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PAC vs. MAC for Decentralized Detection using Noncoherent Modulation Christian R. Berger, Student Member, IEEE, Marco Guerriero, Student Member, IEEE, Shengli Zhou, Member, IEEE, and Peter Willett, Fellow, IEEE
Abstract—In decentralized detection, local sensor observations have to be communicated to a fusion center through the wireless medium, inherently a multiple-access channel (MAC). As communication is bandwidth- and energy-constrained, it has been suggested to use the properties of the MAC to combine the sensor observations directly on the channel. Although this leads to an array-processing gain if the sensors’ transmissions combine coherently on the channel, it has been shown that this is not the case when they combine noncoherently. We review known results for the coherent case and then analyze the noncoherent case based on a simple on/off scheme combined with optimal sensor “censoring”. Since the optimal forwarding function is not available, we also bound the performance using an equivalent communication problem and a centralized estimator to verify trends. We find that for noncoherent modulation, there is no processing gain using the MAC for decentralized detection, but compared to parallel-access channels (PACs) the MAC avoids the noncoherent combining loss. Still the performance of the MAC approach is only of diversity one, as the output of the MAC is approximately a zero-mean complex Gaussian random variable for a large number of sensor. The MAC performance can be increased by using multiple independent channels, each used as a MAC by all sensors, which we term diversity-MAC. This approach always outperforms the PAC scheme on Rayleigh fading channels, where the output is exactly Gaussian, but has inferior performance across random phase channels when few sensors are used, as the PAC does not create “artificial” fading. Index Terms—Wireless sensor network, noncoherent modulation, censoring, multiple-access channel, parallel-access channel.
I. I NTRODUCTION A. Motivation In a wireless sensor network (WSN), observations often are transmitted to a fusion center (FC) for central processing. For simplicity the usual architecture assumes that each sensor transmits through a parallel access channel (PAC), but since the wireless medium is naturally a broadcast medium, these PACs have to be realized through time division multiple access (TDMA), frequency division multiple access (FDMA) or code division multiple access (CDMA). Manuscript submitted August 25, 2008, revised February 11, 2009, accepted March 31, 2009. This work was supported by the Office of Naval Research under contract N00014-07-1-0429 and N00014-07-1-0055. c Copyright 2008 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to
[email protected]. All authors are with the Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Way, U-2157, Storrs, Connecticut 06269 USA (e-mail: {crberger, marco.guerriero, shengli, willett}@engr.uconn.edu) Digital Object Indentifier 00.0000/TSP.2009.000000
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Fig. 1. In wireless sensor networks for decentralized detection the local observations have to be transmitted to a fusion center across adverse channels under energy and bandwidth constraints.
Recently, it has been suggested to exploit the wireless medium directly as a multiple-access channel (MAC) for decentralized detection, as the FC will observe a superposition of the signals sent by the individual sensors and hence perform fusion automatically. In this context most previous work, has assumed that the FC observes the coherent sum of the signals sent by the sensors [1]–[4]. Under this channel model, the received signal-to-noise ratio (SNR) at the FC grows with the number of sensors, due to an array-processing gain, making the noise irrelevant as the number of sensors tends towards infinity. This is clearly not realistic1 , since it assumes phase synchronization between all sensors and the FC. Furthermore, studying a case that leads to noiseless communication is to some extent trivial and any reasonable communication scheme can be applied. It was actually proven, in [5], for type-based multiple access and later in [6] for decentralized inference in more generality, that if the channel is noncoherent, i.e., the random channel coefficients are zero-mean, the performance is in fact limited by the channel noise, even for an infinite number of sensors, as there is no array-processing gain. As the performance of a traditional PAC based scheme is also noise limited, it is not clear how these different approaches will compare in the noncoherent case. Therefore we are motivated to study noncoherent decentralized detection. If the performance is “limited” by the channel properties, a reasonable tradeoff between resources 1 As
also pointed out by one reviewer of our previous work [3].
c 2009 IEEE 0000–0000/00$00.00
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and performance can be found, a trademark of most interesting engineering problems.
In our review of the coherent case, w.l.o.g. we focus on exponentially distributed sensor observations, instead of the commonly used mean-shifted Gaussian distribution. In concurrence with known results for Gaussian observations, we find that for a fixed quality of observations at the sensors and a given energy budget for all sensors, the PAC based performance does not improve beyond a certain point when increasing the number of sensors. This applies to any of the known schemes, like Decode-and-Forward (DnF), Amplifyand-Forward (AnF) or Estimate-and-Forward (EnF), c.f. [3], [4]. This is well reflected in the communication bound, see Table I. A centralized estimator’s performance would increase steadily with more independent observations. On the other hand the corresponding communication problem of parallel AWGN channels depends only on the energy budget - which we purposely fix independent of the number of sensors. In contrast, using the MAC, the performance of all the previously mentioned schemes improves unbounded with the number of sensors, as had been previously reported in [7], [17]. Again this can be also seen in the bound, see Table I, as this time the communication bound drops with the number of sensors, due to increased SNR through an array-processing gain proportional to the number of sensors. Since for noncoherent modulation, optimal forwarding functions are not known for the multiple sensor case, we adopt a DnF-like simple on/off scheme, where each sensor transmits based on a likelihood ratio (LR) test on its observation(s). The sensor LR threshold can be adapted to conserve energy, according to the “censoring” concept [14], [15], [18]–[21], since the “off” symbol consumes no energy resources. For the same exponentially distributed sensor observations and a fixed average energy usage budget, we find the following behavior for increasing number of sensors for the noncoherent case:
B. Related Recent Work The literature on decentralized detection is rich, growing recently especially with focus on WSNs. Several tutorial papers have been published already featuring extensive references [7]–[9]. We discuss recent related work briefly and focus on results needed for our development in the main body of the paper. The communication aspects of decentralized sensing / inference, like fading or energy and bandwidth constraints, have been recently considered in [10], [11]. Although inherently related to our work, the authors consider coherent MAC communication, in the sense that the channel state information is used to derive optimal forwarding rules. Another related work uses space-time-block-codes (STBC) in a realistic noncoherent setup [12], [13]; although different sensors combine noncoherently, the authors assume that the STBC of each user stays coherent, equivalent to a slow fading assumption. In [14], the authors use a general framework to consider various “censoring” sensor networks, including available side information at the sensors, such as channel state or channel statistics; and in [15] “censoring” sensors are considered for robust decentralized detection, where statistical characterization of sensor observations is not fully known. Both “censoring” papers pay only passing attention to the communication aspect, and only parallel channels are considered. C. Our Work We study a decentralized detection scenario, see Fig. 1, where distributed sensors make conditionally independent observations. The sensors have to communicate their information to a FC across a fast varying channel, without any knowledge of the channel state, subject to an average energy constraint. We want to analyze and compare the detection performance using parallel or multiple channel access, therefore we study varying channel models, including a review of the coherent case, using the simple additive white Gaussian noise (AWGN) channel, and the noncoherent case, for both a fast fading Rayleigh channel and a random phase channel. To help our analysis, we first state that the decentralized detection problem is always limited by two bounds: 1) A limiting communication problem; even if the sensors have perfect observations, decision errors at the FC can still occur due to the noisy communication links. 2) The traditional centralized detection problem is always an upper bound on performance, achieved if communication is noiseless. While the centralized detection performance always improves with increasing number of sensors (assuming independent obseravations), the communication bound has to be calculated as the performance of a multi-channel communication system, see e.g. [16, ch. 12]. We list the trends for increasing number of sensors in Table I, together with references to where they are listed explicitly in the body of the paper.
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Using parallel channel access, there is always a finite optimal number of sensors to participate in the detection problem, as including more sensors decreases detection performance; this number is determined by a tradeoff between more sensors giving more information and a combining loss, which is known to exist for parallel channels with noncoherent modulation [16, ch. 12]. Using multiple channel access, after the random superposition of the sensors’ transmissions, the observations of the FC will approach a zero-mean complex Gaussian random variable as the number of transmitting sensors increases, and exactly so for Rayleigh fading. Therefore the only observable information will be in the variance, which is simply the sum of the transmitted energy, and hence the split of the available energy between the sensors is irrelevant and there is no combining gain or loss.
As the performance of a complex Gaussian observation is limited by the communication bound as a diversity one event, to achieve good performance using the MAC, it is necessary to increase diversity by making multiple independent channel observations available to the FC. This amounts to as much as using independent parallel channels, each used as a MAC by all sensors. On a fast-varying channel this can be easily accomplished by a simple repetition code in time, similar to
BERGER et al.: PAC VS. MAC FOR DECENTRALIZED DETECTION USING NONCOHERENT MODULATION
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TABLE I E XAMPLES OF COMMUNICATION BOUNDS ON PROBABILITY OF ERROR FOR INCREASING NUMBER OF Pr(e) vs. N
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using random STBC in [12], [13]. We simply call this scheme diversity-MAC; it has a combining loss like the PAC across parallel channels and therefore an optimal diversity depending on the energy budget. Comparing the outlined diversity-MAC approach with the use of PAC, we find: 1) On the fast-varying Rayleigh fading channel, the diversity-MAC approach always outperforms the PAC based scheme. This is because the MAC can use an optimal number of parallel channels independent of the number of participating sensors. 2) In comparison, on a random phase channel, the PAC scheme is preferable for reliable observations at the sensors or a large energy budget, as the communciation bound is based on nonfading performance. 3) The diversity-MAC is still preferable for large number of sensors with unreliable information and a limited energy budget; also the diversity-MAC can be implemented on slow varying random phase channels, as an independent channel realization only requires the sensors to reset their carrier phase. The rest of the paper is organized as follows: In Section II we formalize our problem and review the coherent case, in Section III we characterize the PAC and MAC performance for noncoherent decentralized detection focussing on Rayleigh fading, in Section IV we extend the scenario to include multiple parallel MACs and the random phase channel, and in Section V we conclude our work. II. S CENARIO D ESCRIPTION & R EVIEW OF C OHERENT C ASE We are interested in a decentralized detection problem using a wireless sensor network (WSN). In the WSN, N sensors make statistically independent observations conditioned on the hypothesis, denoted as {H0 , H1 }, see Fig. 2. The sensors process their observations to a sufficient statistic yn , n = 1, . . . , N , e.g., the log likelihood ratio (LLR) or a monotonic transform thereof. The yn are distributed as f0 (yn ) and f1 (yn ) conditioned on the two hypotheses respectively. Then, the
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sensors forward a function U (yn ) of the sufficient statistic to a fusion center (FC), which combines them to reach a decision. If all sensor observations would be directly accessible at the FC, the optimal hypothesis test is known to be a threshold test on the likelihood ratio (LR). If the sensors calculate the LLRs yn , then the decision at the FC can be calculated as: ( PN H1 , n=1 yn > Γ d(y1 , . . . , yN ) = (1) H0 , otherwise. In decentralized detection, the added challenge is caused by an imperfect communication channel between the sensors and the FC, as the FC observes the channel output zn and not directly the yn , see Fig. 2. We therefore introduce the following: •
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Communication Bound: The performance of any decentralized detection setup is always bounded by an equivalent communication problem. This can be seen as the limiting case when the local probabilites of detection and false alarm at the sensors converge to one and zero respectively, the only uncertainty is then introduced by the communication channel. Observation Bound: The detection performance is always limited by the quality and number of available observations. In decentralized detection this bound is achieved when the communication channel is noiseless.
In decentralized detection imperfect communication links were first modeled by restricting the range of the transmit function U (yn ) to a number of discrete values. In this case the performance can be related to a quantization problem. Under this model, when using a binary function U (yn ), quantization based on an identical threshold test on the LR at all sensors has been shown to achieve optimal performance as the number of sensors approaches infinity [22]. This can also be shown to be optimal under a more general information-theoretic rate constraint on the transmission functions U (yn ) [23]. Although decentralized detection using a MAC and discrete channel models was a minor part of [24], the focus was more on modeling imperfect PAC in terms of channel interference and only a more detailed channel model would motivate actually benefiting from the properties of the MAC.
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Fig. 2. Diagram of the general decentralized detection framework for parallel (PAC) and multiple (MAC) channel access; the channel coefficients hn can be random (noncoherent) or deterministic (coherent); the following processing of the zn will depend on the channel coefficients.
Instead of the rate constraint, in practical wireless communication it makes more sense to accurately model the channel’s effects [8], [9]. In a PAC, the FC receives noisy versions of the sensors’ transmissions, zn , e.g., an additive white Gaussian noise (AWGN) channel (c.f. Fig. 2(a), for hn = 1): zn = U (yn ) + wn
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In this setup, it is most realistic to enforce an average energy usage constraint on the function U (yn ), of the general form: (1 − η)E |U (yn )|2 | H0 + ηE |U (yn )|2 | H1 ≤ Eη /N (4)
This constraint can cover several scenarios, both Bayesian and Neyman-Pearson, where η = 0 only constrains the energy usage under H0 , namely E0 , and for a Bayesian formulation the average energy usage, where η = π1 and 1 − η = π0 are the priors of the hypotheses. The function U (yn ) should optimally converge to a classical modulation scheme when the information at the sensors becomes increasingly reliable – since these are known to be optimal in the equivalent communication problem. Therefore it is practical to define U (yn ) in terms of a binary constellation si , i = 0, 1, e.g., BPSK or OOK, {±1} or {0, 1} respectively. Commonly used approaches for the coherent case are: Decode-and-Forward (DnF), Amplify-and-Forward (AnF), and Estimate-and-Forward (EnF), see [3], [4], [7], [17], [25]. To quantify the performance of a detection problem, the receiver operating characteristic (ROC) is often used, plotting the probability of detection (Pd ) versus the probability of false alarm (Pf a ). For comparison a scalar quantity is preferable, where we choose the probability of error, defined as the
weighted sum of type I and II errors2 : Pr(e) = (1 − η)Pf a + η(1 − Pd )
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Again, if η is chosen according to the priors π1 = 1 − π0 , then this is equivalent to a Bayesian approach. A. Review of Coherent PAC In the coherent case, it is assumed that the FC knows all the channel coefficients and compensates them, rendering communication an additive white Gaussian noise (AWGN) channel (c.f. Fig. 2, for hn = 1), zn = U (yn ) + wn ,
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where for consistency with later cases, we define wn as complex Gaussian of power N0 . Across this channel, the optimal binary signal constellation is BPSK, si = ±1, and the decision statistic at the FC is simply the sum of the real parts, ( PN H1 , n=1 ℜ {zn } > Γ d(z1 , . . . , zN ) = (7) H0 , otherwise. The threshold Γ can be chosen either based on Bayesian or Neyman-Pearson criteria. Mostly Γ = 0 is used, due to the symmetry of the BPSK constellation and this is also optimal in case of equal Bayesian priors π1 = π0 . For additive Gaussian observations at the sensors, the performance of AnF and DnF can be easily defined in closed form, and using the same transmission function U (yn ) at all sensors has been shown to be asymptotically optimal [26]. AnF performs better for unreliable observations at the sensors, while DnF is preferable for more reliable observations. The communication bound can be easily calculated as the performance of N parallel Gaussian channels with coherent detection, see e.g. [16, ch. 12]. The optimum constellation is 2 Equivalently we can also i) apply a Neyman-Pearson formulation, maximizing Pd for a fixed Pf a ; or ii) keep the type I and II errors equal, i.e. maximize Pd while Pf a = 1 − Pd .
BERGER et al.: PAC VS. MAC FOR DECENTRALIZED DETECTION USING NONCOHERENT MODULATION
p ± Eη /N and the real part of the noise has variance N0 /2. The decision statistic in (7) is Gaussian with signal-to-noiseratio (SNR) Eη /(N0 /2), which is independent of N , as both the signal and noise power scale with N . With this, due to the symmetric setup the communication bound is: s ! Eη , (8) Pr(e) = Q N0 /2 R∞ 2 where we define the Q-function as Q(x) = √12π x e−t /2 dt.
B. Review of Coherent MAC Since the wireless medium is inherently a multiple-access channel (MAC) – which is only converted into a PAC using TDMA, FDMA or CDMA for simple demodulation – it is natural to also consider the MAC directly. This way information from distributed sensors can be fused “in the channel”, significantly reducing bandwidth requirements. The coherent MAC model, where signals from all sensors are added coherently, is defined as follows (c.f. Fig. 2(b), for hn = 1): N X z= U (yn ) + w, (9) n=1
with w additive complex Gaussian noise of power N0 . The optimal decision statistic at the FC (assuming BPSK modulation) is a threshold test on the real part of the received signal H1 , ℜ {z} > Γ d(z) = (10) H0 , otherwise To calculate the communication bound, we assume that p under H1 all N sensors send Eη /N . With that the SNR is N Eη /(N0 /2); and due to symmetry the communication bound is: s ! N Eη . (11) Pr(e) = Q N0 /2
Clearly we see that the communication bound is affected by an array processing gain, as the SNR increases linearly with N . As in practice this requires phase synchronization between all sensors, it amounts to distributed beamforming, focusing the transmitted energy at the FC. Based on this fact, it has been shown that for additive Gaussian observations at the sensors, the simple AnF scheme allows optimal inference [7], [17], since as the number of sensors grows, the noise term becomes irrelevant. If the noise term is irrelevant, then any injective transmission function U (yn ) will achieve asymptotically optimal performance. For a finite number of sensors, the same observation as for the PAC-AWGN model about AnF and DnF can be made: DnF outperforms AnF once the observations at the sensors are of sufficient quality. This points towards a generally optimal U (yn ), which has been found for the coherent MAC by maximizing different metrics [3], [4], for various observation models, also including non-Gaussian noise. The simplistic MAC-AWGN model was also used for “type-based” estimation [1], [2], where the FC decides between a finite number of hypotheses (which includes detection as a special case).
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C. Numerical Example Without loss of generality, through out this paper we will consider exponentially distributed observations at the sensors, which could be processed from noncoherent observations, i.e., change in variance. Given a local SNR γobs at the sensors, the nonnegative yn are distributed as, f0 (yn ) = e−yn 1 − yn e γobs +1 . f1 (yn ) = γobs + 1
(12) (13)
If the observations were directly available at the FC, the optimal decision statistic would be the sum, distributed as χ22N . With this we can determine the observation bound for exponentially distributed observations, which will be used throughout this paper: Pd = Pf a =
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where Γ has to be numerically determined such as to minimize Pr(e). In Fig. 3 we show both analytical and Monte-Carlo simulation results for the coherent case. Analytical results for the bounds and the DnF case, while simulated for AnF and EnF (the EnF function for exponentially distributed observations is given in Appendix A). We see that the observation bound is decreasing in N , as the observations are assumed independent this will apply for any distribution of the observations. A strong contrast is the communication bound, which is independent of N for PAC, c.f. (8), but strongly decreasing for MAC, c.f. (11). While the PAC performance is ultimately limited by the communication bound, i.e., by Eη , for large N – the MAC approach “magically” increases the available SNR at the FC by an array-processing gain. In concurrence with other results, [7], [17], we find that for the coherent MAC, any reasonable U (y) has perfect performance as the number of sensors goes to infinity. III. D ECENTRALIZED D ETECTION USING N ONCOHERENT M ODULATION The simple channel model in (9) helped to motivate the direct use of the MAC properties, but it is practically unattractive as it assumes perfect carrier and phase synchronization between all sensors. This can only be achieved by feeding back channel and/or phase estimates from the FC to the sensors, which leads to a communication overhead proportional to the number of sensors, and certainly would require frequent updating. For “type-based” estimation [5] and later for decentralized inference in general [6], it has been shown that if the channel is noncoherent and the total power is fixed, then the performance does not improve with the number of sensors beyond a certain point, in contrast to the coherent case. To be more specific,
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let the channel consist of AWGN and a random channel coefficient hn , c.f. Fig. 2(b): z=
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In this context, the case where hn is distributed zero-mean and the channel state is unknown at the sensor nodes is denoted as noncoherent. Examples are the Rayleigh fading channel, with hn ∝ CN (0, 1), or the unknown phase channel, hn = ejφn , φ ∈ [0, 2π]. In the noncoherent channel model in (16), the FC can only observe the energy on the channel. This detection problem is more challenging compared to the coherent case, analogous to variance change versus mean change detection of a Gaussian random variable. As only the energy of the sensors’ transmissions can hold any information, we adopt a simple on/off scheme and the FC will base its decision on the variance of the channel, indicating the sum of sensors that detected a target. This on-off keying (OOK) scheme is the obvious noncoherent binary modulation and also links to the “censoring” concept [14], [15], [18]–[21], as the “off” symbol doesn’t expend any energy resources. Also in earlier work, [25], using a single sensor for detection, we found that DnF performed close to optimal for noncoherent detection at the FC. Accordingly, we define the function U (yn ) as a simple step function (see Fig. 4), based on an LR test, A, yn ≥ τ (17) U (yn ) = 0, yn < τ .
The local detection threshold τ can be chosen according to the censoring concept to minimize utilized energy. At the FC, we assume an energy detector, H1 , ξ > Γ d(ξ) = (18) H0 , otherwise
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Fig. 4. A binary function similar to on-off-keying (OOK) performs close to optimal; a threshold test on the likelihood ratio is used as in the “censoring” approach. Note that the censoring level and ouput amplitude can be traded-off: (τ, A) and (τ ′ , A′ ) may require the same average energy.
As for the noncoherent case with many sensors an optimal forwarding function U (yn ) is not known, the censoring concept allows us to improve performance to some degree. This concept could also have been applied in the earlier coherent section, but is superseded by the available EnF formulation that performs close to optimal in the coherent case. Due to the censoring approach, we will have to: • jointly choose the optimal τ , A and Γ, in the sense of maximizing detection performance, while adhering to the average energy usage constraint in (4); and • compare to the communication and observation bounds, as the optimal forwarding function is unknown. The output of the MAC always approaches a zero-mean complex Gaussian distribution for large N , by the central limit theorem [27]. The only adjustable parameter is the variance. In case of Rayleigh fading hn is complex Gaussian, and the previous approximation will be exact for all N . Therefore we will first focus on this scenario, as the performance is analytically easier and a comparison to the PAC will be “fair”. By this we mean that the output of each parallel channel will have the same behavior as the output of the MAC, while for
BERGER et al.: PAC VS. MAC FOR DECENTRALIZED DETECTION USING NONCOHERENT MODULATION
other channel models the output of the MAC will produce “artificial” fading, due to the central limit theorem, while the PAC could merely be affected by a random phase. Later on we will extend also to the random phase case and point out differences. As the PAC approach inherently uses N independent channels - an advantage in terms of diversity when faced by fading - we will also consider a case where the MAC approach can use multiple independent channels to increase resistance against fading. This is similar to the approach taken in [12], [13], where STBC were used to increase the performance over fading channels. Since the joint optimization of τ , A and Γ is not available in closed form, we will develop expressions for the performance given a certain parameterization. This way the threedimensional parameter space can be efficiently evaluated, and the optimal parameterization can be found numerically. A. PAC Detection Performance In the PAC, each sensor is received separately see Fig. 2(a), we focus on the Rayleigh fading model: zn = hn U (yn ) + wn ,
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with hn ∝ CN (0, 1) and wn ∝ CN (0, N0 ) We compare the total received energy to a threshold; this is not the optimal decision statistic in this case, but has been shown to be close to optimal for low SNR, c.f. [20], ξP AC =
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We see that |zn |2 given the yn is distributed as an exponential random variable with mean |U (yn )|2 + N0 . Therefore ξ is the sum of N independent non-identically exponentially distributed random variables, with parameters α1 , . . . , αN such that: 1 α1 = . . . = αk = β1 = 2 (21) A + N0 1 (22) αk+1 = . . . = αN = β2 = N0 The received samples are sorted into two groups, one corresponding to the sensors that made a local detection and one corresponding to the others that did not. This means that ξ can be modelled as a sum of two χ2 random variables, one of variance A2 + N0 , degree 2k, and one of variance N0 , degree 2(N − k). By definition k is a binomial random variable: N Pr(k | Hj ) = (1 − pj )N −k pkj , k = 0, . . . , N, (23) k where we defined j = 0, 1 as the index to hypothesis {H0 , H1 }, and pj can be calculated as: Z +∞ p0 = pf a = f0 (y)dy (24) τ Z +∞ p1 = pd = f1 (y)dy (25) τ
7
The probability of detection and probability of false alarm are given by: N X N k Pd = p (1 − pd )N −k FΞ (Γ|k) (26) k d k=0 N X N k Pf a = p (1 − pf a )N −k FΞ (Γ|k) (27) k fa k=0
The conditioned complementary cumulative probability densities FΞ (Γ|k) = Pr(ξ > Γ|k), for k = 1, . . . , N −1, are derived in the appendix, see (59). For k = 0 or k = N , i.e., if all or no sensors send, the output is simply χ2(2N ) with respective variance, m N −1 X Γ 1 (28) e−Γ/N0 FΞ (Γ|0) = m! N 0 m=0 m N −1 X 2 1 Γ FΞ (Γ|N ) = e−Γ/(A +N0 ) (29) 2+N m! A 0 m=0 We also reproduce the optimal decision statistic, developed in [20], which is the LLR: ! 2 2 N X p1 β2 e−β2 |zn | + (1 − p1 )β1 e−β1 |zn | (30) Λ= log p0 β2 e−β2 |zn |2 + (1 − p0 )β1 e−β1 |zn |2 n=1 In [20], it was shown that for vanishing SNR on the sensor-FC link, A2 /N0 → 0 (β1 → β2 ), this combining rule is equivalent to (20). Additionally it is easy to see that for increasingly reliable observations at the sensors, p1 → 1, p0 → 0, the same is true, as (20) is the optimal decision statistic under a regular OOK communication regime. To calculate the communication bound for the PAC scheme, we note that N parallel channels are used, the same transmitted on each of them. This can be linked to noncoherent transmit diversity, but with the less common asymmetric OOK modulation. Still, as can be found in standard textbooks [16, ch. 12], noncoherent parallel channels always have a “combining loss”, i.e., after combining the equivalent SNR is less than if all energy had been sent through a single channel. On the other hand we are considering a fading model, where independent realizations lead to a diversity effect. This means that while one single channel might be in a deep fade, the chance of several independent channels being in a deep fade is much smaller. We will therefore see two trends, and probably a point of optimal tradoff. In any case, the output of a noncoherent fading channel under OOK modulation is simply χ22N distributed, with the probabilities of error already given in the previous section for the observation bound, c.f. (14),(15): n N −1 X Γ Γ 1 − e Eη /(N η)+N0 Pd = (31) n! Eη /(N η) + N0 n=0 n N −1 X Γ 1 (32) e−Γ/N0 Pf a = n! N 0 n=0 The main difference is that the available energy is now divided between N channels, the SNR on each decreasing with 1/N .
8
IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL. X, NO. X., MARCH 2009
1
1
0.95
0.95
0.9
0.9 γobs = 6 dB
0.85
0.85 N = 10
D
0.8
0.75
γ
obs
P
P
D
0.8 = 10 dB
0.7
0.75 0.7
0.65
0.65
MAC(ED) − analytic MAC(ED) − sim PAC(ED) − analytic PAC(ED) − sim PAC(LR) − sim
0.6 0.55 0.5
N = 30
0
0.1
0.2
0.3
0.4
MAC(ED) − analytic MAC(ED) − sim PAC(ED) − analytic PAC(ED) − sim PAC(LR) − sim
0.6 0.55 0.5
0.5
0
0.1
0.2
0.3
P
0.4
0.5
P
FA
FA
(a) N = 5, Eη /N0 = 13 dB
(b) γobs = 3 dB, Eη /N0 = 10 dB
Fig. 5. Comparison of receiver operating characteristics (ROC) of MAC and PAC based noncoherent decentralized detection; in case of few sensors with reliable observations, PAC is preferable, while for many sensors with unreliable observations MAC is superior; LR denotes the likelihood ratio based PAC combining rule in (30); η = 0.5.
B. MAC Detection Performance As in Fig. 2(b), the FC observes the superposition of the waveforms: N X z= hn U (yn ) + w (33) n=1
where we assume the same complex Gaussian fading coefficients and noise of variance N0 . Then z, conditioned on the yn , is zero-mean complex Gaussian with variance N X
with variance either N A2 +N0 or N0 . Since in this case A2 = Eη /(N η), the N cancels and we note that the channel output is actually independent of N . This is because the signals emitted by each sensor add up noncoherently, the observed energy is not affected by its split. Therefore the communication bound is Pd = e Pf a = e
Γ − Eη /η+N
0
−Γ/N0
(37) (38)
which is identical to the PAC communication bound for N = 1, c.f. (31),(32).
|U (yn )|2 + N0 = kA2 + N0 ,
n=1
where k is the number of sensors that claim to have detected a target, according to the censoring rule in Fig. 4. Because of the random phase, clearly the received power, ξMAC = |z|2 ,
(34)
is a sufficient statistic and the threshold test in (18) is the optimal decision rule in this case. To express the probability of exceeding the threshold at the receiver, we see that ξ given the yn is distributed exponentially with mean kA2 +N0 , using the same binomial random variable k defined in (23). With this the detection and false alarm probabilities are given by a mixture of exponentials: N X Γ N k N −k (35) Pd = p (1 − pd ) exp − 2 kA + N0 k d k=0 N X Γ N k Pf a = pf a (1 − pf a )N −k exp − 2 k kA + N0 k=0 (36) The communication bound in the MAC case is simpler, as the output of the channel will always be complex Gaussian,
C. Numerical Example We consider the same exponential observations as in the review of the coherent case, see (12),(13); the local detection and false alarm probabilities are the complementary cumulative density functions: pf a = e−τ pd = e
(39) τ obs +1
−γ
(40)
We will have to choose τ jointly with A and Γ to optimize the performance, this is equivalent to the censoring concept. The joint optimization problem can be implemented as a simple one-dimensional search for the most efficient censoring rule: for any given local detection threshold τ , an amplitude A can be found to satisfy the energy constraint, N A2 [(1 − η)pf a + ηpd ] ≤ Eη
(41)
This is simplified by the fact that a larger A always improves Pr(e), therefore the inequality constraint in (4) is practically an equality constraint. Then varying Γ a receiver operating characterstic (ROC) is generated. The optimal ROC is the
BERGER et al.: PAC VS. MAC FOR DECENTRALIZED DETECTION USING NONCOHERENT MODULATION
0.5
0.5 0.4
PAC(ED) − analytic PAC(ED) − sim PAC(LR) − sim Obs. Bound Comm. Bound
γobs = 3 dB
0.3
γobs = 6 dB
0.4
γ
obs
MAC − analytic MAC − sim Obs. Bound Comm. Bound
= 3 dB
0.3 γ
obs
0.2
0.2
0.15
0.15
Pr(e)
Pr(e)
9
= 6 dB
0.1
0.1
γ
γobs = 10 dB
obs
= 10 dB
0.05
0.05
5
10
15
20
25
30
35
5
10
15
20
25
30
35
N
N (a) PAC
(b) MAC
Fig. 6. Comparing the probability of error for the noncoherent Rayleigh fading case, we see that while the MAC performance is ultimately bounded by the communication bound, independently of N and γobs , the PAC achieves its optimal performance for a finite N that varies with γobs ; Eη /N0 = 10 dB and η = 0.5.
convex closure of all ROCs corresponding to a specific local threshold τ . To get some intuition of the performance behavior, we plot the optimal receiver operating characteristic (ROC) for several configurations of N , Eη and γobs in Fig. 5. We include both closed form results based on the energy detector (ED) and Monte-Carlo simulation, where using simulation we can also evaluate the optimal PAC combining rule based on the likelihood ratio (LR). Interestingly we find that neither the PAC nor the MAC based transmission scheme is always optimal. For certain constellations the PAC outperforms the MAC, e.g., in Fig. 5(a) where there are few sensors with high quality observations. On the other hand, for less reliable observations, see Fig. 5(b), but a large number of sensors, the bandwidth-efficient MAC transmission scheme has superior performance. The optimal combining rule based on the likelihood ratio (LR) increases the PAC performance, but shows generally the same trends. To verify these trends, it is necessary to compare the performance of the MAC and PAC schemes over a large range of N and γobs . Therefore we next plot the probability of error, which can be found by two-dimensional optimization over τ and Γ, where A(τ ) can be always chosen to satisfiy (4), h Pr(e) = min min (1 − η)Pf a (Γ, A(τ ), τ ) τ Γ i (42) + η [1 − Pd (Γ, A(τ ), τ )]
The two dimensions can be decoupled, as Pr(e) is uni-modal in Γ for fixed τ , as Pd and Pf a are monotonically decreasing in Γ. Fig. 6 shows the values for three fixed values of γobs for increasing number of sensors N . Surprisingly MAC and PAC have different trends as N increases: the MAC performance approaches a constant determined by the communication
bound, constant in N and γobs ; the PAC performance decreases after exceeding a certain N , which varies with γobs . This shows the varying tradeoff between diversity gain and combining loss, where the effective SNR on the channel is affected by the local pd and pf a . In comparison to the coherent case, where the PAC peformance was constant for large N and improving for MAC, now the peformance of PAC is affected negatively by N over a certain threshold, while the MAC performance is constant for large N . This points to the fact that in large sensor networks, for reliable observations at the sensors a limited number of sensors should be queried using PAC, while for weak observations all sensors should be combined using the MAC approach. IV. E XTENSIONS
TO
D IVERSITY C HANNEL
AND
R ANDOM P HASE
A. Diversity-MAC We see that only a MAC based scheme can combine a large amount of unreliable sensor measurements efficiently, while PAC includes transmit diversity, but becomes inefficient for large N . If the FC can observe several independent realizations of the MAC, the benefit of diversity is combined with the previously mentioned advantages of the MAC; a similar idea has been suggested in [12], [13] using STBC. Since we assume a fast fading channel, full transmit diversity can be easily achieved by a simple repetition code in our case. Equivalently if multiple FDMA channels that are sufficiently far apart are available, a similar repetition could be executed in this dimension. Following the previous discussion, we suggest giving the FC multiple observations of the MAC to reduce uncertainty. Although many cases can be considered, we limit ourselves to
10
IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL. X, NO. X., MARCH 2009
x
0.5
y1
h11 x
H0 / H1
U
y2
+
γ
obs
MAC: N = 10, analytic MAC: N = 20, analytic MAC: N = 10, sim MAC: N = 20, sim Comm. Bound
= 3 dB
0.3
w1
hN1
0.2
x
0.15
h1B
+
x
wB
U
yN
0.4
z1
Pr(e)
U
zB
hNB
γobs = 6 dB
0.1
0.05
γ
obs
Fig. 7. The main limiting factor of noncoherent transmission across the MAC is that the output is complex Gaussian; to improve the performance the energy should be split between multiple independent realizations of the MAC output, e.g., time diversity in case of fast fading, or possibly frequency diversity.
the case where all sensors send on each parallel channel. This is a straight forward adaption of the previous scheme, as each parallel channel is used as a MAC. Introducing a bandwidth expansion factor B, see Fig. 7, the signal model is similar to (16), but with b = 1, . . . , B realizations, zb =
N X
hnb U (yn ) + wb .
(43)
n=1
The FC uses again an energy detector of the form: ξDMAC =
B X
|zb |2
(44)
b=1
For now, let us assume all the hnb are independent and complex Gaussian, but for large N , the same behavior can also be achieved by randomizing the phase of U (yn ) for arbitrary slowly changing channel coefficients (hnb = hn ). The performance of this scheme is derived in a similar fashion as before, since each zb is distributed CN (0, kA2 +N0 ) and k is constant across b, as the same sensors transmit on each parallel channel. Therefore the result it a mixture of χ2(2B) : N X N k Pd = p (1 − pd )N −k k d k=0 b B−1 X 1 Γ Γ × (45) exp − b! kA2 + N0 kA2 + N0 b=0 N X N k Pf a = p (1 − pf a )N −k k fa k=0 b B−1 X 1 Γ Γ (46) exp − × b! kA2 + N0 kA2 + N0 b=0
Since we kept A as the amplitude used on each parallel channel, the energy usage is scaled by B, Eη = BN A2 [(1 − η)pf a + ηpd ]
(47)
5
10
15
20
25
= 10 dB
30
35
B Fig. 8. When the noncoherent MAC is observed B times, the communication bound is based on B channels, as in the noncoherent PAC, but we still see that for a given γobs the performance can be also improved by increasing N ; Eη /N0 = 10 dB, η = 0.5.
The communication bound is equivalent to the one derived for the noncoherent PAC in (31) and (32), only exchanging B ↔ N. We now evaluate the performance of the diversity-MAC. The results are in Fig. 8; for comparison N is fixed when varying B. We see that when varying the bandwidth expansion factor B, the behavior is similar to the noncoherent PAC scheme, as predicted by the communication bound. The best performance is achieved for a fairly small value of B (between three and six), but the approach still benefits from a large N ; as an example we include results for N = 10 and N = 20, see Fig. 8. Although the PAC plots from Fig. 6(a) are not repeated, the diversity-MAC outperforms the PAC based scheme for any limit on the available sensors N , bandwidth B, energy constraint Eη or observation quality γobs on the fast fading Rayleigh channel. B. Random Phase Channel In the random phase channel, the channel coefficients are simply unity phasors, zn = ejφn U (yn ) + wn ,
φ ∈ [0, 2π],
(48)
and wn is the additive noise. The PAC scheme can use the energy detector in (20) or the optimal combining rule based on the LLR [20]: A2 −N A N 0 + (1 − pd ) X pd I0 N0 /2 |zn | e Λ= log A2 (49) −N 0 + (1 − p |z | e ) pf a I0 NA n=1 n f a 0 /2
where Iα (x) is the αth-order modified Bessel function of first kind, see, e.g., [16, ch. 2]. In the case of the energy detector, the detection performance can be calculated in closed form, as ξ will be a non-central
BERGER et al.: PAC VS. MAC FOR DECENTRALIZED DETECTION USING NONCOHERENT MODULATION
11
0.4 0.3
0.4
γobs = 3 dB
0.3
γobs = 6 dB
0.2 0.15
0.2
Pr(e)
Pr(e)
0.1
γobs = 10 dB
0.05
PAC(ED) − analytic PAC(ED) − sim PAC(LR) − sim Obs. Bound Comm. Bound 0.01
MAC, approx. MAC, sim Obs. bound Comm. Bound
γobs = 3 dB
5
10
15
20
25
30
0.15 0.1
35
γ
γobs = 10 dB
0.05
5
10
obs
15
N
The communication bound can be calculated based on the noncoherent N -parallel channel communication problem, s s ! Eη /η Γ , (53) Pd = QN , N0 /2 N0 /2 n N −1 X 1 Γ (54) e−Γ/N0 . Pf a = n! N 0 n=0 In this case the noncoherent combining loss is obvious, as more channels simply introduce more noise, while the total signal energy stays unchanged.
25
30
35
(a) B = 1
0.4 0.3
The complementary cumulative probability density function of a non-central χ22N variable of even degree is given by the generalized Marcum’s Q function, defined as [16, ch. 2], Z ∞ m−1 2 2 x e−(x +a )/2 Im−1 (ax) dx. (52) Qm (a, b) = x a b
20
N
Fig. 9. For the random phase channel, the communication bound clearly shows the noncoherent combining loss; generally the PAC scheme can achieve better performance than in the Rayleigh fading case, especially for few sensors with large γobs ; Eη /N0 = 10 dB, η = 0.5.
MAC − approx. MAC − sim Comm. Bound
γobs = 3 dB
0.2
Pr(e)
χ22N , with non-centrality parameter kA2 . As in previous cases, k is a binomial random variable, indicating the number of local threshold exceedences. With this, repeating the steps of previous derivations, the performance is a mixture of noncentral χ22N , N X N k Pd = p (1 − pd )N −k k d k=0 s s ! kA2 Γ , (50) , × QN N0 /2 N0 /2 N X N k Pf a = p (1 − pf a )N −k k fa k=0 s s ! kA2 Γ × QN . (51) , N0 /2 N0 /2
= 6 dB
γobs = 6 dB
0.15
0.1
0.05
γobs = 10 dB 5
10
15
20
25
30
35
B (b) N = 20 Fig. 10. (a) On the random phase channel, the MAC performance can be approximated slightly conservatively by the Rayleigh fading case for average to large N , the central limit theorem is accurate for large pd N ; (b) to produce independent observations of the MAC, the sensors only need to randomize their carrier phase each time they generate a new independent channel output; Eη /N0 = 10 dB, η = 0.5
We plot both the theoretically calculated performance using the energy detector (ED) and Monte-Carlo simulation results for both energy detector (ED) and optimal combining rule based on the likelihood ratio (LR) in Fig. 9. First we notice that for the non-fading case, the communication bound is much lower than in Fig. 6(a). This leads to improved performance over the fading case as expected, especially when the observations at the sensors are more reliable. Still, most trends are similar to the Rayleigh fading case, as i) there is an optimal number of sensors that should participate in the decentralized detection, as more sensors would actually lead to a worse performance; and ii) the likelihood ratio (LR) based
IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL. X, NO. X., MARCH 2009
V. C ONCLUSION We analyzed the performance of decentralized detection using parallel or multiple channel access. While for the coherent case the MAC based scheme profits from an array-processing gain proportional to the number of sensors, the performance using PACs is bounded by the available transmit energy, independent of the number of sensors. In comparison, for the noncoherent case both behaviors are different, as the MAC performance is limited by the available energy independent of the number of sensors, and the PAC scheme suffers from a combining loss proportional to the number of sensors. This trend favors the MAC scheme for large numbers of sensors; but to achieve superior performance using noncoherent modulation, the MAC scheme has to be transmitted across multiple independent MACs to increase diversity, as each MAC is basically a fading channel due to the central limit theorem. Across fast-fading Rayleigh channels, the MAC output is exactly complex Gaussian, and ample diversity is available, therefore the MAC approach always outperforms the PAC approach. In contrast, using noncoherent modulation across a random phase channel, the PAC scheme is preferable for reliable sensor observations or large transmit energy budget,
1 f(y|H ) 1
0.8 probability
combining rule only improves the performance slightly, and especially so for unreliable sensor observations, where the general performance is not very enticing. As a comparison we consider the MAC with diversity, as a new random phase realization can be easily generated by resetting the oscillator at the sensor. The Rayleigh fading observations at the FC, which are only generated by the noncoherent superposition of the sensors’ signals, will be completely independent if each sensor randomizes the phase each time. On the other hand the output at the FC will not be exactly Gaussian, especially for small N (also due to censoring, maybe only few sensors will be active, therefore we need pd N > 5 to approximate a Gaussian distribution reasonably). For comparison, we will plot the Monte-Carlo based simulation results together with the theoretically derived expressions for the Rayleigh fast fading channel. Fig. 10 includes the regular MAC without diversity, where we see that the central limit theorem is precise for large pd N , where pd increases with γobs , as then many sensors add up randomly on the channel. Also we show an example of the diversity-MAC, in which we vary B for N = 20 sensors and varying quality of sensor observations γobs , see Fig. 10(b). We see that for this average N , the Gaussian approximation is generally close to the simulation results, where the Gaussian approximation proves to be conservative in terms of perforamnce. This seems reasonable, as the Gaussian distribution is a worst case for most inference and detection problems [28]. The random phase channel therefore has two winners, as both the PAC and MAC benefit from the milder channel conditions, although in completely different ways. While the PAC actually benefits from the reduced uncertainy - which will surely be more so for higher SNR (Eη /N0 ) - the MAC can more easily implement the diversity repetition scheme.
f(y|H0)
0.6 0.4 0.2 0
0
2
4
6
8
10
y
3 2 U(y)
12
1 0
DnF AnF EnF
−1 −2 0
2
4
6
8
10
y
Fig. 11. Plot of fi (y), i = 0, 1 and the varying coherent forwarding functions U (y) for exponential observations of SNR γobs = 10 dB, energy constraint Eη /N0 = 6 dB and η = 0.5.
as then the combining loss is outweighed by the fact that using PACs no “artificial” fading is created. A PPENDIX A O PTIMAL N ON -L INEARITY FOR THE C OHERENT C ASE WITH E XPONENTIAL O BSERVATIONS The SNR optimal non-linearity can be described as Estimate-and-Forward (EnF) As described in [4]. It amounts to forwarding the minimum mean-square error (MMSE) estimate subject to the power constraint, which is equivalent to a linearly scaled version of the conditional expectation: UEnF (y) = λE [s | y] (55) f1 (y) f0 (y) (56) + s1 = λ s0 f0 (y) + f1 (y) f0 (y) + f1 (y) Given the BPSK constellation, the SNR optimal EnF can be determined as, f1 (y) − f0 (y) f1 (y) + f0 (y) γobs y − (1 + γobs ) exp 1+γ obs =λ . γobs y + (1 + γobs ) exp 1+γ obs
UEnF (y) = λ
(57)
(58)
Compared to the case of additive Gaussian observations (c.f. [3], [4]), we note that the function is not symmetric around its zero crossing, as exponential observations are non-negative. We plot one example realization for γobs = 10 dB, energy constraint Eη /N0 = 6 dB and η = 0.5 in Fig. 11, together with the correponding fi (y). All three functions share the same zero crossing (AnF per our own definition), which coincides at the point where f1 (y) = f0 (y). This seems intuitive as the decision at the FC is a threshold test about zero, therefore in a noiseless case transmitting something positive correponds to deciding on H1 and vice versa.
BERGER et al.: PAC VS. MAC FOR DECENTRALIZED DETECTION USING NONCOHERENT MODULATION
13
# k −N +ν ν−1 X X (β1 Γ)k N −ν −1 k−ν (β2 − β1 ) e−β1 Γ (−1) FΞ (Γ|k) = Bk ν β k! k − ν 1 ν=1 k=0 "N −k # −N +ν ν−1 X N − ν − 1 X (β2 Γ)k (β − β ) 1 2 (−1)N −k−ν + Bk e−β2 Γ ν N − k − ν β k! 2 ν=1 "
(59)
k=0
S UM
OF
A PPENDIX B T WO C HI -S QUARE R ANDOM VARIABLES D IFFERENT VARIANCE
R EFERENCES OF
The conditional complementary cdf of ξ when k sensors are transmitting is the sum of N exponential random variables. In case they all have the same rate parameter αi , ξ is simply χ2 distributed; if they have all distinct rate parameters αi 6= αj for i 6= j, the solution is also well known. In our case, k variables have αi = β1 , i = 1, . . . , k and N −k have αi = β2 , i = k + 1, . . . , N . This is the less common case of 1 < a < N distinct rate parameters, each of order rm , m = 1, . . . , a and r1 + . . . + ra = N ; this is given in [29], [30]: rm a X X
FΞ (Γ|k) = B
m=1 l=1
rX m −l (βm Γ)j −βm Γ Φml (−βm ) e (l − 1)!β rm −l+1 j=0 j! (60)
and B=
a Y
m=1 l−1
Φml (t) =
rm
(βm )
d dtl−1
a Y
(61) (βj + t)−rj
(62)
j=1,j6=m
In our case, there are a = 2 distinct rate parameters, r1 = k and r2 = N − k, with this FΞ (Γ|k) = Bk
k X l=1
+ Bk
N −k X l=1
k−l Φ1l (−β1 ) X (β1 Γ)j −β1 Γ e (l − 1)!β k−l+1 j=0 j!
NX −k−l (β2 Γ)j −β2 Γ Φ2l (−β2 ) e (l − 1)!β N −k−l+1 j=0 j!
(63)
and Bk = β1k β2N −k . For a = 2, we can simplify: dl−1 (β2 + t)−r2 dtl−1 dl−2 = l−2 (−r2 )(β2 + t)−r2 −1 dt (−1)l−1 (r2 + l − 2)! (β2 + t)−r2 −l+1 = (r2 − 1)!
Φ1l (t) =
(64) (65) (66)
Inserting the βi and rm : Φ1l (−β1 ) =
(−1)l−1 (N − k + l − 2)! (β2 − β1 )−N +k−l+1 (N − k − 1)! (67)
Φ2l (−β2 ) =
(−1)l−1 (k + l − 2)! (β1 − β2 )−k−l+1 (k − 1)!
(68)
After combining and a slight re-indexing we find the final result in (59).
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Marco Guerriero (S’06) was born in Salerno, Italy, on June 18th, 1981. He received his BASc and MSc (Electrical Engineering) from the University of Salerno, Italy in 2002 and 2005 respectively. During PLACE his MSc he also spent a semester at Delft University PHOTO of Technology (TUDELFT), Netherlands, where he HERE took graduate courses in electrical engineering. He is currently working towards his Ph.D. degree in electrical engineering at the University of Connecticut (UCONN), Storrs. In the summer of 2007, he was as a visiting scientist at the NATO Undersea Research Centre (NURC) La Spezia, Italy. His research interests lie in the areas of signal processing, with particular focus on distributed detection and estimation in sensor networks, target tracking and data fusion.
Christian R. Berger (S’05) was born in Heidelberg, Germany, on September 12th, 1979. He received the Dipl.-Ing. degree in electrical engineering from the Universit¨at Karlsruhe (TH), Karlsruhe, Germany in PLACE 2005. During this degree he also spent a semester at PHOTO the National University of Singapore (NUS), SingaHERE pore, where he took both undergraduate and graduate courses in electrical engineering. He is currently working towards his Ph.D. degree in electrical engineering at the University of Connecticut (UCONN), Storrs. In the summer of 2006, he was as a visiting scientist at the Sensor Networks and Data Fusion Department of the FGAN Research Institute, Wachtberg, Germany. His research interests lie in the areas of communications and signal processing, including distributed estimation in wireless sensor networks, wireless positioning and synchronization, underwater acoustic communications and networking. Mr. Berger has served as a reviewer for the IEEE Transactions on Signal Processing, Wireless Communications, Vehicular Technology, and Aerospacespace and Electronic Systems. In 2008 he was member of the technical program committee and session chair for the 11th International Conference on Information Fusion in Cologne, Germany.
Shengli Zhou (M’03) received the B.S. degree in 1995 and the M.Sc. degree in 1998, from the University of Science and Technology of China (USTC), Hefei, both in electrical engineering and PLACE information science. He received his Ph.D. degree PHOTO in electrical engineering from the University of HERE Minnesota (UMN), Minneapolis, in 2002. He has been an assistant professor with the Department of Electrical and Computer Engineering at the University of Connecticut (UCONN), Storrs, since 2003. His general research interests lie in the areas of wireless communications and signal processing. His recent focus is on underwater acoustic communications and networking. Dr. Zhou has served as an Associate Editor for IEEE Transactions on Wireless Communications from Feb. 2005 to Jan. 2007. He received the ONR Young Investigator award in 2007.
Peter Willett (F’03) received his BASc (Engineering Science) from the University of Toronto in 1982, and his PhD degree from Princeton University in 1986. He has been a faculty member at the UniverPLACE sity of Connecticut ever since, and since 1998 has PHOTO been a Professor. His primary areas of research have HERE been statistical signal processing, detection, machine learning, data fusion and tracking. He has interests in and has published in the areas of change/abnormality detection, optical pattern recognition, communications and industrial/security condition monitoring. He is editor-in-chief for IEEE Transactions on Aerospace and Electronic Systems, and until recently was associate editor for three active journals: IEEE Transactions on Aerospace and Electronic Systems (for Data Fusion and Target Tracking) and IEEE Transactions on Systems, Man, and Cybernetics, parts A and B. He is also associate editor for the IEEE AES Magazine, editor of the AES Magazines periodic Tutorial issues, associate editor for ISIFs electronic Journal of Advances in Information Fusion, and is a member of the editorial board of IEEEs Signal Processing Magazine. He has been a member of the IEEE AESS Board of Governors since 2003. He was General CoChair (with Stefano Coraluppi) for the 2006 ISIF/IEEE Fusion Conference in Florence, Italy, Program Co-Chair (with Eugene Santos) for the 2003 IEEE Conference on Systems, Man, and Cybernetics in Washington DC, and Program Co-Chair (with Pramod Varshney) for the 1999 Fusion Conference in Sunnyvale.
