Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

simple but original diversity approach for the two transmit antenna systems, called the Alamouti algorithm, which does not require information on the transmit channel. Considering that in the first symbol period, two symbols c₁ and c₂ are simultaneously transmitted from antenna 1 and antenna 2, and then two symbols – and are transmitted from antenna 1 and antenna 2 in the second symbol period, it is assumed that the flat fading channel remains unchanged during these two symbol periods, which is expressed as h = [h₁, h₂] (the subscript indicates the antenna number rather than the symbol period). The symbol received in the first symbol period is

      The symbol received in the second symbol period is

      where each symbol is divided by , and then the vector has a unit average energy (assuming that c₁ and c₂ are obtained from the unit average energy constellation). n₁ and n₂ are the corresponding terms of additive noise in each symbol period (in this case, the subscript represents the symbol period rather than the antenna number). Combining Eq. (2.118) with Eq. (2.119), we get

      It can be seen that the two symbols are extended on two antennas over two symbol periods. Therefore, H_eff represents a space–time channel. Adding the matched filter to the received vector y can effectively decouple the transmitted symbols, such as

      

      where n′ satisfies . The average output SNR is

      It shows that the Alamouti algorithm cannot provide array gain due to a lack of information about the transmitting channel (note E{||h||²} = M_T = 2).

      However, for independent and identically distributed Rayleigh channels, the average bit error rate of the above problem has the following upper bound at high SNR.

      It means that despite the lack of transmit channel information, the diversity gain is equal to M_T = 2, which is the same as the transmit maximum ratio combining. From a global perspective, the Alamouti algorithm has a lower performance than the transmit or receive maximum ratio combining due to its zero array gain.

      2.3.3.3Indirect transmit diversity

      The technique of obtaining space diversity by combining or space–time coding described above belongs to the direct transmit diversity technique. By using well-known SISO techniques, converting space diversity to time or frequency diversity can also be realized.

      Assuming M_T = 2, the phase shift is achieved by delaying the signal on the second transmit branch by one symbol period or by selecting the appropriate frequency shift. If the channels h₁ and h₂ are independent and identically distributed Rayleigh channels, the space diversity (using two antennas) is converted to frequency and time diversity, respectively. Indeed, the receiver has a frequency or time fading problem for an effective two-branch summed SISO channel, which can be overcome by conventional diversity techniques, such as forward error correction or interleaving for frequency diversity.

2.3.4MIMO system

      As mentioned above, in order to obtain a sufficiently high transmission rate, we can install multiple antennas on both the transmitter and the receiver to improve the spectral efficiency. The corresponding multi-antenna system is also called the MIMO system. When multiple antennas are used at both ends of the link, in addition to improving diversity gain and array gain, the system’s throughput can also be increased by the spatial multiplexing capability of the MIMO channel. However, it must be pointed out that it is impossible to maximize spatial multiplexing capability and diversity gain simultaneously. Besides, the array gain in the Rayleigh channel is also limited, which is smaller than M_RM_T. In the following, the MIMO technologies will be classified according to the understanding of the channel information by the transmitter.

      2.3.4.1MIMO system with complete transmit channel information

      (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