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.
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
