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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 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 the detection of s_M−1. This successive cancellation process can continue until all data signals are detected sequentially. In other words, the mth signal will be detected after the first M – m signals are detected and the interference is eliminated, which can be described as

where _q represents an estimated value of s_q detected from the received signal u_q. Assuming that there are no errors during all detection process, s_m can be estimated as

where .

Since the successive interference cancellation algorithm introduced above is based on the zero-forcing decision feedback equalizer (ZF-DFE), we call the algorithm a zero-forcing successive interference cancellation (ZF-SIC).^{12, 13} It should be noted that if the channel state matrix H is a fat matrix with M > N, which means the number of rows of the channel matrix is larger than the number of columns, then the successive interference cancellation algorithms cannot be used in this case due to the lack of an upper triangular array after QR decomposition.

In order to improve system performance, background noise should to be considered when performing the detection. To solve this problem, we need to employ a successive interference cancellation algorithm based on minimum mean square error decision equalizer (MMSE-DFE). And two implementation strategies of MMSE-SIC algorithm are introduced in this section.

Strategy 1: Define the extended channel matrix . extended received signal y_ex = [y^T 0^T]^T, and extended background noise vector . Through QR decomposition, we can get

where Q_ex and R_ex represent the unitary matrix and the upper triangular matrix, respectively. Replacing y, n, Q, and R in Eq. (2.158) by y_ex, n_ex, Q_ex, and R_ex, we get

Based on Eq. (2.170), MMSE-SIC detection can be carried out according to Eqs. (2.158)–(2.168).

Strategy 2: Utilizing the minimum mean square error estimator (MMSE Estimator) directly, the MMSE estimator for signal s₁ can be expressed as

where ₁ represents the first column vector of H^H. The hard decision on symbol s₁ can be described as

Assuming that s₁ can be detected correctly and its effect can be removed from y, we can obtain

According to y₁, s₂ can be detected by the MMSE method. And the MMSE-SIC detection of s_m can be carried out by repeated interference cancellation and MMSE estimation.