Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises, 4th Edition

APPENDIX A

Laplace and Fourier Transforms

Both Laplace and Fourier transforms are used frequently in random signal analysis. Laplace transforms are especially useful in analysis of systems that are governed by linear differential equations with constant coefficients. In this application the transformation makes it possible to convert the original problem from the world of differential equations to simpler algebraic equations. Fourier transforms, on the other hand, simply furnish equivalent descriptions of signals in the time and frequency domains. For example, the autocorrelation function (time) and power spectral density function (frequency) are Fourier transform pairs.

Short tables of common Laplace and Fourier transforms will now be presented for reference purposes.

A.1 THE LAPLACE TRANSFORM

Electrical engineers usually first encounter Laplace transforms in circuit analysis, and then again in linear control theory. In both cases the central problem is one of finding the system response to an input initiated at t = 0. Since the time history of the system prior to t = 0 is summarized in the form of the initial conditions, the ordinary one-sided Laplace transform serves us quite well. Recall that it is defined as

The defining integral is, of course, insensitive to f(t) for negative t; but, for reasons that will become apparent shortly, we arbitrarily set f(t) = 0 for t<0 in one-sided ...