B.S.T.J. Briefs: Least-Squares Estimator for Frequency-Shift PositionModulation in White Noise

Frequency-shift-position (FSP) modulation sends a sample x e (0, T) by changing the frequency of a wave from oj0 to wi at the time x. We consider the problem of optimal mean-square estimation of x from an FSP signal to which white noise has been added in transmission. The best estimator, given by a known formula of nonlinear filtering, contains stochastic integrals and is hard to implement. An approximation, obtained by omitting 0(to,-1) terms, is readily implemented over the next interval (0, T) by ordinary differential equations driven by the observed signal. We pose the optimal least-squares demodulation problem for the following simple communications system: The signal to be sent consists of successive i.i.d. random variables x having a density 6 with support (0, T); the channel provides an FM wave, of random initial phase 9, which shifts frequency from co0 to toi at the point x during (0, T); the resulting signal is { R cos(6 -I- wot, t t>x; this signal is observed through white noise, the received signal or observation being y, given by dy, = s, dt + dbt, where b, is a Brownian motion independent of x; the process is repeated over each interval of length T, and the problem is to guess or calculate a good estimate of x from the observation (jy,, 0