I have the following set of $n$ equations:
$ \hspace{4cm}\boxed{y^{(1)}=h_1x_1^{(1)}+ h_2x_2^{(1)} +\ldots+ h_kx_k^{(1)} +w_1\\ y^{(2)}=h_1x_1^{(2)}+ h_2x_2^{(2)} +\ldots+ h_kx_k^{(2)}+w_2\\ \vdots \\ y^{(n)}=h_1x_1^{(n)}+ h_2x_2^{(n)} +\ldots+ h_kx_k^{(n)}+w_n}$
In matrix notation, $ \boxed{\vec{y}= \textbf{X}\vec{h}+\vec{w}},$ where,
$\vec{h}=[h_1,h_2,\ldots, h_k]$ denotes the coefficients where each entry is a complex number.
$\vec{y}=[y_1,y_2,\ldots, y_n]$ is the output vector.
The $ (i,j)^{th}$ entry of matrix $\textbf{X}$ is given by $\textbf{X}_{ij}=x_i^{(j)}$.
$w_j, \forall j \in \{1,2,\ldots,n\}$ is IID Gaussian noise distributed as $\mathcal{CN}(0,1)$.
Also $n>k$.
In my problem setup, $x_i^{(j)} \in \{0,1\}$ and their values are known. My aim is to estimate the coefficients $h_i'^s$.
Question 1: What constraints the 0-1 Binary matrix $\textbf{X}$ needs to satisfy so that I can get proper estimates of $h_i'^s, \forall i \in\{1,2,\ldots, k\}.$
Comment: I have not quantitatively elaborated what a "proper estimate" means. I am happy with any good standard estimates. I would like to know what conditions on $\textbf{X}$ could potentially help me to proceed with finding the standard estimates like MLE/MMSE or Least squares or something else.
