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autocorrelation - HELP!

Started by aron March 12, 2005
Hi!
I am a student who is starting to go over autocorrelation. I am trying
to understand this concept.
I have read several posts here concerning this topic but I cannot find
the answer for my problem.

I have to calculate the autocorrelation of the signal at the receiver 
which is the sum of transmitted signal s(t) and a noise n(t) i.e.
r(t)=x(tau)+n(tau). x(t) and n(t) are statistically independent random
proc. with zero expecd values and autocorrelations respectively :
Rx(tau)=2exp(-|tau|) and Rn(tau)=exp(-|tau|).

Can I solve it as follows:

1)Knowing that                                 

                        +N/2 
Q[k] = lim  1/(N-1) *   SUM { (x[n-k] - x[n])^2 }
       N->inf           n=-N/2

where Q[k] i a "similarity" meassure

I calculate it for signal x(tau) -> further since for both are finite
energy signals (like a single pulse or exponential)

                     N/2           N/2               N/2
Q[k] = lim        { SUM{x[n]^2} + SUM{x[n-k]^2} - 2*SUM{x[n]*x[n-k]} }
       N->inf        n=-N/2       n=-N/2            n=-N/2



and if N>>k

                     N/2           N/2
 Q[k] = lim       2*{ SUM{x[n]^2} - SUM{x[n]*x[n-k]} }
       N->inf        n=-N/2        n=-N/2


comeing to:


                      N/2           N/2
 Q[k] = lim       2*{ SUM{x[n]^2} - SUM{x[n]*x[n-k]} }
       N->inf        n=-N/2        n=-N/2


and finally:

Q[k] = 2 * { E{x[n]^2} - Rx[k] }

and since expectations are 0's then I get Q[k]=-2Rx[k] - which should
be nonnegative!!!

What about the noise? How does it influence the final result?

How to do it? Can someone help?

Thank You in advance for prompt response and hints!

aron
aron wrote:

> Hi! > I am a student who is starting to go over autocorrelation. I am trying > to understand this concept. > I have read several posts here concerning this topic but I cannot find > the answer for my problem. > > I have to calculate the autocorrelation of the signal at the receiver > which is the sum of transmitted signal s(t) and a noise n(t) i.e. > r(t)=x(tau)+n(tau).
Don't you mean r(t) = x(t) + n(t) ?
> x(t) and n(t) are statistically independent random > proc. with zero expecd values and autocorrelations respectively : > Rx(tau)=2exp(-|tau|) and Rn(tau)=exp(-|tau|). > > Can I solve it as follows: > > 1)Knowing that > > +N/2 > Q[k] = lim 1/(N-1) * SUM { (x[n-k] - x[n])^2 } > N->inf n=-N/2
-snip-
> and since expectations are 0's then I get Q[k]=-2Rx[k] - which should > be nonnegative!!! >
Apparently not, since it comes up with the wrong answer -- and you don't appear to have used the noise signal at all. Furthermore you appear to have switched from continuous time (t) to discrete time (k) somewhere in the middle without justification.
> What about the noise? How does it influence the final result?
I believe that's what you are supposed to be discovering.
> > How to do it? Can someone help? >
This should be quite easy if you start with the definition of the autocorrelation function, plug in your expression for x(t) and n(t) into it, and turn the crank. Examine the expanded integrand for familiar looking terms, and see if you can't reduce the thing down to some integrals that you can then name and dispense with individually. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com