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Question relating to Nyquist frequency

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Re: Question relating to Nyquist frequency

Postby ef37a » Wed Feb 06, 2019 10:22 am

Ah now! A sort of passing comment in my new ampliers book by D.Self (you might see a few of these extracts!) he states that the ringing seen on square waves in amplifier reviews are NOT as often stated, a shortcoming of the PA design but more, as Hugh has pointed out, inevitable when square waves are so used.

Not put that quite correctly but will find the exact wording.

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Re: Question relating to Nyquist frequency

Postby Hugh Robjohns » Wed Feb 06, 2019 10:48 am

ef37a wrote:...the ringing seen on square waves in amplifier reviews are NOT as often stated, a shortcoming of the PA design but more, as Hugh has pointed out, inevitable when square waves are so used.

The Fourier analysis of a square wave states that it is comprised of an infinite series of odd harmonics of its fundamental frequency, each with a slightly lower amplitude than the previous one in a specific ratio:

Image

So... if you pass that perfect square wave through any kind of band-limited system, some of the highest harmonic elements will be removed and others strongly attenuated.

It is the removal of these harmonics that result in the square wave developing the apparent 'ringing' artefacts.

As an alternative viewpoint, consider the shape of the waveform as you build up a square wave from a series of harmonic sinewaves, starting with the fundamental, then the first, third, fifth, seventh harmonics and so on.... you'll see that with only a few harmonics present, the squarewave appears to have a lot of ripples (or ringing) on its flat sections, but as more and more harmonics are added the top and bottom of the squarewave become flatter and flatter (with less and less apparent ringing).

Image

Om other words, you need a system with an infinite bandwidth to represent a perfect square wave. Anything less will reproduce a squarewave with ripples (ringing), and the narrower the bandwidth the more pronounced those ripples will be.

And while true ringing after an impulse edge may indicate an unstable system, the presence of ripples on a square wave is more likely to indicate a band-limited system.

This article explains the maths and contains some instructive diagrams:

https://archive.cnx.org/contents/72f90f3a-f72c-4459-b439-1d27bf9d14d2@1/fourier-series-square-wave

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