P1: RPU/XXX
P2: RPU/XXX
CUUK852-Mandal & Asif
QC: RPU/XXX
May 25, 2007
T1: RPU
18:31
Table 6.1. CTFT and Laplace transform pairs for several causal CT signals
CT signals x(t)
CTFT
X ( jω) =
∞
x(t)e−jωt dt
−∞
−∞
(1) Impulse function
x(t) = δ(t)
1
(2) Unit step function
x(t) = u(t)
πδ(ω) +
(3) Causal gate function
x(t) = u(t) − u(t − a)
(1 − e−jaω ) πδ(ω) +
(4) Causal decaying exponential function
x(t) = e−at u(t)
Laplace transform
∞
X (s) =
x(t)e−st dt
1
ROC: entire s-plane
1
s
ROC: Re{s} > 0
1
jω
1
jω
1
a+ s
ROC: Re{s} > −a
1
a + jω
(5) Causal ramp function
x(t) = tu(t)
does not exist
(6) Higher-order causal ramp function
x(t) = t n u(t)
does not exist
1
(1 − e−as )
s
ROC: Re{s} > 0
1
s2
ROC: Re{s} > 0
n!
s n+1
ROC: Re{s} > 0
(7) First-order time-rising causal decaying
exponential function
x(t) = te−at u(t)
1
(a + jω)2
provided a > 0.
1
(a + s)2
ROC: Re{s} > −a
(8) Higher-order time-rising causal
decaying exponential function
x(t) = t n e−at u(t)
n!
(a + jω)n+1
provided a > 0
n!
(a + s)n+1
ROC: Re{s} > −a
s
ω02 + s 2
ROC: Re{s} > 0
ω0
ω02 + s 2
ROC: Re{s} > 0
(9) Causal cosine wave
x(t) = cos(ω0 t)u(t)
(10) Causal sine wave
x(t) = sin(ω0 t)u(t)
(11) Squared causal cosine wave
x(t) = cos2 (ω0 t)u(t)
π [δ(ω − ω0 ) + δ(ω + ω0 )]
jω
+ 2
ω0 − ω2
π
[δ(ω − ω0 ) − δ(ω + ω0 )]
2j
ω0
+ 2
ω0 − ω2
π
[δ(ω) + δ(ω − 2ω0 ) + δ(ω + 2ω0 )]
2
+
1
jω
+
j2ω
2 4ω02 − ω2
2ω02 + s 2
s 4ω02 + s 2
ROC: Re{s} > 0
π
[δ(ω) − δ(ω − 2ω0 ) − δ(ω + 2ω0 )]
2
2ω02
s 4ω02 + s 2
ROC: Re{s} > 0
(13) Causal decaying exponential cosine
function
x(t) = exp(−at) cos(ω0 t)u(t)
1
jω
−
j2ω
2 4ω02 − ω2
a + jω
(a + jω)2 + ω02
provided a > 0
(14) Causal decaying exponential sine
function
x(t) = exp(−at) sin(ω0 t)u(t)
ω0
(a + jω)2 + ω02
provided a > 0
(12) Squared causal sine wave
x(t) = sin2 (ω0 t)u(t)
+
270
a+ s
(a + s)2 + ω02
ROC: Re{s} > −a
ω0
(a + s)2 + ω02
ROC: Re{s} > −a
P1: RPU/XXX
P2: RPU/XXX
CUUK852-Mandal & Asif
QC: RPU/XXX
May 25, 2007
283
T1: RPU
18:31
6 Laplace transform
Table 6.2. Properties of the Laplace transform
The corresponding properties of the CTFT are also listed in the table for comparison
CTFT
Properties in the time domain
Linearity
a1 x1 (t) + a2 x2 (t)
Time scaling
x(at)
∞
X ( jω) =
x(t)e−jωt dt
−∞
Laplace transform
∞
x(t)e−st dt
X (s) =
−∞
a1 X 1 (ω) + a2 X 2 (ω)
a1 X 1 (s) + a2 X 2 (s)
ROC: at least R1 ∩ R2
1 s X
|a|
a
with ROC: a R
1 ω
X
|a|
a
Time shifting
x(t − t0 )
e−jω0 t X (ω)
e−st0 X (s)
with ROC: R
Frequency/s-domain shifting
x(t)e jω0 t or x(t)es0 t
X (ω − ω0 )
Time differentiation
dx/dt
jωX (ω)
X (s − s0 )
with ROC: R + Re{s0 }
Time integration
t
x(τ )dτ
−∞
Frequency/s-domain
differentiation
(−t)x(t)
Duality
X (t)
Time convolution
x1 (t) ∗ x2 (t)
Frequency/s-domain convolution
x1 (t)x2 (t)
Parseval’s relationship
X (ω)
+ π X (0)δ(ω)
jω
s X (s) − x(0− )
with ROC: R
X (s)
s
with ROC: R ∩ Re{s} > 0
−jdX/dω
dX/ds
2π x(ω)
not applicable
X 1 (ω)X 2 (ω)
X 1 (s)X 2 (s)
ROC includes R1 ∩ R2
1
X 1 (s) ∗ X 2 (s)
2π
ROC includes
R1 ∩ R2
1
X 1 (ω) ∗ X 2 (ω)
2π
∞
−∞
Initial value
x(0+ ) if it exists
Final value
x(∞) if it exists
1
2π
1
|x(t)| dt =
2π
2
∞
∞
|X (ω)|2 dω
not applicable
−∞
X (ω)dω
−∞
not applicable
lim sX (s)
s→∞
provided s = ∞ is included
in the ROC of sX(s)
lim sX (s)
s→0
provided s = 0 is included
in the ROC of sX(s)