16 Resulting spatial correlation Envelope correlation coefficient versus the normalized distance = for the two-cluster case shown d/λ in Fig. 15. 3 Effect of the correlation coefficients on MIMO capacity For Rayleigh fading channels, the ergodic MIMO capacity without channel knowledge at the transmitter is: (24) where: and number of receiver and transmitter antennas, respectively nR nT: average received power per antenna p: noise power at each receive antenna σ2: : × identity matrix nR nR ()H and det(): Hermitian and determinant operation, respectively matrix elements are independent identically-distributed complex Gaussian Hw: Random variables with zero mean and unit variance ()½: Hermitian square root of a matrix. The matrices and determine the spatial correlations between the receivers and the RR RT transmitters, respectively, where the channel matrix is defined by , and are positive definite H Hermitian matrices, and finally they are assumed to be normalized such that [RR] for = 1, j,j j K, nR and [RT] for = 1, i,i i K, nT. By assuming that and have full rank and = = then at high (p/2) the capacity can RR RT nR nT n, S/N be approximated as: (25) If we denote the eigenvalues of by = 1, then . From the arithmetic meangeometric RR i K, n, λi, mean inequality: (26) Since , it follows that log2 det(RR) 0, and is zero only if all eigenvalues of are equal, i.e. = RR RR Thus, the correlation determines the MIMO capacity and the loss in the ergodic capacity at high In. is given by (log2 det(RR) + log2 det(RT)) bit/s/Hz. S/N Figure 17 illustrates the effect of spatial correlations on the ergodic capacity of a MIMO channel with = = 2. In the figure = is assumed. The receiver correlation matrix is chosen nR nT RT I2, according to: (27) where denotes the spatial correlation between the receive antennas. ρR 17 Ergodic capacity with low and high receive correlation
16 Resulting spatial correlation Envelope correlation coefficient versus the normalized distance = d/λ for the two-cluster case shown in Fig. 15. 3 Effect of the correlation coefficients on MIMO capacity For Rayleigh fading channels, the ergodic MIMO capacity without channel knowledge at the transmitter is: (24) where: nR and nT: number of receiver and transmitter antennas, respectively p: average received power per antenna σ2: noise power at each receive antenna : nR × nR identity matrix ()H and det(): Hermitian and determinant operation, respectively Hw: matrix elements are independent identically-distributed complex Gaussian Random variables with zero mean and unit variance ()½: Hermitian square root of a matrix. The matrices RR and RT determine the spatial correlations between the receivers and the transmitters, respectively, where the channel matrix H is defined by , and are positive definite Hermitian matrices, and finally they are assumed to be normalized such that [RR] j,j for j = 1, K, nR and [RT] i,i for i = 1, K, nT. By assuming that RR and RT have full rank and nR = nT = n, then at high S/N (p/2) the capacity can be approximated as: (25) If we denote the eigenvalues of RR by λi, i = 1, K, n, then . From the arithmetic meangeometric mean inequality: (26) Since , it follows that log2 det(RR) 0, and is zero only if all eigenvalues of RR are equal, i.e. RR = In. Thus, the correlation determines the MIMO capacity and the loss in the ergodic capacity at high S/N is given by (log2 det(RR) + log2 det(RT)) bit/s/Hz. Figure 17 illustrates the effect of spatial correlations on the ergodic capacity of a MIMO channel with nR = nT = 2. In the figure RT = I2, is assumed. The receiver correlation matrix is chosen according to: (27) where ρR denotes the spatial correlation between the receive antennas. 17 Ergodic capacity with low and high receive correlation
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