This chapter presents the most important decision models considered in detection or estimation problems.
The matched filter is an optimal linear system for the detection of a known signal s(t) corrupted by an independent additive noise. It maximizes the signal-to-noise ratio at the moment when the decision is made. If the noise is white, the impulse response of the matched filter to the signal s(t) has the following expression:
where k is a constant depending on the filter gain, and t0 is a time-delay parameter which usually corresponds to the signal duration T.
The output signal of the matched filter driven by the signal x(t) = s(t) + n(t) becomes thus:
Consequently, the matched filter has the behavior of a correlator with respect to the signal to be detected.
The signal-to-noise ratio is maximum in t = t0 and depends on the signal energy Ws and the noise power spectral density N0/2. For a given noise and different signals having the same energy Ws, the corresponding matched filters give the same signal-to-noise ratio. However, they may be very different in terms of the time resolution, which means the capability of resolving two closely spaced signals.
In fact, without matched filtering ...