Odd+even harmonic wave

Discussion in 'Electronic Design' started by Ethan Border, Aug 3, 2007.

1. Ethan BorderGuest

What is the characteristic wave shape that contains both odd and even
harmonics in equal amplitudes?

I have tried mixing a sawtooth with 1) squarewave and 2) triangle wave
of the same frequency all of the same voltage. This works, to a
degree, but I don't know if it is the best option.

Is it necessary to play around with duty cycles or something else. to
get an optimal result?

Ethan B.

2. D from BCGuest

I got curious and did some spice....

V1 N003 N004 SINE(0 10 3khz)
V2 N004 N005 SINE(0 10 4khz)
V3 N005 N006 SINE(0 10 5khz)
V4 N006 N007 SINE(0 10 6khz)
V5 N007 0 SINE(0 10 7khz)
V6 N002 N003 SINE(0 10 2khz)
V7 N001 N002 SINE(0 10 1khz)
R1 N001 0 10k

7 harmonics..all equal amplitude..all waves starting at 0 degree.
The resulting waveform kinda looks like this |\|\|\|\|\ but bumpy.
I get the impression that with more harmonics it might turn into ramp
wave...

D from BC

3. Guest

The Fourier transform of a repeating spike contains all the harmonics
of the repetition frequency up to a limit set by the width of the
spike. The amplitudes of all the harmonics are equal.

Search on the Dirac spike, which has a finite area, but infinite
height and zero width.

http://en.wikipedia.org/wiki/Dirac_delta_function

Check out the Comb function while you are at it

http://en.wikipedia.org/wiki/Dirac_comb

When I was getting to grips with this, I found it helpful to note that
if you differentiate a square wave you get two interleaved comb
functions, one with positive-going spikes and one with negative-going
spikes.

Differentiating the harmonics of a square wave gives all the odd
harmonics at equal amplitudes - you can see what has happened to the
even harmonics ...

4. meGuest

for sin () you get a series of negative and positive impulses.

| | |
---------------- sorta like this.
| | |

for cos() you get a series of positive impulses.

| | |
---------------- hope this helps....

excellent!

6. Tom BruhnsGuest

Others have pointed out that repeating impulses yield a spectrum of
equal amplitude harmonics; they all equal the fundamental in
amplitude. But that's not the only waveform that results in equal
amplitude harmonics. In fact, there is an infinite set of them--maybe
an infinity of infinities, because the phase of each of an infinite
number of harmonics may take on any of an infinite set of values, and
each set of phases will result in a different wave shape.

An example of a second waveform that has equal amplitude harmonics is
a clip of a white noise that happens to have equal starting and ending
values, so it may be repeated without introducing a discontinuity
(!). (Note that a random has an infinitely small but not zero
probability of having a discontinuity...) So long as you repeat the
same clip over and over to infinity, the result will have only the
fundamental at frequency 1/(clip length) and its harmonics. Since the
clip is limited in length, it won't have exactly equal harmonic
amplitudes. Consider that the clip can have _any_ shape (so long as
the end points are the same value), including an impulse, but also
including one cycle of sine, or one cycle of cosine, and be a valid
clip from a random waveform. But statistically, a clip with harmonics
with greatly different amplitudes will be very rare.

Hope this isn't too confusing...

Using a (pseudo)random clip is much more practical than using an
impulse in many cases, since the peak amplitude is not so outrageously
higher than the RMS. In theory, a random clip _could_ have very high
peak to RMS amplitude, but in practice it's not a problem; it's
statistically exceptionally unlikely, and in any event the methods of
generation guarantee peaks of some maximum value.

One reasonably quick and easy way to play with this and see what
various phases of the harmonics gives you is to use Matlab or Scilab
(free...). You can either build a time domain wave from time domain
sinusoids, or just build a frequency domain signal with equal
amplitude and related or random phases, and do an inverse FFT on it.

Cheers,
Tom

7. Guest

I'd one on file with 20 generators. Just cleans up on the general
shape you're seeing now. Curiously, if you put the first few sources
180degs out of phase, the wave looks identical to one of those heart
beat Alpha waves (or whatever they're called).

8. Don LancasterGuest

While your waveform was fun to plot is is stunningly and mind bogglingly
useless.

It consists of an infinite positive spike just beyond zero phase and an
infinite negative spike just before 360 degree phase.

Truncating to fewer harmonics gives you a narrow positive spike just
beyond zero phase, a bunch of intermediate low level ripple, and a
narrow negative spike just before 360 phase.

The harmonics correllate only near zero and 180 degrees; they
decorrelate otherwise.

There are twice as many zero crossings as the highest harmonic used.
The waveform is overwhelmingly positive from 0 to 180 degrees and
overwhelmingly negative beyond.

"Real" waveforms tend to have diminishing harmonic values.

Lots of fun to do with PostScript.

More useful waveforms appear at http://www.tinaja.com/magsn01.asp
More on PostScript plotting at http://www.tinaja.com/post01.asp

Here is the half cycle code using my Gonzo utilities...

%!PDF

(C:\\Documents and Settings\\don\\Desktop\\gonzo\\gonzo.ps) run % use
internal gonzo

50 50 10 setgrid
40 20 showgrid

0 10 mt

0 0.1 180 {/priang exch store

priang 20 mul 90 div % scale for one cycle

priang sin

2 mul 10 add lineto % should be 20 mul -- scaled in interest of sanity

} for

line1 stroke

showpage

--
Many thanks,

Don Lancaster voice phone: (928)428-4073
Synergetics 3860 West First Street Box 809 Thatcher, AZ 85552

Please visit my GURU's LAIR web site at http://www.tinaja.com

9. Guest

<snip>

I'm not so sure about that - if you take *all* the frequencies
starting from

1/age of the universe

You get a single spike of infinite energy at the Big Bang.

Robin

10. Arie de MuynckGuest

"Ethan Border" ...
A pure sine wave. All harmonics are zero.

Arie de Muijnck

11. LVMarcGuest

I you take the harmonics and adjust the phase so that eacs sucedding
harmonic number has a phase advance, or quad time delay, you get a
chirp! so the chrip and the impulse are closely related in a fourier
sense..whereas the chirp has precise phase advance for each component
and the impulse is just all of then starting at t=0...

the paek t average is way different and thi discussion and technique
moves inot the realm of optimizing waveforms,to produce dsirebale
perfomance. th chirp has some nice rardar propoerites, and a impulse is
"simple" and the core for UWB devices..

Happy waveform generation! BTW, making the chirp is a tricky process!