Signals and Systems - Electrical Engineering

1.4 Representation Using Basic Signals 99

FIGURE 1.12
Eight periods of full-wave rectified signal
x(t)=|cos( 2 πt)|, −∞<t<∞.

− 2 − 1 0 1 2

0

0.2

0.4

0.6

0.8

1

t(sec)

x(

t)

of periodT 0 =0.5. Obtain a representation for a period between 0 and 0.5, and representx(t)in

terms of shifted versions of it. A full-wave rectified signal is used in designing dc sources. It is a first

step in converting an alternating voltage into a dc voltage. See Figure 1.12.

Solution

The period between 0 and 0.5 can be expressed as

p(t)=x(t)[u(t)−u(t−0.5)]=|cos( 2 πt)|[u(t)−u(t−0.5)]

Sincex(t)is a periodic signal of periodT 0 =0.5, we have then that

x(t)=

∑∞

k=−∞

p(t−kT 0 )

n

nExample 1.21

Generate a causal train of pulses that repeats every two units of time using as first period

s(t)=u(t)− 2 u(t− 1 )+u(t− 2 )

Find the derivative of the train of pulses.

Solution

Considering thats(t)is the first period of the train of pulses of period two, then

ρ(t)=

∑∞

k= 0

s(t− 2 k)