# A Low Complexity Reactive Tabu Search Based Constellation Constraints in Signal Detection
## 3 CC-RTS Algorithm
**CC-RTS**: The inspiration to use a CC device to improve traditional RTS algorithm in this paper comes from our previous work [14; 15; 16; 17]. With the analysis of MMSE filter, theoretical research shows that the estimated symbols gradually converge with an increasing value of signal-to-noise ratio. For each transmission, the distance between an estimated symbol and its nearest constellation point can be regarded as an measurement of the reliability of the estimates. For the RTS algorithm, in this work, the proposed CC algorithm [17] is used to reduce the computational complexity for traditional reactive tabu search and the algorithm may prevent the search space from growing exponentially with the increased number of transmitting antennas. In [14; 15; 16; 17], an estimation reliability measuring and refining strategy (CC algorithm) is used to detect MIMO signal with nonlinear decision feedback systems. Inspired by its principle of determining reliability, in this paper, we have modified the CC algorithm to improve heuristic neighborhood search algorithms, such as traditional RTS algorithm. A simplification of the proposed CC structure is shown in Figure 1. The parameters marked in the figure are given as follows, \(\sigma_{\mathbf{s}}\) refers to the signal power, and the threshold \(d_{th}\) represents the device radius which can control the shaded area. It can be designed as a constant, or a joint function of the signal power \(\sigma_{\mathbf{s}}\) and the noise power \(\sigma_{\mathbf{n}}\). The value \(d\) represents the distance between the detection signal and its nearest constellation point. The following equation is given to determine the nearest constellation point for a given signal \(\widehat{\mathbf{s}}_{i}\). \[\mathbf{a}_{i}=\underset{\mathbf{a}_{c}\in\mathbf{A}}{\arg\min}\ |\widehat{ \mathbf{s}}_{i}-\mathbf{a}_{c}| \tag{3}\]  
where \(\mathbf{a}_{c}\) represents the selected element of the constellation. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Modulation Mode** & **BPSK** & **QPSK** & **16-QAM** & **64-QAM** \\ \hline Neighbors & \(N_{t}\) & \(2N_{t}\) & \(4N_{t}\) & \(4N_{t}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number of nearest neighbors for different modulation modes: BSPK, QPSK, 16-QAM, 64-QAM. Figure 1: Constellation constraints structure. \(a_{i}\) is the coordinate point on the constellation. Shaded areas is unreliable area, and bright areas is reliable area. Value \(d_{th}\) is the CC radius and \(d\) denotes the distance between the estimated symbol and its nearest constellation. From Figure 1, we can see that the constellation space is divided into shaded areas and bright areas, we stipulate that if the candidate falls in the shaded area, we determine that this candidate is unreliable. The signal falling in the shadow area to meet the following condition. \[|\Im\{\widehat{\mathbf{s}}_{i}\}|\leq\frac{\sigma_{s}}{2}-d_{th},\qquad\textbf{ when}\qquad|\Re\{\widehat{\mathbf{s}}_{i}\}|\geq\frac{\sigma_{s}}{2} \tag{4}\]  
\[d\geq d_{th},|\Im\{\widehat{\mathbf{s}}_{i}\}|\leq\frac{\sigma_{s}}{2},\qquad \textbf{when}\qquad\frac{\sigma_{s}}{2}-d_{th}\leq|\Re\{\widehat{\mathbf{s}}_{ i}\}|\leq\frac{\sigma_{s}}{2} \tag{5}\]  
\[|\Im\{\widehat{\mathbf{s}}_{i}\}|\in R,\qquad\textbf{when}\qquad|\Re\{ \widehat{\mathbf{s}}_{i}\}|\leq\frac{\sigma_{s}}{2}-d_{th} \tag{6}\]  
If the signal to be detected satisfies the above conditions, the algorithm considers the candidate as unreliable. On the other hand, if the detection signal does not satisfy above conditions, the algorithm considers the candidate as reliable. If it is the case that the proposed CC structure considers the candidate as unreliable, it introduces more candidates as the output for test; While CC structure considers the candidate is reliable, the quantization of the candidate is performed at the output. The detected signals can be MMSE soft output or Zero Force soft output. * **Reliable**: The reliable condition is defined as that if the signal to be detected falls into the bright area of the graph.