h the transmission rate. We define the diversity order for a given transmission rate as: log[Pe(R, Y)] (10-43) where Pe(R,Y) is the PER corresponding to transmission rate R and SNR Y. Thus, the diversity order is the magnitude of the slope of the PER plotted as a function of the SNR on a logarithmic scale. 10.4.4 Precoding and Beamforming The multiple antennas at the transmitter and receiver can be used to achieve array and diversity gain instead of capacity gain. In this case, the same symbol weighted by a complex-valued scale factor is sent from each transmit antenna SO that the input covariance matrix has unit rank. This scheme is referred to as beam- forming. It must be noted that there are two conceptually and practically different classes of beamforming: (1) direction-of-arrival beamforming (i.e., adjustment of transmit or receive antenna directivity); and (2) eigen-beamforming (i.e., a mathematical approach to maximize signal power at the receive antenna based on certain criterion). In this section, we only consider eigen-beamforming schemes. A classic eigen-beamforming scheme usually performs linear, single-layer, complex-valued weighting on the transmitted symbols, such that the same signal is transmitted from each transmit antenna using appropriate weighting factors. In this scheme, the objective is to maximize the signal power at the receiver output. When the receiver has multiple antennas, the single-layer beamforming cannot simultaneously maximize the signal power at every receive antenna, hence, precoding is used for multi-layer beamforming in order to maximize the throughput of a multi-antenna system. Pre- coding is a generalized beamforming scheme to support multi-layer transmission in a MIMO system. Using precoding, multiple streams are transmitted from the transmit antennas with independent and appropriate weighting per antenna such that the throughput is maximized at the receiver output. In a single-user MIMO system, identity matrix precoding (for open-loop) and SVD precoding (for closed-loop) 