ABSTRACT
Integral transform technique can usually be used to map a problem from one domain into another, such that the new problem may easily be solved in the transformed domain. For instance, the Laplace transform can be used to map a linear time-invariant (LTI) ordinary differential equation (ODE) into an algebraic equation. Thus the properties of the original problem, such as stability, can easily be determined, which lays a foundation of the classical control theory. In many real applications, Fourier transforms as well as Mellin transforms and Hankel transforms are all very useful. Thus computer-aided solutions to integral transform problems deserve special attention and are the main topics of this chapter. In Section 5.1, definition and properties of Laplace transform and the inverse Laplace transform are summarized. Our focus is on the MATLAB-based solutions to Laplace transform problems. Section 5.2 presents Fourier transform and its inverse transform, again with a focus on the MATLAB-based solutions. Moreover, sine and cosine Fourier transforms and discrete Fourier transforms are briefly introduced. In Section 5.3, Mellin and Hankel transforms are introduced. Z transform and inverse Z transform are introduced in Section 5.4 with illustrative examples taken from discretetime signals and systems. Finally, problems from complex variable functions such as poles and residues are demonstrated together with Partial fraction expansions for rational functions in Section 5.5, where an evaluation method of closed-path integrals is introduced based on the concepts of residues.
