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lecture19

# lecture19 - EE313 Linear Systems and Signals Fall 2010...

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EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Transfer Functions

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19 - 2 Zero-State Response Linear constant coefficient differential equation Input x ( t ) and output Zero-state response: all initial conditions are zero Laplace transform both sides of differential equation with all initial conditions being zero and solve for Y ( s )/ X ( s ) ) ( ) ( ) ( t y t y t y state zero input zero - - + = ( 29 ( 29 s Y t y ( 29 ( 29 s Y s t y dt d r r r 0 ) 0 ( ) ( ) ( ) ( ' = = + - y t x t y t y ( 29 ( 29 s X t x ( 29 ( 29 s X s t x dt d k k k 1 1 ) ( ) ( ) ( ) ( ) ( ) ( + = = = + s s X s Y s H s X s Y s sY
19 - 3 Transfer Function H ( s ) is called the transfer function because it describes how input is transferred to the output in a transform domain ( s -domain in this case) Y ( s ) = H ( s ) X ( s ) y ( t ) = L -1 { H ( s ) X ( s )} = h ( t ) * x ( t ) H ( s ) = L { h ( t )} Transfer function is Laplace transform of impulse response

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19 - 4 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 T s T s e s X s Y s H e s X s Y T t x t y - - = = = - = Transfer Function Examples Laplace transform Assume input x ( t ) and output y ( t ) are causal Ideal delay of T seconds Initial conditions (initial voltages in delay buffer) are zero ( 29 ( 29 - - = 0 dt e t x s X t s T x ( t ) y ( t )
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lecture19 - EE313 Linear Systems and Signals Fall 2010...

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