s. awad, ph.d. m. corless, m.s.e.e. e.c.e. department university of michigan-dearborn laplace...
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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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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:
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
|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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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:
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
1. Linearity
)()( sXtx LT
)()( sYty LT
)()()()( sbYsaXtbytax LT
The Laplace Transform is a Linear Operation Superposition Principle can be applied
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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)
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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:
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
11. Final Value Theorem
The steady-state value of the signal x(t) can also be determined using the Laplace Transform
)(lim)(lim0
ssXtxst
U of M-Dearborn ECE DepartmentMath Review with Matlab
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Laplace Transform:X(s) Fundamentals
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