Doubt regarding PDF

Hi All

I have some doubt regarding an equation in the following paper:

http://www.lnt.e-technik.tu-muenchen.de/mitarbeiter/goertz/globecom03.pdf

I believe in eqn. (3), the scaling factor should be 1/(2*pi*sigma^2)
instead of 1/(sqrt(2*pi)*sigma). 

Because the authors say that each real dimension of the complex noise has
variance sigma^2.

However, in section 2 just above eqn. (1), they refer to a paper [5] which
says it the way I expect.

However, the as the authors are well known, what is your opinion on this.

Thanks

Chintan

On 14 Okt, 10:37, "cpshah99" <cpsha...@rediffmail.com> wrote:
> Hi All
>
> I have some doubt regarding an equation in the following paper:
>
> http://www.lnt.e-technik.tu-muenchen.de/mitarbeiter/goertz/globecom03...
>
> I believe in eqn. (3), the scaling factor should be 1/(2*pi*sigma^2)
> instead of 1/(sqrt(2*pi)*sigma).
>
> Because the authors say that each real dimension of the complex noise has
> variance sigma^2.
>
> However, in section 2 just above eqn. (1), they refer to a paper [5] which
> says it the way I expect.
>
> However, the as the authors are well known, what is your opinion on this.

First, the philosophical answer:

Maths doesn't care about human authority figures. If you can prove
that a stated mathematical result is wrong, then the stated result
is wrong (assuming of course, that your argument still stands when
scrutinized).

Several years ago I heard an anecdote about a 13-year-old who
came up with an idea about some maths or physics question. His
statements took the professional community by surprise. When they
investigated, they found that the kid was right, so the effect
was named after him.

I don't remember exactly what the topic was, but it *might* have
been something to do with that hot water freezes over faster
than cold water. The explanation was, apparently, that hot water
looses heat through steam in addition to radiation and heat
transport to the surroundings, so it looses a lot more heat a
lot faster than water that doesn't steam. Something like that.

On a more pragmatic level:

There might be several reasons why the stated equation is not
as expected:

1) People make mistakes.
2) Authors do not keep copyrights of published papers.

The paper you refer to is clearly a draft of some journal paper.
The authors can not provide free links to their published paper,
because this would mean they violate the copyrights that they
have signed over to the publishers of the journal. This is a
conflict of interest between the authors and the publishers:

- The publishers want people to pay to get the articles
- The authors have no economical interests in people paying
  for the access (authors don't get any shares of the income),
  but want as many people as possible to read the article

However, authors are free to publish *drafts* of the same papers;
there are no copyrights to be violated on drafts.

So how is a 'draft' different from the 'finished article'?

Well, one difference is that drafts still contain errors and
blunders that will be ironed out during the publishing process.
You have spotted one very obvious such blunder, that will have
been corrected in the published paper.

Rune

>So how is a 'draft' different from the 'finished article'?
>
>Well, one difference is that drafts still contain errors and
>blunders that will be ironed out during the publishing process.
>You have spotted one very obvious such blunder, that will have
>been corrected in the published paper.
>
>Rune
>

HI Rune

Thanks.

The finished version has the same equation thats why I was confused but
now I feel that the authors are wrong.

I just wanted to know if I am not making any mistake(s).

Chintan

On Oct 14, 8:48&#4294967295;am, "cpshah99" <cpsha...@rediffmail.com> wrote:
> >So how is a 'draft' different from the 'finished article'?
>
> >Well, one difference is that drafts still contain errors and
> >blunders that will be ironed out during the publishing process.
> >You have spotted one very obvious such blunder, that will have
> >been corrected in the published paper.
>
> >Rune
>
> HI Rune
>
> Thanks.
>
> The finished version has the same equation thats why I was confused but
> now I feel that the authors are wrong.
>
> I just wanted to know if I am not making any mistake(s).
>
> Chintan

Have you tried to integrate the density function over the valid
domain? It should be equal to 1.

Clay

On Wed, 14 Oct 2009 07:48:33 -0500, cpshah99 wrote:

>>So how is a 'draft' different from the 'finished article'?
>>
>>Well, one difference is that drafts still contain errors and blunders
>>that will be ironed out during the publishing process. You have spotted
>>one very obvious such blunder, that will have been corrected in the
>>published paper.
>>
>>Rune
>>
>>
> HI Rune
> 
> Thanks.
> 
> The finished version has the same equation thats why I was confused but
> now I feel that the authors are wrong.
> 
> I just wanted to know if I am not making any mistake(s).
> 
> Chintan

If you can integrate the PDF over the entire range of the random value, 
it should always ever come up to the probability that there was _some_ 
outcome, i.e. 1.

So if the equation is at all tractable you should be able to check easily.

-- 
www.wescottdesign.com

Ok. Lets try this way:

y=x+n, where x is complex data symbol and n is complex noise.

Now, n=n_I+j*n_Q; where both n_I and n_Q has variance sigma^2 and zero
mean.

As n_I and n_Q are independent and uncorrelated with pdf 

p(n_I)=1/sqrt(2*pi*sigma^2) * (e^(-n_I^2/(2*sigma^2))) and
p(n_Q)=1/sqrt(2*pi*sigma^2) * (e^(-n_Q^2/(2*sigma^2))).

So the pdf of n is:

p(n)=p(n_I)*p(n_Q)=1/(2*pi*sigma^2) e^(-|n|^2/(2*sigma^2))

And hence,

p(y/x)=1/(2*pi*sigma^2) e^(-|y-x|^2/(2*sigma^2))

I guess this is correct.

Please point out any mistakes, if you find.

Thanks

Chintan

>
>Ok. Lets try this way:
>
>y=x+n, where x is complex data symbol and n is complex noise.
>
>Now, n=n_I+j*n_Q; where both n_I and n_Q has variance sigma^2 and zero
>mean.
>
>As n_I and n_Q are independent and uncorrelated with pdf 
>
>p(n_I)=1/sqrt(2*pi*sigma^2) * (e^(-n_I^2/(2*sigma^2))) and
>p(n_Q)=1/sqrt(2*pi*sigma^2) * (e^(-n_Q^2/(2*sigma^2))).
>
>So the pdf of n is:
>
>p(n)=p(n_I)*p(n_Q)=1/(2*pi*sigma^2) e^(-|n|^2/(2*sigma^2))
>
>And hence,
>
>p(y/x)=1/(2*pi*sigma^2) e^(-|y-x|^2/(2*sigma^2))
>
>I guess this is correct.
>
>Please point out any mistakes, if you find.
>

I don't see anything obviously wrong with your approach to a complex
Gaussian RV with IID real/imag parts, but I have not studied their paper. 
Have you considered just emailing the authors?  You can always choose your
words carefully if you're uncertain, but I would not expect them to have to
think much to respond.

On Oct 14, 3:37=A0am, "cpshah99" <cpsha...@rediffmail.com> wrote:
> Hi All
>
> I have some doubt regarding an equation in the following paper:
>
> http://www.lnt.e-technik.tu-muenchen.de/mitarbeiter/goertz/globecom03...
>
> I believe in eqn. (3), the scaling factor should be 1/(2*pi*sigma^2)
> instead of 1/(sqrt(2*pi)*sigma).
>
> Because the authors say that each real dimension of the complex noise has
> variance sigma^2.
>
> However, in section 2 just above eqn. (1), they refer to a paper [5] whic=
h
> says it the way I expect.
>
> However, the as the authors are well known, what is your opinion on this.
>
> Thanks
>
> Chintan

I have not seen the paper being referenced, but I have always seen the
Gaussian density with 1/(sqrt(2*pi)*sigma). as the multiplier.

>On Oct 14, 3:37=A0am, "cpshah99" <cpsha...@rediffmail.com> wrote:
>> Hi All
>>
>> I have some doubt regarding an equation in the following paper:
>>
>>
http://www.lnt.e-technik.tu-muenchen.de/mitarbeiter/goertz/globecom03...
>>
>> I believe in eqn. (3), the scaling factor should be 1/(2*pi*sigma^2)
>> instead of 1/(sqrt(2*pi)*sigma).
>>
>> Because the authors say that each real dimension of the complex noise
has
>> variance sigma^2.
>>
>> However, in section 2 just above eqn. (1), they refer to a paper [5]
whic=
>h
>> says it the way I expect.
>>
>> However, the as the authors are well known, what is your opinion on
this.
>>
>> Thanks
>>
>> Chintan
>
>I have not seen the paper being referenced, but I have always seen the
>Gaussian density with 1/(sqrt(2*pi)*sigma). as the multiplier.
>

For complex Gaussian case????? That must have been for PAM modulation or
BPSK.

Chintan

On Oct 14, 10:37=A0am, "cpshah99" <cpsha...@rediffmail.com> wrote:
> Hi All
>
> I have some doubt regarding an equation in the following paper:
>
> http://www.lnt.e-technik.tu-muenchen.de/mitarbeiter/goertz/globecom03...
>
> I believe in eqn. (3), the scaling factor should be 1/(2*pi*sigma^2)
> instead of 1/(sqrt(2*pi)*sigma).
>
> Because the authors say that each real dimension of the complex noise has
> variance sigma^2.
>
> However, in section 2 just above eqn. (1), they refer to a paper [5] whic=
h
> says it the way I expect.
>
> However, the as the authors are well known, what is your opinion on this.

They made a mistake.

illywhacker;