An algebraic approach to discrete short-time Fourier transform analysis and synthesis
Z. Shpiro, D. Malah · 2005
An algebraic representation of the discrete short-time Fourier transform (DSTFT) is presented for the case in which the analysis window lengthNequals the transform block sizeM. This representation allows the application of algebraic tools for determining an optimal synthesis system which minimizes the mean square error between a given modified DSTFT (which is not necessarily a valid DSTFT sequence) and the DSTFT of the synthesized signal. If no modification is applied, the result is a unity analysis-synthesis system for any given time updateRof the sliding analysis window (provided thatR\leqM). It is shown that the optimal synthesis system can be implemented by the well known weighted overlap-add (WOLA) method using an optimal synthesis window. The algebraic approach enables the extension of some recent results and the relaxation of a constraint on the analysis window. The proposed approach is found also to have a potential for solving the synthesis problem for the more general case ofN>M.