# A Low Complexity Reactive Tabu Search Based Constellation Constraints in Signal Detection

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## Abstract

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## 1. Introduction

- We propose a scheme, namely Constellation Constraints (CC) algorithm for determining the reliability of the initial vector.
- With the proposed CC structure, the algorithm separate the reliable symbol estimates in the initial vector from unreliable symbols.
- A algorithm is developed for generating a new candidate set and neighborhood searching space and their implementation with RTS detection.
- The simulation results of the proposed scheme are given to verify that similar BER performance can be achieved with significant lower computational complexity.

## 2. System Model

## 3. CC-RTS Algorithm

**Maximum Likelihood (ML)**: Maximum likelihood detection algorithm calculates the Euclidean distance between the received signal vector and the product of all possible transmitted signal vectors multiplied by channel

**H**and searching for the one with the minimum distance. Let $\mathit{A}$ denote the whole ML search space with a number of combination candidates which increases exponentially with a raising number of spatial multiplexed transmit antennas modulation level. ML algorithm determines the estimated transmitted signal vector $\widehat{\mathbf{s}}$ as:

**Traditional RTS Algorithm**: The implementation of RTS algorithm in massive MIMO [10,11,12] is introduced in the following. With sufficient number of receiving antennas, the RTS algorithm has a performance closes to the ML algorithm, but with significant lower detection complexity. This algorithm use the MMSE output as the initial solution vector, and find the neighborhood candidates of the initial solution vector; Then, the algorithm computes the respective ML cost with these candidates and searching for the one with the minimum cost. If it is the case that the selected candidate does not appear in the tabu list (Tabu list is used to hold the initial solution of the next iterative process. The elements in the list follow first in first out FIFO rules. The tabu length indicates that the element will not be used again as the initial solution vector in the next few iterations. The length of tabu is divided into variable and immutable. If the tabu length is fixed, then the algorithm is called fixed tabu search. If the length is a variable, then the algorithm is termed reactive tabu search), the selected candidate is used as the initial value for the next iteration of searching, meanwhile the algorithm includes this selected candidate into the tabu list; On the other hand, if the candidate is already included in the tabu list, a suboptimal solution outside the tabu list is selected as the initial value for the next iteration of searching, and the suboptimal solution is added to the tabu list. After several iterations, the candidate with the minimum ML cost can be selected as the final output of the RTS algorithm.

**RTS neighborhood**: The selection criteria of neighborhood candidates in [9] is the metric of Euclidean distance, that is, to find the nearest constellation candidates from a given point. For example: If we use $\widehat{\mathbf{s}}$ to indicate a transmitting signal symbol, and $\mathbf{Neb}\left(\widehat{\mathbf{s}}\right)$ denotes the neighborhood candidates of $\widehat{\mathbf{s}}$. For QPSK modulation, if we consider a $2\times 2$ MIMO model and the initial value is considered to be ${\widehat{\mathbf{s}}}_{i}=\frac{1}{\sqrt{2}}\{-1-j\}$, the number of candidates for ${\widehat{\mathbf{s}}}_{i}$ is 2, by finding the nearest two candidates, we have $\mathbf{Neb}\left({\widehat{\mathbf{s}}}_{i}\right)=\frac{1}{\sqrt{2}}\{-1+j,1-j\}$.

**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 transimitting 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 ${\widehat{\mathbf{s}}}_{i}$.

**Reliable**: The reliable condition is defined as that if the signal to be detected falls into the bright area of the graph. For instance if $\widehat{\mathbf{s}}$ is the initial value of RTS, and the soft output ${\widehat{\mathbf{s}}}_{i}$ falls into the bright area of graph, a hard decision is considered to determine the output ${\widehat{\mathbf{s}}}_{i}$ of the nearest constellation and, then we neither consider generating candidates nor conduct RTS algorithm for this soft output. A quantization operation $k(\u25a0)$ is then performed as$${\widehat{\mathbf{s}}}_{i}=k\left({\widehat{\mathbf{s}}}_{i}\right)$$**Unreliable**: The unreliable condition is that the detection signal falls into the shaded area of the graph. If $\widehat{\mathbf{s}}$ is the initial value of RTS, and the soft output ${\widehat{\mathbf{s}}}_{i}$ falls into the shaded area of the graph, we conduct hard decision for ${\widehat{\mathbf{s}}}_{i}$ and generate candidates of $k\left({\widehat{\mathbf{s}}}_{i}\right)$ for the RTS algorithm. The CC processing is invoked and a candidates set is generated as$$\mathbf{Neb}\left({\widehat{\mathbf{s}}}_{i}\right)=[{c}_{1},{c}_{2},\cdots ,{c}_{m},\cdots ,{c}_{M}]\u03f5\mathbf{A}$$The generated candidates are constrained by the constellation map and the candidate set is a selection of the M nearest constellation points to ${\widehat{\mathbf{s}}}_{i}$. The size of $\mathbf{Neb}\left({\widehat{\mathbf{s}}}_{i}\right)$ can be either fixed or adaptive with the channel condition which introduces a tradeoff between the performance and the detection complexity with a selection algorithm.

## 4. Simulation Results

**Performance**: In Figure 2 and Figure 3, we present the simulated BER performance of the proposed CC-RTS algorithm with various selection of ${d}_{th}$ with Rayleigh channel. Figure 2 shows that the variation of BER as a function of the ${d}_{th}$, with ${N}_{t}$ = ${N}_{r}$ = 4, 8, 16, 24, respectively, and all the simulations are with SNR = 8dB. It can be observed from Figure 2 that, depending on the choice of the value of ${d}_{th}$, BER and ${d}_{th}$ rendering correlation. Figure 3 shows the performances of the proposed CC-RTS and reference RTS, with ${N}_{t}$ = ${N}_{r}$ = 4, 8, 16, 24, and ${d}_{th}$ =0.02. It can be seen that the proposed CC-RTS has a performance close to the traditional RTS algorithm.

**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:

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**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.

**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.)

**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 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, ${d}_{th}$ = 0.02. With the proposed scheme, 50% of the candidates and their corresponding complexity of ML calculation can be saved.

**Figure 6.**Per-symbol-complexity (PSC) in number of real operations and SNR required to achieve ${10}^{-2}$ BER in QPSK for $4\times 4$, $8\times 8$, $16\times 16$, $24\times 24$ MIMO system (Ref: Figure 5 ).

**Table 1.**The number of nearest neighbors for different modulation modes: BSPK, QPSK, 16-QAM, 64-QAM.

Modulation Mode | BPSK | QPSK | 16-QAM | 64-QAM |
---|---|---|---|---|

Neighbors | ${N}_{t}$ | 2${N}_{t}$ | 4${N}_{t}$ | 4${N}_{t}$ |

Input: $\mathit{r}$, $\mathit{H}$, ${d}_{th}$ |

Output: ${\widehat{\mathbf{s}}}_{best}$ |

Initialization: |

${\widehat{\mathbf{s}}}_{soft}\leftarrow $ the output of MMSE |

$\widehat{\mathbf{s}}\leftarrow $ corresponding hard decision of ${\widehat{\mathbf{s}}}_{soft}$ |

${\mathbf{s}}_{stemp}\leftarrow \widehat{\mathbf{s}}$ |

$Item$← Number of iterations |

Best-so-far $\leftarrow \left|\right|\mathit{r}-\mathit{H}{\mathbf{s}}_{stemp}{\left|\right|}^{2}$ |

P ← the length of tabulist |

Constellation Constraints: |

for$i\leftarrow $ 1 to ${N}_{t}$ do |

$j\leftarrow {i}_{th}$ element of ${\widehat{\mathbf{s}}}_{soft}$ |

if j is unreliable |

${\mathbf{sit}}_{i}\leftarrow 1$ |

else |

${\mathbf{sit}}_{i}\leftarrow 0$ |

end if |

end for |

CC-RTS: |

it = 1 |

while it ≤ $Item$ |

$\mathbf{cat}-\mathbf{matrix}\leftarrow \left[\right]$ |

generate tabulist according to P |

for $k\leftarrow $ 1 to ${N}_{t}$ do |

if ${\mathbf{sit}}_{k}==1$ |

$\mathbf{cat}\leftarrow $ the candidates of ${k}_{th}$ element of ${\mathbf{s}}_{temp}$ |

$\mathbf{cat}-\mathbf{matrix}\leftarrow [\mathbf{cat}-\mathbf{matrix};\mathbf{cat}]$ |

end if |

end for |

Using simplified $\mathbf{cat}-\mathbf{matrix}$ to perform traditional RTS algorithm |

it = it+1 |

end while |

Modulation | Channels | Symbol Vectors | Initial Tabu Length | Iterations |
---|---|---|---|---|

QPSK | Rayleigh Fading Channel | 100,000 | 3 | 20 |

**Table 4.**Per-symbol-complexity (PSC) in number of real operations and SNR required to achieve BER = ${10}^{-2}$ with QPSK for $4\times 4$, $8\times 8$, $16\times 16$, $24\times 24$ MIMO systems (Ref: Figure 5 ).

Algorithm | Per-Symbol-Complexity in Number of Real Operations and SNR Required to Achieved ${10}^{-2}$ BER for QPSK | The Computational Complexity of Each Symbol | |||||||
---|---|---|---|---|---|---|---|---|---|

4 × 4 | 8 × 8 | 16 × 16 | 24 × 24 | ||||||

PSC | SNR | PSC | SNR | PSC | SNR | PSC | SNR | ||

RTS | 1320 | 8.3 | 2600 | 7.1 | 20,520 | 6 | 46120 | 5.8 | O(M) + O(${N}_{t}^{2}$) |

CC-RTS | 573 | 8.2 | 1148 | 7.1 | 9168 | 6 | 20662 | 5.8 | O(M) + O(${N}_{t}^{2}$) |

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**MDPI and ACS Style**

Feng, J.; Zhang, X.; Li, P.; Hu, D. A Low Complexity Reactive Tabu Search Based Constellation Constraints in Signal Detection. *Algorithms* **2018**, *11*, 99.
https://doi.org/10.3390/a11070099

**AMA Style**

Feng J, Zhang X, Li P, Hu D. A Low Complexity Reactive Tabu Search Based Constellation Constraints in Signal Detection. *Algorithms*. 2018; 11(7):99.
https://doi.org/10.3390/a11070099

**Chicago/Turabian Style**

Feng, Jiao, Xiaofei Zhang, Peng Li, and Dongshun Hu. 2018. "A Low Complexity Reactive Tabu Search Based Constellation Constraints in Signal Detection" *Algorithms* 11, no. 7: 99.
https://doi.org/10.3390/a11070099