"Roy Lewallen" wrote
It's commonly known that a square wave consists of a sine wave of the
square wave's fundamental frequency, plus all its odd harmonics.
Specifically, all components are in phase, and their amplitudes are the
inverse of the harmonic number. That is, if the amplitude of the
fundamenatal sine wave is 1, the amplitude of the third harmonic is 1/3,
the amplitude of the fifth harmonic is 1/5, and so forth.

So we should be able to create a square wave by adding all those sine
waves -- right?

It turns out that if we add the first few sine wave components, we have
a fairly square looking wave -- but it has an overshoot at the leading
and trailing edges. As we add more and more harmonics, the result gets
more square, and the overshoot gets narrower and narrower -- but it
remains, and with the same amplitude. Although the width approaches zero
as the number of sine waves you've added gets infinite, there's always
an overshoot for any finite number of sine waves.

This is one manifestation of the Gibbs' Phenomenon, which also applies
to other situtations. There's a really nifty demo at
http://klebanov.homeip.net/~pavel/fb...applets/Gibbs/.

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The trouble with Fourier when attempting to use him with waveshapes on
transmission lines is that there is no fundamental frequency or cyclic
repetitions. His infinite series are solely functions of frequency.

Whereas volts and current on lines are functions of time (the recprocal of
frequency) and distance. That's where Heaviside's Operational Calculus comes
in. In special cases (if you can find your particular problem in the long
list of transforms and their inverses) his methods reduce to Laplace
Transforms. But in general, as with Fourier, his answers appear as infinite
series. Of course, infinite series pose no problems with present-day
computers.

The very first problems were encountered by Kelvin with the speed and
distortion and economics of telegraph-code signals on long cables. 0 and 1
signals change shape and merge into each other at high data rates. Fourier
could not provide answers. Exactly the same problems still occur on high
data-rate digital circuits and light-fibers, further aggravated by echos and
reflections. But Heaviside's revolutionary mathematics, which so upset the
old-wives of professors of his day and abolished the need for SWR's, did the
trick.
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Reg, G4FGQ