Abstract
In this chapter we begin the use of transformations for the representation and analysis of continuous-time signals and systems. The Laplace transform is obtained when applying complex exponentials or eigenfunctions to linear time-invariant (LTI) systems. Together with a region of convergence in the Laplace plane, the Laplace transform is uniquely related to a signal. Two-sided and one-sided Laplace transform can be defined to consider different types of signals. A time signal can thus be represented uniquely by poles and zeros, and connected with frequency and damping. The one-sided Laplace transform and its properties allow the computation of ordinary differential equations representing LTI systems, and the computation of two-sided Laplace transforms. The partial fraction is the common inversion method. Special care is needed when exponential terms appear and in the case of two-sided Laplace transforms. The Laplace transform provides the steady-state and transient analysis of LTI systems and their transfer function representation, as well as an algebraic procedure to test for bounded input–bounded output stability. MATLAB is used to compute direct and inverse Laplace transforms.
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