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=] If = 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 E l [ k j = -, 3o < k < x 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 ~ [ kwe uiiit iriipulse drk] by &2[k] spj I= SubBtitxting (12.26) confirms this Q [ k ]- E*[k - 11 (12.29)