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
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This note was uploaded on 01/25/2010 for the course EE 313 taught by Professor Cardwell during the Spring '07 term at University of Texas at Austin.

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

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