Signals and Systems - Electrical Engineering

4.6 Fourier Coefficients from Laplace 263

so that the Fourier coefficients are given by(T 0 =2, 0 =π):

Yk=

1

T 0

Y 1 (s)|s=jok=

1

2 (jπk)^2

[2 cos(πk)− 2 ]

=

1 −cos(πk)

π^2 k^2

=

1 −(− 1 )k

π^2 k^2

k6= 0

This is also equal to

Yk=0.5

[

sin(πk/ 2 )

(πk/ 2 )

] 2

(4.22)

using the identity 1−cos(πk)=2 sin^2 (πk/ 2 ). By observingy(t)we deduce that its DC value is

Y 0 =0.5.

Let us then consider the periodic signalx(t)=dy(t)/dt(shown in Fig. 4.8(b)) with a dc value

X 0 =0. For− 1 ≤t≤1, its period isx 1 (t)=u(t+ 1 )− 2 u(t)+u(t− 1 )and

X 1 (s)=

1

s

[

es− 2 +e−s

]

which gives the Fourier series coefficients (T 0 =2,(the period and the fundamental frequency

are equal to the ones fory(t))

Xk=

sin^2 (kπ/ 2 )

kπ/ 2

j (4.23)

sinceXk=^12 X 1 (s)|s=jπk.

(a) (b)

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

t

Period

y(

t)

(^00204060)
0.2
0.4
0.6
0.8
Ω (rad/sec)
− (^10204060)
−0.5
0
0.5
1
Ω (rad/sec)
|Y
|k
|Y
|k
0 0.5 1 1.5 2
− 1
−0.5
0
0.5
1
t
Period
x(
t)
|Y
|k
(^00204060)
0.2
0.4
0.6
0.8
(^00204060)
0.5
1
1.5
2
Ω (rad/sec) Ω (rad/sec)
|X
|k
FIGURE 4.9
Magnitude and phase line spectra of (a) triangular signaly(t)(top left) and (b) its derivativex(t)(top right).
Ignoring the dc values, the{|Yk|}decay faster to zero than the{|Xk|}, thusy(t)is smoother thanx(t).