# PDF of Quantization Error

Started by December 31, 2004
```hschauhan wrote:

> 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.
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```
```"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.

Clay

```
```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
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Jerry Avins wrote:

> 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
```
```Jerry Avins wrote:
> 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.
```
```Stan Pawlukiewicz wrote:
> Jerry Avins wrote:
>
>> 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.

I didn't say it behaves like Gaussian; very few limited sets of data do.
Since departures from ideal Gaussian are usual for actual observations,
most of us develop a great deal of estimation tolerance for just about
any departures other than pronounced skew or bimodalism. It's not
unusual for Cauchy distributions to fall within that broad tolerance
when looked at casually.

I thought that's what I said. Sorry.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```