ABSTRACT
We design optimal filters to detect a random signal in the presence of random noise. We start by assuming that our filter is time invariant but not necessarily causal. Later we specialize to causal filters. Assume that the input to a linear time invariant (LTI) filter is Y (t) =
S(t)+N(t) where S(t) is the desired random signal and N(t) is the undesired random signal-the noise, and let the output of the filter be R(t). (See Figure 11.1.) An LTI filter is determined by its impulse response. Let us assume that the impulse response of our filter is h(t). For the time being we will not assume anything about causality-h(t) need not be zero for t < 0. (Non-causal systems can be used in practice-if one is willing to introduce an intentional delay into one’s system; by delaying the output, one can have knowledge of a signal’s “future” while processing the signal.) The output of an LTI filter with input Y (t) is:
R(t) = ∫ ∞ −∞
h(τ)Y (t− τ) dτ.
