Wiley signals and systems e book TLFe BO 319

304

12. The Spectrum of Discrete Signals

Using the laws of Iinparity and displacement from Section 12.5, we obtain

F*{&'$T]- & 2 [ k - l ] )

=.F*{Ea[k]}

I]} = F*{EZ[k])-e-3"3*{52[k.j),
( 123 0 )
together with (12.18) we t,heiJ obtain frorn (12.29)
-.F*{E2[k-

F . { E ~ [-~ e]- 7}6 2 ~ 3 , { ~ 2 r ~ ] \ (= 1

(12.31)

and from here we see that

As wc cannot dividr by zero, (12.32) can only he used for 52 # . . . -2n, 0,27r, 4;.r
If .?=*{E~ [ k ] }would contain delta impulses at these fIecpeucies thcy would havt. to
be considered separately. With (12.26), kiowrmi, 5 2 [ k ]has zero wean and therefore
there can be no delta impulses at 62 = 27w, U c Z. It should noxi, be eviclewit why
we split ~ [ kup
] into cl[k] i
~zlA-1. By adding (12.28) and (12.32) togetlier, we
finally obtain from (12.27)

(12.33)

We now con?pare this result with the Iiourier trarisform of the unit step functioii
in (9.92). We also foiind two tPriris there: one delta impulse and a term for wliirh
s = J W citrne from the transfer function of ari integiator. With thc .F* transform,
the discrete unit step fiiriction produces an inipulsc train instead of a11indivichial
impulse. Tlie other trrrri happens to I)t. thc traiisfcr function of a n accumulator.
aiid ~ 7 "will show that it iii fact rtpreseiils the discrete counterpart of a i l integrator
(sce Example 14.5).
12.3.3.4

.E* Transform of a Unilateral ~ x ~ o n ~ n tSequence
ial

\VP will now find the ;L;; transform of a iiiiilateral expoiieiltial sequence x[k! =
I ? E [ ~ ] with a EC from the defining eqrration (12.13):

(12.34)
We know that thc sum of an infinite gcwrietrir series is
c72

(12.35)