Suppose $h(n)$ is a finite impulse response which is unknown. We can feed any input signal $x(n)$ into the system and observe the corresponding output signal $y(n)$.
From this, is it possible to estimate the input signal $x^*(n)$ which when passed through the filter gives the desired output signal $y^*(n)$ ?
I think a Least Mean Squares filter could be used here. Could anyone point out how one proceeds doing this ?
Thanks !
In the most simple case, just to give intuition about the problem, it is really easy.
In the Frequency Domain:
$$ {Y}^{\ast} \left( \omega \right) = H \left( \omega \right) {X}^{\ast} \left( \omega \right) \Rightarrow {X}^{\ast} \left( \omega \right) = \frac{ {Y}^{\ast} \left( \omega \right) }{ H \left( \omega \right) } $$
Since $ {Y}^{\ast} \left( \omega \right) $ is known all needed is $ H \left( \omega \right) $.
Yet since we have access to a black box of $ H \left( \omega \right) $ we can set input of a known signal $ X \left( \omega \right) $ and have the output $ Y \left( \omega \right) $ which will give us $ H \left( \omega \right) $:
$$ Y \left( \omega \right) = H \left( \omega \right) X \left( \omega \right) \Rightarrow H \left( \omega \right) = \frac{ Y \left( \omega \right) }{ X \left( \omega \right) } $$
As others pointed out it is called Deconvolution and as you mentioned it can be done using Least Mean Square (LMS) Filter.
An excellent resource for reading about deconvolution and its implementation is:
https://workspace.imperial.ac.uk/earthscienceandengineering/Public/lecture%20handout%2019Jan09.pdf
