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MA360Handout 4b

# MA360Handout 4b - 1 MA360 Handout 4b(week of Jan 30 Sp2006...

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Unformatted text preview: 1 MA360 Handout 4b (week of Jan 30 ) Sp2006 Kallfelz The Unit Impulse Function 1 Consider square wave of width a and height 1 / a , anchored at t , i.e.: ( ) ( ) ( ) ( ) ( ) ( ) a a t t u t t u a t t u t t u t a a---- = +--- = Δ 1 ) ( Then by definition: ( ) ( ) ( ) ( ) ( ) a a x x u x x u x x x a a a---- = Δ =- → → lim lim δ Lemma 1: ( ) ( ) t t t t u dt d- =- δ Proof: ( ) ( ) ( ) h t t u h t t u t t u h dt d lim-- +- =- → Define: h = - a . Then: ( ) ( ) ( ) ( ) ( ) ( ) lim lim t t a a t t u t t u a t t u a t t u t t u a a dt d- =---- =----- =- → → δ 1 Also known as the ‘Dirac delta-function,’ in homage to physicist Paul Dirac. a 1 t a t + 2 Lemma 2: ( ) { } st e t t L- =- δ Proof: ( ) { } ( ) ( ) ( ) { } ( ) { } { } ( ) ( ) { } ( ) ( ) ( ) s t as a s t as a s t da d as da d a s t as a s t a t s s t a a s t a a a a a e e e s se e as e e as e e s e s e a t t u L s e a t t u L t t u L a a t t u t t u L t t L 1 1 1 lim lim ) ( 1 lim 1 lim lim lim lim lim-- →-- →-- →-- →- +-- →- → → → = = =- =- = - = +-- =---- = ---- =- δ (Note how L’Hopital’s Rule must be invoked to evaluate this above limit)...
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MA360Handout 4b - 1 MA360 Handout 4b(week of Jan 30 Sp2006...

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