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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

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Laplace Transform: X(s) Fundamentals 2 Laplace Transform Fundamentals Introduction to the Laplace Transform Introduction Laplace Transform Definition Laplace Transform Region of Convergence Region of Convergence Inverse Laplace Transform Inverse Laplace Transform Properties of the Laplace Transform Properties

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Laplace Transform: X(s) Fundamentals 3 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: u Linear Differential Equations u Convolutions u Systems with Memory

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Laplace Transform: X(s) Fundamentals 4 Time Domain Operation 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: u Converting the signal to the Complex Domain u Performing Simpler Equivalent Operations u Transforming back to the Time Domain Complex Domain

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Laplace Transform: X(s) Fundamentals 5 Laplace Transform Definition The Laplace Transform of a continuous-time signal is given by: x(t) = Continuous Time Signal X(s) = Laplace Transform of x(t) s = Complex Variable of the form +j

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Laplace Transform: X(s) Fundamentals 6 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

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Laplace Transform: X(s) Fundamentals 7 Magnitude of X(s) For a complex X(s) to exist, it’s magnitude must converge By replacing s with +j , |X(s)| can be rewritten as:

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Laplace Transform: X(s) Fundamentals 8 |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 -j t e - t is a Real number, therefore |e - t | = e - t e -j t is a Complex number with a magnitude of 1 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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Laplace Transform: X(s) Fundamentals 9 Region of Convergence Laplace Transform X(s) exists only for a set of points in the Region of Convergence (ROC) 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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Laplace Transform: X(s) Fundamentals 10 ROC Graphical Depiction In general the ROC is a strip in the complex s-domain jj s-domain The s-domain can be graphically depicted as a 2D plot of the real and imaginary portions of s X(s) Exists Does NOT Exist

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Laplace Transform: X(s) Fundamentals 11 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 Practically, the Inverse Laplace Transform of a rational function is calculated using a method of table look-up

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Laplace Transform: X(s) Fundamentals 12 Properties of Laplace Transform 1.LinearityLinearity 2.Right Time ShiftRight Time Shift 3.Time ScalingTime Scaling 4.Multiplication by a power of tMultiplication by a power of t 5.Multiplication by an ExponentialMultiplication by an Exponential 6.Multiplication by sin( t) or cos( t)Multiplication by sin( t) or cos( t) 7.ConvolutionConvolution 8.Differentiation in Time DomainDifferentiation in Time Domain 9.Integration in Time DomainIntegration in Time Domain 10.Initial Value TheoremInitial Value Theorem 11.Final Value TheoremFinal Value Theorem

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Laplace Transform: X(s) Fundamentals Linearity The Laplace Transform is a Linear Operation Superposition Principle can be applied

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Laplace Transform: X(s) Fundamentals Right Time Shift Given a time domain signal delayed by t 0 seconds The Laplace Transform of the delayed signal is e -t o s multiplied by the Laplace Transform of the original signal

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Laplace Transform: X(s) Fundamentals Time Scaling x(t) Stretched in the time domain x(t) Compressed in the time domain Laplace Transform of compressed or stretched version of x(t)

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Laplace Transform: X(s) Fundamentals Multiplication by a Power of t Multiplying x(n) by t to a Positive Power n is a function of the n th derivative of the Laplace Transform of x(t)

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Laplace Transform: X(s) Fundamentals Multiplication by an Exponential 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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Laplace Transform: X(s) Fundamentals 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

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Laplace Transform: X(s) Fundamentals Convolution 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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Laplace Transform: X(s) Fundamentals 20 In general, the Laplace Transform of the nth derivative of a continuous function x(t) is given by: 8. Differentiation in the Time Domain 1st Derivative Example: 2nd Derivative Example:

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Laplace Transform: X(s) Fundamentals Integration in Time Domain The Laplace Transform of the integral of a time domain function is the functions Laplace Transform divided by s

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Laplace Transform: X(s) Fundamentals Initial Value Theorem For a causal time domain signal, the initial value of x(t) can be found using the Laplace Transform as follows: Assume: u x(t) = 0 for t < 0 u x(t) does not contain impulses or higher order singularities

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Laplace Transform: X(s) Fundamentals Final Value Theorem The steady-state value of the signal x(t) can also be determined using the Laplace Transform

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Laplace Transform: X(s) Fundamentals 24 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

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