In MIMO systems, a transmitter sends multiple streams by multiple transmit antennas. The transmit streams go through a matrix channel which consists of all {displaystyle N_{t}N_{r}} paths between the {displaystyle N_{t}} transmit antennas at the transmitter and {displaystyle N_{r}} 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:[citation needed] {displaystyle mathbf {y} =mathbf {H} mathbf {x} +mathbf {n} } where {displaystyle mathbf {y} } and {displaystyle mathbf {x} } are the receive and transmit vectors, respectively, and {displaystyle mathbf {H} } and {displaystyle mathbf {n} } 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 {displaystyle C_{mathrm {perfect-CSI} }=Eleft[max _{mathbf {Q} ;,{mbox{tr}}(mathbf {Q} )leq 1}log _{2}det left(mathbf {I} +
ho mathbf {H} mathbf {Q} mathbf {H} ^{H}
ight)
ight]=Eleft[log _{2}det left(mathbf {I} +
ho mathbf {D} mathbf {S} mathbf {D}
ight)
ight]} where {displaystyle ()^{H}} denotes Hermitian transpose and {displaystyle
ho } is the ratio between transmit power and noise power (i.e., transmit SNR). The optimal signal covariance {displaystyle mathbf {Q} =mathbf {VSV} ^{H}} is achieved through singular value decomposition of the channel matrix {displaystyle mathbf {UDV} ^{H},=,mathbf {H} } and an optimal diagonal power allocation matrix {displaystyle mathbf {S} ={ extrm {diag}}(s_{1},ldots ,s_{min(N_{t},N_{r})},0,ldots ,0)}. The optimal power allocation is achieved through waterfilling, that is {displaystyle s_{i}=left(mu -{ rac {1}{
ho d_{i}^{2}}}
ight)^{+},quad { extrm {for}},,i=1,ldots ,min(N_{t},N_{r}),} where {displaystyle d_{1},ldots ,d_{min(N_{t},N_{r})}} are the diagonal elements of {displaystyle mathbf {D} }, {displaystyle (cdot )^{+}} is zero if its argument is negative, and {displaystyle mu } is selected such that {displaystyle s_{1}+ldots +s_{min(N_{t},N_{r})}=N_{t}}. If the transmitter has only statistical channel state information, then the ergodic channel capacity will decrease as the signal covariance {displaystyle mathbf {Q} } can only be optimized in terms of the average mutual information as {displaystyle C_{mathrm {statistical-CSI} }=max _{mathbf {Q} }Eleft[log _{2}det left(mathbf {I} +
ho mathbf {H} mathbf {Q} mathbf {H} ^{H}
ight)
ight].} The spatial correlation of the channel has 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 {displaystyle mathbf {Q} } to maximize channel capacity under worst-case statistics, which means {displaystyle mathbf {Q} =1/N_{t}mathbf {I} } and accordingly {displaystyle C_{mathrm {no-CSI} }=Eleft[log _{2}det left(mathbf {I} +{ rac {
ho }{N_{t}}}mathbf {H} mathbf {H} ^{H}
ight)
ight].} Depending on the statistical properties of the channel, the ergodic capacity is no greater than {displaystyle min(N_{t},N_{r})} times larger than that of a SISO system.
