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:

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

where $c\neq 0$ is any constant (always true for $R=1$ ).^10.4

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]

$$\hat{x}(m) = \sum_{k=0}^{N-1} X_m(\omega_k)e^{j\omega_km}$$

$$= \sum_{k=0}^{N-1} \left[ \sum_{n=-\infty}^\infty x(n)w(n-m)e^{-j\omega_kn} \right] e^{j\omega_km}$$

$$= \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} e^{j\omega_k(m-n)}$$

$$= \sum_{n=-\infty}^\infty x(n)w(\underbrace{n-m}_{-rN}) N\sum_{r=-\infty}^\infty \delta(m-n-rN)$$

$$= N\sum_{r=-\infty}^\infty {\tilde w}(rN)x(m-rN)$$

$$= Nw(0) x(m) \qquad \hbox{if $w(rN)=0,\,\forall r\neq 0$} \protect$$ (10.19)

We have thus derived the following sufficient condition for perfect reconstruction [213]:

$\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 $R=1$ .