Page 476 - Advanced engineering mathematics

456 CHAPTER 13 Fourier Series

c /2
n
3

1.5

.725
.36
ω 0 2ω 0 3ω 0 4ω 0 nω 0

FIGURE 13.18 Amplitude spectrum of f
in Example 13.16.

The amplitude spectrum of a periodic function f is a plot of points (nω 0 ,c n /2) for n =
1,2,···, and also the point (0,|c 0 |/2). For the function of Example 13.14, this is a plot of
points (0,3) and, for nonzero integer n, points

2nπ 9 √
, 1 + n π 2 .
2
3 2n π 2
2
The amplitude spectrum for the function of Example 13.16 is shown in Figure 13.18, with
the intervals on the horizontal axis of length ω 0 = 2π/3. This graph displays the relative effects
of the harmonics in the function. This is useful in signal analysis.

SECTION 13.5 PROBLEMS

In Problems 1, 2, and 3, let f be periodic of period p. 5. Let

1for 0 ≤ x < 1,
1. If g is also periodic of period p, show that αf + βg is f (x) = 0for 1 < x < 2,
periodic of period p, for any numbers α and β.
and let f has fundamental period 2.
2. Let α be a positive number. Show that g(t)= f (αt) has
2
period p/α and h(t) = f (t/α) has period αp. 6. Let f (x) = 3x for 0 ≤ x < 4 and let f have funda-
mental period 4.

3. If f is differentiable, show that f has period p.
7. Let

In each of Problems 4 through 12, find the phase angle 1 + x for 0 ≤ x < 3,
f (x) =
form of the Fourier series of the function and plot some 2 for 2 ≤ x < 4,
points of the amplitude spectrum. Some of these functions
are specified by a graph. and suppose f has fundamental period 4.
8. f (x) = cos(πx) for 0 ≤ x < 1and f has fundamental
4. Let f (x)= x for 0≤ x <2, with fundamental period 2. period 1.

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October 14, 2010 14:57 THM/NEIL Page-456 27410_13_ch13_p425-464