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Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder. All rights reserved. ECE 250 Data Structures and Algorithms

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Laplace Transform Outline In this talk, we will: –Definition of the Laplace transform –A few simple transforms –Rules –Demonstrations

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Classical differential equations Laplace Transform Background Time Domain Solve differential equation

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Laplace transforms Laplace Transform Background Time DomainFrequency Domain Solve algebraic equation Laplace transform Inverse Laplace transform

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Laplace Transform Definition The Laplace transform is Common notation:

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Laplace Transform Definition The Laplace transform is the functional equivalent of a matrix-vector product

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Laplace Transform Definition Notation: –Variables in italics t, s –Functions in time space f, g –Functions in frequency space F, G –Specific limits

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Laplace Transform Existence The Laplace transform of f(t) exists if –The function f(t) is piecewise continuous –The function is bound by for some k and M

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Laplace Transform Example Transforms We will look at the Laplace transforms of: –The impulse function (t) –The unit step function u(t) –The ramp function t and monomials t n –Polynomials, Taylor series, and e t –Sine and cosine

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Laplace Transform Example Transforms While deriving these, we will examine certain properties: –Linearity –Damping –Time scaling –Time differentiation –Frequency differentiation

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Laplace Transform Impulse Function The easiest transform is that of the impulse function:

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Next is the unit step function Laplace Transform Unit Step Function

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Laplace Transform Integration by Parts Further cases require integration by parts Usually written as

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Laplace Transform Integration by Parts Product rule Rearrange and integrate

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Laplace Transform Ramp Function The ramp function

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Laplace Transform Monomials By repeated integration-by-parts, it is possible to find the formula for a general monomial for n ≥ 0

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Laplace Transform Linearity Property The Laplace transform is linear If and then

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Laplace Transform Initial and Final Values Given then Note sF(s) is the Laplace transform of f (1) (x)

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Laplace Transform Polynomials The Laplace transform of the polynomial follows:

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Laplace Transform Polynomials This generalizes to Taylor series, e.g.,

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Laplace Transform The Sine Function Sine requires two integration by parts: 1 of 2

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Laplace Transform The Sine Function Consequently: 2 of 2

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Laplace Transform The Cosine Function As does cosine: 1 of 2

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Laplace Transform The Cosine Function Consequently: 2 of 2

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Laplace Transform Periodic Functions If f(t) is periodic with period T then For example,

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Laplace Transform Periodic Functions Here cos(t) is repeated with period

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Consider f(t) below: Laplace Transform Periodic Functions

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Laplace Transform Damping Property Time domain damping ⇔ frequency domain shifting

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Laplace Transform Damping Property Damped monomials A special case:

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Consider cos(t)u(t) Laplace Transform Damping Property

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Time scale by = 2 Laplace Transform Damping Property

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Time scale by = ½ Laplace Transform Damping Property

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Laplace Transform Time-Scaling Property Time domain scaling ⇔ attenuated frequency domain scaling

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Time scaling of trigonometric functions: Laplace Transform Time-Scaling Property

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Consider sin(t)u(t) Laplace Transform Time-Scaling Property

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Time scale by = 2 Laplace Transform Time-Scaling Property

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Time scale by = ½ Laplace Transform Time-Scaling Property

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Laplace Transform Damping Property Damped time-scaled trigonometric functions are also shifted

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Laplace Transform Time Differentiation Property The Laplace transform of the derivative

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Laplace Transform Time Differentiation Property The general case is shown with induction:

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Laplace Transform Time Differentiation Property If g(t) = f(t)u(t) then 0 = g(0 + ) = g (1) (0 + ) = ··· Thus the formula simplifies: Problem: –The derivative is more complex

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Laplace Transform Time Differentiation Property Example: if g(t) = cos(t)u(t) then g(0 – ) = 0 g (1) (t) = sin(t)u(t) + (t)

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Laplace Transform Time Differentiation Property We will demonstrate that –The Laplace transform of a derivative is the Laplace transform times s –The next six slides give examples that f (1) (t) = g(t) implies sF(s) = G(s) 1 of 7

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Laplace Transform Differentiation of Polynomials We now have the following commutative diagram when n > 0 2 of 7

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Laplace Transform Differentiation of Trigonometric Functions We now have the following commutative diagram 3 of 7

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Laplace Transform Differentiation of Trigonometric Functions We now have the following commutative diagram 4 of 7

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Laplace Transform Differentiation of Exponential Functions We now have the following commutative diagram 5 of 7

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Laplace Transform Differentiation of Trigonometric Functions We now have the following commutative diagram 6 of 7

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Laplace Transform Differentiation of Trigonometric Functions We now have the following commutative diagram 7 of 7

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Laplace Transform Frequency Differentiation Property The derivative of the Laplace transform

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Laplace Transform Frequency Differentiation Property Consider monomials

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Laplace Transform Frequency Differentiation Property Consider a sine function We have that but what is ? 1 of 3

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Laplace Transform Frequency Differentiation Property Applying integration by parts 2 of 3

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Laplace Transform Frequency Differentiation Property Substituting 3 of 3

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Laplace Transform Time Integration Property The Laplace transform of an integral

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Laplace Transform Time Integration Property We will demonstrate that –The Laplace transform of an integral is the Laplace transform over s –The next six slides give examples that implies 1 of 7

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Laplace Transform Integration of Polynomials We now have the following commutative diagram 2 of 7

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Laplace Transform Integration of Exponential Functions We now have the following commutative diagram 3 of 7

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Laplace Transform Integration of Trigonometric Functions We now have the following commutative diagram 4 of 7

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Laplace Transform Integration of Trigonometric Functions We now have the following commutative diagram 5 of 7

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Laplace Transform Integration of Trigonometric Functions We now have the following commutative diagram 6 of 7

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Laplace Transform Integration of Trigonometric Functions We now have the following commutative diagram 7 of 7

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Laplace Transform The Convolution Define the convolution to be Then

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Laplace Transform Integration As a special case of the convolution

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Laplace Transform Summary We have seen these Laplace transforms:

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Laplace Transform Summary We have seen these properties: –Linearity –Damping –Time scaling –Time differentiation –Frequency differentiation –Time integration

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Laplace Transform Summary In this topic: –We defined the Laplace transform –Looked at specific transforms –Derived some properties –Applied properties

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Laplace Transform References Lathi, Linear Systems and Signals, 2 nd Ed., Oxford University Press, 2005. Spiegel, Laplace Transforms, McGraw-Hill, Inc., 1965. Wikipedia, http://en.wikipedia.org/wiki/Laplace_Transform

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Usage Notes These slides are made publicly available on the web for anyone to use If you choose to use them, or a part thereof, for a course at another institution, I ask only three things: –that you inform me that you are using the slides, –that you acknowledge my work, and –that you alert me of any mistakes which I made or changes which you make, and allow me the option of incorporating such changes (with an acknowledgment) in my set of slides Sincerely, Douglas Wilhelm Harder, MMath dwharder@alumni.uwaterloo.ca

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