Overlap-Add (OLA) STFT Processing

This chapter discusses use of the Short-Time Fourier Transform
(STFT) to implement *linear filtering* in the frequency domain.
Due to the speed of FFT convolution, the STFT provides the most
efficient single-CPU implementation engine for most FIR filters
encountered in audio signal processing.

Recall from §7.1 the STFT:

where

We noted that if the window
has the
*constant overlap-add property*
at hop-size
,

(9.2) |

then the sum of the successive DTFTs over time equals the DTFT of the whole signal :

(9.3) |

Consequently, the inverse-STFT is simply the inverse-DTFT of this sum:

We may now introduce *spectral modifications* by multiplying each
spectral frame
by some filter frequency response
to get

(9.4) |

Note that can be different for each frame, giving a certain class of

(9.5) |

so that

(9.6) |

where

(9.7) |

This chapter discusses practical implementation of the above relations using a Fast Fourier Transform (FFT). In particular, we use an FFT to compute efficiently what may be regarded as a

(9.8) |

The sum over may be interpreted as adding separately filtered frames . Due to this filtering, the frames must overlap, even when the rectangular window is used. As a result, the overall system is often called an

- Convolution of Short Signals
- Cyclic FFT Convolution
- Acyclic FFT Convolution
- Acyclic FFT Convolution in Matlab
- FFT versus Direct Convolution

- Convolving with Long Signals
- Overlap-Add Decomposition
- COLA Examples
- STFT of COLA Decomposition
- Acyclic Convolution
- Example of Overlap-Add Convolution
- Summary of Overlap-Add FFT Processing

- Dual of Constant Overlap-Add
- Poisson Summation Formula
- Frequency-Domain COLA Constraints
- PSF Dual and Graphical Equalizers
- PSF and Weighted Overlap Add
- Example COLA Windows for WOLA

- Overlap-Save Method
- Time Varying OLA Modifications

- Weighted Overlap Add

- Review of Zero Padding

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University