shown that cluster-based CSS outperforms the conventional CSS especially for larger numbers of cooperating nodes. It is obvious that sensing time is increased in cluster-based CSS scheme. On the other hand, computational complexity of proposed method is less because required number of weights is decreased to the number of clusters.
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Furthermore, an extension to existing approximations is investigated where by the use of joint MGF matching, sum of correlated log-normal variables in different sets is estimated as a log-normal vector. Simulation results confirm the accuracy of our estimation approach. REFERENCES [1]
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