# 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 $x \left( t \right)$?

• One thought: If $G(f)$ is fixed (static), then $Y(f)\approx G(f)$, especially for long time slices, or for the average of many short slices. Then $H(f)=1/G(f)$. – MBaz Jan 26 '16 at 22:48
• @MBaz, of course it is, as it is written. Yet since it is noise we're talking about usually you don't work like that. You do something like LMS. – Royi Jan 27 '16 at 5:11
• I know, but all techniques I'm aware of require either samples from $x(t)$ (training) or some feature that can be exploited (blind estimation). When the input is unknown random noise, the only useful feature I can think of is the PSD; hence my suggestion. But I'm far from an expert in this field. – MBaz Jan 27 '16 at 15:38
• Is this question being asked for the purpose of seeking information, or just so that you can write an answer to show off your own erudition? Why not just accept your own answer (yes, that is allowed) so that your query does not get thrown up by "Community" as an active question that deserves another look? – Dilip Sarwate Sep 25 '16 at 12:10
• @David, Post a specific case and we'll see. – Royi Jun 5 '19 at 18:44

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.