# How to retrieve back the input using Bussgang theorem?

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 $$x$$?

• Something sounds strange. $B$ is obtained as the ratio of two expectations (probably, found empirically on many trials, not just on a single one) to minimize the expected square error, so what "input $x$" are you trying to find? Aug 31, 2020 at 1:09
• @fedja yeah B is the ratio of two expectations, i did mention the formula in the question. Concerning the input x it 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 Aug 31, 2020 at 17:43