Higher Engineering Mathematics, Sixth Edition

652 Higher Engineering Mathematics

i.e.

f(x)=

8

π

(

sinx+

1

3

sin 3x+

1

5

sin 5x

+

1

7

sin 7x+···

)

Hence,

f(x)=

∑∞

n=−∞

−j

2

nπ

( 1 −cosnπ)ejnx

≡

8

π

(

sinx+

1

3

sin3x+

1

5

sin5x

+

1

7

sin7x+···

)

Now try the following exercise

Exercise 237 Further problems on

symmetry 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

cos3x+

1

5

cos5x

−

1

7

cos7x+···

)

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

]

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

f(t)= 4

(

sint−

1

2

sin2t+

1

3

sin3t

−

1

4

sin4t+···

)

71.5 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 frequencies. The ampli-

tude and phase of these components are obtained from

the Fourier coefficientsanandbn; this is known as a

frequency domain. Plots of amplitude/frequency and

phase/frequency are together known as thespectrum

of 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 71.5.

L 510

f(t)

21 1 t

20

0

Figure 71.5