# Good fft algorithms repeat this trick until the nal

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Unformatted text preview: [Xk] 0 0.5 1 −0.2 0 5 10 15 25 n 20 30 35 40 45 50 0 x 0.2 0.4 3.1. THE Z TRANSFORM AND THE DFT 67 CHAPTER 3. DISCRETE FOURIER TRANSFORM " £ 68  3©  X" X I£ ) 4" " ) £ ) I£ ) £ )  £ First, we will assume that one wants to compute the DFT of the time series . Using the deﬁnition  ) BGEF4 1 ¡¢' § 7 ) 6© '     h & gX De5  dcX % £ 3© X § 5 (3.22) we can rewrite the last equation in terms of two time series composed of even samples and odd samples , respectively.   4   h  ¦h e X & X  " 6© gX eP5 hi ¨¥ X cdX &' % ¨© XI£ ¨ © §6© gX P5  IdcX ' %  I£ 3 © § 5 g  "" ¤ 44" ¥£ ) £ ) £ § £ "" p X £ 44" C£ ) I£ )  § (3.23) It is clear that the RHS term can be written in terms of the DFTs of (even samples) and (odd samples) £ B E 4¡ h &5 " RQ' § 7 ) T© gX DIdcX ' % ¦ ¤ § 5 e5 (3.24) £ samples of the DFT of The last equation provides a formula to compute the ﬁrst based on the samples of the DFT of and . Now, note that we need another formula to retrie...
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