Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

      First, we consider zero forcing (ZF) detection. The ZF detection linear filter is defined as

      And the corresponding ZF signal is estimated as

      With and , the hard decision of the transmitted signal vector s can be made by the symbol-level estimation.

      It should be noted that since the noise term, namely the effect of (H^HH)⁻¹H^Hn in Eq. (2.148), will be amplified, the equivalent noise will be amplified when the channel matrix H is nearly singular. Therefore, the performance of the ZF detection cannot be well guaranteed. In order to reduce the influence caused by the equivalent noise being amplified in the ZF detection, the MMSE detection utilizes the statistical property of the noise to improve the ZF detection method. The calculation of the MMSE filter matrix is based on the minimum mean square error criterion.

      where E_s represents the signal energy. The corresponding estimate of the transmitted signal vector can be expressed as

      

      Therefore, the MMSE hard decision of the signal vector s can be obtained.

      2.4.2.3Successive interference cancellation (SIC) detection

      With the consideration of the existence of interference signals, how to realize high-performance signal detection has become a key issue that modern wireless communication needs to solve. For example, assume that the signal received by the receiver is

      where s_i and h_i represent the ith signal and the channel gain experienced by the signal, respectively, and n represents the background noise. When detecting the signal s₁, the signal-to-interference plus noise ratio can be expressed as

      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