# PDF of Quantization Error

Started by December 31, 2004
```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?

Regards
--Himanshu

```
```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.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```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.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
```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

```
```hschauhan wrote:

> 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
```
```Tim Wescott wrote:

> 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.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```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.

```
```"Jerry Avins" <jya@ieee.org> wrote in message
news:33lmu9F40e1dsU1@individual.net...
> 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.
> &#4294967295;

If you use  the central limit theorem everything is Guassian eventually!

```
```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

```
```Country_Chiel wrote:

> "Jerry Avins" <jya@ieee.org> wrote in message

...

>>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.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```