This is a Laplace course of System.

1. The Laplace Transform and
Application to Continuous-
Time System Analysis

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The Laplace Transform
• The Laplace transform of a continuous-time signal
x(t) is defined as:
in which, s is a complex variable: s = σ +jω.
• The inverse Laplace transform:
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Example: Find the Laplace transforms of signal
a. x(t)=(t)
b. x(t)= u(t)
c. x(t)= t.u(t)
d. x(t) = t2u(t)
e. x(t)= e-at.u(t)
f. x(t)= cos(wt).u(t)
g. x(t) = sin(wt).u(t)
The Laplace Transform

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• The region of convergence (ROC)
A region in the s-space such that for any value of s in
this region the Laplace transform always converges.
 ROC of the Laplace transform depends only on the real
part of s.
ROC of the Laplace transform must not contain the poles.
If a signal has finite length and there exists at least a value
of s for which the Laplace transform of the signal
converges, then the ROC of the Laplace transform is the
entire s-plane.
The Laplace Transform

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Properties
 Linearity:
with the ROC containing ROC[X1(s)]ROC[X2(s)
Time shifting
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The Laplace Transform

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Properties
Shifting in s-plane:
Convolution
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The Laplace Transform

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• Properties
Differentiation:
• with the ROC containing ROC[X(s)].
Integration:
• with the ROC containing ROC[X(s)]{σ > 0}.
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The Laplace Transform

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Properties
Time scaling
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The Laplace Transform

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Inverse Laplace Transform
• Inverse Laplace Transform
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Partial-fraction expansion method (1)
 Without loss of generalization, assume that X(s) is
represented in the form of a proper rational function
N(s)/D(s) (N(s) and D(s) are polynomials and the degree
of N(s) is less than the degree of D(s)).
 Denote{spk}the poles of X(s):{spk}are roots of the
equation D(s) = 0.
Inverse Laplace Transform

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• Partial-fraction expansion method (2)
 If all{spk}are distinct, the partial-fraction expansion of X(s) is:
 in which, the coefficients{Ak}are computed by:
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Inverse Laplace Transform

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• Partial-fraction expansion method (3)
 In case X(s) has repeated poles, denote mk the
number of repetitions of the pole spk, then we have the
following expansion of X(s):
 in which, the coefficients{Akm}are computed by:
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Inverse Laplace Transform

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The impulse response of LTI system
• Consider a continuous-time LTI system
• If then
• h(t) is call impulse response
h(t)
x(t) y(t)
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The transfer function of system
 Perform the Laplace transform for both sides of the
above equation and apply the convolution property of the
Laplace transform to obtain:
• H(s) is called the transfer function of the system.
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Connection
• Cascade connection:
• Parallel connection:
h1(t) h2(t)
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Biến đổi Laplace ngược
Bậc tử nhỏ hơn bậc mẫu
Ví dụ: Tìm biến đổi Laplace ngược của F(s)
Giải: Điểm cực
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Biến đổi Laplace ngược
Bậc tử nhỏ hơn bậc mẫu
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Bậc tử nhỏ hơn bậc mẫu
Ví dụ:
• Cực phức
•
•
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