How to find the envelope of a wide-band signal?

Question

I would like some feedback on possible techniques that one may use to determine the envelope of a broad-band time domain signal. I have heard anecdotally, that it is not as straight-forward as it seems, but I am not sure. (Some previous discussions about similar subjects exist here and here but I would like an explicit discussion on broad band signals here).

For narrow-band signals, one may compute the envelope via the absolute value of the analytical signal as such:

Let $s(t) = A cos(2\pi ft)$ be the real signal. Then, the analytical signal is:

$$ a(t) = Ae^{j2\pi ft } = Acos(2\pi ft) + jH(s(t)) = Acos(2\pi ft) + jAsin(2\pi ft) $$

where $H(.)$ denotes the hilbert transform of the real signal. Then the envelope is given by $E = |a(t)|$.

What we are really doing is converting a real signal into phasor form, and taking that phasors absolute magnitude. In other words, since any real signal can be described as two phasors of equal magnitude rotating at both negative and positive angular frequencies, we simply discard of those phasors and work with the remaining one, and take its absolute magnitude.

Now to my understanding, this method works well for narrow-band signals, and not for broad-band signals. Thus my questions are as follows:

---

EDIT: I have edited and narrowed down my questions based on the feedback I have received:

Question: I am wondering, what ways one may use to compute the energy-envelope of the following broad-band signals, assuming the above analytical approach does not work well. (If it doesnt, why doesnt it?)

i) A chirp signal, at some carrier. (linear, exponential, etc)

ii) A very short duration windowed sinusoid, of length $T_p$. (So bandwidth is $2/T_p$)

iii) An OFDM signal at some carrier. OFDM is composed of multiple orthogonal (I do not care about it as much, but put it for completion).

iv) @Andrey's 'mixture of tones'. (I am not clear on this, but I am guessing it means simple summation of multiple sinusoids around a carrier?)

I cannot think of any other types of broad-band signals, but answering those cases might go far in grounding my understanding here.

Thanks.

Answer 1

The analytical signal approach works very well for chirp signals. It stops working for mixtures of different tones. Taking the absolute value is a nonlinear operation that introduces cross terms and thus distorts you envelope.

I'd say that the envelope of a broad band signal as a property that is not well defined. First you need to define what you actually mean by that. Than you can come up with a method. For instance, the phase of a signal is not well defined for broad-band signals as well, because there exist two or more phases simultaneously. But depending on the actual problem you can come up with a special solution.

Just some notes to the phase example, because I'm an expert there. In physics there is the problem of defining a phase to measure phase synchronization of two or more broad-band systems. One solution is to compute the Lyapunov spectrum which measure phase synchronization implicitly without ever defining a concrete phase.

I just write that to make you think about what you actually want to know/measure/manipulate in order to step a bit back and have a more general look at your problem.