Laplace Transform

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

2. Calculating theLaplace Transform • Definitionof Laplace Transform • Basic Examples(Unit Step, Exponential, and Impulse) • Matlab Verification(Unit Step, Exponential, and Impulse) • Multiplication by Power of t Example • Sine Example • Linearity Example with Matlab Verification of Region of Convergence

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

4. Basic Examples • Find the Laplace Transform and it’s Region of Convergence for the following functions of time: • Unit Step • Exponential • Impulse

5. 0 t Unit Step Example • Find the Laplace Transform of the unit step function u(t) Must find ROC

6. ROC s-domain jw s ROC U(s) ROC • For a complete answer, the Region of Convergence must be specified • ROC exists where:

7. ROC Exponential Example • Find the Laplace Transform of the exponential function:

8. ROC ROC jw jw s -b s -b ROC X(s) ROC For Negative b For Positive b

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

10. Matlab Basic Verifications • Use Matlab to verify the the Laplace Transform for the following functions of time: • Unit Step • Exponential • Impulse

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

12. Matlab Unit Step Verification » syms X x_unitstep » x_unitstep = sym('1'); • Create a unit step symbolic variable • 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 » X=laplace( x_unitstep ) X = 1/s

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

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

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

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:

17. LT{ t3u(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

18. Verify T3 Using Matlab • The Matlab verification is straight forward: » syms X t » X=laplace(t^3) X = 6/s^4

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

20. Euler’s Identity • Use Euler’s identity to expand sin(bt) • X(s) is the sum of the Laplace Transforms of each part

21. 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 ROC

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

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

24. LT LT LT LT Linearity Property • Using the Linearity Property, sum the Laplace Transform of each term to get X(s)

25. LT LT LT LT Intersection of ROCs • ROC of X(s) is the Intersection of the ROCs of the Summed Components of X(s)

26. Linear ROC ROC

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

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 • To converge, s must be greater than largest pole Poles are at s = -1 and s = 2 • Thus verifying the ROC is s > 2

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 tn to calculate the Laplace Transform • Verifying Region of Convergence