I got a simple modulating signal $x(t)=\sin(2\pi\alpha t)\sin(2 \pi \beta t)$ with carrier frequency $\alpha$ and modulation frequency $\beta$. The spectral correlation will obviously have components at the zero modulation frequency ($\beta = 0$). This is not surprising as the envelope signal is always positive.
I am aware that components at $\beta = 0$ also represent the average energy of the signal (Welch) but in the means of components separation/energy conservation, I would like to have the energy of $x(t)$ concentrated at $S_x(\alpha, \beta)$ only.
What am I missing here?
For the current case, do I have a complete separation between the two components having the same carrier and different modulation frequencies?
To me, it seems that part of the energy "leaks" to the zero modulation frequency component.