The predictability of certain optimum finite-impulse-response digital filters

Some of the properties of optimal solutions to the finite-impulse-response low-pass filter design problem are discussed. These solutions are optimum in the sense of discrete Chebyshev approximation over a union of closed compact sets, i.e., the error of approximation exhibits at least
(N + 3)/2
alternations (of equal amplitude) over the frequency ranges of interest, where
N
is the duration of the filter impulse response in samples. It has been shown that, in certain special cases, the solution can exhibit
(N + 5)/2
alternations of equal amplitude. These solutions have been called extraripple filters because of the extra alternation that is present. How these extraripple solutions can, within bounds, be scaled to yield additional solutions, which are still optimal over new frequency ranges, is shown.' Thus an infinite number of optimal low-pass filters may be obtained directly from a finite number of extraripple solutions. An interpretation of the various types of optimal filters, in terms of locations of the zeros of the
z
- transform polynomial, is also given.