The Laplace Transform

1
The Laplace Transform
ECE 2221/MCT 2210 Signals and Systems
(Analysis) Sem. I (03/04)

• Br. Shahrul Naim Sidek
• Department of Mechatronics Engineering
• International Islamic University Malaysia

Lecture Chapter 5
http//eng.iiu.edu.my/snaim
2
Unilateral Laplace Transform

• More general than Fourier, where wider class of
  signals can be applied.
• This is accomplished by multiplying a signal with
  an exponential convergence factor, exp(-st)
• . where s sjw
• Forward transform lower limit of integration is
  0- (i.e. just before 0) to avoid ambiguity that
  may arise if x(t) contains an impulse at origin.
• Inverse transform we will use table instead of
  using definition.

3
Existence of Laplace Transform

• As long as e-s t decays at a faster rate than
  rate x(t) explodes, Laplace transform converges
• for some M and s0,there exists s0 gt s to make
  the Laplace transform integral finite
• We cannot always do this, e.g.
  does not have a Laplace transform

4
Convergence

• The condition Res gt -Rea is the region of
  convergence, which is the region of s for which
  the Laplace transform integral converges
• Res -Rea is not allowed (see next slide)

5
Regions of Convergence

• What happens to X(s) 1/(sa) at s -a? (1/0)
• -ea t u(-t) and e-a t u(t) have same transform
  function but different regions of convergence

x(t)
x(t)
1
t
-1
t
x(t) -ea t u(-t)
x(t) e-a t u(t)
non-causal
causal
6
Key Transform Pairs
7
Key Transform Pairs
8
Laplace Transform Properties

• Linearity
• Time shifting
• Frequency shifting
• Differentiationin time

9
Differentiation in Time Property
10
Laplace Transform Properties

• Differentiation in frequency
• Integration in time
• Example x(t) d(t)
• Integration in frequency

11
Laplace Transform Properties

• Scaling in time/frequency
• Under integration,
• Convolution in time
• Convolution in frequency

x(2 t)
t
1
-1
Area reduced by factor 2
12
Example

• Compute y(t) e a t u(t) e b t u(t) , where a
  ? b

13
Inverse Laplace Transform
14
Inverse Laplace Transform

• Definition has integration in complex plane
• We will use lookup tables instead
• Table 5.3
• Many Laplace transform expressions are ratios of
  two polynomials, a.k.a. rational functions
• Convert complicated rational functions into
  simpler forms
• Apply partial fractions decomposition
• Use lookup tables

15
Partial Fractions Example 1
16
Partial Fractions Example 2
17
Partial Fractions Example 3
18
Final and Initial Values Example

• Transfer function
• Poles at s 0, s -1 ? j2
• Zero at s -3/2

Initial Value
Final Value