An algebraic approach to discrete short-time Fourier transform analysis and synthesis

An algebraic representation of the discrete short-time Fourier transform (DSTFT) is presented for the case in which the analysis window length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> equals the transform block size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> . 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 update <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> of the sliding analysis window (provided that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> ). 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 of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N>M</tex> .