Continuous Time Fourier Transform and Its Extension to Laplace Transform

Summary

This chapter extends the Fourier series representation of continuous time periodic signals to aperiodic signals. The generalized form of Fourier series, which enables us to represent both continuous time periodic and aperiodic functions in terms of their frequency content, is called the Fourier transform. Fourier transform enables us to represent the time domain signals in the frequency domain. The existence of frequency domain representation is assured by Dirichlet conditions. We study the basic properties of Fourier transforms, such as linearity, time shifting, time scaling, derivation, and integration. We prove that convolution operation in the time domain is transformed into multiplication in the frequency domain. We also study an important concept, called duality. We observe that differential and integral equations become algebraic equations in the frequency domain. Thus, solving them in the frequency domain is rather easier than solving them in the time domain. Finally, we define a new domain, called the Laplace domain, where the purely imaginary frequency domain variable is extended to a complex variable. This generalization enables us to find the Laplace transforms of the functions, which do not exist in the frequency domain, and the Laplace transform exists in some regions of the complex plane, called the region of convergence.