Asymptotically optimal quantization for detection in power constrained decentralized sensor networks Akshay Kashyap Tamer Bas ¸ar R. Srikant Abstract— We study a Bayesian decentralized binary hypoth- esis testing problem, in which N sensors make observations related to a two-valued hypothesis, and send messages based on the observations to a fusion center, with the objective of enabling the fusion center to accurately reconstruct the realized hypothesis. We assume that the observations at the sensors are independent and identically distributed conditioned on the hypothesis, and that the sensors transmit their messages over independent, parallel channels to the fusion center. We also make the natural assumption that the sensors have identical, finite message sets at their disposal, that each message has some cost associated with it, and that there is a constraint on the average cost incurred by a sensor. We study the large N asymptote, and therefore are interested in schemes that optimize the error exponent at the fusion center. Our main contributions are 1) proving that the problem of finding the optimal schemes is a finite dimensional optimization problem, 2) a description of the structure of optimal rules: we prove that optimal sensor rules are randomized likelihood ratio quantizers (LRQs), with randomization being over at most two deterministic LRQs. We further show that under some conditions, randomization is not required. I. I NTRODUCTION In a typical decentralized detection problem, sensors ob- serve data, the distribution of which is dependent in some way on the state of the environment in which the sensors are stationed. In this paper, we study such a binary Bayesian detection problem, in which we wish to detect the realization of some state H of the environment, which takes one of two values, H ∈{H 0 ,H 1 }, with known probabilities P [H 0 ] and P [H 1 ] respectively. There are N sensors, numbered 1 through N , and for each i =1,...,N , sensor i observes random variable Y i . The variables Y i are assumed to be independent and identically distributed conditioned on H . We consider the case where the sensor observations are communicated over parallel, independent noisy channels to a fusion center, which then attempts to give an estimate ˆ H of the realized hypothesis. We assume that each sensor can transmit, corresponding to each observation it makes, one of a finite, pre-specified set of symbols on the channel connecting it to the fusion center. Further, we assume that some non-negative power is expended when using any one of the symbols. These assumptions are motivated by wireless sensor networks, where sensors have finite constellation sizes for transmission on wireless channels, and there is some This work was supported in part by the NSF ITR Grant CCR 00-85917. The authors are with the Dept. of ECE and the Coordinated Science Lab, UIUC 1308 W. Main St, Urbana, IL 61801 Emails: kashyap@uiuc.edu basar1@uiuc.edu, rsrikant@uiuc.edu power expense associated with using each constellation point (in the examples, we also assume that there is at least one symbol that has zero cost, which corresponds to “no transmission”). We are interested in a situation in which at each time instant, the state H is realized independently of all previous time instants, measurements of variables Y i are made at each sensor synchronously, messages are transmitted from the sensors to the fusion center, and an estimate ˆ H is generated by the fusion center. Further, this operation is continued over a long period of time, and we wish to minimize the probability of error P [ ˆ H = H ], subject to a constraint on the average power used by any sensor. This is in general a difficult functional optimization prob- lem, involving a search over all functions from the set of observations Y i to the set of messages that a sensor can output. In [1] a similar problem is studied, though with a sum (over all sensors) power constraint and a general set of messages (which might be infinite and uncountable) available to the sensors, and a metric to compare the performance of sensor rules against is provided. However, the problem of which rules would optimize this metric has not been addressed. In detection theory, however, functional optimization prob- lems of the above kind can often be simplified to finite- dimensional optimization problems by proving that the op- timal sensor rule has to be a threshold function (with the number of thresholds being one less than the number of messages available to the sensor) of the likelihood ratio of the observations. In this paper, using results from [2], we prove that the power-constrained optimal sensor rule can be obtained by randomizing between at most two likelihood ratio quantizers. Further, for detection problems in which 1) the sensors choose between one of two symbols to transmit on the channel, and 2) the likelihood ratio has a density which is positive on some interval and zero elsewhere (under either hypothesis), we show that the optimal power constrained sensor rule does not require randomization. We also provide an example of a case when randomization is necessary for optimality. A note on the organization of this paper: we state the problem precisely in Section II, and review some prior work that is relevant to this paper in Section III. We present our main results in Section IV, and then study problems with continuous likelihood ratio in Section V. In Section VI we present an example of a problem in which the optimizing quantizer is randomized. We end with some concluding Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006 WeC20.2 1-4244-0210-7/06/$20.00 ©2006 IEEE 2060