Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

      It can be seen that the diversity gain is still M_R.

      For other constellations, using maximum likelihood detection,⁵ the error probability is

      where and d_min are the nearest neighbor and the minimum separation distance of the constellation, respectively. The analytical solution of the above expression can be obtained by following Eq. (2.106). The upper bound of the bit error rate is usually obtained using the Chernoff bound. Equation (2.108) can also be written as

      Since u is a χ² variable, the average upper bound above is

      when the SNR is large, Eq. (2.110) is simplified to

      Similar to the case of BPSK, the diversity gain is equal to the number of receiving branches in an independent and identically distributed Rayleigh channel.

      (3) Minimum mean square error combining

      When the noise is spatially correlated or non-Gaussian interference occurs, the maximum ratio combining is no longer optimal. In this case, the minimum mean-squared error combining is an optimal gain combining, from which the weight is obtained by minimizing the mean square error between the transmitted symbol c and the combiner output z, namely

      And it is easy to get the optimal weight vector

      where R_ni is the correlation matrix of noise and interference. When there is no interference, R_ni = E{nn^H}. If the noise across the antenna is white noise, then , and the minimum mean square error combining diversity is simplified to a maximum ratio combining diversity with only one coefficient difference.

      2.3.2.3Reception diversity through hybrid selection combining or gain combining

      A hybrid approach combines the selection algorithm with the maximum ratio combining. At each moment, the receiver first selects branches with the largest SNR among M_R branches and then combines them by the maximum ratio combining algorithm. This strategy is called augmentation selection.

      Obviously, it can be concluded that the average SNR of the combiner output is the sum of the two items. The first item corresponds to the maximum ratio combining branches and the second item is generated from the branches of the M_R branches, which is the extension of Eq. (2.95). Therefore, the overall array gain is

      Similarly, for selective combining ( = 1), the diversity gain of the hybrid selection combining and maximum ratio combining is equal to M_R instead of .

2.3.3MISO system

      The MISO system utilizes M_T transmitting antennas with pre-processing or precoding to perform diversity at the transmitting end. And the obvious difference from reception diversity is that the transmitter may not have knowledge of the MISO system channel. Since the channel characteristics can be estimated at the receiving end, while the channel information needs to be fed back from the receiver to the transmitter at the transmitting end, there are basically two ways to obtain direct transmit diversity.

      •When the transmitter has complete channel knowledge, beamforming can be achieved by various optimization metrics such as SNR and SINR to obtain diversity and array gain.

      •When the transmitter has no channel information, the so-called space–time coding pre-processing can be used to obtain the diversity gain, but the array gain cannot be obtained.

      In the following, different beamformers will be evaluated and several indirect transmit diversity techniques will be discussed, which can convert space diversity into time or frequency diversity.

      2.3.3.1Transmit diversity formed by matched beamforming

      This beamforming technique is also known as transmitting maximum ratio combining and it assumes that the transmitter knows all the information about the channel. In order to use diversity, signal c is appropriately weighted before being transmitted to each antenna. At the receiving end, the signal can be expressed as

      where h = [h₁ , . . . , h_MR] denotes the MISO channel vector and W is the weight vector. And the weight vector that maximizes the received SNR is

      where the denominator guarantees that the average total transmitting power remains unchanged and is equal to E_S. This vector makes the transmission in the direction of the matched channel and is therefore also called the matched beamforming or conventional beamforming. Similarly, for reception maximum ratio combining, the average output SNR is ρ_out = M_Tρ, and therefore, the array gain is equal to the number of transmitting antennas M_T. If the bit error rate has the following upper bound at a high SNR, the diversity gain is also equal to M_R.

      Therefore, the matched beamformer exhibits the same performance as the reception maximum ratio combining. It requires knowledge of the complete information of the transmitting channel, which means there is feedback from the receiver in the time duplex system. If frequency duplexing is adopted, the interchangeability of the upper and lower channels is no longer guaranteed, and the understanding of the channel information at the transmitting end is greatly reduced. In addition, the matched beamformer is optimal in the absence of interfering signals but cannot cancel the interference.

      Similar to the aforementioned augmented selection algorithm for SIMO systems, the matched beamformer can be combined with the selective combining algorithm. In the beamformer, the transmitter selects antennas out of M_T antennas. Obviously, this technique yields a full-diversity gain M_T, but reduces the transmit array gain.

      2.3.3.2Space–time coded transmit diversity

      The beamforming technique described previously requires channel information for the transmitter to obtain optimal weights. Conversely,