Detecting the Occurrence of an Event by FM Through Noise

The theory of frequency modulation has always been beset by analytical difficulties, and nowhere have these been more in evidence than in the area of optimal demodulation of noisy FM signals. Recent advances in nonlinear filtering, however, make it possible to solve certain problems of detection and estimation quite explicitly. We report on such a class of problems here. The basic problem setup is this: an event of interest occurs at a random time T. Its occurrence is signaled by sending a pulse of shape h(-), starting at T; that is, we send where h{-) is some causal, integrable pulse. The signal s(t) is transmitted by FM; the waveform is cos 6 + ut + s(u)du for a carrier frequency to and initial phase 6. In transmission this wave suffers the degradation of having white noise added to it; thus, we observe a signal y, defined by 2227 dy, = cos with bt a Brownian motion independent of t. We would like to construct a nonlinear filter acting causally on yt to estimate optimally at each time t whether r t or not, and if so, by how much. This filter will be obtained by solving the nonlinear filtering problem of determining the conditional probability p0(t) = P{r> tys,0 s t) that is relevant to whether T occurred by time t, and if so, how far back. In particular, the filter (po, pi) yields least-squares estimates of r, by integration over u, according to the formula E{Ty8,0