Wiley signals and systems e book TLFe BO 317

302 12 The Sgec*trumof Discrete Signals iscrete-Time Fonrier This section containb the Fourier transforms of the simple sequences discussed in Section 12.2 Thcse were the uriit impulse, unit stcp tunction and bilateral exponential series As a concluding example we will deterrilirre the spectrum of a rectangle ftmction +F*Transform of a 3.1 iserete Unit Impulse 'li, calculate the 7% traiisform of a unit iiiipirlse x [ k ] = [ k ]we start with the defining equation of the ;C, transform, and insert the unit impulse The result caii he given immediately, using the selcetive property of the unit impiilse: X(&) = C G [ k j e - ' ~ ~ k= I (12.17) k We thus obtain the transform pair (12 18) I Ark - K ] leads to a linear change T h e transform of a shifted unit impiilse L[/C) of phase as for a contiiiuous-time signal: q/C- fFje-3SZk = e- J $ Z h , (12.19) k and so the transform pair is We can PP 3.3.2 that, tbe transform pAr (12.18) is B special rase for K = .F* Transform of an Undamped Complex Exponential Series To find the F*transform of an uiidaniped bilateral exponcntial scries start wit,h the interesting relationship e3"Ok, we (12.21) fc U (,hat we had derived from equation (11.7) in thc last clmpter Now we are using it to determine the spcctruni of ,r[k]= eJnrlk: