[SI-LIST] Re: Causality
- From: <colin_warwick@xxxxxxxxxxx>
- To: <vladimir_dmitriev-zdorov@xxxxxxxxxx>, <si-list@xxxxxxxxxxxxx>
- Date: Fri, 15 May 2009 11:21:27 -0600
Hi Vladimir,
On your PS:
> P.S. On thing I'm curious, how exactly you observed non-causality from time
> domain simulation? Since time domain simulators all work in sequential
> manner (step after step, with increasing time) the response can never come
> ahead of the input. Although, I agree that time response could be very
> different from what you'd expect from you frequency dependence.
I don't know about Jennifer's case, but a bad time-domain response sometimes
manifests itself as "bleed in" of the response from the next period of an IFFT
into the one (one hopes) represents the impulse response. (It doesn't: see
below)
Consider the following MATLAB code which models a lossless delay by its
amplitude and phase frequency response, then applies an inverse discrete
Fourier transform, then plots one period of the time domain response:
close all
clear all
npts = 256;
delta_t = 1e-9; % s
t = 0:delta_t:delta_t*(npts-1);
f = linspace(-(npts-1)/(2*npts*delta_t),1/(2*delta_t),npts); % Hz
amplitude = ones(size(f));
delay = 10.5e-9; %s
phase = -2 * pi * f * delay;
fresp = amplitude .* exp(j * phase);
tresp = ifft(ifftshift(fresp));
plot(t,tresp)
The resulting plot shows that this method cannot be used to create an accurate
impulse response:
(see image on my blog posting at http://bit.ly/ifft-kk )
The pulse is spread out so badly that the skirt of the next period leaks into
the end of our candidate.
A fundamental issue is that to get an impulse response, you have to do an
inverse Laplace transform, not an inverse Fourier transform. (The output of an
inverse Fourier transform isn't an impulse response at all: it's one period of
the repeated pulse train response.)
"But," you may say, "I don't have the frequency response in the complex plane s
= alpha + j * omega, I only have the steady state response on the upper half of
the f = j * omega line." Kramers-Kronig relation to the rescue! This relation
says that if you have a real physical system i.e. a causal system the frequency
response it is possible to construct the impulse response (causal of course)
from the steady state data alone.
I put a link to our DesignCon 2008 paper and our patent application in the post.
Hth
-- Colin
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