From ???@0x0000114C Tue Jan 22 12:05:26 2002 Path: pitt.edu!newsflash.concordia.ca!sunqbc.risq.qc.ca!nf3.bellglobal.com!peer1-sjc1.usenetserver.com!usenetserver.com!e3500-atl1.usenetserver.com.POSTED!53ab2750!not-for-mail Message-ID: <3C4D88DD.5E7090B3@bellsouth.net> From: "Clay S. Turner" Organization: Wireless Systems Engineering, Inc. X-Mailer: Mozilla 4.7 [en] (Win98; I) X-Accept-Language: en MIME-Version: 1.0 Newsgroups: comp.dsp,comp.soft-sys.matlab,rec.audio.tech,alt.engineering.electrical Subject: Re: Physical meaning of Group-Delay ? References: <3C47544E.595CF956@rtp.ericsson.com> <67ef4d54.0201180947.406d5458@posting.google.com> <3C4862FE.5B54FB54@rtp.ericsson.com> <67ef4d54.0201181757.17b6fb84@posting.google.com> <3C48E372.40176D4C@ieee.org> <67ef4d54.0201191119.550e3208@posting.google.com> <3C49D8BD.E5A2F791@ieee.org> <67ef4d54.0201191947.36a2d4b8@posting.google.com> <3C4AC479.2DB87E12@ieee.org> <67ef4d54.0201201605.2a6b07b@posting.google.com> <3C4B8E09.5040A0BB@ieee.org> <67ef4d54.0201211625.67a2af55@posting.google.com> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Lines: 86 X-Complaints-To: abuse@usenetserver.com X-Abuse-Info: Please be sure to forward a copy of ALL headers X-Abuse-Info: Otherwise we will be unable to process your complaint properly. NNTP-Posting-Date: Tue, 22 Jan 2002 10:32:26 EST Date: Tue, 22 Jan 2002 10:44:29 -0500 Xref: pitt.edu comp.dsp:120613 comp.soft-sys.matlab:113318 rec.audio.tech:157925 Status: N Bob_Stanton wrote: > The slope of the phase curve, *directly* indicates the amount of time > for a sine wave, of frequency w, to propagate. > My statement was and is correct. > > Bob Stanton Bob, Your above statement is half true. The negative of the slope of the phase curve gives the group delay. But is not the time that a single sine takes to go through the filter. For your RC circuit E_out 1 ----- = ------- where s=jw and w is freq in rads/sec E_in 1 + sRC Phase shift = arg(E_out/E_in) = -atan(wRC) group delay = -d phase/d w = (RC) / (1+(wRC)^2) With R=1000 Ohms and C=1E-6 Farads The phase shift at 1000 rads/sec = -pi/4 radians The Group delay at 1000 rads/sec = 500 uSec If you put a dual trace scope on the circuit with one trace looking at the input and the other looking at the output, you will see the 1000 radian per second signal has a -pi/4 phase shift between the input and output - this is 785 uSec of delay not 500uSec! On the other hand if you put two very closely spaced frequencies say at 1000 +- e (e is an epsilon which will become small later), you will see these two combine to produce a modulated signal. The "carrier" is at 1000 rads/sec and the modulation is at e rads/sec. Use the following classical trig identity to convince yourself of this cos(a)+cos(b) = 2 cos((a+b)/2)*cos((a-b)/2) Now what is the delay of the modulation? Since each original freq is phase shifted, the output is cos((1000-e)t+p1)+cos(1000+e)t+p2) where p1 and p2 are the phase shifts The input is of course cos((1000-e)t)+cos((1000+e)t) By the aforementioned trig identity the following carrier-modulation form is found for the filter's output: 2*cos(1000t+(p1+p2)/2)*cos(et+(p2-p1)/2) This can be interpreted as a carrier at the average of the two original frequencies with a phase shift that is also the average of the original phase shifts modulated by a sinusoid whose frequency is one half of the frequency difference and has a phase shift that is one half of the differences in the phase shifts of the original two frequencies. Now look at the modulation portion cos(et+(p2-p1)/2) It has a frequency of e radians per sec. and a phase shift of (p2-p1)/2 rads. Thus the modulation has a delay of -(p2-p1)/(2e) seconds. This is simply -delta(phase)/delta(frequency) If you let e->0 (delta frequency) in a limiting fashion, you can see the group delay is just the negative derivative of phase with resect to frequency! The group delay is the delay of the modulation resulting from a mix of frequencies. A single sinusoid's delay can and will be usually different from the group delay except for the case of linear phase. I hope this clears up some of the confusion. Clay