ome MIMO channels + n . Since U and Vare unitary matrices and = diag( (A1, A2, ..., ^min(N,,N,), 0,0,...,0), it is clear that the capacity of this model is the same as the capacity of the model y = Hx + n. However, is a diagonal matrix with K non-zero elements on the main diagonal, thus y 1 = 11 1 1 1 n 1, , y k = k+1 = +1. The latter equations are conceptually equivalent to K parallel SISO eigen-channels, each with a signal power of 22, = 1,2, min(Nt, Nr). Hence, the MIMO channel capacity can be rewritten in terms of the eigenvalues of the input signal covariance matrix E. When the channel knowledge is available at the transmitter and receiver, then H is known and we can optimize the capacity over E, subject to the power constraint tr(E)<r It is shown in the literature CHAPTER 10 The IEEE 802. 16m Physical Layer (Part II) optimal E in this case exists and is known as a water-filling solution. The channel capacity in this case is given by: (10-32) a non-linear function of eigenvalues of the channel input covariance matrix. The effect of various channel conditions on the channel capacity has been extensively studied in the literature. For example, increasing the LoS signal strength at fixed SNR reduces the capacity in Ricean channels [38]. This can be explained in terms of the channel matrix rank or through various eigenvalue properties. The issue of correlated fading is of considerable importance for implementations where the antennas are required to be closely spaced. The optimal water-filling allocation strategy is obtained when the power allocated to each spatial sub-channel is non-negative. Figure 10-26 illustrates an example where the water-filling algorithm is applied to an OFDM system assuming Number of Sub-channels = 16, Total Power = -20 dBm, No = -80 dBm, and X 10-6 Unused Eigen-modes/ Used Eigen-modes/ Sub-channels Sub-channels Fraction of Allocated Power to each Sub-channel Noise to Carrier Ratio Sub-channel Index FIGURE 10-26 An example illustration of a water-filling algorithm in an OFDM system 10.4