Design of filters for discrete short-time Fourier transform synthesis

In this paper we present a method for computing the synthesis filter (window) needed in a weighted overlap-add (WOLA) scheme for the reconstruction of signals in analysis/synthesis systems used to implement the discrete short time Fourier transform (DSTFT). The method is based on an algebraic representation of the analysis synthesis process and assume that the analysis filter (window) is known, that its length N is larger than or equal to the transform size (i.e. the number of frequency bands) M, that no modification of the DSTFT is performed, and that exact signal reconstruction (unity system) is to be achieved. The last condition can be achieved only if the shift R of the sliding analysis window satisfiesl \leq R < M. The solution presented in this paper extends an earlier result obtained for N = M, and is of practical importance, since using N > M results in analysis filter banks with improved frequency band separation. The algebraic method presented is simple and efficient as it reduces the large dimensionality of the problem into a solution of R sets of linear equations of reduced dimensions. The solutions of these equations are the individual synthesis polyphase filters from which the synthesis filter is constructed.