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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, higherfrequency 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 timeshifting property, show that the Fourier transform of
sinc[wo(t  T)) is ~rect(2~o)ejwT. 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 alttol.
T his function, shown in Fig. 4.21a, is a timeshifted version of e  altl (depicted in Fig.
4.19a). From Eqs. (4.36) and (4.37) we have
 alttol e = 2a a 2 + w2 e jwto Fig. 4 .22
signal. Another example of timeshifting and its effect on the Fourier spectrum of a (4.38) The spectrum of e alttol (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 .35 The FrequencyShifting Property If
J (t) ¢ =} F (w) J (t)e iwot ¢ =} F (w  wo) t hen (4.39) 4 ContinuousTime 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  <
"7 < 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)ejwotejwtdt = [ : f (t)ej(wwo)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 timeshifting
a nd t he frequencyshifting properties.
C hanging wo t o  wo in Eq. (4.39) yields
f (t)e jwot <==* 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)ejwotl
From Eqs. (4.39) a nd (4.40), i t follows t hat f (t) cos wot <==* ~ [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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 Spring '13
 Bayliss
 Signal Processing, The Land

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