004_BME343_Laplace_Transform

004_BME343_Laplace_Transform - Continuous-Time System...

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Continuous-Time ystem Analysis using System Analysis using the Laplace Transform Why bother? • Analysis of continuous-time systems • Easier way to solve differential (system) equations Laplace Transform 4-1 • Time-domain vs. frequency-domain
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Objectives Be able to calculate the Laplace transform of a function directly from the integral, sketch the region of convergence e able to calc late the Laplace transform and its in erse sing the Be able to calculate the Laplace transform and its inverse using the table of transforms and the properties of the Laplace transform e able to use partial fraction expansion to help calculate the inverse Be able to use partial fraction expansion to help calculate the inverse Laplace transform of a ratio of two polynomials Be able to compute the impulse response of a linear constant coefficient ifferential equation using the Laplace transform differential equation using the Laplace transform Be able to use the Laplace transform to simplify convolutions Be able to determine the frequency response of a system from the Laplace transform and sketch it (Bode plots) Laplace Transform Be able to explain tradeoffs between choosing a particular filter as a function of the number of poles. 4-2
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Motivation • Differential equation • Simple math – solve the second-order linear differential equation – multiply VIII –by XI    2 56 ( ) 1 ( ) DD y tDx t   – for the initial conditions (0 ) 2 ) 1 y – and the input (0 ) y 4 ) () t t e ut Laplace Transform 4-3 xt
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Roman numbers Laplace Transform 4-4
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Motivation: the Laplace Transform Time-Domain Time-Domain t () y t 0 t t dt Model Solution y Integration and Convolution requency requency [.] L 1 [.] L lgebraic Frequency- Domain odel Frequency- Domain olution s Ys () () Ys GsFs Algebraic Techniques ime omain vs Frequency omain analysis Model Solution Laplace Transform 4-5 Time-domain vs. Frequency-domain analysis • Forward and inverse Laplace transform
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The Laplace Transform Laplace transform is transformation from time-domain to frequency- domain: for a signal x ( t ), its Laplace transform X ( s ) is defined by where s is the complex frequency ( () ) st X xte d t s  The signal x ( t ) is the inverse Laplace transform of X ( s ) 1 cj st Xsed x ts  Bilateral Laplace transform 2 c j j   Note that 1 ( ) [ ( )] and [ ( )] X s xt  LL 1 x tX s Laplace Transform 4-6 11 [() ] } and { { [ () ]} x t X x t X ss  L L
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Linearity of the Laplace Transform •I f ) () and t X s xt Then 11 2 2 () and  roof 2 2 1 1 ax t aX s  Proof [( ) ( ) ] = ) ( ) ] st a t a t e dt xx   L 2 2 1 2 2 st st ax t e d t a x t e d t      Laplace Transform 4-7
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The Region of Convergence (ROC) ( () ) st X x te d t s • Region of convergence (also called the region of xistence) specifies the set of values of r region existence) specifies the set of values of s
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This note was uploaded on 09/28/2009 for the course BME 343 taught by Professor Emelianov during the Fall '09 term at University of Texas at Austin.

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004_BME343_Laplace_Transform - Continuous-Time System...

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