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

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.

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Given Andre's answer, what kind of signal are you talking about? – Jim Clay Dec 21 '12 at 13:34
@JimClay I have edited the question to give more information, thanks. – Mohammad Dec 21 '12 at 16:51
Short signals are usually problematic due to the border artefacts. Up to my experience chirps work very well (they are locally narrow band, broad band just globally). OFDM is what I mean with a mixture (summation of frequencies). And as written in my post for mixtures you get distortion due to nonlinear cross terms. You would need to apply band pass filters to the frequency regions of interest before extracting the envelope or better use an analytical band pass filter, so you have the envelope signal right away. – André Bergner Dec 27 '12 at 12:36
@AndréBergner I see, in other words, narrow-band filtering apriori is necessary to extract some narrow band frequency, before then using the envelope. What about something like speech? – Mohammad Dec 27 '12 at 16:49