Signals and Systems - Electrical Engineering

188 C H A P T E R 3: The Laplace Transform

0 5 10

0

0.5

1

t

u(t)

0 5 10

− 1

0

1

t

cos(5t)u(t)

0 5 10

0

0.5

1

t

exp(−0.5t)u(t)

0 5 10

− 1

0

1

t

cos(5t)exp(−0.5t)u(t)

− 2 − 1 0 1

− 10

0

10

σ

jΩ

FIGURE 3.10

For poles shown in the middle, possible signals are displayed around them anti–clockwise from bottom right.

The poles= 0 corresponds to a unit-step signal; the complex conjugate poles on thejaxis correspond to a

sinusoid; the pair of complex conjugate poles with a negative real part provides a sinusoid multiplied by an

exponential; and the pole in the negative real axis gives a decaying exponential. The actual amplitudes and

phases are determined by the other poles and by the zeros.

3.3.2 Differentiation

For a signalf(t)with Laplace transformF(s)its one-sided Laplace transform of its first-and second-order

derivatives are

L

[

df(t)

dt

u(t)

]

=sF(s)−f( 0 −) (3.11)