Signals and Systems - Electrical Engineering

1.4 Representation Using Basic Signals 91

where the first termx 1 a(t)is continuous and the secondx 1 b(t)is discontinuous. The derivative is

dx 1 (t)

dt

=− 2 πsin( 2 πt)[u(t)−u(t− 1 )]+(cos( 2 πt)− 1 )[δ(t)−δ(t− 1 )]+δ(t)−δ(t− 1 )

=− 2 πsin( 2 πt)[u(t)−u(t− 1 )]+δ(t)−δ(t− 1 )

since

(cos( 2 πt)− 1 )[δ(t)−δ(t− 1 )]=(cos( 2 πt)− 1 )δ(t)−(cos( 2 πt)− 1 )δ(t− 1 )

=(cos( 0 )− 1 )δ(t)−(cos( 2 π)− 1 )δ(t− 1 )

= 0 δ(t)+ 0 δ(t− 1 )= 0

The termδ(t)in the derivative indicates that there is a jump from 0 to 1 inx 1 (t)att=0 and that

in−δ(t− 1 )there is a jump of−1 (from 1 to 0) att=1. See Figure 1.7.

0 0.2 0.4 0.6 0.8 1

− 2

0

2

x^1

(t

)

0 0.2 0.4 0.6 0.8 1

− 2

0

2

x^1

(a

t)

0 0.2 0.4 0.6 0.8 1

− 2

0

2

x^1

(b

t)

t

(c)

t

(b)

t

(a)

FIGURE 1.7
(a) Decomposition ofx 1 (t)=cos( 2 πt)[u(t)−u(t− 1 )]into (b) a continuous and (c) a discontinuous signal
(a pulse).