iple streams are transmitted from the transmit antennas with independent and appro- priate weighting per each antenna such that the throughput is maximized at the receiver output. Let us begin our concise study of eigen-beamforming and MIMO precoding using a simpli- fied model, where we have two transmit antennas at the base station and a UE with a single receive antenna. The goal is to find the complex-valued precoding weights such that the SNR at the receiver is maximized. The channel in this example is a vector h = (h1, h2) where h1 and h2 are channel coefficients. It can be shown that the complex-valued weight- ing factors p1 and P2 can be calculated as shown in Fig. 4.65. This example illustrates the concept of digital precoding and how in theory the weighting vectors/matrices are calculated to maximize the SNR at the receiver. 568 Chapter 4 Figure 4.65 Concept of the digital precoding. In an SU-MIMO system, the identity matrix precoding (for open-loop) and SVD precoding (for closed-loop) can be used to achieve link-level MIMO channel capacity. In addition, random unitary precoding can achieve the open-loop MIMO channel capacity with no sig- naling overhead in the uplink. The SVD precoding, on the other hand, has been shown to achieve the MIMO channel capacity when CSI is known at the transmitter. In a precoded SU-MIMO system with NTX transmit antennas and NRX receive antennas, the input-output relationship can be described as y = HWs + n where S = is an MX1 vector of normalized complex-valued modulated symbols, and = (n1,n2,..., nnrx) are the NRX X 1 vectors of received signal and noise, respectively, H is the NRX NTX complex-valued channel matrix, and W is the NTX X M linear precoding matrix. In the receiver, a hard-decoded symbol vector S is obtained by decoding the received vector y using a vector decoder, assuming perfect knowledge of the channel and the optimal selec- tion of precoding matrices. We assume that the entries of H are independent and identically distributed according to C(0, 1) (complex-va