| Topic: |
Science > Physics |
| User: |
"bill" |
| Date: |
03 Nov 2005 09:44:35 AM |
| Object: |
Fourier analysis Q |
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
I don't know much about Fourier analysis, but it seems to me that
I should be able to apply some type of Fourier technique to this
data to measure the quantification more precisely, and in a way
that is not at all affected by the arbitrary choices made in making
the histogram.
What throws me off is that, in my little experience with Fourier
analysis, all the techniques are applied to a "signal" relative to
some variable (usually time). In this case all I have is a list
of about 100,000 non-negative numbers, which, when histogrammed,
appear to bunch up at regular intervals. Something *similar* to
126.380819738467
126.283380093841
125.748352912939
42.4897817871943
168.131819727191
168.163828641399
41.769198680611
0.0370432374476124
83.828042075852
42.2004219487581
42.1663080612675
167.85268740098
84.1654003775314
126.075822588658
41.6125736396355
where all the numbers are close to a multiple of 42 (actually the
periodicity become less obvious as the magnitude of the numbers
increases).
Given that this is nothing like a time-dependent (or space-dependent)
signal, I'm a bit of at a loss as to how to apply a Fourier
transformation to the data.
Any suggestions would be much appreciated!
bill
.
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| User: "Old Man" |
|
| Title: Re: Fourier analysis Q |
03 Nov 2005 02:29:16 PM |
|
|
"bill" <please_post@nomail.edu> wrote in message
news:dkdb93$ohb$1@reader2.panix.com...
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
I don't know much about Fourier analysis, but it seems to me that
I should be able to apply some type of Fourier technique to this
data to measure the quantification more precisely, and in a way
that is not at all affected by the arbitrary choices made in making
the histogram.
What throws me off is that, in my little experience with Fourier
analysis, all the techniques are applied to a "signal" relative to
some variable (usually time). In this case all I have is a list
of about 100,000 non-negative numbers, which, when histogrammed,
appear to bunch up at regular intervals. Something *similar* to
126.380819738467
126.283380093841
125.748352912939
42.4897817871943
168.131819727191
168.163828641399
41.769198680611
0.0370432374476124
83.828042075852
42.2004219487581
42.1663080612675
167.85268740098
84.1654003775314
126.075822588658
41.6125736396355
where all the numbers are close to a multiple of 42 (actually the
periodicity become less obvious as the magnitude of the numbers
increases).
Given that this is nothing like a time-dependent (or space-dependent)
signal, I'm a bit of at a loss as to how to apply a Fourier
transformation to the data.
Any suggestions would be much appreciated!
You really have to be more specific about the physics.
If the data represents an energy spectrum, such as that obtained
from nuclear inelastic scattering data, then a Fourier transform of
the entire spectrum isn't useful. However, a Fourier transform
of the histogram of an individual 'spike' will give you the decay
time for the nuclear excited state represented by that 'spike'.
bill
[Old Man]
.
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| User: "Gregory L. Hansen" |
|
| Title: Re: Fourier analysis Q |
04 Nov 2005 12:11:25 PM |
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In article <eMKdnfPk-qq97vfenZ2dnUVZ_tGdnZ2d@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"bill" <please_post@nomail.edu> wrote in message
news:dkdb93$ohb$1@reader2.panix.com...
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
....
You really have to be more specific about the physics.
If the data represents an energy spectrum, such as that obtained
from nuclear inelastic scattering data, then a Fourier transform of
the entire spectrum isn't useful. However, a Fourier transform
of the histogram of an individual 'spike' will give you the decay
time for the nuclear excited state represented by that 'spike'.
It will? I'd find the decay time by fitting an exponential to it. How
would a Fourier transform help?
--
"A nice adaptation of conditions will make almost any hypothesis agree
with the phenomena. This will please the imagination but does not advance
our knowledge." -- J. Black, 1803.
.
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| User: "Old Man" |
|
| Title: Re: Fourier analysis Q |
04 Nov 2005 03:39:20 PM |
|
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"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
news:dkg88d$jcn$1@rainier.uits.indiana.edu...
In article <eMKdnfPk-qq97vfenZ2dnUVZ_tGdnZ2d@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"bill" <please_post@nomail.edu> wrote in message
news:dkdb93$ohb$1@reader2.panix.com...
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
...
You really have to be more specific about the physics.
If the data represents an energy spectrum, such as that obtained
from nuclear inelastic scattering data, then a Fourier transform of
the entire spectrum isn't useful. However, a Fourier transform
of the histogram of an individual 'spike' will give you the decay
time for the nuclear excited state represented by that 'spike'.
It will? I'd find the decay time by fitting an exponential to it. How
would a Fourier transform help?
The energy width; delta_E, of a peak which represents
inelastic scattering from a nuclear excited state is related
to the state's life time, delta_t, by
delta_E * delta_t = hbar
One gets delta_E from fitting a Gaussian to the peak and
setting delta_E = 2 * sigma.
The Fourier transform of a Gaussian is another Gaussian,
such that sigma_1 * sigma_2 = 1
[Old Man]
.
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| User: "Gregory L. Hansen" |
|
| Title: Re: Fourier analysis Q |
04 Nov 2005 06:27:41 PM |
|
|
In article <m7CdnWM7m4KWSPbeRVn-pw@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
news:dkg88d$jcn$1@rainier.uits.indiana.edu...
In article <eMKdnfPk-qq97vfenZ2dnUVZ_tGdnZ2d@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"bill" <please_post@nomail.edu> wrote in message
news:dkdb93$ohb$1@reader2.panix.com...
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
...
You really have to be more specific about the physics.
If the data represents an energy spectrum, such as that obtained
from nuclear inelastic scattering data, then a Fourier transform of
the entire spectrum isn't useful. However, a Fourier transform
of the histogram of an individual 'spike' will give you the decay
time for the nuclear excited state represented by that 'spike'.
It will? I'd find the decay time by fitting an exponential to it. How
would a Fourier transform help?
The energy width; delta_E, of a peak which represents
inelastic scattering from a nuclear excited state is related
to the state's life time, delta_t, by
delta_E * delta_t = hbar
One gets delta_E from fitting a Gaussian to the peak and
setting delta_E = 2 * sigma.
The Fourier transform of a Gaussian is another Gaussian,
such that sigma_1 * sigma_2 = 1
[Old Man]
Uh... you're way out there, Old Man. delta t isn't a decay time, it's a
standard deviation. And it looked to me like he has spikes that decay.
Decay time constants aren't an uncertainty thing, and they aren't
gaussians. They're something like
f(t) = 0, t<0
= A*expt(-kt), t>0
--
"For every problem there is a solution which is simple, clean and wrong."
-- Henry Louis Mencken
.
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| User: "Old Man" |
|
| Title: Re: Fourier analysis Q |
05 Nov 2005 05:18:22 PM |
|
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"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
news:dkgu9t$q5p$2@rainier.uits.indiana.edu...
In article <m7CdnWM7m4KWSPbeRVn-pw@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
news:dkg88d$jcn$1@rainier.uits.indiana.edu...
In article <eMKdnfPk-qq97vfenZ2dnUVZ_tGdnZ2d@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"bill" <please_post@nomail.edu> wrote in message
news:dkdb93$ohb$1@reader2.panix.com...
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
...
You really have to be more specific about the physics.
If the data represents an energy spectrum, such as that obtained
from nuclear inelastic scattering data, then a Fourier transform of
the entire spectrum isn't useful. However, a Fourier transform
of the histogram of an individual 'spike' will give you the decay
time for the nuclear excited state represented by that 'spike'.
It will? I'd find the decay time by fitting an exponential to it. How
would a Fourier transform help?
The energy width; delta_E, of a peak which represents
inelastic scattering from a nuclear excited state is related
to the state's life time, delta_t, by
delta_E * delta_t = hbar
One gets delta_E from fitting a Gaussian to the peak and
setting delta_E = 2 * sigma.
The Fourier transform of a Gaussian is another Gaussian,
such that sigma_1 * sigma_2 = 1
[Old Man]
Uh... you're way out there, Old Man. delta t isn't a decay time, it's a
standard deviation. And it looked to me like he has spikes that decay.
Decay time constants aren't an uncertainty thing, and they aren't
gaussians. They're something like
f(t) = 0, t<0
= A*expt(-kt), t>0
--
The OP said that his abscissa wasn't time or distance. That the
peaks have a (high energy ?) tail is probably instrumental in
origin. In inelastic nuclear scattering, peaks usually have a low
energy tail due to scattering from slit edges.
And Yes: Old Man is "way out there" where Gregory hasn't
bothered to look. Gregory might take the time to wonder
how nuclear physicists measure the life time of a nuclear
excited state that's on the order of 10^(-21) seconds. Hint: it
isn't measured directly.
[Old Man]
.
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| User: "Gregory L. Hansen" |
|
| Title: Re: Fourier analysis Q |
06 Nov 2005 08:54:14 AM |
|
|
In article <wLWdnZBcptVcoPDeRVn-gw@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
news:dkgu9t$q5p$2@rainier.uits.indiana.edu...
In article <m7CdnWM7m4KWSPbeRVn-pw@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
news:dkg88d$jcn$1@rainier.uits.indiana.edu...
In article <eMKdnfPk-qq97vfenZ2dnUVZ_tGdnZ2d@prairiewave.com>,
Old Man <nomail@nomail.net> wrote:
"bill" <please_post@nomail.edu> wrote in message
news:dkdb93$ohb$1@reader2.panix.com...
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
...
You really have to be more specific about the physics.
If the data represents an energy spectrum, such as that obtained
from nuclear inelastic scattering data, then a Fourier transform of
the entire spectrum isn't useful. However, a Fourier transform
of the histogram of an individual 'spike' will give you the decay
time for the nuclear excited state represented by that 'spike'.
It will? I'd find the decay time by fitting an exponential to it. How
would a Fourier transform help?
The energy width; delta_E, of a peak which represents
inelastic scattering from a nuclear excited state is related
to the state's life time, delta_t, by
delta_E * delta_t = hbar
One gets delta_E from fitting a Gaussian to the peak and
setting delta_E = 2 * sigma.
The Fourier transform of a Gaussian is another Gaussian,
such that sigma_1 * sigma_2 = 1
[Old Man]
Uh... you're way out there, Old Man. delta t isn't a decay time, it's a
standard deviation. And it looked to me like he has spikes that decay.
Decay time constants aren't an uncertainty thing, and they aren't
gaussians. They're something like
f(t) = 0, t<0
= A*expt(-kt), t>0
--
The OP said that his abscissa wasn't time or distance. That the
peaks have a (high energy ?) tail is probably instrumental in
origin. In inelastic nuclear scattering, peaks usually have a low
energy tail due to scattering from slit edges.
And Yes: Old Man is "way out there" where Gregory hasn't
bothered to look. Gregory might take the time to wonder
how nuclear physicists measure the life time of a nuclear
excited state that's on the order of 10^(-21) seconds. Hint: it
isn't measured directly.
He said that he has data with spikes at regular intervals that decay
exponentially. I haven't seen any indication that he's looking at
short-lived nuclear excited states. In fact, barring further information,
if he has spikes at regular intervals then it's more likely he's looking
at crosstalk from an oscillator, or a signal from something spinning.
There's just no way to know how to meaningfully analyze it without more
information, but pulling out Heisenberg is very likely unhelpful.
--
"We don't grow up hearing stories around the camp fire anymore about
cultural figures. Instead we get them from books, TV or movies, so the
characters that today provide us a common language are corporate
creatures" -- Rebecca Tushnet
.
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| User: "Gregory L. Hansen" |
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| Title: Re: Fourier analysis Q |
03 Nov 2005 09:52:33 AM |
|
|
In article <dkdb93$ohb$1@reader2.panix.com>,
bill <please_post@nomail.edu> wrote:
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
I don't know much about Fourier analysis, but it seems to me that
I should be able to apply some type of Fourier technique to this
data to measure the quantification more precisely, and in a way
that is not at all affected by the arbitrary choices made in making
the histogram.
What throws me off is that, in my little experience with Fourier
analysis, all the techniques are applied to a "signal" relative to
some variable (usually time). In this case all I have is a list
of about 100,000 non-negative numbers, which, when histogrammed,
appear to bunch up at regular intervals. Something *similar* to
126.380819738467
126.283380093841
125.748352912939
42.4897817871943
168.131819727191
168.163828641399
41.769198680611
0.0370432374476124
83.828042075852
42.2004219487581
42.1663080612675
167.85268740098
84.1654003775314
126.075822588658
41.6125736396355
where all the numbers are close to a multiple of 42 (actually the
periodicity become less obvious as the magnitude of the numbers
increases).
Given that this is nothing like a time-dependent (or space-dependent)
signal, I'm a bit of at a loss as to how to apply a Fourier
transformation to the data.
Any suggestions would be much appreciated!
bill
You really need to start by knowing which questions you want answered.
The horizontal axis has to be SOMETHING, or else you might just as well
plot the data in any random order. Fourier analysis will represent the
data in terms of frequencies. Does that make sense? Does that give you
some insight into the system you're studying? If spikes are clearly
visible, you might be better just looking at the average period between
them. If you're interested in the decay, you might be better off fitting
a decay curve to each one to find a decay time constant. Or trying
randomness tests.
--
"We need to remember that when we are faced with an unstructured problem
it is we who create the model in the form of a quantitative metaphor;
there is no correct model waiting in the wings for us to call onstage." --
Thomas L. Saaty, "Mathematical Methods of Operations Research" (1988)
.
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| User: "tadchem" |
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| Title: Re: Fourier analysis Q |
04 Nov 2005 12:57:22 PM |
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bill wrote:
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
I don't know much about Fourier analysis, but it seems to me that
I should be able to apply some type of Fourier technique to this
data to measure the quantification more precisely, and in a way
that is not at all affected by the arbitrary choices made in making
the histogram.
What throws me off is that, in my little experience with Fourier
analysis, all the techniques are applied to a "signal" relative to
some variable (usually time). In this case all I have is a list
of about 100,000 non-negative numbers, which, when histogrammed,
appear to bunch up at regular intervals. Something *similar* to
126.380819738467
126.283380093841
125.748352912939
42.4897817871943
168.131819727191
168.163828641399
41.769198680611
0.0370432374476124
83.828042075852
42.2004219487581
42.1663080612675
167.85268740098
84.1654003775314
126.075822588658
41.6125736396355
where all the numbers are close to a multiple of 42 (actually the
periodicity become less obvious as the magnitude of the numbers
increases).
Given that this is nothing like a time-dependent (or space-dependent)
signal, I'm a bit of at a loss as to how to apply a Fourier
transformation to the data.
Any suggestions would be much appreciated!
All you have presented to us is a list of numbers. There is no way to
make any realistic appraisal of their relationships to anything (even
to each other) unless there is some context that provides meaning to
those numbers.
Are they an ordered set or a simple collection?
Do they have any units? Are they frequencies or voltages or currents
or what?
Do they represent measurements or calculations?
In a cursory examination I notice only that all 14 that you have given
us round off to exact multiples of 42, and that the fractional parts
appear to be randomly distributed.
The best analysis would start with a model that describes what the
expected behavior is.
Given a model, the characteristics of the subsequent analysis could be
established.
Tom Davidson
Richmond, VA
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| User: "Puppet_Sock" |
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| Title: Re: Fourier analysis Q |
03 Nov 2005 10:05:44 AM |
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bill wrote:
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
I don't know much about Fourier analysis, but it seems to me that
I should be able to apply some type of Fourier technique to this
data to measure the quantification more precisely, and in a way
that is not at all affected by the arbitrary choices made in making
the histogram.
What throws me off is that, in my little experience with Fourier
analysis, all the techniques are applied to a "signal" relative to
some variable (usually time). In this case all I have is a list
of about 100,000 non-negative numbers, which, when histogrammed,
appear to bunch up at regular intervals. Something *similar* to
126.380819738467
126.283380093841
125.748352912939
42.4897817871943
168.131819727191
168.163828641399
41.769198680611
0.0370432374476124
83.828042075852
42.2004219487581
42.1663080612675
167.85268740098
84.1654003775314
126.075822588658
41.6125736396355
where all the numbers are close to a multiple of 42 (actually the
periodicity become less obvious as the magnitude of the numbers
increases).
Given that this is nothing like a time-dependent (or space-dependent)
signal, I'm a bit of at a loss as to how to apply a Fourier
transformation to the data.
Any suggestions would be much appreciated!
You've opened a large can of worms here.
First, the general subject you are talking about is data analysis, not
just
Fourier analysis. It is possible you've got some kind of periodic thing
here,
but it's also possible you've got a lot of things.
The ideal would be to have some idea what to expect based on physics
of the situation. "I have some data" is a very nebulous start. Unless
it's
a puzzle that you are supposed to unravel based on internal clues in
the data, you really want some notion of where to start.
Consider: If this were energy of emitted photons, you might be seeing
something of a band structure. If the emitting source had several
equally spaced bands, you might see a lot of photons at the first band,
a lot jumping two bands, a lot jumping three, and so on. Then the
rest of the signal might be accounted for on the basis of processes
that added noise, such as thermal vibration, scattering before the
photon hit the detector, etc. And the band structure might only extend
for part of the spectrum, after which you started to see some other
form. In that case, a Fourier analysis might not be very useful.
Or as another alternative: If you were looking at time of signal
return,
you might be seeing echoes from different parts of your device. This
might show some limited periodicity, followed by a range of
progressively
more chaotic values, followed by a range that looked nearly continuous.
In this case again, a Fourier analysis might not be very useful.
Particularly
without some idea of why different frequencies were not equally likely.
Suggestion: Work with somebody who understands your system. See
if you can't come up with some kind of indication of what the structure
in the data ought to look like. Then take that model and see if it fits
the
data. That will give you a start on whether you understand your data.
Socks
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| User: "Randy Poe" |
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| Title: Re: Fourier analysis Q |
04 Nov 2005 01:10:04 PM |
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bill wrote:
I have some data that, when I histogram it, shows some degree of
quantization (i.e. the histogram has definite spikes at regular
intervals, and this is not an artifact of the histogramming
procedure). The spikes decay in magnitude roughly exponentially.
(In between the spikes the hits are few but not zero.)
As I read this, what you are really asking is whether there's a
good statistical test for the question "is my data from a quantized
source?" and wondering whether the tools of Fourier analysis or
time-series analysis could be brought to bear to answer that question.
Also I gather these are to be regarded as random samples from some
unknown distribution rather than having any particular order.
I'd be inclined to do a (nonlinear) least squares fit to the histogram
itself, of a model that included evenly spaced discrete values, plus
gaussian noise (I suspect that's what your "exponential" decay
between the spikes is).
Free parameters: spacing (approximately 42)
probability data value = k*spacing, k=1,...,maxspike
sigma of additive gaussian noise
- Randy
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