# A Low Complexity Reactive Tabu Search Based Constellation Constraints in Signal Detection
## 4 Simulation Results
It can be seen that the proposed CC-RTS has a performance close to the traditional RTS algorithm. Besides, in the case that the channel is time-varying, we also make a model simulation. The channel model adopted is Doppler Frequency Shift, and the value of \(d_{th}\) is 0.02. In this simulation process, we used rayleighchan MATLAB build in function to generate channel with Doppler Frequency Shift. The sampling frequency shift was 1/15,000 and the maximum Doppler Frequency Shift is 500. The performance of the proposed CC-RTS and reference RTS with Channel with Doppler Frequency Shift is shown in Figure 4. From Figure 4, we can get a conclusion that: With Doppler Frequency Shift, CC-RTS algorithm also have the advantages of greatly reducing the computational complexity of RTS  
Figure 3: Bit error rate performance of CC-RTS taking MMSE solution as the initial vector for \(4\times 4\), \(8\times 8\), \(16\times 16\), \(24\times 24\) MIMO system with Rayleigh channel, QPSK modulation, \(d_{th}=\) 0.02. Figure 2: Bit error rate performance of different \(d_{th}\) for \(4\times 4\), \(8\times 8\), \(16\times 16\), \(24\times 24\) MIMO systems with QPSK modulation. (The dashed line indicates the performance of the reference traditional RTS.) algorithm without affecting the performance of the traditional RTS algorithm, although the bit error rate is slightly different. **Complexity**: In order to further verify the proposed algorithm, we exam the complexity reduction over the traditional RTS algorithm. The complexity analysis is based on the assumption that the radius of the CC device is given by 0.02. We can see from Figure 5 that with the proposed scheme, more than half number of the candidates are excluded under different antenna configurations compare to the traditional RTS algorithm. For same number of antennas, with an increasing value of SNR, the number of candidates is gradually reduced and the subsequently calculation of ML cost and overall detection complexity can be saved. In addition, from Figure 6 and Table 4, we can see that for a given target BER performance, such as \(10^{-2}\), the per-symbol-complexity (PSC) of the CC-RTS algorithm is smaller than that of the traditional RTS algorithm. The following is a detailed derivation of the mathematical expression of PSC:  
It is known that the most expensive part of the RTS algorithm is to calculate the Euclidean distance. For the \(N_{t}*N_{r}\) MIMO models, \(N_{t}*N_{r}\) multiplication and \(N_{t}*N_{r}\) addition are needed in this process. If the number of iterations is \(Item\), each symbol's neighbor is \(M\), then the computational complexity of each symbol in the RTS algorithm is \(\frac{Item*M*N_{t}*(2*N_{t}*N_{r}-1)}{N_{t}}\). As an improvement of RTS algorithm, CC-RTS algorithm could significantly reduce the number of search candidates without reducing the ML cost of each set of candidates. Therefore, the computational complexity of CC-RTS algorithm and RTS algorithm on a vector is consistent, but the overall computational complexity of CC-RTS algorithm is smaller than that of RTS algorithm. Figure 4: Bit error performance of CC-RTS taking MMSE solution as the initial vector for \(4\times 4\), \(8\times 8\), \(16\times 16\), \(24\times 24\) MIMO system with Channel with Doppler Frequency Shift, QPSK modulation, \(d_{th}=0.02\). Figure 5: Candidates reduction: percentage of candidates over traditional RTS algorithm for \(4\times 4\), \(8\times 8\), \(16\times 16\), \(24\times 24\) MIMO systems with QPSK modulation, \(d_{hh}=0.02\). With the proposed scheme, \(50\%\) of the candidates and their corresponding complexity of ML calculation can be saved.