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Space-Time Coding phần 2

Vì vậy, các kênh là tương đương với các kênh biểu diễn như là r = Dx + n (1,15) (1,14)Số lượng các giá trị riêng khác không của ma trận HHH bằng thứ hạng của ma trận H, ký hiệu là r. Đối với × NR nT ma trận H, xếp hạng tối đa là m = min (KL, nT), điều đó có nghĩa rằng m hầu hết các giá trị từ của nó là khác không. | MIMO System Capacity Derivation 5 Let us introduce the following transformations r UH r x VH x 1.14 n UH n as U and V are invertible. Clearly multiplication of vectors r x and n by the corresponding matrices as defined in 1.14 has only a scaling effect. Vector n is a zero mean Gaussian random variable with i.i.d real and imaginary parts. Thus the original channel is equivalent to the channel represented as r Dx n 1.15 The number of nonzero eigenvalues of matrix HHh is equal to the rank of matrix H denoted by r. For the nR X nT matrix H the rank is at most m min nR nT which means that at most m of its singular values are nonzero. Let us denote the singular values of H by ỰXĨ i 1 2 . r .By substituting the entries s ki in 1.15 we get for the received signal components ri x ixi ni i 1 2 . r . . 1.16 ri ni i r 1 r 2 . nR As 1.16 indicates received components ri i r 1 r 2 . nR do not depend on the transmitted signal i.e. the channel gain is zero. On the other hand received components ri for i 1 2 . r depend only on the transmitted component xi. Thus the equivalent MIMO channel from 1.15 can be considered as consisting of r uncoupled parallel subchannels. Each sub-channel is assigned to a singular value of matrix H which corresponds to the amplitude channel gain. The channel power gain is thus equal to the eigenvalue of matrix HHH. For example if nT nR as the rank of H cannot be higher than nR Eq. 1.16 shows that there will be at most nR nonzero gain sub-channels in the equivalent MIMO channel as shown in Fig. 1.2. On the other hand if nR nT there will be at most nT nonzero gain sub-channels in the equivalent MIMO channel as shown in Fig. 1.3. The eigenvalue spectrum is a MIMO channel representation which is suitable for evaluation of the best transmission paths. The covariance matrices and their traces for signals r x and n can be derived from 1.14 as Rr r UH Rrr U Rx x VH Rxx V 1.17 Rnn UH R U tr Rr r tr Rrr tr Rx x tr Rxx 1.18 tr R w tr Rnn 6 Performance Limits of .