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:

sin

^{2}( 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.