g matrix UH immediately after reception, a diagonalization of the channel matrix is attained. The vector of symbols after receiver shaping (y) can be expressed by considering Equation (3.2) as follows: (3.3) y=Dx+U"n where x is the vector of modulation symbols and X is the vector of precoded sym- bols to be transmitted on the antenna ports. It is clear from Equation (3.3) that the Orthonormal columns of each U and V imply UUH=IN and VHV=] IM. © 2009 by Taylor & Francis Group, LLC Long Term Evolution: 3GPP LTE Radio and Cellular Technology Complex modulation symbols Stream 1 Stream 1 Stream 2 Pre-coding MIMO Channel Rx Shaping Stream 2 [x = v .x] [y = Hx + n] [y=UH.y] Stream K Stream K PMI (V) H Estimation and V = Q(V) PMI feedback SVD Processing Quantization and pre-coding matrix selection Figure 3.25 Basic idea of codebook-based unitary precoding. equivalent channel matrix is the diagonal matrix D, which implies R independent parallel channels for data transmission. The eigenvalues of HHH will determine the gain of each parallel channel. The overall process, including transmit precod- ing, propagation through MIMO channel, and receiver shaping, is illustrated in Figure 3.25. As stated before, both the transmitter and the receiver need knowledge of the downlink channel in order to perform the precoding and shaping processes. However, in FDD systems, the downlink channel cannot be estimated at the transmitter, SO the receiver estimates should be reported via uplink transmissions. Furthermore, the uplink bandwidth is severely constrained in real systems and the limited feedback does not allow for reporting the true estimates of H. Therefore, the SVD processing is carried out at the receiver and a quantized version of the precoding matrix (v) is reported to the transmitter, as shown in Figure 3.25. The precoding matrix is selected from a predefined set of precoding matrices referred to as a codebook. LTE employs a precoded spatial multiplexing scheme with codebook-based feedback from users. Both the UE and the eNode