Signals and Systems - Electrical Engineering

98 C H A P T E R 1: Continuous-Time Signals

FIGURE 1.11

(a) The triangular signal3(t)and (b) its derivative.

dt

0 0

(^11)
− 1
1 2 1 2
t t
Λ(t)
dΛ(t)
(a) (b)
In fact, sincer(t− 1 )andr(t− 2 )have values different from 0 fort≥1 andt≥2, respectively,
then
3(t)=r(t)=t for 0≤t≤ 1
and that for 1≤t≤2,
3(t)=r(t)− 2 r(t− 1 )=t− 2 (t− 1 )=−t+ 2
Finally, fort>2 the three ramp signals are different from zero, so
3(t)=r(t)− 2 r(t− 1 )+r(t− 2 )
=t− 2 (t− 1 )+(t− 2 )
= 0 t> 2
and by definition3(t)is zero fort<0. So the given expression for3(t)in terms of the ramp
functions is identical to its given mathematical definition.
Using the mathematical definition of the triangular function, its derivative is given by
d3(t)
dt

=







1 0 ≤t≤ 1

−1 1<t≤ 2

0 otherwise

Using the representation in Equation (1.32) this derivative is also given by

d3(t)

dt

=u(t)− 2 u(t− 1 )+u(t− 2 )

which are two unit pulses, as shown in Figure 1.11. n

nExample 1.20

Consider a full-wave rectified signal,

x(t)=|cos( 2 πt)| −∞<t<∞