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Overlap-add Method; Overlap-add; Overlap add; Overlap-add method

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The following is an episode list for the Australian television show H2O: Just Add Water, which first aired on Network Ten in Australia and has since been broadcast in more than 120 countries worldwide. Series one premiered in Australia on 7 July 2006 and series two began there on 28 September 2007.
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Wikipedia

Overlap–add method

In signal processing, the overlap–add method is an efficient way to evaluate the discrete convolution of a very long signal x [ n ] {\displaystyle x[n]} with a finite impulse response (FIR) filter h [ n ] {\displaystyle h[n]} :

where h[m] = 0 for m outside the region [1, M]. This article uses common abstract notations, such as y ( t ) = x ( t ) h ( t ) , {\textstyle y(t)=x(t)*h(t),} or y ( t ) = H { x ( t ) } , {\textstyle y(t)={\mathcal {H}}\{x(t)\},} in which it is understood that the functions should be thought of in their totality, rather than at specific instants t {\textstyle t} (see Convolution#Notation).

The concept is to divide the problem into multiple convolutions of h[n] with short segments of x [ n ] {\displaystyle x[n]} :

x k [ n ]     { x [ n + k L ] , n = 1 , 2 , , L 0 , otherwise , {\displaystyle x_{k}[n]\ \triangleq \ {\begin{cases}x[n+kL],&n=1,2,\ldots ,L\\0,&{\text{otherwise}},\end{cases}}}

where L is an arbitrary segment length. Then:

x [ n ] = k x k [ n k L ] , {\displaystyle x[n]=\sum _{k}x_{k}[n-kL],\,}

and y[n] can be written as a sum of short convolutions:

y [ n ] = ( k x k [ n k L ] ) h [ n ] = k ( x k [ n k L ] h [ n ] ) = k y k [ n k L ] , {\displaystyle {\begin{aligned}y[n]=\left(\sum _{k}x_{k}[n-kL]\right)*h[n]&=\sum _{k}\left(x_{k}[n-kL]*h[n]\right)\\&=\sum _{k}y_{k}[n-kL],\end{aligned}}}

where the linear convolution y k [ n ]     x k [ n ] h [ n ] {\displaystyle y_{k}[n]\ \triangleq \ x_{k}[n]*h[n]\,} is zero outside the region [1, L + M − 1]. And for any parameter N L + M 1 , {\displaystyle N\geq L+M-1,\,} it is equivalent to the N-point circular convolution of x k [ n ] {\displaystyle x_{k}[n]\,} with h [ n ] {\displaystyle h[n]\,} in the region [1, N].  The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem:

where:

  • DFTN and IDFTN refer to the Discrete Fourier transform and its inverse, evaluated over N discrete points, and
  • L is customarily chosen such that N = L+M-1 is an integer power-of-2, and the transforms are implemented with the FFT algorithm, for efficiency.