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

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

Classical differential equations Laplace Transform Background Time Domain Solve differential equation

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

Laplace Transform Definition The Laplace transform is Common notation:

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

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

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

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

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

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

Next is the unit step function Laplace Transform Unit Step Function

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

Laplace Transform Integration by Parts Product rule Rearrange and integrate

Laplace Transform Ramp Function The ramp function

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

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

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

Laplace Transform Polynomials The Laplace transform of the polynomial follows:

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

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

Laplace Transform The Sine Function Consequently: 2 of 2

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

Laplace Transform The Cosine Function Consequently: 2 of 2

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

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

Consider f(t) below: Laplace Transform Periodic Functions

Laplace Transform Damping Property Time domain damping ⇔ frequency domain shifting

Laplace Transform Damping Property Damped monomials A special case:

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

Time scale by  = 2 Laplace Transform Damping Property

Time scale by  = ½ Laplace Transform Damping Property

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

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

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

Time scale by  = 2 Laplace Transform Time-Scaling Property

Time scale by  = ½ Laplace Transform Time-Scaling Property

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

Laplace Transform Time Differentiation Property The Laplace transform of the derivative

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

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

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)

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

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

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

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

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

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

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

Laplace Transform Frequency Differentiation Property The derivative of the Laplace transform

Laplace Transform Frequency Differentiation Property Consider monomials

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

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

Laplace Transform Frequency Differentiation Property Substituting 3 of 3

Laplace Transform Time Integration Property The Laplace transform of an integral

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

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

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

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

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

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

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

Laplace Transform The Convolution Define the convolution to be Then

Laplace Transform Integration As a special case of the convolution

Laplace Transform Summary We have seen these Laplace transforms:

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

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

Laplace Transform References Lathi, Linear Systems and Signals, 2 nd Ed., Oxford University Press, Spiegel, Laplace Transforms, McGraw-Hill, Inc., Wikipedia,

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