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Unformatted text preview: n, we should have denoted t he Fourier transform by F (jw) r ather t han F (w)
in Eq. (4.3). In fact, t he n otation F (jw) for t he Fourier transform is o ften used
in t he l iterature. I t is, however, a bit clumsy a nd does not lend itself so easily
t o m anipulation a s t he n otation F (w). For this reason we shall continue with t he
d ual notation, while remembering t hat b oth F (w) a nd F (jw) represent t he s ame
entity. This fact is p articularly i mportant w hen we discuss t he Laplace transform
a nd filtering in t he future, a nd should be kept in mind throughout t he r est of the
book. In t he same way, we m ust remember t hat H (w) a nd H ( jw) r epresent t he
same entity. 4 .12 LT IC System Response Using the Fourier Transform We wanted t o r epresent a signal f (t) as a sum of (everlasting) exponentials so
t hat we could find a system response to f (t) as a sum of t he s ystem's responses
t o t he e xponential components of f (t). Consider a n a symptotically stable LTIC
system with transfer function H (s). T he response of this system t o everlasting
exponential eiwt is H (w )e iwt . Such a n i nputoutput p air will be denoted by t he
directed arrow representation as b M arvel o f t he F ourier t ransform. Therefore A Marvelous Balancing Act ei(n£:;w)t An i mportant p oint t o r emember here is t hat f (t) is r epresented (or synthesized) by exponentials or sinusoids t hat a re everlasting (not causal). Such conceptualization l eads t o a r ather fascinating picture when we t ry t o visualize t he synthesis
of a timelimited pulse signal f (t) (Fig. 4.6) by t he sinusoidal components in its
Fourier s pectrum. T he signal f (t) exists only over an interval (a, b) a nd is zero
outside this interval. T he s pectrum of f (t) c ontains an infinite number of exponentials (or sinusoids) which s tart a t t =  00 a nd continue forever. T he a mplitudes
a nd phases o f t hese components are such t hat t hey add up exactly t o f (t) over the
finite interval (a, b) a nd add u p to zero everywhere outside this interval. Juggling
with a mplitudes a nd phases of a n infinite number of components t o achieve such
a perfect a nd delicate balance boggles t he h uman imagination. Yet t he Fourier
transform accomplishes i t routinely, without much thinking on our p art. Indeed,
we become s o involved in mathematical manipulations t hat we fail to notice this
marvel. a nd = ==} [F(n~; )l::.w] e(jn£:;w)t = ==} H (nl::.w )ei(n£:;w)t
[F(nl::.W )~;nl::.w )l::.w] ei(n£:;w)t Using t he l inearity property ~ l im
~w_o ~ [ F(n6w)6w] e (jn6w)t 27r = lim
. 6.w+o ' ""'
~ [F(n6w)~;n6w)6W] ej (n6w)t n =oo ~~v~
o utput y et) v ; nput f (t) T he lefthand side is t he i nput f (t) [see Eqs. (4.6a) a nd (4.6b)], a nd t he r ighthand
side is t he response y (t). T hus
1 00 ' ""' F(nl::.w)H (nl::.w )ei(n£:;w)t l::.w
211" £:;w~O L...J y(t) =  lim n =oo 1
1 00 = 1 A Word A bout Notation 1:
1: 211" I n C hapter 2 [Eq. (2.48)], we defined t he s ystem transfer function H (s) as
H (s) =
S etting s = j w in this equation yields
H (jw) = 00 = 1 h (t)estdt (4.16) F(w)H(w)eiwtdw  00 211" Y(w)eiwtdw (4.18)...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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