Pranavi Boyalakuntla Jonathan Kelley
Multiple antennas help support multiple users sending streams of data in the same frequency band. We see this with technologies like IoT and bluetooth where multiple devices frequently communicate in close range. Systems like these are called "multiple input multiple output" (MIMO) systems.
In this lab, we were given a simulated 4x4 MIMO channel in MATLAB to pass constructed data through. If we were to implement this in hardware, this system might look like Fig. 1.
Fig. 1: A visual representation of a 4x4 MIMO system.
In Fig. 1, we can see that each of the receivers is "hearing" a combination of all of the signals sent in this system. We processed the received data at each receiver for the original signal sent. In order to do this, we considered two cases.
Case 1: We assume we do not know the channels between transmitter and receiver.
Case 2: We assume we do know the channels between transmitter and receiver.
We assumed a flat fading and timing synchronized model. A flat fading model assumes that the signal fades evenly across all frequencies. A timing synchronized model assumes that the cosine signals at the transmitters and receivers are synchronized. These assumptions are reasonable because we are simulating the channel.
There are 16 channels in a 4x4 MIMO system because each transmitter sends signals to each receiver. We can send training signals through the channels and watch how the signal changes once it is received to estimate channel responses. To implement this, we sent a known signal from one transmitter at a time and compared it to what was received on all four receivers.
Eqn. 1 shows the calculation of the channel response, $h$, by dividing the received signal, $y$, with the sent signal, $x$.
$$ h = y./x \tag{1} $$
From the 16 channels we estimated, we formed the $H$ matrix. Eqn. 2 shows the structure of the $H$ matrix where the rows increment by transmitter and the columns increment by receiver.
$$ H =
\begin{bmatrix} h_{11} & h_{12} & h_{13} & h_{14} \\ h_{21} & h_{22} & h_{23} & h_{24} \\ h_{31} & h_{32} & h_{33} & h_{34} \\ h_{41} & h_{42} & h_{43} & h_{44} \end{bmatrix}
\tag{2} $$
Eqn. 3 and Eqn. 4 illustrate how the channels changed the transmitted signals, $x$, to the received signals, $y$.
$$ y = Hx \tag{3} $$
$$ \begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{bmatrix} =
\begin{bmatrix} h_{11} & h_{12} & h_{13} & h_{14} \\ h_{21} & h_{22} & h_{23} & h_{24} \\ h_{31} & h_{32} & h_{33} & h_{34} \\ h_{41} & h_{42} & h_{43} & h_{44} \end{bmatrix}
\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4}\end{bmatrix}
\tag{4} $$