# Spread Spectrum Modulation with Grassmannian Constellations for Mobile Multiple Access Underwater Acoustic Channels

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## Abstract

**:**

## 1. Introduction

^{−1}) is five orders of magnitude lower than the speed of light in air, and this causes long propagation delays and multipath channels. The communication system can be considered as ultra-wideband, since the center frequency is comparable with the available bandwidth (as an example, in this paper, we consider a center frequency of 27 kHz with a bandwidth equal to 4 kHz). Sound attenuation in water is frequency dependent and time-varying, and together with background acoustic noise, they limit the achievable data rate considerably as indicated in [1,2]. Because of these drawbacks, realizing a communication system that makes a fleet of Autonomous Underwater Vehicles (AUVs) a mobile connected network is a challenging task. An example of the considered scenario consists of multiple AUVs that communicate with a receiver array deployed over a surface control unit, which could be a boat or a buoy. In the following, we consider the access channel between a fleet of ${N}_{u}$ AUVs transmitting data and a receiver equipped with ${N}_{r}$ hydrophones situated at the sea surface.

- The implementation of Doppler shift estimation and frame synchronization processes at the receiver side for more realistic communication and a comparison of the MU-HFM spreading sequence against Pseudo-Noise (PN) sequence as preamble for Doppler shift estimation.

## 2. System Model

#### 2.1. Mathematical Notations

#### 2.2. Transmitter

#### 2.3. Receiver Architecture

## 3. Grassmannian Constellations

#### 3.1. Transmitter Modulation Scheme Design

#### 3.2. Receiver Demodulation Scheme Design

## 4. Results and Discussion

#### 4.1. Watermark Replay Channel

#### 4.2. Performance Metrics

#### 4.3. Performance Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

AUV | Autonomous Underwater Vehicle |

AWGN | Additive White Gaussian Noise |

BER | Bit Error Rate |

CP | Cyclic Prefix |

CDMA | Code-Division Multiple Access |

CIR | Channel Impulse Response |

CSI | Channel State Information |

CSMA | Carrier Sense Multiple Access |

PSK | Phase Shift Keying |

DFE | Decision Feedback Equalizer |

DPSK | Differential Phase Shift Keying |

DBPSK | Differential Binary Phase Shift Keying |

DQPSK | Differential Quaternary Phase Shift Keying |

DSSS | Direct Sequence Spread Spectrum |

FDMA | Frequency Divsion Multiple Access |

FEC | Forward Error Correction |

FER | Frame Error Rate |

FFT | Fast Fourier Transform |

FSK | Frequency Shift Keying |

HFM | Hyperbolically Frequency Modulation |

ISI | Inter Symbol Interference |

MC-CDMA | Multi-Carrier Code-Division Multiple Access |

MLS | Maximal Length Sequence |

CSS | Chirp Spread Spectrum |

LFM | Linear Frequency Modulation |

MU-CSS | MultiUser Chirp Spread Spectrum |

MU-HFM | MultiUser Hyperbolically Frequency Modulation |

MU-MIMO | multiuser Multiple-Input Multiple-Output |

OFDM | Orthogonal Frequency Division Multiplex |

PN | Pseudo-Noise |

PPC | Passive Phase Conjugation |

QAM | Quadrature Amplitude Modulation |

R_c | FEC rate |

RMS | Root Mean Square |

SIMO | Single Input Multiple Output |

SISO | Single Input Single Output |

SINR | Signal-to-Interference-plus-Noise Ratio |

SNR | Signal-to-Noise Ratio |

SRRC | Square Root Raised Cosine |

TDMA | Time-Division Multiple Access |

UAC | Underwater Acoustic channel |

UWA | Underwater Acoustic |

Watermark | underWater AcousTic channEl Replay benchMARK |

VTRM | Virtual Time Reversal Mirror |

FrFT | Fractional Fourier Transform |

RMSE | Root Mean Square Error |

## Appendix A. Calculation of γ_{i,k,p}, η_{i,k,p} and w_{i,k,p}

## Appendix B. Definition of the Bijective Mapping **ϕ**_{l} for the Cube-Split Modulation

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**Figure 2.**Transmitter diagram, where ${N}_{SF}$ denotes the spreading factor, and ${\mathsf{\Pi}}_{{T}_{s}}\left(t\right)$ is the rectangular function of duration ${T}_{s}$.

**Figure 4.**Delay-Doppler spread function extracted from channel sounding for ${D}_{1}=200$ m, roadstead of Brest, France.

**Figure 5.**RMSE for Doppler shift estimation based on the number of users for different types of preamble of a 63.75 ms duration for 5000 frames.

**Figure 6.**AverageFrame Error Rate (FER) performance (

**up**) and effective data rate per user (

**down**) versus the number of users for the replayed channel of the roadstead of Brest, where ${N}_{r}=1$ sensor, and average SNR = 10 dB. The abbreviation PN means that a PN code type preamble was used; in the other cases, a MU-HFM type preamble was considered.

**Figure 7.**AverageFER performance (

**up**) and effective data rate per user (

**down**) versus the number of users for the replayed channel of the roadstead of Brest, where ${N}_{r}=5$ sensors, and average SNR = 10 dB. The abbreviation PN means that a PN code type preamble was used; in the other cases, a MU-HFM type preamble was considered.

Symbol | Signification | Value |
---|---|---|

${f}_{c}$ | Center frequency | 27 kHz |

${f}_{s}$ | Sampling frequency | 96 kHz |

B | Signal bandwidth | 4 kHz |

${D}_{i}$ | Transmission range | [65, 540] m |

${z}_{w}$ | Water depth | 10 m |

SNR | Signal to noise ratio | 10 dB |

${\tau}_{max}$ | RMS channel delay spread [30] | [8.85, 26.49] ms |

${\sigma}_{max}$ | RMS channel Doppler spread [30] | [0.85, 2.9] Hz |

Symbol | Signification | Value |
---|---|---|

M | Grassmannian modulation order | 4, 8 |

${L}_{0}$ | Number of local coordinates | 1, 2 |

${N}_{r}$ | Number of hydrophone receiving | 5 |

${N}_{s}$ | Number of symbols per frame | 200 |

${N}_{f}$ | Number of frames | 5000 |

$\mathcal{C}$ | FEC code type | Convolutive code |

${g}_{\mathcal{C}}$ | FEC code generator | (133, 171)_{o} |

${R}_{\mathcal{C}}$ | FEC code rate | $\frac{1}{2}$ |

${T}_{g}$ | Guard interval time TDMA | 31.3 ms |

${T}_{c}$ | Chip duration | 0.25 ms |

${f}_{l}$, ${f}_{h}$ | Bounds of HFM signal | 6 kHz, 10 kHz |

$\alpha $ | Pulse shaping filter roll-off factor | 0.25 |

${T}_{s}$ | Symbol duration | 31.75 ms |

${N}_{SF}$ | Spreading factor | 127 |

${T}_{pr}$ | Preamble duration | 63.75 ms |

${N}_{S{F}_{pr}}$ | Spreading factor for the preamble | 255 |

${T}_{{g}_{pr}}$ | Guard interval time between the preamble and the message | 100 ms |

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**MDPI and ACS Style**

Bernard, C.; Bouvet, P.-J.; Tomasi, B.
Spread Spectrum Modulation with Grassmannian Constellations for Mobile Multiple Access Underwater Acoustic Channels. *Sensors* **2022**, *22*, 8518.
https://doi.org/10.3390/s22218518

**AMA Style**

Bernard C, Bouvet P-J, Tomasi B.
Spread Spectrum Modulation with Grassmannian Constellations for Mobile Multiple Access Underwater Acoustic Channels. *Sensors*. 2022; 22(21):8518.
https://doi.org/10.3390/s22218518

**Chicago/Turabian Style**

Bernard, Christophe, Pierre-Jean Bouvet, and Beatrice Tomasi.
2022. "Spread Spectrum Modulation with Grassmannian Constellations for Mobile Multiple Access Underwater Acoustic Channels" *Sensors* 22, no. 21: 8518.
https://doi.org/10.3390/s22218518