The Laplace Transform

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by Dr. Robert Akl

1

Generalizing the Fourier Transform

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2

Generalizing the Fourier Transform

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3

Generalizing the Fourier Transform

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4

Complex Exponential Excitation

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5

Complex Exponential Excitation

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6

Pierre-Simon Laplace

3/23/1749 - 3/2/1827

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7

The Transfer Function

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8

The Transfer Function

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9

Cascade-Connected Systems

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10

Direct Form II Realization

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11

Direct Form II Realization

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12

Direct Form II Realization

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13

Direct Form II Realization

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14

Direct Form II Realization

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15

Direct Form II Realization

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16

Inverse Laplace Transform

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17

Existence of the Laplace Transform

Time Limited Signals

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18

Existence of the Laplace Transform

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19

Existence of the Laplace Transform

Right- and Left-Sided Signals

Right-Sided Left-Sided

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20

Existence of the Laplace Transform

Right- and Left-Sided Exponentials

Right-Sided Left-Sided

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21

Existence of the Laplace Transform

Right-Sided Exponential

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22

Existence of the Laplace Transform

Left-Sided Exponential

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23

Existence of the Laplace Transform

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24

Existence of the Laplace Transform

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25

Laplace Transform Pairs

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26

Region of Convergence

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27

Region of Convergence

Some of the most common Laplace transform pairs(There is more extensive table in the book.)

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28

Laplace Transform Example

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29

Laplace Transform Example

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30

Laplace Transform Example

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31

Laplace Transform Example

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32

MATLAB System Objects

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33

MATLAB System Objects

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34

Partial-Fraction Expansion

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35

Partial-Fraction Expansion

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36

Partial-Fraction Expansion

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37

Partial-Fraction Expansion

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38

Partial-Fraction Expansion

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39

Partial-Fraction Expansion

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40

Partial-Fraction Expansion

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41

Partial-Fraction Expansion

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42

Partial-Fraction Expansion

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43

Inverse Laplace Transform Example

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44

Inverse Laplace Transform Example

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45

Inverse Laplace Transform Example

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46

Inverse Laplace Transform Example

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47

Inverse Laplace Transform Example

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48

Inverse Laplace Transform Example

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49

Inverse Laplace Transform Example

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50

Use of MATLAB in Partial Fraction Expansion

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51

Laplace Transform Properties

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52

Laplace Transform Properties

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53

Laplace Transform Properties

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54

Laplace Transform Properties

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55

Final Value Theorem

Laplace Transform Properties

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56

Use of Laplace Transform Properties

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57

Use of Laplace Transform Properties

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58

The Unilateral Laplace Transform

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The Unilateral Laplace Transform

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The Unilateral Laplace Transform

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The Unilateral Laplace Transform

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The Unilateral Laplace Transform

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The Unilateral Laplace Transform

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64

The Unilateral Laplace Transform

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65

Pole-Zero Diagrams and Frequency Response

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66

Pole-Zero Diagrams and Frequency Response

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67

Pole-Zero Diagrams and Frequency Response

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68

Pole-Zero Diagrams and Frequency Response

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69

Pole-Zero Diagrams and Frequency Response

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70

Pole-Zero Diagrams and Frequency Response

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71

Pole-Zero Diagrams and Frequency Response

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72

Pole-Zero Diagrams and Frequency Response

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73

Pole-Zero Diagrams and Frequency Response

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74

Pole-Zero Diagrams and Frequency Response

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75

Pole-Zero Diagrams and Frequency Response

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76

Pole-Zero Diagrams and Frequency Response

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77