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lecture2 - EE 313 Linear Systems and Signals Spring 2009...

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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 Spring 2009 Continuous-Time System Properties
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2 - 2 ( 29 ( 29 ( 29 ( 29 t x f a t x a f = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t x f t x f t x t x f 2 1 2 1 + = + Linearity A system is linear if it is both Homogeneous : If we scale the input, then the output is scaled by the same amount: Additive : If we add two input signals, then the output will be the sum of their respective outputs Response of a linear system to zero input?
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2 - 3 ( 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?
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2 - 4 ( 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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2 - 5 ( 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 : Nonlinear (in fact, fails both tests) Differentiation is linear Homogeneity test: Additivity test: ( 29 ( 29 ( 29 t x t y cos = ( 29 ( 29 t x t y 2 = ( 29 dt d x ( t ) y ( t )
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