Higher Engineering Mathematics

698 FOURIER SERIES

Now try the following exercise.

Exercise 249 Further problems on symme-

try relationships

1. Determine the exponential form of the
   Fourier series for the periodic function
   defined by:

f(x)=

⎧

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎩

−2, when−π≤x≤−

π

2

2, when−

π

2

≤x≤+

π

2

−2, when+

π

2

≤x≤+π

and has a period of 2π

[

f(x)=

∑∞

n=−∞

(

4

nπ

sin

nπ

2

)

ejnx

]

2. Show that the exponential form of the Fourier
   series in problem 1 above is equivalent to:

f(x)=

8

π

(

cosx−

1

3

cos 3x+

1

5

cos5x

−

1

7

cos 7x+···

)

3. Determine the complex Fourier series to rep-
   resent the functionf(t)= 2 tin the range−π
   to+π.
   [

f(t)=

∑∞

n=−∞

(

j 2

n

cosnπ

)

ejnt

]

f(t)

20

− 1 01 t

L = 10

Figure 74.5

4. Show that the complex Fourier series in
   problem 3 above is equivalent to:

f(t)= 4

(

sint−

1

2

sin 2t+

1

3

sin 3t

−

1

4

sin 4t+···

)

74.5 The frequency spectrum

In the Fourier analysis of periodic waveforms seen

in previous chapters, although waveforms physically

exist in the time domain, they can be regarded as

comprising components with a variety of frequen-

cies. The amplitude and phase of these components

are obtained from the Fourier coefficientsanandbn;

this is known as afrequency domain. Plots of ampli-

tude/frequency and phase/frequency are together

known as thespectrumof a waveform. A simple

example is demonstrated in Problem 6 following.

Problem 6. A pulse of height 20 and width 2

has a period of 10. Sketch the spectrum of the

waveform.

The pulse is shown in Figure 74.5.

The complex coefficient is given by equation (12):

cn=

1

L

∫L

2

−L 2

f(t)e−j

2 πnt

L dt

=

1

10

∫ 1

− 1

20e−j

2 πnt

(^10) dt=
20
10
[
e−j
πnt
5
−jπn
5
] 1
− 1