Multiple Symbol Differential Decision Fusion for Mobile Wireless Sensor Networks

Transcription

1 Multiple Symbol Differential Decision Fusion for Mobile Wireless Sensor Networks Andre Lei and Robert Schober University of British Columbia, alei, Abstract We consider the problem of decision fusion in mobile wireless sensor networks where the channels between the sensors and the fusion center are time variant. We assume that the sensors make independent local decisions on the M hypotheses under test and report these decisions to the fusion center using differential phase shift keying DPSK), so as to avoid the channel estimation overhead entailed by coherent decision fusion. For this setup we derive the optimal and three low complexity, suboptimal fusion rules which do not require knowledge of the instantaneous fading gains. Since all these fusion rules exploit an observation window of at least two symbol intervals, we refer to them collectively as multiple symbol differential MSD) fusion rules. Simulation and analytical results confirm the excellent performance of the proposed MSD fusion rules and show that in fast fading channels significant performance gains can be achieved by increasing the observation window to more than two symbol intervals. I. INTRODUCTION Decentralized detection is an important task in wireless sensor networks WSNs) [1], [2]. To limit complexity, the sensors usually make independent decisions based on their respective observations and forward these decisions over the wireless channel to a fusion center which forms a final decision on the hypothesis under test. Most of the existing literature on the decentralized detection problem assumes ideal error free communication between the sensors and the fusion center. While this is a reasonable assumption for wired sensors, it may lead to significant performance degradations if wireless sensors are employed. Therefore, the problem of fusing sensor decisions transmitted over noisy fading channels has received considerable interest recently. For example, channel aware decision fusion for phase coherent WSNs employing phase shift keying PSK) modulation was investigated in [3], [4]. In [5], channel statistics based fusion rules for WSNs employing on/off keying OOK) modulation were considered. The impact of fading on the performance of power constrained WSNs was studied in [6]. Most recently, the impact of channel errors on decentralized detection was studied for PSK, OOK, and frequency shift keying FSK) modulation in [7]. Interestingly, existing work on decision fusion for noisy fading channels has mainly considered coherent e.g. PSK) and noncoherent e.g. OOK, FSK) modulation schemes. While the former are suitable for static fading channels, the latter are appropriate for extremely fast fading channels, where the fading gain changes from symbol to symbol due to e.g. fast frequency hopping. However, for applications where the fading gains change slowly over time due to the mobility of the sensors and/or fusion center, noncoherent modulation may not be a preferred choice due to the inherent loss in power efficiency compared to coherent modulation. On the other hand, coherent modulation requires the insertion of pilot symbols for channel estimation which reduces spectral efficiency and complicates system design. Thus, for conventional point to point communication systems differential PSK DPSK) is often preferred for signaling over time varying fading channels [8]. While DPSK does not require instantaneous channel state information CSI) for detection, the performance loss compared to coherent PSK can be mitigated by multiple symbol differential detection MSDD) if statistical CSI is available at the receiver [9], [10], [11]. This motivates the investigation of DPSK for transmission in WSNs and the design of corresponding fusion rules. In this paper, we consider the decentralized M ary hypothesis testing problem in time variant fading channels. We assume that the sensors employ M DPSK to report their local decisions to the fusion center and derive corresponding multiple symbol differential MSD) fusion rules. Since the complexity of the optimal fusion rule is exponential in both the number of sensors and the observation window size used for MSD decision fusion, we propose three suboptimal fusion rules with significantly lower complexity and good performance. All considered fusion rules only require statistical CSI but not any knowledge about the instantaneous channel gains. For the special case of binary hypothesis testing M = 2), we provide performance bounds for the optimal fusion rule, and analytical expressions for the probabilities of false alarm and detection for the so called max log fusion rule, which achieves the best performance among the considered suboptimal fusion rules. Our analytical and simulation results show that significant performance gains can be achieved by increasing the observation window size of the MSD fusion rules to more than two symbols. This paper is organized as follows. In Section II, we introduce the system model. The optimal and suboptimal fusion rules are derived in Section III, and the performance analysis is presented in Section IV. In Section V, simulation and numerical results are provided, and conclusions are drawn in Section VI. Notation: In this paper, bold upper case and lower case letters denote matrices and vectors, respectively. [ ] T, [ ] H, ), and E denote transposition, Hermitian transposition, complex conjugation, and statistical expectation, respectively. δ[ ] and ū[ ] refer to the discrete time Delta and unit step functions, respectively. In addition, [X] i,j, det ), I X,, and diagx stand for the element of matrix X in row i and column j, the determinant of a matrix, the X X identity matrix, the Kronecker product, and a diagonal matrix with the entries of vector x on its main diagonal, respectively. Finally, P ) and p ) are used to denote probabilities and probability

2 density functions pdf), respectively. In particular, PA B) and pa b) denote the probability of event A conditioned on event B and the pdf of random variable a conditioned on random variable b, respectively. II. SYSTEM MODEL In this paper, we consider the distributed multiple hypothesis testing problem where a set K 1, 2,..., K of K sensors are used to decide which one out of M possible hypotheses H i, i M, M 0, 1,..., M 1, is present. The a priori probability of hypothesis H i is denoted by PH i ), i M. A. Processing at Sensors At time n Z each sensor k K makes an M ary decision u k [n] based on its own noisy observation x k [n]. We assume that the K observations x k [n], k K, are independent of each other, conditioned on the M different hypotheses. The sensors map their local decisions to M ary PSK M PSK) symbols a k [n] w i i M, w i e j2πi/m, such that hypothesis H i corresponds to the PSK symbol w i. The differential phase symbols a k [n] are differentially encoded before transmission over the wireless channel to obtain the absolute phase symbols s k [n] = a k [n]s k [n 1], 1) where s k [n] w i i M. This differential encoding operation facilitates detection without CSI at the receiver which is particularly useful for transmission over time variant fading channels [8]. In the context of WSNs, such time variant channels may arise for example in vehicular WSNs with mobile sensors and/or mobile fusion centers [12], battlefield surveillance [13], or collaborative spectrum sensing with mobile nodes [14]. To keep our model general, we quantify the quality of the local decisions made by the sensors in terms of conditional probabilities P k a k [n] = w j H i ), i M, j M, k K. B. Channel Model The sensors communicate with the fusion center over orthogonal flat fading channels using e.g. a time division multiple access TDMA) protocol. The received signal from sensor k at time n is given by y k [n] = PK h k [n]s k [n] + n k [n], 2) where P K P/K with total transmitted power P, and h k [n] and n k [n] denote the fading gain and zero mean complex valued additive white Gaussian noise AWGN), respectively. The noise is independent, identically distributed i.i.d.) with respect to both the sensors, k, and time, n, and has variance σ 2 n E n k[n] 2. We assume independent, non identically distributed i.n.d.) Rayleigh fading with fading gain variances σ 2 k E h k[n] 2, k K. For the temporal correlation of the fading gains, we adopt Clarke s model with ϕ hh,k [λ] Eh k [n+λ]h k [n] = σ2 k J 02πB k Tλ), where B k denotes the Doppler shift of sensor k and T denotes the time interval between two observations y k [n] and y k [n + 1]. Note that if the sensors use TDMA to report their observations in a round robin fashion to the fusion center, T is equal to T = KT s, where T s is the symbol duration. It is also interesting to observe that the effective Doppler shift B k T increases with decreasing data rate since T increases with decreasing data rate. C. Fusion Center Processing Since the differential encoding operation in 1) introduces memory, symbol by symbol information fusion is not optimum. Instead, results from point to point communication systems suggest that the received signals should be processed on a block by block basis [9], [10], [11]. If blocks of received signals are properly processed, performance improves as the block size N 2 increases and approaches the performance of coherent detection for N [9], [15]. Here, we adopt the same philosophy for information fusion and process blocks of N received signals y k [y k [n N + 1] y k [n N + 2]... y k [n]] T corresponding to blocks of N 1 differential symbols a k [a k [n N+2] a k [n N+3]... a k [n]] T, k K. Based on these blocks of received signals any one of the N 1 differential symbols in a k can be detected and corresponding fusion rules will be discussed in the next section. III. MSD DECISION FUSION In this section, we will derive the optimal and several suboptimal fusion rules for the system model introduced in Section II. For derivation of the considered fusion rules, we assume that the fusion center has knowledge of both the statistical properties of the channel and the performance indices P k a k [n] = w j H i ), i M, j M, of the sensors k K. To simplify our notation, we will address in the following the νth element of vectors y k and a k by y k ν), 1 ν N, and a k ν), 1 ν N 1, respectively. We denote the index of the differential symbol considered for detection by ν 0, ν 0 1, 2,..., N 1. To simplify the notation further, we will drop the index of the differential symbol considered for detection wherever possible and denote it by a k = a k ν 0 ). A. Optimal Fusion Rule The optimal fusion rule based on the observations y [y T 1 y T 2... y T K ]T can be formulated as Hî = argmax logph i y)) + α i, 3) H i, i M where α i is a bias term which allows the prioritization of certain hypotheses. Since we assume that fading and noise are independent across different sensors, the conditional probability PH i y) can be rewritten as PH i y) = py H i)ph i ) py) = K p k y k H i )PH i ). 4) p k y k ) Furthermore, the conditional pdf p k y k H i ) of sensor k can be expanded as p k y k H i ) = = M 1 j=0 p k y k a k = w j )P k a k = w j H i ) 5) M 1 1 M N 2 p k y k a k )P k a k = w j H i ), j=0 a k A j

3 where A j contains all M N 2 possible vectors a k with a k = w j and the conditional pdf p k y k a k ) is given by [10] p k y k a k ) = 1 π N detr k ) exp r H k R 1 k r k). 6) Here, r k [r k [n N + 1] r k [n N + 2]... r k [n]] T with r k [n] y k [n]s k [n] and R k Er k r H k = P K R hh,k +σni 2 N, where [R hh,k ] i,j = ϕ hh,k [i j]. Combining 3) 6) and omitting all irrelevant terms yields the optimal MSD fusion rule K M 1 Hî = argmax log p k y k a k ) H i i M j=0 a k A j ) + β i P k a k = w j H i ) where β i α i + logph i )) denotes the new bias term. Discussion: Despite its optimal performance, the MSD fusion rule in 7) has several short comings: a) The complexity of the fusion rule in 7) is exponential in both K and N. b) Because of the large dynamic range of the exponential functions in 6), especially for high channel SNRs i.e., PK σk 2/σ2 n 1), the optimal fusion rule in 7) causes numerical problems, especially in fixed point implementations. c) The optimal fusion rule requires statistical CSI in form of R k ) and knowledge of the sensor performance in form of P k a k = w j H i )). The above listed drawbacks of the optimal fusion rule motivate the search for suboptimal fusion rules, which overcome most of these problems but still provide good performance. B. Chair Varshney CV) Fusion Rule The complexity of the optimal fusion rule can be tremendously reduced by assuming that the double sum in the second line of 5) is dominated by the maximum likelihood ML) vector â k [â k 1)... â k N 1)] T, which maximizes p k y k a k ), i.e., p k y k â k ) p k y k a k ), a k â k, k K. This assumption is valid for high channel SNR. In this case, the optimal fusion rule can be simplified to K Hî = argmax H i, i M logp k y k â k )P k â k H i )) + β i 7), 8) where â k = â k ν 0 ) denotes the element of â k which is considered for detection. We note that the ML vectors â k, k K, can be efficiently obtained from y k by applying the multiple symbol differential sphere decoding MSDSD) algorithm in [16, Fig. 1]. Eq. 8) can be regarded as the MSD version of the familiar CV fusion rule [2]. Discussion: The complexity of the suboptimal fusion rule in 8) grows only linearly in the number of sensors K. Furthermore, for sufficiently high channel SNR the average complexity of MSDSD is polynomial in N [16], and thus, the complexity of the proposed fusion rule is also polynomial in N. Similar to the optimal fusion rule, both statistical CSI and knowledge of the sensor performance are required for the CV fusion rule. C. Fusion Rule for Ideal Local Sensors ILS) The derivation of the CV fusion rule implicitly assumed that the uncertainty about the hypothesis at the fusion center originates only from the local sensor decisions, whereas the channel between the sensors and the fusion center was assumed ideal. Another extreme case is when we assume that the local sensor decisions are ideal, i.e., P k a k = w i H i ) = 1 and P k a k = w j H i ) = 0 for j i, and the uncertainty at the fusion center is due to the noisy transmission channel only. In this case, a k = a, k K, is valid and the optimal ML block decision rule for a is given by K â = argmax logp k y k a)) + β i, 9) a where the bias β i is determined by the trial symbol a = aν 0 ) = w i, i M, and the hypothesis estimate Hî can be directly obtained from the relevant element â = âν 0 ) = wî of â. The computational complexity of the fusion rule in 9) is only linear in K but still exponential in N if a brute force search over all possible a is conducted. Similar to the CV fusion rule, the application of sphere decoding is the key to reducing complexity further. For this purpose, we rewrite 9) as K ŝ = argmin s H U H k U k s β i, 10) s, sn)=1 where U k L H k diagy k ) is an upper triangular matrix and L k is a lower triangular matrix obtained from the Cholesky factorization of R 1 k L k L H k. s [s1) s2)... sn)] T contains the absolute phase symbols from which the elements of a are obtained as aj) = sj + 1)s j). Because of the phase ambiguity inherent to 10), we can set sn) = 1 without loss of generality. The sum over k and the bias term β i in 10) make a direct application of the MSDSD algorithm in [16] impossible. However, as will be explained in the following, a modified version of MSDSD can be used to solve 10) efficiently provided that β i 0, i M. The latter condition can always be fulfilled by properly choosing α i, i M. The modified MSDSD only examines candidate vectors that meet K s H U H k U ks β i R 2, 11) where R is a pre defined radius. Assuming we have found preliminary) decisions ŝl) for the last N l components sl), ν + 1 l N, we can define an equivalent squared length N 2 d 2 ν+1 = K N u k lµŝµ) βî δ[ν 0 ν 1], 12) l=ν+1 µ=l where u k νµ [U k ] ν,µ and βî is obtained from ŝν 0 + 1)ŝ ν 0 ) = â = wî. Comparing 11) and 12), possible values for sν) have to satisfy d 2 ν R2 with K d 2 ν = N uk νν sν) + 2 u k νµŝµ) β i δ[ν 0 ν] + d 2 ν+1 µ=ν+1 13)

4 where β i is determined by ŝν 0 +1)s ν 0 ) = a = w i. Once a valid vector ŝ is found, i.e., ν = 1 is reached, the radius R is dynamically updated by R := d 1, and the search is repeated starting with ν = 2 and the new radius R. If condition 13) cannot be met for some index ν, ν is incremented and another value of sν) is tested. Based on 11) 13) a modified MSDSD algorithm for ILS decision fusion can be implemented similar to the conventional MSDSD algorithm in [16, Fig. 1]. Discussion: The complexity of the ILS fusion rule is linear in K and for high channel SNR polynomial in N. While statistical CSI is still required, the local sensor performance P k a k = w j H i ), i M, j M, does not have to be known at the fusion center. Of course, the price to be paid for this advantage is a loss in performance compared to the optimal fusion rule. D. Max Log Fusion Rule For high channel SNR i.e., PK σk 2/σ2 n 1, k K) one of the exponentials in 7) will be dominant and the max log approximation, which is well known from the Turbo coding literature [17], can be applied K Hî = argmax max logp k y k a k )) H i, i M j M,a k A j + logp k a k = w j H i )) + β i. 14) The max log fusion rule in 14) is computationally more efficient and numerically more stable than the optimal fusion rule in 7) since exponential functions are avoided in 14). However, if 14) is implemented in a straightforward fashion, its computational complexity is still exponential in N, since for every test hypothesis H i, i M, the maximum of logp k y k a k )) has to be found over all a k A j, j M. Fortunately, the max log fusion rule can be rewritten as Hî = argmax H i, i M K max j M logp k y k â j k )) ) + logp k â k = w j H i )) + β i 15) where â j k is that a k A j which maximizes p k y k a k ). â j k can be efficiently computed using sphere decoding. In particular, the MSDSD algorithm in [16, Fig. 1] can be slightly modified to account for the fact that the search is constrained to those a k with aν 0 ) = a = w j. For the binary case, M = 2, 15) is equivalent to choosing H 1 if likelihood ratio Λ m exceeds threshold γ 0 = β 0 β 1, and H 0 otherwise, where Λ m is defined as ) K Λ m max min p k y log k â j k )P kâ k = w j H 1 ) j M i M p k y k â i k )P. kâ k = w i H 0 ) 16) Discussion: The complexity of the max log fusion rule is linear in K and for high channel SNR polynomial in N. The implementation of 15) and 16) requires both statistical CSI and knowledge of the sensor performance P k a k = w j H i ), i M, j M. Furthermore, the complexity of the max log fusion rule is higher than that of the CV and ILS fusion rules discussed in Sections III-B and III-C, respectively. In particular, for the max log fusion rule, MSDSD has to be performed MK times, whereas the CV and ILS fusion rules require only K and one MSDSD operations, respectively. On the other hand, the max log fusion rule achieves a superior performance compared to the CV and ILS fusion rules, cf. Section V. IV. PERFORMANCE ANALYSIS OF FUSION RULES An analysis of the optimal fusion rule does not seem to be possible. For the CV and the ILS fusion rules exact expressions for the probabilities of false alarm and detection can be obtained for N = 2, while for N > 2 accurate approximations can be derived for the CV fusion rule. However, because of space limitation, we omit the analysis for the CV and ILS fusion rules here and focus on the analysis of the max log fusion rule since a) the analysis of the max log fusion rule is more involved and b) the max log fusion rule has a superior performance. To make the analysis tractable, we assume M = 2, i.e., M = 0, 1, i.i.d. channels, i.e., σ 2 k = σ2, B k = B, R k = R, and p k y k a k ) = py k a k ), k K, and identical sensors with probability of false alarm P f Pa k = 1 H 0 ) = P k a k = 1 H 0 ) and probability of detection P d Pa k = 1 H 1 ) = P k a k = 1 H 1 ), k K. A. Performance Bounds Before considering the max log fusion rule, we briefly discuss two performance upper bounds valid for any fusion rule including the optimal one. 1) Bound I: For the first bound, we assume that all sensors make correct decisions and decision errors at the fusion center are due to transmission errors only, i.e., a k = a, k K, and zero bias, i.e., β 0 = β 1 = 0. In this case, the sensor network is equivalent to a point to point transmission with K fold receive diversity and conventional MSDD [10] is the optimal fusion rule. Thus, the probabilities of false alarm and detection are given by P f0 = BER ν0 and P d0 = 1 BER ν0, 17) where BER ν0 denotes the probability that a = aν 0 ) was transmitted and â a, â ±1, 1 ν 0 N 1, was detected, i.e., BER ν0 is the bit error rate BER) for 2 PSK symbol aν 0 ) for point to point transmission and MSDD at the receiver. Hence, BER ν0 can be straightforwardly calculated using the results in [15]. The corresponding expression is omitted here. Eq. 17) constitutes a performance upper bound for any fusion rule with noisy sensors. This bound becomes tight for optimal decision fusion if transmission errors dominate the overall performance, which is the case for example at low channel SNRs and for highly reliable local sensors. 2) Bound II: For the second bound, we assume a noise free transmission channel, i.e., the decision errors at the fusion center are caused by local decision errors at the sensors only. In this case, the CV fusion rule is optimum and the

5 corresponding probabilities of false alarm and detection are given by [2] K ) K P yi = Px κ κ i 1 P xi ) K κ, i M, 18) κ= K where y 0 = f 0, y 1 = d 0, x 0 = f, x 1 = d, and K, 0 K K, is a parameter that allows to find a desired trade off between P f0 and P d0. For realistic, noisy transmission channels, 18) constitutes a performance upper bound which becomes tight for high channel SNRs. B. Analysis of Max Log Fusion Rule For the max log fusion rule, the probabilities of false alarm and detection can be expressed as P yi = Pr Λ m < γ 0 H i, i M, 19) cf. 16). Denoting the Laplace transform of the pdf of the negative log likelihood ratio Λ m by Φ m s H i ), 19) can be rewritten as P yi = 1 2πj c+j c j Φ m s H i )e γ0s ds, i M, 20) s where c is as small positive constant that lies in the region of convergence of the integral. Since P f0 and P d0 can be obtained by numerical integration from 20) if Φ m s H i ) is known, the remainder of this section will be devoted to the calculation of this Laplace transform. As the fading gains and noise samples in the different diversity branches are i.i.d., respectively, Φ m s H i ) can be expressed as Φ m s H i ) = Φ z s H i )) K, i M, 21) where Φ z s H i ) denotes the Laplace transform of the pdf of ) z k max min py log k â j k )Pâ k = w j H 0 ) j M i M py k â i k )Pâ. 22) k = w i H 1 ) Φ z s H i ) can be rewritten as Φ z s H i ) = 1 P xi )Φ z s â = 1, a = 1) +P xi Φ z s â = 1, a = 1), 23) where Φ z s â, a) denotes the Laplace transform of the pdf of z k given a k = a and â. For calculation of Φ z s â, a) it is useful to note that for M = 2, 22) can be expressed as ) maxpy z k = log k â 0 k )1 P f), py k â 1 k )P f maxpy k â 0 k )1 P d), py k â 1 k )P. 24) d Using the definition y logpy k â 0 k )/py k â 1 k )) a=1 and assuming P f < 0.5 and P d > 0.5, we can show that 24) can be rewritten as z k = β 1, ay < b 1 ay + β 2, b 1 ay b 2 β 3, ay > b 2, 25) where β 1 logp f /P d ), β 2 log1 P f )/P d ), β 3 log1 P f )/1 P d )), b 1 logp f /1 P f )), and b 1 β 3 z k β 1 Fig. 1. Illustration of relationship between z k and y for a = 1. Note that with the definitions in Section IV-B, b 1 < 0, b 2 > 0, β 1 < 0, and β 3 > 0 as long as P d > 0.5 and P f < 0.5. b 2 b 2 logp d /1 P d )). To arrive at 25) for a = 1, we have exploited logpy k â 0 k )/py k â 1 k )) a=1 = logpy k â 0 k )/py k â1 k )) a= 1. For convenience 25) is illustrated in Fig. 1 for the case a = 1. Fig. 1 reveals that the max log fusion rule soft limits the log likelihood ratios y of the individual sensors at the fusion center by taking into account the a priori values P f and P d. Denoting the pdf of y by p y y) and exploiting 25), we can express Φ z s â = 1, a) as Φ z s â = 1, a) = b1 b2 + e sy+β2) p y ay)dy + b 1 e sβ1 p y ay)dy b 2 e sβ3 p y ay)dy.26) For calculation of p y y), we distinguish in the following the cases N = 2 and N > 2. N = 2: For N = 2, the only possible error event leading to â = 1 is ŝ = [ s1) 1] T and y can be expressed as y = r H k R 1 k r k s + r H k R 1 k r k ŝ, where s = [s1) 1] T. In other words, y is simply the decision variable for conventional differential detection. Thus, the Laplace transform of p y y) is given by [15, Eq. 27)] Φ y s) = v 1 v 2 s + v 1 )s v 2 ), 27) where v 1 2 = 1 + 1/ρ ν0 1)/2, ρ ν [ R 1] νν P K + σ 2 n) 1, with ν 0 = 1. From 27) we can calculate p y y) as p y y) = c v e v 1yūy) + e v2y ū y) ), 28) where c v v 1 v 2 /v 1 + v 2 ). Combining 26) and 28) we obtain e sβ1+vjb1 Φ z s â = 1, a) = c v + e sβ2 1 e s+vj)b1 s + v j v j + 1 e s+vi)b2 s + v i ) y + e sβ3 vib2 v i ) 29) where i, j) = 1, 2) and i, j) = 2, 1) for a = 1 and a = 1, respectively. N > 2: For N > 2 the problem is more difficult, since there are more than one possible error events that lead to âν 0 ) = â = 1. The most likely error events

6 are ŝ ν0 and ŝ ν0+1 which differ from s only in positions ν 0 and ν 0 + 1, respectively. The corresponding likelihood ratios are denoted by y 1 logpy k â 0 k )/py k â1 k )) ŝ ν0 and y 2 logpy k â 0 k)/py k â 1 k)) ŝν0. To make the problem +1 tractable, we assume that ŝ ν0 and ŝ ν0+1 are the only relevant error events, i.e., we neglect all other error events, which is a valid approximation for high channel SNRs. In this case, y is given by y = miny 1, y 2. In order to get closed form results, we make the following two additional approximations: a) y 1 and y 2 are independent and b) y 1 and y 2 are identically distributed. Both assumptions are justified for high channel SNRs. By exploiting results from order statistics [18] and [15], we obtain for the pdf of y p y y) = 2p y1 y)1 P y1 y)), 30) where p y1 y) = c v e v1y ūy) + e v2y ū y)), cf. 28), P y1 y) = y p y 1 x)dx, and v 1 2 = 1 + 1/ρ ν0 1)/2 with 1 ν 0 N 1. Combining 26) and 30) leads to a closed form expression for Φ z s â = 1, a) similar to 29). We do not provide this expression here because of space limitation. Combining 20), 21), 23), 29), and the corresponding expression for Φ z s â = 1, a) for N > 2, the probabilities of false alarm and detection can be exactly approximately) computed for N = 2 N > 2). We note that a direct numerical integration of 20) is problematic since the inverse Laplace transform of Φ m s H i ) has discontinuities reflected e.g. by the first and last term in the sum on the right hand side of 29)). However, the terms corresponding to the discontinuities can be easily inverted in closed form, and the remaining terms without discontinuities can then be inverted numerically using the methods in [19]. V. NUMERICAL AND SIMULATION RESULTS In this section, we present numerical and simulation results for the proposed optimal and suboptimal fusion rules. For all results shown in this section, the middle symbol of the observation window is used for detection, i.e., ν 0 = N/2, since this leads to the best performance. In order to confirm our simulation results with the analytical results from Section IV, we assume M = 2, i.i.d. Rayleigh fading channels, identical sensors, and PH 0 ) = PH 1 ) = 1/2. All curves labeled with Theory for N = 2) and Approximation for N = 6) were generated using the analytical methods presented in Section IV for the max log fusion rule and similar methods not presented in this paper) for the CV and ILS fusion rules, while the remaining curves were obtained by computer simulation. In Fig. 2, we consider the error probability P e P f0 PH 0 ) + 1 P d0 )PH 1 ) of the considered suboptimal MSD fusion rules vs. E b /N 0 E b : total average received energy per bit from all sensors); N 0 : one sided power spectral density of underlying continuous time noise process). Zero bias β 0 = β 1 = 0) was used for all fusion rules and K = 8, BT = 0.1, P d = 0.8, and P f = In addition to the suboptimal MSD fusion rules, Fig. 2 also contains the two performance upper bounds introduced in Section IV-A and the performance of the optimal fusion rule for N = 2 the optimal fusion rule is computationally not feasible for N = 6) and the error probability of the coherent max log fusion rule for DPSK. We note that the coherent fusion rule requires perfect knowledge of the fading channel gains. Fig. 2 shows that while the ILS fusion rule has the best performance for very low E b /N 0, where transmission errors dominate the overall performance, the CV and the max log fusion rule yield a superior performance for medium to high E b /N 0. For N = 2, the performance of both the CV and the max log fusion rules is limited by the high error floor caused by the fast fading. This error floor is also not overcome by the optimal fusion rule which yields a negligible performance gain compared to the computationally simpler max log fusion rule for N = 2. For N = 6 this error floor is mitigated and both the CV and the max log fusion rules approach Bound II for high E b /N 0, i.e., performance is limited by local sensor decision errors in this case and not by transmission errors. For the ILS fusion rule increasing the observation window to N > 2 is not beneficial. This somewhat surprising behavior is caused by the local decision errors at the sensors, which were ignored for derivation of the ILS fusion rule. For N = 2, theoretical and simulation results in Fig. 2 match perfectly confirming the analysis in Section IV. As expected from the discussions in Section IV, for N = 6, there is a good agreement between theoretical and simulation results for the CV and the max log fusion rules at high E b /N 0 ratios. At low E b /N 0 ratios, the analytical results overestimate the actual P e since the assumptions leading to the analytical result for N > 2 are less justified. In Fig. 3, we show P d0 as a function of E b /N 0 for a fixed probability of false alarm of P f0 = 0.001, which is achieved by adjusting the bias terms, β 0 and β 1, accordingly. Furthermore, K = 8, P d = 0.7, P f = 0.05, and BT = 0.1. In Fig. 3, the max log fusion rule yields a superior performance compared to the other suboptimal MSD fusion rules but the CV and ILS fusion rules approach the max log performance for high and low E b /N 0, respectively. For N = 6, both the max log and the CV fusion rules approach Bound II for high enough E b /N 0, whereas for N = 2, these fusion rules as well as the optimal fusion rule are limited by transmission errors caused by the fast fading. In contrast, the ILS fusion rule achieves a better performance for N = 2 than for N = 6. Fig. 3 shows again a good agreement between analytical and simulation results. In Fig. 4, we show the receiver operating curve ROC) for the considered MSD fusion rules and the coherent max log fusion rule for DPSK. K = 8, P d = 0.7, P f = 0.05, BT = 0.1, and E b /N 0 = 20 db. Fig. 4 shows the superiority of the max log fusion rule especially if low probabilities of false alarm are desired. Increasing N from two to six yields significant gains for both the max log and the CV fusion rules. In fact, the max log fusion rules with N = 6 bridges half of the performance gap between the coherent max log fusion rule and the MSD max log fusion rule with N = 2. On the other hand, for N = 2 the optimal fusion rule performs only slightly better than the max log fusion rule. We note that the gap between the coherent and the MSD max log fusion rules is even smaller for smaller value of BT.

7 10 0 N = 2 Simulation) N = 2 Theory) N = 6 Simulation) N = 6 Approximation) Optimal Fusion Rule N = 2) Coherent Max Log Max Log 10 1 ILS 0.8 CV 0.7 Pe 10 2 Pd0 0.6 CV Max Log 0.5 ILS 10 3 Bound I Bound II E b/n 0 [db] Fig. 2. Probability of error P e vs. E b /N 0 for zero bias β 0 = β 1 = 0). K = 8, M = 2, P d = 0.8, P f = 0.01, BT = 0.1, and i.i.d. Rayleigh fading N = 2 Simulation) N = 2 Theory) N = 6 Simulation) N = 6 Approximation) Optimal Fusion Rule N = 2) Coherent Max Log P f0 Fig. 4. Probability of detection P d0 vs. probability of false alarm P f0. K = 8, M = 2, P d = 0.7, P f = 0.05, BT = 0.1, E b /N 0 = 20 db, and i.i.d. Rayleigh fading. Pd N = 2 Simulation) N = 2 Theory) N = 6 Simulation) N = 6 Approximation) Optimal Fusion Rule N = 2) Coherent Max Log Max Log Bound II E b/n 0 [db] Fig. 3. Probability of detection P d0 vs. E b /N 0 for a probability of false alarm of P f0 = K = 8, M = 2, P d = 0.7, P f = 0.05, BT = 0.1, and i.i.d. Rayleigh fading. VI. CONCLUSIONS In this paper, we have considered the distributed multiple hypothesis testing problem for mobile wireless sensor networks where sensors employ DPSK to cope with time variant fading. We have shown that since the differential modulation introduces memory, it is advantageous to consider fusion rules that base their decisions on an observation window of multiple symbol intervals. Specifically, we have derived the optimal MSD fusion rule, whose complexity is exponential in the number of sensors and the observation window size, and three suboptimal MSD fusion rules, whose complexity is linear in the number of sensors and, at high SNR, polynomial in the observation window size. Our simulation and analytical results show that the CV and ILS fusion rules approach the performance of the optimal fusion rule for high and low channel SNRs, respectively. The proposed max log fusion rule achieves a close to optimal performance over the entire SNR range but has a higher complexity than the CV and ILS fusion rules. CV ILS REFERENCES [1] R. Tenney and N. Sandell, Jr.. Detection with distributed sensors. IEEE Trans. on Aerospace and Electronics Systems, 174): , July [2] P. K. Varshney. Distributed Detection and Data Fusion. Springer Verlag, New York, [3] B. Chen, R. Jiang, T. Kasetkasem, and R. K. Varshney. Channel aware decision fusion in wireless sensor networks. IEEE Trans. Signal Processing, 52: , December [4] R. Niu, B. Chen, and P. K. Varshney. Fusion of decisions transmitted over Rayleigh fading channels in wireless sensor networks. IEEE Trans. Signal Processing, 54: , March [5] R. Jiang and B. Chen. Fusion of censored decisions in wireless sensor networks. IEEE Trans. Wireless Commun., 4: , Nov [6] J.-F. Chamberland and V. Veeravalli. The impact of fading on decentralized detection in power constrained wireless sensor networks. In Proc. IEEE Intern. Conf. Acoustics, Speech, and Signal Processing ICASSP), pages 17 21, May [7] V. Kanchumarthy, R. Viswanathan, and M. Madishetty. Impact of channel errors on decentralized detection performance of wireless sensor networks: A study of binary modulations, Rayleigh-fading and nonfading channels, and fusion-combiners. IEEE Trans. Signal Processing, 56: , May [8] J.G. Proakis. Digital Communications. McGraw Hill, New York, forth edition, [9] D. Divsalar and M. K. Simon. Multiple-symbol differential detection of MPSK. IEEE Trans. Commun., 38: , March [10] D. Fung and P. Ho. Error performance of multiple-symbol differential detection of PSK signals transmitted over correlated Rayleigh fading channels. IEEE Trans. Commun., 40: , October [11] D. Divsalar and M. K. Simon. Maximum-likelihood differential detection of uncoded and trellis coded amplitude phase modulation over AWGN and fading channels Metrics and performance. IEEE Trans. Commun., 42:76 89, January [12] L. Song and D. Hatzinakos. Architecture of wireless sensor networks with mobile sinks: Sparsely deployed sensors. IEEE Trans. Veh. Technol., 56: , July [13] A. Durresi, M. Durresi, and L. Barolli. Secure mobile communications for battlefields. In Proc. Intern. Conf. Complex, Intelligent and Software Intensive Systems, pages , March [14] J. Unnikrishnan and V.V. Veeravalli. Cooperative sensing for primary detection in cognitive radio. IEEE Trans. Signal Processing, 2:18 27, February [15] V. Pauli, R. Schober, and L. Lampe. A unified performance analysis framework for differential detection in MIMO Rayleigh fading channels. IEEE Trans. Commun., 56: , November [16] L. Lampe, R. Schober, V. Pauli, and C. Windpassinger. Multiple-symbol differential sphere decoding. IEEE Trans. Commun., 53: , December [17] P. Hoeher, P. Robertson, and E. Villebrun. A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain. In Proc. IEEE Intern. Conf. Commun. ICC), pages , Seattle, June [18] H. David and H. Nagaraja. Order Statistics. Wiley, [19] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset. Computing error probabilities over fading channels: A unified approach. European Transactions on Telecommunications, 9:15 25, January-Feburary 1998.