Block Diagram Interpretation of Time-Varying STFT Modifications

Assuming ${\hat h}$ is causal gives

$$\begin{eqnarray*} y(n) &=& \sum_{r=0}^\infty x(n-r) {\hat h}_{n-r}(r) \\ &=& x(n) {\hat h}_n(0) + x(n-1) {\hat h}_{n-1}(1) + x(n-2) {\hat h}_{n-2}(2) + \cdots \end{eqnarray*}$$

This is depicted in Fig.8.17.

The term $h_n(k)$ can be interpreted as the FIR filter tap $k$ at time $n$ . Note how each tap is lowpass filtered by the FFT window $w$ . The window thus enforces bandlimiting each filter tap to the bandwidth of the window's main lobe. For an $L$ -term length-$M$ Blackman-Harris window, for example, the main-lobe reaches zero at frequency $L\Omega_M=2\pi L/M$ (see Table 5.2 in §5.5.2 for other examples). This bandlimiting places a limit on the bandwidth expansion caused by time-variation of the filter coefficients, which in turn places a limit on the maximum STFT hop-size that can be used without frequency-domain aliasing. See Allen and Rabiner 1977 [9] for further details on the bandlimiting property.