Signals and Systems - Electrical Engineering

260 C H A P T E R 4: Frequency Analysis: The Fourier Series

Notice that about 11 of them (including the zero values), or the dc value and 5 harmon-

ics, provide a very good approximation of the pulse train, and would occupy a bandwidth

of approximately 10πrad/sec. The power contribution, as indicated byX^2 kafterk=±6, is

relatively small. n

nExample 4.6

Find the Fourier series of the full-wave rectified signalx(t)=|cos(πt)|shown in Figure 4.6. This

signal is used in the design of dc sources. The rectification of an ac signal is the first step in this

design.

Solution

The integral to find the Fourier coefficients is

Xk=

∫0.5

−0.5

cos(πt)e−j^2 πktdt

which can be computed by using Euler’s identity or any other method. We want to show that this

can be avoided by using the Laplace transform.

A periodx 1 (t)ofx(t)can be expressed as

x 1 (t−0.5)=sin(πt)u(t)+sin(π(t− 1 ))u(t− 1 )

− 2 − 1 0 1 2

−0.2

0

0.2

0.4

0.6

0.8

1

t

x(

t)

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

1

t

x^1

(t

)

(a) (b)

FIGURE 4.6

(a) Full-wave rectified signalx(t)and (b) one of its periodsx 1 (t).