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?
Thank you for any advice.
Ethan B.
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
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.
Check out the Comb function while you are at it
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 ...
-- Bill Sloman, Nijmegen
D from BC wrote in news: snipped-for-privacy@4ax.com:
for sin () you get a series of negative and positive impulses.
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---------------- sorta like this. | | |
for cos() you get a series of positive impulses.
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---------------- hope this helps....
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A train of narrow impulses contains all harmonics at equal amplitudes.
John
excellent!
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
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).
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
Here is the half cycle code using my Gonzo utilities...
(C:\\Documents and Settings\\don\\Desktop\\gonzo\\gonzo.ps) run % use internal gonzo
50 50 10 setgrid 40 20 showgrid0 10 mt
0 0.1 180 {/priang exch storepriang 20 mul 90 div % scale for one cycle
priang sin priang 2 mul sin add priang 3 mul sin add priang 4 mul sin add priang 5 mul sin add priang 6 mul sin add priang 7 mul sin add priang 8 mul sin add
priang 9 mul sin add priang 10 mul sin add priang 11 mul sin add priang 12 mul sin add priang 13 mul sin add priang 14 mul sin add
priang 15 mul sin add priang 16 mul sin add priang 17 mul sin add priang 18 mul sin add priang 19 mul sin add priang 20 mul sin add
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 rss: http://www.tinaja.com/whtnu.xml email: don@tinaja.com Please visit my GURU's LAIR web site at http://www.tinaja.com
I'm not so sure about that - if you take *all* the frequencies starting from
1/age of the universeYou get a single spike of infinite energy at the Big Bang.
Robin
"Ethan Border" ...
A pure sine wave. All harmonics are zero. :-)
Arie de Muijnck
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!
snipped-for-privacy@ieee.org snipped-for-privacy@ieee.org posted to sci.electronics.design:
OK, i knew about this for some time. The kicker is does this represent something physical or is it just an interesting mathematical artifact. What are the social consequences of each case?
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