From: MATGBB@LURE.LATROBE.EDU.AU (BYRNES,Graham)
Newsgroups: rec.audio.pro
Subject: Re: Rudy Van Gelder: Digital vs Analog
Date: 26 Aug 92 00:57:20 GMT
Organization: La Trobe University
X-News-Reader: VMS NEWS 1.22

In <23547@alice.att.com> jj@alice.att.com writes:

> In article <714723353snx@black.demon.co.uk> tchannon@black.demon.co.uk (Tim Channon) writes:
> >>Would you care to explain what the Gibbs phenomenon is?
> >
> >Given a brick wall filter with near perfect group delay (the ideal filter) the 
> >transient response will be found to exhibit damped ringing which extends in 
> >_both_ directions in time from the transient edge in question. This is called 
> >the Gibbs phenomenon.
> NO!
> 
> You are referring here to the ringing one sees due to the filter
> response.
> 
> The "Gibbs Phenomina", strictly speaking, refers ONLY to the "ears"
> one gets when one tries to reproduce, say, a square wave with a fourier
> series.

Umm, so far so good... the definition of Gibb's phenomenon can probably
be stretched a bit though. Strictly it refers to the fact that, although
the Fourier series of any bounded function converges rms (e the energy
of the difference goes to zero as you take more terms), it doesn't converge
uniformly, that is not at every point (to be precise, not at discontinuities
of the original function.) So you get increasingly high narrow spikes at
those points like the corners of square waves. If you check the F. coeffs
for a square wave, they only drop off as 1/n. However the series
--
\
/__ 1/n diverges.

  This **non-linearity**
Whoa, in the case of a signal reconstructed from a finite set of terms,
the wriggles are best thought of as resulting from a truncation of the
Fourier series. Since any real signal has finite rise time, and hence
a uniformly convergent F.S., it really IS that. Now truncation is a vector
space projection, and fits every decent definition of linearity I've heard
(I've heard some indecent ones too, though(8->  ).
I guess jj meant it in a colloquial sense, but to avoid future confusion...

> comes about from signals thave have non
> zero energy at infinity,
Energy is an integral over time...the criteria for (continuous) Fourier
transforms is that the integral of the square of the signal over all time be
finite...ie that the signal have finite energy...maybe finite power at infinity
would scrape through...

 or have discontinuous first derivitives (those
> two statements are rather, um, redundant).
Not at all. A triangle wave has discontinuous derivatives, but it's coeff's 
go as 1/n^2 and so converge absolutely (No Gibbs phen.)
Or more strongly, consider the non-periodic signal
-inf ____________/\_______________ +inf
which obviously has finite energy.
> 
> The ringing one sees from a filter is not, strictly speaking, the
> Gibbs Phenomina.   Look to the reproduction and convergence capabilities
> of a fourier series.
I'll second that.
> -- 
>  Bush    *Copyright alice!jj 1992, all rights reserved, except transmission
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>  Clinton  --    *granted only for complete copies that include this notice.
>  Just Say !NO!       *Use on pay-for-read services specifically disallowed.
                 Graham Byrnes
                 Dept of Maths
                 Latrobe Uni

PS I could still be wrong!
From: myers@hpfcso.FC.HP.COM (Bob Myers)
Date: Tue, 25 Aug 1992 19:06:44 GMT
Subject: Re: Rudy Van Gelder: Digital vs Analog
Organization: Hewlett-Packard, Fort Collins, CO, USA