{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MA360Handout 4b

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)...
View Full Document

{[ snackBarMessage ]}

Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online