| Author |
Message |
hschauhan
Guest
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 Posted:
Sat Jan 01, 2005 12:05 am Post subject:
PDF of Quantization Error |
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Hello!
I am newbee in DSP and am reading the book by Steven Smith. In chapter
three,
the PDF of quantization error is not a gaussian. And I am wondering
why?
The errors are random then why isn't it a gaussian?
Can somebody help me on this?
Thanx in advance
Regards
--Himanshu |
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Jerry Avins
Guest
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| Posted:
Sat Jan 01, 2005 12:17 am Post subject:
Re: PDF of Quantization Error |
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hschauhan wrote:
| Quote: | Hello!
I am newbee in DSP and am reading the book by Steven Smith. In chapter
three,
the PDF of quantization error is not a gaussian. And I am wondering
why?
The errors are random then why isn't it a gaussian?
Can somebody help me on this?
Thanx in advance
|
What makes you believe that ever random distribution is Gaussian? That's
clearly not even approximately true. Most simple pseudo-random number
generators have uniform distributions, as does an honest roulette wheel.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Tim Wescott
Guest
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| Posted:
Sat Jan 01, 2005 12:19 am Post subject:
Re: PDF of Quantization Error |
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hschauhan wrote:
| Quote: | Hello!
I am newbee in DSP and am reading the book by Steven Smith. In chapter
three,
the PDF of quantization error is not a gaussian. And I am wondering
why?
The errors are random then why isn't it a gaussian?
Can somebody help me on this?
Thanx in advance
Regards
--Himanshu
If I take a coin and assign a value of 1 to heads and -1 to tails, and |
flip the coin, the PDF of the resulting value is not Gaussian. Why?
The coin flips are random, why isn't the PDF a Gaussian?
If I take the instantaneous power of a random process with a zero-mean
Gaussian distribution the resulting PDF is Rayleigh, not Gaussian. Why?
The instantaneous power is random, why isn't it's PDF Gaussian?
If I look at the PDF of the values of a low-light detector that only
collects a few photons each sampling period it's PDF is not Gaussian.
Why? The process is random (it's a Poisson, by the way), why isn't it's
PDF Gaussian?
Answer these questions, and you will have a very good start on answering
yours.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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hschauhan
Guest
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| Posted:
Sat Jan 01, 2005 12:40 am Post subject:
Re: PDF of Quantization Error |
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Hi!
I think it is because of the uniform distribution of the QEs. It is not
gaussian
because the probablity of occurence is same.
Am I true?
Regards
--Himanshu |
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Tim Wescott
Guest
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| Posted:
Sat Jan 01, 2005 1:06 am Post subject:
Re: PDF of Quantization Error |
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hschauhan wrote:
| Quote: | Hi!
I think it is because of the uniform distribution of the QEs. It is not
gaussian
because the probablity of occurence is same.
Am I true?
Regards
--Himanshu
Yes, that is correct. |
Jerry's point (and mine, for that matter), was that you have no cause to
be surprised that a particular PDF is not Gaussian. In fact, I suspect
that there are very few, if any, underlying processes that actually have
PDF's that are Gaussian if you dig deep enough.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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Jerry Avins
Guest
|
| Posted:
Sat Jan 01, 2005 2:15 am Post subject:
Re: PDF of Quantization Error |
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Tim Wescott wrote:
| Quote: | hschauhan wrote:
Hello!
I am newbee in DSP and am reading the book by Steven Smith. In chapter
three,
the PDF of quantization error is not a gaussian. And I am wondering
why?
The errors are random then why isn't it a gaussian?
Can somebody help me on this?
Thanx in advance
Regards
--Himanshu
If I take a coin and assign a value of 1 to heads and -1 to tails, and
flip the coin, the PDF of the resulting value is not Gaussian. Why? The
coin flips are random, why isn't the PDF a Gaussian?
If I take the instantaneous power of a random process with a zero-mean
Gaussian distribution the resulting PDF is Rayleigh, not Gaussian. Why?
The instantaneous power is random, why isn't it's PDF Gaussian?
If I look at the PDF of the values of a low-light detector that only
collects a few photons each sampling period it's PDF is not Gaussian.
Why? The process is random (it's a Poisson, by the way), why isn't it's
PDF Gaussian?
Answer these questions, and you will have a very good start on answering
yours.
|
Just as Gaussian is the limiting case of binomial, Rayleigh is the
limiting case of Poisson. That insight has served me well a few times.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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maTheMatic
Guest
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| Posted:
Sat Jan 01, 2005 5:42 am Post subject:
Re: PDF of Quantization Error |
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The distribution of the QE is related the distribution of the input
signal. But we can conclude that this distribution must not be Gasssian
strictly because the QE can't reach the infinity while Gassian random
variable will do.
Gassian is regular is part becasue the central limited theorem. But
here if the input signal is uniform distribution, I can't see any
reasion that the output QE is Gaussian distribution.
and Yes ,as had beed said as above, it helps that you think about the
physical meaning of some mathematic assumption. |
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Country_Chiel
Guest
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| Posted:
Sat Jan 01, 2005 7:57 am Post subject:
Re: PDF of Quantization Error |
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"Jerry Avins" <jya@ieee.org> wrote in message
news:33lmu9F40e1dsU1@individual.net...
| Quote: | hschauhan wrote:
Hello!
I am newbee in DSP and am reading the book by Steven Smith. In chapter
three,
the PDF of quantization error is not a gaussian. And I am wondering
why?
The errors are random then why isn't it a gaussian?
Can somebody help me on this?
Thanx in advance
What makes you believe that ever random distribution is Gaussian? That's
clearly not even approximately true. Most simple pseudo-random number
generators have uniform distributions, as does an honest roulette wheel.
Jerry
--
Engineering is the art of making what you want from things you can get.
Ż
|
If you use the central limit theorem everything is Guassian eventually! |
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hschauhan
Guest
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| Posted:
Sat Jan 01, 2005 4:20 pm Post subject:
Re: PDF of Quantization Error |
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Hi!!
I also think that more the random components, more is the Gaussian
nature.
Because random ness of all forces change the distribution from uniform
to normal one.
is that right?
Regards
--Himanshu |
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Jerry Avins
Guest
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| Posted:
Sat Jan 01, 2005 8:35 pm Post subject:
Re: PDF of Quantization Error |
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|
Country_Chiel wrote:
| Quote: | "Jerry Avins" <jya@ieee.org> wrote in message
|
...
| Quote: | What makes you believe that ever random distribution is Gaussian? That's
clearly not even approximately true. Most simple pseudo-random number
generators have uniform distributions, as does an honest roulette wheel.
If you use the central limit theorem everything is Guassian eventually!
|
Most (but not all) distributions become Gaussian when enough instances
are summed. No matter how long you wait for the typical PRNG to exhibit
a Gaussian distribution, your "eventually" will never come.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Jerry Avins
Guest
|
| Posted:
Sat Jan 01, 2005 8:40 pm Post subject:
Re: PDF of Quantization Error |
|
|
hschauhan wrote:
| Quote: | Hi!!
I also think that more the random components, more is the Gaussian
nature.
Because random ness of all forces change the distribution from uniform
to normal one.
is that right?
Regards
--Himanshu
|
That's true of the great majority of distributions. The distribution of
sum of two variables uniformly distributed over the same range is
triangular; of six, close enough to Gaussian for many uses; of twelve,
nearly indistinguishable from Gaussian by most tests.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Clay S. Turner
Guest
|
| Posted:
Sat Jan 01, 2005 8:43 pm Post subject:
Re: PDF of Quantization Error |
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"Country_Chiel" <Chiel@bothy.nichts.co.uk> wrote in message
news:1104565931.539467@ftpsrv1...
| Quote: |
If you use the central limit theorem everything is Guassian eventually!
|
Actually no.
You can add Cauchy random vars together to your heart's content, and the
answer will always be a Cauchy random varible. This distribution is also
known as a Lorentzian distribution which shows up in cases of resonance.
I.e. the amplitude verses frequency function for a emission line produced by
an atom as an electron moves from a high to a low energy level.
Clay |
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Jerry Avins
Guest
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| Posted:
Sat Jan 01, 2005 9:30 pm Post subject:
Re: PDF of Quantization Error |
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Clay S. Turner wrote:
| Quote: | "Country_Chiel" <Chiel@bothy.nichts.co.uk> wrote in message
news:1104565931.539467@ftpsrv1...
If you use the central limit theorem everything is Guassian eventually!
Actually no.
You can add Cauchy random vars together to your heart's content, and the
answer will always be a Cauchy random varible. This distribution is also
known as a Lorentzian distribution which shows up in cases of resonance.
I.e. the amplitude verses frequency function for a emission line produced by
an atom as an electron moves from a high to a low energy level.
|
The shape of the PDF is close enough to Gaussian so the difference falls
within the tolerance we normally allow when matching practice to theory.
Because of that, the exception is easily overlooked. Peruse
http://www.math.hope.edu/~tanis/Phoenix/Cauchy1.html
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Tim Wescott
Guest
|
| Posted:
Sat Jan 01, 2005 11:43 pm Post subject:
Re: PDF of Quantization Error |
|
|
Jerry Avins wrote:
| Quote: | Clay S. Turner wrote:
"Country_Chiel" <Chiel@bothy.nichts.co.uk> wrote in message
news:1104565931.539467@ftpsrv1...
If you use the central limit theorem everything is Guassian eventually!
Actually no.
You can add Cauchy random vars together to your heart's content, and the
answer will always be a Cauchy random varible. This distribution is also
known as a Lorentzian distribution which shows up in cases of resonance.
I.e. the amplitude verses frequency function for a emission line produced by
an atom as an electron moves from a high to a low energy level.
The shape of the PDF is close enough to Gaussian so the difference falls
within the tolerance we normally allow when matching practice to theory.
Because of that, the exception is easily overlooked. Peruse
http://www.math.hope.edu/~tanis/Phoenix/Cauchy1.html
Jerry
|
The PDF of atmospheric noise being received by an LF or MF (300kHz or
so) radio receiver is Cauchy-like in that it has an infinite variance,
and it cannot be approximated well enough by a Gaussian to approach an
optimal demodulator for PSK-encoded digital. You _must_ take the
probability distribution into account when making a decoder. As of
1990, when I last paid attention to it, a demonstrably optimal decoder
has not been found, only decoders that significantly outperformed linear
decoders based on the assumption of Gaussian noise.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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Stan Pawlukiewicz
Guest
|
| Posted:
Mon Jan 03, 2005 8:03 pm Post subject:
Re: PDF of Quantization Error |
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Jerry Avins wrote:
| Quote: | Clay S. Turner wrote:
"Country_Chiel" <Chiel@bothy.nichts.co.uk> wrote in message
news:1104565931.539467@ftpsrv1...
If you use the central limit theorem everything is Guassian eventually!
Actually no.
You can add Cauchy random vars together to your heart's content, and the
answer will always be a Cauchy random varible. This distribution is also
known as a Lorentzian distribution which shows up in cases of resonance.
I.e. the amplitude verses frequency function for a emission line produced by
an atom as an electron moves from a high to a low energy level.
The shape of the PDF is close enough to Gaussian so the difference falls
within the tolerance we normally allow when matching practice to theory.
Because of that, the exception is easily overlooked. Peruse
http://www.math.hope.edu/~tanis/Phoenix/Cauchy1.html
Jerry
|
I don't agree Jerry. You have to clip observations to generate
histogram. It really doesn't behave like a Gaussian. |
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