Math Review with Matlab Laplace Transform Calculating the

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

X(s) Laplace Transform: Calculating the Laplace Transform § § § Definition of 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 2

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

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

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

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

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

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

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

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

X(s) Laplace Transform: Calculating the Laplace Transform 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 11

X(s) Laplace Transform: Calculating the Laplace Transform Matlab Unit Step Verification § Create a unit step symbolic variable » syms X x_unitstep » x_unitstep = sym('1'); § 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 12

X(s) Laplace Transform: Calculating the Laplace Transform 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) 13

X(s) Laplace Transform: Calculating the Laplace Transform 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 14

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

X(s) Laplace Transform: Calculating the Laplace Transform 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: 16

X(s) Laplace Transform: Calculating the Laplace Transform LT{ 3 t 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 17

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

X(s) Laplace Transform: Calculating the Laplace Transform 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 19

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

X(s) Laplace Transform: Calculating the Laplace Transform 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: ROC is the entire s-domain except s = ±jb 21

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

X(s) Laplace Transform: Calculating the Laplace Transform 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 23

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

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

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

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

X(s) Laplace Transform: Calculating the Laplace Transform 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 28

X(s) Laplace Transform: Calculating the Laplace Transform Summary § Calculating Laplace Transformation of the Basic Functions unit step, exponential, and impulse done by hand 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 29