Fourier and Laplace Transforms: Concepts and Applications

This lecture covers the fundamental concepts of Fourier and Laplace transforms, focusing on their definitions, properties, and applications in analyzing signals. The instructor begins with a recap of the Fourier transform, explaining its definition for piecewise continuous functions and the conditions for its well-defined nature. The inverse Fourier transform is also introduced, emphasizing its role in recovering the original function. The lecture then transitions to the Laplace transform, detailing its definition for signals defined on the positive real line and its relationship to differential equations. The instructor highlights the similarities and differences between the two transforms, particularly in their applications to signal processing and system analysis. Key properties such as linearity, time shifting, and frequency shifting are discussed, along with theorems that underpin these concepts. The lecture concludes with practical examples illustrating how these transforms are utilized in engineering and physics, providing students with a comprehensive understanding of these essential mathematical tools.