Some thoguhts on Fourier holography.

I wanted to point out something which undercuts effort like SETI in trying
to find radio evidence of intelligent life.

A notion that some people may have, is that white noise generally consists
of signal energy which is distributed evenly over the spectrum, and that to
find a real pattern embedded within the noise is just a matter of finding
frequency peaks.

What some people do not realize, is that using computers, the frequency
spectrum of sound, or of some radio signals, can be divided so finely, that
there will be as many frequency coefficients as there were original data
points. This type of Fourier transform is sometimes also referred to as a
Fourier hologram, because the original data points can be reconstructed as
easily (or with similar difficulty) from the frequency spectrum.

But why am I saying this? Because each sample of white noise is different
from the other samples. And this implies, that the highly accurate Fourier
transforms of white noise must also be different. Indeed, each sample of
white noise can be analyzed into frequencies which move closer together if
the original sample gets longer, and this spectrum is itself filled with
randomness when this is done.

When the general definition is considered for white noise, as having a flat
frequency-response curve, hen this should be regarded as an average. As an
average of many samples of white noise.

Figuratively, if one sampled one second of sound accurately enough, there
would not be as much energy at 2,000 Hertz as there would be at 2,001 Hertz.

I do not believe that this concept is in any way too abstract for SETI to
understand. After all, their Scientists studied higher Math as did many
others. And in fact, I would expect that SETI researchers already use a
strategy to deal with this. If after all, frequency spikes in the noise are
themselves due to randomness, then the strategy at a general level could be
to make the sampling window larger. More precisely, SETI researchers may be
using this axiom first to look at larger spectra using relatively small
windows of data, and then, if they think they may have found something, to
zoom in on one small subset of data.

If the frequency peaks were random, then doubling or quadrupling the size of
the sample, or expanding it by a factor of 10, should not reveal a
systematic continuation of the original frequency. Instead, the strategy
should behave as if a gambling bet were repeated and the results would
become *in*consistent. But, since the frequency peaks were first regarded as
'conspicuous', the need to analyze too much data might be hedged against,
since the larger window of data is not used ubiquitously.

But I do believe that this turns the entire effort to find meaningful
patterns in radio noises, frankly less meaningful. It implies that when one
sees patterns, they are ultimately not patterns by themselves. Indeed, other
information would need to be brought into the equation than just frequency
response curves, or just a repeated measurement one Earth-rotation (day)
later. After all, if small windows can't be trusted, slightly larger windows
are not more meaningful in themselves either, because one does not propose
to perform a general Fourier analysis, with any real bandwidth, even of
short-wave radio signals which would be one second long, let alone a day
long. And machines tend to repeat the noises they make, only after
completing some complex, repetitive task which might require such a long
window. Or after completing a message as long.

A car engine makes noises that repeat after each cylinder's valves operate,
thus being hundreds of milliseconds long, but extending to tens of Kilohertz
in frequency. A computer produces noise which may only repeat itself
thousands of times per second, due to the multitasking OS, but which has
Gigahertz of bandwidth...

Dirk

Dirk Mittler wrote:

When the general definition is considered for white noise, as having a flat
frequency-response curve, hen this should be regarded as an average. As an
average of many samples of white noise.

Figuratively, if one sampled one second of sound accurately enough, there
would not be as much energy at 2,000 Hertz as there would be at 2,001 Hertz.

I do not believe that this concept is in any way too abstract for SETI to
understand. After all, their Scientists studied higher Math as did many
others.

Not being one of them I am certain they would be very happy if I
choose to speak only for myself.

This appears a bit analogous to knowing the particle emission rate of
a sample of radioactive material but it is possible to watch one atom
for all eternity and never see an emission.

Or perhaps it is more like Xeno's paradox.

In theory one could monitor a sufficiently narrow spectrum for all
eternity and never detect anything. There would also be an infinite
number of such narrow slices in every Hertz and in every picohertz and
at which point we cheat and kick the hare across the finish.

Real world noise sources are not infinitely narrow emmissions. If
they were then in our finite universe the number of filled spectra
slices would be smaller than the empty ones. The empty slices could be
used without concern for the concept of signal to noise ratio.

In the real world noise sources are not at absolute zero so thermal
motion is always imposed upon the emitted frequency. Any reasonable
interstellar noise source has 10^lots of zeros of emitting sources
moving at all the possible relative velocities in the smooth thermal
spectrum. Even to the point observer all the points sum and to the
non-point observer the antenna also sums the sources.

In the real world of non-infinitely narrow spectrum slices measured
over a time much greater than the mythical "instant in time" of
calculus have a near zero chance, even less chance than a negligable
chance, of being empty. In the practical world where the minimum
detectable narrow frequency has an inverse relationship between
bandwidth and observation time these considerations do not arise as
the practical limits of the narrowness of the frequency bins is
already considered before the analysis begins.

That last sentence could have been the only sentence in the reply.
Given the length of the sample you can determine the best that can be
done with it before you start.

--
The question is not if Bush compares to Hilter
the question is if there are any differences.
-- The Iron Webmaster, 3213

Dirk Mittler wrote:

When the general definition is considered for white noise, as having a flat
frequency-response curve, hen this should be regarded as an average. As an
average of many samples of white noise.

Figuratively, if one sampled one second of sound accurately enough, there
would not be as much energy at 2,000 Hertz as there would be at 2,001 Hertz.

I do not believe that this concept is in any way too abstract for SETI to
understand. After all, their Scientists studied higher Math as did many
others.

Not being one of them I am certain they would be very happy if I
choose to speak only for myself.

This appears a bit analogous to knowing the particle emission rate of
a sample of radioactive material but it is possible to watch one atom
for all eternity and never see an emission.

Or perhaps it is more like Xeno's paradox.

In theory one could monitor a sufficiently narrow spectrum for all
eternity and never detect anything. There would also be an infinite
number of such narrow slices in every Hertz and in every picohertz and
at which point we cheat and kick the hare across the finish.

Real world noise sources are not infinitely narrow emmissions. If
they were then in our finite universe the number of filled spectra
slices would be smaller than the empty ones. The empty slices could be
used without concern for the concept of signal to noise ratio.

In the real world noise sources are not at absolute zero so thermal
motion is always imposed upon the emitted frequency. Any reasonable
interstellar noise source has 10^lots of zeros of emitting sources
moving at all the possible relative velocities in the smooth thermal
spectrum. Even to the point observer all the points sum and to the
non-point observer the antenna also sums the sources.

In the real world of non-infinitely narrow spectrum slices measured
over a time much greater than the mythical "instant in time" of
calculus have a near zero chance, even less chance than a negligable
chance, of being empty. In the practical world where the minimum
detectable narrow frequency has an inverse relationship between
bandwidth and observation time these considerations do not arise as
the practical limits of the narrowness of the frequency bins is
already considered before the analysis begins.

That last sentence could have been the only sentence in the reply.
Given the length of the sample you can determine the best that can be
done with it before you start.

--
The question is not if Bush compares to Hilter
the question is if there are any differences.
-- The Iron Webmaster, 3213

In article ,
Dirk Mittler wrote:

I do not believe that this concept is in any way too abstract for SETI to

As SETI is a process, not an organisation, it can't understand!

understand. After all, their Scientists studied higher Math as did many

Reading "their" as "Berkeley Space Sciences, SETI@Home project's"...

others. And in fact, I would expect that SETI researchers already use a
strategy to deal with this. If after all, frequency spikes in the noise are

They do. They use a threshold of 22 times mean noise power, which
results in only 1 or 2 false positive single detections in a work unit,
even after allowing for all the different chirp rates tried, rather than
using a threshold close to mean noise power. (The probability density
function is exponential.) (In general, all the thresholds are calibrated
to produce this sort of false positive rate.)

Combined with requiring two compatible detections at different times, this
reduces the false positive to manageable levels, that can be handled by
doing targetted follow up observations.

themselves due to randomness, then the strategy at a general level could be
to make the sampling window larger. More precisely, SETI researchers may be
using this axiom first to look at larger spectra using relatively small
windows of data, and then, if they think they may have found something, to
zoom in on one small subset of data.

In practice, this strategy doesn't work, and they have to compute all the
intervals.