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
![$ c\neq 0$](http://www.dsprelated.com/josimages_new/sasp2/img1611.png)
![$ R=1$](http://www.dsprelated.com/josimages_new/sasp2/img69.png)
The Filter Bank Summation (FBS) is interpreted as a demodulation
(frequency shift by
) and subsequent lowpass filtering by
. Therefore, to
resynthesize our original signal, we need to remodulate each
baseband signal and sum up the channels. For
(no
downsampling), this sum is given by [9]
We have thus derived the following sufficient condition for perfect reconstruction [213]:
Since normally our windows are shorter than![]()
![$ N$](http://www.dsprelated.com/josimages_new/sasp2/img61.png)
![$ R=1$](http://www.dsprelated.com/josimages_new/sasp2/img69.png)
![$ w$](http://www.dsprelated.com/josimages_new/sasp2/img580.png)
![$ R=1$](http://www.dsprelated.com/josimages_new/sasp2/img69.png)
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Nyquist(N) Windows
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The DFT Filter Bank