� ---------------------------- Original Message ---------------------------- Subject: Re: [music-dsp] � 45� Hilbert transformer using pair of IIR APFs From: "Eric Brombaugh" <ebrombau...@cox.net> Date: Mon, February 6, 2017 11:37 am To: music-dsp@music.columbia.edu -------------------------------------------------------------------------- > 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? -- r b-j � � � � � � � � �r...@audioimagination.com "Imagination is more important than knowledge."
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