Wiley signals and systems e book TLFe BO 316

30 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 ( $ + 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 ~ ~ ) t m r one period of the spectriun (12.16) I This relat iorisliip represents the znverse dzso ehe-time Pbnnw transforni t ~ ~ g