Understanding convolution using a moving average filter

Convolution can be quite a difficult concept to get to grips with. This article uses a simple application, a moving average filter, to help visualise the process.

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An example signal

Here is a function that represents our “signal”. Convolution, of course, is a mathematical process that operates on mathematical functions, but in many practical applications, those functions might represent physical signals. For example, this function might represent a sound:

We might see this signal on an oscilloscope measuring a voltage in an electronic amplifier that will eventually be converted to sound waves by a speaker. Although in this case, the signal shown has an oscillation period of several seconds, so humans would not detect it as a sound.

Since this is a time-varying signal, the horizontal axis represents time, t, rather than x. We will call the function above f(t).

A simple filter

Let’s suppose we wanted to modify this signal by smoothing out some of the smaller bumps in the function. This would have the audible effect of increasing the bass and decreasing the treble, sometimes called a lowpass filter.