Downsampled STFT Filter Bank

So far we have considered only $R=1$ (the ``sliding'' DFT) in our filter-bank interpretation of the STFT. For $R>1$ we obtain a downsampled version of $X_m(\omega_k)$:

$$\begin{eqnarray*} X_{mR}(\omega_k) &=& \sum_{n=-\infty}^\infty [x(n)e^{-j\omega_... ...Delta}{=}}\hbox{\sc Flip}(w)) \\ &=& (x_k \ast {\tilde w})(mR) \end{eqnarray*}$$

Let us define the downsampled time index as $\tilde{m} \mathrel{\stackrel{\Delta}{=}}mR$ so that

$$X_{\tilde{m}}(\omega_k) = \sum_{n=-\infty}^\infty [x(n)e^{-j\omeg... ...}-n) \mathrel{\stackrel{\Delta}{=}}\left(x_k \ast {\tilde w}\right)(\tilde{m})$$

i.e., $X_{\tilde{m}}$ is simply $X_m$ evaluated at every $R^{th}$ sample, as shown in Fig.9.17.

Note that this can be considered an implementation of a phase vocoder filter bank [195].