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# The time series is the dft is given by 330 r 0 e

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Unformatted text preview: ve the second half of the samples of the DFT of , 4 £ £ 4  E 1 "" hh &  © ¨  " BGF4 D) 44" ) 4 § 7 ) 6© T© '    egX ¨ g  dcX % £ ¨© X §  4E7 §  h &  " B E 4 1D) "4" " ) 4 § 7 ) T© '    gX De5  dcX % £ ¨© X § 5 (3.25) In the last equation we apply the following substitution: and we end up with the following for- £ After rewriting the last expression in terms of mula: (3.26) (3.27) B E 4) h &  © " GFDA' §  )  T© gX DIdcX ' % E  § ¨  e5 as a function Now we have two expression to compute the DFT of a series of length of two time series of length . Good FFT algorithms repeat this trick until the ﬁnal time series are series of length . The recursions given in (3.24) and (3.27) are applied 41 B 4 3.1. THE Z TRANSFORM AND THE DFT 69 to recover the DFT of the original time series. It can be proved that the total number of operations of the FFT is proportional to (for a time series of length ). This is an impo...
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