Signal Processing and Linear Systems-B.P.Lathi copy

To be able t o use t he d ft t echnique of circular

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Unformatted text preview: gorithm, known as t he F ast F ourier t ransform ( FFT), reduces the number of computations from s omething o f t he o rder of NJ t o No log N o. To compute one sample Fr from Eq. (5.18a), w e require No complex multiplications and No - 1 complex additions. To compute No such values (Fr for r = 0 ,1, . .. , No - 1), we require a t otal of NJ F, I I I I n5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.5 I I I • • : I'(Hz)_ 23 24 25 2. 27 28 29 30 31 353 H, 8.000 F,H, I I I I I I I I 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.5 I I I I I I I 7.119 5.027 2.331 0.000 -1.323 -1.497 -.8616 0.000 .5803 .6682 .3778 0.000 -.2145 - .)989 -.06964 0.000 -.06964 -.1989 -.2145 0.000 .3778 .6682 .5803 0.000 -.8616 -1.497 -1.323 0.000 2.331 5.027 7.179 Y, 8.000 7.179 5.027 .9285 2.331 . 9123 .4847 . 08884 - .05698 - .0\383 .02933 .004837 - .01966 - .002156 . 0153' .0009828 - .0\338 - .0002876 .01280 - .000287. - .0\338 .0009828 . 0153' - .002156 - .01966 .004837 .03933 - .0\383 - .05698 . 0888' . '847 .9123 0.000 -1.323 -1.497 -.8616 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -.8616 - 1.497 -1.323 0.000 2.331 5.027 7.179 \.009 \.090 \.090 \.009 complex ~ultiplications a nd No(No - 1) complex additions. For a large No, these computatIOns c an be prohibitively time-consuming, even for a high-speed computer. Although there ~re m any variations of t he original Tukey-Cooley algorithm, these can be grouped mto two basic types: d ecimation-in-time a nd d ecimationin-frequency. T he a lgorithm is simplified if we choose N o t o b e a power of 2 although such a choice is n ot essential. For convenience, we define ' (5.34) so t hat N o-l L Fr = ° ~ r ~ No - 1 ikWt!;, k=O a nd (5.35a) ° (5.35b) 1 N o-l kr ik = N '"' Fr WNo .L.... o ~ k ~ No - 1 r=O T he D ecimation-in- Time A lgorithm .H~re we divide t he No-point d ata sequence i k i nto two (~)-point sequences conslstmg of even- and o dd-numbered samples respectively, as follows: f o, , 12, f 4, . .. , f No -2, .... s equence 9 k ' f t, 13, f s, " . .. , f No-l " " s equence h k T hen, from Eq. (5.35a), ~-l Fr = ~-l 2kr 'L.... f 2k W No +.L.... ' "' . "' k=O k=O f 2 k+l W (2k+l)r No (5.36) 5.3 The Fast Fourier Transform ( FFT) 5 Sampling 354 fo Fo 355 fo Go f2 N.0 ~4 ' DFf F ig. 5 .19 14 Butterfly. F2 we have Fr = L k =O 0) fl F4 + WNo L fs F6 f) F7 fJ Ho h k+1W;' k =O 2 fl Fs ~-l h kW;' f2 f6 f6 (5.37) ;a-I f4 °2 Also, since fJ 2 0 ::; r ::; No - 1 N o=4 (5.38) where Gr a nd H r a re the (!:fJ' )-point D FTs o f the even- and o dd-numbered sequences, 9k a nd h k, respectively. Also, G r a nd H r , being the (!:fJ' )-point DFTs, are ( !:fJ' )-periodic. FI °1 Is HI D Ff H2 fJ H) Hence - l W8 D Ff - G r+(!:f) = G r (5.39) H r+(!:f) = H r Moreover, (a) ( b) fo Fo f4 FI f2 F2 f6 F3 fl F4 fs Fs f) F6 (5.40) From Eqs. (5.38), (5.39), and (5.40), we o btain (5.41) This property c an b e used t o reduce the number of computations. We c an compu...
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