Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

the maximum signal-to-noise ratio (MSNR) can be expressed as

      Comparing Eq. (2.14) with Eq. (2.6), we see that the MMSE combination is essentially an MSNR combination.

      (3) Maximum likelihood combining

      If the transmitted signal s is considered as an estimated parameter, the signal s can be estimated by the maximum likelihood (ML) estimation algorithm. Assuming that vector n in Eq. (2.2) is a zero-mean CSCG random variable, namely , then for vector r given in Eq. (2.2), the probability density of the signal s is

      Therefore, the maximum likelihood estimate for signal s is

      where the maximum likelihood combining vector is

      It is easy to know that the maximum likelihood estimation is actually achieved by adjusting the weight vector w_ML through a linear combination operation, which we call the maximum likelihood combining.

      Comparing Eq. (2.17) with Eq. (2.14), the maximum likelihood combining is essentially an MSNR combination.

      (4) Maximum ratio combining

      Maximum ratio combining (MRC) is a linear combination technology often used in the fading channel environment, which can effectively improve the system performance of the fading channel. In fact, MRC can be seen as a special case of MSNR combination.

      In order to derive the MRC algorithm, we assume that the noise terms in Eq. (2.1) are uncorrelated and their variances are equal, namely E(nn^H) = N₀I. In this case, the SNR can be obtained according to Eq. (2.12).

      According to the Cauchy–Schwartz inequality, it is easy to prove that the combined vector that maximizes SNR is w = αh. This is also a special case of the MSNR combination vector in Eq. (2.14). The linear combination of linear combination vectors w = αh is called MRC.

      When the MRC algorithm is applied, the SNR can be obtained.

      (5) Generalized selection diversity combining

      In wireless communication systems, selection diversity (SD) is also a common spatial diversity technology. Different from the principle that the MRC algorithm combines all the received signals to maximize SNR, the SD algorithm picks out only the strongest signal of the N received signals for processing, which makes it easy to implement.

      The SNR of the SD algorithm is

      where

      In order to improve the performance of SD, a generalized SD combining (GSDC) algorithm is proposed in related literatures. The generalized SD combining selects M signals from the N received signals. When M = N, GSDC is equivalent to the optimal MRC combining. When M = 1, GSDC is a common SD algorithm. It is easy to see that GSDC algorithm has a good balance between performance and computational complexity.

      If M signals are obtained under the MSNR standard, the final SNR is

      where SNR_(k) is the kth maximum SNR of SNR_k, k = 1, 2, . . . , N.

      2.1.1.2The combining method of unknown channels

      The signal combining technologies we discussed above are all based on the assumption that the channel transmission vector h is known. However, in some cases, the channel transmission vector h is difficult to accurately measure, or h is a random variable and there is no accurate value. In this case, when it is necessary to perform a mathematical estimate of the channel transmission vector, the channel transmission vector is regarded as a random variable by considering of the inevitable estimation error.

      In the following, the MMSE combining method where the channel transmission vector is a random variable is further analyzed.

      Assuming that the expectation and variance of the channel transmission vector h are known, they are given as

      When estimating the channel transmission vector h, we can replace its mean vector with the estimated value of h and replace its covariance matrix C with the estimated error covariance of h. In this case, the MMSE combining vector is

      If = 0, the MMSE combining vector is also 0, and this indicates the failure of the MMSE combining method. In order to avoid such problems, we need to correct the MMSE combining method, and the correction methods are different based on different applications.

      In the following, we will illustrate the correction of the MMSE combining method.

      Assume that the channel transmission vector h = e^jϕh₀, where ϕ is a random phase vector and h₀ is a non-zero constant vector. If ϕ is uniformly distributed, = 0, then the amplitude of the channel transmission vector gain is also constant, but its phase is time-varying. If the time-varying phase changes slowly, some conventional signal modulation technologies can be used to detect the received signal with an unknown phase estimation.

      

      If the received signal is expressed as

      and c = e^jϕs is regarded as a new detection signal, then the MMSE combining vector is

      However, it should be noted that the MMSE combining vector here is not time-varying. The output of the MMSE combiner is

      According to , the receiver can detect the signal.

      In general, if the time-varying random channel transmission vector can be decomposed into the known constant and the random variable, then we can realize the application of MMSE in the actual production process by modifying the known signals.

2.1.2Received signal detection