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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 $h$?

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    $\begingroup$ Do you know $T$? in that case, no need for optimization. It's straight forward writing down the linear system of equations giving $y[3]$ and $y[9]$ and solving that. $\endgroup$ Jan 9 at 10:12
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    $\begingroup$ Also, if $T$ is invertible, then you know $X_1, X_2$, and then it's really trivial: $y[3]=[X_1,X_2]*[h_1, h_2]$, and $y[9]=[X_7, X_8] *[h_1,h2]$; write that down as matrix system $$\begin{pmatrix}y[3]\\y[9]\end{pmatrix}=\begin{pmatrix}X_1&X_2\\X_7&X_8\end{pmatrix}\begin{pmatrix}h_2\\h_1\end{pmatrix}$$ and solve it. $\endgroup$ Jan 9 at 10:25
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    $\begingroup$ If you don't know $T$, well, your prior information is worth nothing (you can always find a $T$ that fulfills that condition) and your problem can't be optimized at all. $\endgroup$ Jan 9 at 10:28
  • $\begingroup$ Yes, $T$ is well-known and invertible, do you mean by $*$ the convolution operation as in my question? or it's multiplication ? $\endgroup$
    – Fatima_Ali
    Jan 9 at 10:41
  • $\begingroup$ and please write down the details as an answer in order to accept it and close the question. $\endgroup$
    – Fatima_Ali
    Jan 9 at 10:44
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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

$$\begin{pmatrix}y[3]\\y[9]\end{pmatrix}=\begin{pmatrix}X_1&X_2\\X_7&X_8\end{pmatrix}\begin{pmatrix}h_2\\h_1\end{pmatrix}$$

and solve it.

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