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Unformatted text preview: 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: f ( t ) * g ( t ) = Z + - f ( ) g ( t- ) d = Z + - 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 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 lim T R + iT - iT e st 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...
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