From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Wavelets, coherent states 11: windows -> wavelets.
Organization: Whole Earth 'Lectronic Link
Date: Tue, 12 Jan 1993 04:48:25 GMT


11. "suppose we are given an arbitrary signal without any idea of the type
of information it carries. The choice of a window h(t) then defines a scale
in time given by &, and it is this scale that distinguishes what will be
perceived as frequency and what as time. Thus, variations of f(t) in time
intervales much smaller than & are reflected in the v-behavior of F, while
variations at scales much larger than & survive in the s-dependence of F.
What an elephant perceives as a tone may appear as a rhythm to a mouse."
p.42

The Nyquist-Shannon sampling theorem gives a reconstruction for band-
limited signals F(v) -> 0 for |v| > W in terms of samples of f(t) taken at
instants t = n/2W. In comparison, the windowed Fourier transform
reconstruction need not assume that f(t) is band-limited. It need only be
square-integrable.

Suppose, however, that f(t) is band-limited (i.e. 2W = bandwidth about v =
0). Choose a window h(t) such that its Fourier transform H(v) is peaked
about v = 0 with width @ < W. Fs(v) -> 0 for |v| > W + @. Choose k = W + @.
Fs(mk) -> 0 whenever m=/ 0. To apply the reconstruction formula, choose
T < &. The uncertainty principle &@ > 1/2 implies & > 1/2W. The double-sum
reconstruction formula then limits to the Nyquist/Shannon form

f(t) -> g(t)^-1S(n)[h(t-nT)FnT(0)]  (83)

this approximation is more accurate as @/W -> 0. But a small @ means a
large &, so samples of FnT(0) are smeared over a long time interval (p.42)

Wavelet Transforms
The frame vectors for windowed Fourier transforms were the wave packets

h[v,s] = e^-i2pivt h(t-s)       (84)

The window function h(t) is assumed to vanish outside of -& < t < 0, and to
be smooth, not steep, so that the window's Fourier transform H(v) is also
centered about a narrow interval @ about zero frequency (i.e., v = 0). But
if h(t) has compact support, H(v) cannot vanish on any interval and cannot
have compact support.However, we can have H(v) decay rapidly outside of @.
The phase factor e^-i2pivt translates the window in frequency space so that
h[v,s] is a window in the time-frequency plane centered about (v,s) with
area &@ whose minimum "Planck phase space cell" is 1/2.  Thus, the
frequency components of the classical signal (time-series) f(t) were picked
out by rigidly translating the basic windows in both time and frequency. It
is this rigidity that links the windowed Fourier transform to the Weyl-
Heisenberg simply periodic "phase space crystal" group.

The Weyl-Heisenberg group is the group of translations in classical phase
space amended with multiplication by phase factors required to close the
Lie algebra. Kaiser explains this in more detail in his Ch.3. The important
idea for now is that the simple windowed Fourier transform has the same
time widt & for all frequencies. The number of wavelengths admitted for
analysis is v&. For low frequencies ("infra-red limit") v& << 1 and this is
no good because we cannot gain any meaningul frequency information by
looking at a small fraction of a single wavelength. In the opposite "ultra-
violet limit" v& >> 1, too many wavelengths are admitted. In this limit, a
time interval & seems effecitvely infinite, thus destroying the "locality"
which was the intent of the windowed Fourier transform to start with.

*To what extent are the infinities in standard Feynman QED an artifact of
using the wrong transform?* (Sarfatti half-baked thought flash from
elsewhere.)*

This over-rigidity of the windowed Fourier transform is fixed by the
wavelet transform!

to be continued.