Lesson_27 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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EEL 3135: Dr. Fred J. Taylor, Professor Lesson Title: Laplace Transform Lesson Number: (Supplemental) Background: Continuous-time signals where introduced in Chapter 9 along with the introduction of the concept of convolution, impulses, and unit step functions. The study hovered around the study of convolution of simple cases that involved un-delayed and delayed impulses. It was claimed that continuous-time convolution, defined by: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 τ d h t x d t h x t x t h t y - - - = - = = 1. where h(t) is the linear system’s impulse response, x(t) the system input, and y(t) is the system’s output. Equation 1 and convolution has been claimed to be fundamentally import to the study of signals and systems but little has been done to provide the reader with the ability to make intelligent use of convolution. In this lesson, this question will be addressed and hopefully resolved. The enabling tool will be the Laplace transform. Laplace Transform Given for one instant an intelligence that could comprehend all the forces by which nature is animated and ,,, sufficiently vast to submit these data to analysis – it would embrace in the same formula the movement of the greatest bodies of the universe and those of the lightest atom. For it, nothing would be uncertain and the future as the past, would be present to its eyes. - - Pierre Simon de Laplace At one point in Laplace’s career, his interests included solving ordinary differential equations (ODE). He realized that the solution to an ODE has the general form: ( 29 = = n i t i s i e a t x 1 2. where s i is called the eigenvalue of the ODE. Laplace invented a tool that would “scan” x(t), looking to terms of the form e s i t , and from this test determine a i (to some this would be called correlation). The tool is called the Laplace transform which is defined by: ( 29 ( 29 - - = st e t x s X 3. 1 Pierre-Simon Laplace
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Dr. Fred J. Taylor, Professor provided the integral exists. Example What is the Laplace transform of x(t)=e σ 0 t u(t). It follows that: ( 29 ( 29 0 0 ) 0 ( 0 0 1 σ - = = = - - s dt e dt e e s X t s st t for Re(s)> σ 0 . End of example --------------------------------------- As with z-transforms, common Laplace transforms have been reduced to tables (see Table 1) which eliminates the need to formally perform a Laplace transform of mathematically known signal x(t). Table 1: Common Laplace Transforms x(t) Laplace Transform δ (t) 1 u(t) 1/s tu(t) 1/s 2 t n u(t) n!/s n+1 e at u(t) 1/(s-a) t e at u(t) 1/(s-a) 2 t n e at u(t) n!/(s-a) n+1 sin( ϖ 0 t) u(t) ϖ 0 /(s 2 + ϖ 0 2 ) cos( ϖ 0 t) u(t) s/(s 2 + ϖ 0 2 ) e at sin( ϖ 0 t) u(t) ϖ 0 /((s-a) 2 + ϖ 0 2 ) e at cos( ϖ 0 t) u(t) (s-a)/((s-a) 2 + ϖ 0 2 ) The basic properties of Laplace transforms are also tabled (see Table 2), Their utility is in synthesizing the Laplace transform of complex signal by assembling the transform using Laplace Transform 2
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.

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Lesson_27 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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