Why do the spectra of distorted waves have humps?

[danbullard]

TL;DR Summary

distorted waves spectra humps

When I distort a sine wave by clipping or other mechanisms, I get humps in the spectrum after doing an FFT. They don't appear to be random, so where did they come from? Here is the time domain picture of a sine wave clipped on the top of a 16 bit ADC.

And here is the spectrum, note the humps.

 

[jrmichler]

Start by searching spectrum of square wave. The spectrum of a clipped sine wave is somewhere between that of a sine wave and that of a square wave. It depends on how much the sine wave is clipped.

You can get more information by searching Fourier series. The Wikipedia hit has some good diagrams. If you are really curious after the above, search Fourier transform. If the above leaves you confused, that's because there is a lot that is not covered in these simple internet searches. If you are curious enough to want to get a solid base, you would need a class in signal processing. Such a class would normally be a graduate level class for a person majoring in physics or engineering, although it might be available to a senior level undergrad.

 

[tech99]

When you clipped only the positive peaks you created a zero frequency, or DC, component, which can be seen. I suspect the dips in your spectrum come from the finite duration of the flat top you have created on each sine wave.

 

[sophiecentaur]

TL;DR Summary: distorted waves spectra humps

And here is the spectrum, note the humps.

That spectral graph can be misleading if you don't realise that the vertical scale is in dB. All those components are a few percent or less of the main (original) component. The actual energy that's not at the fundamental frequency is quite low. On a linear scale, they could easily be ignored.

I think the humps are due to the asymmetrical limiting of the waveform and the sampling rate relative to the fundamental. I would expect those humps to move about as the fundamental frequency is changed slightly. Sampling can be looked upon as a form of modulation which produces extra products that you didn't expect. The term "quantisation noise" is not totally accurate because it's actually distortion and not random noise. Every element of the so-called noise is harmonically related to the sampled waveform (sidebands, if you like).

 

[Baluncore]

If the clipping was symmetrical, the odd harmonics would be present.
Because the top is clipped, and the bottom is not, that asymmetry raises the even harmonics, and the DC component.

The humps may be due to the duty-cycle of the minor clipping that is present. Change the clipping and the sample rate to see if the humps are part of the signal, or an artefact of the processing.

What anti-aliasing filter did you use?
What window function did you use?
Does the A-D sample contain an integer number of cycles?