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Unformatted text preview: ContinuousTime Convolution EE 313 Linear Systems and Signals Spring 2009 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 4  2 Useful Functions • Unit gate function (a.k.a. unit pulse function) • What does rect(x / a) look like? • Unit triangle function ( 29 < = = 2 1 1 2 1 2 1 2 1 rect x x x x ( 29 < = ∆ 2 1 2 1 2 1 x x x x 1 1/21/2 x rect( x ) 1 1/21/2 x ∆ ( x ) 4  3 Unit Impulse (Functional) • Mathematical idealism for an instantaneous event • Dirac delta as generalized function (a.k.a. functional) Unit area : Sifting provided g(t) is defined at t = 0 Scaling : • Note that δ (0) is undefined ∫ ∞ ∞ = 1 ) ( dt t δ ∫ ∞ ∞ = ) ( ) ( ) ( g dt t t g δ ∫ ∞ ∞ ≠ = if 1 ) ( a a dt at δ 1 2 2 lim Area = = → ε ε ε ( 29 ( 29 t P t ε ε δ lim → = ε ε t ε 2 1 = ε ε ε 2 rect 2 1 ) ( t t P ( 29 ( 29 t P t ε ε δ lim → = ε ε t ε 1 ∆ = ε ε ε t t P 1 ) ( 1 lim Area = = → ε ε ε 4  4 Unit Impulse (Functional) • Generalized sifting Assuming that a > 0 • By convention, plot Dirac delta as arrow at origin Undefined amplitude at origin Denote area at origin as (area) Height of arrow is irrelevant Direction of arrow indicates sign of area • With δ ( t ) = 0 for t ≠ , it is tempting to think f ( t ) δ ( t ) = f (0) δ ( t ) f ( t ) δ ( t T ) = f ( T ) δ ( t T ) t ( 29 t δ (1) Simplify unit impulse under integration only  < < < = ∫ a T a T a T a dt T t a a or if if 1 ) ( δ 4  5 Unit Impulse (Functional) • We can simplify δ (t) under integration • What about?...
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This note was uploaded on 01/22/2012 for the course EE 312 taught by Professor Shafer during the Spring '08 term at University of Texas.
 Spring '08
 Shafer

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