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 This message was sent using the Comp.DSP web interface on www.DSPRelated.com

# Cross-Correlation Function of Band-limited Noise

Started by ●March 30, 2005

Reply by ●March 31, 20052005-03-31

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

Reply by ●March 31, 20052005-03-31

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."

Reply by ●April 2, 20052005-04-02

>the cross-correlation of two independent random processes is the productof>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

Reply by ●April 6, 20052005-04-06

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 This message was sent using the Comp.DSP web interface on www.DSPRelated.com