# Effect of band limiting on cross correlation of 2 white noise timestreams

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#### phys314

##### Newbie level 3
I know that band limiting a single timestream of white noise will turn the autocorrelation function into a sinc function from a delta function.

But suppose I have 2 timestreams of band limited white noise, what effect does the band limiting have on their cross correlation? If I collect an infinite amount of data, the cross correlation will go to zero with or without band limiting, but I have some data suggesting that the cross correlation of 2 band limited white noise timestreams averages to zero slower that that of 2 non-filtered white noise timestreams. Does anyone know how to explain this quantitatively.

I'm a physics major, so I'm new at signal processing.

I would intuitively expect a similar behaviour, but don't feel a need to prove a general validity. Do you?

I'm trying to understand how the rms of cross-correlation $\rho_{x,y}(\tau)$ between 2 timestreams (both with mean zero) x(t) and y(t) of white noise decreases with the number of samples N in each timestream.

$\rho_{x,y}(\tau) = \frac{\sigma_{x,y}^2}{\sigma_x \sigma_y} = \frac{\frac{1}{N}\sum_t x_t y_{t+\tau}}{\sigma_x \sigma_y}$

With no filtering of the timestreams, the numerator is just a random walk whose rms position is $\sigma_x \sigma_y \sqrt{N}$, so the rms of the cross correlation is just $1/\sqrt{N}$. But I find that when I band pass filter both timestreams, the rms still goes down as $1/\sqrt{N}$, but with a coefficient larger that 1. I'm trying to understand what that coefficient means.

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