# Low Power Consumption Signal Detector Based on Adaptive DFSD in MIMO-OFDM Systems
## 4 Proposed Adaptive DFSD Detection Scheme
### The Thresholds Based on Channel Condition  
The channel condition number is defined as follows,  
\[\Gamma(m)\simeq\frac{\sigma_{\max}(m)}{\sigma_{\min}(m)}, \tag{13}\]  
where \(\Gamma(m)\) is the condition number of subcarrier channel matrix, \(\sigma_{\max}(m)\) is maximum singular value, \(\sigma_{\min}(m)\) is minimum singular value, and \(m\) is the subcarrier index. The channel condition is determined by the condition number. When \(\Gamma(m)\) is close to one, it is the best channel condition but when \(\Gamma(m)\) is larger than one, it is the worst channel condition [18; 19]. Through the above conditions, the complexity depends on thresholds,  
\[\theta_{i}=\sum_{m=1}^{N}\frac{2\Gamma(m)i}{MN} \tag{14}\]  
Equation (14) shows the process of determining thresholds through statistical analysis. \(M\) denotes the modulation order, \(i\) denotes the index of threshold, \(N\) is the number of subcarriers and the number of thresholds is \(M-1\). Since \(\Gamma(m)\) is different depending on the channel condition, the thresholds should be distributed evenly according to the number of \(i\) indexes based on minimum \(\Gamma(m)\) and maximum \(\Gamma(m)\). For example, in QPSK modulation, \(\theta_{1}\) is present at a quarter of the all \(\Gamma(m)\). Also, \(\theta_{2}\) is present at a half of the all \(\Gamma(m)\), and \(\theta_{3}\) is present at three quarters of the all \(\Gamma(m)\). It means that there are thresholds for each interval on the channel condition number. Therefore, the DFSD detection scheme, having a different number of nodes at the \(N_{t}\)-th layer, is adaptively used when \(\Gamma(m)\) changes.  
Figure 4 shows the relationship between the channel condition number and the thresholds in QPSK modulation. In the adaptive DFSD detection scheme, when \(\Gamma(m)\) is smaller than \(\theta_{1}\), that channel has the best condition and uses one node at the \(N_{t}\)-th layer. Then, the DFSD detection scheme is processed from the \((N_{t}-1)\)-th layer to the last layer as shown in Figure 3a. When \(\Gamma(m)\) is between \(\theta_{1}\) and \(\theta_{2}\), the channel has a worse condition than the former channel and uses two nodes at the \(N_{t}\)-th layer. Then, the DFSD detection scheme is also processed from the \((N_{t}-1)\)-th layer to the last layer. In this way, when \(\Gamma(m)\) is between \(\theta_{2}\) and \(\theta_{3}\), the adaptive DFSD detection scheme uses three nodes at the \(N_{t}\)-th layer. Finally, when \(\Gamma(m)\) is larger than \(\theta_{3}\), the channel has the worst condition and uses all nodes at the \(N_{t}\)-th layer as shown in Figure 3b. It means that the adaptive DFSD detection scheme uses a few number of nodes at the \(N_{t}\)-th layer in a good channel condition to reduce the complexity. In contrast, the adaptive DFSD detection scheme uses a large number of nodes at the \(N_{t}\)-th layer in a bad channel condition to improve the error performance.  
Figure 4: The relationship between the channel condition number and the thresholds.  
Figure 5 shows the flowchart for the adaptive DFSD detection scheme. The flowchart contains the method for determining the number of nodes at the \(N_{t}\)-th layer and for reducing the path.