Fourier Transform

Lesson Title: Fourier Transform

Lesson Number: 26

Background:

In Chapter 11 the basic form of the Fourier transform and series was established. The Fourier transform pair was defined as:

The Fourier transform was said to exist if the signal is absolutely integrable (although this is not a necessary condition). At times, the production of a Fourier transform of series can be a straightforward calculation (e.g., pulse), at other times it requires great finesse. The difficult cases can often be resolved by properly interpreting the inverse transform equation as illustrated below. Fortunately many of the important transforms are summarized in Tables 11.2 with properties given in Table 11.3.

Challenge:

1. What is the inverse Fourier transform of d(w-w₀)?

2. Suppose x(t)=sgn(t)= (see Figure 1), what is X(jw)?

Figure 1: sgn function.

Response:

1. Inverse Fourier transform if a line spectrum d(w-w₀).

Therefore, fince , . Notice computing the Fourier transform of directly is far more challenging as suggested below.

You need to evaluate this integral as w w₀ from both sides. Your on your own from this point on.

2. Suppose x(t)=sgn(t).

Notice that x(t) is technically not absolutely integrable. Nevertheless, its Fourier transform exists and can be computed as shown below. Approximate the sgn function using the decaying exponentials shown in Figure 1. That is:

Then

which has a magnitude of 2/w, which peaks at w=0, and a phase shift equal to -90 for w³0 and 90 for w<0.