Correlation and Expected value

Hi,

I have a doubt wrt Correlation and Expected value:

1. In many signal processing applications, there is a usage that the input
signal and noise are uncorrelated. This leads to E(Signal.Noise) = 0, that
is, expected value of the product of signal and noise is taken as zero.
How?

It is not given that Signal and Noise are independent. I am only aware of
the property of independent random variables that E(XY) = E(X).E(Y) and
that Independent means Uncorrelated and NOT otherwise.

But how is it derived that E(XY) = 0 in the case of Uncorrelated X and Y
?????

Anyone who is knowledgeable on this concept, Please explain. 

Thanks,
Padmi

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aarthis wrote:

> Hi,
> 
> I have a doubt wrt Correlation and Expected value:
> 
> 1. In many signal processing applications, there is a usage that the input
> signal and noise are uncorrelated. This leads to E(Signal.Noise) = 0,

Vice versa.
If the correlation of signal and noise == 0, then they say that the 
signal and noise are uncorrelated.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

"aarthis" <aarthi.sagi@gmail.com> writes:

> Hi,
>
> I have a doubt wrt Correlation and Expected value:
>
> 1. In many signal processing applications, there is a usage that the input
> signal and noise are uncorrelated. This leads to E(Signal.Noise) = 0, that
> is, expected value of the product of signal and noise is taken as zero.
> How?
>
> It is not given that Signal and Noise are independent. I am only aware of
> the property of independent random variables that E(XY) = E(X).E(Y) and
> that Independent means Uncorrelated and NOT otherwise.
>
> But how is it derived that E(XY) = 0 in the case of Uncorrelated X and Y
> ?????

It isn't derived, because it isn't true in general. 

If X and Y are uncorrelated, then, by definition, E[XY] = E[X]E[Y]. So
if X and Y are uncorrelated, then the only way E[XY] = 0 is if E[X] = 0 or
E[Y] = 0 (since the reals are an integral domain).

Usually, in communications, the noise is assumed to be zero-mean, so if
X is any random signal and N is the zero-mean noise, then E[XN] = 0. 
-- 
%  Randy Yates                  % "Though you ride on the wheels of tomorrow,
%% Fuquay-Varina, NC            %  you still wander the fields of your
%%% 919-577-9882                %  sorrow."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO
http://home.earthlink.net/~yatescr

On May 30, 7:21 pm, "aarthis" <aarthi.s...@gmail.com> wrote:
> Hi,
>
> I have a doubt wrt Correlation and Expected value:
>
> 1. In many signal processing applications, there is a usage that the input
> signal and noise are uncorrelated. This leads to E{Signal*Noise} = 0,
> that is, expected value of the product of signal and noise is taken
> as zero.  How?
>
> It is not given that Signal and Noise are independent. I am only aware of
> the property of independent random variables that E{X*Y} = E{X}*E{Y}

that property is not the defining property of independent random
variables.  it the defining property of uncorrelated and a consequence
of independence.  the defining property of independent random
variables has to do with either their conditional probability
functions (like the p.d.f.) or their joint probability functions.

    p{XY} = p{X|Y}*p{Y} = p{Y|X}*p{X}

where p{XY} means "the probability that both X and Y occur" and p{X|Y}
means "the probability of X occuring given that Y has occured".

anyway, if X and Y are independent r.v.'s, then knowing that Y occured
tells you nothing additionally about how likely X will occur.  (and
vice-versa.)  that means

    p{X|Y} = p{X}

and that

    p{XY} = p{X} * p{Y}

(you can switch around X and Y and get the same result with the same X
and Y.)

> and that Independent means Uncorrelated and NOT otherwise.

that is true.

> But how is it derived that E{X*Y} = 0 in the case of Uncorrelated X and Y
> ?????

it is *defined* to be that E{X*Y} = E{X}*E{Y} means that X and Y are
uncorrelated (i am allowing for the possibility that X and/or Y have a
DC component or non-zero mean).  if the mean of either X or Y is zero
and they are uncorrelated, then E{X*Y} = 0.

now, maybe the question is: how is it that independent random
variables are always uncorrelated?  or what is a situation where two
r.v.'s are not correlated but still dependent (as a counter example to
any claim that these two conditions are the same)?

dunno but i speculate as to what the question is.

r b-j

Hi Yates, 
Thank you very much. Your reply cleared my doubts. It was very simple yet
I couldn't see it. I forgot that the noise is assumed to be of zero mean.

Thank you. 

_____________________________________
Do you know a company who employs DSP engineers?  
Is it already listed at http://dsprelated.com/employers.php ?