# Cross-Correlation Function of Band-limited Noise

Started by 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

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:

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

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....

tak

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```
```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....
>
>

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.

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

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```
```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.
***