Overlap-Add Signal Decomposition

Consider breaking the input signal $x$ , into frames using a finite, zero-centered, length $M$ (odd) window. Let $x_m$ denote the $m^{th}$ frame.

$$x_m(n) \mathrel{\stackrel{\Delta}{=}}x(n)w(n-mR) \hspace{1cm} n \in (-\infty, +\infty)$$

or

$$x_m \mathrel{\stackrel{\Delta}{=}}x \cdot \hbox{\sc Shift}_{mR}(w)$$

where

$$R \mathrel{\stackrel{\Delta}{=}}\hbox{frame step (hop size)} \hspace{2cm} m \mathrel{\stackrel{\Delta}{=}}\hbox{frame index}$$

The hop size is the number of samples between adjacent frames. Specifically, it is the number of samples by which we advance each succesive window.

For fast convolution only, choose

• $w=w_R=$ rectangular window
• $R=M$ (hop size = window length)