Let's have the vector $y = h * x$ where $*$ is the convolution operation, $h$ is the channel with length $N$ and $x$ is a symmetry vector which means $x = [x_M, x_{M-1}, ....,x_0, 0 , x_0, x_1, .... x_{M-1}, x_M]$. If we explained that convolution operation as:
$y = X h_z$,
where $X$ is a $(2M+1) \times (2M+1)$ Toeplitz matrix with first column $x$, and first row $[x_M, 0, 0, . . . , 0]$ and $h_z$ is a vector of $(2M+1) \times 1$ such that $h_z = [h, \ 0, \ 0, \ 0 .... \ 0, \ 0]$. It's known that $M > N$
My question, is it possible, in that case, to recover the vector $h$ based only on the second half of the vector $y$ ? it means if $y^T = \begin{bmatrix}y_0 & y_1& \ldots & y_{2M-2}& y_{2M-1} \end{bmatrix}$, can we estimate the vector $h^T$ based on $\begin{bmatrix}y_{(2M-1) / 2} & y_{(2M-1) / 2 + 1 } &\ldots & y_{2M-1} &y_{2M} \end{bmatrix}$? Remember $x$ is a symmetry vector as explained above.
