Block Diagram Interpretation

Assuming $\hat{h}^w$ is causal gives

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

This is depicted in the following diagram:

• $h_m(l)=$ FIR filter tap $l$ at time $m$
• Each tap is lowpass filtered by the FFT window $w$
• The window thus enforces bandlimiting of the filter taps (see the text for further details)