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Unformatted text preview: is f ascinating behavior, let us define F(w), a c ontinuous function
of w, as
(4.3) 1 Ll.w = 21r
To
a nd Eq. (4.5) becomes fro(t) = A glance a t E qs. (4.2c) a nd (4.3) shows t hat Dn =  F(nwo)
To As To  + 00, Wo becomes infinitesimal (wo  t 0). Hence, we shall replace wo b y a
more a ppropriate n otation, Ll.w. I n t erms o f t his new n otation, E q. (4.2b) becomes (4.4) T his m eans t hat t he F ourier coefficients Dn a re ( liTo) t imes t he s amples of F(w)
u niformly s paced a t i ntervals of wo, a s depicted in Fig. 4.2a.t Therefore, ( lITo)F(w)
is t he envelope for t he coefficients Dn. We now let To  + 00 by doubling To r epeatedly. Doubling To halves t he f undamental frequency wo [Eq. (4.2b)], so t hat t here
a re now twice a s m any c omponents (samples) in t he s pectrum. However, by doubling To, t he e nvelope ( liTo) F(w) is halved, as shown in Fig. 4.2b. I f we continue
t his p rocess of doubling To r epeatedly, t he s pectrum progressively becomes denser
while i ts m agnitude becomes smaller. Note, however, t hat t he r elative s hape of
t he envelope r emains t he s ame [proportional t o F(w) in Eq. (4.3)]. In t he l imit as
To ~ 00, wo ~ 0 a nd Dn ~ O. T his r esult means t he s pectrum is so dense t hat
t he s pectral c omponents a re spaced a t zero (infinitesimal) intervals. A t t he s ame
t ime, t he a mplitude o f each component is zero (infinitesimal). We have n othing o f
e verything, y et we have something! T his p aradox s ounds like A lice in Wonderland,
t For t he sake of simplicity, we assume D n , a nd therefore F(w), in Fig. 4.2, t o b e real. T he
a rgument, however, is also valid for complex D n [or F(w)). nf;oo [F(n~:)Ll.w] e(jnt>.w)t (4.6a) Equation (4.6a) shows t hat fTo(t) c an b e e xpressed as a s um o f everlasting exponentials of frequencies 0, ±Ll.w, ±2Ll.w, ±3Ll.w,. '" ( the F ourier series). T he a mount
of t he c omponent o f frequency nLl.w is [F(nLl.w)Ll.wl/21r. I n t he l imit as To ~
0 0, Ll.w  t 0 a nd f ro(t)  t f (t). T herefore
00 f (t) = lim fro(t) = Jim 2.. "
F(nLl.w)e(jnt>.w)tLl.w
t>.wo 21r n ~
=oo Tooo (4.6b) T he s um o n t he r ighthand side of Eq. (4.6b) c an b e viewed as t he a rea u nder t he
function F(w )e jwt , as i llustrated i n Fig. 4.3. Therefore 1 00 f (t) =  1 21r F(w)ejwtdw (4.7)  00 T he i ntegral o n t he r ight h and side is called t he F ourier i ntegral. We have now
succeeded in representing a n a periodic signal f (t) by a Fourier integral ( rather tIf nothing else, t he reader now has a n irrefutable p roof of t he proposition t hat 0% ownership of
everything is b etter t han 100% o wnership of nothing. 238 4 C ontinuousTime Signal Analysis: T he F ourier Transform 4.1 239 Aperiodic Signal R epresentation by Fourier Integral t han a F ourier s eries).t T his i ntegral is b asically a Fourier series (in t he limit)
w ith f undamental frequency ~w  t 0, a s seen from Eq. (4.6). T he a mount o f t he
e xponential ejnL1wt is F(n~w)~w/21':. T hus, t he function F (w) given by Eq...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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