[SI-LIST] Re: Square wave harmonics
- From: "Zhou, Xingling (Mick)" <xlzhou@xxxxxxxxx>
- To: "'Daniel.Erik@xxxxxxxx'" <Daniel.Erik@xxxxxxxx>,lucien_op@xxxxxxxxx
- Date: Tue, 27 Nov 2001 11:40:00 -0500
Hi, folks
Very interesting discussions ! I like to share some ideas (may not be 100%
correct).
First of all, it is somewhat philosophical. The answer may depend on the
observers like
other general problems. Fortunately, in this case, there is no doulbt that
the observers are
ourselves, (electrical) engineers.
Second, in reality, there is no "frequency domain", everything is in time
domain for us (human beings). Since great man Fourier described his
transformation (skip the history and details),
everybody started thinking of frequency problems. It is so popular now.
However, we still live
in time domain. Let us think about how we describe a pure sine (or others)
wave, the x-axes is still
time, from -infinity to +infinity. This implies it does not purely exist.
Furthermore, basically, the transformation itself contains infinite terms
(or integrals), which implies to me
we are not able to implement it anyway to make the equalty really true. From
this sense, Fourier
transformation is a mathematical description.
However, it is really a nice description as we have been experiencing. Back
to electrical engineering, people have invented many "frequency" devices
like resonators etc. But we should say they are still in time domain even we
can approximately describe them using frequency domain terminologies. If
the approximation is good enough and practical, we always accept it.
Even in the quantum world, considering the uncertainty principles (not only
between distance and time, but also between other pairs), we never have
pure frequency domain waves.
Last I want to mention is, we can decompose any time domain signals into
other convenient forms.
Wavelet is another example. If wavelet was before Fourier, we may talk about
wavelet domain.
Of course, Fourier is still the most convenient one and we are used to it.
It has become real !
Let's stick to it now, not go too far.
Thanks.
Mick
-----Original Message-----
From: Daniel, Erik S., Ph.D. [mailto:Daniel.Erik@xxxxxxxx]
Sent: Tuesday, November 27, 2001 8:20 AM
To: lucien_op@xxxxxxxxx
Cc: si-list@xxxxxxxxxxxxx; Daniel, Erik S., Ph.D.
Subject: [SI-LIST] Re: Square wave harmonics
Lucien-
Although there have already been many very good responses to your question
posted, I'll offer yet another.
The bottom line is, yes, Fourier series or Fourier transform components are
"real" and "physical" in every sense that matters.
In the world of electrical engineering, you will come across two primary
ways of thinking about signals: "time domain" and "frequency domain".
You'll meet engineers who may be more comfortable with one "domain" than the
other. In the time domain camp, the primary tool for signal analysis is the
oscilloscope, and signals are described by how voltages (or currents, or
fields ...) change in time. In the frequency domain camp, the primary tool
for signal analysis is the spectrum analyzer, and signals are described by
frequency content. In my opinion, to be a really good enginner or
scientist, one needs to become comfortable with both camps, learning how to
mentally flip between domains, because both domains are really needed to
describe many complex (useful) signals.
For example, consider your square wave example. In reality, the wave will
not be perfectly "square". In the time domain, this shows up as a finite
"risetime" (e.g., time it takes for the voltage to switch between limits).
In the frequency domain, this shows up as a decrease in amplitude of the
higher harmonics. The square wave will not continue on forever. In the
time domain, the signal will have a definite start and end point (e.g.,
defined by when you turn the source on and off). In the frequency domain,
this shows up as a broadening of the harmonic tones (which you can easily
see if you can turn the source on and off fast enough). The square wave
will not be perfectly periodic. In the time domain, this shows up as slight
differences in the time delays between adjacent rising edges. In the
frequency domain, this shows up as "noise" -- energy at frequencies other
than the fundamental or the harmonics. I don't know if this is common
practice in university training these days, but I think it is *EXTREMELY
INSTRUCTIVE* to set up a complex waveform generator, run it into both an
oscilloscope and a spectrum analyzer, and get a feel for what different
signals look like in both domains.
- Erik
P.S. If the philosophical ideas of time and frequency domain duality are
nauseating to you, better steer clear of a quantum mechanics course!
==================================================================
Erik Daniel, Ph.D. Voice: (507) 538-5461
Mayo Foundation Fax: (507) 284-9171
200 First Street SW E-mail: daniel.erik@xxxxxxxx
Rochester, MN 55905 Web: www.mayo.edu/sppdg/
==================================================================
> -----Original Message-----
> From: lucien_op@xxxxxxxxx [mailto:lucien_op@xxxxxxxxx]
> Sent: Monday, November 26, 2001 4:00 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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