Computing detection filter transfer functions


Another problem of common interest is the detection of the presence or absence of a signal in noise. The ideal filter detects a desired signal d(t) by producing an output signal pulse so(0) in the total output c(t) such that the peak output signal-to-rms output noise SNRo is maximized, has

H(jw) = E{D*(jw)}/E{N(jw)N*(jw)}
= Conjugate spectrum of desired signal to be detected/Noise autocorrelation spectrum / Input autocorrelation spectrum
This filter produces a narrow output pulse (which reduces temporal or time ambiquity) centered at time t=0 with undefined shape.


An inverse detection filter outputs an impulse Uo(t) when only signal s(t) and no noise is applied. It outputs a narrow pulse when a signal similar to s(t) is applied. Using Eq. 4.4.9a, the transfer function of an inverse detection filter equals

H(jw) = E{F(Uo(t))}/E{S(jw)} = 1/E{S(jw)}
When signal plus noise is applied to the filter, its input equals R = S+N. The output of the filter then equals C + HR = 1 + N(jw)/E{S(jw)}. The undesired output component due to nonzero input noise is F^(-1)(N(jw)/E{S(jw)}) ~ F^(-1)(1/SNRi). When the spectral signal-to-noise is large, then this undesired output component is negligible. However SNRi is seldom large so this component is present which produces output error. This error can be significantly reduced using the high-resolution filter.


Another conceptually useful filter is the high-resolution detection filter which outputs an impulse when both the signal s(t) and noise n(t) are applied. It outputs a narrow pulse when a signal similar to s(t)+n(t) is applied. Setting d(t) = Uo(t) or D(jw)=1 in the uncorrelated estimation filter gives

H(jw) = E{[S*(jw)+N*(jw)]}/E{|S(jw)+N(jw)|^2}
When the signal and noise are uncorrelated and the noise has zero mean, then
H(jw) = E{S*(jw)}/E{|S(jw)|^2 + |N(jw)|^2}
The high-resolution detection filter of Eq. 4.4.42 functions as a matched detection filter (Eq. 4.4.37) at frequencies where the spectral SNRi is low. It acts as an inverse or deconvolution filter (Eq. 4.4.39) at high spectral SNRi. Effectively this filter is the combination of a Wiener estimation filter followed by an inverse filter. The estimation filter best estimates the signal from the noisy input and the inverse filter converts it into an impulse.

© C.S. Lindquist, Adaptive and Digital Signal Processing with Digital Filtering Applications, vol. 2, pp. 287-288, 292-293, Steward & Sons, 1989.