Channel Estimation / Equalization - Estimate Channel Inverse Using White Noise Statistics Only
A Regular Random Process is is the the result of White Noise Going through a Minimum Phase LTI System.

A Non Perfectly Predictable Random Process can be defined as (See Wold Theorem / Wold Decomposition):

$$ x \left[ n \right] = \sum_{k = 0}^{\infty} {g}_{k} u \left[ n - k \right] + z \left[ n \right] $$

Where $ {\left\{ {g}_{k} \right\}}_{k = 1}^{\infty} $ is a Minimum Phase LTI System, $ u \left[ n \right] $ is White Noise and $ z \left[ n \right] $ is a Perfectly Predictable Process.

Since the questions deals with colored noise, $ z \left[ n \right] = 0 $.

Hence the process can be defined as following:



The optimal linear predictor is given by:

$$ \hat{x} \left[ n \mid n - 1 \right] = \sum_{k = 0}^{\infty} {h}_{k} x \left[ n - 1 - k \right] $$

Where the filter $ H \left( f \right) $ is given by:

$$ H \left( f \right) = 1 - \frac{1}{G \left( f \right)} $$

Hence, given the Linear Prediction Filter one could define the Whitening Filter as:

$$ W \left( f \right) = 1 - H \left( f \right) = \frac{1}{G \left( f \right)} $$

The Linear Predictor Filter can be approximated by the LMS Filter:



Which now, the Whitening Filter can be easily defined and approximated using the LMS Framework.