# Low Power Consumption Signal Detector Based on Adaptive DFSD in MIMO-OFDM Systems
## 3 Conventional Signal Detection Schemes
### Conventional Depth-First Sphere Decoding  
The DFSD detection scheme has similar error performance to the ML detection scheme with a low complexity. The DFSD detection scheme is restricted to the search area of the ML detection scheme in a sphere with radius of \(C\) and centers around the received symbol. The DFSD detection scheme inequality is as follows [14; 15; 16],  
\[\left\|\mathbf{Y}-\mathbf{HX}\right\|^{2}\leq C^{2}, \tag{4}\]  
where \(C\) is sphere radius. Because it is hard to determine which lattice nodes lay inside of the \(m\)-dimensional sphere, the DFSD detection scheme is represented at the tree structure. Before the DFSD detection scheme searches the received symbol \(\mathbf{Y}\), QR-decomposition is performed from the complex channel matrix \(\mathbf{H}\). The equation of channel matrix is as follows,  
\[\mathbf{H}=\mathbf{Q}\mathbf{R}, \tag{5}\]  
where \(\mathbf{Q}\) is the \(N_{r}\times N_{r}\) unitary matrix and \(\mathbf{R}\) is the \(N_{r}\times N_{t}\) upper triangular matrix. To remove the elements of \(\mathbf{Q}\), Equation (1) is multiplied by \(\mathbf{Q}^{H}\) as follows,  
\[\mathbf{Z}=\mathbf{Q}^{H}\mathbf{Y}=\mathbf{Q}^{H}(\mathbf{HX}+\mathbf{N})= \mathbf{RX}+\mathbf{Q}^{H}\mathbf{N}, \tag{6}\]  
\[\mathbf{Z}=\left[ \begin{array}{cccc}r_{11}&r_{12}&\cdots&r_{1N_{t}}\\ 0&r_{22}&\cdots&r_{2N_{t}}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&r_{N_{t}N_{t}}\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&0\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{N_{t}}\end{array} \right]+\left[ \begin{array}{c}\hat{n}_{1}\\ \hat{n}_{2}\\ \vdots\\ \hat{n}_{N_{t}}\end{array} \right], \tag{7}\]  
where \((\cdot)^{H}\) denotes the Hermitian transpose, \(\mathbf{Z}\) denotes the altered received signal vector and \(\hat{n}_{i}\,(i\,=\,1,\,2,\,\cdots\,,N_{r})\) denotes the elements of the altered noise vector. The object of DFSD detection scheme is to find the first radius in a depth direction at the tree structure. So, this detection scheme uses a squared Euclidean distance (ED). The squared ED is represented as follows,  
\[d_{N_{t}}(k)=\|z_{N_{t}}-r_{N_{t}N_{t}}s(k)\|^{2}, \tag{8}\]  
where \(d_{N_{t}}(k)\) is the \(k\)-th squared ED at the \(N_{t}\)-th layer, \(z_{N_{t}}\) is the element of altered received signal vector, \(r_{N_{t}N_{t}}\) is the element of \(\mathbf{R}\) matrix and \(s(k)\,(k=1,2,\,\cdots\,,\,|S|)\) is the reference symbol. At the \((N_{t}-1)\)-thlayer, to consider the exact signal estimation, the DFSD detection scheme uses the accumulated squared ED as follows,  
\[d_{N_{t}-1}^{t}(k) =\left\|z_{N_{t}-1}-\left\{r_{N_{t}-1N_{t}-1}s(k)+r_{N_{t}-1N_{t}} \hat{x}_{N_{t}}^{t}\right\}\right\|^{2} \tag{9}\] \[\quad+d_{N_{t}}(k),\]  
where \(\hat{x}_{N_{t}}^{t}\) is the \(t\)-th estimated symbol among the reference symbols at the \(N_{t}\)-th layer. The Equation (9) is generalized to express the \(n\)-th layer \((1\leq n\leq N_{t}-1)\) as follows,  
\[d_{n}^{t}(k)=\left\|z_{n}-\left\{r_{nn}s(k)+\sum_{l=n+1}^{N_{t}}r_{nl}\hat{x}_ {l}^{t}\right\}\right\|^{2}+d_{n+1}^{t}(k_{n+1}), \tag{10}\]  
where \(\hat{x}_{l}^{t}\) is the \(t\)-th estimated symbol among the reference symbols from the \(N_{t}\)-th layer to the \((n+1)\)-th layer and \(d_{n+1}^{t}(k_{n+1})\) is the \(k_{n+1}\)-th accumulated squared ED from the \(N_{t}\)-th layer to the \((n+1)\)-th layer. Figure 2 shows DFSD tree structure in a \(4\times 4\) MIMO-OFDM system with quadrature phase shift keying (QPSK) modulation. There are too many metrics to include in Figure 2. So, several metrics are omitted through the dotted lines. Real numbers at the side of each metric are accumulated squared ED. At the first stage, squared ED of Equation (8) is calculated at the \(N_{t}\)-th layer, then, the DFSD detection scheme selects the node which has the shortest squared ED. At the next layer, this detection scheme calculates the accumulated squared ED from the selected node of the \(N_{t}\)-th layer to the candidate symbol nodes of the \((N_{t}-1)\)-th layer.