Sampled STFT

$$\begin{eqnarray*} X_m(\omega_k) &=& e^{-j\omega_k mR}\sum_{n=-M/2}^{M/2-1} x(n+mR) w(n) e^{-j\omega_k n}\nonumber \\ &=& e^{-j\omega_k mR}\hbox{\sc DFT}_{N,\omega_k}(\hbox{\sc Shift}_{-mR}(x) \cdot w). \end{eqnarray*}$$

• Since indexing in the DFT is modulo $N$ , the sum over $n$ can be ``rotated'' to a sum from 0 to $N-1$ as is conventionally implemented for the DFT

• In practice, this means that the right half of the windowed data frame goes at the beginning of the FFT input buffer, and the left half of the windowed frame goes at the end, with zero-padding in the middle (``zero-phase zero-padding'')