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I'm a little confused about how 'rich' the scattering environment should be for MIMO to work (i.e. for the channel matrix to be full rank). For instance, let's say we have a Tx and Rx with two antennas each, and under the two-ray model (or the flat/plane earth model), one path is direct from the Tx to the Rx and another path follows a reflection from the ground.
Can MIMO be used here? In other words, can two separate spatial streams be obtained?
The Channel Impulse Response (CIR) for a beamforming vector $\mathbf{w} \in \mathbb{C}^{N_T \times 1}$ (which corresponds to your spatial streams) is the following
$$\mathbf{c}(n) = \sum\limits_{k=1}^q \alpha_k \mathbf{a}_{R,k}\mathbf{a}_{T,k}^\top \mathbf{w} \delta(n - \tau_k) \tag{1}$$ where $q$ is the number of paths (is $2$ due to the two-ray propagation model), $\alpha_k$ is the complex gain of the $k^{th}$ path, $ \mathbf{a}_{R,k}$ and $\mathbf{a}_{T,k}$ are the receive and transmit steering vectors of the $k^{th}$ path (depending on the antenna placement and the Angles-of-Arrival and Angles-of-Departures of the path) and finally $\tau_k$ is the propagation delay of the $k^{th}$ path.
I can see that in your case $\mathbf{a}_{T,k} \in \mathbb{C}^{2 \times 1},\mathbf{a}_{R,k} \in \mathbb{C}^{2 \times 1},\mathbf{w} \in \mathbb{C}^{2 \times 1}$. Note that equation $1$ could be written as $$\mathbf{c}(n) = \Big(\sum\limits_{k=1}^q \alpha_k \mathbf{a}_{R,k}\mathbf{a}_{T,k}^\top \delta(n - \tau_k) \Big) \mathbf{w}$$ So in order to "equalize" to get $\mathbf{w}$, you need to know the matrix $\mathbf{H} = \sum\limits_{k=1}^q \alpha_k \mathbf{a}_{R,k}\mathbf{a}_{T,k}^\top \delta(n - \tau_k) $.
