Abstract
An ideal lowpass filter permits no signal transmission outside of a prescribed frequency band centered around the origin. In this paper, it is shown that the ideal lowpass filter which provides the fastest monotonic step-response for a prescribed bandwidth is the prolate filter. The system-function of this circuit is obtained from the autocorrelation of the zero-order, zero degree angular prolate spheroidal wave function. The bandwidth and risetimes of the filter are related through the largest eigenvalue of the same wave function. The computation and realization of the optimum frequency- and time-responses is discussed in detail and is illustrated by a numerical example. It is also shown how the asymptotic forms of the optimum system function degenerate, for very small or very large risetimes, into the well-known Hadamard and Gaussian filter functions, respectively.
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