Maybe the misconception here is that although the whole amplitude envelope signal "a" depends on the whole signal "x", we cannot really say that instantaneous value "a(t)" depends only on "x(t)". Maybe it's a bit like making a conscious notational distinction between a mathematical function "f" which represents the function in its entirety vs its application to a particular value "f(x)" which only represents the evaluation of "f" at one single point? In a somewhat sloppy notation, these things ("f" and "f(x)") may get confused. The whole signal "a" depends on the whole signal "x" only globally, but locally "a(t)" cannot be computed from "x(t)" alone. The global information contained in x is gathered by the convolution integral with the Hilbert impulse response that is used to produce y. Every y(t) depends on the whole function x and not only on x(t).G-Spot wrote:Since the imaginary component y has been made up with a Hilbert transform, it depends on x and doesn't contain any additional information, so your gain a actually depends on x only.
By the way, I have my Hilbert filter (prototype) up and running (thanks A_SN for the formula - that really saved me some time) and I can now reproduce the effect on the sawtooth. Well, sort of - I have still some artifact - I see some small parasitic rippling/oscillation at the Nyquist freq - probably due to a non-ideal Hilbert filter, I guess. But generally, it seems to work. Now the experimentation fun can begin...