Pre-Class Music

Keith Kothman,

Interludes

video by John Fillwalk

Convolution

Convolution Background

Fundamental operation in digital audio processing.

Even if you don’t specifically know it, you know its effects (through filtering, modulation, reverberation, cross synthesis). A filter convolves its IR with the input signal to

produce filtered output.

Uses of Convolution

Reverberation obtain the IR of a room, and convolve it

with an arbitrary signal to make it sound as if the arbitrary signal has been played in that room.

Filtering arbitrary signals to model the characteristics of an audio

system, such as a microphone or guitar amp.

The Math of Convolution

The equation (the * denotes convolution)

for every sample in the arbitrary signal a, multiply it by every sample in the IR b, and sum the results (offset by each sample in a)

length(output) = length(a) + length(b) - 1 in samples

€

a[n]∗b[n] = output[k] = a[n]×b[k − n]n=0

N−1

∑

Convolution is not Multiplication

Multiplication of two audio signals is amplitude modulation for each point in time, one sample is

multiplied by another sample.Convolution of two audio signals is a

series of multiplications, and a summation of those results. every sample in one signal is multiplied by

the entire set of samples in the second signal.

The Law of Convolution

Convolution in the time domain is equal to multiplication in the frequency domain, and vice versa. (btw, that’s an important concept—it will

be on the test.) convolution does not distinguish between

samples and spectra. Both are series of discrete values.

Implementation of Convolution

Direct Convolution of amplitude samples is computationally intensive.

Fast Convolution is preferred. an FFT is performed on each audio signal,

and their corresponding spectra are multiplied.

an inverse FFT is applied to the result.

Musical Uses of Convolution

Filtering: frequencies present with high amplitude levels in both signals are reinforced; frequencies only present in one signal are eliminated.

Reverberation: convolution has time domain results, including echo and time smearing, which can be used to recreate natural or artificial reverb, or to otherwise distort and blur the original source material.