# Spectral correlation zero modulation frequency $S_x(\alpha, 0)$

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.

• I might not be familiar with your terminology: could you define what $\beta$ means? Is $x$ a sinusoidal with that frequency, which gets mixed up to a frequency $\alpha$? – Marcus Müller Aug 11 '19 at 15:48
• Edited the question... – Gideon Genadi Kogan Aug 12 '19 at 1:48
• @MarcusMüller, do you find the edition helpful? – Gideon Genadi Kogan Aug 12 '19 at 11:43