# Pilot-Based Adaptive Channel Estimation for Underwater Spatial Modulation Technologies

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## Abstract

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## 1. Introduction

## 2. Spatial Modulation Technologies

#### 2.1. Spatial Modulation

**x**is nonzero, where g represents the index of active antenna. Then, at the ${k}^{th}$ symbol interval, at the SM-MIMO transmitter the resultant output transmit vector can be represented as:

#### 2.2. Generalized Spatial Modulation

#### 2.3. Fully Generalized Spatial Modulation

## 3. Optimal Detector

## 4. Underwater Acoustic Channel Model

## 5. Recursive Least-Squares Adaptive Channel Estimation

^{th}transmitted antennas and received antennas r

_{x}which are recursively estimated to minimize the sum of square errors denoted as:

Algorithm 1 Recursive Least-Squares |

Input: The individual entries of $\mathbf{y}$ are represented by ${d}_{i}$ as $\mathbf{y}={[{d}_{1},{d}_{2},\dots ,{d}_{i}]}^{T}$, and the individual rows of $\mathbf{H}$ can be denoted by ${u}_{i}$ as $\mathbf{H}={[{\mathbf{u}}_{1},{\mathbf{u}}_{2},\dots ,{\mathbf{u}}_{i}]}^{T}$.Initialization: ${\mathbf{w}}_{0}=[\text{\hspace{1em}}]$, ${\mathbf{P}}_{0}=\epsilon \mathbf{I}$ the initial covariance matrix $\mathbf{P}$, here $\epsilon $ is a large positive number and $\lambda =0.995$.Repeat: 1: ${\alpha}_{i}=\frac{1}{(1+{\lambda}^{-1}{\mathbf{u}}_{i}{\mathbf{P}}_{i-1}{\mathbf{u}}_{i}^{T})}$ 2: ${\mathbf{l}}_{i}={\lambda}^{-1}{\mathbf{P}}_{i-1}{\mathbf{u}}_{i}{\alpha}_{i}$; 3: ${e}_{i}={d}_{i}-{\mathbf{u}}_{i}{\mathbf{w}}_{i-1}$; % minimizing the error 4: ${\mathbf{w}}_{i}={\mathbf{w}}_{i-1}+{\mathbf{l}}_{i}{e}_{i}$; % update the filter value 5: ${\mathbf{P}}_{i}={\lambda}^{-1}{\mathbf{P}}_{i-1}-{\lambda}^{-1}{\mathbf{l}}_{i}{\mathbf{P}}_{i-1}$; % updating the $\mathbf{P}$ value End |

## 6. Simulation Results and Discussion

#### 6.1. Spatial Modulation

#### 6.2. Generalized Spatial Modulation

#### 6.3. Fully Generalized Spatial Modulation

#### 6.4. Spectral Efficiency

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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AWGN | Additive White Gaussian Noise |

BER | Bit Error Rate |

bpcu | Bit Per Channel Use |

CSI | Channel State Information |

EFGSM | Enhanced Fully Generalized Spatial Modulation |

FGSM | Fully Generalized Spatial Modulation |

GSM | Generalized Spatial Modulation |

GSSK | Generalized Spatial Shift Keying |

IAS | Inter-antenna Synchronization |

ICI | Inter-carrier Interference |

LMS | Least Mean Square |

LoS | Line of Sight |

LS | Least-Squares |

MIMO | Multiple-Input Multiple-Output |

ML | Maximum Likelihood |

M-PSK | M-ary Phase Shift Keying |

M-QAM | M-ary Quadrature Amplitude Modulation |

MSE | Mean Square Error |

P-CSI | Perfect Channel State Information |

Probability Density Function | |

PSAM | Pilot Symbol Assisted Modulation |

RF | Radio Frequency |

RLS | Recursive Least-Squares |

SIMO | Single-Input Multiple-Output |

SM | Spatial Modulation |

SMT | Spatial Modulation Technology |

SNR | Signal to Noise Ratio |

SSK | Spatial Shift Keying |

UWA | Underwater Acoustic |

UWAC | Underwater Acoustic Communication |

UWC | Underwater Communication |

V-BLAST | Vertical-Bell Laboratories Layered Space-Time |

Block Bits | Spatial Bits | Data Bits | Antenna Index |
---|---|---|---|

00 b1b2 | 00 | b1b2 | 1 |

01 b1b2 | 01 | b1b2 | 2 |

10 b1b2 | 10 | b1b2 | 3 |

11 b1b2 | 11 | b1b2 | 4 |

Block Bits | Spatial Bits | Data Bits | Antenna Index |
---|---|---|---|

00 b1b2 | 00 | b1b2 | 1,2 |

01 b1b2 | 01 | b1b2 | 1,3 |

10 b1b2 | 10 | b1b2 | 1,4 |

11 b1b2 | 11 | b1b2 | 2,3 |

Block Bits | Spatial Bits | Data Bits | Antenna Index |
---|---|---|---|

000 b1b2 | 000 | b1b2 | 1 |

001 b1b2 | 001 | b1b2 | 2 |

010 b1b2 | 010 | b1b2 | 3 |

011 b1b2 | 011 | b1b2 | 4 |

100 b1b2 | 100 | b1b2 | 1,2 |

101 b1b2 | 101 | b1b2 | 1,3 |

110 b1b2 | 110 | b1b2 | 1,4 |

111 b1b2 | 111 | b1b2 | 2,3 |

Number of Pilots | Used Spectral Efficiency | MSE | BER FGSM 7 pbcu) | BER GSM (6 pbcu) | BER SM (6 pbcu) |
---|---|---|---|---|---|

2% | 98% | 0.5678 | 0.3155 | 0.2003 | 0.2939 |

5% | 95% | 0.0953 | 0.1219 | 0.0436 | 0.1025 |

10% | 90% | 0.0180 | 0.0781 | 0.0235 | 0.0757 |

15% | 85% | 0.0091 | 0.0730 | 0.0220 | 0.0729 |

20% | 80% | 0.0060 | 0.0679 | 0.0195 | 0.0697 |

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**MDPI and ACS Style**

Junejo, N.U.R.; Esmaiel, H.; Sun, H.; Qasem, Z.A.H.; Wang, J.
Pilot-Based Adaptive Channel Estimation for Underwater Spatial Modulation Technologies. *Symmetry* **2019**, *11*, 711.
https://doi.org/10.3390/sym11050711

**AMA Style**

Junejo NUR, Esmaiel H, Sun H, Qasem ZAH, Wang J.
Pilot-Based Adaptive Channel Estimation for Underwater Spatial Modulation Technologies. *Symmetry*. 2019; 11(5):711.
https://doi.org/10.3390/sym11050711

**Chicago/Turabian Style**

Junejo, Naveed Ur Rehman, Hamada Esmaiel, Haixin Sun, Zeyad A. H. Qasem, and Junfeng Wang.
2019. "Pilot-Based Adaptive Channel Estimation for Underwater Spatial Modulation Technologies" *Symmetry* 11, no. 5: 711.
https://doi.org/10.3390/sym11050711