Wiley signals and systems e book TLFe BO 318

12.3. Discrete-Time Fourier Trarislorrri

303

The trnnsform pair is

and states that tlie spectrum of an rxponential series eJCiok
is in fa(
at L
I = Ro. The ambiguousness we discwsed in Section 12.2.3 likewise applies to
the spect rum of tlie delta impulse at (1 =
2nw, v E 22. The rwulting impulse
train in the frequency-spectrurri c m be elegantly represented using the sha-symbol
we idroducetl in Chapter 11. The correctness of the transform pair (12.23) can
also 17e confirmed simply by using the inverse F, t ransforrn on LLL
A special case ot (12.23) is the spectmm of the sequence ~ [ k=] 1. If
= 0 is
put into (12.23), the transformation yields

+

(9).

12.3.3.3 F..Transform of the Discrete Unit Step F'unction

Thc discrete mit step function ~ [ kcan
] be expressed as the sum of
term
1
E l [ k j = -,
--3o < k < x

2
a n d a bilateral step series with no middle value

it

constant
(12.25)

(12.26)
E[k]= E l !k]I- &a[k].

The Fortrier transfclrm of the constanl term
from (12.24)

EI

(12.27)
[ k ] can be obtainetl directly

( 12.28)
] express the
To determine the Fourier transforiii of the second term ~ 2 [ kwe
uiiit iriipulse drk] by &2[k]

spj

I=

SubBtitxting (12.26) confirms this.

Q [ k ]- E*[k - 11.

(12.29)