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From: gregr@tekig1.UUCP (Greg Rogers)
Newsgroups: net.audio
Subject: Re: Re: The stuff square waves are made of ....
Message-ID: <1696@tekig1.UUCP>
Date: Tue, 3-Jul-84 00:32:50 EDT
Article-I.D.: tekig1.1696
Posted: Tue Jul  3 00:32:50 1984
Date-Received: Sat, 30-Jun-84 03:34:25 EDT
References: <494@drutx.UUCP> <2546@allegra.UUCP>, <139@brl-vgr.ARPA> <1692@tekig1.UUCP>, <449@ihu1g.UUCP>
Organization: Tektronix, Beaverton OR
Lines: 42


Bob I don't know who you are now arguing with or why you posted a reply
to my explaination about rectangular waveforms.  As I agreed a square wave
has only odd harmonics of the fundamental.  However the point I made was
that often when dealing with so-called squarewaves produced by real
generators the duty factor is not exactly 1/2.  In that case the even
harmonics are ALSO present and if this is not taken into account then
errors will be made when using these signals to test audio gear.  A 
true square wave is a special case of a periodic gate function (or 
rectangular wave) and in fact can only be approximated to some arbitrary
degree by real world generators.  The Fourier coefficients for periodic
pulses of width a, occuring at period T = 2*pi/w is

               c  =  a/T * [sin (nwa/2)] / [nwa/2]
                n

where n = 1 is the fundamental, n = 2 the second harmonic, and so forth,
which has been normalized for a unit amplitude.  Obviously for a square
wave where a = T/2, 

               c  =  a/T * [sin (n * pi/2)] / [n * pi/2]
                n

which has nonzero values only when n is odd, hence has only odd harmonics.

However if the rectangular wave is not perfectly "square" (i.e. a <> T/2)
then the equation reduces to

               c  =  a/T * [sin (n * pi * a/T)] / [n * pi * a/T]
                n

at which point harmonics are only missing at n = T/a, 2T/a, 3T/a ....
If T/a is not an integer then no harmonics are missing at all since n
is of course always an integer.  Hence you can see if the "square wave"
is only slightly non-perfect then T/a will not be exactly 2 but rather
a non-integer close to 2, and all harmonics, even and odd will be present.

Well I hope THIS is the LAST word on the subject, surely you don't disagree
with any of this Bob.
				Greg Rogers
				Tektronix