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# lecture3 - EE 313 Linear Systems and Signals Fall 2010...

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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 EE 313 Linear Systems and Signals Fall 2010 Continuous-Time Systems

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3 - 2 Systems A system is a transformation from One signal (called the input) to Another signal (called the output or the response) Continuous-time systems with input signal x and output signal y (a.k.a. the response): y ( t ) = x ( t ) + x ( t -1) y ( t ) = x 2 ( t ) Discrete-time examples y [ n ] = x [ n ] + x [ n -1] y [ n ] = x 2 [ n ] x ( t ) y ( t ) x [ n ] y [ n ]
3 - 3 ( 29 ( 29 ( 29 ( 29 ) ( ) ( t y a t x f a t x a f t y scaled = = = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ) ( ) ( ) ( 2 1 2 1 2 1 t y t y t x f t x f t x t x f t y additive + = + = + = Linearity Given a system y ( t ) = f ( x ( t ) ) System is linear if it is both Homogeneous : If we scale the input signal by constant a , output signal is scaled by a for all possible values of a Additive : If we add two input signals, output signal will be the sum of their respective outputs Response of a linear system to all-zero input? x ( t ) y ( t )

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3 - 4 ( 29 ) ( T t x t y - = Examples Identity system. Linear? Ideal delay by T seconds. Linear? Scale by a constant (a.k.a. gain block). Linear? T x ( t ) y ( t ) 0 a x ( t ) y ( t ) ( 29 ) ( 0 t x a t y = 0 a x ( t ) y ( t ) ( 29 ( 29 t x t y = Two different but equivalent graphical syntaxes x ( t ) y ( t ) Role of initial conditions?
3 - 5 ( 29 t x T T T Σ ( 29 t y 0 a 1 - N a 2 - N a 1 a Each T represents a delay of T time units Examples Tapped delay line Linear? There are N-1 delays ( 29 ( 29 ( 29 ( 29 ( 29 - = - - = - - + + - + = 1 0 1 1 0 ) 1 ( N k k N kT t x a T N t x a T t x a t x a t y

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3 - 6 ( 29 ( 29 t x dt d t y = ( 29 ( 29 ( 29 t x dt d a t x a dt d = ( 29 ( 29 ( 29 ( 29 ( 29 t x dt d t x dt d t x t x dt d 2 1 2 1 + = + Examples Transcendental system Answer : Nonlinear (in fact, fails both tests) Squarer Answer :
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