Example Overlap-Add Decomposition

We desire the sum of overlapping frames $x_m$ to give back the original input signal $x$ :

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

Hence, $x= \sum_{m} x_m$ if and only if

$$\zbox{\sum_m w(n-mR) = 1}$$   (COLA condition)