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).

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?

Thanks,
---Joel Kolstad

Suppose a 1 MHz signal is frequency modulated by a 1 kHz sine wave. The
frequency deviates over some frequency range, for example +/- 5 kHz. If we
put that signal into a times-ten multiplier the FM signal is at 10 MHz and
the frequency deviatioon is ten times greater (+/- 50 kHz). However, the
*rate* at which the 10 MHz signal traverses the greater frequency deviation
is still 1 kHz. That does not change and that is what the FM detector output
delivers.

Bill W0IYH

"Joel Kolstad" wrote in message
...
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).

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?

Thanks,
---Joel Kolstad

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...:-) ]

Thanks for the detailed reply, Len... I'll sit down and work through some of
the math if/when I get a moment.

Could you tell which of the following is correct? -- Say I amplitude modulate
my voice onto a carrier and feed it through a frequency multiplier. Besides
shifting the carrier, does it just make me sound like a chipmunk (i.e., if I
manage to hum a perfect 1kHz sine wave, the demodulated signal is me humming a
2kHz sine wave with a 2x mulitplier)? Or is there far more serious distortion
present?

---Joel

"Joel Kolstad" ) writes:
Thanks for the detailed reply, Len... I'll sit down and work through some of
the math if/when I get a moment.

Could you tell which of the following is correct? -- Say I amplitude modulate
my voice onto a carrier and feed it through a frequency multiplier. Besides
shifting the carrier, does it just make me sound like a chipmunk (i.e., if I
manage to hum a perfect 1kHz sine wave, the demodulated signal is me humming a
2kHz sine wave with a 2x mulitplier)? Or is there far more serious distortion
present?

Virtually all multipliers we see work by overdriving a stage. This strips
off any amplitude modulation, so there's no modulation at the output of
the multiplier.

Michael VE2BVW

and it will NOT work with QPSK

Mark

Mark wrote:

and it will NOT work with QPSK

Mark

It would work to the extent that if you go 4x you could phase lock for a
reference -- but you wouldn't have QPSK any more.

--
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

From: (Michael Black) on Thurs 14 Jul 2005
16:24

"Joel Kolstad" ) writes:
Thanks for the detailed reply, Len... I'll sit down and work through some of
the math if/when I get a moment.

Could you tell which of the following is correct? -- Say I amplitude modulate
my voice onto a carrier and feed it through a frequency multiplier. Besides
shifting the carrier, does it just make me sound like a chipmunk (i.e., if I
manage to hum a perfect 1kHz sine wave, the demodulated signal is me humming a
2kHz sine wave with a 2x mulitplier)? Or is there far more serious distortion
present?

Virtually all multipliers we see work by overdriving a stage. This strips
off any amplitude modulation, so there's no modulation at the output of
the multiplier.

Well, that IS rather serious distortion...! :-)

True, the multipliers (or any following over-driven stage) will
strip off all the amplitude variations/modulations. But FM and
PM don't "modulate" the amplitude...even if the match APPEARS
that there are various amplitude sidebands present. Those
various resulting sidebands of FM/PM are the result of the
carrier "swinging" due to modulation and the various amplitude
sidebands are the RESULT of the constant swinging.

One of the ways of calibrating the DEVIATION of FM or PM, as
in checking a modulation monitor meter, is to (actually) look
for ZERO carrier frequency amplitude content, either by a
spectrum analyzer or a narrowband receiver. That's been an
accepted method of calibrating broadcast transmitter air
monitor meters for years and years. [for a hint on how that
is done, check them Bessel Function "zero-crossings" found in
many texts to derive the modulation test frequency and the
deviation]

Multiplying the carrier of FM/PM changes the DEVIATION of the
carrier (frequency for FM, phase for PM, a little of both
actually when there is some distortion happening). The rate of
change of the modulation itself has NOT changed, hence no
"chipmunk" voice sounds. Note that, all youse guys what is
locked into AM thinking. :-)

For a good treatment of the various modulation processes, I
suggest "Electronic Designers' Handbook" by Robert W. Landee,
Donovan C. Davis, Albert P. Albrecht, McGraw-Hill 1957, section
Five (45 pages). Vacuum tube oriented (note copyright of 1957)
but the modulation basics are there...the basics apply to all
active devices.

For "chipmunk sounds" (which I used to do for funsies at WRRR
many yarns ago) just use a tape deck with a different speed
capstan for play-back. The rate of change of modulation is
altered by the fractional (not integer) tape transport speed,
thus the change in voice frequencies. The same thing can be
done with recorded digital audio recovered with a DIFFERENT
clock rate on read-out compared to sampling/recording. That's
not just "multiplying frequencies" but rather changing the RATE
OF CHANGE of the WHOLE sound. Not the same as multiplying just
the carrier wave.

"Chipmunk" secret, the easy way: Use a 50 Hz capstan on the
Magnecord reel-to-reel to record, play back with the standard
60 Hz capstan to get the increased rate of change of sound.
Done in 1956, hardly any transistors on the market then,
certainly NO integrated circuits having on-board A-D things.
:-) Alas, reel-to-reel analog tape units are things of
antiquity in 2005. :-)