# Convolution between a vector and another symmetric vector

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.