I feel like this should not be so hard, but I am somehow stuck.
I would like to decompose the signal $$a\sin(\varphi t)+b\sin(\vartheta t)$$ into an amplitude modulation and a periodic carrier signal. For $a=b$, the solution is $$\underbrace{2a\cos\left(\frac{\varphi - \vartheta}{2}t\right)}_{AM}\underbrace{\sin\left(\frac{\varphi +\vartheta}{2}t\right)}_{carrier}.$$
However, for $a\not=b$ I have problems deriving the closed form solution for the carrier. The AM in that case can be seen to be $$A(t)=\sqrt{a^{2}+b^{2}+2ab\cos\left(\left(\varphi-\vartheta\right)t\right)}.$$ This is the same as $$A(t)=\sqrt{\left(a-b\right)^{2}+4ab\cos^{2}\left(\frac{\left(\varphi-\vartheta\right)}{2}t\right)},$$ which reduces to the above case when $a=b$.
The carrier is obviously periodic, but after I did a few numeric simulations, I am not so sure anymore whether it is a single sinusoid. What I did is to look at the numeric Fourier spectrum of $$A(t)^{-1}\left(a\sin(\varphi t)+b\sin(\vartheta t)\right)$$ which clearly showed no single peak, but several peaks around the (sub)harmonics of $\varphi$ and $\vartheta$.
My questions are
- Is there a close form solution of the carrier? Ideally I would like to have a function of a single carrier frequency: e.g. something like $f(\sin(\xi t+\eta))$. If would be great, of course, if $f$ turned out to be a polynomial.
- If there is no close form solution, is there at least a close form term for the frequency of the carrier?
If you write $a$ and $b$ as
$$ a = {a+b \over 2} + {a-b \over 2} $$
and
$$ b = {a+b \over 2} - {a-b \over 2}, $$
then the signal have the form of the sum of two signals which you know how to analyze:
$$ {a+b \over 2} \left( \sin \varphi t + \sin \vartheta t \right) + {a-b \over 2} \left( \sin \varphi t - \sin \vartheta t \right). $$
If you still have questions after approaching the problem this way, let me know.