Wavelets[1]
Notes by Jack Sarfatti in MathType5
The Invisible College[2]
Periodic functions in 1-dimension
The squeezed-shifted wavelet: Zoom in and zoom out!
Windows on the world above and below.
Oroborus Narcissus: The Self-Excited Wavelet Godelian Strange Loop?
Note, click on the equations in your web browser to see them larger.
Assume for a complex function of a dimensionless real
variable ,
the boundary condition
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(1.1) |
This is a transformation from the real line to the complex plane.
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(1.2) |
The Fourier series is
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(1.3) |
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(1.4) |
Orthogonality of the basis set of functions
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(1.5) |
Mother pre-wavelet of Fourier series is obviously
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(1.6) |
The basis is generated from it by integer dilation of the phase x.
Parseval identity of signal conservation of energy under basis transformations
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(1.7) |
We now allow non-periodic functions over the whole real line
without the periodic boundary condition (1.1). Replace the sinusoidal wave
mother pre-wavelet by a localized mother wavelet packet
or window function. We must shift
it to cover whole real line. One obvious way is to use integer shifts
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(1.8) |
We also need to dilate the mother wavelet. Communication engineers like to partition the frequencies into bands of consecutive “octaves” using integer powers of 2. The convention is
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(1.9) |
This is a combined operation of squeezing by a factor
of with a translation not of
,
but of rescaled
.
That is
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(1.10) |
Therefore, squeezes the domain of support (where
) into a smaller region as well as decreasing
the shift. This is a zoom-out transformation like decreasing the magnification
of a microscope showing less detail, less resolution of the image. On the
contrary,
does the opposite zoom-in increasing the
resolution of the image or increasing the magnification of the microscope. Note
that ordinary Fourier analysis in physics does not have this capability that is
particularly suitable for nonstationary statistical processes with fast changes
especially for open non-equilibrium systems.
A wavelet series can be of the form
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(1.11) |
provided that we have a complete orthonormal basis[3] or “frame of reference”
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(1.12) |
In terms of quantum measurement theory, the squeezed-shifted wavelet base functions are formally “filters” or eigenfunctions of some “wavelet observable” if there is also a directly measurable eigenvalue by some constructable physical detector. This formal filter structure may or may not translate to something objectively real depending on more specific physical information.
Just like in Fourier analysis we can generalize a discrete integer wavelet series to a continuous integral wavelet transform where squeeze-shift integers j,k are replaced by real parameters a, b
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(1.13) |
The notation is that is the integral wavelet transform, whilst
is the orthodox integral Fourier transform.
We can, of course, still keep as a special case
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(1.14) |
consistent with (1.10). Note that the wavelet literature
calls the binary dilation, and
the dyadic position when the octave
algorithm (1.14) is used.
“while the two components of Fourier analysis… the Fourier series and the Fourier transform are basically unrelated; the two corresponding components of wavelet analysis, namely the wavelet series (1.11) and the integral wavelet transform (1.13) have an intimate relationship as described in (1.14)” p. 6 Chui
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(1.15) |
Think of as an analog signal. This would be some kind
of stochastic time series coming out of a detector. You can generalize this to
a classical gauge force field configuration
with spatial frequencies or wave numbers as
well as temporal frequencies. We want to do better than Fourier analysis, which
is only really best, suited for stationary uniform stochastic processes. The
familiar Fourier transform
is the spectrum of the temporal signal.
Parseval’s theorem (1.7) is a special form of what is called a unitary
transformation in quantum physics in which inner products are invariant under
change of frame of reference (basis). The integral of the squared modulus of
the spectrum is a measure of the total signal power.
“to extract the spectral information from the analog signal
… takes an infinite amount of time, using
both past and future information just to evaluate the spectrum at a single
frequency
.. the [Fourier] formula does not even
reflect frequencies that change with time. What is really needed is … to be
able to determine the time intervals what yield the spectral information on any
desirable range of frequencies (or frequency band) … for high-frequency
spectral information, the time interval should be relatively small to give
better accuracy, and for low-frequency spectral information, the time interval
should be relatively wide to give complete information… it is important to have
a flexible time-frequency window that automatically narrows at high “center-frequency”
and widens at low-center frequency … the integral wavelet transform
[(1.13)] relative to some basic wavelet
… has this
… zoom-in and zoom-out capability.” Pp. 6-7 Chui
Both basic “Mother” wavelet window function and its classical Fourier transform
must decay to zero sufficiently fast in order
to define both a center and a width in both time and frequency. The product of the complementary widths of
course still obey the Heisenberg uncertainty noise of a quantized area in phase
space. However, we will see later how W. Zurek shows substructure inside this
quantized area under special conditions. Therefore, we can, it appears, in
principle, pierce the veil and see inside h per conjugate pair degree of
freedom, where h is Planck’s quantum of action.[4] Extending Zurek’s idea to a Bose-Einstein
condensate shows why the latter is protected against decoherence of the quantum
phasing. The argument is essentially, a single particle state whose Wigner
phase space density extends over a large phase space “area”
shows “spotty” sub-quantum pilot qubit wave
structure on the scale
.
In which
is the sensitivity of the large
single-particle nonlocal “Schrodinger Cat” state to environmental quantum
decoherence. This is why it is impossible to see single particles in many
places at once, so to speak, on a large scale. The quantum phase coherence is
rapidly destroyed the larger
.
My new idea not found in Zurek’s Nature paper is that Bose-Einstein condensates
work exactly the dual opposite way. Now the effective phase space volume per
particle inside the superfluid is
,
where
is the number of Bose-Einstein condensed
identical quanta of spin 0 or spin 1, real or virtual. Therefore,
is large which means London rigidity. The
macro-quantum order parameter
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(1.16) |
is robust, rigid, insensitive to decoherence. Zurek’s argument is turned topsy-turvy upside-down.
Let be a window function. Its center
is defined as
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(1.17) |
The window function’s width is defined as twice where
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(1.18) |
All windows functions must obey
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(1.19) |
Let Mother wavelet and its Fourier transform
both be kosher windows functions[5]
in their time and frequency domains respectively with centers
and widths
.
Heisenberg’s uncertainty principle, even in the classical domain, sans
of course, is that
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(1.20) |