where di = 1,2, hin(N,Nr) are the non-zero eigenvalues of HHH We can decompose the MIMO channel into K < min(N,) equivalent parallel SISO channels using the Singular Value CHAPTER 10 The IEEE 802. 16m Physical Layer (Part II) H11(k) H12(k) Frequency Index OFDM Symbol Index Frequency Index OFDM Symbol Index H21(k) H22(k) Frequency Index OFDM Symbol Index Frequency Index OFDM Symbol Index FIGURE 10-24 An example of the multipath frequency responses of Nt=2, Nr = 2 MIMO channel Decomposition (SVD) theorem. [15,31,32] Let y = Hx + n describe the input-output relationship of the MIMO channel where y is the output vector with Nr components, X is the input vector with N components, n is the additive noise vector with Nr components, and H is the Nr X N channel matrix. Using SVD, we can show H = UEVH Let x = VHx,y = UHy, and n = UHn denote the unitary transformation of the channel input and output and noise vectors, it can be shown that The concept of decomposition of an N x N Hermitian matrix in terms of quadratic product of the N x N unitary matrix composed of eigenvectors, an N X N diagonal matrix of eigenvalues can be generalized to M x N complex-valued matrices of rank K. If A is an M X N (M > N) complex-valued matrix of rank K, then A = UVH denotes the Singular Value Decomposition of A where the M x M unitary matrix U is composed of the eigenvectors of AAH; i.e., U = (U1,U2,...Um) and the N X N unitary matrix V is composed of eigenvectors of AHA (i.e., V = V1,V2,...Vn), AHA) = The elements of the M X N matrix = A = diag(a1, 02, OK) the square of the eigenvalues of matrix AHA are roots which are referred to as the singular values of matrix A. Therefore, matrix A may be written as A = The number of non-zero singular values of matrix A is the rank of A. The singular values of matrix A are positive real numbers which satisfy 01>02> 10.4 Multi-antenna transmission schemes Nt = 1, Nr r=1 Nt = 2, Nr=2 Nt = 2, Nr. = 4 Nt = 4, Nr = 2 Nt = 4, Nr = 4 SNR (dB) FIGURE 10-25 A comparison of the information theoretic capacity of s