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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 opoint 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 oddnumbered samples. Then we continue this process f7
1  wsl  (e) F ig. 5 .20 Successive steps in 8point F FT. W3
S F7 W 1
S 5 Sampling 356 until we reach t he o nepoint 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 twopoint 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 opomt 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 decimationinfr~quency .algonth.m, IS sImilar t o t he d ecimationintime algorithm. T he only dIfference 18 t hat m stead of
dividing fk i nto two sequences o f even and o ddnumbered 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 singlepoint 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 declmatiOnmtIme
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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 Spring '13
 Bayliss
 Signal Processing, The Land

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