Compromises in FIR matched filter design

With duration-limited signals there is the opportunity for perfect matched filtering by a suitable FIR filter, providing the noise can be construed as being supplied through an all-pole coloration filter. The popular matrix solution formulation does not make it obvious just what size the optimal filter length must be, and the signal vector zeropadding mechanism needed for coaxing out the optimal coefficient vector is also unclear. Difficulties are compounded when the filter length allowable for implementation falls short of the optimal length. Worse yet, if it happens that the noise is shaped by a coloration filter which has some zeros (i.e., is MA or ARMA instead of just AR), then any FIR filter can only be an approximation to the ideal IIR matched filter. In either case, decreasing filter length requires compromise strategies that are not at all transparent. We base our analysis approach to the FIR problem setup in terms of (time) correlations and convolutions in which the whitening filter has the central role. It is then easy to see that both "pole-only" and "some-zero" noise cases yield optimal SNR values that are exactly calculatable by a time-domain scalar product. The inevitable degradations of SNR with decreasing FIR filter lengths are, in turn, readily quantifiable. We study several compromise strategies arising from the whitening filter convolution approach and find (albeit with a very limited set of test cases) that they are not attractive when contrasted to the common matrix solution. The matrix solution itself, meanwhile, is shown to demand close attention to zeropadding patterns employed in it if best performance is to be obtained as filter length is reduced. Fortunately, exhaustive zero-padding assessment is a practical proposition, and this is our recommended procedure at this early stage of investigation