Estimation of Input Signal to Obtain the Desired Output Signal for an Unknown Filter 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.