Formula Sheets: Laplace Transform in Signals & Systems | Signals and Systems - Electrical Engineering (EE) PDF Download

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Formula Sheet for Laplace Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Laplace Transform: Transforms a time-domain signal x(t) to the s-domainX(s),
used for analyzing linear time-invariant (LTI) systems.
• Complex Frequency: s =s +j?, where s: Real part, ?: Imaginary part.
• Region of Convergence (ROC): Set of s values where the transform converges.
2. Laplace Transform De?nition
• Unilateral Laplace Transform:
X(s) =
Z
8
0
-
x(t)e
-st
dt
• Bilateral Laplace Transform:
X(s) =
Z
8
-8
x(t)e
-st
dt
3. Inverse Laplace Transform
• Inverse Transform:
x(t) =
1
2pj
Z
s+j8
s-j8
X(s)e
st
ds
(Typically evaluated using partial fraction expansion and tables).
4. Properties of Laplace Transform
• Linearity:
ax
1
(t)+bx
2
(t)?aX
1
(s)+bX
2
(s)
• Time Shifting:
x(t-t
0
)u(t-t
0
)?e
-st
0
X(s)
• Frequency Shifting:
e
s
0
t
x(t)?X(s-s
0
)
• Time Scaling:
x(at)?
1
|a|
X

s
a

• Di?erentiation in Time:
dx(t)
dt
?sX(s)-x(0
-
)
1
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Formula Sheet for Laplace Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Laplace Transform: Transforms a time-domain signal x(t) to the s-domainX(s),
used for analyzing linear time-invariant (LTI) systems.
• Complex Frequency: s =s +j?, where s: Real part, ?: Imaginary part.
• Region of Convergence (ROC): Set of s values where the transform converges.
2. Laplace Transform De?nition
• Unilateral Laplace Transform:
X(s) =
Z
8
0
-
x(t)e
-st
dt
• Bilateral Laplace Transform:
X(s) =
Z
8
-8
x(t)e
-st
dt
3. Inverse Laplace Transform
• Inverse Transform:
x(t) =
1
2pj
Z
s+j8
s-j8
X(s)e
st
ds
(Typically evaluated using partial fraction expansion and tables).
4. Properties of Laplace Transform
• Linearity:
ax
1
(t)+bx
2
(t)?aX
1
(s)+bX
2
(s)
• Time Shifting:
x(t-t
0
)u(t-t
0
)?e
-st
0
X(s)
• Frequency Shifting:
e
s
0
t
x(t)?X(s-s
0
)
• Time Scaling:
x(at)?
1
|a|
X

s
a

• Di?erentiation in Time:
dx(t)
dt
?sX(s)-x(0
-
)
1
d
n
x(t)
dt
n
?s
n
X(s)-
n-1
X
k=0
s
n-1-k
d
k
x(0
-
)
dt
k
• Integration in Time:
Z
t
0
-
x(t)dt ?
X(s)
s
• Convolution:
x(t)*h(t)?X(s)H(s)
• Initial Value Theorem:
x(0
+
) = lim
s?8
sX(s)
• Final Value Theorem (if poles of sX(s) in left half-plane):
lim
t?8
x(t) = lim
s?0
sX(s)
5. Common Laplace Transform Pairs
• Unit Impulse:
d(t)? 1
• Unit Step:
u(t)?
1
s
, Re(s) > 0
• Unit Ramp:
tu(t)?
1
s
2
, Re(s) > 0
• Exponential:
e
-at
u(t)?
1
s+a
, Re(s) >-a
• Sinusoid:
sin(?t)u(t)?
?
s
2
+?
2
, Re(s) > 0
cos(?t)u(t)?
s
s
2
+?
2
, Re(s) > 0
• Damped Sinusoid:
e
-at
sin(?t)u(t)?
?
(s+a)
2
+?
2
, Re(s) >-a
6. System Analysis
• Transfer Function (LTI system):
H(s) =
Y(s)
X(s)
2
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Formula Sheet for Laplace Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Laplace Transform: Transforms a time-domain signal x(t) to the s-domainX(s),
used for analyzing linear time-invariant (LTI) systems.
• Complex Frequency: s =s +j?, where s: Real part, ?: Imaginary part.
• Region of Convergence (ROC): Set of s values where the transform converges.
2. Laplace Transform De?nition
• Unilateral Laplace Transform:
X(s) =
Z
8
0
-
x(t)e
-st
dt
• Bilateral Laplace Transform:
X(s) =
Z
8
-8
x(t)e
-st
dt
3. Inverse Laplace Transform
• Inverse Transform:
x(t) =
1
2pj
Z
s+j8
s-j8
X(s)e
st
ds
(Typically evaluated using partial fraction expansion and tables).
4. Properties of Laplace Transform
• Linearity:
ax
1
(t)+bx
2
(t)?aX
1
(s)+bX
2
(s)
• Time Shifting:
x(t-t
0
)u(t-t
0
)?e
-st
0
X(s)
• Frequency Shifting:
e
s
0
t
x(t)?X(s-s
0
)
• Time Scaling:
x(at)?
1
|a|
X

s
a

• Di?erentiation in Time:
dx(t)
dt
?sX(s)-x(0
-
)
1
d
n
x(t)
dt
n
?s
n
X(s)-
n-1
X
k=0
s
n-1-k
d
k
x(0
-
)
dt
k
• Integration in Time:
Z
t
0
-
x(t)dt ?
X(s)
s
• Convolution:
x(t)*h(t)?X(s)H(s)
• Initial Value Theorem:
x(0
+
) = lim
s?8
sX(s)
• Final Value Theorem (if poles of sX(s) in left half-plane):
lim
t?8
x(t) = lim
s?0
sX(s)
5. Common Laplace Transform Pairs
• Unit Impulse:
d(t)? 1
• Unit Step:
u(t)?
1
s
, Re(s) > 0
• Unit Ramp:
tu(t)?
1
s
2
, Re(s) > 0
• Exponential:
e
-at
u(t)?
1
s+a
, Re(s) >-a
• Sinusoid:
sin(?t)u(t)?
?
s
2
+?
2
, Re(s) > 0
cos(?t)u(t)?
s
s
2
+?
2
, Re(s) > 0
• Damped Sinusoid:
e
-at
sin(?t)u(t)?
?
(s+a)
2
+?
2
, Re(s) >-a
6. System Analysis
• Transfer Function (LTI system):
H(s) =
Y(s)
X(s)
2
• Impulse Response:
h(t) =L
-1
{H(s)}
• Output Response:
Y(s) = H(s)X(s)
• Stability: System stable if all poles of H(s) have Re(s) < 0.
7. Region of Convergence (ROC)
• Causal Signal: ROC is to the right of the rightmost pole.
• Anti-Causal Signal: ROC is to the left of the leftmost pole.
• Two-Sided Signal: ROC is a strip between poles.
• Stability: ROC must include the j?-axis for a stable system.
8. Partial Fraction Expansion
• For Distinct Poles X(s) =
N(s)
Q
n
i=1
(s-p
i
)
:
X(s) =
n
X
i=1
A
i
s-p
i
, A
i
= lim
s?p
i
(s-p
i
)X(s)
• Repeated Poles (order m):
X(s) =
m
X
k=1
A
k
(s-p)
k
+other terms, A
k
=
1
(m-k)!
lim
s?p
d
m-k
ds
m-k
[(s-p)
m
X(s)]
9. Frequency Response from Laplace Transform
• Frequency Response:
H(j?) = H(s)



s=j?
(Valid if ROC includes j?-axis).
• Magnitude and Phase:
|H(j?)|, ?H(j?)
10. Design Considerations
• System Analysis: Use transfer function to determine stability, zeros, and poles.
• Transient Response: Analyze using inverse Laplace transform.
• Applications: Control systems, circuit analysis, signal processing.
• ROC: Ensure correct ROC to determine causality and stability.
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