Overlap–add method: Difference between revisions

Revision as of 01:24, 27 September 2019

Revision as of 13:40, 11 October 2019

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:<math>

y[n] = x[n] * h[n]

\ \stackrel{\text{def}}{=}triangleq\ \sum_{m=-\infty}^{\infty} h[m] \cdot x[n - m]

= \sum_{m=1}^{M} h[m] \cdot x[n - m],

</math>

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The concept is to divide the problem into multiple convolutions of ''h''[''n''] with short segments of <math>x[n]</math>:

:<math>x_k[n]\ \stackrel{\text{def}}{=}triangleq\ \begin{cases}

x[n + kL] & n = 1, 2, \ldots, L\\

0 & \text{otherwise},

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\end{align}</math>

where <math>y_k[n]\ \stackrel{\mathrm{def}}{=}triangleq\ x_k[n] * h[n]\,</math> is zero outside the region {{nowrap|[1, ''L'' + ''M'' − 1]}}. And for any parameter <math>N \ge L + M - 1,\,</math> it is equivalent to the <math>N\,</math>-point [[circular convolution]] of <math>x_k[n]\,</math> with <math>h[n]\,</math> in the {{nowrap|region [1, ''N'']}}.

The advantage is that the [[circular convolution]] can be computed very efficiently as follows, according to the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]]:

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== See also ==

*[[Overlap–save method]]

== References ==

*{{Cite book

| author1=Rabiner, Lawrence R.