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EUSIPCO 3 569736677 FULLY ISTRIBUTE SIGNAL ETECTION: APPLICATION TO COGNITIVE RAIO Franc Iutzeler Philippe Ciblat Telecom ParisTech, 46 rue Barrault 753 Paris, France email: firstnamelastname@telecom-paristechfr ABSTRACT Cooperative detection out fusion center has many applications such as spectrum sensing in cognitive radio or intrusion detection in ad hoc networ In this paper, we propose a new asynchronous fully-distributed cooperative algorithm which does not require any nowledge on the underlying nodes networ iscussion about the threshold choice the respective duration of the sensing step the gossip step is conducted Index Terms Cooperative detection, spectrum sensing, fully-distributed decision INTROUCTION There are some applications where agents have to detect rapidly the presence or absence of a signal of interest For instance, one can mention the spectrum sensing in cognitive radio or the intrusion detection in military mobile ad hoc networs In order to mae an accurate decision, these agents/nodes may have to cooperate each other The traditional way to fix this problem consists in providing hard or soft detection decisions to a fusion center This centralized approach is clearly sensitive to fusion center failure Moreover, in ad hoc networs context, fusion center election protocol routing protocol have to be carried out which is costful in terms of overhead time Therefore, designing fully distributed decision algorithms is of great interest Such algorithms rely on decision test function decision threshold computed in a distributed way, ie, when only exchanges of local data neighbor are allowed Cooperative detection has recently received a lot of attention see [] references therein Nevertheless, most wors assume the existence of a fusion center finally focus on the design of operations done at each node in order to help the fusion center to mae the right decision In the literature, only a few algorithms are fully distributed in the sense defined above [, 3, 4] An important difference is that sensing gossiping steps are alternated in [, 3] whereas sensing steps come before gossiping steps in [4]These algorithms are well adapted to time-varying environments but they suffer The wor has been partially funded by GA French efense Agency from difficulties in computing the threshold distributively Indeed, in [, 4], the threshold is chosen in an asymptotic regime performances especially, false alarm probability are not ensured in finite time In [3], the threshold is chosen assuming the absence of diffusion/gossiping step Hence, threshold distributed computation remains an open issue We thus propose a new fully distributed signal decision algorithm based on a recently-developed gossiping algorithm [7] where sensing steps are followed by gossiping steps, where the threshold is chosen adequately In addition, thans to the separation of both steps, we are able to optimize their durations at the expense of less adaptivity compared to [, 3] This paper is organized as follows: in Section, we introduce the signal model In Section 3, we remind some results of centralized cooperative detection In Section 4, we propose a new fully-distributed cooperative detection algorithm Threshold distributed computation is discussed, ROC curves are derived In Section 5, numerical results confirm our claims SYSTEM MOEL We consider a networ of K nodes collaborating to detect the presence or absence of a signal The received signal at time n on node writes y n We assume that the duration of sensing is the same for all nodes equal to N s Let y = [y y N s ] T be the sensing data associated node, where the superscript T sts for the transposition operator The signal to potentially detect is denoted by x = [x x N s ] T at node where x n corresponds to its value at time n Finally, an additive noise can disturbed the detection is denoted by n = [n n N s ] T at node where n n corresponds to its value at time n Let N m, Σ be a Gaussian vector mean m covariance matrix Σ n is a iid Gaussian vector of distribution N, σ I N s where I Ns sts for the identity matrix of size N s Throughout the paper, we assume that the statistics of x n are nown at node The hypothesis test dealing our problem can thus be written as follows { H :, y = n H :, y = x + n

In the context of cognitive radio which is our main application of interest, we prefer to consider hypothesis test targeting a fixed probability of detection so a variable false alarm probability that we wish to minimize rather than the stard Neyman-Pearson approach Indeed, in this context, the secondary users the nodes in our framewor should not disturb the primary user the signal to detect in our framewor up to a pre-defined probability Moreover, a high false alarm probability only implies that the secondary users do not use the white spaces while they could Therefore we would lie to minimize this false alarm probability According to the approach developed in [5], one can prove that our optimal test minimizing the false alarm probability given a target probability of detection still boils down to the so-called Lielihood Ratio Test LRT Λy := log py H py H λ where py H is the probability density of y given the tested hypothesis H where λ is chosen such that the target probability of detection, denoted by P target, is ensured 3 REVIEW ON CENTRALIZE COOPERATIVE SPECTRUM SENSING Before going further, we remind some important results about centralized cooperative spectrum sensing We focus, on the one h, on an energy-based detector when the sought signal is unnown, on the other h, on a training-based detector when the sought signal is nown thus corresponds to a training sequence [3] 3 Energy-based detector When the sought signal is unnown, it is usual to assume x is an zero-mean iid Gaussian vector covariance matrix γ I N s Then the Signal-to-Noise Ratio SNR at node is equal to SNR := γ /σ is assumed to be nown at node Assuming independence of the received signals at different nodes this assumption is reasonable since even if the same signal is transmitted by the primary user, the rom wireless channel leads to independent received signals between nodes, the test given in Eq achieved at the fusion center can be decomposed as follows: Λy = = Λ y Λ y = logpy H /py As x N, γ I N s n N, σ I N s, we obtain the following test by removing the constant terms T y := K K y H γ + SNR η 3 σ = where η must be chosen such as PT y > η H = P target In order to compute the threshold η, we need to exhibit the probability density of T Unfortunately, due to unequal SNRs, T is not χ -distributed In [8], it is advocated that the density of T can be approximated a Gamma distribution, denoted by Γκ, θ, whose the probability density function is equal to g κ,θ defined by g κ,θ x = Γκθ κ xκ e x/θ, x, 4 otherwise The terms κ θ are chosen in order to match the mean the variance of T In the following, we denote its cumulative distributive function by G κ,θ its inverse by G κ,θ After some algebraic manipulations, we obtain that T Γκ T, θ T κ T = KN s K = SNR K K = SNR θ T = K K = SNR K K = SNR 6 One can then deduce that the optimal threshold given the target probability of detection is 3 Training-based detector η = G κ T,θ T P target We now assume that each node has the nowledge of the possible transmit signal x Typically, the signal x may decomposed as h x where h corresponds to the nown channel fading between the node the sought transmitter x is a training sequence [3] Here, the signal power is γ = x /N s Then the test given in Eq taes the following form T y := K K = y T x σ 5 H η 7 As x is deterministic, T is Gaussian-distributed mean m T variance ςt given by K m T = N s SNR ςt = N s K SNR K K K = = As a consequence, the threshold is obtained as follows η = ς T Q P target + mt where Q is the inverse of the Gaussian tail function 4 FULLY ISTRIBUTE ALGORITHM The purpose of this paper is to perform detection in a distributed way, ie, out fusion center Obviously, tests described in Eqs 3-7 are not computable since a node may

not have the contribution of the others To overcome this problem, we propose to introduce a gossiping step in order to compute the involved averages Prior to this gossiping step, the nodes perform a sensing step so that each node can provide the term { y t y = SNR /γ + σ if energy detector y Tx /σ if training detector Let N g be the duration of the gossiping step T = N s + N g be the total duration of the processing Most gossiping algorithms for averaging can tae the following form T y T K y = P t y t K y K where T y is the final test function at node, where P = p l,l=,,k corresponds to the considered gossiping algorithm matrix after N g iterations see [6] for more details 4 Energy-based detector When an energy-based detection is carried out, the final test function at node is T y = K l= p l y l γ l + σ l SNR l H η, where η is the threshold at node Actually, the main issue in this paper is to find a distributive way for selecting a good threshold at any time, ie, ensuring the target probability of detection P target as close as possible at any step of the algorithm Once again, by assuming that T is well approximated by a Gamma distribution, we obtain that η = G κ,θ P target where P target is the target probability of detection associated node, K κ = N s l= p lsnr l 8 l= p l SNR l θ = l= p l SNR l l= p lsnr l 9 Then, we obtain the Receiver Operating Characteristic ROC curve is equal to P F A = G κ,θ G κ,θ P Actually, the decision is made before the convergence of the gossip algorithm Indeed, assuming a primary user is present, T could be above the threshold whereas the gossip has not still converged to the consensus κ = N s θ = K l= p SNR l l +SNR l l= p l l= p l l= p l SNRl +SNR l SNRl +SNR l SNR l +SNR l Unfortunately, the terms involving p l in Eqs 8-9 prevent to obtain the threshold η in a distributed way at node for ensuring the probability of detection P target To overcome this issue, we propose hereafter two approaches Approach : distributed nowledge of K Actually, on the centralized scheme, the threshold depends on the average of the SNR the square SNR through Eqs5-6 A simple idea is to replace these exact averages the averages obtained thans to the considered gossip algorithm It is clear that if N g is large enough, the obtained thresholds will be close to those of the centralized case also to those described in Eqs 8-9 since p,l is close to /K so p l can be well approximated by p l /K As a consequence, the new threshold is η κ = G κ,θ = KN s P target K l= p lsnr l l= p lsnr l θ = K l= p lsnr l l= p lsnr l This algorithm is still not fully distributed since the nowledge of the number of nodes is required Furthermore, the target probability of detection is not ensured since the real probability of detection, denoted by P, is given by P = G κ,θ G κ,θ P target In contrast, we prove that the ROC curve is the following one P F A = G κ,θ G κ,θ P which is the same as in Eq Consequently, the ROC curve is not degraded due to our approximate threshold, but depends on the gossip algorithm In addition, the operating point in the ROC curve can not fixed a priori Approach : fully distributed In this approach, the nowledge of the number of nodes will not be required anymore Recently, new gossip algorithms, based on the sumweight principle see [7] references therein have been 3

introduced in order to perform fast estimation of the average the sum Let v = [v, v K ] T be the vector whose component v is the value of the node before gossiping eg, SNR or SNR for threshold computation, or t for test function computation For computing simultaneously the average the sum of z, these algorithms rely on the three following variables, given by, z := Qv, w := Q, w e := Qe where the matrix Q represents the gossip algorithm after N g iterations see [7] for more details, is the K-sized vector whose elements are, e is the K-sized vector whose the first component is equal to the others Only the -th component of all involved vectors is available at node Then, each node calculates the -th component of z p = z w z s = z w e where is the elementwise division In [7], it is proven that z p z s converge to the average the sum of v respectively for large N g In addition, we remar that z p = Pv z s = Sv P = diag Q Q S = diag Qe Q Consequently, the final test function is computed the gossip algorithm related to matrix P given in Eq The threshold are then obtained as follows η κ = G κ,θ = N s P target K l= s lsnr l l= s lsnr l θ l= = p lsnr l l= s lsnr l As T is Gaussian distributed mean m variance ς, we have η = ς Q P target + m m = N s K l= p l SNR l ς = N s p lsnr l l= Once again, this algorithm can not be computed in a distributed way due to the presence of the terms p l in the variance To overcome this issue, previous proposed approaches can be applied straightforwardly 5 NUMERICAL RESULTS Except otherwise stated, the energy-based detector is carried out T = 8, N s = N g P target = 99, performance are averaged over rom geographical graphs K = nodes The SNRs at each node are exponentiallydistributed mean SNR Only performance for the node exhibiting the smallest SNR realization will be plotted Hereafter, we always test the following algorithm configurations: i the centralized one, ii the pairwise gossip PG [6] centralized threshold, iii the pairwise gossip approach based threshold, iv the broadcast sum-weight gossip BWG [7] approach based threshold In Fig, we plot the ROC curve for the above-mentioned algorithm configurations We remar that the ROC curves are 9 8 7 6 Centralized etector PG: centralized threshold PG: Approach threshold BWG: Approach threshold The algorithm is fully distributed since even the number of nodes is not required Once again, the new threshold does not ensure the target probability of detection, the ROC curve is still described by Eq but P given by Eq Finally, for both approaches, the ROC curves converge to the ROC curve related to the centralized case when N g is large enough since the parameters κ, κ, θ θ converge to those of the centralized case P 5 4 3 3 4 5 6 7 8 9 Fig ROC curve P vs P F A 4 Training-based detector When the training-based detector is implemented, the test function at node after N g gossiping iterations is T y = K l= p l y T l x l σ l H η, before gossipping, the second third variables of each node must be initialized to match e respectively For the second variable, each node is initialized to For the third variable, only the first node is initialized to whereas the others to In cognitive radio context, the first node is the secondary user launching the sensing, ie wanting to access the medium very close to each other In addition, when the same gossip algorithm is used, the ROC curve is identical regardless of the threshold technique computation In Figs 3, we display empirical P F A P versus SNR N s respectively So, the loss in false alarm probability for the fully-distributed approach is reasonable Moreover its probability of detection is higher than the target one We also remar that both steps spectrum sensing gossip should have similar durations In Fig 5, these algorithms have been evaluated when the hidden terminal practical configuration described in Fig 4 has been simulated 4

9 9 8 7 6 or P 5 4 3 Centralized etector P Centralized etector PG: centralized threshold P 8 PG: centralized threshold 7 PG: Approach threshold P 6 PG: Approach threshold BWG: Approach threshold P BWG: Approach threshold Target P or P 5 4 3 Centralized detector P Centralized detector PG: centralized threshold P PG: centralized threshold PG: Approach threshold P PG: Approach threshold BWG: Approach threshold P BWG: Approach threshold Target P 5 5 5 Mean SNR db 4 6 8 T Fig P F A P versus SNR Fig 5 P F A P versus T for the hidden terminal 9 Centralized detector P 9 Centralized etector P 8 Centralized detector 8 Centralized etector 7 PG: centralized threshold P PG: centralized threshold 7 PG: Approach threshold P PG: Approach threshold 6 PG: Approach threshold P 6 BWG: Approach threshold P or P 5 4 PG: Approach threshold BWG: Approach threshold P BWG: Approach threshold Target P P or 5 4 BWG: Approach threshold iffusion LMS etector P iffusion LMS etector Target P 3 3 4 6 8 N S 5 5 5 Mean SNR Fig 3 P F A P versus N s Fig 6 P F A P versus SNR for proposed algorithms diffusion LMS [3] [] P Braca, et al, Asymptotic optimality of running consensus in testing binary hypotheses, IEEE Trans on Signal Processing, vol 58, no, Feb [3] F Cattivelli, A Sayed, istributed detection over adaptive networs using diffusion adaptation, IEEE Trans on Signal Processing, vol 59, no 5, May Fig 4 Hidden terminal configuration Finally, in Fig 6, we compare our training-based algorithms to the diffusion LMS one described in [3] Notice that N s N g is incremented by at each iteration of the diffusion LMS We remar that our algorithms outperform the diffusion LMS Actually, our bloc processing for the sensing step is much more efficient that the adaptive LMS one in [3] Moreover, unlie diffusion LMS, our algorithms are asynchronous which simplifies the networ management 6 REFERENCES [4] W Zhang, et al, istributed Cooperative Spectrum Sensing based on Weighted Average Consensus, Globecom [5] H Van Trees, etection, Estimation Modulation Theory : Part I, Wiley, 4 [6] S Boyd, et al, Romized Gossip Algorithms, IEEE Trans on Information Theory, vol 5, no 6, June 6 [7] F Iutzeler, et al, Analysis of Sum-Weight-lie algorithms for averaging in Wireless Sensor Networs, IEEE Trans on Signal Processing, vol 6, no, June 3 [8] P Ciblat, et al, Training Sequence Optimization for joint Channel Frequency Offset estimation, IEEE Trans on Signal Processing, vol 56, no 8, Aug 8 [] E Axell, et al, Spectrum sensing for cognitive radio, IEEE Signal Processing Mag, vol 9, no 3, May 5