overlap-and-add

Overlap-and-add convolution in Python

Overview

This module was built aiming at applying reverberation on audio signals. It contains several files related to speed improvement requirements. If you are looking for a quick result, then you should use function convolve in file ola3.py:

import ola3

ola3.convolve(x, y, hop_size)

Underlying theory

When sound is produced within a room, it propagates towards the listener. This is called direct sound. However, sound also propagates towards the wall, then it reflects and propagates towards both the listener and the other walls. This results in a series of reflections, which essentially depend on the room shape, size and material.

A room, in regular conditions, can be considered a linear system, which means that if two sources produce sound within the room then the listener will hear the sum of the individual sounds. This is similar to playing two instruments together and, if you like other forms of art, it is also similar to overlaying semi-transparent photographs. Because of the linear property, the sum of the direct sound with the reflections - that is, the reverberated sound - is equal to the convolution between the original sound signal and the room's impulse response, which is literaly the sound that is heard when an impulse is produced within the room.

The convolution is an operation that is very costly to calculate directly - its complexity is O(N^2), which makes it very slow, even for offline applications. However (and you might want to refer to a good DSP book for this demonstration), the convolution in the time domain corresponds to a multiplication (which is O(N)) in the frequency domain. Hence, using the FFT (O(N logN)) and its inverse, we can transform our time-domain signals into the frequency domain, multiply them and then transform them back to the time domain, which is much faster than the direct, time-domain operation.

One disadvantage of the frequency-domain convolution is that it requires the whole audio signal to be known before yielding the results. This prevents the convolution to be performed in real-time. The solution for this is to divide the input signal into blocks of known length and then calculating the convolution in each of the blocks. The convolution is a linear operation, hence the sum of the blockwise convolutions is equal to the convolution of the sum of all blocks.

Doing this with a very long impulse response signal requires a large FFT to be computed, which is problematic because it requires a lot of memory. Hence, we can also divide the impulse response into blocks and use these blocks to compute the convolution with each block of the input signal.

This package contains some code to perform blockwise convolutions and some code to measure performance of each variation, which are shown below.

Performance measurements

• Time-domain convolution
• Frequency-domain convolution (no blocks)
• Frequency-domain convolution (only input uses blocks)
• Frequency-domain convolution (both input and impulse response use blocks)