Repeat Example 4.1, but using the function f(t) = A for 0 ≤ t ≤ W and f(t) = 0 for all oth...

Repeat Example 4.1, but using the function f(t) = A for 0 ≤ t ≤ W and f(t) = 0 for all other values of t. Explain the reason for any differences between your results and the results in the example.

EXAMPLE 4.1: Obtaining the Fourier transform of a simple function.

The Fourier transform of the function in Fig. 4.4(a) follows from Eq. (4.2-16):

where we used the trigonometric identity sin θ = (e jθ − e−jθ)/2j In this case the complex terms of the Fourier transform combined nicely into a real sine

FIGURE 4.4 (a) A simple function; (b) its Fourier transform; and (c) the spectrum. All functions extend to infinity in both directions.

function.The result in the last step of the preceding expression is known as the sinc function:

(4.2-19)

where sinc(0) = 1, and sinc(m) = 0 for all other integer values of m.Figure 4.4(b) shows a plot of F(μ).

In general, the Fourier transform contains complex terms, and it is customary for display purposes to work with the magnitude of the transform (a real quantity), which is called the Fourier spectrum or the frequency spectrum:

Figure 4.4(c) shows a plot of |F(μ)| as a function of frequency.The key properties to note are that the locations of the zeros of both F(μ) and |F(μ)| are inversely proportional to the width,W, of the “box” function, that the height of the lobes decreases as a function of distance from the origin, and that the function extends to infinity for both positive and negative values of μ As you will see later, these properties are quite helpful in interpreting the spectra of two-dimensional Fourier transforms of images.

(4.2-16)