From: "Doug MacMynowski"
Subject: RE: transfer functions
Date: Thu, 6 Jan 2011 16:01:15 -0800

Attached are 3 files:
 - gest_simple has all the matlab code to create transfer function
 - gest is the version I use, that has a lot more control over plotting (as
well as some archival commented out chunks of code to try different things
for testing purposes, so its impossible to read)
 - TFexample is a piece of code I just wrote to illustrate with a simple
example.

The example being the system \tau\dot{T}=\mu H - T + n

Where H is a known input to the system, and n is an unknown input, and I
low-pass filtered H so that at low frequencies most of the response results
from known input, and at high frequencies, most of the response comes from
unknown inputs

Though right now gest returns the parameters "mu" being mu/tau and "epsilon"
being 1/tau


Note that the code is currently set up to plot the magnitude, the
phase, and the coherence.  Due to aliasing, the phase in particular
becomes garbage at high frequencies regardless of noise levels; in
practice, with all the data we have, the phase becomes garbage due to
poor coherence long before aliasing becomes an issue.

Also, it seems that sometimes when I run this it gets fairly lousy estimates
of the parameters; even with long time records, low noise, and large nfft,
the parameter estimates can still be off a bit (e.g. change Tmax=1000 years,
Namp=0.01; and nfft=10*Fsamp...)

The phase is quite useful info too, though it seems to look noisier than
magnitude (the errors are uniform in a real vs imaginary axis plot, but
while a 20% error might look small in magnitude on a log plot, the
corresponding phase errors don't look small on that plot; also looks like
I'm getting a small nonzero bias on the phase... again, this is small
compared to errors in real data, but I should figure out why its there).
Some useful things on the phase are to see which signal "leads" the other
signal, or whether they're both in phase; that can diagnose whether the
correlation between the signals is x causes y, y causes x, or both x & y are
caused by some third factor.

doug