Wiley signals and systems e book TLFe BO 321

306

12. The Spectrum of Uiscretc Signals

that can be rearranged as

The resull is shown in Figure 12.8

Figure 12.8: Magnitiide spectrum of a rectangle scries for clifevent

IV

Up t o IIOW. wc have discussed seyitences in gerternl, without consicleririg where
they come lrorn or what they represent. Kow we will examine wqucnccs x i k ] ,
that have tome about; from the sampling of continuous-time signals 2(L). The
~
intiividuaI members of a series xjkl are h e sarnpld valurs of t h continuous-time
signal B ( k T ) . as in (12.1).
According to this simple relationship in the timc-domain. a corresponding relationship in the frec~iimcy-dornaiiishotiltl exist. that mcaris bvtwcen the Fouricr
transform of t h r sampled continuous-time signal and the spectrum of the discretetime signal.
The sigiial .T, ( t )tlerived from the c.oiitimious-timr + p a l F ( t ) with ideal s;timpliiig (see Figure 12.9) will first of all be represented by ifii impulse train

Ir(kT)S(t-- k T ) .

(12.40)

W e expiess t h e weighting of the delta impulses by the values ,c[k]of the discrete-

time signal (see (12.1))
(12.41)