tailieunhanh - Mobile and Wireless Communications-Physical layer development and implementation 2012 Part 9

Tham khảo tài liệu 'mobile and wireless communications-physical layer development and implementation 2012 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | MIMO Channel Characteristics in Line-of-Sight Environments 151 side the four antennas numbered clockwise are distributed on the corners of a square with the antenna spacing d. R represents the distance between the transmitter and the receiver. Fig. 2. Arrangement of 4 Rx and 4 Tx antennas model We assume the distance R is much larger than the antenna spacing d. This assumption results in a plane wave from the transmitter to the receiver. In addition the effect of path loss differences among antennas can be ignored only the phase differences will be considered. From the geometrical antenna arrangement we have the different path lengths rmn from transmitting antenna n to receive antenna m 1 R r21 r4 1 JR2 d2 R d2 2R r31 yỊR2 2d2 R d2 R . All the approximations above are made use of first order Taylor series expansion which becomes applicable when the distance is much larger than antenna spacing. Denoting the received vector from transmitting antenna n as hn exp -j Tn . exp -j 1 n T n A A where A is the wavelength and - T denotes the vector transpose. Thus the channel matrix is given as 12 H hi h 2 h3 h4 13 The best situation for the channel matrix is that its condition number 11 equals to one. It is satisfied when H is the full orthogonality matrix which means all the columns or rows are orthogonal. Orthogonality between different columns in 13 is obtained if the inner product of two received vectors from the adjacent transmitting antennas equals to zero 2 d2 w d2 hk hk 1 2 exP -j - 2R VR- 1 exP -j 0 14 A 2R A R which results in AR d2 k2- 2k 1 k 0 1. 15 152 Mobile and Wireless Communications Physical layer development and implementation To get practical values of d we choose k 0 to update 15 . The optimal design constraint therefore becomes d2 R 16 From 16 we can see the optimal antenna spacing is a function of carrier frequency and propagation distance as well as the geometrical arrangement. As Ả c f shows higher frequency results in smaller antenna spacing .