Compressed Sensing in Li-Fi and Wi-Fi Networks

Malek Benslama, Hatem Mokhtari

• Publisher: ISTE Press - Elsevier
• Year: 2017
• Print ISBN: 9781785482007
• eBook ISBN: 9780081019689
• Topic: Tecnologia e ingegneria
• Subtopic: Comunicazioni mobili e wireless

Compressed Sensing in Li-Fi and Wi-Fi Networks features coverage of the first applications in optical telecommunications and wireless. After extensive development of basic theory, many techniques are presented, such as non-asymptotic analysis of random matrices, adaptive detection, greedy algorithms, and the use of graphical models. The book can be used as a comprehensive manual for teaching and research in courses covering advanced signal processing, efficient data processing algorithms, and telecommunications. After a thorough review of the basic theory of compressed sensing, many mathematical techniques are presented, including advanced signal modeling, Nyquist sub-sampling of analog signals, the non-asymptotic analysis of random matrices, adaptive detection, greedy algorithms, and the use of graphical models.- Offers extensive development of basic theory behind telecommunications and wireless networks- Contains broad coverage of treat compressed sensing, including electromagnetism signals- Provides insights into the two key areas of telecommunications, WIFI and LIFI- Includes information on advanced signal modeling, Nyquist sub-sampling of analog signals, the non-asymptotic analysis of random matrices, adaptive detection, greedy algorithms, and more

1

Shannon’s Theorem in Classic Data Processing

Abstract

Lombardi et al. give a comprehensive explanation of Shannon information. The question asked in their overview is: what is Shannon information? This chapter’s approach seems useful in measuring or understanding compressed sensing and a prelude on Shannon’s theorem is a necessary preliminary. Lombardi relies on numerous references: we highlight the work of Bell, Fisher, Khinchin, Kolmogorov and MacKay on data in general, without forgetting the depth of insight developed by Landauer.

Keywords

Capacity channel; Cryptography; Data Processing; Entropy; Sampling; Shannon’s theorem

1.1 Introduction

Lombardi et al. give a comprehensive explanation of Shannon information [LOM 16]. The question asked in their overview is: what is Shannon information? This chapter’s approach seems useful in measuring or understanding compressed sensing and a prelude on Shannon’s theorem is a necessary preliminary. Lombardi relies on numerous references: we highlight the work of Bell [BEL 57], Fisher [FIS 25], Khinchin [KHI 57], Kolmogorov [KOL 65, KOL 68, KOL 83] and MacKay [MAC 69] on data in general, without forgetting the depth of insight developed by Landauer [LAN 91, LAN 96].

It is traditionally accepted that Shannon’s seminal work on the data point in math is the paper in which he introduces a specific formalism designed to solve some specific technical problems [SHA 48] . Shannon entropy is considered with a given system’s statistical properties and the correlation between the states of two systems independently of their meaning or their semantic content. Shannon’s theorem is the basic ingredient of digital communication studies. Currently, the philosophy of data has posed a number of problems linked to the concept of data. The possibility of unifying these data theorems, the question of the logic of information, the relationship between data and thermodynamics, the significance of quantum data and links between data and calculation are crucial to a complete understanding of data.

We could assume that all questions about the meaning and interpretation of Shannon data have had clear and undisputable answers. However, this is not the case.

1.2 Shannon’s theorem

In his article on mathematical data theory [SHA 48], Shannon gave us specific results on the resources needed for optimal coding and error-free communication. Shannon’s theorem was then formalized by Khinchin [KHI 57].

According to Shannon [SHA 48], a general communication system is formed of the following five parts:

• – a source S that generates the message that should be received at the destination;
• – a transmitter T that transmits the message generated at the source into a signal that should be transmitted. Where data are coded, the coding is also implanted into this system;
• – a channel CH that is the medium used to transmit the signal from the transmitter to the receiver;
• – a receiver R that reconstructs the message from the signal;
• – a destination D that receives the message.

In Figure 1.1, we give an explicit schema of the system.

Figure 1.1 Schema of the system

The source S is a system with an interval of possible states s ₁,…s_n usually called letters, whose respective probabilities of occurrence are p(s₁),…,p(s_n).

The quantity of information generated at the source through the occurrence of s_i can be defined as [SHA 48]:

$I\left(s_i\right)=log\left(\frac{1}{p\left(s_i\right)}\right)=-logp\left(s_i\right)$

[1.1]

Since S produces sequences of states usually called messages, the source’s entropy s is defined as [SHA 48]:

$H\left(S\right)={\sum }_{i=1}^{n}p\left(s_i\right)log\left(\frac{1}{p\left(s_i\right)}\right)=-{\sum }_{i=1}^{n}p\left(s_i\right)logp\left(s_i\right)$

[1.2]

By analogy, the destination D is a sy...

Table of contents

• Cover image
• Title page
• Table of Contents
• Dedication
• Copyright
• Preface
• List of Acronyms
• Introduction
• 1: Shannon’s Theorem in Classic Data Processing
• 2: Shannon’s Theorem in Quantum Data
• 3: Sparse Signals and Compressed Sensing
• 4: Compressed Sensing and the Fourier Transform
• 5: Compressed Sensing and Entanglement
• 6: [WITHDRAWN] Compressed Sensing and Intelligent LiFi Systems
• 7: Compressed Sensing in LiFi Systems in Mobile Communications and Cryptography
• 8: Compressed Sensing in WiFi Systems
• 9: Compressed Sensing in Interconnections Covering WiMAX, UMTS and MANET Satellite Networks
• 10: Compressed Sensing in Radar Interferometry
• 11: Compressed Sensing in Radars
• 12: Compressed Sensing in Electromagnetism
• Conclusion
• Bibliography
• Index