Mathematical Description
In MIMO systems, a transmitter sends multiple streams by multiple transmit antennas. The transmit streams go through a matrix channel which consists of all paths between the transmit antennas at the transmitter and receive antennas at the receiver. Then, the receiver gets the received signal vectors by the multiple receive antennas and decodes the received signal vectors into the original information. A narrowband flat fading MIMO system is modelled as
where and are the receive and transmit vectors, respectively, and and are the channel matrix and the noise vector, respectively.
Referring to information theory, the ergodic channel capacity of MIMO systems where both the transmitter and the receiver have perfect instantaneous channel state information is
where denotes Hermitian transpose and is the ratio between transmit power and noise power (i.e., transmit SNR). The optimal signal covariance is achieved through singular value decomposition of the channel matrix and an optimal diagonal power allocation matrix . The optimal power allocation is achieved through waterfilling, that is
where are the diagonal elements of, is zero if its argument is negative, and is selected such that .
If the transmitter has only statistical channel state information, then the ergodic channel capacity will decrease as the signal covariance can only be optimized in terms of the average mutual information as
The spatial correlation of the channel have a strong impact on the ergodic channel capacity with statistical information.
If the transmitter has no channel state information it can select the signal covariance to maximize channel capacity under worst-case statistics, which means and accordingly
Depending on the statistical properties of the channel, the ergodic capacity is no greater than times larger than that of a SISO system.
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