Preface

A wavelet is an important mathematical tool that appears in many fields of science and technology. It refers to analyzing data with special features and different scales depending on the application requirements.

The name "wavelet" first appeared in the early 1980s. It comes from the French word "ondelette," meaning "small wave." The original idea is rooted in many separate thoughts including Jean-Baptiste Joseph Fourier and his theory of approximation in which a complex function can be approximated as a weighted sum of simpler functions, Alfréd Haar and his theory of a sequence of rescaled "squareshaped" functions, and Dennis Gabor and his function for minimizing the deviation in the time and frequency domains.

The application of wavelet theory is rapidly growing in diverse fields and disciplines. As such, this book examines some of the most creative and popular applications including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations.

My personal interest in wavelet theory lies in its features, which are in contrast to Fourier transform, and its application in converting time domain signals to frequency domain signals and vice versa. The wavelet transform (WT) identifies what frequencies are present in a signal as well as when the signal experiences changes in the time domain, and thus the wavelet has information about where, what scale, and when the change occurred. This makes it very interesting for the study of high peaks in the time domain, which causes distortions in the frequency domain.

The mentioned research topics are also known as peak to average power ratio (PAPR) reduction or crest factor reduction (CRF) techniques. Any improvement of time domain peaks is directly related to the power efficiency of the amplification stage of telecommunication systems. Therefore, the study of the nonlinear behavior and distortion in high power amplifiers (HPA) is relative to the use of wavelets in this field. These topics are relevant to existing and future wireless telecommunication systems such as 5G and beyond.

I would like to thank all my family and friends for their encouragement and support. I also acknowledge that this publication is associated with CONNECT - the Science Foundation Ireland Research Centre for Future Networks and Communications.

> **Somayeh, Mohammady** School of Electrical and Electronic Engineering, Technological University Dublin (TU Dublin), Dublin, Republic of Ireland

Section 1

Wavelet Algorithm

and Overview

**1**

Section 1
