[SI-LIST] Re: Square wave harmonics
- From: jan.vercammen.jv1@xxxxxxxxxxxxxxxx
- To: si-list@xxxxxxxxxxxxx
- Date: Tue, 27 Nov 2001 17:20:59 +0100
Lucien, si-list,
I have followed the responses to Lucien's question with interest.
I do understand the problems he is facing, text books are often ambiguous. The
term "modeled" is indeed a bad choice.
First note that we only know of functions or signals in the time-domain.
Although one often use the term
frequency domain, one should be aware of the fact that sinoids are time-domain
functions. However, one
of their interesting and very useful engineering attributes is their frequency.
So engineers and scientists
use it all the time in their work.
Time domain signals are but an extension of vectors, albeit that continuous
signals are vectors of an infinite dimensional nature.
Consider first a 2-dimensional vector V in a rectangular coordinate system. We
can decompose V into two components
along the 1x and 1y unit basis vectors by means of the scalar or dot product:
V=a1*1x + b1*1y, where a1 and b1 are the elements
of the vector under the rectangular basis.
Now if we take another coordinate system (e.g. cylindrical) then the same
vector is written V=a2*1r+b2*1phy.
Here 1r and 1phy are the cylindrical unit vectors (radius and angle). Using
linear algebra we can find out
how the unit vectors are related (and transform) so that we can find a relation
between the vector elemens (a1, b1) and (a2,b2).
So the same vector has two different representations, but both are equal and
dependent on the chosen basis.
You might wonder were this discussing is heading for. Actually it is quiet
simple, but it requires a bit of abstract thinking.
As vectors need unit basis vectors for a proper representation one can also
state that a signal S needs (possibly an
infinite number of) basis functions for a proper representation. For example we
can state that a signal S uses sinoids as basis
functions and the linear sum of weigthed (amplitude) sinuoids (frequency &
phaze) is just the signal S, just as the sum of
the basis vectors make up the vector.
There exist many useful basis functions: delta functions, polynomials, bessel
functions, pulse functions, triangular ....
Obvious basis functions need to fullfil some fundamental requirements
(orthogonality) to be useful as a basis function.
Not all functions have the same capabilities, for example, sinoids cannot be
used to represent signals with discontinuities
as sinusoids have continuous derivatives, because even an infinite sum of
sinusoids cannot represent a jump.
This leads to Gibs phenomena.
In science and engineering is the choice of the basis functions important
because it can simplify the solution of a problem.
The answer to your question: "Is a square wave physically composed of harmonic
sinusoids?" is negative. It depends on
the chosen basis functions used for the signal representation!
Physics is based on isomorphism (translation=same shape), that is, we have
models for the "real world" that behave the same
as the real world (have the same shape --> are isomorph). Take for example some
voltage in an electric circuit. We can
represent it as an infinte set of real numbers from simulations and compare this
to a finite set of measured real numbers. Often
both sets agree well. Because we can use whatever basis to represent the
simulation or measured set we cannot state that
the signal is physically composed of sinoids or other basis functions.
If one uses a frequency domain measurement tool then one projects the measured
signal onto sinoid basis functions (one-by-one)
to extract its amplitude (and possible phase). The electrical circuits used for
that are narrow band filters that just ring to a specific
basis function frequency.
In the time-domain one uses a gate- or sampling function to approximate a delta
function. The gate function projects the signal
onto the delta function basis functions to extract its amplitude, which is
easily displayed on a screen.
It does not make sense to state that a "real world" signal is physically
composed of sinuoids. First we need to state which basis
functions one uses, which allows us to extract a set of real numbers (by
projecting). We can use whatever basis functions and
extract a set of real numbers, that multiplied by the basis functions, and
summed represent the signal. All representations give
equal results. No basis funtion is better or more fundamental than others.
I hope that this answers somewhat your question. It is about perception.
Currently I am at my desk looking at a picture of my 8 month
old son. If I change position he looks different, even though it is the same
little guy. It is all about how you look at things. The
same applies to the frequency and time domain representation of square
waveforms, your looking from a different perspective
at the same thing!
See you,
Jan Vercammen
Agfa-Gevaert Belgium
lucien_op@xxxxxxxxx@freelists.org on 11/26/2001 10:59:32 PM
Please respond to lucien_op@xxxxxxxxx
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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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