# A Novel Linear Antenna Synthesis for Linear Dispersion Codes Based on an Innovative HYBRID Genetic Algorithm

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Linear Dispersion Code

_{r}receiving antennae [39]. After matched filtering and symbol level sampling, the received signal matrix in the receiving antenna array is:

_{r}× T matrix is $V$, in which all elements are not dependent and identically distributed complex Gaussian random variables in a complex domain with a distribution is $CN(0,{N}_{0})$. E

_{s}is the total power transmitted on the

**M**transmitting antenna. The SNR (signal to noise ratio) of those receiving antennae is written by $SNR:={E}_{s}/{N}_{0}$. At this point, the spectrum efficiency of the system is defined as:

## 3. System Model

_{t}transmitting antennae and M

_{r}receiving antennae in it, but just M transmit antennae are selected by the transmitter to transmit. $H$ being a ${M}_{r}\times {M}_{t}$ channel coefficient array whose elements are absolutely independent and its distribution is $CN\left(0,1\right)$. At the same time, supposing the receiver has known an ideal CSI (channel status information), and given the channel state $H$ and ${\left\{{M}_{n}\right\}}_{n=0}^{N-1}$, then the ${M}_{r}\times {M}_{t}$ channel array Hs can be selected by one function $\hslash $ below:

## 4. Maximum Minimum Posterior SNR Criterion for the LDC-TAS

^{−2}, moreover, with the increase of SNR, the performance gain was higher.

## 5. Antenna Synthesis Via an Innovative HYBRID Genetic Algorithm

_{k}, k = 0, 1, 2, 3, …, M − 1.

_{k}being the weighting coefficients of the kth element, λ means the wavelength of the propagation channel, u = sin θ − sin θ

_{0}, and θ

_{0}and θ are respectively the steering angles of these arrays and the incident angle of the impinging plane wave. It is very essential to combine an array configuring to produce a BP that meets some requirements (e.g., side lobes level (SLL)) or rebuilds a special shape (p

^{ref}(u)).

_{0}, …, x

_{k}, …, x

_{M−1}; w

_{0}, …, w

_{k}, …, w

_{M−1}]

^{t}being the uncertain quantity arrays and where:

_{dB}(u)/Q} > ${p}_{dB}^{ref}\left(u\right)$, but ${p}_{dB}^{ref}\left(u\right)$ is a desired BP shape. Finally, based on the optimization method, k

_{1}, k

_{2}, k

_{3}, and k

_{4}were all selected as the normalized parameters.

- These SGAs cooperate with the parameters’ coding over computation, not these parameters themselves;
- These SGAs being a multiple-agent hunting process (i.e., a multiple sampling of those hunting spaces);
- These SGAs do not have to make use of those derivatives indeed;
- These SGAs do not apply the deterministic transition schemes, but use some random ones instead.

- The application for a hybrid coding way;
- The non-correlation among the chromosomes’ genes (i.e., the power weights parameters and the genes, which representing the position of those array elements should be optimized synchronously);
- The conception over a prior-knowledge-enhanced operator;
- The design of the scheme for a mind of the adaptive evolution;
- The definition of a hybridization design using the local hunting way.

_{2}(L), and L being the number of these values that the discrete parameters are able to suppose [11]. However, due to the time consuming coding/decoding processes [12] and the quantization noise, to use the binary coding is disadvantageous and unpractical once the real unknowns are taken in account. Consequently, the real-valued expression is suggested to overtake those drawbacks above [13].

_{k}; k = 0, 1, 2, …, M − 1}, and number of the active elements, M. In this part, a hybrid coding based on the GA is adopted to address this problem properly. The chromosome is supposed as the structure below:

_{k}is a Boolean value (i.e., true or false) giving a status (off or on) of the kth array elements, k being an integer number of (λ/2) intervals between the left array limit (x

_{k}= k λ/2) and the kth element, w

_{k}is the related excitation coefficient, and the parameter N is the number of intervals (λ/2 in length) in which the array length has been dispersed.

**The Solution for Genetic Operators**

#### 5.1. The Selection

#### 5.2. The Crossover

_{c}).

#### 5.3. The Mutation

_{bm}. To let the variations into the chromosome, these values of the string position are changed. The mutation is executed under the different schemes on the basis of the type of the gene to change. Once those genes that are randomly chosen are two-dimensionally-valued, b

_{k}, and the standard binary mutation is performed [17] by using the different probabilities for the birth of the array element or the death.

#### 5.4. Elitism

## 6. Simulation Analyses

#### 6.1. Result Analyses

_{slp}) via the changing element weight was solved. As for the problem, in [22] the combining process was utilized to the linear matrix of 25 isotropic elements D = 50 λ in length. The power weight parameters were set to change inside the range [0.2, 2.0] and the parameter u

_{start}was fixed to 0.04 to compare the performances of the results achieved via normal GA with the IGA optimization.

_{slp}= −14.77 dB. However, to the authors’ knowledge, the best threshold for the side-lobes level obtained in article [26] was just −14.45 dB approximately, and u

_{ml}= 0.191 was the width of the half-beam width.

_{slp}= −14.67 dB and the main-lobe width value u

_{ml}= 0.0204.

_{k}; k = 0, 1, 2, 3, …, N − 2, N − 1) and an optimization procedure took these array element positions into account. Figure 8 draws the beam pattern and the specific location position layout in contact with the best optimal ways, according to these peak values of the minimum side-lobe level (Φ

_{slp}= −13.06 dB as well as u

_{ml}= 0.0170; IGA (a)) and according to the optimal trade-off among the side-lobe level value and those main-lobe widths (Φ

_{slp}= −12.32 dB as well as u

_{ml}= 0.0126; IGA (b)), achieved via the ways of those IGA-based methods. These specific characters of the array synthesized using the SA-based procedure are given (Φ

_{slp}= −12.07 dB as well as u

_{ml}= 0.0133; SA).

#### 6.2. Complexity Analyses

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Performance of linear dispersion code transmitting antenna selection (LDC-TAS) vs. space–time block code (STBC)-TAS.

**Figure 6.**Optimization of the element power weights—(

**a**) beam pattern (BP) with a side-lobe peak value of −14.77 dB; and (

**b**) positions and power weights of those array elements.

**Figure 7.**Optimization of the element power weights—peak side-lobe value level as a specific function of the main-lobe width ((dB) units; simulated innovative genetic algorithm (IGA); O: Simulated SA).

**Figure 8.**Optimization of the element location positions and power weights—optimal combinations for isophoric arrays. (

**a**) Beam patterns; and (

**b**) array placement.

Region | BP Amplitude Factors |
---|---|

0 < $\mu $ ≤ 0.0418 | 0 dB |

0.0419 < $\mu $ ≤ 0.307 | −13.368 dB |

0.308 < $\mu $ ≤ 0.446 | −26.876 dB |

0.447 < $\mu $ ≤ 0.797 | 13.395 dB |

0.798 < $\mu $ ≤ 1 | 0 dB |

**Table 2.**Statistical behavior of the quantity of those active elements, side lobe peak value as well as the width of main-lobe behind several dozens of the procedure implement.

Quantity of the Active Elements (M) | |||

The Best | The Worst | Average | Std. Dev |

149 (73.5%) | 156 (76.5%) | 148.9 (74.8%) | 1.0358 |

Side-Lobe Peak Value (Φ_{slp}; dB) | |||

The Best | The Worst | Average | Std. Dev |

−23.19 | −22.61 | −22.91 | 0.2322 |

Main-Lobe Width (u_{ml}) | |||

The Best | The Worst | Average | Std. Dev |

0.0050 | 0.0052 | 0.00508 | ~0 |

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**MDPI and ACS Style**

Wang, J.; Ye, Z.; M. Sanders, T.; Li, B.; Zou, N.
A Novel Linear Antenna Synthesis for Linear Dispersion Codes Based on an Innovative HYBRID Genetic Algorithm. *Symmetry* **2019**, *11*, 1176.
https://doi.org/10.3390/sym11091176

**AMA Style**

Wang J, Ye Z, M. Sanders T, Li B, Zou N.
A Novel Linear Antenna Synthesis for Linear Dispersion Codes Based on an Innovative HYBRID Genetic Algorithm. *Symmetry*. 2019; 11(9):1176.
https://doi.org/10.3390/sym11091176

**Chicago/Turabian Style**

Wang, Jinpeng, Zhengpeng Ye, Tarun M. Sanders, Bo Li, and Nianyu Zou.
2019. "A Novel Linear Antenna Synthesis for Linear Dispersion Codes Based on an Innovative HYBRID Genetic Algorithm" *Symmetry* 11, no. 9: 1176.
https://doi.org/10.3390/sym11091176