Estimating parameters using 0-1 matrix system of equations

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.

• Well, it seems that $\boldsymbol{X}^{H}\boldsymbol{X}$ should be invertible. Then, you can use (complex) least squares to get an estimator of $\vec{h}$. See this question for details.
– user140541
Dec 21, 2021 at 19:12
• @d.k.o. Could you elaborate why $\textbf{X}^H\textbf{X}$ seems to be invertible? Is it right to say that the matrix $\textbf{X}$ needs to be a disjunct matrix for $\textbf{X}^H\textbf{X}$ to be invertible? (en.wikipedia.org/wiki/Disjunct_matrix) Dec 21, 2021 at 23:32