# Linear Approximation Signal Detection Scheme in MIMO-OFDM Systems
## 3 Conventional Detection Schemes
### Dfe  
For higher error performance than the linear detection, the DFE uses ordered decision feedback and successive interference cancellation at each layer. The DFE has three stages, i.e., ordering, QR decomposition, and finally, decision feedback and IAI cancellation. At first, the DFE decides the cancelling order with respect to the norm value of each row of the \(\mathbf{G}\) in Equation (3) to minimize the error propagation. Likewise, each column of the \(\mathbf{H}\) is also ordered with respect to above order and the ordered channel matrix \(\mathbf{H}_{\text{S}}\) is defined as follows,  
\[\mathbf{H}_{\text{S}}=\mathbf{H}\mathbf{U} \tag{5}\]  
where \(\mathbf{U}\) is column change matrix, which is composed of 1 and 0.  
Then, \(\mathbf{H}_{\text{S}}\) is decomposed by using QR decomposition. In the QR decomposition, Q denotes unitary quadrature matrix, which is satisfied with \(\mathbf{Q}^{H}\mathbf{Q}=\mathbf{I}\) and R denotes the upper triangular matrix. The QR decomposition of the \(\mathbf{H}_{\text{S}}\) is as follows,  
\[\begin{split}\mathbf{H}_{\text{S}}&=\mathbf{Q} \mathbf{R}\\ &=\left[\begin{array}{cccc}Q_{11}&Q_{12}&\cdots&Q_{1N}\\ Q_{21}&Q_{22}&\cdots&Q_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ Q_{N1}&Q_{N2}&\cdots&Q_{NN}\end{array}\right]\left[\begin{array}{cccc}R_{1 1}&R_{12}&\cdots&R_{1N}\\ 0&R_{22}&\cdots&R_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&R_{NN}\end{array}\right].\end{split} \tag{6}\]  
Using Equation (6), the MIMO received vector in Equation (1) can be rewritten as follows,  
\[\mathbf{Y}=\mathbf{H}\mathbf{X}+\mathbf{N}=\mathbf{H}_{\text{S}}\mathbf{X}_{ \text{S}}+\mathbf{N}=\mathbf{Q}\mathbf{R}\mathbf{X}_{\text{S}}+\mathbf{N}, \tag{7}\]  
where \(\mathbf{X}_{\text{S}}=\mathbf{U}^{-1}\mathbf{X}\) is ordered transmit vector.  
Multiplying \(\mathbf{Q}^{H}\) on both sides of Equation (7), the resulted \(N\times 1\) vector \(\mathbf{V}\) is as follows,  
\[\mathbf{V}=\mathbf{Q}^{H}\mathbf{Y}=\mathbf{R}\mathbf{X}_{\text{S}}+\mathbf{ \tilde{N}}, \tag{8}\]  
where \(\mathbf{\tilde{N}}=\mathbf{Q}^{H}\mathbf{N}\) is modified noise vector which has the same statistical property as existing noise vector \(\mathbf{N}\) because \(\mathbf{Q}\) is unitary orthogonal matrix.  
The vector form of \(\mathbf{V}\) without noise is as follows,  
\[\left[\begin{array}{c}V_{1}\\ V_{2}\\ \vdots\\ V_{N}\end{array}\right]=\left[\begin{array}{cccc}R_{11}&R_{12}&\cdots&R_{1N}\\ 0&R_{22}&\cdots&R_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&R_{NN}\end{array}\right]\left[\begin{array}{c}X_{\text{S},1}\\ X_{\text{S},2}\\ \vdots\\ X_{\text{S},N}\end{array}\right]. \tag{9}\]  
Finally, decision feedback and interference cancellation are operated from the bottom layer to the top layer in Equation (9). The estimated MIMO transmit symbol at the \(n\)-th layer \(\hat{X}_{n}\) is as follows,  
\[\hat{X}_{\text{S},n}=Q\left(\left(V_{n}-\sum_{i=n+1}^{N}R_{ni}\hat{X}_{i} \right)/R_{nn}\right), \tag{10}\]  
where \(Q(\cdot)\) denotes quantization operator.  
To estimate the MIMO transmit vector, \(\hat{X}_{\text{S}}\) is multiplied by \(\mathbf{U}\) as follows,  
\[\hat{\mathbf{X}}=\mathbf{U}\hat{X}_{\text{S}}. \tag{11}\]  
The DFE has higher error performance than the ZF due to detection ordering. However, the complexity of the DFE is higher than the ZF.