12.4 Sayripling Continuous Signals 30'7 Transfoming xLl(t)with ttra E'ouricr integral (9.1 ) yields (12.42) k We see fk)iil cornparisorr with the tPe:firlitiort of thc: F*transforln ( I 2.13): that the spectra agree if It = u!T is understood to be c,he normalised arigtdar freqiiency: ( I 2.13) The periodic Fom ie1 traiisforrrr of the sariqilcd c.ontiiinous-tizrrc signal x, ( ) is the saim as the spectrum of' (tic discrete signal r[k].Tlie cliinensionlcss angular frequency it of ~ ( r - ' "WII ) of t h r angular frequency of X ( cl") rrorrrr&rtl wit 11 jarxiplc ixitexval T The rclat iondiips h t w w n the cwntinuous-tinrc signal ?i'(t),the sampled 1.oiitiriiioiis-timc sigiia\ c a ( f ) , lhe series of: sampled vahies ~ [ k ] i ~ i i dtlicir spectra arc tlepicled iri Figure 12.9 discrete signals continuous signals \;,it) Figure l2.1): R.&i,io~~ship between the and 3; spectrum The relat ioiiship lictcvecii F arid Fk spectra gives a n irnpor l m t insight that allows us to transfe'er rnmy imp per ties aiitl thwrenis th;it apply t o the spectra of rontinuoiis-time sigria -a of discretc-(iirie signals In the following section tlie niobt important givrii It is casy to recognis;e t h a t soixrr th~urcnis,for emiiiplv, thr sim ern, miinot he applicd to discret c>-tiriie sigiids bet.ausc of the sampling pi0