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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. 4.34 The TimeShifting Property
If
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)ejwtdt f (x)ejw(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 ontinuousTime 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 timeshifting 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 timeshifting property. F(w) = Tsinc (~T) ejw~ 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 igherfrequency c omponents
m ust u ndergo p roportionately h...
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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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