I've been reading and playing with the Hilbert transform in the context of extracting the envelope of functions, and I noticed something when playing around with a a simple case.
If we consider the solution for the undamped and resonantly driven oscillator, $$x(t) = \frac{F_{0}}{2 m \omega_{0}} t \sin(\omega_{0} t) + A_{0} \sin(\omega_{0} t + \phi_{0}) \text{,}$$ where $F_{0}$ is the amplitude driving force, $m$, the mass, $\omega_{0}$ is the frequency of the oscillator and of the oscillatory driving force, while, $A_{0}$ and $\phi_{0}$ are the initial amplitude and phase of the system.
We can find the envelope of $x(t)$ by calculating $$\sqrt{\left( x(t) \right)^{2} + \left(\dot{x}(t) / \omega_{0}\right)^{2}} \text{,}$$
or, we can find the Hilbert transform of $x(t)$ and calculate
$$\left|x(t) + i \mathcal{H}(x(t)) \right|$$
which will only extract the modulating signal component of $\sqrt{\left( x(t) \right)^{2} + \left(\dot{x}(t) / \omega_{0}\right)^{2}}$. As shown below:
Here, the blue line is $x(t)$, the yellow-orange line is $\left|x(t) + i \mathcal{H}(x(t)) \right|$ while the olive-green line is $\sqrt{\left( x(t) \right)^{2} + \left(\dot{x}(t) / \omega_{0}\right)^{2}}$.
One can extract the same result as $\left|x(t) + i \mathcal{H}(x(t)) \right|$ from $\sqrt{\left( x(t) \right)^{2} + \left(\dot{x}(t) / \omega_{0}\right)^{2}}$ by choosing the dominating components by hand -- what is interesting to me is that the Hilbert transform method seems to do this implicitly.
So my questions are:
- does the Hilbert Transform always only extract the modulating signal, or have I just "found" a specific case?
- If the above is true, why? My guess would be that the oscillatory components get integrated out, leaving only the modulated component
- What constitutes an envelope of a function, as both examples I have shown would surely qualify, or in DSP does this always mean the modulating signal?