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Unformatted text preview: r transform o f t he s tep function. Note t hat u (t) i s n ot a " true" d c signal because i t is n ot c onstant over t he i nterval  00
t o 0 0. To synthesize a " true" d c, we r equire only one everlasting exponential w ith w = 0
(impulse a t w = 0). T he s ignal u (t) h as a j ump d iscontinuity a t t = O. I t is impossible
to synthesize such a signal w ith a single everlasting exponential ejw ,. To synthesize this
signal from everlasting exponentials, we need, in addition t o a n i mpulse a t w = 0, all
frequency components, a s i ndicated b y t he t erm l /jw i n Eq. (4.29). • 0 Observe t hat t he u pper limit of e  jw' as t  > 0 0 yields a n i ndeterminate answer. So we
a pproach this problem by considering u (t) as a decaying exponential e a'u(t) in t he limit
as a  > 0 (Fig. 4.14a). Thus
u (t) Fig. 4 .14 6. 00 = :~ e  jw ' 1 0 t (4.27) T he s pectrum o f cos wot consists of two impulses a t Wo a nd  wo, as shown in Fig. 4.13.
T he result also follows from qualitative reasoning. An everlasting sinusoid cos wot can
be synthesized by two everlasting exponentials, e jwo ' a nd e  jwo '. Therefore t he Fourier
s pectrum c onsists of only two components o f frequencies Wo a nd woo • U(w) 01_ . .......... . ........... Adding Eqs. (4.26a) and (4.26b), and using t he above formula, we o btain (4.29) tThe second term on the righthand side of Eq. (4.28b), being an odd function of w, has zero area
regardless of the value of a. As a  + 0, the second term approaches 1 / j w. 4.3 Some properties of the Fourier Transform W e n ow s tudy s ome o f t he i mportant p roperties o f t he F ourier t ransform a nd
t heir i mplications a s w ell a s a pplications. B efore e mbarking o n t his s tudy, w e s hall
e xplain a n i mportant a nd p ervasive a spect o f t he F ourier t ransform: t he t imefrequency d uality. 4 ContinuousTime Signal Analysis: T he Fourier Transform 252 4.3 Some P roperties of t he Fourier Transform 253 T able 4 .1
A S hort T able o f F ourier T ransforms F(w) j (t) ! (t) a> 0 a + jw 1
a  jw F(ro ) a> 0 J
~ 2a
a2 +w 2 1 (a a>O + j w)2
n! (a
6 4.31 E quations (4.8) show a n i nteresting fact: t he d irect a nd t he inverse transform
operations are remarkably similar. These operations, required to go from j (t) t o
F(w) a nd t hen from F(w) t o j (t), a re depicted graphically in Fig. 4.16. T here a re
only two minor differences in these operations: t he factor 27r a ppears only in t he
inverse operator, a nd t he e xponential indices in t he two operations have opposite
signs. Otherwise t he two operations are symmetrical. t T his observation has farreaching consequences in t he s tudy of t he Fourier transform. I t is t he basis of t he
socalled duality of time a nd frequency. T he duality principle m ay be compared with
a photograph a nd i ts negative. A photograph can be obtained from i ts negative, a nd
by using a n i dentical procedure, t he negative can be obtained from the phot...
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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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