STFT of COLA Decomposition

To represent practical FFT implementations, it is preferable to shift the $m^{th}$ frame back to the time origin:

$${\tilde x}_m(n) \isdef x_m(n+mR) \eqsp \hbox{\sc Shift}_{-mR,n}(x_m)$$ (9.20)

This is summarized in Fig.8.11. Zero-based frames are needed because the leftmost input sample is assigned to time zero by FFT algorithms. In other words, a hopping FFT effectively redefines time zero on each hop. Thus, a practical STFT is a sequence of FFTs of the zero-based frames ${\tilde x}_0, {\tilde x}_1, \ldots$ . On the other hand, papers in the literature (such as [7,9]) work with the fixed time-origin case ( $x_0, x_1, \ldots$ ). Since they differ only by a time shift, it is not hard to translate back and forth.

Note that we may sample the DTFT of both $x_m$ and ${\tilde x}_m$ , because both are time-limited to $M$ nonzero samples. The minimum information-preserving sampling interval along the unit circle in both cases is $\Omega_M \isdeftext 2\pi/M$ . In practice, we often oversample to some extent, using $\Omega_N$ with $N>M$ instead. For ${\tilde x}_m$ , we get

$${\tilde X}_m(\omega_k) \isdef \hbox{\sc Sample}_{\Omega_N,k}\left(\hbox{\sc DTFT}\left({\tilde x}_m\right)\right)$$

$$= \hbox{\sc DFT}_{N,k}({\tilde x}_m), \protect$$ (9.21)

where $\omega_k \isdef 2\pi k/N = k\Omega_N$ . For $x_m$ we have

$$\begin{eqnarray*} X_m(\omega_k) &\isdef & \hbox{\sc Sample}_{\Omega_N,k}\left(\hbox{\sc DTFT}(x_m)\right)\\ &\longleftrightarrow& \hbox{\sc Alias}_N(x_m)\\ &\neq& {\tilde x}_m \; \hbox{(in general).} \end{eqnarray*}$$

Since ${\tilde x}_m = \hbox{\sc Shift}_{-mR}(x_m)$ , their transforms are related by the shift theorem:

$$\begin{eqnarray*} {\tilde X}_m(\omega_k) &=& e^{jmR\omega_k} X_m(\omega_k) \\ \longleftrightarrow\quad {\tilde x}_m(n) &=& \hbox{\sc Alias}_{N,n+mR}(x_m)\\ &=& x_m(n+mR)_N \end{eqnarray*}$$

where $(n+mR)_N$ denotes modulo $N$ indexing (appropriate since the DTFTs have been sampled at intervals of $\Omega_N = 2\pi/N$ ).