# OpenStax-CNX

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# Fast Convolution

Module by: Douglas L. Jones. E-mail the author

Summary: Efficient computation of convolution using FFTs.

## Fast Circular Convolution

Since, m=0N1xmhnmmodN=yn is equivalent to Yk=XkHk m 0 N 1 x m h n m N y n is equivalent to Y k X k H k yn y n can be computed as yn=IDFTDFTxnDFThn y n IDFT DFT x n DFT h n

### Cost

• #### Direct

• N2 N 2 complex multiplies.
• N(N1) N N 1 complex adds.
• #### Via FFTs

• 3 FFTs + NN multipies.
• N+3N2log2N N 3 N 2 2 N complex multiplies.
• 3(Nlog2N) 3 N 2 N complex adds.
If Hk H k can be precomputed, cost is only 2 FFts + NN multiplies.

## Fast Linear Convolution

DFT produces cicular convolution. For linear convolution, we must zero-pad sequences so that circular wrap-around always wraps over zeros.

To achieve linear convolution using fast circular convolution, we must use zero-padded DFTs of length NL+M1 N L M 1

Choose shortest convenient N N (usually smallest power-of-two greater than or equal to L+M1 L M 1 ) yn= IDFT N DFT N xn DFT N hn y n IDFT N DFT N x n DFT N h n

### note:

There is some inefficiency when compared to circular convolution due to longer zero-padded DFTs. Still, ONlog2N O N 2 N savings over direct computation.

## Running Convolution

Suppose L= L , as in a real time filter application, or LM L M . There are efficient block methods for computing fast convolution.

### Overlap-Save (OLS) Method

Note that if a length-MM filter hn h n is circularly convulved with a length-NN segment of a signal xn x n ,

the first M1 M 1 samples are wrapped around and thus is incorrect. However, for M1nN1 M 1 n N 1 ,the convolution is linear convolution, so these samples are correct. Thus NM+1 N M 1 good outputs are produced for each length-NN circular convolution.

The Overlap-Save Method: Break long signal into successive blocks of N N samples, each block overlapping the previous block by M1 M 1 samples. Perform circular convolution of each block with filter hm h m . Discard first M1 M 1 points in each output block, and concatenate the remaining points to create yn y n .

Computation cost for a length-NN equals 2n 2 n FFT per output sample is (assuming precomputed Hk H k ) 2 FFTs and NN multiplies 2(N2log2N)+NNM+1=N(log2N+1)NM+1 complex multiplies 2 N 2 2 N N N M 1 N 2 N 1 N M 1 complex multiplies 2(Nlog2N)NM+1=2Nlog2NNM+1 complex adds 2 N 2 N N M 1 2 N 2 N N M 1 complex adds

Compare to M M mults, M1 M 1 adds per output point for direct method. For a given M M, optimal N N can be determined by finding N N minimizing operation counts. Usualy, optimal N N is 4MNopt8M 4 M Nopt 8 M .

Zero-pad length-LL blocks by M1 M 1 samples.

Add successive blocks, overlapped by M1 M 1 samples, so that the tails sum to produce the complete linear convolution.

Computational Cost: Two length N=L+M1 N L M 1 FFTs and M M mults and M1 M 1 adds per L L output points; essentially the sames as OLS method.

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