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Fast Convolution

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

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Summary: Efficient computation of convolution using FFTs.

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Fast Circular Convolution

Since, ∑m=0N−1xmhn−mmodN=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.
    • NN−1 N N 1 complex adds.

  • Via FFTs

    • 3 FFTs + NN multipies.
    • N+3N2log2N N 3 N 2 2 N complex multiplies.
    • 3Nlog2N 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.

Figure 1
Figure 1 (figure6.png)

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

Figure 2
Figure 2 (Figure7.png)

Choose shortest convenient N N (usually smallest power-of-two greater than or equal to L+M−1 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 L≫M ≫ 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 ,

Figure 3
Figure 3 (figure4.png)
the first M−1 M 1 samples are wrapped around and thus is incorrect. However, for M−1≤n≤N−1 M 1 n N 1 ,the convolution is linear convolution, so these samples are correct. Thus N−M+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 M−1 M 1 samples. Perform circular convolution of each block with filter hm h m . Discard first M−1 M 1 points in each output block, and concatenate the remaining points to create yn y n .

Figure 4
Figure 4 (Figure1.png)

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 2N2log2N+NN−M+1=Nlog2N+1N−M+1 complex multiplies 2 N 2 2 N N N M 1 N 2 N 1 N M 1 complex multiplies 2Nlog2NN−M+1=2Nlog2NN−M+1 complex adds 2 N 2 N N M 1 2 N 2 N N M 1 complex adds

Compare to M M mults, M−1 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 4M≤Nopt≤8M 4 M Nopt 8 M .

Overlap-Add (OLA) Method

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

Figure 5
Figure 5 (figure5.png)

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

Figure 6
Figure 6 (Figure2.png)
Computational Cost: Two length N=L+M−1 N L M 1 FFTs and M M mults and M−1 M 1 adds per L L output points; essentially the sames as OLS method.

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