Fourier, Laplace, and the Tangled Love Affair with Transforms

Unlock the intricate language of signals and systems with this in-depth exploration of Fourier and Laplace transforms. Designed for advanced undergraduates, graduate students, and professionals in engineering, physics, and applied mathematics, this book unravels the foundations of signal processing with a rigorous yet engaging approach.

Beginning with the fundamentals and building to advanced topics, each chapter guides you through the Fourier series, Fourier, and Laplace transform and into the realms of discrete Fourier and Z transforms, multi-dimensional analysis, and applications of the Fourier Transform in solving PDE, ODE, and Integral equations. The text brings mathematical theory to life through real-world applications in signal synthesis, systems engineering, and differential equations, making complex topics accessible and inspiring.

With its unique blend of historical insights, practical applications, and intuitive explanations, this book offers a comprehensive yet approachable journey into the world of transforms. Whether you're a student building your foundation or a professional seeking to deepen your expertise, this book invites you to discover the elegance and utility of transforms in a way that bridges theory with the demands of modern engineering and science.

Offers an exploration of the mathematical foundations of signal processing, using Fourier and Laplace transforms Breaks down concepts progressively, from fundamentals to advanced topics, making it ideal for students and professionals Demystifies complex topics, weaving theory with application in a way that is both comprehensive and accessible

Autorentext

Dr. Sofen Kumar Jena is currently a researcher at the Process & Energy Laboratory in the Department of Mechanical Engineering at the Technical University of Delft, Netherlands. Dr. Jena holds a PhD in Mechanical Engineering and has over a decade of experience in research and professional work, particularly in applied mathematics. His expertise spans various domains, but his primary focus has been on turbulence modelling and the analysis of turbulence signals, which are key components of his research. This book emerges from his core research, reflecting his deep engagement with the mathematical principles behind signal analysis in turbulent systems. Dr. Jena's unique combination of profound theoretical understanding and practical experience allows him to present intricate mathematical concepts in an accessible and engaging manner. His work consistently bridges the gap between complex theoretical constructs and their real-world applications, offering readers a comprehensive understanding and practical insights.

Inhalt

Introduction.- Historical Development of The Subject.- Signals & Systems.- Fourier Series.- Fourier Analysis.- Fourier Transforms of Generalized Functions.- The Laplace Transform.- Discrete Fourier Transform.- Convolution, Cross-correlation & Stochastic Analysis.- Linear Systems.- Z Transform.- Higher Dimensional Fourier Analysis.- Fourier Analysis for The Solution of Differential and Integral.- Equations.- Conclusion.