e high chordal distance. Obtain householder vectors. From each vector in the rank-one codebook, obtain a set of householder-basis vectors. Suppose X is a valid precoding © 2009 by Taylor & Francis Group, LLC Long Term Evolution: 3GPP LTE Radio and Cellular Technology Table 5.6 Codebook for Transmission on Antenna Ports {0,1,2,3} Number of layers V Codebook index u==[1 -j 1 j] uz=[1 1 -1 1] uz=[1 j 1 - j] WWW14 w11114 u = [1 -1 1 1] WwW24 W,w4W13 " = [1 1 1 -1] Ww1 10 " =[1 j -1 j] u3=[1 -1 1 1] W¹ 13 /2 us=[1111] W! 15 vector; the corresponding householder vector u is obtained SO that the first column of (I - 2uuH) is equal to X. Note that this is always possible for all unit norms X. Further, the resulting u is unique up to a phase rotation and is guaranteed to have norm one. Obtain rank-four codebook. From the set of householder vectors u, construct the 4 X 4 unitary matrices H(u) = (I - 2uuH). These matrices form the rank- four codebook. © 2009 by Taylor & Francis Group, LLC MIMO OFDM Schemes for 3GPP LTE Obtain rank-two and rank-three codebooks by column selection. Codebooks for ranks two and three are obtained by selecting submatrices from the rank- four codebook. One exemplary construction is that the rank-R codebook just contains the first R columns of each matrix in the rank-four codebook. However, more sophisticated construction techniques would pick the sub- matrices in order to optimize the chordal distance or some other perfor- mance-related criterion. Some salient features of the preceding construction include: Optimum performance for rank one. This feature ensures optimum cell-edge throughput performance. In addition, multiple rank-one codebooks can be defined to fit different channel scenarios. Optimum chordal distance for rank three. One possible construction of rank- three codebooks is to pick the last three columns of the X 4 matrices H(u). It can be shown that, in this case, the chordal distance is exactly equal to that of the rank-one codebook. Thus, if the rank-one codebook has the optimum ch