Relation between Fourier Series and Transform

Transcription

1 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Rlion bwn ourir Sris n Trnsform Th ourir Trnsform T is riv from h finiion of h ourir Sris S. Consir, for xmpl, h prioic complx sinl To wih prio T. Th xponnil S of h sinl llows h rprsnion of To s n To Dn, n Whr h complx cofficins D n r vlu s D n T T To n for ny prio T. or convninc, w will us h prio T T, which ivs D n T T To T n, No h h mnius n phss of h cofficins D n r, rspcivly, known s h mpliu n phs spcrums of To, or D n is mpliu spcrum of To, n D n is phs spcrum of To. Now, l us spr h iffrn prios of To wih zros such h h iffrn prios of To mov wy from ch ohr by holin on prio of To n incrsin h prio urion T unil i bcoms infini. Whn T, h sinl To cn b rnm sinc i is no lonr prioic bu only conins on prio of To. In his cs, T, n h iscr-im sinl rprsn by h cofficins D n vs. n bcoms in h limi coninuous-im sinl in rms of nw vribl n. Throfr, D cn b wrin s n n T, whr n, n D n n T.

2 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su ourir Trnsform T n Invrs ourir Trnsform IT Th T of sinl is no ] n is fin s ], n h IT of is no ] is fin s ] W sy h n for T pir, or. Noic h h xponn rm in hs niv sin bu no niv sin xiss in h xponn in. Also, noic h h inrion in is in rms of n i is in rms of in. Also noic h in nrl is complx sinl h hs boh mniu n phs θ, or θ. nrl Propris of h T Symmry or REAL * * θ θ or REAL wih vn funcions or REAL wih o funcions is PURLEY REAL is PURELY IMAINARY Exisnc of h T or sinl, if <, T ] xiss.

3 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Th opposi is no ncssrily ru. This bov xisnc coniion coms from h fc h. < Thrfor, if h inrion of h mniu of funcion is fini, hn h T inrl is lso fini n hrfor, h T xiss. 3 Linriy If n f, hn b f b. Mnin of Niv rquncy On p 57 of your xbook, h uhor xplins h mnin of niv frquncy. Hr I ll rphrs wh h mnion in slihly iffrn wy. W know h h frquncy of ny sinl is lwys ivn in rms of posiiv numbr. You, for xmpl, woul sy h h frquncy of rio chnnl is 65 khz n nvr sy h i is 65 khz. So, wh is h mnin of h pr of h T h flls o h lf of h y xis h pr wih <? I is known h you cn scrib h frquncy spcrum of ny rl sinl such s ll h sinls h you cn nr in h lb usin only on hlf of h frquncy rn ihr posiiv or niv, bu no n for boh. So, wh hppns on h ohr si? Th mniu of h frquncy rspons of ny rl funcion is n vn funcion, n h phs of h frquncy rspons is n o funcion, so if w know ihr h niv or h posiiv hlvs, w cn h ohr hlf usin hs symmry propris. or purly iminry sinl, similr hin hppns n hrfor, only on hlf of h frquncy spcrum is n. Now, complx vlu sinl is combinion of rl sinl n n iminry sinl n hrfor, i crris wic h moun of informion s rl vlu or n iminry sinl. So, complx vlu sinl woul rquir wic h frquncy rn o scrib is conns s rl or iminry sinl. This oubl h moun of informion is bsiclly scrib on h wo hlvs of h spcrum. In fc, you cn us only h posiiv hlf, bu you woul n wo frquncy spcrums o scrib h frquncy conns of h complx sinl. T of Imporn uncions

4 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su T of h Uni Impuls uncion ] T of h uncion > < rc sin rc rc Th funcion sinxx is cll sincx rc sinc 3 IT of ] o 4 T of cos ] ] ] ] cos cos cos ] Similrly,

5 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su sin ] Wh os Spcrum of Sinl Mn? Now w know how o h T from sinl n how o bck from usin h IT formul. Bu, wh os h T of sinl physiclly mn? Th T of sinl rprsns h frquncy conns of h sinl. Th is, wh sins or cosins up ohr o form h sinl. So, i pprs h h T os h sm hin s h S. In fc, h is ru. Th iffrnc bwn h S n h T is h h S shows wh sins n cosins wih frquncis h r mulipls of som funmnl frquncy combin o prouc h PERIODIC sinl. Th prioiciy of ny sinl h h S simuls vn if h sinl w r pplyin h S o is no prioic, h S uomiclly ssums h prioiciy of h sinl h is ivn in h prio h w r inrin ovr cuss h spcrum of h sinl o b iscr frquncy sinl h is fin only mulipls of. or nrl sinls h r no prioic, h T which is form of h S bcoms coninuous frquncy sinl h is fin for ll vlus of. So, how much nry or powr os sin funcion, for xmpl, wih spcific frquncy conribu o sinl h is no prioic? Th nswr is nrlly zro. Only if w k rn of frquncis, such s h rn of 5 o 6 Hz, w cn sy h h conribuion of h sin wvs wih frquncis in his rn is J his coms from h fc h h inrion of sinl wih fini mniu bwn h poins 5 us bfor 5 o 5 us fr 5 is lwys zro. Propris of h ourir Trnsform If 3 Symmry bwn h T n IT L s b im vribl n β b frquncy vribl. ] H By comprin h rm bwn h brcks wih Equion 3 bov, w

6 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su ] H b Tim Sclin ] if > : L ] if < : L ] c Tim n rquncy Rvrsls ], L This cn lso b obin usin b bov wih Tim Diffrniion

7 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Sinc h iffrniion is wih rspc o n h inrion is wih rspc o, w cn brin riviv insi h inrl s ] - n n n n Tim Inrion Also hr, h wo inrls r in rms of iffrn vribls, so w cn swich h orr of inrion s Sinc h prio of inrion in h innr inrl is, w cn insr uni sp funcion u h is qul o in his rion n ousi n chn h limis of inrion o b from o, which ivs u. By sin s, w

8 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su s s u s s u s s Noic h h innr inrl is nohin bu h T of us, ] - - f Tim n rquncy Shifin n Muliplyin by Sinusoi Usin h frquncy shifin propry in bov, cos ] sin ] h Convoluion of Two Sinls Th convoluion of wo sinls n f is fin s f f f * * *

9 EE Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Similrly, * s * s s s * s s Th T of h convoluion of wo sinls is h prouc of h wo Ts, n h IT of h convoluion of wo sinls is h prouc of h wo ITs * f n f *