FBS Window Constraints for R=1

Recall that in overlap-add (Chapter 8), perfect reconstruction required only that the analysis window meet a constant overlap-add (COLA) constraint:

 (10.18)

where is any constant (always true for ).10.3

The Filter Bank Summation (FBS) is interpreted as a demodulation (frequency shift by ) and subsequent lowpass filtering by . Therefore, to resynthesize our original signal, we need to remodulate each baseband signal and sum up the channels. For (no downsampling), this sum is given by [9]

 (10.19)

We have thus derived the following sufficient condition for perfect reconstruction [213]:
Since normally our windows are shorter than , this condition holds trivially. In the overlap-add context, we also had guaranteed perfect reconstruction in this case ( ), because every window overlap-adds to a constant at displacement .

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Nyquist(N) Windows
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The DFT Filter Bank