FBS Window Constraints for R=1

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

$\displaystyle \sum_{m=-\infty}^\infty w(n-mR) = c$ (10.18)

where $ c\neq 0$ is any constant (always true for $ R=1$ ).10.3

The Filter Bank Summation (FBS) is interpreted as a demodulation (frequency shift by $ -\omega_k$ ) and subsequent lowpass filtering by $ w$ . Therefore, to resynthesize our original signal, we need to remodulate each baseband signal and sum up the channels. For $ R=1$ (no downsampling), this sum is given by [9]

$\displaystyle \hat{x}(m)$ $\displaystyle =$ $\displaystyle \sum_{k=0}^{N-1} X_m(\omega_k)e^{j\omega_km}$  
  $\displaystyle =$ $\displaystyle \sum_{k=0}^{N-1} \left[ \sum_{n=-\infty}^\infty
x(n)w(n-m)e^{-j\omega_kn} \right] e^{j\omega_km}$  
  $\displaystyle =$ $\displaystyle \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} e^{j\omega_k(m-n)}$  
  $\displaystyle =$ $\displaystyle \sum_{n=-\infty}^\infty x(n)w(\underbrace{n-m}_{-rN})
N\sum_{r=-\infty}^\infty \delta(m-n-rN)$  
  $\displaystyle =$ $\displaystyle N\sum_{r=-\infty}^\infty {\tilde w}(rN)x(m-rN)$  
  $\displaystyle =$ $\displaystyle Nw(0) x(m) \qquad \hbox{if $w(rN)=0,\,\forall r\neq 0$}
\protect$ (10.19)

We have thus derived the following sufficient condition for perfect reconstruction [213]:
$\textstyle \parbox{0.8\textwidth}{Perfect
reconstruction is assured in the sliding STFT remodulated filter-bank
sum provided the analysis window is zero at all nonzero multiples of
$N$.}$
Since normally our windows are shorter than $ N$ , this condition holds trivially. In the overlap-add context, we also had guaranteed perfect reconstruction in this case ($ R=1$ ), because every window $ w$ overlap-adds to a constant at displacement $ R=1$ .


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Nyquist(N) Windows
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The DFT Filter Bank