Practical Signals Theory with MATLAB Applications

The Illustrated Fourier Transform

In the pages that follow, a series of Fourier transform pairs are presented, each showing a function of time s(t) on the left and the corresponding Fourier transform function of frequency S( f ) on the right, where:

In each graph, real components are shown with a solid line, while imaginary components are plotted with a dashed line. The time- and frequency-domain sketches may be switched directly with at most a change of sign in the imaginary component, since:

This illustrated list provides a ready reference for common Fourier transform pairs from which more complicated Fourier transforms may be derived by applying the appropriate rules, often by inspection and without the need for further computations.

As a study guide, careful examination of each transform pair confirms and reveals a wealth of information about the general properties characterizing the Fourier transform, particularly those relating to odd and even functions, real and imaginary components, and both periodic and nonperiodic signals. For example, in each case, expect that the value at origin in one domain equals the area in the other.

FIGURE A.1 δ(t) ⇔ 1.

FIGURE A.2 δ(t − 5)⇔cos(10 ...