University of Central Florida School of Electrical Engineering and Computer Science EGN 3420 Engineering Analysis Fall 2009 dcm Laplace Transform and its application for solving differential equations Fourier and Z Transforms Motivation Transform methods are widely used in many areas of science and engineering For example transform methods are used in signal processing and circuit analysis in applications of probability theory The basic idea is to transform a function from its original domain into a transform domain where certain operations can be carried out more efficiently carrying out the operation in the transform domain and then carrying out an inverse transform of the result from the transform domain to the original domain For example the convolution operation of two functions of time t f t and g t is defined as Z Z f t g t f g t d f t g d with a real number The convolution in the time domain becomes multiplication in the Laplace or Fourier domain As an application consider a linear circuit with the impulse response h t and with the signal x t as input Then the output of the linear circuit is y t x t h t If H s X s and Y s are the Laplace transforms of the impulse response the input and the output respectively then Y s H s X s as seen in Figure 1 Once we know Y s we can apply the inverse Laplace Transform to obtain the response of the circuit y t function of time t Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations While in some ways similar to separation of variables transform methods can be effective for a wider class of problems Even when the inverse of the transform cannot be found analytically numeric and asymptotic techniques now exist for their inversion Laplace Transform Let R be the field of real numbers and C the field of complex numbers Consider a function f R 7 R such that f t t R t 0 Then the Laplace Transform of f t is denoted as L f t and it is defined as F s with s C F s L f t R 0 e st f t dt The Laplace transform F s typically exists for all complex numbers s such that Re s a where a R is a constant which depends on the behavior of f t The Inverse Laplace Transform is given by the following complex integral f t L 1 F s 1 2 i limT R iT iT est F s ds Figure 1 A circuit with the impulse response h t and x t as input Then the output is y t x t h t If H s X s and Y s are respectively the Laplace transforms of the impulse response the input and the output then Y s H s X s where is a real number so that the contour path of integration is in the region of convergence of F s normally requiring s Re s for every singularity s of F s and with i 1 If all singularities are in the left half plane in this case Re s 0 for every s then can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform This integral is known as the Fourier Mellin integral The Bilateral Laplace Transform is defined as Z F s L f t e st f t dt In probability theory the Laplace transform is defined by means of an expectation value If X is a random variable with probability density function fX then the Laplace transform of fX is given by the expectation LfX s E e sX By abuse of language one often refers instead to this as the Laplace transform of the random variable X Replacing the variable s by t gives the moment generating function of X The Laplace transform has applications throughout probability theory including first passage times of stochastic processes such as Markov chains and renewal theory Laplace Transforms of a few functions f t In each case we start from the definition For example f t k Z t7 k k F s L k e st kdt e st t 0 s s Now f t e t t F s L e Z e st e t dt 0 1 s t t7 1 e t 0 s s Properties of the Laplace Transform The properties of the Laplace Transform summarized in Table 1 can be derived easily starting from the definition Table 1 Properties of the Laplace Transform The function f t is assumed to be n times differentiable with n th derivative of exponential type Notations F s L f t G s n L g t f n is the n th derivative R t of f t F s is the n th derivative of F s u t is the Heaviside step function u t d with the Dirac delta function f g t is the convolution of f t and g t R Property Time domain s domain Linearity f t g t F s G s 1 Scaling f t F s Frequency shifting e t f t F s Time shifting f t u t e s F s n n Frequency differentiation tn f t 1 R F s Frequency integration f t t F r dr s Differentiation f n t sn F s sn 1 f 0 f n 1 0 Rt Integration u f t 1s F s f d 0 Convolution f g t F s G s R T st 1 Periodic function f t f t T f t e f t dt 1 e T s 0 For example to prove linearity consider two functions f t and g t and their Laplace Transforms Z Z st F s L f t e f t dt and G s L g t e st g t dt 0 0 From the definition of the Laplace Transform it follows that Z Z Z st st L f t g t e f t g t dt e f t dt 0 0 e st g t dt F s G s 0 It is also easy to see that F 0 represents the area under the curve f t Z F s 0 f t dt 0 The Laplace Transform can be expressed as L f t f 0 s f 0 0 s2 f 00 0 s3 f 000 0 s4 Proof This important property of the Laplace Transform is a consequence of the following equality Z e x f 0 x f 00 x f 000 x x e f x dx f x 2 3 This is easy to prove by applying the derivation operator of both sides then the left hand side becomes A e x f x The right hand side is the sum of two terms B and C f 0 x f 00 x f 000 …