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

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

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

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

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 

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

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:

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

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:

Laplace Transform: X(s) Fundamentals 10 ROC Graphical Depiction  In general the ROC is a strip in the complex s-domain jj  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

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

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

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

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

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)

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)

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

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

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

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:

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

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

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

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