Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

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.

      2.3.4.2MIMO system without transmit channel information

      When the transmitter has no channel information, multiple antennas can be used at the transmitter and receiver ends to achieve diversity and increase the system capacity. This can be realized by spreading the symbols over the antenna (space) and time using the so-called space–time coding. In the following, the space–time block code will be briefly introduced.

      Similar to MISO system, two symbols c₁ and c₂ are simultaneously transmitted from antenna 1 and antenna 2 in the first symbol period, and the symbols – and are transmitted from antenna 1 and antenna 2 in the next symbol period.

      Assuming that the flat fading channel remains unchanged in the two consecutive symbol periods, the 2 × 2 channel matrix can be expressed as

      It is worth noting that the subscripts here represent the receive and transmit antenna labels instead of the symbol periods. The signal vector received by the receiving array in the first symbol period is

      The signal vector received in the second symbol period is

      where n₁ and n₂ are additive noise components per symbol period of the receive antenna array (the subscripts represent symbol periods instead of antenna labels). Therefore, the receiver produces a mixed signal vector

      Similar to the MISO system, two symbols c₁ and c₂ are transmitted during two symbol periods of two transmit antennas. Therefore, matrix H_eff is orthogonal to all channel information, namely .

      If , then

      where n′ satisfies E{n′} = 0_2×1 and . The above equation shows that the transmission of the symbols c₁ and c₂ is completely decoupled, which means

      The average output SNR is