WIRELESS Sensor Networks (WSNs) consist of. Performance Analysis of Likelihood-Based Multiple Access for Detection Over Fading Channels

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1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 4, APRIL Performance Analysis of Likelihood-Based Multiple Access for Detection Over Fading Channels Kobi Cohen and Amir Leshem, Senior Member, IEEE Abstract In this paper, we consider the binary hypothesis testing problem using wireless sensor networks. We analyze the case where sensors transmit their local log-likelihood ratio (LLR) directly to a fusion center (FC) using an analog transmission scheme over multiple-access fading channels. Due to the nature of the wireless medium, the FC receives a superposition of sensor transmissions. The decision is made by the FC and is based on received data from the sensors. In contrast to the case of identical channels and i.i.d observations, the analog transmission of the LLR over multiple-access fading channels does not achieve the centralized error exponent. Large deviation tools are used in this paper to characterize the error exponent in the asymptotic regime (when the number of sensors approaches infinity) in the case of non-i.i.d observations and non-i.i.d fading channels. Chernoff bounding techniques are used to provide bounds on the error probability for a finite number of sensors when the observations and the fading channels are independent across sensors. Specific performance analysis is provided for detection over both i.i.d and spatially correlated Markovian fading channels. Simulation results then illustrate the detector s performance. Index Terms Chernoff bound, Gartner Ellis theorem, Hoeffding bound, large deviations, multiple-access channel (MAC), signal detection, wireless sensor networks (WSNs). I. INTRODUCTION WIRELESS Sensor Networks (WSNs) consist of low-power sensor nodes with limited computational capabilities. A good survey of available technology appears in [1] [3]. In this paper, we consider a binary hypothesis testing problem, where the fusion center (FC) decides whether an unknown hypothesis is or based on messages received directly from the sensors. These single-hop WSNs are studied in [4] [16]. Typically, the sensors are low-power and low-cost nodes, while the FC is a powerful hardware unit. The FC can be a static cluster head or a mobile access point [4]. Most transmissions in ad hoc sensor networks are between low-lying antennas, where signal intensity drops as the fourth power of distance due to partial cancelation by ground-reflected rays [17], [18]. In the case of a mobile access point, however, there may be a free space between the sensor field and the AP (for example, flying airplanes) where the signal intensity only drops Manuscript received February 14, 2012; revised September 29, 2012; accepted October 25, Date of publication December 11, 2012; date of current version March 13, This work was supported in part by the MAGNET consortium of the Israeli MAGNET Program. The authors are with the Faculty of Engineering, Bar-Ilan University, Ramat-Gan 52900, Israel ( kobi.cohen10@gmail.com; leshem.amir1@gmail.com). Communicated by D. Palomar, Associate Editor for Detection and Estimation. Digital Object Identifier /TIT as the second power of distance [4], [19]. Therefore, in many cases, communication between sensors and FC is preferable in terms of energy efficiency. Furthermore, the communication protocol layout in single-hop networks is much simpler, since the sensors are not required to discover and maintain routes, store and relay packets, etc. In the traditional communication approach for detection in WSN, sensors transmit some function of their observations over parallel channels (for instance, FDM/TDM fashion) [7], [20]. However, the bandwidth increases linearly with the number of sensors in this scheme. Therefore, for a large-scale WSN, transmission over multiple-access channels (MAC) is generally preferred. Using MAC, all sensors transmit simultaneously in one dimension (or a small number of dimensions). As a result, the bandwidth requirement does not depend on the number of sensors. It is well known that digital communication (where sensor nodes convert their observations into a bit stream) does not lead to optimal performance in general network problems. The correct way of understanding thenatureofinformationisinan analog form, whereas bits are inappropriate [6]. In [21], joint source channel strategies over MAC were developed that often outperform separation-based strategies. Analog transmission schemes over MAC for detection and estimation using WSN have been investigated in [5] [8], [13] [16], [22], and [23]. The key idea in these schemes is to transmit some functions of the observed data using predetermined waveforms. Due to thenatureofthewireless channel, the FC receives a superposition of sensor transmissions. In the case where sensor nodes are not aware of their environment statistics, the type-based multiple-access (TBMA) scheme can be used [5], [7], [24]. In TBMA, the observations are quantized before communication and each sensor transmits the waveform corresponding to its quantized observation. As a result, the bandwidth depends on the number of quantization levels (which is typically much smaller than the number of sensors in a large-scale WSN). The FC receives a superposition of sensor transmissions and matched-filters it by the corresponding waveforms. In the case of independent and identically distributed (i.i.d) observations and identical channels, the FC receives a noisy version of the type of sensor observations in each dimension, which is a sufficient statistic for detection. However, the error exponent is affected by the quantization operation. Here, we look at analog transmission of the log-likelihood ratio (LLR) over multiple-access fading channels. The observation statistics are available to the sensors and the sensors can compute their local likelihood ratio. The sensors deliver their local LLR over multiple-access fading channels using a single predetermined waveform. In [7], Liu and Sayeed considered the /$ IEEE

2 2472 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 4, APRIL 2013 case of Gaussian multiple-access identical channels, i.e., with no fading and i.i.d observations under each hypothesis. They showed that in this scenario, the analog transmission of the LLR achieves the centralized error exponent as the number of sensors approaches infinity. In this paper, we analyze this scheme over multiple-access fading channels and non-i.i.d observations. The decision is made by the FC and is based on received data from sensors. Throughout this paper, we present a transmission scheme in which sensors transmit their LLRs without directly referringtocorrectingthechannel gain (although correcting the phase is assumed to avoid zero-mean channels). As a result, the received LLRs at the FC are multiplied by random channel gains. Nevertheless, this transmission scheme is applicable to many common applications. 1) The case where the channel gain is not corrected at the transmitter to make the scheme robust against changes in the channel statistics. Thus, the average transmission energy of the signal is determined according to the observations statistics purely to satisfy the average energy constraint. This transmission scheme is very simple to implement and is generally preferred in WSNs with a mobile access point [4], for instance, where the channel statistics may vary rapidly and are not available at the sensors. Note that correcting the channel phase (by transmitting a signal with the complex conjugate channel phase) is assumed to avoid zero-mean channels, as was done in [13]; however, the signal energy is not affected by this operation. Correcting the channel phase can be done by transmitting a pilot signal by the FC before the sensor transmissions to estimate the channel phase [9], [10]. In fact, estimating the channel phase by sensors with an estimation error of less than is sufficient to correct the phase at the transmitter to guarantee positive components at the receiver. 2) The case where sensors exploit the channel state to correct the fading effect (for instance, by dividing the LLRs at the transmitters by the channel state to obtain identical channels at the FC). However, due to channel estimation errors, the LLR values are still multiplied by random gains at the FC. 3) The case where the sensors adapt their transmission power according to the channel state to obtain discrete channels at the FC. For example, consider a transmission scheme where each sensor transmits its LLR divided by the channel state only if the channel gain is greater than a predetermined threshold (to satisfy a power constraint). Otherwise, the sensor does not transmit (to save energy). As a result, the FC receives the transmitted LLRs multiplied by 1 (good channel) with probability,where is the probability that the channel gain is greater than the predetermined threshold. All other LLRs are multiplied by 0 (bad channel) with probability. This scenario is called a transmission scheme over ON/OFF fading channels. In the case where the fading channels form a Markov chain across sensors we obtain the well-known spatial Gilbert Elliot channels. In Section VI, we examine the detector s performance in these scenarios. In contrast to the case of identical channels and i.i.d observations, the analog transmission of the LLR over multiple-access fading channels does not achieve the centralized error exponent. However, in the following, we summarize the main results of this paper in this respect. 1) We use large deviation (LD) theory to characterize the detector s error exponent in the asymptotic regime (when the number of sensors approaches infinity) in the case of non-i.i.d observations and non-i.i.d fading channels across sensors. 2) We provide bounds on the error probability for a finite number of sensors using some extensions of the Hoeffding bound, in the case where the observations and the fading channels are independent across sensors. 3) We examine the detector s performance for detection over i.i.d fading channels. A closed-form expression is obtained for a linear model detection over ON/OFF channels. 4) We examine the detector s performance for detection over spatially correlated Markovian fading channels. Closedform bounds are obtained for a linear model detection over spatial Gilbert Elliot channels. The rest of this paper is organized as follows. In Section II, we present the network model. In SectionIII,wepresentthe transmission scheme and the proposed detector. In Section IV, we provide the analysis tools that were used in this paper. In Section V, we analyze the detector s performance. In Section VI, we examine the detector s performance in special cases. In Section VII, we provide numerical examples to illustrate the detector s performance. II. NETWORK MODEL In this paper, we consider a binary detection problem using a WSN containing sensors. The sensors measure a certain phenomenon and deliver some function of their observations to an FC through a multiple-access block-fading channel. We assume that sensor experiences a block-fading channel with a nonzero channel mean 1. The FC determines whether an unknown hypothesis is or based on the received data from sensors. The aprioriprobabilities of the two hypotheses, are denoted by,, respectively. Let and be the random observation at sensor and the probability density function (PDF) of conditioned on, respectively. Let and be the likelihood ratio (LR) and the LLR at sensor, respectively. III. DETECTION USING AN ANALOG LLR TRANSMISSION SCHEME In this section, we present the analog LLR transmission scheme and the proposed detector. A. Transmission Scheme Let sensor be a predetermined normalized waveform. Every transmits 1 As explained in Section I, this is done by correcting the channel-phase at the transmitter.

3 COHEN AND LESHEM: PERFORMANCE ANALYSIS OF LIKELIHOOD-BASED MULTIPLE ACCESS FOR DETECTION OVER FADING CHANNELS 2473 where can be any fixed constant or a function of the number of sensors, such that energy constraint is satisfied. The average energy of the signal is given by where is the expectation of and is the Kullback Leibler (KL) divergence between hypothesis to hypothesis at sensor, definedin(12). denotes the variance of. We consider a transmission scheme where is equal for all sensors, and sensor transmissions are limited by an average energy constraint, as was done in [7] and [14]. is chosen such that for all,where is the maximal average energy allowed for each sensor. The advantages of this scheme are twofold. First, in the following, we show that transmitting LLRs with equal gain for all sensors approaches the optimal LLR test as the fading variance approaches zero and the observation dependences vanish. Second, in the case of nonidentically distributed observations, sensors with a higher second moment of the LLR transmit with higher average energy. Since a higher magnitude of LLRs typically implies more informative observations [25] [27], it is reasonable to invest more energy in better observations. The received signal at the FC is given by where is the channel AWGN with zero mean and a power spectrum density. After matched-filtering by the corresponding waveform at the FC, we have where. B. Detector Let where. Note that the noise variance decreases with for 2,forall. As a result, performance can be improved by increasing the number of sensors in the network without increasing the total transmission energy. We discuss the decay rate of the error probability in Section IV. We propose the following detector: Decide if Otherwise, decide, where is the detector s threshold. (1) (2) (3) In a Bayesian setting, an LLR test of independent observations (in which the global likelihood ratio can be replaced by the sumofthellrs)with is optimal under the MAP criterion. In Theorem 1, we characterize the error exponent of the detector over fading channels and non-i.i.d observations, as well. To simplify the presentation, throughout this paper, we focus on the Bayesian setting. Under the Neyman Pearson criterion, is determined according to the desired false-alarm probability. Note that the proposed detector is the optimal ML detector in the case of independent observations and identical noise-free channels ( ). In [7], the authors considered the case of Gaussian multiple-access identical channels (i.e., with no fading) and i.i.d observations under each hypothesis. They showed that in this special case, the analog transmission of the LLR achieves the centralized error exponent as.in this paper, we analyze this scheme over multiple-access fading nonzero mean channels and non-i.i.d observations. We use Chernoff bounding and LD tools to show exponential decaying of the error probability and to characterize the error exponent. IV. DETECTION ERROR EXPONENT The error probability of the detector is given by where is the probability to decide when is true (Type-I error probability), and is the probability to decide when is true (Type-II error probability). Note that depend on the number of sensors. However, for convenience, we remove the index throughout the paper. We are interested in characterizing the rate at which approaches zero, as increases. In this section, we provide the tools that were used in analyzing the detector s performance. A. Large Deviations Throughout this paper, we use the large deviation principle (LDP) to characterize the limiting behavior of the error probability. Assuming that, we are interested in evaluating the rate function (termed error exponent) of the error probability. Definition 1 [28]: Let, be the interior and closure of aset, respectively. We say that satisfy the LDP with a rate function if, for any,wehave where. The effective domain of is defined by. In a hypothesis testing property [28]: (4) (5) mostly satisfies the I-continuity 2 The term refers to for some.

4 2474 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 4, APRIL 2013 Then The Gartner Ellis theorem [28] is used throughout this paper to characterize the rate function: Let and let Theorem (Gartner Ellis): If (8) exists as an extended real number, smooth, and continuous, then satisfy theldpwitharatefunction is the Fenchel Legendre transform of. In this paper, we focus on a threshold-based detector for binary hypothesis testing (3). We are interested in characterizing the error exponent of the detector. Let and denote the decision regions. The detector decides if or decides if. Under hypothesis, an error occurs if,thus in (5) (9). Therefore,. Under hypothesis, an error occurs if ; thus in (5) (9). Therefore,. Assume that (8) exists as an extended real number, smooth, and continuous. Then, applying the Gartner Ellis theorem to characterize the error exponent of the detector yields: (6) (7) (8) (9) the well-known Chernoff and Hoeffding bounds. The Chernoff bound uses the Markov inequality to show exponential decreasing of the error probability. Theorem (Chernoff Bound): For any, the probability that a random variable (r.v) is greater than is upper bounded by The Hoeffding bound [29] upper bounds the probability that the sum of bounded r.v deviates from its expected value. be indepen-.let be the expected value Theorem (Hoeffding Bound): Let dent r.v, such that be the empirical mean and of.then The Chernoff-based bounds that are used in this paper are the Hoeffding bound and some extensions applied to the problems in this paper. V. DETECTOR PERFORMANCE ANALYSIS In this section, we analyze the detector s performance in both the finite number of sensors scenario and the asymptotic case. The KL divergence between hypothesis to hypothesis at sensor is defined by Let (12) (13) where under hypothesis in (5) (9). Typically, in hypothesis testing,. In this case, we have (10) (11) B. Bounding Techniques Chernoff bounding techniques are used to upper bound the error probability for any finite. In the following, we review First, we overview the result reportedin[7].in[7],theauthors considered the case of Gaussian multiple-access identical channels (i.e., with no fading) and i.i.d observations under each hypothesis. In this special case, the analog transmission of the LLR achieves the centralized error exponent as. Theorem [7]: Consider the case of identical channels and i.i.d observations under.if,forany, then the proposed detector (3) achieves the centralized error exponent as. Next, we analyze the detector s performance over multipleaccess nonzero mean fading channels and non-i.i.d observations. For a finite, we assume bounded LLRs and bounded fading channels. We use some extensions on the Hoeffdingbound to upper bound the error probability for a finite.inthe asymptotic regime ( ), we use the Gartner Ellis theorem to characterize the error exponent. In the case of i.i.d observations under and i.i.d fading channels, it suffices to assume

5 COHEN AND LESHEM: PERFORMANCE ANALYSIS OF LIKELIHOOD-BASED MULTIPLE ACCESS FOR DETECTION OVER FADING CHANNELS 2475 that the moment generating function of the r.v is finite to characterize the asymptotic error exponent. However, in the case of non-i.i.d observations or non-i.i.d fading channels, we assume that (8) exists as an extended real number, smooth, and continuous to apply the Gartner Ellis theorem. Theorem 1: Assume that the proposed detector (3) is implemented. Let be the minimal number of sensors such that and. Then: a) Consider the case of independent observations under and independent fading channels. Assume that the fading channels and the LLR values are bounded, i.e.,,where are finite constants. For all,the error probability is upper bounded by: (14) Furthermore, if,forany,thenin the asymptotic regime ( ), the following holds. b) Consider the case of i.i.d observations under and i.i.d fading channels. Assume that the moment generating function of the r.v is finite. Then, satisfies the LDP with a rate function: under, and the error probability decays exponentially with. c) Consider the case of non-i.i.d observations under and non-i.i.d fading channels. Assume that (8) exists as an extended real number, smooth and continuous. Then, satisfies the LDP with a rate function:, under,where isgivenin(9)andtheerrorprobability decays exponentially with. Proof: a) The expectation of the r.v under is given by Since, in a noise-free channel scenario ( ), the Hoeffding bound yields The bound in (14) is a generalization of the Hoeffding bound that includes the additive channel noise. With minor changes on the generalized Hoeffding bound that was shown in [7], (14) follows. b) We use the Gartner Ellis theorem to prove the theorem. Due to the i.i.d property, under each hypothesis, we have Hence, satisfies the LDP with a rate function under. Since the expectation of in the i.i.d scenario under is given by,,bothtype1andtype 2 error probabilities (4) decay exponentially with. c) We assume that (8) exists as an extended real number, smooth, and continuous. Hence, the Gartner Ellis theorem can be applied. The Gartner Ellis theorem states that satisfiestheldpwitharatefunction under,where is given in (8). Since the expectation of the r.v under is given by,,thenboth type 1 and type 2 error probabilities (4) decay exponentially with. Remark 1: In Section III-B, we inferred from (2) that the detector s performance could be improved by increasing the number of sensors in the network without increasing the total transmission energy. Theorem 1 characterizes the decay rate of the error probability. Assume that. According to the proof of Theorem 1.b, for all, the noise term does not affect the asymptotic rate and the error probability decays exponentially with. Note that for, the error probability still decays exponentially with. However, the asymptotic rate is now affected by a finite noise term for some. The same explanation holds when applying the Gartner Ellis theorem in Theorem 1.c. The asymptotic exponential rate can be verified in these cases ( and ) from Theorem 1.a as well. On the other hand, for, according to Theorem 1.a, the error probability decays only subexponentially with. Remark 2: In Theorem 1.c (i.e., in the non-i.i.d scenario), we assumed that (8) exists as an extended real number, smooth and continuous to apply the Gartner Ellis theorem. In the following, we discuss two common scenarios where the theorem holds. First, consider a common scenario where sensors are located in different areas ( sensors in area )andtheir observations are independent but not necessarily identically distributed under. However, sensor observations in the same area areassumedtobei.i.dunder.inthis common scenario, we have

6 2476 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 4, APRIL 2013 Hence, the Gartner Ellis theorem can be applied and Theorem 1.c holds. Note that denotes the expectation with respect to a random scattering of sensors over the field, where denotes the expectation in area with respect to hypothesis. Furthermore, note that the Gartner Ellis theorem can be appliedwhenther.varecorrelated and obey a Markov chain [28]. Hence, another common scenario where Theorem 1.c holds is when sensors transmit their LLRs over spatially correlated Markovian fading channels. We examine the detector performance in this scenario in Section VI-B. In the non-i.i.d scenario, we need to verify that (8) exists as an extended real number, smooth, and continuous to apply the Gartner Ellis theorem. However, here we consider the case of i.i.d observations and i.i.d fading channels across sensors. Thus, the conditions on hold immediately by Theorem 1.b. From Theorem 1.b, under,wehave (18) VI. APPLICATIONS In this section, we examine the detector s performance in the Gaussian linear model, due to its general applicability to real-world applications and its tractable mathematical analysis. In the classical Gaussian linear model, the random observation data vector at sensor can be written under and as [30] (15) where is an vector of received data samples, is a known observation full rank matrix with, is a vector of parameters which is assumed to be known, and is an random vector with PDF, i.i.d across sensors. Note that under,wehave, and under we have. In this case, the LLR is given by In the following, we examine the detector performance for the Bayesian probability of error. Due to the symmetry of the problem evaluating the Fenchel Legendre transform at zero maximizes the error exponent. As a result, the error exponent achieved by the proposed detector (3) is given by. Solving can be done numerically. Note that we can extend this application to the case of continuous value fading channels, by replacing the summation in (18) by an integration over the fading channel domain. Next, we show that in the ON/OFF channel scenario, a closed-form solution is obtained. Consider the case where with probability or with probability. Rewriting (18) yields As a result, the LLR of sensor under is an r.v with PDF: The error exponent is given by. Differentiating and equating to zero yields.asa result, the error exponent is given by where (16) (17) under both hypotheses. Note that the last expression follows since. (19) A. Detection Over I.I.D Fading Channels First, we examine the problem of the linear model detection (15) over i.i.d fading channels. Let beasetof possible values of the channel fading. We assume that each sensor experiences a channel fading with probability i.i.d across sensors, where. To simplify the presentation, we consider discrete fading channels. However, these applications can be extended to the case of continuous value channels, as well. Since,wehave under conditioned on. B. Detection Over Spatially Correlated Markovian Fading Channels In this section, we examine the problem of the linear model detection (15) over spatially correlated Markovian fading channels. We consider a 1-D field of sensors, as was done in [5], [31]. Let be a set of possible values of channel fading. We assume that the fading channels form a Markov chain across sensors, as depicted in Fig. 1. Let be the transition probability to experience given that the neighbor node experiences.let be the transition probability matrix of the fading channel across

7 COHEN AND LESHEM: PERFORMANCE ANALYSIS OF LIKELIHOOD-BASED MULTIPLE ACCESS FOR DETECTION OVER FADING CHANNELS 2477 Fig. 1. One-dimensional field of sensors, as discussed in Section VI-B. The fading channels form a Markov chain across sensors,. the sensors. Let initial state : be the Markov probability measure with the Fig. 2. Illustration of the spatial Gilbert Elliot channel, as discussed in Section VI-B. where is a diagonal matrix:... Using the Perron Frobenius theorem [28] yields (21) Note that the spatial Gilbert Elliot channel is a special case of this problem, where (bad state) and (good state). Next, we compute, where is the expectation of with respect to, to use the Gartner Ellis theorem. Note that under,wehave where denotes the Perron Frobenius eigenvalue of the matrix. Note that is the isolated root of the characteristic equation of the matrix, positive, finite, and differentiable with respect to [32]. Therefore, we can apply the Gartner Ellis theorem and Theorem 1.c holds. In the following, we examine the detector performance for the Bayesian probability of error. Due to the symmetry of the problem, evaluating the Fenchel Legendre transform at zero maximizes the error exponent. As a result, the error exponent is given by, under. Next, we provide useful bounds for the error exponent. Note that the Perron Frobenius eigenvalue of is bounded by [33] Hence, the error exponent, under, is bounded by (22) where is a nonnegative matrix, whose elements are,under. denotes the power of the matrix. can be rewritten as (20) We can view these bounds as the Fenchel Legendre transforms of,where is the logarithm of the moment generating function given the current channel state. Note that this application can be extended to the case of a continuous value finite-state Markov channel (FSMC) [34] [36]. In FSMC, the PDF of the fading channel is partitioned into intervals. When a channel is in a state it experiences a continuous value in the corresponding interval. The channel moves to a state from a state according to the transition probability. The extension to the case of FSMCs follows by replacing the entries in the matrix by their expected values in the corresponding interval. Next, we show that in the case of the spatial Gilbert Elliot channel, closed-form bounds are obtained. Consider the case where (bad channel) and (good channel), as

8 2478 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 4, APRIL 2013 Fig. 3. Detecting over i.i.d ON/OFF channels: Simulation parameters:,. (a) Theoretical error exponent as a function of the success probability. (b) Error probability as a function of the number of sensors (, ). depicted in Fig. 2. The spatial transition matrix between adjacent sensors is given by if,and (24) Summing over the rows of yields and,under. Similar to the optimization of the i.i.d ON/OFF channel in Section VI-A, we can show that (23) if. The error exponent achieved over the spatial Gilbert Elliot channel is lower bounded by the error exponent achieved over the inferior i.i.d ON/OFF channel (corresponding to the inferior row in ) and upper bounded by the error exponent achieved over the better i.i.d ON/OFF channel (corresponding to the better row in ).

9 COHEN AND LESHEM: PERFORMANCE ANALYSIS OF LIKELIHOOD-BASED MULTIPLE ACCESS FOR DETECTION OVER FADING CHANNELS 2479 Fig. 4. Detecting over i.i.d ON/OFF channels using truncated LLRs. The error probability is presented as a function of the number of sensors (,,, ). VII. SIMULATION RESULTS In this section, we provide numerical examples to illustrate the detector s performance. We examine the problem of detecting the parameter in an observation AWGN. Detecting dc in AWGN is a special case of the linear model detection (15), where and is a scalar parameter. The random observation at sensor can be written under and as where is the additive Gaussian observation noise, i.i.d across sensors. By (17), we have,,and.inthesimulations,weset,. A. Detecting DC in AWGN Over I.I.D ON/OFF Channels First, we consider the problem of detecting the parameter in an observation AWGN over i.i.d ON/OFF channels, as discussed in Section VI-A. We consider the case where with probability or with probability. By placing, in (19), the error exponent is given by under both hypotheses. The theoretical error exponent is depicted in Fig. 3(a) as a function of. As expected, increasing increases the error exponent. In Fig. 3(b), we present the error probability as a function of the number of sensors. We compare the achieved error probability to the theoretical LD characterization. It can be seen that the empirical error probability decays exponentially with and approaches the theoretical error exponent (i.e., the rate of the error probability,, approaches as increases.). Next, we compare the empirical error probability to the upper bound that was given in (14) and to the theoretical LD characterization. The upper bound (14) requires independent and bounded LLRs and fading channels. Hence, we consider the case where the transmitted LLRs are bounded in the range. In Fig. 4, we present the error probability as a function of the number of sensors. It can be seen that the empirical error probability is upper bounded by (14) for all.onthe other hand, the theoretical LD,, characterizes its asymptotical rate (for sufficiently large ). It can be seen that the empirical error probability decays exponentially with and the rate,, approaches as increases. B. Detecting DC in AWGN Over Spatially Correlated Fading Channels In this section, we examine the problem of detecting the parameter in an observation AWGN over correlated Markovian fading channels, as discussed in Section VI-B. We illustrate the detector s performance with the following parameters: the channel realizations are,,,withthe following transition matrix: where. The transition matrix was chosen such that the channel has aprobability0.5tostayinthecurrent state. The channel has a probability to reach a channel state and a probability to reach a channel state, given that the current channel state is. The theoretical error exponent is depicted in Fig. 5(a) as a function of. As expected, increasing increases the error exponent. In Fig. 5(b), we present the error probability as a function of the number of sensors. Again, we compared the achieved error probability to the theoretical LD characterization. It can be seen that the empirical error probability decays exponentially with and the rate,, approaches as increases.

10 2480 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 4, APRIL 2013 Fig. 5. Detecting over Markovian channels: Simulation parameters:,. (a) Theoretical error exponent as a function of the transition probability. (b) Error probability as a function of the number of sensors (, ). VIII. CONCLUSION In this paper, we investigated the binary hypothesis testing problem where the decision is made by an FC and is based on received data from sensors. We focused on analog transmission of the LLRs over multiple-access fading channels and assumed non-i.i.d observations. Due to the nature of the wireless channel, the FC receives a superposition of the LLRs from all sensors. We used LD and Chernoff bounding techniques to characterize the error exponent in the asymptotic regime and to provide bounds on the error probability in the case of a finite number of sensors. Simulation results were presented to illustrate the detector s performance. ACKNOWLEDGMENT We would like to thank the anonymous reviewers for comments that significantly improved the presentation of this paper. REFERENCES [1] J. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, A survey on sensor networks, IEEE Commun. Mag., vol. 40, no. 8, pp , Aug [2] C.-Y. Chong and S. P. Kumar, Sensor networks: Evolution, opportunities, and challenge, Proc. IEEE, vol. 91, no. 8, pp , Aug [3] A.Swami,Q.Zhao,Y.-W.Hong,andL.Tong, Wireless Sensor Networks: Signal Processing and Communications Perspectives. New York: Wiley, 2007.

11 COHEN AND LESHEM: PERFORMANCE ANALYSIS OF LIKELIHOOD-BASED MULTIPLE ACCESS FOR DETECTION OVER FADING CHANNELS 2481 [4] L. Tong, Q. Zhao, and S. Adireddi, Sensor networks with mobile agents, in Proc. Int. Symp. Military Commun.,Boston,MA,Oct.2003, pp [5] G. Mergen, V. Naware, and L. Tong, Asymptotic detection performance of type-based multiple access over multiaccess fading channels, IEEE Trans. Signal Process., vol. 55, no. 3, pp , Mar [6] M. Gastpar, Uncoded transmission is exactly optimal for a simple Gaussian sensor network, IEEE Trans. Inf. Theory, vol. 54, no. 11, pp , Nov [7] K. Liu and A. Sayeed, Type-based decentralized detection in wireless sensor networks, IEEE Trans. Signal Process., vol. 55, no. 5, pp , May [8] A. Anandkumar and L. Tong, Type-based random access for distributed detection over multiaccess fading channels, IEEE Trans. Signal Process., vol. 55, no. 10, pp , Oct [9] Y. Chen, Q. Zhao, V. Krishnamurthy, and D. Djonin, Transmission scheduling for optimizing sensor network lifetime: A stochastic shortest path approach, IEEE Trans. Signal Process., vol. 55, no. 5, pp , May [10] K. Cohen and A. Leshem, A time-varying opportunistic approach to lifetime maximization of wireless sensor networks, IEEE Trans. Signal Process., vol. 58, no. 10, pp , Oct [11] K. Cohen and A. Leshem, Energy-efficient detection in wireless sensor networks using likelihood ratio and channel state information, IEEE J. Sel. Areas Commun., vol. 29, no. 8, pp , Sep [12] C. Yang and L. Shi, Deterministic sensor data scheduling under limited communication resource, IEEE Trans. Signal Process., vol. 59, no. 10, pp , Oct [13] G. Mergen and L. Tong, Type based estimation over multiaccess channels, IEEE Trans. Signal Process., vol. 54, no. 2, pp , Feb [14] S. Marano, V. Matta, L. Tong, and P. Willett, A likelihood-based multiple access for estimation in sensor networks, IEEE Trans. Signal Process., vol. 55, no. 11, pp , Nov [15] A. Sarwate and M. Gastpar, A little feedback can simplify sensor network cooperation, IEEE J. Sel. Areas Commun., vol. 28, no. 7, pp , Sep [16] K. Cohen and A. Leshem, Density-based multiple access for energyefficient detection in wireless sensor networks, in preparation, to be published. [17] K. Sohrabi, B. Manriquez, and G. Pottie, Near ground wideband channel measurement, in Proc. 49th Veh. Technol. Conf., Houston, TX, May 1999, pp [18] G. Pottie and W. Kaiser, Wireless integrated network sensors, Commun. ACM, vol. 43, pp , May [19] T. Rappaport, Wireless Communications Principles and Practice. Englewood Cliffs, NJ: Prentice-Hall, [20] R. Viswanathan and P. Varshney, Distributed detection with multiple sensors: Part I Fundamentals, Proc. IEEE, vol. 85, no. 1, pp , Jan [21] B. Nazer and M. Gastpar, Computation over multiple-access channels, IEEE Trans. Inf. Theory, vol. 53, no. 10, pp , Oct [22] M. Banavar, A. Smith, C. Tepedelenlioglu, and A. Spanias, Distributed detection over fading MACs with multiple antennas at the fusion center, in Proc. IEEE Int. Conf. Acoust., Speech Signal Process., Mar. 2010, pp [23] A. Leong, S. Dey, and J. Evans, Asymptotics and power allocation for state estimation over fading channels, IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 1, pp , Jan [24] K. Liu and A. Sayeed, Asymptotically optimal decentralized typebased detection in wireless sensor networks, in Proc. IEEE Int. Conf. Acoust., Speech Signal Process., May 2004, vol. 3, pp [25] C. Rago, P. Willett, and Y. Bar-Shalom, Censoring sensors: A low communication-rate scheme for distributed detection, IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 2, pp , Apr [26]R.S.BlumandB.M.Sadler, Energyefficient signal detection in sensor networks using ordered transmissions, IEEE Trans. Signal Process., vol. 56, no. 7, pp , Jul [27] V. Cheng and T. Wang, Performance analysis of distributed decision fusion using a censoring scheme in wireless sensor networks, IEEE Trans. Veh. Technol., vol. 59, no. 6, pp , Jul [28] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. New York: Springer-Verlag, [29] W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., vol. 58, pp , [30] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Englewood Cliffs, NJ: Prentice-Hall, 1998, vol. 2. [31] M. Dong, L. Tong, and B. Sadler, Impact of data retrieval pattern on homogeneous signal field reconstruction in dense sensor networks, IEEE Trans. Signal Process., vol. 54, no. 11, pp , Nov [32] P. Lancaster, Theory of Matrices. New York: Academic, [33] C. Meyer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000, vol. 2. [34] C. Tan and N. Beaulieu, On first-order Markov modeling for the Rayleigh fading channel, IEEE Trans. Commun., vol. 48, no. 12, pp , Dec [35] C. Pimentel and I. Blake, Modeling burst channels using partitioned Fritchman s Markov models, IEEE Trans. Veh. Technol., vol. 47, no. 3, pp , Aug [36] Y. Guan and L. Turner, Generalised FSMC model for radio channels with correlated fading, IEEE Proc. Commun., vol. 146, no. 2, pp , Apr Kobi Cohen received the B.Sc.(cum laude) in electrical engineering from Bar- Ilan University, Ramat Gan, Israel, in He is currently working towards the Ph.D. degree in the Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel. His main research interests include resource-constrained signal processing techniques, communication protocols design, application of game theory to dynamic spectrum access of communication networks and dynamic spectrum management for wireless and wireline multichannel networks. Amir Leshem (SM 06) received the B.Sc.(cum laude) in mathematics and physics, the M.Sc. (cum laude) in mathematics, and the Ph.D. in mathematics all from the Hebrew University, Jerusalem, Israel, in 1986,1990 and 1998 respectively. From 1998 to 2000 he was with Faculty of Information Technology and Systems, Delft university of technology, The Netherlands, as a postdoctoral fellow working on algorithms for the reduction of terrestrial electromagnetic interference in radio-astronomical radio-telescope antenna arrays and signal processing for communication. From 2000 to 2003 he was director of advanced technologies with Metalink Broadband where he was responsible for research and development of new DSL and wireless MIMO modem technologies and served as a member of ITU-T SG15, ETSI TM06, NIPP-NAI, IEEE and From 2000 to 2002 he was also a visiting researcher at Delft University of Technology. He is one of the founders of the new school of electrical and computer engineering at Bar-Ilan university where he is currently an Associate Professor and head of the Signal Processing track. From 2003 to 2005 he also was the technical manager of the U-BROAD consortium developing technologies to provide 100 Mbps and beyond over DSL lines. He was the leading guest editor a special issue of the IEEE JOURNAL ON SELECTED TOPICS IN SIGNAL PROCESSING, dedicated to signal processing for space research and for a special issue of the Signal Processing Magazine, dedicated to signal processing in astronomy and cosmology. Since 2008 he is an associate editor for IEEE TRANSACTIONS ON SIGNAL PROCESSING. His main research interests include multichannel wireless and wireline communication, applications of game theory to dynamic and adaptive spectrum management of communication and sensor networks, array and statistical signal processing with applications to multiple element sensor arrays and networks in radio-astronomy, brain research, wireless communications and radio-astronomical imaging, set theory, logic and foundations of mathematics.