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Unformatted text preview: implies t hat if f (t) is wider, its s pectrum is narrower, F ig. 4 .19 e - a1tl a nd i ts Fourier spectrum. 4.3-4 The Time-Shifting Property
f (t) - ¢:=} F (w) t hen
f (t - to) Proof: B y definition,
F [J(t - to)] = L etting t - to = x , we have
F [J(t - to)] = i: - ¢:=} i: F (w)e- jwto (4.37a) f (t - to)e-jwtdt f (x)e-jw(x+t o) dx i: a nd vice versa. Doubling t he s ignal d uration halves its b andwidth a nd vice versa.
T his s uggests t hat t he b andwidth o f a signal is inversely p roportional t o t he signal
d uration o r w idth (in seconds). We have already verified t his fact for t he g ate pulse,
where we f ound t hat t he b andwidth of a g ate p ulse of w idth T seconds is ~ Hz. More
discussion o f t his i nteresting topic a ppears in t he l iterature. 2
B y l etting a = - 1 in Eq. (4.34), we o btain t he t ime a nd f requency i nversion
p roperty This r esult shows t hat d elaying a s ignal b y to s econds does n ot c hange i ts a mplitude
s pectrum. T he p hase s pectrum, h owever, i s c hanged b y - wto. (4.35) T ime d elay in a signal causes a linear p hase s hift in i ts s pectrum. T his r esult
can also b e d erived by heuristic reasoning. Imagine f (t) b eing synthesized by its
Fourier components, which are sinusoids o f c ertain a mplitudes a nd p hases. T he
delayed signal f (t - to) c an b e s ynthesized by t he s ame s inusoidal components, f (-t) - ¢:=} F (-w) = e- jwto f (x)e- jwx d x = F (w)e- jwto (4.37b) Physical Explanation o f t he Linear Phase 258 4 C ontinuous-Time S ignal Analysis: T he F ourier T ransform 4.3 S ome P roperties o f t he F ourier T ransform 259 ~ (a) o t~ L F (ro) = -rot~··· F ig. 4 .21
• Effect of time-shifting on the Fourier spectrum of a signal. E xample 4 .11 F ind the Fourier transform of t he g ate pulse J (t) illustrated in Fig. 4.22a.
T he pulse J (t) is t he gate pulse rect (~) in Fig. 4.10a delayed by T / 2 seconds. Hence,
according t o Eq. (4.37a), its Fourier transform is t he Fourier transform of rect ( i) multiT
plied by e - JW ~. Therefore
F ig. 4 .20 Physical explanation of the time-shifting property. F(w) = Tsinc (~T) e-jw~ each delayed b y to s econds. T he a mplitudes o f t he c omponents r emain u nchanged.
T herefore, t he a mplitude s pectrum o f J (t - to) is identical t o t hat o f J (t). T he t ime
d elay o f to i n e ach s inusoid, however, d oes c hange t he p hase o f e ach c omponent.
Now, a s inusoid cos w t d elayed b y to is given b y T he amplitude spectrum IF(w)1 (depicted in Fig. 4.22b) of this pulse is the same as t hat
indicated in Fig. 4.lOc. But the phase spectrum has an added linear term - WT / 2. Hence,
the phase spectrum of J (t) is identical to t hat in Fig. 4.lOb plus a linear term - WT / 2, as
indicated in Fig. 4.22c.
6 E xercise E 4.6 cos wet - to) = cos (wt - wto)
T herefore a t ime d elay to in a sinusoid o f f requency w m anifests a s a p hase d elay
o f wto. T his is a linear f unction o f w, m eaning t hat h igher-frequency c omponents
m ust u ndergo p roportionately h...
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