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]
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