 largest singular value of A or the square root of largest eigenvalue of the positive-semi-definite matrix AH A, that is, (AHA) where AH denotes the conjugate transpose of A [22]. 570 Chapter 4 complex-valued downlink channel matrix and Wk is the kth NTX X 1 normalized linear pre- coding vector. The mathematical relationship for the input and output of a precoded MU-MIMO system in the uplink (alternatively known as MAC in the literature) with NRX receive antennas at the base station and one transmit antenna at each user terminal can be written as y = NISE Skvkhk + n where SkVk is the weighted complex-valued modulated symbol from the kth user, y = and n=(m1, = are the NRX X 1 vectors of received signal and noise, respectively, hk is the kth NRX X 1 channel vector, where matrix H = (h1, h Nuser) is the NRX X Nuser complex-valued uplink channel matrix. As men- tioned earlier, perfect knowledge of the CSI is necessary at the transmitter in order to achieve the capacity of a multi-user MIMO channel. However, in practical systems, the receiver only provides partial CSI through uplink feedback channels to the transmitter, that is, the multi-user MIMO precoding with limited feedback. The received signal in the down- link of a MU-MIMO system with limited feedback precoding is mathematically expressed as yk = Nk, = 1, 2, Nuser. The transmit vector for limited feedback pre- coding is modeled as Wi = Wi where Ei is the error vector generated as a result of the limited feedback and vector quantization, the expression for the received signal can be rewritten as + Nk, k = 1, 2, Nuser where is the residual interference due to the limited-feedback precoding. To reduce the residual interference term, one should use more accurate CSI feedback which results in the use of more uplink resources for the feedback. It is shown in the literature that the number of feedback bits per user Nfeedback must be increased linearly with the SNR YdB (in decibels) at the rate of Nfeedback = (NTX 1)log2Y = VaB(NTX - 1)/3 in order to achieve the full 