Signal Processing and Linear Systems-B.P.Lathi copy

# 1 a periodic signal r epresentation b y fourier

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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 requency-domain 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) &lt; ==} F{w) • T o r ecapitulate, E xample 4 .1 Find the Fourier transform of e-atu(t). By definition [Eq. (4.8a)], (4.8a) F(w) a nd 1 = 00 e -atu(t)e-jwt dt = F{w)ejwtdw (4.8b) - 00 But le-jwtl = 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) = e-ate-jwt = 0 if a &gt; 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 &gt;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 &lt; 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 &lt; 0 (growing exponential). Clearly, n ot all signals are Fourier-transformable. T he 4 Continuous-Time 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.

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