If we have a non-linear function $f$, that is applied to input $x$, we have then the output $y=f(x)$

Using Bussgang decomposition we can linearize this nonlinearity and express $y$ as $y=Bx+ η$,

Where B could be found as per the below formula below:

$B= \frac{E(yx^{*})}{E(|x|^2)}$

We can use input pilots at the beginning, so assume we we know $x,y$ and we get $B$, from the just above formula.

Then we obtain $\eta= y-Bx$, and this was for the training phase.

But now for the testing , I already have $B$ that I obtained previously from the training phase, but I don’t have $\eta$ nor $x$, so how can I find back the input $x$?

is the ratio of two expectations, i did mention the formula in the question. Concerning the inputBxit is the variable of linearization, may be i am not understanding Bussgang properly, i would appreciate if you can refer me a good source code imlementation, or resource to understanding Bussgang decompostion better :) thanks in advance $\endgroup$