Laplace transform - Continuous and Discrete Time Signals and Systems - page 283

Table 6.2. Properties of the Laplace transform
The corresponding properties of the CTFT are also listed in the table for comparison

CTFT Laplace transform

∞

∞

x ( t )e − j ω t d t X (j ω ) =

x ( t )e − st d t X ( s ) =

Properties in the time domain

−∞

−∞

Linearity
a 1 x 1 ( t ) + a 2 x 2 ( t )

a 1 X 1 ( ω ) + a 2 X 2 ( ω ) a 1 X 1 ( s ) + a 2 X 2 ( s )

ROC: at least R 1 ∩ R 2

1
a X

ω
a

1
a X

s
a

Time scaling
x ( at )

with ROC: aR

Time shifting
x ( t − t 0 )

e − j ω 0 t X ( ω ) e − st 0 X ( s )

with ROC: R

Frequency / s-domain shifting
x ( t )e j ω 0 t or x ( t )e s 0 t

X ( ω − ω 0 ) X ( s − s 0 )

with ROC: R + Re s 0

sX ( s ) − x (0 − )
with ROC: R

Time differentiation
d x / d t

j ω X ( ω )

X ( ω )
j ω

X ( s )
s
with ROC: R

Time integration
t

+ π X (0) δ ( ω )

∩ Re s > 0

x ( τ )d τ

−∞

Frequency / s-domain
differentiation
( − t ) x ( t )

− jd X / d ω d X / d s

Duality
X ( t )

2 π x ( ω ) not applicable

Time convolution
x 1 ( t ) ∗ x 2 ( t )

X 1 ( ω ) X 2 ( ω ) X 1 ( s ) X 2 ( s )

ROC includes R 1 ∩ R 2

1
2 π X 1 ( ω ) ∗ X 2 ( ω )

1
2 π X 1 ( s ) ∗ X 2 ( s )
ROC includes
R 1 ∩ R 2

Frequency/s-domain convolution
x 1 ( t ) x 2 ( t )

∞ ∞

= 1
2 π

x ( t ) 2 d t X ( ω ) 2 d ω

Parseval's relationship not applicable

−∞ −∞

∞

1
2 π

Initial value
x (0 + ) if it exists

X ( ω )d ω

s →∞ sX ( s )
provided s =∞ is included
in the ROC of sX ( s )

lim

−∞

Final value
x ( ∞ ) if it exists

not applicable lim

s → 0 sX ( s )
provided s = 0 is included
in the ROC of sX ( s )