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Unformatted text preview: o f H(w), t hat is , H(w) = R(w) + j X(w), t hen show t hat R (w) = e -(kw'+jwto) -00 sinc2 ( kx) d x = ~
k Hint: Recognize t hat t he i ntegral is t he e nergy of j (t) = sine (kt). F ind t his energy
by using P arseval's t heorem.
4.6-3 A lowpass signal j (t) is a pplied t o a s quaring device. T he s quarer o utput j 2(t) is
applied t o a lowpass filter of b andwidth l l.F Hz (Fig. P4.6-3). Show t hat if l l.F is
very small ( ll.F - > 0), t hen t he filter o utput is a dc signal y (t) "" 2 Ejll.F. 4 316 C ontinuous-Time S ignal A nalysis: T he F ourier T ransform
f \t) f (t) Lowpass
Filter • 317 P roblems y (t) = 2 Ef l:>.f m (t) F ig. P 4.6-3 = (a) Hint: I f f 2(t)
A(w), t hen show t hat Y (w) <;; [411'A(0)ll.F]8(w) i f A F --> O. Now,
show t hat A(O) = E f.
1 .6-4 Generalize P arseval's t heorem t o show t hat for real, Fourier transformable signals
h (t) a nd h (t) /
1 .6-5 x (t) x(t) 00 h (t)h(t)dt = 211' / "" FI(-w)F2(W)dw = 211' /00 FI(W)F2(-w)dw
1 -00 (b) - 00 For t he s ignal f (t) = F ig. P 4.7-3 2a t2 + a2
4 .7-3 d etermine t he essential b andwidth B Hz of f (t) such t hat t he energy contained in
t he s pectral c omponents of f (t) o ffrequencies below B Hz is 99% of t he signal energy
E f. H int: S ee Exercise E4.5b.
1 .7-1 For each of t he following 3 baseband signals ( i) m (t) = cos lOOOt ( ii) m (t) =
2 cos 1000t + cos 2000t ( iii) m (t) = cos 1000t cos 3000t ( a) Sketch t he s pectrum of m (t).
( b) Sketch t he s pectrum o f t he DSB-SC signal m (t) cos 10,000t.
( c) I dentify t he u pper sideband (USB) a nd t he lower sideband (LSB) spectra.
( d) I dentify t he frequencies in t he b aseband, and t he c orresponding frequencies in
t he D SB-SC, USB a nd LSB s pectra. Explain t he n ature of frequency shifting in each
case. I n p ractice, t he a nalog multiplication o peration is difficult a nd expensive. For this reason, in ~plitude m odulators, i t is necessary t o find some alternative t o m ultiplication
of ~(t! w Ith cos ,":,et. F ortunately, for this purpose, we c an replace multiplication with
sWItchmg operatIOn. A s imilar observation applies to demodulators. I n t he scheme
d~picted i n F.ig. P 4.7-3a, t he p eriod of t he r ectangular periodic pulse x (t) s hown in
Fig. P 4.7-3b IS To = 211'/w e. T he b andpass filter is centered a t ± w . Note t hat multip!ication by a square periodic pulse x (t) in Fig. P 4.7-3b a mounts eto p eriodic on-off
sWltchmg of m (t). T his is a relatively simple a nd inexpensive operation.
Show t hat t his scheme can generate a mplitude m odulated signal k cos wet. D etermine
t he value of k. Show t hat t he s ame scheme can also be used for demodulation provided
t he b andpass filter in Fig. P 4.7-3a is replaced by a lowpass (or baseband) filter. [ A+m(t)]cosoV k m ( t)cos (J)c t m (t) (a) F ig. P 4.7-4
(b) cos 3 roc!
(Carrier) F ig. P 4.7-2
4 .7-2 You a re a sked t o design a DSB-SC m odulator t o g enerate a modulated signal
k m(t) cos wet, where m (t) is...
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