Signal synthesis from modified discrete short-time transform

The discrete short-time transform (DSTT) is a generalization of the discrete short-time Fourier transform (DSTFT). The necessary and sufficient conditions on the analysis filter, under which perfect reconstruction of the input signal is possible (when the DSTT is not modified), are presented. The class of linear modifications for which the original input can be reconstructed when the modification is applied is characterized. The synthesis of an optimal (in the minimum-mean-square-error sense) signal from a modified DSTT (MDSTT) of finite duration is presented. It is shown that for an analysis filter length that does not exceed a given value, the optimal synthesis scheme is independent of the duration of the given MDSTT and is an extension of the weighted overlap add (WOLA) synthesis method. For longer analysis filters, the optimal synthesis scheme becomes quite cumbersome, and therefore, a steady-state solution (as the duration of the MDSTT approaches infinity) is presented for this case. It is shown that this solution can be approximated with arbitrarily small reconstruction error