Signals and Systems Analysis

Chapter 2: Laplace Transform

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OUTLINE

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Laplace Transform

• Applications of Laplace Transform

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INTRODUCTION

Why Laplace transform?

• Laplace transform is used to transform circuits from a time domain to the
frequency domain and after obtaining the result we apply inverse Laplace
transform to find the result in the time domain.
Advantages
• It exists for most common signals.

• Easier to solve circuit problems with inductors and capacitors

• Provides the total response containing the natural and forced response

• Can be applied to a wider variety of inputs than phasor analysis .

• Convolution in time domain ➔ Multiplication in frequency domain.

• It doesn’t have any physical meaning; just a mathematical tool to facilitate analysis.

– Fourier transform gives us the frequency domain representation of signal.

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OUTLINE

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Lapalace Transform

• Applications of Fourier Transform

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LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM

• Laplace transform is the integral transformation of a function f(t) from time

domain to complex frequency domain

• General form of integral transformation:


F (U) = − f(t) k(t,u)dt,

input kernel function

• Bilateral Laplace transform (two-sided Laplace transform)

+∞ s =  + j
F(S)=‫׬‬−∞ 𝑓(𝑡)𝑒 −𝑠𝑡 𝑑𝑡
– s =  + j is a complex frequency variable damping factor angular
– Notations: frequency
F (s) = L[f(t)]

f(t)  F (s)
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LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

• Unilateral Laplace transform (one-sided Laplace transform)

+
+
F (s) = 0− x(t)
X (−𝑠𝑡)
f(t)𝑒exp(−st)dt
dt

▪ 0− :The value of f(t) at t = 0 is considered.

▪ Useful when we deal with causal signals or causal systems.

• Causal signal: f(t) = 0, t < 0.

• Causal system: h(t) = 0, t < 0.

▪ We are going to simply call unilateral Laplace transform as Laplace

transform.
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INVERSE LAPLACE TRANSFORM

 + j
L-1[F(S)] = f(t) =  (F(S) est dt
 − j

– Requires knowledge about complex analysis and contour

integration.(Beyond the scope of this course )

– We will see simpler methods to find the inverse Laplace transform.

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LAPLACE TRANSFORM
• Time domain v.s. s-domain
o f(t) : a function of time t and it is called the time domain signal
o F (s): a function of S and it is called the S-domain signal
• S-domain is also called the complex frequency domain

o By converting the time domain signal into the s-domain, we can

usually, greatly simplify the analysis of the LTI system.

o S-domain system analysis:

1. Convert the time domain signals to the s-domain with the Laplace
transform
2. Perform system analysis in the s-domain
3. Convert the s-domain results back to the time-domain
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LAPLACE TRANSFORM: REGION OF CONVERGENCE

• Region of Convergence (ROC)

• For a function f(t) to have Laplace transform the integral must converge
to a finite value
∞
න 𝑓(𝑡)𝑒 −𝑠𝑡 𝑑𝑡 < ∞
0

• ROC is the range of s that the Laplace transform of a signal converges.

• Fortunately, all function of interest in circuit analysis satisfy the

convergence criteria and have Laplace transform.
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LAPLACE TRANSFORM
• Example
– Find the Laplace transform of x(t) = −e (−5t) u(t)
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LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

• Example: find the unilateral Laplace transform of the

following signals.
– 1. x(t) = U(t)

– 2. x(t) =  (t)
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LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

• Example
– 3. x(t) = e( -at)u(t)

– 4. x(t) = cos(wt) u(t)

– 5. x(t) = sin( wt) u(t)

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LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM
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OUTLINE

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Lapalace Transform

• Applications of Fourier Transform

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PROPERTIES: LINEARITY
• Linearity
– If f1 (t)  F 1 (s) f2 (t)  F 2 (s)
– Then a1 f 1 (t) + a2 f2 (t)  a1F 1 (s) + a2 F 2 (s)

Exercise: Find the ROC

• Example
– Find the Laplace transfrom of A + B e (−bt) u(t)
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PROPERTIES: TIME SCALING

• Time scaling
– If f(t)  F (s)
– Then 1 𝑠
f(at)  𝐹( )
𝑎 𝑎

Exercise: Find the ROC

• Example
– Find the Laplace transform of x(t) = u(at)
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PROPERTIES: TIME SHIFTING

• Time shifting
– If f(t)  F (s) and For a  0

– Then
f(t − a )u(t − a )  𝑒 −𝑎𝑠 F (s)

Exercise: Find the ROC

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PROPERTIES: SHIFTING IN THE s DOMAIN

• Shifting in the s domain (Frequency Shifting)
– If f(t)  F(s)
– Then 𝑒−𝑎𝑡f (t)  F( s + a )

Exercise: Find the ROC

• Example
– Find the Laplace transform of x(t) = A𝑒−𝑎𝑡cos(0t)u(t)
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PROPERTIES: DIFFERENTIATION IN TIME DOMAIN

• Differentiation in time domain

– If f(t)  F(s)
– Then df(t)
 sF(s) − f(0 − )
dt
d 2 f(t)
 s 2F(s) − sf(0 − ) − f' (0 − )
dt2
d n f(t)
 s nF(s) − s n−1 f(0 − ) −. . . − sf (n−2)
(0 − ) − f (n−1)
(0 − )
dt n
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PROPERTIES: DIFFERENTIATION IN S DOMAIN

• Differentiation in s domain (Frequency Differentiation)

– If x(t)  X (s)

𝑑
𝑡𝑓 𝑡 − 𝐹(𝑠)
𝑑𝑠

– Then d n X (s)
(−t) x(t) 
n

dsn
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PROPERTIES: INTEGRATION IN TIME DOMAIN

• Integration in time domain

– If f(t)  F (s)
– Then 1
0
t
f( )d   F (s)
s
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PROPERTIES: CONVOLUTION

• Convolution
– If f(t)  F(s) h(t)  H (s)
– Then f(t)  h(t)  F(s)H (s)
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PROPERTIES: MODULATION
• Modulation: (Reading Assignment)
– If f(t)  F(s)

– Then f(t) cos( 0t) 

1
F(s + j 0) + F(s − j 0)
2

f(t) sin(  0t) 

j
F(s + j 0) − F(s − j 0 )
2
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PROPERTIES: INITIAL VALUE THEOREM

• Initial value theorem
– If the signal f(t) is infinitely differentiable on an interval around f(0 + )
then
f(0 − ) = lim sF (s)
s→

– The behavior of f(t) for small t is determined by the behavior of F( s)

for large s.
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PROPERTIES: FINAL VALUE THEOREM

• Final value theorem
– If f(t)  F (s)
– Then: lim f(t) = lim sF (s)
t→ s→0
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PROPERTIES
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OUTLINE

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Lapalace Transform

• Applications of Laplace Transform

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INVERSE LAPLACE TRANSFORM

• Inverse Laplace transform
1  + j
f(t) = 
2j  − j
F(s) e(st)ds

– Evaluation of the above integral requires the use of contour

integration in the complex plan ➔ difficult.
• Inverse Laplace transform: special case
– In many cases, the Laplace transform can be expressed as a
rational function of s
bm s m + bm−1 s m−1 +. . . + b1 s + b0
F(s) =
an s n + an−1 s n−1 +. . . + a1 s + a0

– Procedure of Inverse Laplace Transform

1. Partial fraction expansion of F(s)
2. Find the inverse Laplace transform through the Laplace
transform table.
Partial Fraction Expansion

• m < n proper rational function

• m > n improper rational function

• In a proper rational function, the roots of the numerator are called the
zeros of F(S).

• The roots of the denominator are called the poles of F(S).

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Partial Fraction Expansion with Distinct Poles

If all the poles of F(s)are distinct (different from each another), we can
factor the denominator of F(s) in the form