Distrubution of noise times noise

If I have two signals that are corrupted with Gaussian random noise and
I multiply them, what is the noise distribution called?

x1 = s+n1
x2 = s+n2
x1*x2 = s*s+(n1+n2)*s+n1*n2

Is noise times noise (n1*n2) the focus of any study? I would be
interested to read about it.

Thanks.

Hi,

On 30 nov, 05:40, "spasmous" <spasm...@gmail.com> wrote:
> If I have two signals that are corrupted with Gaussian random noise and
> I multiply them, what is the noise distribution called?
>
> x1 = s+n1
> x2 = s+n2
> x1*x2 = s*s+(n1+n2)*s+n1*n2
>
> Is noise times noise (n1*n2) the focus of any study? I would be
> interested to read about it.

As n1 and n2 are independent random gaussian noise, their product 's
distribution is a Bessel distribution.

I only found an old french article about it :
http://archive.numdam.org/ARCHIVE/RSA/RSA_1968__16_4/RSA_1968__16_4_65_0/RSA_1968__16_4_65_0.pdf

> 
> Thanks.

Hope this helps you,
Matthieu

Matthieu Puigt skrev:
> Hi,
>
> On 30 nov, 05:40, "spasmous" <spasm...@gmail.com> wrote:
> > If I have two signals that are corrupted with Gaussian random noise and
> > I multiply them, what is the noise distribution called?
> >
> > x1 = s+n1
> > x2 = s+n2
> > x1*x2 = s*s+(n1+n2)*s+n1*n2
> >
> > Is noise times noise (n1*n2) the focus of any study? I would be
> > interested to read about it.

I don't know. n1*n1 has been studied, though. Look for the
Rayleigh distribution.

> As n1 and n2 are independent random gaussian noise, their product 's
> distribution is a Bessel distribution.

Is that the same as the Rayleigh distribution? I know French
nomenclature can differ from English nomenclature. What is
termed "Snell's law" in English books is alledgedly termed
"Descarte's law" in the French literaure.

Rune

> > As n1 and n2 are independent random gaussian noise, their product 's
> > distribution is a Bessel distribution.
>
> Is that the same as the Rayleigh distribution? I know French
> nomenclature can differ from English nomenclature. What is
> termed "Snell's law" in English books is alledgedly termed
> "Descarte's law" in the French literaure.

They are different distributions. If n1 and n2 are independent and
Gaussian, then their product's PDF is:

f(z) = (1/(pi*sigma1*sigma2)) * Ko(abs(z)/(sigma1*sigma2)), where Ko(z)
is a modified Bessel function of the second kind. Pretty messy.

And, n1^2 does not have a Rayleigh distribution; a Rayleigh PDF arises
if you have two independent Gaussian random variables n1 and n2, and
form a new variable n3 = sqrt(n1^2+n2^2). In this case, n3 will have a
Rayleigh distribution. The n1^2 case can be thought of as giving a
random variable with a chi-square distribution with one degree of
freedom, or a Rayleigh random variable. For the one-degree-of-freedom
case, they only differ by a factor of a sqrt(2).

Some links:
http://en.wikipedia.org/wiki/Gaussian_distribution (where the above
function is specified, under "Properties")
http://en.wikipedia.org/wiki/Rayleigh_distribution

Jason

cincydsp@gmail.com wrote:
> ...If n1 and n2 are independent and
> Gaussian, then their product's PDF is:
>
> f(z) = (1/(pi*sigma1*sigma2)) * Ko(abs(z)/(sigma1*sigma2)), where Ko(z)
> is a modified Bessel function of the second kind. Pretty messy.
>

Ouch. That's not nice. But thank you for sharing your knowledge.

Do you know the "rules" for combining two signals with such a
distribution? Eg. would summing n1*n2 + n3*n4 (where all are Gaussian,
independent noise with the same variance) have the same distrubution as
n1*n2 but reduced... variance? I want to know the best way to combine
multiple signals that have this distribution.

> Do you know the "rules" for combining two signals with such a
> distribution? Eg. would summing n1*n2 + n3*n4 (where all are Gaussian,
> independent noise with the same variance) have the same distrubution as
> n1*n2 but reduced... variance? I want to know the best way to combine
> multiple signals that have this distribution.

That would really come down to the properties of the Bessel
distribution that results, as n1*n2 + n3*n4 would be the sum of two
such random variables. I don't have a good reference for that
particular distribution, so I'm not sure.

Jason

spasmous,

What you have written in the equations is the same signal corrupted by
two different gaussian noise signals and multiplied, which is different
than what is asked in the question  ("..two signals that are corrupted
..").

Then you ask about what the noise distibution of the product of the
noisy signals, which is not the (n1*n2) that you subsequently ask
about. The noise distribution of the product of the noisy signals has a
's' signal dependent term not included in (n1*n2).

So what is the real question?

Dirk

spasmous wrote:
> If I have two signals that are corrupted with Gaussian random noise and
> I multiply them, what is the noise distribution called?
>
> x1 = s+n1
> x2 = s+n2
> x1*x2 = s*s+(n1+n2)*s+n1*n2
>
> Is noise times noise (n1*n2) the focus of any study? I would be
> interested to read about it.
> 
> Thanks.

dbell wrote:

> spasmous,
> 
> What you have written in the equations is the same signal corrupted by
> two different gaussian noise signals and multiplied, which is different
> than what is asked in the question  ("..two signals that are corrupted
> ..").
> 
> Then you ask about what the noise distibution of the product of the
> noisy signals, which is not the (n1*n2) that you subsequently ask
> about. The noise distribution of the product of the noisy signals has a
> 's' signal dependent term not included in (n1*n2).
> 
> So what is the real question?

The real question is "Will you do my homework for me".

VLV

spasmous wrote:

> Ouch. That's not nice. But thank you for sharing your knowledge.
>
> Do you know the "rules" for combining two signals with such a
> distribution? Eg. would summing n1*n2 + n3*n4 (where all are Gaussian,
> independent noise with the same variance) have the same distrubution as
> n1*n2 but reduced... variance? I want to know the best way to combine
> multiple signals that have this distribution.

I'm not following what you want to get out of this.  Are you trying to
estimate s?  Are you trying to estimate some property of the noise?

What are your observations, and what is it that you want?  I have a
feeling that it will be a lot simpler if you rethink your problem
statement.

Julius

dbell wrote:
> spasmous,
>
> What you have written in the equations is the same signal corrupted by
> two different gaussian noise signals and multiplied, which is different
> than what is asked in the question  ("..two signals that are corrupted
> ..").
>
> Then you ask about what the noise distibution of the product of the
> noisy signals, which is not the (n1*n2) that you subsequently ask
> about. The noise distribution of the product of the noisy signals has a
> 's' signal dependent term not included in (n1*n2).
>
> So what is the real question?
>
> Dirk
>

Alright - I didn't want to be accused of asking you to do my "homework"
;) So I just asked a little snippet. Here is the full problem:

x1 = A*s + n1
x2 = B*s + n2
x3 = C*s + n3
etc.

y1 = A*t + n4
y2 = B*t + n5
y3 = C*t + n6
etc.

A,B,C are known and n is Gaussian, independent noise.

I want to find the best estimate of s*t. One way is to solve for s and
t then form s*t. This is what I'm currently doing.

Another way is to form products first, like x1*y1, x1*y2, etc, and add
them in some way. I was trying to think what the best way to do this
might be. Perhaps it is the minimizer of the least squares difference
between s*t and the noise corrupted estimate - I can't figure out what
this is. The use of least squares might not be appropriate for this
noise distribution either.

I'm just looking for hints - hopefully someone is interested in this
too or has done it before. Your comments much appreciated.