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

# T his possibility c an give valuable insights a bout

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

This is the end of the preview. Sign up to access the rest of the document.

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 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)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...
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