Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

      where E[|s_i|²] = E_i and E[|n|²] = N₀. If the received power of the two signals is assumed to be the same, namely E₁ = E₂, and the channel gain is also assumed to be the same, namely |h₁|² = |h₂|², then the SINR of signal s₁ will be less than 0 dB, which brings great difficulty to the signal detection.

      Successive interference cancellation is an alternative method to improve the performance of signal detection. Assuming E₁ > E₂, the SINR of s₁ is relatively high at this time, and it is possible to detect s₁ first. Let ₁ be the detection value of s₁, and if the detection of ₁ is correct, then it is possible to eliminate the interference of s₁ during the detection process of s₂. Therefore, the interference-free detection of s₂ can be realized, which is expressed as

      This detection method is called Successive Interference Cancellation (SIC) and can be applied to MIMO joint signal detection.

      In order to realize SIC, QR decomposition plays an important role in the SIC-based detection process.^{10, 11} QR decomposition is a common method of matrix decomposition, which can be decomposed into the product of an orthogonal matrix and an upper triangular matrix. And how the 2 × 2 MIMO system performs QR decomposition will be introduced first in this section.

      Assume that there is a 2 × 2 channel matrix H = [h₁ h₂], where h_i represents the ith column vector of H. Define the inner product of the two vectors to be . In order to find an orthogonal vector with the same lattice as H, we define

      where

      According to the linear relationship described by Eq. (2.154), it can be judged that [h₁ h₂] and [r₁ r₂] can span the same subspace. And if r_i is a non-zero vector (i = 1, 2), it can be found that

      where q_i = r_i/||r_i||. Based on Eq. (2.156), an orthogonal matrix Q = [q₁ q₂] and an upper triangular matrix can be obtained, and thus, the QR decomposition of H is achieved. It is worth noting that if r₂ = h₂ and r₁ = h₁ – ωh₂, another QR decomposition result of H can be obtained.

      The successive interference cancellation of the received signal can be performed according to the QR decomposition of the channel matrix. This section only discusses the case where the channel matrix H is a square matrix or a thin matrix (M ≤ N) whose number of rows is larger than the number of columns.

       1. H is a square matrix

      H can be decomposed into an M × M unitary matrix Q and an M × M upper triangular matrix R.

      where r_p,q is defined as the (p, q)th element of R. By the premultiplication of Q^H, the received signal can be expressed as

      where Q^Hn is a zero-mean CSCG random vector. Q^Hn has the same statistical property as n, and hence, n can be used directly to replace Q^Hn. As a result, Eq. (2.158) can be transformed into

      If x_k and n_k are defined as the kth element of x and n, the above equation can be expanded as follows:

      Therefore, SIC detection can be carried out

       2. H is a thin matrix

      The QR decomposition of H is

      where M < N and Q is an unitary matrix. We have , with denoting an M × M upper triangular matrix. According to Eq. (2.162), the received signal vector can be expressed as

      Furthermore,

      

      Since the received signal {x_M+1, x_M+2, . . . , x_N} does not contain any useful information, it can be ignored directly. On this basis, Eqs. (2.161) and (2.164) are identical in form, and thus, successive interference cancellation can be realized. First, s_M can be detected based on x_M.

      If is used as the K-QAM constellation symbol set of the signal, the detection expression of s_M is

      The above equation shows that since there is no interference term in the detection of s_M, the influence of s_M can be eliminated during

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