Clipped sine waves

One source of distortion in electronic music is clipping. The highest and lowest portions of a wave form are truncated due to limitations of equipment. As the gain is increased, the sound doesn't simply get louder but also becomes more distorted as more of the signal is clipped off.

Clipping 0.2

For example, here is what a sine wave looks like when clipped 20%, i.e. cut off to be between -0.8 and 0.8.

A simple sine wave has only one Fourier component, itself. But when we clip the sine wave, we move energy into higher frequency components. We can see that in the Fourier components below.

You can show by symmetry that the even-numbered coefficients are exactly zero.

Clipping 0.6

Here are the corresponding plots for 60% clipping, i.e. the absolute value of the signal is cut off to be 0.4. First the signal

and then its Fourier components.

Here are the first five sine waves with the amplitudes given by the Fourier coefficients.

And here we see how the of the sines above do a pretty good job of reconstructing the original clipped sine. We'd need an infinite number of Fourier components to exactly reconstruct the original signal, but the first five components do most of the work.

Continuous range of clipping

Next let's look at the ratio of the energy in the 3rd component to that of the 1st component as we continuously vary the amount of clipping.

Now for the 5th harmonic. This one is interesting because it's not strictly increasing but rather has a little bump before it starts increasing.

Finally, here's the ratio of the energy in all higher frequencies to the energy in the fundamental.

More Fourier series posts

• Sine series for a sine
• Relating Fourier series and Fourier transforms
• Fourier-Bessel series and Gibbs phenomenon