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Unformatted text preview: . (4.3)
a cts as a s pectral function. IF(w)1 = IF(w)1 (4.lOa) L F(w) =  LF(w) (4.10b) T hus, for real f (t), t he a mplitude s pectrum IF(w)1 is a n even function, a nd t he
p hase s pectrum L F(w) is a n o dd f unction o f w. T his p roperty t he ( conjugate
s ymmetry p roperty) is valid only for real f {t). T hese r esults were derived earlier
for t he F ourier s pectrum o f a periodic signal [Eq. (3.77)] a nd s hould come as no
surprise. T he transform F(w) is t he f requencydomain specincation o f f (t).
lIa.... f (t) o
F ig. 4 .3 T he Fourier series becomes the Fourier integral in the limit as To t 0 0. We call F (w) t he d irect F ourier t ransform o f f (t), a nd f (t) t he i nverse F ourier
t ransform o f F (w ). T he s ame information is conveyed by t he s tatement t hat f (t)
a nd F (w) a re a F ourier t ransform p air. Symbolically, this s tatement is expressed as o
(a)
 It '2 F {w) = .1'[J(t)] f {t) = .1'1 [F{w)] a nd F ig. 4 .4 e atu(t) and its Fourier spectra. or f {t) < ==} F{w) • T o r ecapitulate, E xample 4 .1
Find the Fourier transform of eatu(t).
By definition [Eq. (4.8a)], (4.8a) F(w) a nd 1 = 00 e atu(t)ejwt dt = F{w)ejwtdw (4.8b)  00 But lejwtl = 1. roo e(a+jw)t dt = ~e_(a+jw)tIOO
Jo
a+ JW
0 00 00 f (t)=1
21': 1 Therefore, as t  t 0 0, e(a+jw)t
1 I t is h elpful t o keep in m ind t hat t he F ourier integral in Eq. (4.8b) is of t he
n ature o f a F ourier series w ith f undamental frequency ~w a pproaching zero [Eq.
(4.6b)]. T herefore, m ost o f t he discussion a nd p roperties of Fourier series apply t o
t he F ourier t ransform as well. We c an p lot t he s pectrum F (w) a s a function of w.
Since F{w) i s complex, we have b oth a mplitude a nd angle (or phase) s pectra Expressing F (w) = IF(w)lejLF(w) = eatejwt = 0 if a > O. Therefore Therefore (4.9) in which IF(w)1 is t he a mplitude a nd LF(w) is t he angle (or phase) of F (w). According t o E g. (4.8a), F (w)= l :f(t)ejwtdt
F rom t his e quation a nd Eq. (4.8a), i t follows t hat if f (t) is a r eal function of t, t hen
F (w) a nd F (  w) a re complex conjugates. Therefore
tThis derivation should not be considered as a rigorous proof of Eq. (4.7). The situation is not
simple as we have made it appear. l as F (w)=. a + JW a >O (4.11a) a+ j w in the polar form as va 2 + w 2 ej tan'(';;'), we obtain
(4.11b) 1 IF(w)I=~ and LF(w) =  tan l (~) (4.12) The amplitude spectrum IF(w)1 and the phase spectrum LF(w) are depicted in Fig. 4.4b.
Observe t hat IF(w)1 is an even function of w, and LF(w) is an odd function of w, as
expected.
•
Existence o f the Fourier Transform I n E xample 4.1 we o bserved t hat w hen a < 0, t he F ourier integral for e atu{t)
does n ot converge. Hence, t he F ourier t ransform for e atu(t) does n ot exist if
a < 0 (growing exponential). Clearly, n ot all signals are Fouriertransformable. T he 4 ContinuousTime Signal Analysis: T he Fourier Transform 240 existence of t he Fourier transform is assured...
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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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