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

S. Awad, Ph.D.

M. Corless, M.S.E.E.

E.C.E. Department

University of Michigan-Dearborn

LaplaceTransform

Math Review with Matlab:

Fundamentals

U of M-Dearborn ECE DepartmentMath Review with Matlab

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Laplace Transform:X(s) Fundamentals

Laplace Transform Fundamentals

Introduction to the Laplace Transform Laplace Transform Definition Region of Convergence Inverse Laplace Transform Properties of the Laplace Transform


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Introduction The Laplace Transform is a tool used to convert an

operation of a real time domain variable (t) into an operation of a complex domain variable (s)

By operating on the transformed complex signal rather than the original real signal it is often possible to Substantially Simplify a problem involving:

Linear Differential Equations Convolutions Systems with Memory


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Time Domain Time Domain

)(tfin )(tfoutOperation

Signal Analysis Operations on signals involving linear differential equations

may be difficult to perform strictly in the time domain

These operations may be Simplified by: Converting the signal to the Complex Domain Performing Simpler Equivalent Operations Transforming back to the Time Domain

Complex Domain

)()( sFoperationsF outin


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Laplace Transform Definition The Laplace Transform of a continuous-time

signal is given by:

0

)()()( dtetxtxLTsX st

x(t) = Continuous Time Signal

X(s) = Laplace Transform of x(t)

s = Complex Variable of the form +j


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Finding the Laplace Transform requires integration of the function from zero to infinity

For X(s) to exist, the integral must converge Convergence means that the area under the integral is finite

Laplace Transform, X(s), exists only for a set of points in the s domain called the Region of Convergence (ROC)

Convergence

0

)()()( dtetxtxLTsX st


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Magnitude of X(s) For a complex X(s) to exist, it’s magnitude must converge

)(sX

0

)(

0

)()()( dtetxdtetxsX tjst

0

)()( dteetxsX tjt

By replacing s with +j, |X(s)| can be rewritten as:


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|X(s)| Depends on The Magnitude of X(s) is bounded by the integral of the multiplied

magnitudes of x(t), e-t, and e-jt

000

)()()()( dteetxdteetxdteetxsX tjttjttjt

e-t is a Real number, therefore |e-t| = e-t e-jt is a Complex number with a magnitude of 1

0

)()( dtetxsX t

Therefore the Magnitude Bound of X(s) is dependent only upon the magnitude of x(t) and the Real Part of s


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Region of Convergence Laplace Transform X(s) exists only for a set of points in the

Region of Convergence (ROC)

0

)( dtetx t

X(s) only exists when the above integral is finite

The Region of Convergence is defined as the region where the Real Portion of s ( meets the following criteria:


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ROC Graphical Depiction

In general the ROC is a strip in the complex s-domain

0

)( dtetx t

j

s-domain js

The s-domain can be graphically depicted as a 2D plot of the real and imaginary portions of s

0

)()( dtetxsX st

X(s) Exists Does NOT ExistDoes NOT Exist


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Inverse Laplace Transform Inverse Laplace Transform is used to compute x(t) from X(s)

The Inverse Laplace Transform is strictly defined as:

Strict computation is complicated and rarely used in engineering

jc

jc

stdsesXj

sXLTtx )(2

1)()( 1

Practically, the Inverse Laplace Transform of a rational function is calculated using a method of table look-up


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Properties of Laplace Transform1. Linearity

2. Right Time Shift

3. Time Scaling

4. Multiplication by a power of t

5. Multiplication by an Exponential

6. Multiplication by sin(t) or cos(t)

7. Convolution

8. Differentiation in Time Domain

9. Integration in Time Domain

10. Initial Value Theorem

11. Final Value Theorem


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1. Linearity

)()( sXtx LT

)()( sYty LT

)()()()( sbYsaXtbytax LT

The Laplace Transform is a Linear Operation Superposition Principle can be applied


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2. Right Time Shift Given a time domain signal delayed by t0 seconds

The Laplace Transform of the delayed signal is e-tos

multiplied by the Laplace Transform of the original

signal

)()( sXettx stLTo

o

)()( sXtx LT


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3. Time Scaling

a

sX

aatx LT 1

)(

x(t) Stretched in the time domain

x(t) Compressed in the time domain

1 )( )( aifatxtostretchedtx

1 )( )( aifatxtocompressedtx

Laplace Transform of compressed or stretched version of x(t)


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4. Multiplication by a Power of t

Multiplying x(n) by t to a Positive Power n is a function of the nth derivative of the Laplace Transform of x(t)

)()( sXtx LT

)()1()( sX

ds

dtxt

n

nnLTn


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5. Multiplication by an Exponential

)()( sXtx LT

)()( asXtxe LTat

A time domain signal x(t) multiplied by an exponential function of t, results in the Laplace Transform of x(t) being a shifted in the s-domain


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6. Multiplication by sin(t) or cos(t)

A time domain signal x(t) multiplied by a sine or cosine wave results in an amplitude modulated signal

)()(2

1)cos()(

)()(2

)sin()(

jsXjsXttx

jsXjsXj

ttx

LT

LT


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7. Convolution

)()()(*)( sYsXtytxLT

0

)()()(*)( dtyxtytx

The convolution of two signals in the time domain is equivalent to a multiplication of their Laplace Transforms in the s-domain

* is the sign for convolution


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In general, the Laplace Transform of the nth derivative of a continuous function x(t) is given by:

8. Differentiation in the Time Domain

)0()()(

xssXdt

tdxLT

dt

tdxsxsXs

dt

txdLT

)()0()(

)( 22

2

)1(

)1(

)2(

)2(21 )0()0(

...)0(

)0()()(

n

n

n

nnnn

n

n

dt

xd

dt

xds

dt

dxsxssXs

dt

txdLT

1st Derivative Example:

2nd Derivative Example:


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9. Integration in Time Domain

)(1

)(0

sXs

dxLTt

The Laplace Transform of the integral of a time domain function is the functions Laplace Transform divided by s


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10. Initial Value Theorem For a causal time domain signal, the initial value

of x(t) can be found using the Laplace Transform as follows:

)(lim)0( ssXxs

Assume: x(t) = 0 for t < 0 x(t) does not contain impulses or higher order singularities


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11. Final Value Theorem

The steady-state value of the signal x(t) can also be determined using the Laplace Transform

)(lim)(lim0

ssXtxst


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Properties of the Laplace Transform that can be used to simplify difficult time domain operations such as differentiation and convolution

Summary Laplace Transform Definition

Region of Convergence where Laplace Transform is valid

Inverse Laplace Transform Definition

Documents

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