### Acyclic Convolution

Getting back to acyclic convolution, we may write it as(9.22) |

or less along the unit circle. This is the

*dual*of the usual sampling theorem. We conclude that practical FFT acyclic convolution may be carried out using an FFT of any length satisfying

(9.23) |

where is the frame size and is the filter length. Our final expression for is

*overlap*when even if . In general, an LTI filtering by increases the amount of overlap among the frames. This completes our derivation of FFT convolution between an indefinitely long signal and a reasonably short FIR filter (short enough that its zero-padded DFT can be practically computed using one FFT). The fast-convolution processor we have derived is a special case of the

*Overlap-Add*(OLA) method for short-time Fourier analysis, modification, and resynthesis. See [7,9] for more details.

**Next Section:**

Example of Overlap-Add Convolution

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STFT of COLA Decomposition