Constant Overlap Add (COLA)

Mathematical definition of the Short-time Fourier Transform (STFT) is given by

$$X_m(\omega) = \sum_{n=-\infty}^{\infty}x(n)w(n-mR)e^{-j\omega n},$$

where $R$ is the hopsize, and $m$ is the length of the window.

The window used in the STFT, $w(n)$, must satisfy the Constant Overlap-Add (COLA) property:

$$\sum_{m=-\infty}^{\infty}w(n-mR) = 1.$$

If COLA is satisfied, then the sum of successive DTFTs over time equals the DTFT of the whole signal $X(\omega)$, that is:

$$\sum_{m=-\infty}^{\infty}X_m(\omega) \triangleq \sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}x(n)w(n-mR)e^{-j\omega n}$$

$$= \sum_{n=-\infty}^{\infty}x(n)e^{-j\omega n}\sum_{m=-\infty}^{\infty}w(n-mR)$$

$$= \sum_{n=-\infty}^{\infty}x(n)e^{-j\omega n} = X(\omega),$$

Rectangle window is COLA if there is no overlap. Bartlett window, and all the Hamming family are COLA with 50 % overlap (when end points are handled correctly).