Choosing nonuniform tap spacings for a tapped-delay-line filter

The family of frequency responses that can be generated by adjusting the weights on a tapped-delay-line filter with nonuniformly spaced taps is just the span of a basis set whose elements are arbitrary (as long as linear independence is preserved) frequency shifts of the characterizing function of the set of delays. The characterizing function is just the frequency response that results when uniform weights are applied at those delays. If the delays are chosen to provide a unimodal characterizing function with low sidelobes (probability theory provides one approach), then it tends to be easy to approximate a desired frequency response from linear combinations of various frequency shifts of the characterizing function. If the characterizing function is severely multimodal, it is relatively difficult to get the linear combination of the characterizing-function shifts to approximate a desired response. Though this intuition about what is possible comes from visualizing the manual construction of the desired response, the conclusions reached generally also predict the degree of success that will be obtained from an optimal design of the filter weights. Further and most importantly, the characterizing-function idea offers guidance in the choice of the delays themselves.