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

518a is passed through an ideal lowpass filter of

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Unformatted text preview: te the first (!:fJ') p oints (0 ::; n ::; !:fJ' - 1) of Fr using Eq. (5.38) and compute t he last !:fJ' points using Eq. (5.41) as No (5.42a) No (5.42b) O <-r < - - l 2 O<r<--l -2 Thus, an N o-point D FT c an be computed by combining the two (!:fJ' )-point DFTs, as in Eq. (5.42) _ These equations can be represented conveniently by the s ignal f low graph depicted in Fig. 5.19. This structure is known as a b utterfly. Figure 5.20a shows the implementation of Eqs. (5.39) for t he case of No = 8. T he n ext s tep is t o c ompute the {!:fJ')-point D FTs G r a nd H r. We repeat t he same procedure by dividing 9k a nd h k i nto two (~)-point sequences corresponding t o t he even- and odd-numbered samples. Then we continue this process f7 -1 - wsl - (e) F ig. 5 .20 Successive steps in 8-point F FT. W3 S F7 W 1 S 5 Sampling 356 until we reach t he o ne-point D FT. F igures 5.20a, 5.20b, a nd 5.20c show these steps for t he case o f No = 8. Figure 5.20c shows t hat t he two-point D FTs require no multiplication. To count t he n umber o f c omputations required in t he first step, assume t hat G a nd H a re known. Equations (5.42) clearly show t hat t o c ompute all t he No p ;ints o f t he F r , w e require No complex additions a nd !ft complex multiplicationst (corresponding t o W IVoHr). I n t he s econd s tep, t o c ompute t he (!ft ) -point D FT Gr from t he (~)-point D FT, we require 1ft complex additions a nd ~ complex multiplications. We require a n e qual number of computations for Hr. Hence, in t he second step, t here are. No c omplex a dditions a nd & complex multiplications. T he n umber o f computatiOns required remains t he sa~e in each step. S ince a t otal of log2 No s teps is needed t o a rrive a t a one- point D F T, we require, conservatively, a t otal of No log2 No. complex additions a nd (1!.IJ.) log2 No complex multiplications, t o c ompute t he N o-pomt D FT. T he proced~re for obtaining I DFT is identical t o t hat ~~d t o o btain t he ?~T e xcept t hat W No = ei (21f/No) i nstead of e - 3 (21f/No) (in a ddItIon t o t~e mu~tIp~ler l /No). A nother F FT a lgorithm, t he decimation-in-fr~quency .algonth.m, IS sImilar t o t he d ecimation-in-time algorithm. T he only dIfference 18 t hat m stead of dividing fk i nto two sequences o f even- and o dd-numbered samples, we divide fk i nto two sequences formed by t he first !ft a nd t he l ast !ft digits, proceeding in t he s ame way until a single-point D FT is reached in log2 No s teps. T~e to~al ~um.ber o f c omputations in this algorithm is t he s ame as t hat in t he declmatiOn-m-tIme algorithm. 5 .4 A ppendix 5.1 P roblems 5 .5 357 Summary A signal bandlimited t o B Hz c an b e r econstructed exactly from its samples if t he sampling r ate ;::. > 2 B Hz (the sampling theorem). Such a r econstruction, although possible theoretically, poses practical problems such as t he need for filters with zero gain over a b and (or bands) of frequencies. Such filters are unrealizabl...
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