# Decorrelating Stationary Colored Gaussian Noise — Effect On The Desired Signal

So given stationary colored gaussian noise $\mathbf{n}$, I know that I can decorrelate it by first finding it's autocorrelation $R_{nn}$ and performing $R^{-\frac{1}{2}}_{nn} \mathbf{n}$.

In practice of course, I need to estimate $R_{nn}$, which I can do via averaging $\mathbf{r}_{nn}$ or by fitting it to an AR model.

So, suppose I have a good estimate $\hat{R}_{nn}$, for a finite data record of length $N$. Such that the resulting spectrum of the noise frame is white enough, according to some metric.

Now, suppose I have a signal $\mathbf{x} = \mathbf{s} + \mathbf{n}$. I want to decorrelate the noise, such that after the decorrelation, I get a signal in white noise. From what I have read, the way to do this is to also apply $R^{-\frac{1}{2}}_{nn} \mathbf{x}$.

However, won't this distort the desired signal $\mathbf{s}$? Can anyone point me to some references regarding this?

Also: How else can I decorrelate colored noise in order to get my signal embedded in white noise?

Yes, the "Whitening" is basically filtering the signal using an LTI System.

The Result would be the signal $\mathbf{s}$ filtered by the system which whitens the noise.

In the framework of Matched Filter this is OK since you will be able to filter the Reference Signal as well and hence have the same performance.

Yet for other usages you must take under consideration the effect of whitening system.

• I figured. The autocorrelation of colored noise decays exponentially. Hence, if we were to whiten it, we would need a function that increases exponentially. Therefore, it seems to me that a whitening filter will act to boost the high frequencies of the signal, while nulling the lower frequencies. – The Dude May 16 '16 at 18:15
• Well, that depends how the Function which colored the noise looked like. If it was "High Pass Filter" the result would be exactly the opposite of what you described. – Royi May 16 '16 at 19:09