S. Awad, Ph.D.

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LaplaceTransform
Math Review with Matlab
Calculating the Laplace Transform

• S. Awad, Ph.D.
• M. Corless, M.S.E.E.
• E.C.E. Department
• University of Michigan-Dearborn

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Calculating theLaplace 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

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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 Res X(s) is FINITE
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Basic Examples

• Find the Laplace Transform and its Region of
  Convergence for the following functions of time

• Unit Step
• Exponential
• Impulse

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Unit Step Example

• Find the Laplace Transform of the unit step
  function u(t)

Must find ROC
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U(s) ROC

• For a complete answer, the Region of Convergence
  must be specified

• ROC exists where

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Exponential Example

• Find the Laplace Transform of the exponential
  function

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X(s) ROC

• For Positive b

For Negative b
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Impulse Example

• Find the Laplace Transform of the Unit Impulse
  Function

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Matlab Basic Verifications

• Use Matlab to verify the the Laplace Transform
  for the following functions of time

• Unit Step
• Exponential
• Impulse

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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
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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

Xlaplace( x_unitstep ) X 1/s
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Matlab Exponential Verification

• Create an Exponential Right-Sided symbolic
  variable

syms x_exp b t X x_exp exp(-bt)

• Verify Laplace Transform of Exponential function

Xlaplace( x_exp ) X 1/(sb)
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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
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Multiplication by a Power of t Example

• Given

• Numerically Calculate the Laplace Transform X(s)
• Verify the result using Matlab

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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

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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

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Verify T3 Using Matlab

• The Matlab verification is straight forward

syms X t
Xlaplace(t3) X 6/s4
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sin(bt) Example

• Given

• Numerically Calculate the Laplace Transform X(s)
• Verify the result using Matlab

• Use the following form of Eulers Identity to
  expand sin(bt) into a sum of complex exponentials

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Eulers Identity

• Use Eulers identity to expand sin(bt)

• X(s) is the sum of the Laplace Transforms of each
  part

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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
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Matlab Verification

• Use Matlab to verify the result

syms b t xlaplace(sin(bt)) X b/(s2b2)
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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 Matlabs symbolic toolbox to verify both the
  Laplace Transform X(s) AND verify the Region of
  Convergence

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Linearity Property

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

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Intersection of ROCs

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

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Linear ROC
ROC
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Verify Linear Example

• The linear example can be verified using Matlab

syms x1 x2 x3 t X x1sym('Dirac(t)')
x2-(4/3)exp(-t) x3(1/3)exp(2t)
Xlaplace(x1x2x3) X 1-4/3/(1s)1/3/(s-2)
LT
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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 gt 2

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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