Construction of time-limited functions with specified autocorrelation functions

A necessary and sufficient condition is derived for a function to be the autocorrelation function of a time-limited function of specified length. A constructive procedure for determining all possible time-limited functions having the same autocorrelation function as a given time4imited function is presented. The procedure consists of passing the given time function through one out of a collection of specially tailored all-pass filters. The class of functions so generated may be either finite or infinite depending on the original time function. Specific examples of the construction procedure are given.