On 2017-02-06, robert bristow-johnson wrote:
[...] and analytic signala(t) = x(t) + j y(t) = g(t) cos(w t) + j g(t) sin(w t) = g(t) e^(j w t) the analytic envelope is |a(t)| = sqrt( x(t)^2 + y(t)^2 ) = g(t) so that works great for a single sinusoid.
Indeed. Originally the formula was derived for a single sinusoid. At the outset it's very much like you derive group delay from phase delay, given a single sinusoid.
However, the formula generalizes just as well. What you have there is a single complex sinusoid, with a gain function g(t) (ostensibly an exponentially decaying one). Leave that out, and you have a single complex sinusoid. Which you can then integrate over frequency to yield your typical Fourier transform, in the s-plane, or sum over the unit circle in the z-plane to yield the discrete version.
In either case, you have the complex version of Parseval's theorem backing you up. Now that we went to the Fourier domain, suddenly even those complex sums/integrals work out just fine, as (approximable) sums of complex exponentials (over a discrete basis, by the sampling theorem and such). Suddenly what you get for an arbitrary sum of exponentials, is something whose backwards transform's norm is by Parseval very much the same as you had before for a single frequency.
That is why Hilbert-transforms are used to get "instantaneous envelopes" even with wideband signals. Basically the pointwise modulus of a Fourier transform of an analytical signal does what you did above for a single frequency, a sinusoid, while the whole of the Fourier transform does it for all frequencies at the same time, and the finally by Parseval even the sum/integral actually, first locally but then by extension globally, guarantees that the modulus actually follows the whole signal envelope as well.
What fucks you up, then, is that a proper Hilbert transform which lets you access the analytical continuation of a real signal, is not only an acausal operator, but also one which converges rather slowly in (future) time. It does so in o(1/t) time/shift, just as the ideal sinc(x) interpolation function does; which is rather slow as it goes, especially in asymptote.
That then implies that if you try to do "instantaneous envelopes" using Hilbert transforms and the analytical signals they imply, you might seem to get "something for free". A better and more reactive estimator of envelopes. The theory is sound as well, because of what I said above about Parseval's theorem. It's just that there's *still* no free lunch: if you want to approximate an Hilbert transform pair for real, for any given accuracy you'll have to expend o(1/t) delay in order to derive an o(1^-at) accuracy class Hilbert transformer (or a pair).)
But once you toss in additional frequency components into x(t), then that creates high-frequency components into |a(t)|^2 that don't belong in any definition of envelope, no?
If you look at it that way, yes. But isn't that the very reason Parseval's theorem is so great? It actually deals away with this kind of stuff, and returns the whole deal towards first order analysis over a function space.
what other meaning of "envelope" are you working with, Eric?
Finally, there are other meanings. The psychoacoustical ones in particular. Which I'm not too well versed with.
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