Signals, Systems, and Signal Processing

This chapter presents the Laplace transform, which is as fundamental to continuous-time systems as the z-transform is to discrete-time systems. Several properties and examples are presented. Similar to the z-transform, the Laplace transform can be regarded as a generalization of the appropriate Fourier transform. In continuous time, the Laplace transform is very useful in the study of systems represented by linear constant-coefficient differential equations (i.e., rational LTI systems). Frequency responses, resonances, and oscillations in electric circuits (and in mechanical systems) can be studied using the Laplace transform. The application in electrical circuit analysis is demonstrated with the help of an LCR circuit. The inverse Laplace transformation is also discussed, and it is shown that the inverse is unique only when the region of convergence (ROC) of the Laplace transform is specified. Depending on the ROC, the inverse of a given Laplace transform expression may be causal, noncausal, two-sided, bounded, or unbounded. This is very similar to the theory of inverse z-transformation. Because of these similarities, the discussion of the Laplace transform in this chapter is brief.