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

1 a periodic signal r epresentation b y fourier

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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) < ==} 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 > 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 Fourier-transformable. T he 4 Continuous-Time Signal Analysis: T he Fourier Transform 240 existence of t he Fourier transform is assured...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online