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From: tcapizzi@world.std.com (tom capizzi)
Subject: Re: Stapp, PK & Physics Today
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Date: Fri, 18 Aug 1995 06:24:41 GMT
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        The nature of the binary signal described in a previous post is
precisely the format for spectral analysis by means of Walsh functions.
These are a set of orthogonal waveforms which are the fastest way to get
a spectrum from a set of data points. It is at least an order of magni-
tude faster to compute than a fast Fourier transform. The Walsh functions
are square waves that are easily generated from a simple binary counter.
This permits the mathematical process of finding a spectrum to be imple-
mented in analog hardware for even faster results. The process of finding
a spectrum involves multiplying a sample waveform by one of the reference
square waves. In the case of a Fourier spectrum, it means 4 quadrant
analog multiplication. This is quite tedious in analog hardware, and also
cumbersome in digital hardware. Since the reference wave in a Walsh spec-
trum is a square wave with a value of +1 or -1, multiplying the sample
involves only the sign of the sample. This is easily accomplished with
linear hardware, and reduces the complexity of the software algorithm as
well. Since only the sign bit is involved, there are no multiplications
required. There are n log n additions/subtractions needed to perform the
spectral decomposition. I don't believe any process is more efficient at
obtaining a spectrum.
        Each square wave in the set is characterized by the number of 
zero-crossings of the waveform. The product of any two square waves is
just another waveform in the set. There are log n primaries which are 50%
duty cycle octaves of some fundamental frequency. All the other waves in
a set are composed of products of these primaries. The content of a Walsh
frequency can be determined by taking the gray code of the frequency. It
involves taking the exclusive or of the frequency number and half the num-
ber. The resulting 1-bits represent the primary factors. Gray code is a
cyclic operator which returns to the original value after 2^n operations.
It has very interesting number theory relations to pi and the trig func-
tions. It is also used in digital positioning schemes because consecutive
number pairs differ by only a single bit. With ordinary counters there can
be carry errors, since so many bits must change simultaneously. With gray
code, only one bit 'carries' at a time. So if there is any uncertainty in
a gray code reader, it means that it is precisely on a transition of the
square wave from plus-to-minus or minus-to-plus.
        So far we have been looking at the one dimensional case - a string
of samples of some source with respect to time. We specified Walsh func-
tions of time. We could just as easily define them as functions of space
in one dimensional analysis. To expand a Fourier analysis into three dimen-
sions is a rather messy triple integral. Using Walsh functions it makes no
difference in computing effort to be one dimensional or three dimensional.
Of course, there is a limited number of ways the one dimensional waveform
can be bent and twisted into a three dimensional waveform, and these are
related by symmetry operations. As long as the sample sequence number is
mapped to the proper location in the x-y-z grid, the spectral analysis is
transparent to the number of dimensions. Another unique feature of a Walsh
spectrum is the property of being auto-inverse. To recover a data set from
a spectrum merely repeat the identical operation on the spectra. Other
kinds of spectra also have inverses, but none are auto-inverse, that I'm
aware of.
        In order to make use of the Walsh functions it is necessary to
have a real-time input. The simplest I have found uses a standard game
port on a PC. It's useful for biofeedback experiments, and consists of
probes made out of a string of common 1N914 diodes. It takes about 8
diodes in series to bias the game port input to about 50%. At this level
of current through the diode string a high resistance results. Small
changes in current produce significant changes in voltage. Similarly, a
very small change in temperature results in a significant shift in the
operating point of the diode junction. In addition to temperature, the
sensor can be used to measure ambient fields. Paranormal events are fre-
quently associated with temperature 'chills' which would be detectable.
There are claims that capacitors pick up charge from gravity and other
fields. If a large number of these detectors were in use. There is a
way of combining all their results so as to reveal any geographic pat-
terns using the Walsh functions in two dimensions. 
        It would be interesting to see if ufo sightings could be cor-
related to the spectral readings. It would also be quite interesting
to detect unconscious esp or to register paranormal appearances. The
detector hardware is very cheap and software is simple. Volunteers?

Tom Capizzi


