Practical STFT Computation

• Our STFT definition is standard, but ``theoretical''

• In practice, the STFT is a sequence of FFTs of windowed data frames, where the window ``slides'' or ``hops'' forward through time.

The STFT can be rewritten, adding $mR$ to $n$ , as

$$\begin{eqnarray*} X_m(\omega) &=& \sum_{n=-\infty}^{\infty} x(n+mR) w(n) e^{-j\omega (n+mR)}\nonumber \\ &=& e^{-j\omega mR} \sum_{n=-\infty}^{\infty} x(n+mR) w(n) e^{-j\omega n} \nonumber \\ &=& e^{-j\omega mR}\hbox{\sc DTFT}_\omega(\hbox{\sc Shift}_{-mR}(x) \cdot w). \end{eqnarray*}$$

• In this form, the data centered about time $mR$ are translated to time 0, multiplied by the zero-phase window $w$ , and finally the DTFT is performed

• Since the nonzero portion of the windowed data is centered on time zero, the DTFT can be sampled to yield the DFT (or FFT)

• Sampling will not cause (time) aliasing if the number of samples around the unit circle is greater than the width (in samples) of the time window