From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Wavelets, coherent states 10, sampling-reconstruction
Organization: Whole Earth 'Lectronic Link
Date: Mon, 11 Jan 1993 04:04:32 GMT


10.Reconstruction of signal by sampling.
The h[v,s] are highly redundant (over-complete, not linearly independent as
kets etc - see earlier notes). We want to find discrete subsets that are
complete frames. Kaiser gives an algorithm on p.38.

Let T > 0 be a fixed time interval. Sample output signal Fs(v) only at
instants s = nT, n is integer. Discretize frequency as well.

fnT(t) = hnT(t)*f(t)    (74)  (* is cc of h not a "convolution" command)

has compact support in interval nT-& <=t<= nT, hence expand it in a Fourier
series

fnT(t) = S(m)[e^-i2pimt/& cmn]    (75)

cmn = (1/&)Integral(from nT-& to nT)[e^i2pimt/& h(t-nT)* f(t)

    = (1/&)F(m/&|nT)  (76)

It can be shown (p.39) that

&hnT(t)fnT(t) = S(m)h[m/&,nT](t)<h[m/&,nT]|f>  (77)

sum over n, assume convergence limit g(t) where

g(t) = @ S(n)[|h(t-nT)|^2] (78)

0< A <= g(t) <= B        (79)

then double discrete sum

S^2(n,m)[|<h[m/&,nT]|f>|^2 = Integral dt g(t)|f(t)|^2   (80)

hence the subset

H(M)[T,&} = {h[m/&,nT]|m,n in Z}   (81)

forms a "discrete subframe" with frame constants A nd B. This subframe is
not generally tight, but the "metric" operator G is simply multiplication
by g(t) . Note how the metric is built from sampled windows in equation
(76).  The time signal is reconstructed as


f(t) = g(t)^-1 S^2(n,m)[h[m/&,nT](t) F(m/&|nT)]   (82)

*The idea here is that we can recover the continuous signal by discrete
sampling at time intervals T and frequency intervals 1/&. The Heisenberg
uncertainty principle is here. Heisenberg's principle is the condition that
the discrete subset h[m/&,nT] forms a frame. We cannot satisfy g(t) +> A >0
unless the supports of successive windows like h(t) and h(t-T) overlap so
that T < &  which is the uncertainty principle going the wrong way! It is
the same as the condition on "virtual particles" fluctuating out of and
back into the vacuum in Yukawa force picture. It is really consistent with
Heisenberg because, if we use quantum pictures, we are asking the
"complementary" question. We are not measuring conjugate (i.e.
incompatible) observables on real particles but estimating the minimal
noise levels of virtual particles that allow us to measure the real
particles.

The closer T gets to &, the more difficult for the window function h to be
smooth. Does it break into a kind of chaotic fractal at a T-& resonance?
When T = & h(t) must be discontinuous if the frame condition is obeyed.
This means that the Fourier transform of the window function H(v) is no
longer concentrated near v = 0. This suggests some kind of a phase
transition on computer analysis of natural time series of complex systems
in which we vary T on the computer and see if there are any natural time
scales & in the complex system!

**Above two paragraphs are Sarfatti's half-baked conjectures not Kaisers!**

This means that for nice windows h(t) we can get a good "perturbative"
solution to the reconstruction of the signal by truncating the discrete
double sampling sum in eq.(82) (like Feynman diagrams in QED? Sarfatti
note). But the convergence of the perturbation series breaks down at a T-&
resonance.

If we are only interested in a particular time interval and a particular
frequency interval because we want to eliminate high frequency static in a
sound system, for example, then the appropriate truncation includes an area
in the time-frequency plane only slightly larger than the area of interest.
In effect we are getting an least-squares approximation f1 to the original
signal. (p.41)

Newsgroups: sci.physics