Unformatted text preview: 350 5.2 Numerical C omputation o f t he F ourier Transform: T he D FT 351 where we define t he l inear convolution s um o f two discrete sequences f k a nd 9k as
00 i k * 9k = 2:: f n9k-n n =-oo F ig. 5 .17 Graphical picture of circular convolution. E quation ( 5.32b) c an b e proved in t he s ame way.
For non p eriodic sequences, t he c onvolution c an b e visualized in t erms o f two
sequences, w here one sequence is fixed a nd t he o ther sequence is inverted a nd moved
p ast t he fixed sequence, one digit a t a time. I f t he two sequences a re No-periodic,
t he s ame c onfiguration will r epeat a fter N o s hifts of the sequence. C learly t he
c onvolution i k ® 9k becomes No-periodic (circular), a nd s uch convolution c an b e
conveniently visualized as i llustrated in Fig. 5.17 for t he case of N o = 4. T he i nner
sequence i k is clockwise a nd fixed. T he o uter sequence 9k is i nverted so t hat i t
b ecomes counterclockwise. T his s equence is now r otated clockwise one u nit a t a
t ime. We m ultiply t he o verlapping numbers a nd a dd. For example, t he v alue of
i k ® 9k a t k = 0 (Fig. 5.17) is B ecause of t he w idth p roperty o f t he c onvolution, Ck e xists for 0 ::; k ::; N 1 + N 2 - 1.
To be able t o use t he D FT t echnique of circular convolution, we m ust m ake s ure
t hat t he c ircular convolution will yield t he s ame r esult as t he l inear convolution.
In o ther words, t he r esulting signal of t he c ircular convolution m ust have t he s ame
l ength ( Nl + N2 - 1) a s t hat o f t he s ignal resulting from linear convolution. T his
s tep c an b e a ccomplished by adding N2 - 1 d ummy s amples of zero value t o i k a nd
N l - 1 d ummy s amples of zero value t o 9k (zero padding). T his p rocedure changes
t he l ength of b oth i k a nd 9k t o b e N l + N2 - 1. T he c ircular convolution now is
identical t o t he l inear convolution except t hat it r epeats p eriodically with period
N l + N2 - 1. T he rigorous p roof o f t his s tatement is provided in Sec. 10.6-3.
We c an use t he D FT t o find t he c onvolution i k * 9k i n t hree s teps, as follows:
1. F ind t he D FTs Fr a nd G r c orresponding t o s uitably p added i k a nd 9 k.
2. M ultiply F r b y G r.
3. F ind t he I DFT o f F rG r . T his p rocedure of convolution, when implemented
by t he fast Fourier t ransform a lgorithm (discussed later), is known as t he f ast
Filtering f090 + 1193 + / 292 + 1391 a nd t he v alue o f i k ® 9k a t k = 1 is (Fig. 5.17)
f091 + 1190 + / 293 + / 392 a nd so on.
Applications o f O FT T he D FT i s useful n ot o nly in t he c omputation of direct a nd inverse Fourier
transforms, b ut also in o ther a pplications such as convolution, correlation, a nd filtering. Use o f t he efficient F FT a lgorithm discussed in Sec. 5.3 makes i t p articularly
L et f (t) a nd 9 (t) b e t he two signals t o b e convolved. I n general, t hese signals
may have different t ime d urations. To convolve t hem by using t he...
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