Читать книгу Spatial Multidimensional Cooperative Transmission Theories And Key Technologies - Lin Bai - Страница 49

(1) The dominant eigenmode transmission

First, the diversity gain of the M_R × M_T MIMO system is maximized, which can be realized by selecting M_T × 1 weight vector W_T and transmitting the same signal from all transmit antennas. In the receiving array, the antenna outputs are combined into a scalar signal z according to the M_R × 1 weight vector W_R. Thereafter, the transmission can be expressed as

By maximizing , the maximized received SNR can be achieved. In order to solve this optimization problem, it is necessary to perform singular value decomposition for H.

where U_H and V_H are M_R × r(H) and M_T × r(H) dimensional unitary matrices, respectively. r(H) is the rank of matrix H and Σ_H = diag{σ₁, σ₂, . . . , σ_r(H)} is a singular value diagonal matrix containing matrix H. By the decomposition of the channel matrix, it can be clearly seen that when W_T and W_R are the transmitting and receiving singular vectors corresponding to the maximum singular value σ_max = max{σ₁, σ₂, . . . , σ_r(H)} of H, the received SNR is maximized.⁶ This technique is known as the dominant eigenmode transmission, and Eq. (2.125) can be rewritten as

where the variance of .

As can be seen from Eq. (2.127), the array gain is equal to = E{λ_max} with λ_max representing the maximum eigenvalue of HH^H. For an independent and identically distributed Rayleigh channel, the upper bound of the array gain is

The asymptotic array gain of the dominant eigenmode transmission (when M_T and M_R are large) is given by

Finally, the diversity gain has upper and lower bounds at high SNR⁷ (Chernoff bound is a good approximation of SER at high SNR)

It means that the error rate is a function of the SNR and the slope of the curve is M_TM_R. The full diversity gain M_TM_R is obtained by the dominant eigenmode transmission.

(2) The dominant eigenmode transmission with antenna selection

The principle of the dominant eigenmode transmission with antenna selection is as follows. First, the matrix set H′ consisting of columns of matrix H is removed according to the definition. The set of all possible H′ is S{H′}, and its potential is . At each instantaneous time, the selection algorithm uses the matrix to provide the largest singular value for a dominant eigenmode transmission. Therefore, the output SNR becomes

The average SNR can be calculated according to the method provided in Ref. 7, and the corresponding array gain is

where

where a_S is the coefficient of u^m of .

Similar to the traditional dominant eigenmode transmission, if all transmit antennas are used, the antenna selection algorithm can obtain the same diversity gain, which means the diversity gain is M_TM_R.

(3) Multi-eigenmode transmission

The eigenmode transmission will not achieve multiplexing gain when the same symbol is sent to all transmit antennas. As an alternative, the system throughput can be increased by maximizing spatial multiplexing gain. For this purpose, the symbols are spread over the non-zero eigenmode of all channels. Assuming M_R ≥ M_T, the channel matrix is an independent and identically distributed Rayleigh channel, and singular value decomposition is made for the channel matrix by Eq. (2.125). If the transmitter uses the precoding matrix V_H to multiply the input vector c(M_T × 1) and the receiver uses matrix to multiply the received vector, the input–output relationship can be written as

It can be seen that the channel has been decomposed into M_T parallel SISO channels given by {σ₁, . . . , σ_nt}. It should be noted that if M_T virtual data channels are established, all of these channels will be fully decoupled. Therefore, the mutual information of the MIMO channel is the sum of the SISO channel capacities.

where {p₁, . . . , p_MT} is the eigenmode power allocation for each channel, satisfying the normalization condition . The capacity is linear with M_T, so the spatial multiplexing gain is equal to M_T. This transmission mode might not achieve full diversity gain M_TM_R, but at least provides a M_R-times array and diversity gain. Multi-eigenmode transmission can also be combined with antenna selection at the receiving end. As long as ≥ M_T, the multiplexing gain is still M_T, but the array gain and diversity gain are reduced.