Frequency Doubling
I'm using an ADAU144X at 192kHz in an application to double a 25kHz sinewave sync signal to obtain a 50kHz operating frequency. Multiplying the signal by itself and DC-blocking the output does the job just fine, but intuitively I have this feeling that multiplying something by itself should just increase amplitude exponentially as the waveform rises from the zero crossing. I am having trouble visualizing this process in four quadrants however. I get about a 12dB loss through the multiplier, and although everything works I would dearly love to understand circuit action in non-mathematical terms. Can someone explain this simply or point me to a source? Thanks!
I'm not sure I understand your reference to a circuit since you are describing DSP. But you can look at an analog circuit like AD633, which was the old-school method of building a modulator effect. That datasheet has some theory on how modulation works.
http://www.analog.com/static/imported-files/data_sheets/AD633.pdf
When you multiply two signals, you get sum-and-difference tones. This is used commonly in RF transmission; for example FM transmission modulates the "side" audio signal with a 38kHz carrier, which yields sidebands at 23 - 38kHz (15kHz bandwidth) and 38-53kHz. Thus the full 53kHz bandwidth can transmit an encoded stereo signal.
Back to frequency doubling, when you modulate a signal with itself, you also get sum-and-difference tones, except your difference tone is always zero (25kHz - 25kHz). So you are left with the sum tone of 50kHz with a DC offset, which is the difference tone. You didn't ask for math, but this equation gives you a hint:
sin2 ( x ) = 1/2 - 1/2 cos ( 2x )
Consider that a full-scale digital sine wave varies from 1 to -1; if you square that, zero is still zero but that is now your minimum excursion as -1 becomes 1. So your signal has peak-to-peak amplitude of 1 rather than 2 (1 - -1), but a frequency twice as fast, because you have twice as many 1s as you used to have.
Thus, you'll lose 6dB amplitude as a result of modulating a full-scale sine with itself. You could be starting without a full-scale sine wave; if you start at -6dBFS peak then you'll get -12dB attenuation: peak-to-peak of 0.25 after modulation vs. 1 before.
Hey, Jon,
That all makes perfect sense, and I now understand why I'm losing 12dB through the multiplier. Many thanks for the useful and simple tutorial!
Hello,
Jon, excellent description of what happens when frequency doubling by squaring the input signal.
Electrojim, If your 25 KHz input signal is "dirty" or drifts in amplitude and it would help to clean it up for a stable 50 KHz reference, have a look at the PLL technique described at A -1701 FM MPX Demodulator.
Best regards,
Bob
Hey, Bob,
Thanks for that citation, a very clean DSP implementation of the classic stereo demod and a good example of how to do a PLL in Sigma. My 25kHz pilot did have amplitude perturbations, which I cleaned up, first with a clipper and follow-up filter, and second with an amplitude leveler with a somewhat unique character.
I never cease to be amazed at what can be done in Sigma by using very basic arithmetic elements, sometimes surpassing the performance of the more complex 'building blocks.'
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