Overlap-Add Synthesis

A less computationally expensive alternative to sinusoidal summation is called overlap-add reconstruction [1,3] which consists of the following steps:

6. Apply any desired modification to the spectra, such as multiplying by a filter frequency response function, to obtain the modified frame spectrum $\hat{X}_m^\prime$. Additionally, desired spectral components can be added to the FFT buffer [4,21].

7. Inverse FFT $\hat{X}_m^\prime$ to obtain the windowed output frame:

$$\hat{x}_m^\prime (n)=\frac{1}{N}\sum_{k=-N/2}^{N/2-1}\hat{X}_m^\prime (e^{j\omega_k })e^{j\omega_k n}$$

8. Reconstruct the final output by overlapping and adding the windowed output frames:

$$\hat{x}(n)=\sum_m \hat{x}_m^\prime (n-mR)$$

Analysis and resynthesis by overlap-add (in the absence of spectral modifications) is an identity operation if the overlapped and added analysis windows sum to unity, i.e., if

$$A_w(n) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\sum_{m=-\infty}^{\infty} w(n-mR) = 1 \protect$$ (6)

for every $n$. If the overlap-added window function $A_w(n)$ is not constant, it is then an amplitude modulation envelope with period $R$. That is, when the analysis window does not displace and add to a constant, the output is amplitude modulated by a periodic signal having its fundamental frequency at the frame rate $f_s/R$. Frame rate distortion of this kind may be seen as AM sidebands with spacing $f_s/R$ in a spectrogram of the output signal. Not too surprisingly, condition Eq. (6) can be shown (by means of the ``digital Poisson summation formula'' [16]) to be equivalent to the condition that $W(e^{j\omega_k })$ be 0 at all harmonics of the frame rate $f_s/R$.