| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
14 Apr 2007 06:29:29 PM |
| Object: |
Frequency retrieval from noise |
Hi group,
I tried this one on sci.math but nobody was able to offer an answer.
Is there any meaningful way to assign a frequency to a given
probability distribution?
Typically when one obtains noisy data X(t) = x(t) + n(t), where n(t)
represents the noise component drawn from some probability
distribution, one performs fourier transform to identify the
frequencies contained in x(t) by examining the power distribution.
What if X(t) = n(t) so the input is e.g. normally distributed data
n(t) ~ N(mu,sigma).
Thanks in advance,
James
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| User: "Ron Baker, Pluralitas!" |
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| Title: Re: Frequency retrieval from noise |
15 Apr 2007 01:43:29 PM |
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<dkjk@bigpond.net.au> wrote in message
news:1176593369.329996.190210@b75g2000hsg.googlegroups.com...
Hi group,
I tried this one on sci.math but nobody was able to offer an answer.
Is there any meaningful way to assign a frequency to a given
probability distribution?
No. A signal or noise with a certain probability distribution
can have nearly any power spectral density (PSD).
E.g. lowpass filter white Gaussian noise. Both the input and output
values are Gaussian distributed but have different PSDs.
Typically when one obtains noisy data X(t) = x(t) + n(t), where n(t)
represents the noise component drawn from some probability
distribution, one performs fourier transform to identify the
frequencies contained in x(t) by examining the power distribution.
What if X(t) = n(t) so the input is e.g. normally distributed data
n(t) ~ N(mu,sigma).
Thanks in advance,
James
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| User: "Androcles" |
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| Title: Re: Frequency retrieval from noise |
15 Apr 2007 02:13:31 AM |
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<dkjk@bigpond.net.au> wrote in message =
news:1176593369.329996.190210@b75g2000hsg.googlegroups.com...
Hi group,
=20
I tried this one on sci.math but nobody was able to offer an answer.
=20
Is there any meaningful way to assign a frequency to a given
probability distribution?
=20
Typically when one obtains noisy data X(t) =3D x(t) + n(t), where n(t)
represents the noise component drawn from some probability
distribution, one performs fourier transform to identify the
frequencies contained in x(t) by examining the power distribution.
=20
What if X(t) =3D n(t) so the input is e.g. normally distributed data
n(t) ~ N(mu,sigma).
=20
Thanks in advance,
If X(t) =3D x(t) + n(t) and X(t) =3D n(t) then x(t) =3D X(t)-n(t) =3D 0.
If you found the Fourier transform of noise and subtracted
that from the signal you'd be no better off than if you found
X(t) and discounted the higher frequencies. A low frequency
noise is indistinguishable from signal anyway, for example
the hum (unwanted noise) that comes from a 50Hz or 60Hz supply=20
can be filtered out but that would also filter out a 50Hz or 60Hz=20
signal, so best not to let it get into the amplifier in the first place.
White noise, as in snow on a screen, is all frequencies. Pink noise
has a higher amplitude at lower frequencies but is still noise.
http://hyperphysics.phy-astr.gsu.edu/hbase/audio/tape5.html
Since then sound has gone digital, of course, and CD-ROMS,
MP3 players and such are virtually noise free.
If an amplifier amplifies noise it is doing its job, if it introduces
noise then that is unwanted.=20
Depending on what you want, noise is a rather like a weed.
A weed is a plant a gardener doesn't want. Poppies are
weeds in Britain because they are plentiful and flowers in the
USA because they are not.=20
=20
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| User: "Randy Poe" |
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| Title: Re: Frequency retrieval from noise |
14 Apr 2007 08:30:30 PM |
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On Apr 14, 7:29 pm, wrote:
Hi group,
I tried this one on sci.math but nobody was able to offer an answer.
Is there any meaningful way to assign a frequency to a given
probability distribution?
Typically when one obtains noisy data X(t) = x(t) + n(t), where n(t)
represents the noise component drawn from some probability
distribution, one performs fourier transform to identify the
frequencies contained in x(t) by examining the power distribution.
What if X(t) = n(t) so the input is e.g. normally distributed data
n(t) ~ N(mu,sigma).
You mean, what if the data is pure gaussian noise?
What about it?
The typical model for additive noise is AWGN, additive white gaussian
noise, which means that the the spectrum of the noise is flat.
Same energy at all frequencies.
Not sure what you mean by "assign a frequency". Most signals,
like noise, contain a range of frequencies.
- Randy
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| User: "" |
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| Title: Re: Frequency retrieval from noise |
14 Apr 2007 09:06:14 PM |
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"Randy Poe" wrote in message
news:<1176600630.309801.211320@w1g2000hsg.googlegroups.com>...
On Apr 14, 7:29 pm, wrote:
Hi group,
I tried this one on sci.math but nobody was able to offer an answer.
Is there any meaningful way to assign a frequency to a given
probability distribution?
Typically when one obtains noisy data X(t) = x(t) + n(t), where n(t)
represents the noise component drawn from some probability
distribution, one performs fourier transform to identify the
frequencies contained in x(t) by examining the power distribution.
What if X(t) = n(t) so the input is e.g. normally distributed data
n(t) ~ N(mu,sigma).
You mean, what if the data is pure gaussian noise?
What about it?
The typical model for additive noise is AWGN, additive white gaussian
noise, which means that the the spectrum of the noise is flat.
Same energy at all frequencies.
Not sure what you mean by "assign a frequency". Most signals,
like noise, contain a range of frequencies.
- Randy
Hi Randy,
Thanks for your response.
Yes, I'm asking what are the peaks in the power spectrum of pure
gaussian noise, or noise pulled from any distribution for that matter.
For a uniform distribution I would expect that the power spectrum is
flat. I was kind of hoping that other distributions (such as gaussian)
may reveal additional structure, but you claim it is still flat.
Does this result hold for _any_ probability distribution? Is there a
simple argument why this should always be the case?
Regards,
James
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| User: "Randy Poe" |
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| Title: Re: Frequency retrieval from noise |
15 Apr 2007 11:05:34 AM |
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On Apr 14, 10:06 pm, wrote:
"Randy Poe" wrote in message
<news:1176600630.309801.211320@w1g2000hsg.googlegroups.com>...
On Apr 14, 7:29 pm, wrote:
Hi group,
I tried this one on sci.math but nobody was able to offer an answer.
Is there any meaningful way to assign a frequency to a given
probability distribution?
Typically when one obtains noisy data X(t) = x(t) + n(t), where n(t)
represents the noise component drawn from some probability
distribution, one performs fourier transform to identify the
frequencies contained in x(t) by examining the power distribution.
What if X(t) = n(t) so the input is e.g. normally distributed data
n(t) ~ N(mu,sigma).
You mean, what if the data is pure gaussian noise?
What about it?
The typical model for additive noise is AWGN, additive white gaussian
noise, which means that the the spectrum of the noise is flat.
Same energy at all frequencies.
Not sure what you mean by "assign a frequency". Most signals,
like noise, contain a range of frequencies.
- Randy
Hi Randy,
Thanks for your response.
Yes, I'm asking what are the peaks in the power spectrum of pure
gaussian noise, or noise pulled from any distribution for that matter.
Your question still is not very meaningful, but I'll provide a little
more info and see if it helps.
AWGN is gaussian in amplitude distribution, white in power
spectrum. Those are independent parameters. Noise can also be
"pink", "blue", "brown" or have other power spectra. These names
don't refer to actual colors, but to whether the frequency content
is stronger at lower or higher frequency.
In any given sample of noise, the "peaks" are all but meaningless.
Take a look at the sample of white noise here for instance:
http://en.wikipedia.org/wiki/Colors_of_noise
There are lots of small peaks across the the spectrum. They're
just random. Five seconds later they'd be in different places.
For a uniform distribution I would expect that the power spectrum is
flat.
You've got to start by being clear on whether your distribution
is in the time or frequency domain. Do you mean a uniform
amplitude distribution? I'm not so sure. It seems to me that
a sawtooth spends exactly as much time at every amplitude
level, and so its amplitude distribution is uniform. But I sure
wouldn't expect the spectrum of a sawtooth to be flat.
I was kind of hoping that other distributions (such as gaussian)
may reveal additional structure, but you claim it is still flat.
I claim that WHITE gaussian noise has a WHITE spectrum, but
that other kinds of noise don't necessarily have a white spectrum.
The truth is there is no simple relationship such as you are looking
for.
Does this result hold for _any_ probability distribution?
What result?
The point is there is no fixed relationship between amplitude
distribution and power spectrum.
Is there a
simple argument why this should always be the case?
You haven't said what "this" is. One reason why there is no
fixed relationship is that there are two other degrees of freedom:
(a) in the time domain, two different signals could have the same
distribution but totally different correlation functions. As I
mentioned,
the sequence 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, ... has a
uniform distribution on 1-5. And
(b) In the frequency domain, two signals can have identical
power spectra but totally different phase structure, which means
their time-domain signals are completely different.
- Randy
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| User: "" |
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| Title: Re: Frequency retrieval from noise |
15 Apr 2007 12:26:46 AM |
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The typical model for additive noise is AWGN, additive white gaussian
noise, which means that the the spectrum of the noise is flat.
Same energy at all frequencies.
Wouldn't that lead to an infra-red divergence paradox?
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| User: "Randy Poe" |
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| Title: Re: Frequency retrieval from noise |
15 Apr 2007 11:08:27 AM |
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On Apr 15, 1:26 am, wrote:
The typical model for additive noise is AWGN, additive white gaussian
noise, which means that the the spectrum of the noise is flat.
Same energy at all frequencies.
Wouldn't that lead to an infra-red divergence paradox?
It leads to various infinite-energy problems, which is why it's only
a
theoretical ideal. An easy fix is to have band-limited white noise,
which is white across all frequencies of interest, but not from DC
to light. You're only ever interested in the part of the noise that
makes it into your receiver anyway (so it is band-limited by your
antenna and receiver filter).
I used to study lightning-induced white noise. It's white across a
pretty broad chunk of the radio spectrum.
- Randy
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