Inverse STFT

If the window $w(n)$ has the Constant OverLap-Add (COLA) property at hop-size $R$ :

$$\zbox{\sum_{m=-\infty}^{\infty} w(n-mR) = 1, \; \forall n\in\mathbb{Z}} \quad\mbox{($w\in\hbox{\sc Cola}(R)$)}$$

then the individual DTFTs will sum to the DTFT of all $x$ :

$$\begin{eqnarray*} \sum_{m=-\infty}^\infty X_m(\omega) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{m=-\infty}^\infty\sum_{n=-\infty}^{\infty} x(n) w(n-mR) e^{-j\omega n}\\ &=& \sum_{n=-\infty}^{\infty} x(n) e^{-j\omega n} \underbrace{\sum_{m=-\infty}^\infty w(n-mR)}_{1\hbox{ if }w\in\hbox{\sc Cola}(R)} \\ &=& \sum_{n=-\infty}^{\infty} x(n) e^{-j\omega n} \\ &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{\sc DTFT}_\omega(x) = X(\omega)\\ [20pt] \longleftrightarrow \quad x(n) %= \IDTFT_n(X) &=& \frac{1}{2\pi}\int_{-\pi}^{\pi} X(\omega) e^{j\omega n}d\omega \end{eqnarray*}$$