Channel Estimation / Equalization - Estimate Channel Inverse Using White Noise Statistics Only Given the system defined in the following figure:



We have a system $ G \left( f \right) $ which is unknown yet can be defined by $ {N}_{p} $ poles and $ {N}_{z} $ zeros.  

The signal $ x \left( t \right) $ is an IID White Noise.
This signal samples can not be accessed.

The signal $ y \left( t \right) $ is the colored noise (Filtered).
This signal samples are available.

The filter $ H \left( f \right) $ is adaptive (Time changing).
Its model, which is to be determined, is given by $ {N}_{z} $ poles and $ {N}_{p} $ zeros. 

The signal $ z \left( t \right) $ is the result of the $ H \left( f \right) $ Filter.
This signal samples are available.

How can one set the parameters (Poles and Zeros) of $ H \left( f \right) $ such that $ z \left( t \right) $ is White Noise?
This means how to adaptively set the parameters of $ H \left( f \right) $ to be the inverse filter of $ G \left( f \right) $.

Usually those kind of problems are solved using the Least Mean Square Filter yet since the samples of $ x \left( t \right) $ aren't accessible it is not the classic model.

Is there any other way to set the parameters of $ H \left( f \right) $ s.t. its output, $ z \left( t \right) $, is white noise without requiring access to $ x \left( t \right) $?
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.