I have a linear model which is given as
$$ \begin{align} \mathbf{x} &= A\mathbf{s} + \mathbf{H}{\bf\unicode[Times]{x3b8}} \\ &= [\mathbf{s} ~ \mathbf{H}][A ~ {\bf\unicode[Times]{x3b8}}^{\rm{T}}]^{\rm{T}} \\ &= [\mathbf{s} ~ \mathbf{H}]\tilde{{\bf\unicode[Times]{x3b8}}} \end{align} $$
The result is a vector. If $\mathbf{H} \in \mathbb{R}^{N \times K}$, ${\bf\unicode[Times]{x3b8}} \in \mathbb{R}^{K \times 1}$ and $\mathbf{s} \in \mathbb{R}^{N}$, $\mathbf{x} \ \in \mathbb{R}^{N}$.
Now in my work, I want to formulate a multichannel linear model. This means that I have $L$ channels such that
$$ \mathbf{x}{_{l}} = A\mathbf{s}{_{l}} + \mathbf{H}{\bf\unicode[Times]{x3b8}}_{l} $$
If the term $\mathbf{s}$ was similar for all channels (i.e. $\mathbf{s}_{1} = \mathbf{s}_{2} = \cdots = \mathbf{s}_{L} = \mathbf{s}$) then it is simple to write the multichannel linear model
$$ \mathbf{X} = [\mathbf{s} ~ \mathbf{H}]\tilde{{\bf\Theta}} $$
where
$$\tilde{{\bf\Theta}} = \begin{bmatrix} A & A & \cdots & A \\ {\bf\unicode[Times]{x3b8}}_{1} & {\bf\unicode[Times]{x3b8}}_{2} & \cdots & {\bf\unicode[Times]{x3b8}}_{L} \end{bmatrix}$$
Knowing that each ${\mathbf{s}}_{l}$ is different, how can I write the multichannel linear model?
I need to know this in order to evaluate the model for use in a maximum likelihood estimator.
Well, after some work, I found one way.
Suppose $\tilde{\mathbf{H}}_{l}$ to be $[\mathbf{s}_{l} ~ \mathbf{H}]$, and $\mathbf{e}_{l} = [0 \cdots 1 \cdots 0]^{\rm{T}}$ with the $1$ at the $l$-th position (i.e. the unit vector for dimension $l$).
A way to write the multichannel linear model is
$$ \mathbf{X} = \sum^{L}_{l=1} \tilde{\mathbf{H}}_{l} \tilde{\bf\unicode[Times]{x3b8}}_{l} \mathbf{e}_{l}^{\rm{T}} $$
If all $\mathbf{s}_{l}$ are similar, then the expression becomes the simple multichannel linear model in the original question post.