# 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 $$h$$?

• 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. Commented Jan 9, 2021 at 10:12
• 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. Commented Jan 9, 2021 at 10:25
• 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. Commented Jan 9, 2021 at 10:28
• Yes, $T$ is well-known and invertible, do you mean by $*$ the convolution operation as in my question? or it's multiplication ? Commented Jan 9, 2021 at 10:41
• and please write down the details as an answer in order to accept it and close the question. Commented Jan 9, 2021 at 10:44

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.