S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn Laplace Transform Math Review with Matlab: Calculating the Laplace Transform

Laplace Transform: X(s) Calculating the Laplace Transform 2  Definition of Laplace Transform Definition  Basic Examples (Unit Step, Exponential, and Impulse) Basic Examples  Matlab Verification (Unit Step, Exponential, and Impulse) Matlab Verification  Multiplication by Power of t Example Multiplication by Power of t  Sine Example Sine  Linearity Example with Matlab Verification of Region of Convergence Linearity Example

Laplace Transform: X(s) Calculating the Laplace Transform 3 Fundamentals  The Laplace Transform of a continuous-time signal is defined as:  The Laplace Transform is only valid for a Region of Convergence (ROC) in the s-domain where:  = Re{s} X(s) is FINITE s is COMPLEX s =  +j 

Laplace Transform: X(s) Calculating the Laplace Transform 4 Basic Examples  Find the Laplace Transform and it’s Region of Convergence for the following functions of time:  Unit Step Unit Step  Exponential Exponential  Impulse Impulse

Laplace Transform: X(s) Calculating the Laplace Transform 5 Unit Step Example  Find the Laplace Transform of the unit step function u(t) Must find ROC t 0

Laplace Transform: X(s) Calculating the Laplace Transform 6 U(s) ROC  For a complete answer, the Region of Convergence must be specified ROC  ROC exists where: jj  ROC s-domain

Laplace Transform: X(s) Calculating the Laplace Transform 7 Exponential Example  Find the Laplace Transform of the exponential function: ROC

Laplace Transform: X(s) Calculating the Laplace Transform 8 X(s) ROC For Positive b ROC For Negative b jj  ROC -b jj  ROC -b

Laplace Transform: X(s) Calculating the Laplace Transform 9 Impulse Example t 0  Find the Laplace Transform of the Unit Impulse Function: ROC is the entire s-domain X(s) is not dependent on the value of s, therefore the region of convergence is the entire s-domain

Laplace Transform: X(s) Calculating the Laplace Transform 10 Matlab Basic Verifications  Use Matlab to verify the the Laplace Transform for the following functions of time:  Unit Step Unit Step  Exponential Exponential  Impulse Impulse

Laplace Transform: X(s) Calculating the Laplace Transform 11 Laplace Matlab Command  The Matlab Symbolic Toolbox command laplace can be used to evaluate the Laplace Transform of a function of t L = laplace(F) F = scalar sym variable with default independent variable t L = Laplace transform of F. By default, L is a function of s

Laplace Transform: X(s) Calculating the Laplace Transform 12 Matlab Unit Step Verification  Create a unit step symbolic variable » syms X x_unitstep » x_unitstep = sym('1'); » X=laplace( x_unitstep ) X = 1/s  Note that all inputs into the laplace function are right-sided thus x_unitstep = 1 implies 1 for all positive t and 0 for all negative t  Verify Laplace Transform of Unit Step

Laplace Transform: X(s) Calculating the Laplace Transform 13 Matlab Exponential Verification  Create an Exponential Right-Sided symbolic variable » syms x_exp b t X » x_exp = exp(-b*t); » X=laplace( x_exp ) X = 1/(s+b)  Verify Laplace Transform of Exponential function

Laplace Transform: X(s) Calculating the Laplace Transform 14 Matlab Impulse Verification  Create a symbolic impulse variable using Dirac(t) » syms x_impulse » x_impulse = sym( 'Dirac(t)' ); » X = laplace( x_impulse ) X = 1  Verify Laplace Transform of Impulse (Delta-Dirac)

Laplace Transform: X(s) Calculating the Laplace Transform 15 Multiplication by a Power of t Example  Given:  Numerically Calculate the Laplace Transform X(s)  Verify the result using Matlab

Laplace Transform: X(s) Calculating the Laplace Transform 16 Approach  The Laplace Transform could be calculated directly using Integration by Parts in 3 stages  It is easier to use the Multiplication by a Power of t Property of the Laplace Transform to solve since t is raised to a positive n:

Laplace Transform: X(s) Calculating the Laplace Transform 17 LT{ t 3 u(t) }  Using the multiplication by a power of t property:  X(s) is directly calculated by taking the third derivative of U(s)=1/s and multiplying by (-1) 3

Laplace Transform: X(s) Calculating the Laplace Transform 18 Verify T 3 Using Matlab  The Matlab verification is straight forward: » syms X t » X=laplace(t^3) X = 6/s^4

Laplace Transform: X(s) Calculating the Laplace Transform 19 sin(bt) Example  Given:  Numerically Calculate the Laplace Transform X(s)  Verify the result using Matlab  Use the following form of Euler’s Identity to expand sin(bt) into a sum of complex exponentials

Laplace Transform: X(s) Calculating the Laplace Transform 20 Euler’s Identity  Use Euler’s identity to expand sin(bt)  X(s) is the sum of the Laplace Transforms of each part

Laplace Transform: X(s) Calculating the Laplace Transform 21 ROC Result of LT{ sin(bt) }  Multiply by complex conjugates to get common denominators  Simplify the expression  Because the Magnitude of sine is always Bounded by 1: is the entire s-domain except s =  jb

Laplace Transform: X(s) Calculating the Laplace Transform 22 Matlab Verification » syms b t » x=laplace(sin(b*t)) X = b/(s^2+b^2)  Use Matlab to verify the result:

Laplace Transform: X(s) Calculating the Laplace Transform 23 Linear Example  Building upon the previous examples and the Linearity Property, find the Laplace Transform of the function  Also determine the Region of Convergence by hand  Use Matlab’s symbolic toolbox to verify both the Laplace Transform X(s) AND verify the Region of Convergence

Laplace Transform: X(s) Calculating the Laplace Transform 24 Linearity Property  Using the Linearity Property, sum the Laplace Transform of each term to get X(s) LT

Laplace Transform: X(s) Calculating the Laplace Transform 25 Intersection of ROCs  ROC of X(s) is the Intersection of the ROCs of the Summed Components of X(s) LT

Laplace Transform: X(s) Calculating the Laplace Transform 26 Linear ROC ROC

Laplace Transform: X(s) Calculating the Laplace Transform 27 Verify Linear Example  The linear example can be verified using Matlab » syms x1 x2 x3 t X » x1=sym('Dirac(t)'); » x2=-(4/3)*exp(-t); » x3=(1/3)*exp(2*t); » X=laplace(x1+x2+x3) X = 1-4/3/(1+s)+1/3/(s-2) LT

Laplace Transform: X(s) Calculating the Laplace Transform 28 Verify ROC  No Matlab function exists to directly determine Region of Convergence  To verify the ROC in the Laplace Domain, look at the poles of the transformed function Poles are at s = -1 and s = 2  To converge,  must be greater than largest pole  Thus verifying the ROC is  > 2

Laplace Transform: X(s) Calculating the Laplace Transform 29 Summary  Calculating Laplace Transformation of the Basic Functions unit step, exponential, and impulse done by hand and using Matlab  Using some of the Properties of the Laplace Transform such as linearity and multiplication by t n to calculate the Laplace Transform  Verifying Region of Convergence