lued normal distribution) and the entries of noise vector n are independent and identically distributed according to C(0,No) The input vector S is assumed to be normalized, thus E[ssH] = I where I is an identity matrix. Let us further assume that precoding matrix W is unitary, thus WWH = I. The receiver selects a precoding matrix W, = 1,2,... Ncodebook from a finite set of quantized precoding matrices N = {W1, W2, W N codebook ) and sends the index of the chosen precoding matrix back to the transmitter over a low-delay feedback channel. There are two important issues concerning the above precoding scheme: (1) optimal selection criterion for choosing a precoding matrix from set N and (2) design of codebook L. The matrix W, i = 1, 2, ., Ncodebook can be selected from N by using either of the following optimization criteria [75]: (1) minimizing the trace of the mean squared error (MMSE-trace selection), (2) minimizing the determinant of the mean squared error (MMSE-determinant selection), (3) maximizing the minimum sin- gular value of HW (singular value selection), (4) maximizing the instantaneous capacity (capacity selection), or (5) maximizing the minimum received symbol vector distance (mini- mum distance selection). The above selection criteria may be evaluated at the receiver using New Radio Access Physical Layer Aspects (Part 2) 569 a full search over all matrices in L. Using distortion functions based on the selection crite- ria, it can be shown that the codebook N is designed using Grassmannian subspace packing [75]. If MMSE-trace, singular value, or minimum distance selection is used, the codebook is designed such that E =minwwWWH-WjW216is = maximized. If MMSE- determinant or capacity selection optimization method is used, the codebook is designed such that E = is maximized. The precoding matrices (MIMO codebooks) are designed based on a trade-off between per- formance and complexity. The following are some desirable properties of the codebooks: 1. Low-complexity codebooks can be designed by choosing the el