Re: gibbs phenomenon

[John Denker <jsd@MONMOUTH.COM>]

At 06:48 PM 6/25/00 +0800, Jayasanker_J wrote:

> We have gibbs oscillations due to , for eg. , poor convergence
> of fourier terms at discontinuities in a square wave; we also see
> overshoots/undershoots in electrical networks due to impedance
> mismatches; i am confused as to whether the two are related ?

They are not related in any useful way.

1) If you send a square pulse into a mismatch, you get a transmitted and
reflected pulse.  The resulting waveforms don't look anything like Gibbs
oscillations.

2) The equations that describe these phenomena don't look anything like
each other, either.

3) Mismatch applies to traveling waves f(x,t) only.  Gibbs applies just
fine to localized oscillations f(t).

4) If I splice a 50-ohm coax to a 75-ohm coax, it will be mismatched over a
wide range of frequencies;  at least 4 orders of magnitude will be readily
observable.  The Gibbs phenomenon results from narrowbanding the signal
with a particular type of ill-behaved filter.

I suppose you could contrive a low-pass filter using mismatched stubs, so
that reflected waves interfere with the original waves, and then argue that
the resulting filter was "related" to Gibbsian truncation of the Fourier
series, but that seems like quite a stretch.