A recursive approach to signal detection

The classical problem of detecting known signals in additive nonstationary nonwhite noise is treated using state-variable methods. The noise is assumed to be generated as the solution of a linear time-varying differential equation driven by white noise. The Markov property of the noise is used to derive a difference equation for the likelihood ratio. The terms in the difference equation are given directly in terms of the coefficients of the differential equation defining the noise. These results, being in recursive form, reduce considerably the computational effort when discrete samples are used. The signal-to-noise ratio is also derived. Explicit expressions for the likelihood ratio are also obtained for continuous processing. These results are given directly in terms of the signals and the noise-generating equation. Thus, there are no integral equations to solve. The signal-to-noise ratio is calculated explicitly for both stationary and nonstationary noise.