Harmonics question

[Magnus Danielson]

>>>>> "SC" == Sean Costello <costello at costello.seanet.com> writes:

 SC> At 04:34 PM 5/19/98 -0600, you wrote:
 >> 
 >> Simple waveforms are expressible in terms of the strength of each of these
 >> harmonics.  A sine wave is simple - it's all fundamental.  A sawtooth wave
 >> has all of the harmonics, but the strength of each successive one decreases
 >> exponentially.  A triangle wave has only the even harmonics.  A square wave
 >> has only the odd harmonics. 

 SC> Actually, a triangle wave has only odd harmonics, just at lower amplitudes
 SC> than the square wave.

 SC> To have all even harmonics would be weird - would that mean that the
 SC> fundamental wouldn't be included? :)  The only waveform I know of with all
 SC> even harmonics would be a full-wave rectified sine.

Simple experiment to get a waveform with only even harmonics:

Take an odd harmonics waveform (Say at 880 Hz) add a sine at 440 Hz
and voila! You just got yourself an all-even harmonics waveform.
I don't think that it will sound particularly earth-moving, but yeat
usefull. If you synk or PLL the odd-harmonics waveform to be at the
double frequency of the sine you can get it to track up better.

 SC> Is there any simple function that contains all even harmonics?
 SC> If so, what does the waveform look like?

I recall seeing diagrams of such unlinear waveshapers. I even think
that we have discussed this several times. For instance do I recall
that the Serge waveshapers would do odd and even harmonics and that
the schematic is very simple indeed.

The absolute function is BOUND to create even harmonics. Just look at
it:

 \   |   /
  \  |  /
   \ | /
    \|/
-----+-----
     |

You will get two positive going ends of a balanced signal. Thus, a
precision rectifier will act as a even-harmonics waveshaper.
The interesting thing is that as long as it is DC balanced (allways
appear on both positive and negative sides) it will always have even
only harmonics, totally independent on waveform or signal strength
(given in reallity that you have precission in your precission
rectifier ;).

You may alter the harmonic distribution by varying the offset of the
waveform before hitting the abs function.

You can get an odd harmonic function in similar manor by doing this:

     |
     | /\
     |/  \
-----+-----
 \  /|
  \/ |
     |

This one is level sensitive, but when you get up to a certain level it
will fold back, and the distorsion will when DC balanced and level
adjusted allways appear twice. You can easilly see how a sine suddenly
get a 3rd harmonic out of this...

There are other ways to accomblish similar unlinear effects.

One trick is to do weak amplifiers that will flatten out or even fold
back. One could even device them so that another weak amp is pulling
the opposite direction. Things like that. Bipolars, Fets, tubes can be
used in such ways. Bringing a coil to saturation etc. depending on the
magnetic material you will get diffrent nonlinearities.

Mixing the output of an schmitt-trigger with the signal etc.

Another trick that can be interesting is by quantising (A/D) a signal
and then start doing tricks with the bits, like inverting a bit,
droping a bit (in the middle), having a bit inverting another (XOR).
The output of this bit manipulation is then D/A converted. One could
possibly mix this with the original signal. A friend of mine
experienced "interesting" phenomenes of digital distorsions when the
old 3M digital 32-channel tapemachnies lost a bit every now and
then...

Cheers,
Magnus