The Laplace Transform and the z-transform

Abstract

The Laplace Transform and the z-transform are closely related to the Fourier Transform, and to our work in the two preceding chapters. There are several good reasons for covering these additional transforms in a book on electronic signals and systems. The Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems. It is particularly valuable for analysing signal flow through causal LTI systems with nonzero initial conditions. The Laplace Transform also overcomes some of the convergence problems associated with the continuous-time Fourier Transform, and can handle a broader class of signal waveforms. The z-transform, on the other hand, is especially suitable for dealing with discrete signals and systems. It offers a more compact and convenient notation than the discrete-time Fourier Transform. The Laplace Transform and the z-transform should be viewed as complementary to the Fourier Transform, rather than essentially different. Since we have already covered much of the relevant conceptual framework, we will concentrate on those features of the transforms which shed additional light on electronic systems.

“I have a little shadow That goes in and out with me, And what can be the use of him Is more than I can see.”

Robert Louis Stevenson (1850–1894)

“Keeping time, time, time, In a sort of Runic rhyme, To the tintinabulation that so musically wells From the bells, bells, bells, bells.”

Edgar Allen Poe (1809–1849)