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

1 3 2 similarly a f unction f r epresents t he f

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Unformatted text preview: igher p hase s hifts t o achieve t he s ame t ime delay. T his effect is d epicted i n F ig. 4.20 w ith t wo sinusoids, t he f requency o f t he lower sinusoid b eing t wice t hat o f t he u pper. T he s ame t ime d elay to a mounts t o a p hase s hift of IT /2 i n t he u pper s inusoid a nd a p hase s hift of IT in t he lower sinusoid. T his verifies t he f act t hat t o a chieve t he s ame t ime delay, higher-frequency s inusoids m ust undergo proportionately higher phase s hifts. T he p rinciple o f l inear p hase s hift is v ery i mportant a nd we shall e ncounter i t a gain i n d istortionless signal t ransmission a nd f iltering a pplications. • Using pair 18 and the time-shifting property, show that the Fourier transform of sinc[wo(t - T)) is ~rect(2~o)e-jwT. Sketch the amplitude and phase spectra of the Fourier transform. 'V (a) t . .... ( b) E xample 4 .10 r o-- F ind the Fourier transform of e -alt-tol. T his function, shown in Fig. 4.21a, is a time-shifted version of e - altl (depicted in Fig. 4.19a). From Eqs. (4.36) and (4.37) we have - alt-tol e = 2a a 2 + w2 e -jwto Fig. 4 .22 signal. Another example of time-shifting and its effect on the Fourier spectrum of a (4.38) The spectrum of e -alt-tol (Fig. 4.21b) is t he same as t hat of e -altl (Fig. 4.19b), except for an added phase shift of - wto . Observe t hat the time delay to causes a linear phase spectrum - wto. This example clearly demonstrates the effect of time shift. • 4 .3-5 The Frequency-Shifting Property If J (t) ¢ =} F (w) J (t)e iwot ¢ =} F (w - wo) t hen (4.39) 4 Continuous-Time Signal Analysis: T he Fourier Transform 260 4.3 Some Properties of the Fourier Transform 261 4 F (w) (a) 2 -w w (b) w- t -- -&lt; &quot;7 &lt; T wcos lOt ! (t) F ig. 4 .24 A n example o f s pectral s hifting by a mplitude m odulation. a nd demodulation appears in Secs. 4 .7 a nd 4..8. To sketch a signal f (t) cos wot, we observe t hat Fig. 4 .23 f (t) f (t) cos wot = { - f(t) A mplitude m odulation of a signal causes s pectral s hifting. Proof: B y definition, F[J(t)ejWotl = [ : f (t)ejwote-jwtdt = [ : f (t)e-j(w-wo)tdt = F(w - wo) jwot According t o t his property, t he multiplication of a signal by a factor e shifts the spectrum of t hat signal by w = woo Note t he d uality between t he time-shifting a nd t he frequency-shifting properties. C hanging wo t o - wo in Eq. (4.39) yields f (t)e- jwot -&lt;==* F(w + wo) (4.40) Because ejwot is n ot a real function t hat c an be generated, frequency shifting in practice is achieved by multiplying f (t) by a sinusoid. This assertion follows from the fact t hat f (t) cos wot = ~[J(t)ejWot + f (t)e-jwotl From Eqs. (4.39) a nd (4.40), i t follows t hat f (t) cos wot -&lt;==* ~ [F(w - wo) + F(w + wo)l (4.41 ) T his result shows t hat t he multiplication of a signal f (t) by a sinusoid of frequency wo shifts t he s pectrum F(w) by ±wo, as depicted in Fig. 4.23. Multiplication of a sinusoid cos wot b y f (t) amounts to modulating t he sinusoid amplitude. T his t ype...
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