Detection Of a Random Walk Signal in the Regime of Low Signal to Noise Ratio and Long Observation Time

This paper considers the detection of a Markov signal in additive white Gaussian noise (AWGN). Here, the Markov signal is taken to be a certain class of random walk processes. A closed form expression of the likelihood ratio (LR) is derived for a general Markov signal in AWGN. Then, under the conditions of low signal to noise ratio (SNR) and long observation time, necessary conditions are derived for the LR of the random walk to be approximated by a bank of filtered energy (FE) detectors, as well as by a single FE detector. The FE detector is an intuitive way to perform detection; however, it is not necessarily optimal. The results are applicable to the detection of an electron spin in a magnetic resonance force microscopy (MRFM) experiment