time-domain channel estimation based on two vectors optimization Let's the input data vector $X = [X_1, X_2, X_3, X_4, X_5,X_6,X_7,X_8];$ where $[X_7,X_8]$ are well known, and the vector $y = h*X$ where $*$ is the convolution operation and $h = [h_1,h_2]$ is the channel vector.
I have a priory information that  $[X_7,X_8]$ = $T[X_1,X_2]$ where $T$ is a $2$x$2$ square matrix.
The issue I am trying to solve is can we get the channel vector $h$ based on optimizing $([y_7,y_8]$ - $T[y_1,y_2])$ ?  Is there any optimization algorithm we can use to estimate the vector $h$?
You specify that $T$ is known and invertible, so you know $X_1, X_2$, and then it's really trivial: $y[3]=[X_1,X_2]*[h_1, h_2]$, and $y[9]=[X_7, X_8] *[h_1,h_2]$; write that down as matrix system
$$\begin{pmatrix}y[3]\\y[9]\end{pmatrix}=\begin{pmatrix}X_1&amp;X_2\\X_7&amp;X_8\end{pmatrix}\begin{pmatrix}h_2\\h_1\end{pmatrix}$$
and solve it.