In MIMO systems, a transmitter transmits multiple streams through multiple transmit antennas and a receiver receives the multiple stream signals arrived via air.

The transmit streams go through a matrix channel which consists of multiple paths between multiple transmit antennas at the transmitter and multiple receive antennas at the receiver. Then, the receiver receives the wireless signal vectors by the multiple receive antennas and decodes the received signal vectors into the original information. Here is a MIMO system model

$ \mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n} $

where $ \mathbf{y} $ and $ \mathbf{x} $ are the $ N_r \times 1 $ receive and $ N_t \times 1 $ transmit vectors, respectively. In addition, $ \mathbf{H} $ and $ \mathbf{n} $ are the channel matrix and the noise vector, respectively.

Referring to information theory, the average capacity of a MIMO system is as follows:

  • Closed loop MIMO can achieve
    $ C_\mathrm{CL} = E[\max_{\mathbf{Q}} \log_2 (\mathbf{I} + \mathbf{H}\mathbf{Q}\mathbf{H}^{H})] = E[\log_2 (\mathbf{I} + \mathbf{U}\mathbf{S}\mathbf{U}^{H})] $
    where we have used that $ \mathbf{UDV}^H = \mathrm{svd}(\mathbf{H}) $ and $ \mathbf{S} = \mathrm{waterfilling(\mathbf{D}^2)} $. The functions of svd() and waterfilling() represent singular value decomposition and power allocation by the water filling rule, respectively.
  • Open loop MIMO can achieve
    $ C_\mathrm{OL} = \max_{\mathbf{Q}} E[\log_2 (\mathbf{I} + \mathbf{H}\mathbf{Q}\mathbf{H}^{H})] = E[\log_2 (\mathbf{I} + \mathbf{H}\mathbf{H}^{H})] $
    since any unitary matrix of $ \mathbf{Q} $ can achieve the capacity of a open-loop MIMO system, which is mostly $ \min(N_t, N_r) $ times larger than that of a SISO system