[SI-LIST] Re: Square wave harmonics

• To: si-list@xxxxxxxxxxxxx
• Date: Fri, 30 Nov 2001 15:12:49 -0800

```Lucien,

You certainly got a lot of excellent responses!  I would
like to add something that I haven't seen mentioned yet.

I like to understand and explain how things work in practical
ways.  In school we learn about all the theories, and
mathematical derivations for solving things, but these
techniques often leave only a mystery in the student's
(and my) mind.

I could never understand how in the world does the Fourier
integral extract the harmonics from a function.  I.e., I
could not see what was the mechanism by which the harmonics
were pulled out of a time domain signal by doing all that
math.

Interestingly enough, none of my teachers mentioned this in
school, and when I interviewed a person with a math second
major for an opening we had, he couldn't tell me the answer
either.  Hmmmm...

After some thinking I realized that the whole thing is based
on the fact that the integral of a sinusoidal signal is zero
when integrated over an integer multiple of its period.
However, the same signal raised to the second power, i.e.
multiplied by itself will yield a nonzero value after
integration!  This is what makes the Fourier integral work!
In the integral we have our (unknown) signal that can be thought
of as a sum of sinusoids and another (known) sinusoid that
multiplies it.

If our signal contains a component that has the same frequency
as the sinusoid we multiply it with, we will get a non zero
value after integration.  All other components of the analyzed
signal will not get "squared", so their contribution to the result
of the integration will be zero.  In the Fourier series we
multiply the signal with integer multiples of frequencies and
only use full periods (or rules of symmetry) for the integration.
In the Fourier transform we allow anything in-between, which means
that the sinusoids will not always be integrated over complete
periods.  This is why the result of the Fourier transform does
not look like impulse functions at the frequencies which make
up a signal.

I know this sounds like a bad hand waving explanation, but it
helped me to gain a lot of insight to what really is happening
there...

Intel Corporation
================================================================

-----Original Message-----
From: lucien_op@xxxxxxxxx [mailto:lucien_op@xxxxxxxxx]
Sent: Monday, November 26, 2001 2:00 PM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Square wave harmonics

I'm an undergrad at UW, Seattle.  I have a question concerning square
waves.

My signals text book says, a square wave is modeled by a sum of
harmonically-related sinusoids (the Fourier series).  Mark the word
"modeled."  Another source uses the term "represented."

Recently, I've been told at my workplace by several senior
engineers that a physical square wave is PHYSICALLY composed of
harmonics.  In other words, they say that the Fourier series is not
just a mathematical tool describing square waves, but is indeed an
accurate description of the physical square wave.  They tell me all
physical square waves contain harmonics.

The two ideas above seem in conflict.  My undergraduate brain is
growing frustrated, and all I can conclude with certainty is that a
square wave BEHAVES AS a Fourier series, regardless of how it is
created.

I know from reading HP manuals for signal/pulse generators that these
devices do not build square waves by adding sinusoids.  So in my mind,
it doesn't seem possible that these square waves can contain
harmonics.  As for how a spectrum analyzer gives Fourier Coefficients
I have no idea.  I don't know if it just calculates and displays the
Fourier coefficients, or if it actually detects physical harmonics
within a signal and displays their magnitudes.

Can anyone give me the low-down on square waves!  Basically, my
question is:  In our physical reality, do square waves contain
harmonics?  Or does the idea of square wave harmonics only exist on
paper as a mathematical model, used to PREDICT the natural behavior of
the square wave?

Lucien Opperman
Seattle

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