Coninuous im ourir rnsform Rviw. or coninuous-im priodic signl x h ourir sris rprsnion is x x j, j 2 d wih priod, ourir rnsform Wh bou priodic signls? W willl considr n priodic signl s priodic signl wih n infini priod. h rsul is h ourir rnsform of coninuous-im signls. or coninuous-im signl x, h ourir rnsform pir is x 2 jw j j d x j d Invrs ourir rnsform ourir rnsform
2 rom ourir Sris o ourir rnsform Considr signl x for x of fini durion. Consruc priodic signl x h is sm s x during on priod.. x x j, j h ourir sris rprsnion of x d. is 2 W will s how h infinily long priod urn h ourir rsprsnion ino h ourir rnsform. x x x j j. j j d choosing h priod in which x d sinc x d ousid, dfining j x j dd, x, ourir rnsform 2
3 Subsiuing 2 ino, x 2 2 j j j j j j j d 2 noing, s, As, x x for ny fini im inrvl. j x j d Invrs ourir rnsform 2 or coninuous-im signl x, h ourir rnsform pir is j x j d Invrs ourir rnsform 2 j jw x d ourir rnsform No. Alhough w hv drivd h ourir rnsform pir bsd on durion limid signls, w will us h rnsform lso for signls which r no limid in durion. Mos prcicl priodic signls vnish s h im gos o infiniy.
4 of n Exponnil u j R L x u. h ourir rnsform of x is jw x j, j j d d R. Whn is rl nd posiiv, j 2 j n 2 nd
5 of h Uni Impuls Signl h ourir rnsform of is j j jw d ll frquncis.
6 of Consn 2 Sinc, using h invrs ourir rnsform, 2 j d is usful idniy for. Inrchnging nd in, 2 2 which sys 2. j d j d Howvr from h lmm follwing,.
7 Lmm = rl, Proof of h cs. or ny funcion g coninuous, g d subsiuing, On h ohr hnd, g d g. d Combining q. nd 2, g d g ( d2) hrfor g d g d g for ny..
8 of Boxcr Signl x,, ohrwis sin 2 h ourir rnsform of x is jw x j d j d or, jw or, 2 jw j 2 j j sin 2 j 2 j j No sin sin = for. sin is clld sinc crosss zro whnvr is nonzro mulipl of. As dcrss, h signl bcoms nrrowr in h im domin, nd h firs zro crossing poin incrss in h frquncy domin. In h limi s, x 2 nd ( j). 2
9 of Sinc Signl x W sin W W, j, W W Considr h signl x whos ourirr rnsform is, j, W W. ind x using h invrs ourir rnsform. x 2 j j d 2 W W 2 j j jw W sin W W d jw No h duliy propry of h ourir rnsform. No h h impuls rspons of n idl lowpss filr is sinc funcion.
of Complx Exponnil j 2 W now. Using h invrs ourir rnsform, j dw. 2 Inrchnging nd, 2 j d, rplcing by 2 Susiuing, 2 j d. j d which implis j j d, j 2
of Priodic Signls j 2 Considr priodic signl x wih priod. h ourir sris rprsnion of h signl x is j x Rclling 2, j 2. 2 j
2 of sin sin j j j x sin Rclling 2, 2j 2j j j 2j 2j j 2 2 j j Purly Imginry nd Odd j j j
3 of cos cos x j cos Rclling 2, Rl nd Evn 2 2 j j j 2 2 2 2
4 of Impuls rin 2 2, Priodic Impuls rin wih Priod Priodic Impuls rin wih Priod h ourir cofficins of x r j 2 for ll. d Rclling 2, h ourir sris rprsnion of x x j j. 2 2 j.. Rl nd Evn Priodic Impuls rin wih Priod is
5 Rlion bwn h ourir rnsform of im Durion Limid Signl nd h ourir Cofficins of Priodic Signl jw 2 As n xmpl, considr boxcr funcion. x is im durion limid, nd is ourir rnsform is sin jw 2 x is priodic wih priod. sin 2. for, 2,. In his xmpl, jw sin 2 sin 2 2 sin jw 2 Rmmbr his propry is ru for ny durion limid signl x W drivd h ourir rnsform from h ourir sris bsdd on h propry..
6 jw h firs zro crossing occurs Whn 4, 2 2 Whn 8, sin 2 h firs zro crossing poin coincids wih 2 2 4., nd, nd h firs zro crossing poin coincids wih 4 in jw...