Wiley signals and systems e book TLFe BO 322

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, ( 1 ) 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 7 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