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Cross-Correlation Function of Band-limited Noise

Started by tk229 March 30, 2005
Hi

I'm looking at an instrument used in radio astronomy and hoped that you
guys might be able to clarify something for me...

1) What is the cross-correlation function of two sets independent sets of
band-limited white noise? 

2) Peyton & Peebles says that if the two sets of band-limited noise are
"even", Rxy(t) = 0 for all t  because X and Y are orthogonal. How can you
start talking about band-limited noise being even or odd? What's it going
to be in a real RF system?

            --------------------------------------
For reference, our system is 6-12GHz IF. I found an interesting plot in
Peyton & Peebles ("Probability, Random Variables and Random Signal
Principles", 3rd ed, p257):

It derives the cross correlation function of two band-limited function
centred about 

           w0 - W1  to  w0 + W2
and        -w0 -W1  to  -w0 -W2

Basically the bandwidth is  W = W1 + W2. If the two random processes are
even, W1 = W2 = W/2 (ie bands centred about +/-w0)  and the two processes
are orthogonal. In this case, the cross-correlation function is naturally
zero. But are band-limited noise even? How can you talk about noise being
even or odd?  W1 and W2 define the odd-ness of the noise.

They go on to show that the cross-correlation function is:

Rxy(t) = WP/pi  sinc(Wt/2) sin[(W-2W1)t / 2],


where t is the lag. If W1 = W/2 (even case), Rxy(t) = 0. But this result
is kind of interesting because the correlated signal that we get is:

Rss(t)  = W/pi  sinc(Wt) cos(Wt)

                ------------------------------------------

Many thanks in advance
tak



		
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Correction: 

>Peyton & Peebles ("Probability, Random Variables and Random Signal >Principles", 3rd ed, p257):
.....Should be Peyton Z Peebles -one person. On further thoughts, am I right in thinking that "a noise N(t) with an even spectrum" is equivalent to the cosine parts of the narrowband noise N(t) = A(t) [cos( Q(t) ) cos(w0 t) - sin( Q(t) ) sin(w0 t)] where the amplitude A(t) has a Reyleight prob distribution, the phase Q(t) has uniform prob distribution and w0 is the narrowband frequency? I still find it hard to believe how the cross correlation of two independent noise processes could have non-zero cross-correlation.... Any comments? tak This message was sent using the Comp.DSP web interface on www.DSPRelated.com
in article 4d2dnXsG5PTz1tHfRVn-pQ@giganews.com, tk229 at tk229@cam.ac.uk
wrote on 03/31/2005 14:11:

> Correction: > >> Peyton & Peebles ("Probability, Random Variables and Random Signal >> Principles", 3rd ed, p257): > > .....Should be Peyton Z Peebles -one person. > > > On further thoughts, am I right in thinking that "a noise N(t) with an > even spectrum" is equivalent to the cosine parts of the narrowband noise > > N(t) = A(t) [cos( Q(t) ) cos(w0 t) - sin( Q(t) ) sin(w0 t)] > > where the amplitude A(t) has a Reyleight prob distribution, the phase Q(t) > has uniform prob distribution and w0 is the narrowband frequency? > > > I still find it hard to believe how the cross correlation of two > independent noise processes could have non-zero cross-correlation.... > > > Any comments?
the cross-correlation of two independent random processes is the product of each mean. i'm pretty sure of that. sometimes a random process is input to a linear system and the output of that is cross-correlated to the input and that is not generally zero. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
>the cross-correlation of two independent random processes is the product
of
>each mean. i'm pretty sure of that.
Very good point! The cross-correlation func, Rxy(T) = E[ X(t)Y(t+T) ] can be written separately if X and Y are statistically independent and wide-sense stationary Rxy(T) = E[X(t)] E[Y(t+T)] = MEAN[X] MEAN[Y], as you say. Since X and Y have zero means, the cross-correlation Rxy(T) = 0. This is also the condition for X and Y to be orthogonal. Thanks for clearing that up! Now the question remains about the cross-correlation function of band-limited noise... I've made a little bit of pregress with this problem. Here's how far I've got: According to p. 176 of Peebles (3rd ed), it's possible to set an upper limit on the cross correlation function: ABS( Rxy(T) ) <= SQRT[ Rxx(0) Ryy(0) ] If X and Y have the same bandwidth and statistical properties, Rxx(0) = Ryy(0). If X and Y are band-limited noise, the autocorrelation Rxx(T) is a cos function with a sinc envelope. Rxx(T) peaks at T=0 so I guess at least this sets an upper limit. One of the other question I had was about how you can talk about noise being even or odd. Well, I think it was talking about the odd or evenness of the power density spectrum (PDS). The PDS is given by, p_xx(w) = Lim(T->inf) E[ ABS( X(w) )^2 ] / 2T This means that p_xx(w) must be real and even. This seems to suggest that the cross-correlation of band-limited noise Rxy(T), must be zero from the arguments summarised in the original posting. From the Wiener-Khinchin relation, the cross-power density spectrum p_xy(w) = 0. Does this conclusion seem reasonable?? tak This message was sent using the Comp.DSP web interface on www.DSPRelated.com
I'm still worried about the cross-correlation of band-limited noise and my
colleague suggested a reductio ad absurdum argument to show that there
must be some correlation at different delays:

Compare the cross-correlation at two delays Rxy(t1) and Rxy(t2), where X
and Y are band-limited noise. Rxy(t) is zero -that's true when integrated
for an infinite length of time. When it's integrated over a finite time,
Rxy(t) is a zero-mean random number. 


But what are the correlation between the cross-correlations Rxy(t1) and
Rxy(t2)?

When t1 and t2 are different, Rxy(t1) and Rxy(t2) are completely
independent random numbers  -as you'd expect.  If you now move t2 towards
t1, in the infinitesimal limit, Rxy(t1) and Rxy(t2 = t1 + dt) must
increasingly become dependent. 

The bandlimited noise X and Y are generated by convolving white Gaussian
noise sequences with the filter function h(t)  -a sinc function with a cos
carrier wave. In a hand-wavy kind of way, the random variable Rxy(t1) is
correlated with Rxy(t2) by the tail of h(t). When t2 = t1, the correlation
is 1 at the central peak of the sinc function. 

With White Gaussian Noise (WGN), there'll be no correlation between the
finite cross-correlation functions unless t1 = t2 because the filter
function of h(t) of WGN is a delta function.

   ***
In summary, the measured cross-correlation over finite time at different
delays are not independent for band-limited noise.
   ***
  
Any comments? 

tak
		
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