## 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 .

**Next Section:**

Nyquist(N) Windows

**Previous Section:**

The DFT Filter Bank