Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

      In order to overcome the negative impact of fading on the bit error rate, diversity techniques are often employed. The principle of diversity is to provide the receiver with multiple copies of the same transmitted signal, and each replica acts as a diversity branch. If this replication is affected by independent fading conditions, the probability that all branches will be in a fading state at the same time can be greatly reduced. Therefore, diversity stabilizes the link by channel enhancement, which improves the bit error rate performance of the system.

      As fading can occur in time, frequency, and space domains, diversity techniques can be used in these domains. For example, time diversity can be obtained by coding and interleaving, and frequency diversity can exploit the time spread of the channel (in the τ domain) by equalization techniques or multi-carrier modulation. Obviously, time and frequency diversity techniques can result in a loss of time or bandwidth due to the introduction of redundancy. Conversely, because multiple antennas are used at one or both ends of the link, space or polarization diversity does not sacrifice time and bandwidth.

2.3.2SIMO system

      The SIMO system relies on the number of antennas at the receiving end M_R ≥ 2 to realize diversity. If these antennas are sufficiently spaced (such as a wavelength), then when the physical channel exhibits good characteristics, the system will fade independent of the diversity of each branch. Reception diversity can be achieved in two different ways: selective combining and gain combining.

      2.3.2.1Reception diversity through selective combining

      Among M_R received signals, the combiner selects a branch having the largest SNR (or the highest absolute power, bit error rate, etc.) for signal detection. Suppose that M_R channels are subject to unit Rayleigh energy independent and identical distribution and the noise level is equal on each antenna. At this time, the selection algorithm compares the instantaneous amplitude of each channel s_n(n = 1, . . . , M_R) and selects the branch having the largest amplitude s_max = max(s₁, . . . , s_Mn}. The probability that s_max is below a certain threshold S² is given by

      The distribution corresponding to s_max can be obtained by the differentiation of Eq. (2.93).

      The average SNR of the combiner output is³

      when M_R is large, the array gain is approximately

      where γ ≈ 0.577 215 66 is the Euler constant.

      The diversity obtained by the selective combining can be estimated by calculating the bit error rate using the fading distribution given by Eq. (2.94). For the system using BPSK modulation and owning a two-branch diversity, the bit error rate as a function of the average SNR can be expressed corresponding to each channel ρ⁴ as follows:

      when the SNR is high,

      The slope of the bit error rate curve is 2. In general, the diversity gain of the M_R-branch selection diversity scheme is equal to M_R, which indicates that the selection diversity collects all possible diversity from the channel.

      2.3.2.2Reception diversity based on gain combining

      In gain combining, the signal z used for detection is a linear combination of all branches.

      where W_n denotes the combined weight and W = [W₁, . . . , W_MR]^T. According to the selection of these weights, there are different gain combining methods. It is assumed that the data symbol c is transmitted by the channel and received by M_R antennas. Each antenna is described by the channel h_n = |h_n|e^jϕn (n = 1, . . . , M_R). Suppose they obey the unit variance Rayleigh distribution and all channels are independent. Combining signals from all antennas, the detected variable can be expressed as

      where h = [h₁ , . . . , h_MR]^T.

      (1) Equal gain combining

      The weight of equal gain combining is W_n = e^−jϕn, indicating that the signals from different antennas are in phase and can be added together. This approach requires the combiner to have a complete knowledge of the known signal phase. And the post-combiner signal of Eq. (2.100) becomes

      where is the Gaussian white noise.

      When the channel is a Rayleigh distribution, the mean value of the output SNR can be obtained.

      It can be seen that the array gain increases linearly with M_R, and it is greater than the array gain of selective combining. In addition, the diversity gain of equal gain combining is M_R, which is similar to that of the selective combining.

      (2) Maximum ratio combining

      The selection weight of maximum ratio combining is , and its post-combiner signal

      where n′ = h^Hn. Because it maximizes the output SNR ρ_out, this strategy is called maximum ratio combining. And

      In the maximum ratio combining diversity scheme, the array gain g_a is always equal to M_R.

      Consider the case of transmitting with BPSK modulation. It is well known that when u = ||h||² and different channels are independently distributed Rayleigh channels, u obeys 2M_R degrees of freedom χ² distribution.

      The bit error rate can be given by

      when the SNR is large, the above equation becomes