3
$\begingroup$

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.

enter image description here

Can MIMO be used here? In other words, can two separate spatial streams be obtained?

$\endgroup$
1
$\begingroup$

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) $.

$\endgroup$
2
  • $\begingroup$ @Ahmad Perhaps two spatial streams can be obtained based on the nature of H? In other words, if its rank is 2, then 2 streams can be resolved separately? $\endgroup$ – V-Red Aug 1 '19 at 12:21
  • $\begingroup$ @V-Red indeed. You could see the spatial streams in the vector $\mathbf{w}$ as in a precoding matrix $\mathbf{P}$ multiplied by vector of streams $\mathbf{s}$, i.e.$\mathbf{w} = \mathbf{P}\mathbf{s}$. Hence $$\mathbf{c}(n) = \Big(\sum\limits_{k=1}^q \alpha_k \mathbf{a}_{R,k}\mathbf{a}_{T,k}^\top\mathbf{P} \delta(n - \tau_k) \Big) \mathbf{s}$$ Then your $\mathbf{H} = \sum\limits_{k=1}^q \alpha_k \mathbf{a}_{R,k}\mathbf{a}_{T,k}^\top \mathbf{P} \delta(n - \tau_k)$ and now you can start arguing on the rank of $\mathbf{H}$. $\endgroup$ – Ahmad Bazzi Aug 1 '19 at 12:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.