3.7

GENERALIZED FOURIER TRANSFORM

In Example 3.8 it was shown that the unit-step function () does not have a Fourier transform in the ordinary sense. It is also easy to see that cos(at) and sin(wo) do not have a Fourier transform in the ordinary sense. Since the step function and sinusoidal functions often arise in the study of signals and systems, it is very desirable to be able to define the Fourier transform of these signals. We can do this by defining the notion of the generalized Fourier transform, which is considered in this section.

First, the Fourier transform of the unit impulse (1) will be computed. Recall that (t) is defined by