Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

Spatial Multidimensional Cooperative Transmission Theories And Key Technologies

Spatial Multidimensional Cooperative Transmission Theories And Key Technologies - Lin Bai

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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 –figure and figure 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

figure

      The symbol received in the second symbol period is

figure

      where each symbol is divided by figure, and then the vector figure 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

figure

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

figure

      

      where n′ satisfies figure. The average output SNR is

figure

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

figure

      It means that despite the lack of transmit channel information, the diversity gain is equal to MT = 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 MT = 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.

      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 MRMT. 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 MR × MT MIMO system is maximized, which can be realized by selecting MT × 1 weight vector WT 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 MR × 1 weight vector WR. Thereafter, the transmission can be expressed as

figure

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

figure

      where UH and VH are MR × r(H) and MT × r(H) dimensional unitary matrices, respectively. r(H) is the rank of matrix H and ΣH = diag{σ1, σ2, . . . , σr(H)} is a singular value diagonal matrix containing matrix H. By the decomposition of the channel matrix, it can be clearly seen that when WT and WR are the transmitting and receiving singular vectors corresponding to the maximum singular value σmax = max{σ1, σ2, . . . , σr(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

figure

      where the variance of figure.

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

figure

      The

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