With the above model (parallel orthogonal channels),the fundamental capacity of ar as follows. 2.2 Channel Known to the Transmitter When the perfect channel knowledge is available at both the transmitter and the the transmiter can optimize its ower input covariancematri across antennas according to the"water-filling"rule (in space)to maximize the capacity formula(0.3).Substituting the matrix SVD(0.5)into (0.3)and using properties of unitary matrices,we get the MIMO capacity with CSIT and CSIR as C-恩+K,刃 -2e+ optimization leads to a 1sism,is given parametrically by (0.11) where denotes max(0,a),and uis chosen to satisfy EA-P (0.12) The resulting capacity is then bits/channel use (0.13) which is achieved by choosing each component according to an independent Gaussian distribution with power P The covariance matrix of the capacity-achieving transmitted signal is given by K,=VPV# where P=diag(P.P.P.0.0)is an NxN matrix. Water-filling algorithm: The power allocation in (0.11)can be determined iteratively using the water-filling algorithm.We now describe it. We first set the iteration count p to 1 and assume that所有(m-p叶l)个并行子信道都 66 With the above model (parallel orthogonal channels), the fundamental capacity of an MIMO channel can be calculated in terms of the positive eigenvalues of the matrix HHH as follows. 2.2 Channel Known to the Transmitter When the perfect channel knowledge is available at both the transmitter and the receiver, the transmitter can optimize its power allocation or input covariance matrix across antennas according to the “water-filling” rule (in space) to maximize the capacity formula (0.3). Substituting the matrix SVD (0.5) into (0.3) and using properties of unitary matrices, we get the MIMO capacity with CSIT and CSIR as 2 :( ) 0 1 max log det x x H N x Tr P C = N ⎛ ⎞ ⎛ ⎞ = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ I Λ Λ K K K 2 : 1 0 max log 1 i i i m i i P PP i P N λ ≤ = ⎛ ⎞ = + ⎜ ⎟ ∑ ⎝ ⎠ ∑ where Pi is the transmit power in the ith sub-channel. Solving the optimization leads to a water-filling power allocation over the parallel channels. The power allocated to channel i, 1≤i≤m, is given parametrically by 0 i i N P μ λ + ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ (0.11) where a+ denotes max(0, a), and μ is chosen to satisfy 1 m i i P P = ∑ = (0.12) The resulting capacity is then WF 2 0 2 ( ) 1 1 0 0 1 1 log 1 log m m i i i i C N N N μλ μλ + + = = ⎡ ⎤ ⎛⎞ = + −= ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝⎠ ∑ ∑ bits/channel use (0.13) which is achieved by choosing each component i x according to an independent Gaussian distribution with power Pi. The covariance matrix of the capacity-achieving transmitted signal is given by H Kx = VPV where ( ) 1 2 diag , ,., ,0,.,0 P = PP Pm is an N×N matrix. „ Water-filling algorithm: The power allocation in (0.11) can be determined iteratively using the water-filling algorithm. We now describe it. We first set the iteration count p to 1 and assume that 所有(m-p+1)个并行子信道都