From: "Joel Kolstad" on Wed 13 Jul 2005 08:13

I was over on comp.dsp exposing my ignorance the other day when it eventually
dawned on me that a frequency multiplier will, in the frequency domain, just
convolve whatever the input signal is with itself. This got me to thinking...
why is it that frequency multipliers work as well as they do for something
like FM? Assuming a sine wave modulating signal, the FM spectra is a sum
harmonics with amplitudes dictated by a Bessel function; frequency multiplying
this would seem to add new harmonic content to the mix besides just doubling
the frequency of what's already present. So... does it turn out,
mathematically, that frequency multiplying an FM signal just so happens to end
up what nothing more than a "frequency scaled" spectra of what was originally
present? Or is some amount of distortion added in the process (assuming
perfect mixers used as the frequency multipliers and the DC component of the
mixers' outputs removed).

No distortion per se, but, in multiplying the carrier by an
integer number, you also multiply the deviation by that same
number. You do the math, I did that a long, long time ago. :-)

I've been told that, in general, frequency multiplier can be effectively
applied to most any modulation scheme that has a reasonably constant envelope,
e.g., FM, PM, FSK, even QPSK. Is this generally accepted knowledge?

It has been accepted - and done - since at least 1939 for
FM and PM.

The first dedicated radio relay sets for the U.S. military,
using Type C Carrier equipment in/out (four voice channels
frequency-multiplexed within 12 KHz bandwidth), did that
with crystal control, a reactance modulator, and lots of
multiplication to reach VHF. See AN/TRC-1, -3, -4 70-90
MHz sets. Long-lived sturdy-bird radios they were.

Had there been distortion in the multiplying process, the
Carrier equipment it worked with wouldn't have functioned
properly. However, it worked just dandy. :-)

Most of the early FM radio transmitters were of the crystal
oscillator into a reactance modulator and then into Class
C frequency multipliers...broadcasting to police car
radios. About the only way to get FM or PM in the very
early days was by a reactance modulator. In the post-war
period there were some specialized tubes made for FM/PM
modulation but those are collector's pieces now. Took lots
more years before the variable-capacitance diodes appeared
to mess with an oscillator's tank for FM.

See "Carson's Rule" and things of that nature in a search.
[John R. Carson, not of the old Tonight Show...:-) ]