On Feb 6, 2017, at 8:24 PM, robert bristow-johnson wrote: > > On 02/05/2017 07:52 PM, robert bristow-johnson wrote: > >> > >> > I'm curious what aspects of a music make the complex magnitude of the > >> analytic signal inappropriate for estimating the envelope? In > >> communications signal processing we use this often, even for signals > >> that are fairly wide-band with respect to the sample rate and it seems > >> to work. > >> > > >> > >> well, with a single sinusoid, there should be no intermodulation product > >> so the analytic envelope should be exactly correct. but consider: > >> > >> x(t) = g1(t) cos(w1 t) + g2(t) cos(w2 t) > >> > >> which has for it's Hilbert > >> > >> y(t) = g1(t) sin(w1 t) + g2(t) sin(w2 t) > >> > >> and analytic signal > >> a(t) = x(t) + j y(t) > >> > >> a(t) = g1(t) cos(w1 t) + g2(t) cos(w2 t) + j( g1(t) sin(w1 t) + g2(t) > >> sin(w2 t) ) > >> > >> |a(t)|^2 = |g1(t)|^2 + |g2(t)|^2 + 2 g1(t) g2(t) cos( (w1-w2) t ) > > > >> the last term on the right needs to be sorta filtered out with a LPF to > >> get the correct square of envelope, no? > > > > Ah OK - you have a different definition of "envelope" than I do. That's > > fine. > > i'm trying to be as inclusive in the definition of "envelope" as i can be. > see, if you have a single sinusoid: > > x(t) = g(t) cos(w t) > > and Hilbert transform > > y(t) = g(t) sin(w t) > > and analytic signal > > a(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. 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? what > other meaning of "envelope" are you working with, Eric?
I agree with all of your math. I don't agree that the 2 g1(t) g2(t) cos( (w1-w2) t ) term is not part of the square of the envelope. If you have an analytic signal which is the sum of two complex exponentials as you described further above and the g1(t) and g2(t) terms are constant, then when you compute the magnitude of the sum you actually do see the result vary due to the beating of the two frequencies which is the frequency difference term you've proposed to discard. In the communications processing I've worked with that variation is a critical part of the envelope so using just the magnitude of the analytic signal is sufficient to define the envelope. Perhaps in audio processing this isn't the case and the filtering you describe is necessary. Eric _______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
