# Determine modulation function from spectrum?

I have a two periodic signals $x(t)$ and $y(t)$ which are related to each other by a periodic modulation function $\alpha(t)$: $$y(t) = \alpha(t) x(t)$$ If $x(t)$ is a sinusoid, is it possible to determine $\alpha(t)$ from the spectrum of $y(t)$? How might I do this?

• Your formulation is a bit strange. So all these are periodic. Do you mean your message is repeated in time? or you just talk about one period? – msm Nov 5 '16 at 22:22
• This looks like what every generic AM radio demodulator accomplishes (during those times when some solo musical instrument is playing some constant pitch). – hotpaw2 Nov 5 '16 at 22:27
• Also, is it a strict requirement to look at the spectrum? A product detector can also do what you want... – msm Nov 5 '16 at 22:38

Now, in general, the problem of detecting a signal and its modulation is pretty hard – governments pay loads of money for well-working signal classificators, because in many cases, you're looking at a very weak $y(t)$ within a mixture of hundreds, and it's still hard to guess whether you're looking at an 16-QAM or an 8-PSK if you don't have timing information.
Thus: without a lot of additional info on what you're looking at, guessing the modulation class of $\alpha$ from $|Y(f)|$ is very hard. And it's harder to actually estimate $\alpha(t)$ from that.