The sequential decoding metric for detection in sensor networks B. Narayanaswamy, Yaron Rachlin, Rohit Negi and Pradeep Khosla Department of ECE Carnegie Mellon University Pittsburgh, PA, 15213 Email: {bnarayan,rachlin,negi,pkk}@ece.cmu.edu Abstract— Prior work motivated the use of sequential decoding for the problem of large-scale detection in sensor networks. In this paper we develop the metric for sequential decoding from first principles, different from the Fano metric which is conventionally used in sequential decoding. The difference in the metric arises due to the dependence between codewords, which is inherent in sensing problems. We analyze the behavior of this metric and show that it has the requisite properties for use in sequential decoding, i.e., the metric is, 1) expected to increase if decoding proceeds correctly, and 2) expected to decrease if more than a certain number of decoding errors are made. Through simulations, we show that the metric behaves according to theory and results in much higher accuracies than the Fano metric. We also show that due to an empirically-observed computational cutoff rate, we can perform accurate detection in large scale sensor networks, even when the optimal Viterbi decoding is not computationally feasible. I. I NTRODUCTION The term “large scale detection” characterizes sensor net- work detection problems where the number of hypotheses is exponentially large. Examples of such applications include the use of seismic sensors to detect vehicles [5], the use of sonar to map a large room [6] and the use of thermal sensors for high resolution imaging [10]. In these applications the environment can be modeled as a discrete grid, and each sensor measurement is effected by a large number of grid blocks simultaneously (‘field of view’), sometimes by more than 100 grid blocks [10]. Conventionally, there have been two types of approaches to such problems. The first set of algorithms are computationally expensive such as Viterbi decoding and belief propagation [7]. These algorithms are at least exponential in the size of the field of view of the sensor, making them infeasible for many common sensing problems. The second set of algorithms make approximations (such as independence of sensor measurements [6]) to make the problem computationally feasible, at the cost of accuracy. Thus, there has existed a trade-off between computational complexity and accuracy of detection. Our previous work in [8] has defined the concept of ‘sensing capacity’ as the ratio of target positions to number of sensors, required to detect an environment to within a specified accu- racy. Based on parallels between communication and sensing established by this work, [10] built on an analogy between sensor networks and convolutional codes, to apply a heuristic algorithm similar to the sequential decoding algorithm of convolutional codes, for the problem of detection in sensor networks. The average decoding effort of sequential decoding is independent of the code memory, if the rate is below the computational cutoff rate of the channel. Therefore, by analogy, it is expected that the same independence applies to the problem of sensor network decoding, so long as enough measurements are collected (to keep the rate below the cutoff rate of the sensor network.) Such a computational cutoff rate behavior was indeed empirically observed in [10]. Thus, the paper demonstrated that the trade-off between computational complexity and detection accuracy can be altered by collecting additional sensor measurements. While [10] developed the possibility of using sequential decoding in sensor networks, the metric used there (the Fano metric) originated in the decoding of convolutional codes. The Fano metric is not justifiable for sensor networks, except perhaps in special cases. The main contribution of the present paper is the derivation of a sequential decoding metric for sensor networks from first principles, following arguments analogous to those used by Fano [3] for convolutional codes. The derived metric differs significantly from the Fano metric, due to the dependence between the ‘codewords’ that is inherent in sensor networks [8]. We analyze the behavior of this metric and show that it has the requisite properties for use with sequential decoding. i.e., the metric is expected to, 1) increase if decoding proceeds correctly, and 2) decrease if more than a certain number of decoding errors are made. In simulations, the metric behaves as predicted by theory and results in significantly lower error probability than the Fano metric. We also show that, due to an empirically-observed computational cutoff rate, we can perform accurate detection in large scale sensor networks, even in cases where the optimal Viterbi decoding is not computationally feasible due to the wide field of view of the sensors. II. SENSOR NETWORK MODEL We consider the problem of detection in one-dimensional sensor networks. While a heuristic sequential decoding pro- cedure has been applied to complex 2-D problems [10] using essentially the same model as the one described below, we present the 1-D case for ease of understanding and analysis. Motivated by parallels to communication theory and prior work, we model a contiguous sensor network as shown in Fig. 1. In this model, the environment is modeled as a k- dimensional discrete vector v. Each position in the vector can represent any binary phenomenon such as presence or absence of a target. Possible target vectors are denoted by