Near-Optimal Signal Detection for Finite-State Markov Signals with Application to Magnetic Resonance Force Microscopy

Abstract

Detection of a finite-state Markov signal in additive white Gaussian noise (AWGN) can be done in an intuitive manner by applying an appropriate filter and using an energy detector. One might not expect this heuristic approach to signal detection to be optimal. However, in this paper, we show that for a certain type of finite-state Markov signal, namely, the discrete-time (DT) random telegraph, this filtered energy detector is approximately optimal under the following conditions of: symmetric transition probabilities, low signal-to-noise ratio (SNR), long observation time, and small probability of transition between two consecutive time instances. When these last three conditions hold, but the transition probabilities are not symmetric, we show that a variant of the filtered energy detector is approximately optimal. It is also shown, under low SNR conditions, that the likelihood ratio test (LRT) for a finite-state DT Markov signal in AWGN reduces to the matched filter statistic with the minimum mean-squared-error (MMSE) predictor signal values used in place of the known signal values. Using this result, we propose a general methodology for obtaining an approximation to the LRT of a finite-state DT Markov signal in AWGN. Specifically, instead of the conditional mean (also MMSE) estimators, affine estimators with lowest mean squared error are used. This work is relevant to magnetic resonance force microscopy, an emerging technology that uses ultrasensitive force sensing to detect magnetic resonance. Sensitivity down to the single spin level was demonstrated in a recent experiment.