Fusion of Decisions Transmitted Over Fading Channels in Wireless Sensor Networks

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1 Fusion of Decisions Transmitted Over Fading Channels in Wireless Sensor Networks Biao Chen, Ruixiang Jiang, Teerasit Kasetkasem, and Pramod K. Varshney Syracuse University, Department of EECS, Syracuse, NY 3244 Abstract Information fusion by utilizing multiple distributed sensors is studied in this work. We derive the optimal likelihood based fusion statistic for a parallel decision fusion problem with fading channel assumption. This optimum fusion rule, however, requires perfect knowledge of the local decision performance indices as well as the fading channel. Several alternatives are presented that alleviate these requirements. At low SNR, the likelihood based fusion rule reduces to a form analogous to a maximum ratio combining statistic; while at high SNR, it leads to a two-stage approach using the well known Chair-Varshney fusion rule. Athird alternative in the form of an equal gain combiner is also proposed that requires the least amount of information regarding the sensor/channel. Simulation shows that the two-stage approach, which considers the communication and decision fusion as two independent stages, suffers performance loss compared with the other two alternatives for practical SNR range. Introduction Wireless sensor networks (WSN) have generated intensive interest from the research community with current and future envisioned applications ranging from battlefield surveillance, environment and structure monitoring, to telemedicine. Much of current research effort on WSN is focused on the development of energy efficient routing protocols, distributed data compression and transmission schemes, and collaborative signal processing algorithms as documented in [] and references therein. The fact that locally processed information is transmitted through a fading channel has not attracted much attention. While channel fades may be treated purely as a communication issue and thus be dealt with exclusively through transceiver design, we contend in this paper that a more cohesive approach that integrate the communication with the information processing should be adopted. Consider a distributed detection problem in the context of WSN where a number of sensors are used to detect the possible presence of a target. The problem of distributed detection, and in particular, decision fusion, has been studied extensively in the past decades. Indeed, in the absence of fading channel consideration, optimum fusion rules have been obtained for both binary and multibit (soft) local sensor output under the conditional independence assumption [2, 3]. Fusion rules with statistically correlated observations have also been investigated though the results are considerably more involved [4, 5]. Decision fusion with uncertainty has also been investigated and a Bayesian sampling approach has been proposed to address this issue [6]. Decision fusion under a communication constraint has also been considered [7, 8, 9,, ]. The constraint, however, is often in the form of the total number of bits allowed [9,, ]. The actual transmission is still idealized, i.e., the information sent from local sensors is assumed intact at the fusion center. While this assumption may be reasonable for some applications, it may not be realistic for many wireless sensor networks where the transmitted informaiton has to endure both channel fading and noise/interference. Decision fusion with non-ideal channels has been investigated by Thomopoulos and Zhang [2]. The local decisions are transmitted over noisy channels so that they may not be correctly received at the fusion center. Yet the channel model is simplified as a binary channel thus does not allow a full integration of signal transmission into information processing. In this paper, we study the practical WSN scenario where decisions are transmitted over fading channels. The emphasis is on the integration of the communication and information fusion. The organization of the paper is as follows. In the next section we formulate the parallel fusion problem with noisy and fading communication channel layer and derive the optimal LR based fusion rule with binary local decisions. Section 3 provides three alternative fusion rules: a two-stage approach using the Chair-Varshney fusion rule, a maximum ratio combining (MRC) fusion statistic, and an equal gain combiner (EGC) fusion statistic. Performance analysis is given in Section 4 where we find that under the Rayleigh fading assumption, the last alternative, namely the EGC statistic, is the most robust fusion alternative even though it requires minimum information. We conclude in Section 5. 2 Decision fusion under fading channel assumption 2. Problem formulation Fig. depicts a typical parallel fusion structure where a number of sensors observe data generated accordingtoeitherh or H, the two hypotheses un-

2 der test. Each sensor processes its observations and makes a preliminary decision about the hypothesis before sending it to a fusion center. In the conventional parallel fusion paradigm, the fading channel layer is not considered and the information sent from individual sensors is assumed to be received intact at the fusion center. For WSN with limited resources, the effect of channel fade and noise renders the information received at the fusion center to be unreliable. While increased transmission power along with channel coding may be implemented, they may not be enough (or desirable) for many WSN application with limited resources. Incorrectly received local decisions will lead to performance loss. This loss can be minimized by properly considering and including the effect of channel impairments in the derivation of the decision fusion rule. Extending the classical parallel fusion problem by incorporating the fading channel layer, we consider the decision fusion problem as described in Fig.. The following assumptions are made to simplify our analysis and allow us to gain better insights from the results.. The k th local sensor make a binary decision u k {+, }, with false alarm and detection probabilities P fk and P dk respectively. That is, P [u k = H ]=P fk and P [u k =H ]=P dk.noticethat these performance indices for local sensors may vary from sensor to sensor. 2. Each local decision u k is transmitted through a fading channel and the output of the channel (or input to the fusion center) for the k th sensor is y k = h k u k + n k () where h k is some non-negative number and n k is zero mean Gaussian with variance σ 2. This flat fading channel model is reasonable for many lowpower WSN operating at short ranges (hence small delay spread) with low bit rate. Our goal now is to derive a fusion rule based on y k, for k =,,K that balances the performance with its requirement on aprioriinformation. 2.2 Optimal LR based fusion rule We derive, using the model specified in the previous section, the optimal likelihood ratio (LR) based fusion rule by assuming complete knowledge regarding the fading channel and the local sensor performance indices, i.e., the P fk and P dk values. Assuming conditional independence of observations at the sensors and that each local sensor makes a binary decision, the final LR test statistic can be derived in a straightforward manner as Λ(y) = K P dk e (y k h k ) 2 2σ 2 +( P dk )e (y k +h k )2 2σ 2 P fk e (y k h k )2 2σ 2 +( P fk )e (y k +h k )2 2σ 2 (2) where y =[y,,y K ] T is a vector containing observations received from all K sensors and σ 2 is the variance of additive white Gaussian noise for all channels. An H /H P d /P f P d2 /P f2 P dk /P fk Sensor Sensor 2 Sensor K u u 2 u K h h 2 h K n n 2 n K y y2 y K Fusion Center Figure : Parallel fusion model in the presence of fading and noisy channel between local sensors and the fusion center. implicit assumption is that all the channel outputs are co-phased. This assumption allows us to deal exclusively with real observations. While the form of the LR based fusion rule is straightforward to implement, it does need both the local sensor performance indices and complete channel knowledge. Suboptimum fusion rules that relieve the above requirements are more desirable. u 3 Suboptimum fusion rules The first two alternatives are proposed as high and low SNR approximations (or equivalent, to be more precise) to the LR fusion rule that partially relieve some of the requirements associated with the LR based fusion rule. Further, motivated by the form of the second alternative, we propose another alternative that requires minimum aprioriinformation, and, as it turns out, enjoys the most robust performance among the three suboptimal approaches for a wide range of channel SNR values. 3. A two stage approximation using the Chair-Varshney fusion rule The fusion rule specified in (2) jointly considers the effects of the fading channel and the local sensor output to achieve optimal performance. A direct alternative is to separate this into a two stage process first y k is used to infer about u k, and then, the optimum fusion rule based on u k (assuming that the estimates are reliable), as derived in [2] can be applied. Given () and with u k {, } and h k >, the maximum likelihood (ML) estimate for u k is simply û k = sign(y k ). Applying the fusion rule derived in [2], herein termed

3 the Chair-Varshney fusion rule, we obtain Λ = log P dk + log P dk (3) P fk P fk sign(y k )= sign(y k )= Clearly, this statistic is a good approximation for Λ for large channel SNR when the ML estimate on u k tends to be most likely correct. Indeed, Λ is mathematically equivalent to Λ for the large SNR case. Proposition As the channel noise variance σ 2, i.e., SNR,thelogarithmofΛ defined in (2) reduces to Λ defined in (3), i.e., lim log Λ = Λ σ 2 Notice that Λ does not require any knowledge regarding the channel gain but does require P dk and P fk for all k. Further, this two-stage approach falls into the conventional thinking that separates the communication and signal processing aspects. We show later through numerical examples that this two-stage approach suffers significant performance loss at low to moderate channel SNR. 3.2 Fusion rule using a maximum ratio combining statistic At low SNR, we have the following result. Proposition 2 As σ 2, Λ in (2) reduces to K ˆΛ 2 = (P dk P fk )h k y k Further, if the local sensors are identical, i.e., P dk and P fk are the same for all k s, then Λ further reduces to a form analogous to a maximum ratio combiner [3]: Λ 2 = K h k y k (4) K The factor K in Λ 2 does not affect the detection performance but is introduced for the convenience of performance analysis. Notice that the form of Λ 2 in (4) does not require the knowledge of P dk and P fk provided P dk P fk >, i.e., the local detectors are unbiased. Knowledge of the channel gain is, however, required. 3.3 Fusion rule using an equal gain combining statistic While Λ and ˆΛ 2 (or Λ 2 with identical sensors) relieve some of the requirements compared with the optimal likelihood ratio based fusion rule, they still need some information either about the local sensors or the channel statistics. Further, we note that these fusion statistics, Λ and Λ 2, as approximations to the optimal LR based fusion rule at high and low SNR cases, may suffer performance loss for SNR outside those ranges. It would be very important to investigate other robust alternatives that operate well for the non-extreme SNR range while requiring the same or even less amount of information regarding the channel and/or the sensors. Motivated by the fact that Λ 2 resembles a MRC statistic for diversity combining, we propose a third alternative in the simple form of an equal gain combiner (EGC) that requires minimum amount of information: Λ 3 = K y k K Since y k s are assumed to be phase coherent outputs of each channel, we still require the phase of the fading channel but no other information regarding the channel/sensor is needed. While this heuristic and the simple fusion rule in the form of an EGC statistic does relieve most of the requirements compared with the optimal LR based fusion rule, its usefulness largely depends on its performance compared with the optimum fusion rule as well as the first two alternatives. 4 Performance evaluation While it is clear that the LR based fusion rule provides the best detection performance, it is interesting to see how much performance degradation the other three simple alternatives suffer, and among these three, which one provides the best and most robust detection performance. While analytical results are most desirable, the problem is, in general, intractable. The MRC and EGC fusion rules, however, are amenable to asymptotic analysis because of their simple expression in the form of a sum of some random variables that are independent of each other. In the case of identical sensors and fading statistics, these independent random variables are also identical to each other, which leads to the direct application of the central limit theorem (CLT) for asymptotic analysis. We emphasize that the EGC and MRC are perhaps more desirable because of their performance advantage compared with the twostage approach for low to medium SNR values. Most WSN operating using on-board battery supply are energy limited. Given that RF communication is the most energy consuming function of a sensor node, it is, therefore, imperative to use as little power as possible for data transmission, which usually results in modest SNR values at the fusion center receiver. Throughout this section, we will assume a Rayleigh fading channel for both analysis and numerical simulation. Other fading types, such as Ricean fading, can be used instead though the analysis is more involved. 4. Asymptotic analysis of MRC/EGC While it is well known that MRC is optimal in output SNR, it relies on an assumption that is taken for granted in wireless communications, that is, the sources for multiple independently faded channels are identical to each other. Under this condition, MRC achieves maximum output SNR as it involves full coherent combining. In the context of sensor networks, this is not necessarily the case the local sensors are prone to make decision errors due to the nature of the problem. Without identical input to the multiple fading channels, there is no guarantee that MRC is

4 MRC µ MRC = 2P H f σmrc 2 = [ + K σ2 +4P f ( P f )] µ MRC = 2P H d σmrc 2 = [ + K σ2 +4P d ( P d )] EGC µ EGC = π (2P 2 f ) σegc 2 = 4 π + σ 2 + πp K 4 f ( P f ) Λ µ EGC = π (2P 2 d ) σegc 2 = 4 π + σ 2 + πp K 4 d ( P d ) Λ Table : Mean and variance of MRC and EGC under H and H with K sensors. still preferable compared with other alternatives such as EGC. In the following, we call the performance at the network level as system level detection probability and false alarm rate, denoted by P d and P f,todistinguish them from sensor level P dk and P fk. We assume in this section that the sensors are identical to each other, thus P fk = P f and P dk = P d for all k. Therefore, both MRC and EGC fusion statistics are sums of independent and identically distributed (i.i.d.) random variables which allows direct application of the CLT. This converts the decision fusion problem into hypothesizing between two Gaussian distributions which can lead to a lot of insight. In order to use the CLT, we need the first and second order statistics which are derived and summarized in Table. Given the above statistics, the probabilities of detection and false alarm can be easily obtained using the Q( ) function, defined as the complimentary distribution function of standard Gaussian. Fig. 2 presents the receiver operating characteristic (ROC) curves obtained both by Monte Carlo simulation and numerical approximation using CLT. In this example, the total number of sensors is 8 with sensor level P f =.5 and P d =.5 and channel SNR equal to 5dB. While some discrepancy exists, the approximations using CLT match relatively well to the corresponding simulation results, Probability of detection sensors with sensor level P f =.5 and P d =.5, channel SNR=5dB Monte Carlo for MRC Approximation for MRC Monte Carlo for EGC Approximation for EGC Probability of false alarm Figure 2: ROC curves for MRC and EGC obtained by simulation and numerical approximation using CLT. Application of CLT also allows more intuitive explanation and analysis. From Stein s lemma [4], the relative entropy (Kullback-Leibler distance) between the two distributions under test is directly related to the detection performance in an asymptotic regime. The relative entropy between two Gaussian distributions can be calculated readily as D(P P 2 )=log σ + (σ2 σ2 )+(µ µ ) 2 σ 2σ 2 We can therefore obtain the asymptotic relative entropy as a function of channel SNR for both MRC and EGC statistics by plugging in the corresponding mean and variance from Table. Plotted in Fig. 3 are the results for both MRC and EGC for the same parameter setting. While the MRC has slight advantage over the EGC statistic for very low SNR, the EGC is a better statistic for a wide range of SNR values that are of practical importance. Kullback Leibler Divergence sensors with sensor level P f =.5 and P d =.5 Kullback Leibler divergence for MRC Kullback Leibler divergence for EGC SNR (db) Figure 3: Kullback Leibler distance (relative entropy) between the two hypotheses for both MRC and EGC using Gaussian approximations. 4.2 Simulation results We assume Rayleigh fading with unit power (i.e., E[h 2 k ] = ) for easy SNR calculation. Binary decision u k {+, } is made at the local sensors and we assume that the sensors have identical performance. Specifically, the sensor level false alarm rate is assumed to be P f =.5 while the detection probability is P d =.5 with the total number of sensors fixed at 8. Fig. 4 gives the probability of detection as a function of channel SNR for a fixed system false alarm at P f =.. The parameter setting is identical to the above example. From this figure, it is easy toseethatatverylowandhighsnr,mrcandthe Chair-Varshney statistic can approach the LR performance quite well, while the EGC fusion rule provides the most robust performance among the three suboptimal approaches. The tradeoff between detection performance and the requirement on aprioriinformation for each of the fusion schemes is summarized in Table 2.

5 Fusion rule aprioriinformation required Performance Λ(LR) Channel SNR and sensor performance indices Optimum Λ (Chair-Varshney) Sensor performance indices Near-optimal for large SNR Λ 2 (MRC) Channel SNR Near-optimal for low SNR Λ 3 (EGC) None Robust for most SNR range Table 2: Comparison among the four different fusion rules for binary decisions transmitted through fading channels. 5 Conclusions Fusion of binary decisions transmitted over fading channels in the context of WSN was studied in this paper. Based on a parallel fusion structure that incorporates the fading channel, a likelihood ratio based fusion rule has been derived. In the absence of prior knowledge regarding the local sensors and/or fading channels, several alternatives were proposed. The twostage implementation using the Chair-Varshney fusion rule provides high SNR approximation to the LR fusion rule, while the statistic in the form of a MRC gives low SNR approximation. Another heuristic scheme in the form of an EGC was proposed and we demonstrated that it performs better than both the Chair-Varshney approximation and MRC for a wide range of SNR values. Asymptotic analysis and numerical simulation are carried out for performance comparison. References [] S. Kumar, F. Zhao, and D. Shepherd edts., Special issue on collaborative signal and information processing in microsensor networks, IEEE Signal Processing Magazine, vol. 9, Mar. 22. [2] Z. Chair and P.K. Varshney, Optimal data fusion in multiple sensor detection systems, IEEE Trans. Aerospace Electron. Sys., vol. 22, pp. 98, Jan [3] Pramod K. Varshney, Distributed Detection and Data Fusion, Springer, New York, 997. [8] C. Rago, P.K. Willett, and Y. Bar-Shalom, Censoring sensors: a low-communication-rate scheme for distributed detection, IEEE Trans. AES, vol. 32, pp , Apr [9] J. Hu and R. Blum, On the optimality of finite-level quantization for distributed signal detection, IEEE Trans. Information Theory, vol. 47, pp , May 2. [] J. Chamberland and V.V. Veeravalli, Decentralized detection in wireless sensor networks, in Proc. of the 36th Annual Conference on Information Science and Systems, Princeton, NJ, Mar. 22. [] Q. Cheng, P.K. Varshney, K. Mehrotra, and C.K. Mohan, Optimal bandwidth assignment for distributed sequential detection, in Proc. 5th International Conference on Information Fusion, Annapolis, MD, July 22. [2] S.C.A. Thomopoulos and L. Zhang, Distributed decision fusion with networking delays and channel errors, Information Science, vol. 66, pp. 9 8, Dec [3] G.L. Stüber, Principles of Mobile Communication, Kluwer, Boston, MA, 2nd edition, 2. [4] T.M. Cover and J.A. Thomas, Elements of Information Theory, Wiley, New York, 99. [4] E. Drakopoulos and C.C. Lee, Optimum multisensor fusion of correlated local decisions, IEEE Trans. on Aerospace and Elect. Syst., vol. 27, no. 4, pp , July 99. [5] M.Kam,Q.Zhu,andW.S.Gray, Optimaldata fusion of correlated local decisions in multiple sensor detection systems, IEEE Trans. Aerospace Elect. Syst., vol. 28, pp , July 992. [6] B. Chen and P. Varshney, A Bayesian sampling approach for decision fusion using hierarchical models, IEEE Trans. Signal Processing, vol. 5, pp , Aug. 22. [7] F. Gini, F. Lombardini, and L. Verrazzani, Decentralised detection stratigies under communication constraints, IEE Proceedings, Part F: Radar, Sonar, and Navigation, vol. 45, pp , Aug 998. Probability of detection System level P f =., 8 sensors with sensor level P f =.5 and P d =.5 LR Chair Varshney MRC EGC SNR (db) Figure 4: Probability of detection as a function of channel SNR for Rayleigh fading channels with 8 sensors. The system false alarm rate is fixed at P f =..