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Unformatted text preview: is clear t hat over the i ntervallwl ::0 T, F (w),
t he Fourier transform of f (t) is
assuming negligible aliasing. Hence ¥, fk = T f(kT)
= To f (kT)
No L k=O Derivation o f t he Discrete Fourier transform (OFT)
I f f (kT) a nd F(rwo) are t he k th a nd r th samples of f (t) a nd F (w), respectively,
t hen we define new variables fk a nd Fr as 341 N o-l F (w) (5.17a) = T F(w) = T L f (kT)e- jkwT k=O and and Ws Iwl::0 2 N o-l Fr = F(rwo) (5.17b) 27r
w o=27r:Fo=To Fr = F(rwo) = T L f (kT)e-jkrwoT (5.21) (5.17c) k=O where I f we let woT = flo, t hen from Eqs. (5.16a) and (5.16b) (5.22) We shall now show t hat fk a nd Fr are related by the following equations:t
Also, from Eq. (5.17a),
t In E qs. (5.18a) a nd (5.18b), t he s ummation is performed fro~ 0 t o No - 1. I t is s hown i n Eq.
(10.12) t hat t he s ummation m ay b e p erformed over a ny successlVe No values of k o r r . T f(kT) = h 5 Sampling 342 5.2 Numerical C omputation o f t he Fourier Transform: T he D FT Therefore, Eq. (5.21) becomes To N o-l Fr - no = - f k e - jrflok ""'
L.J (5.23) No k=O Points o f Discontinuity T he inverse transform relationship (5.18b) could be derived by using a similar
procedure with t he roles of t a nd w reversed, b ut here we s~all use a ~ore direct
proof. To prove t he inverse relation in Eq. (5.18b), we multiply b oth sIdes o f Eq.
(5.23) by e jmflor a nd s um over r as L N o-l F rejmflor = r =O LL N o-l [ NO-l r=O 1 i ke-jrflok e jmflor k=O By interchanging t he o rder of summation on t he r ight hand side
N o-l ~ F rejmflor = t; No-l [ NO-l ~ e j(m-k)flor ik } A ppendix 5.1 shows t hat t he i nner sum on t he r ight hand side is. zero for k i= m ,
a nd t hat t he s um is No when k = m . T herefore t he o uter sum WIll have only one
nonzero t erm w hen k = m , a nd i t is N oik = N ofrn. Therefore
1 L N o-l fm = F rejmflor
N o r =O no = 211" - (5.24) No Because F r is No-periodic, we need t o d etermine t he values of F r over a nyone
period. I t is c ustomary t o d etermine F r over t he range (0, No - 1) r ather t han over
t he r ange ( -,*, '* (5.27) N o=y 211" 343 - 1).:j: I f f (t) has a j ump d iscontinuity a t a s ampling point, t he sample value should
be taken as t he average of t he values on t he two sides of t he discontinuity because
t he Fourier representation a t a p oint o f d iscontinuity converges t o t he average value.
Zero Padding Recall t hat observing F r is like observing t he s pectrum F (w) t hrough a "picketfence." I f t he frequency sampling i nterval:Fo is n ot sufficiently small, we could miss
o ut o n some significant details a nd o btain a misleading picture. To obtain a higher
number o f samples, we need to reduce :Fo. B ecause:Fo = l/To, a higher number
of samples requires us t o increase t he value of To, t he p eriod o f r epetition for f (t).
T his option increases No, t he n umber of samples of f (t), by adding dummy samples
with a value of O. T his addition of dummy samples is known as z ero p adding.
T hus, zero padding increases t he n umber o f samples and may help in...
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