How to make colored noise white?
Given the linear model $$y = hx + z$$ where $z \sim \mathcal{N}(0,K_z)$. We say that noise is colored when $K_z \neq \Sigma$, a diagonal matrix. To uncolor the noise, you can pre-multiply $y$ by $K_z^{-\frac{1}{2}}$, so 
$$K_z^{-\frac{1}{2}}y = K_z^{-\frac{1}{2}}hx +K_z^{-\frac{1}{2}}z = K_z^{-\frac{1}{2}}hx +\tilde{z}$$
Notice that the noise is now $\tilde{z} = K_z^{-\frac{1}{2}}z$, which is uncolored since $$E(\tilde{z} \tilde{z} ^T) =  K_z^{-\frac{1}{2}}E({z} {z} ^T)K_z^{-\frac{1}{2}} = K_z^{-\frac{1}{2}}K_zK_z^{-\frac{1}{2}} =  K_z^{-\frac{1}{2}}K_z^{+\frac{1}{2}}K_z^{+\frac{1}{2}}K_z^{-\frac{1}{2}} = I$$
So $\tilde{z} \sim \mathcal{N}(0,I)$