We study the problem of rate-constrained robust estimation of noisy sensor measurements in an unreliable sensor network. Noisy measurements of physical process X (e.g., temperature) made by individual sensors need to be communicated to a central observer over unreliable channels. Limited power resources of the sensors place a strong limit on the resolution with which the noisy sensor readings can be transmitted. Both the sensors and the communication links can break down. However, the central observer is guaranteed to reliably receive readings from a minimum of k out of n sensors. The goal of the central observer is to estimate the observed physical process from the received transmissions where a specified distortion metric provides an objective measure of estimation quality. In this work, we derive an information-theoretic achievable rate-distortion region for this problem when the sensor measurement noise statistics are symmetric with respect to the sensors.
In the special case of a Gaussian sensor network, i.e., when the source is i.i.d. Gaussian, the sensor noise processes are additive i.i.d. Gaussian, and the distortion metric is mean squared error, we have the following remarkable result. When any k out of n unreliable sensor transmissions are received, the central decoder's estimation quality can be as good as the best reconstruction quality that can be achieved by deploying only k reliable sensors and the central decoding unit is able to receive transmissions from all k sensors. Furthermore, when more than k out of the n sensor transmissions are received, the estimation quality strictly improves.
Figure 1: An unreliable sensor network. A noisy version of source X is observed at each of the n sensors. The noisy observations Y1,...,Yn are encoded and the encoded indices I1,...,In are transmitted. The network delivers some m >= k indices. The decoder obtains the best possible reconstruction of X from the received indices.
Figure 2: Random code construction. n independent random codebooks of block length l are constructed, each containing 2lR' codewords. Each codebook is randomly partitioned into 2lR bins each with approximately 2l(R'-R) codewords.
Figure 3: Structure of sensor encoders that can achieve rate-distortion performance. The encoder of each sensor consists of a rate R' quantizer, for block length l, followed by a rate R random binning function.
Figure 4: Structure of the central observer decoder that can achieve rate-distortion performance. The central observer first decodes the received indices to intermediate representations of length l, and then forms the best estimate of the source. Here, m >= k.
3Professor, University of Michigan, Ann Arbor