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 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
By maximizing
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
where the variance of
As can be seen from Eq. (2.127), the array gain is equal to
The asymptotic array gain of the dominant eigenmode transmission (when MT and MR 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 MTMR. The full diversity gain MTMR 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
The average SNR can be calculated according to the method provided in Ref. 7, and the corresponding array gain is
where
where aS is the coefficient of um 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 MTMR.
(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 MR ≥ MT, 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 VH to multiply the input vector c(MT × 1) and the receiver uses matrix
It can be seen that the channel has been decomposed into MT parallel SISO channels given by {σ1,
