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
- 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+···
)
- 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
]
- 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