Validity criteria for filters matched to delayed, Doppler-shifted signals

Abstract

The response of a filter matched to a delayed, Doppler-shifted version of a narrow-band transmitted signal in white noise is considered. Conditions under which the response is the one expected are arrived at by (1) setting up the response to a more accurate representation of the echo waveform, (2) applying Taylor's theorem, with the response considered a function of v/c (range rate over the speed of propagation), and (3) requiring that the remainder be small relative to the leading term (expected response). The resulting validity criteria differ from that put forward by several authors, namely, 2(|v|/c)B T ≪ 1 (one also sees <), where B is a signal bandwidth, and T is the signal duration. It is shown that, to good approximation, the old criterion implies the new, and that modulations exist such that one of the new criteria is satisfied but the old is not.