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Unformatted text preview: Fortu~ately, every practical signal f [k] m ust d ecay with k (because
o f a filllte energy r~Ulrement), a nd s uch signal becomes negligibly small beyond
~ 2: .No for some SUitable value o f No. For instance, t he signal f [k] = (0.6)k has
l~fimte length. However, f [k] ::; 0.00028 a t k 2: 16. Hence, we m ay t runcate t his
~Igna.l beyond k = 16 (or even a little earlier). Straightforward signal t runcation
I II thl~ m anner amounts t o using a rectangular window w ith a u nit weight for t he
d ata I II t he r ange 0 ::; k ::; No, a nd zero weight for t he d ata beyond k = No.
S uch a t runcation r esults in Gibbs phenomenon with consequent oscillations in the
s pectrum o f t he t runcated signal as demonstrated in t he following example.
• E xample 1 0.11
.The signal f [k] = (D.8)ku[k] has infinite length. Find the DFT of this signal using an
8-pomt rectangular window.
The DTFT of this signal obtained in Eq. (10.37) is 1 0.6-3 D iscrete T ime Filtering (Convolution) Using O FT T he D FT is useful n ot only in t he c omputation o f direct a nd inverse Fourier
transforms, b ut also in o ther a pplications such as filtering, convolution, a nd correlation. Use o f t he efficient F FT a lgorithm makes D FT p articularly a ppealing.
We generally t hink o f filtering in t erms o f some hardware-oriented solution (using
summers, multipliers, a nd delay elements). However, filtering also h as a softwareoriented solution [a c omputer a lgorithm t hat yields t he filtered o utput y(t) for a
given i nput f(t)]. Such filtering c an b e conveniently accomplished by using t he
F iltering can b e accomplished either in t he frequency d omain o r i n t he t ime
domain. I n t he frequency domain approach, for a given i nput J[kJ, we are required
t o find t he o utput y[k] o f a filter w ith a given transfer function H (n). I n t he t ime
d omain approach, for a given i nput f [k], we a re required t o find t he o utput y[k] o f a
filter w ith a given impulse response h[k]. I n t he t ime d omain, t he o utput is o btained
by (linear) convolution o f J[k] w ith h[k]. I n t he frequency domain, t he o utput is
o btained as a n I DFT o f F rHr w ith F r = F (rno) a nd H r = H (rno). B ecause t he
frequency domain m ethod a ppears as a substep of t he t ime d omain method, we
s hall consider here only t he t ime d omain (convolution) method.
We would like t o p erform linear convolution required in filtering o peration using
D FT (utilizing F FT a lgorithm) because of its computational efficiency. However,
D FT c an b e used t o e valuate only t he c ircular convolution, n ot t he l inear convolution. Fortunately, linear convolution can b e m ade equivalent t o c ircular convolution
b y s uitably padding t he two sequences w ith zeros. T his s tatement, i ntroduced in
C hapter 5, will now b e proved.
W hen is Linear C onvolution E quivalent t o C ircular Convolution? F(!1) _ 1 - 1 _ D.8e- i r! and IF(!1)1 = 1 v'1.64 - 1.6 cos !1 T he 8-point rectangular window function is T he circular ( or periodic) conv...
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