Convrgnc of ourir Trnsform Rding Assignmn Oppnhim Sc 42 pp289 Propris of Coninuous im ourir Trnsform Rviw Rviw or coninuous-im priodic signl x, j x j d Invrs ourir Trnsform 2 j j x d ourir Trnsform Linriy x j y Y j x by j by j
2 Tim Shifing x j 0 j 0 j j j j j 0 0 x mgniud unchngd phs shifd by j 0 j 0 proof x 0 x 2 2 2 j j d j j rplcing by 0 d j 0 j j d 0 Exmpl @49 ind h ourir rnsform of x x 5 x x 2 2 2 5 2 jw sin 2 2 2 jw 3 sin 2 2 jw 5 j 2 3 sin sin 2 2 2
3 Rfr sinn T jw 2T T
4 Tim nd rquncy Scling x j x j x j Th Invrs Rlionshio bwn h Tim nd rquncy Domins: If h signl is comprssd in h im domin, is T is srchd in h frquncy domin proof j j x x d x Whn 0, rplcing by j d x Whn 0, rplcing by x j j d j d
5 Conjug Symmry x j x j If x is rl, j = j If x is rl nd vn, j j = j, Rl nd Evn If x is rl nd odd, j j j, Purly Imginry nd Odd proof j x j d 2 king complx conjug on boh sids j x j d 2 2 2 j j j d j d j x j d () 2 Eq sys j is h T of x rplcing by -
6 If x is rl, x x Thrfor i mus b h Eq 2 is quivln o = 2 j j = 3 j j Wri j in Crsin form: jw R j jim jw nd jw R jw jim jw Eq 3 sys R jw R j nd Im jw Im jw ( 4) Eq 4 sys h, if x is rl, jw jw R is n vn funcion of nd Im is n odd funcion of Wri j j j j nd j jw jw jw jw Eq 3 sys in polr form jw jw nd j j ( 5) Eq 5 sys h, if x is rl, jw is n vn funcion of nd j is n odd funcion of
7 Exmpl - T of n Exponnil Rcll u j R 0 Assum is rl so h x is rl Thn R Im j j j j 2 2 is vn is odd 2 2 No j j n is vn 2 2 is odd x u j j R 0 Combining h wo sids, x x 2 2 2 0 is rl nd vn Is T is rl nd vn x x j j j j j 2 2 2
8 Diffrniion x j x j j proof Using h invrs ourir rnsform, j x j d 2 Diffrniing wih rspc o, j x j j d 2 j j is h T of x
9 T of Uni Sp j u proof L u U j Thn u Uj Combining h wo sids, Tking T on boh sids, u u xcp 0 2 U j U j ( ) L U j k V j, whr V j is n ordinry funcion Thn q sys Sinc, k k V j Vj 2 2 2 k V j V j ( 2) Eq 2 sys k nd V j is n odd funcion of 3 U j V j To find V q4 sys j, considr h rlion: u 4 d d Howvr j u ju j
0 Thus w hv 5 ju j Muliplying j o boh sids of q3, Sinc 0, j jv j V j j inlly w hv U j j
Convoluion y x h Y j j H j y is h oupu of h LTI sysm wih h uni impuls rspons h h inpu is x whn proof Rcll h pproch w ook from S o T jk 0 j x j d 2 lim 0 2 0 k jk0 jk 0 0 r ignfuncions of n LTI sysm Whn h LTI sysm hs h impuls rsposn h, h oupu is jk y jk0 H jk0 0 0 2 whr h H j H lim 0 ( ) 0 k j is h frquncy rsposn of h LTI sysm Eq is quivln o j y jh j d (2) 2 Eq 2 is n invrs T nd ss Y j j H j
2 Propris of LTI Sysms Rvisid Commuiv Propry x h h x, Disribuiv Propry x h 2 h x h x h 2 Associiv Propry Th ovrll sysm rspons dos no dpnd on h ordr of h sysms in cscd of LTI sysms
3 Ingrion x d j 00 j proof L y x d y is h oupu of h ingror LTI sysm whn h inpu is x y x Th uni impuls rspons of h ingror is u Thrfor w w cn s u rom h convoluion propry, Y j j U j, whr U j u j Y Noing j j j j 0, j Y j j j 0 Exmpl @42 ind h T of x soluion W will firs find h T of g d x d nd hn pply h ingrionn ppropry
4 On wy o driv x Thn x x u Rclling v x u u is o wri u v 0 0 0 u u providd v is coninuous 0, G 2sin j j 2sin 2cos j j G j G j 0 G 0 22 0 j 2sin 2cos j Rfr jw 2T sin On cn lso us h T of u T T u o g h idnicl rsul
5 Duliy in h Signls 2 x j j x j x j d 2 j 2 x j d Rplcing by, j 2 x j d Inrchnging nd, j 2 x j d d Eq d ss 2 x is h T of j
6 Exmpl - T of Boxcr Signl Knownn x, T 0, ohrwis jw sin T 2T T T of Sinc Signl Considr h signl h 2 j Thn from h duliy, Rplc T by W Thn H j 2 x 2 x h W sinw W is h uni impuls rspons of h idl low pss filr
7 Duliy in h Propris Exmpl Diffrniion Diffrni h signl in h im domin Thn w hd h propry: x j x j j Wh will hppn if w diffrni h T in h frquncy domin? Diffrniing wr, j j x d d j j jx d d Now w hv h following propry d x j jx j d
8 Exmpl Shif Shif h signl in h im domin Thn w hd h propry: j 0 x j x j 0 Wh will hppn if w shif h T in h frquncy domin? Rplcing by, j x j d 2 0 j0 x j 0 d 2 j j 0 2 0 x j d Now w hv h following propry j0 x j x j 0
9 Prsvl s Rlion 2 2 2 2 x d j d 2 lhs h ol nrgy in h im domin j 2 h nrgy pr uni frquncy j is rfrrd o s h nrgy - dnsiy spcrum proof W know j x j d 2 2 j j d rplcing by 2 Sinc x xx, j x d x j d d 2 2 2 2 2 inrchnging h ordr of ingrions j x j dd j j x d d j j d 2 2 j d
20 Muliplicion S j P j r s p Rj S jp jd 2 2 Homwork Prov h muliplicion propry