Error exponents for bayesian detection with randomly spaced sensors

Abstract

We study the detection of Gauss-Markov signals using randomly spaced sensors. We derive a lower bound on the Bayesian detection error based on the Kullback-Leibler divergence, and from this, define an error exponent. We then evaluate the error exponent for stationary and non-stationary Gauss-Markov models where the sensor spacings, d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , . . ., are drawn independently from a common distribution F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> . In both models, error exponents take on simple forms involving the parameters of the Markov process and expectations over F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> of certain functions of d1. These expressions are evaluated explicitly when F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> corresponds to (i) exponentially distributed sensors with placement density λ (ii) equally spaced sensors, and (iii) the proceeding cases when sensors also fail with probability q. Many insights follow. For example, in the non-stationary case, we determine the optimal λ as a function of q. Numerical simulations show that the error exponent, based on an asymptotic analysis of the lower bound, predicts trends of the actual error rate accurately, even for small data sizes.