Assume we have a random signal $h$ convoluted with another signal $s$ which is assumed to be Walsh code represented by one column of Hadamard-matrix, i.e.,
$$s = \begin{bmatrix} 1\\ -1\\ -1\\ 1\end{bmatrix}$$
which means that all code of Walsh are have the same amplitude but they are only different in their sign. If we have $$x = h * s$$ where $*$ denotes convolution, is it possible to estimate $h$ based on $x$?
here is an example:
$$h = \begin{bmatrix} 0.84\\ 1.24\\ 1.742\\ 0.012\end{bmatrix}$$
and
$$s = \begin{bmatrix} 1\\ -1\\ -1\\ 1\end{bmatrix}$$
Hence $x = h*s = :$
$$x = \begin{bmatrix} 0.84\\ 0.40\\ -0.3380\\ -2.13\\ -0.5140\\ 1.73\\ 0.0120 \end{bmatrix}$$
So, how can we solve $x$ in order to get $h$ based on $x$ ?
Thank you very much in advance.