Wiley signals and systems e book TLFe BO 316

30 1

12.3. Discrete-Time Fourier Transform.

e
It wonld be comwiierit to iise the advantages oflooking a t continuous-time signals in the frequency-clomain with ciiscrrte-the signa
To this end we ad1 be
introducing t h r discrete counterpart 1o the Fourier transform, the discrete-time
Fourier lransfvrrri (DTFT). As with the Fourier and Laplace transforms, its ust
in the frequency-dornain requires an inverse transform, t r aiisform pairs. theorems
and symmetry properties.

e ~ ~ i tofi the
o ~

iscrete-Time Fourier
As a series z [ k ]is only dcfincd for discrete values of k E Z,we cannot use the
Fourier iritegral (9.1) introduced in Chapt,er I). We therefore ctefine t,he discretetzme Fourzer transform or the
transform as:
(12.13)

It transforms a series .r[X]into a coritimious complex fiinct?ionof a real variable $2.
X ( e J " ) is also called the speetrum of U senes. In contrast to a coiilinuoiis signal,
il is periodic wit li 2n, so
X ( e 3 ( $ 2 + 2 q= SjeJ").
(12.14)
, as each term of the sum in (12.13) contains a 27i-periodie term
i27k'L
111 order to see this more cleaily. w~ miritc elf' as the argument of l,he .E*
transform arid d&nc the Fourier transform over the unity circle of the complex
plane. This coiivcntion will make Ihe transfer t o the z-transforiii easier.

A sufw~errt corrditzn?~to shorn7 the existence of the spc rum .7== { x [ k ]). is that
the sum of the series s [ k ] is finite:
(12.15)

iserete-Time
The defiriitioii of the spectrum of a sequence from (12.13) rcpreserrts a Fuurier
series of X(e-'"'). The period is 2;.r and tlic Fourier coeficieiits arp the valu
In order to rccover the series s [ k ]Goni the spectrum X ( e ? " ) , we haw t o iise the
formula for finding Foiirier coefficients. It consisls of air inlegration of ~ ( r 7 ~ ~ )
t m r one period of the spectriun
(12.16)
I

This relat iorisliip represents the znverse dzso ehe-time Pbnnw transforni

t ~ ~ g 1