# Theoretical Derivation and Optimization Verification of BER for Indoor SWIPT Environments

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## Abstract

**:**

## 1. Introduction

## 2. System Model

#### 2.1. System Description

#### 2.2. Transmitter Architecture Design

_{n}. ${\alpha}_{n}\left(f\right)=-\frac{2\pi f}{c}\sqrt{{\epsilon}_{r}}\xb7{l}_{n}$, where C is the light speed and ${\epsilon}_{r}$ is the permittivity of the feed line. We exploited eight antennas at the transmitter, positioned at φ

_{n}= 0, 45, 90, 135, 180, 225, 270, and 315 degrees in the circle. Equation (4) derives the array factor of the transmitter as the following equation:

#### 2.3. Impulse Response

_{1}to f

_{2}can be expressed as

#### 2.4. Receiver Architecture Design

## 3. Resource Allocation Problem Formulation

#### 3.1. Quality of Service (QoS) Factor for Indoor SWIPT System

_{d}is the time span of the signal. The pulse-amplitude modulation symbols $d\in \left\{\pm 1\right\}$ are considered to be independently identically distributed, and the Gaussian waveform can be expressed as

#### 3.2. Optimization Problem Formulations

## 4. Evolution Algorithms

#### 4.1. APSO

#### 4.2. Self-Adaptive Dynamic Differential Evolution

## 5. Numerical Results

#### 5.1. Environment Settings

#### 5.2. Simulation Result

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

List of mathematical symbols and abbreviations | |

Proper noun/mathematical symbols | Abbreviations/Description |

Simultaneous wireless information and power transfer | SWIPT |

Internet of Things | IoT |

Multiple-input multiple-output | MIMO |

Energy efficiency | EE |

Self-adaptive dynamic differential evolution | SADDE |

Asynchronous particle swarm optimization | APSO |

Bit error rate | BER |

Ultra-wideband | UWB |

Quality of service | QoS |

Federal Communications Commission | FCC |

${E}_{tot}\left(r,\theta ,\varphi \right)$ | Electrical field |

$\theta $ | The spherical coordinate system |

$\varphi $ | The spherical coordinate system |

${\theta}_{n}$ | The spherical coordinates of the transmitting antenna |

${\varphi}_{n}$ | The spherical coordinates of the transmitting antenna |

${A}_{n}$ | The phase term |

$A{F}_{}\left(\theta ,\phi ,f\right)$ | The array factor |

${I}_{n}$ | The excitation current voltage |

$\lambda $ | The wavelength |

$k$ | The wave-number |

$R$ | the radius of circle array |

${\epsilon}_{r}$ | permittivity |

${l}_{n}$ | the feed line length |

$\mathrm{N}\left(\theta ,\phi ,f\right)$ | the three-dimensional radiation field vector |

$H\left(f\right)$ | the channel frequency response |

${b}_{i}$ | the ith receiving magnitude |

$f$ | the frequency of sinusoidal wave |

${\theta}_{i}$ | the ith phase shift |

${f}_{1}$ | Start frequency |

${f}_{2}$ | Stop frequency |

${H}_{\mathrm{UWB}}\left(f\right)$ | The frequency response of the ultra-wideband |

${P}_{i,j}$ | The received RF signals power |

${\mu}_{i}$ | The fraction of RF signals |

$\eta $ | The portion of RF signals |

${G}_{k}$ | The channel vectors between the transmitter and the kth idle receiver |

${W}_{n}$ | The corresponding beam-forming vector |

${h}_{\mathrm{UWB}}\left(t\right)$ | The multi-path channel with an impulse response |

$x\left(t\right)$ | The pulse stream |

${\tau}_{i}$ | The time delay for the ith path |

$p\left(t\right)$ | The Gaussian waveform |

$r\left(t\right)$ | The received signal |

$q\left(t\right)$ | The receiver samples the signal with suitably delayed references |

$Z\left(n\right)$ | The output of the correlator |

$\sigma $ | The variance of the output noise |

${\alpha}_{i}$ | The attenuation of the pat |

${T}_{d}$ | The duration of the signal |

$d$ | The pulse-amplitude modulation symbols |

${E}_{k}$ | The total amount of energy harvested by the K idle receivers |

$HP{\left(w\right)}_{total}$ | The total energy harvested |

$TP\left(w\right)$ | The energy radiated by the transmitter |

${\eta}_{eff}\left(w\right)$ | The energy harvesting efficiency |

${W}_{BER}$ | The constraint for the system parameter |

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**Figure 5.**Objective value generations for the self-adaptive dynamic differential evolution (SADDE) and asynchronous particle swarm optimization (APSO) algorithms.

**Figure 6.**Radiation pattern for Tx to Rx by SADDE in different generations: (

**a**) 100, (

**b**) 150, (

**c**) 250, and (

**d**) 600.

**Figure 7.**Radiation pattern for Tx to Rx by APSO in different generations: (

**a**) 100, (

**b**) 150, (

**c**) 250, and (

**d**) 600.

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**MDPI and ACS Style**

Chien, W.; Hsieh, T.-T.; Chiu, C.-C.; Cheng, Y.-T.; Lee, Y.-H.; Chen, Q.
Theoretical Derivation and Optimization Verification of BER for Indoor SWIPT Environments. *Symmetry* **2020**, *12*, 1185.
https://doi.org/10.3390/sym12071185

**AMA Style**

Chien W, Hsieh T-T, Chiu C-C, Cheng Y-T, Lee Y-H, Chen Q.
Theoretical Derivation and Optimization Verification of BER for Indoor SWIPT Environments. *Symmetry*. 2020; 12(7):1185.
https://doi.org/10.3390/sym12071185

**Chicago/Turabian Style**

Chien, Wei, Tzong-Tyng Hsieh, Chien-Ching Chiu, Yu-Ting Cheng, Yang-Han Lee, and Qiang Chen.
2020. "Theoretical Derivation and Optimization Verification of BER for Indoor SWIPT Environments" *Symmetry* 12, no. 7: 1185.
https://doi.org/10.3390/sym12071185