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

Broadly speaking there are two types of modulation

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Unformatted text preview: ry p roperty t o t he a ppropriate p air in Table 4.1 t o show t hat ( a) ~[o(t) + f,) { =} u(w) ( b) o (t + T ) + o(t - T) { =} 2 cos Tw ( c)o(t + T ) - o(t - T) { =} 2 jsin Tw. j Im[F(w)] ( b) Verify t hese r esults by finding t he F ourier transforms of t he even a nd o dd components of t he following signals: (i) u (t) ( ii) e -atu(t). t .1-4 t .1-5 F rom definition (4.8a), find t he Fourier transforms of the signals depicted in Fig. P4.1-5. t .1-6 Using Eq. (4.8b), find t he inverse Fourier transforms of the s pectra in Fig. P4.1-6 t .1-7 Using Eq. (4.8b), find t he inverse Fourier transforms of the s pectra in Fig. P4.1-7. t .2-l Sketch t he following functions: (a)rect(~) (b)~(:~) ( c)rect(t-g,O) ( d) sinc("sW) ( e) sinc(W-iO,,) ( f) sinc(!)rect(,~,,), Hint: f (x'b a ) is f(~) r ight-shifted by a. t .2-2 F rom definition (4.8a), show t hat t he Fourier transform of rect (t - 5) is sinc (~)e-jSw. Sketch t he r esulting a mplitude a nd p hase spectra. 4.3-1 F rom definition (4.8a), find t he Fourier transforms of t he signals f it) in Fig. P4.1-4. 4 .3-2 T he Fourier transform of t he t riangular pulse f it) in Fig. P4.3-2a is expressed as F(w) = ...!..(e jW - jwe jw - 1) 2 w Using t his information, a nd t he t ime-shifting a nd t ime-scaling properties, find t he Fourier transforms o f t he signals f i(t) (i = 1, 2, 3, 4, 5) shown in Fig. P4.3-2. Hint: See Sec. 1.3 for explanation o f various signal operations. Pulses J;(t) (i = 2 ,3,4 c an b e expressed as a combination of f it) a nd / lit) w ith suitable time shift (which may be positive or negative). 4.3-3 Using only t he t ime-shifting property a nd Table 4.1, find t he F ourier transforms of t he signals depicted in Fig. P4.3-3. J' -T P roblems C ontinuous-Time S ignal A nalysis: T he F ourier T ransform 4 3 12 o'~ T (a) t -> o o 1fI2 -':V ·V v. VfV V:V\T" It It '~ (c) t~ a I (b) 3 13 T ( b) (a) (d) ~fiAAAAAfiAA~ 4J VVlJ VlJ'V VlJ V31t 'I t_ t_ (e) F ig. P 4.3-3 F ig. P 4.3-6 o -4 -3 F(oo) 2 -2 3 -3 -4 4 234 -2 (b) (a) D -5 o -3 Using t he t ime-shifting property, show t hat i f f (t) + T ) + f (t - T) { =} { =} F(w), t hen 2F(w) cos Tw T his is t he d ual o f Eq. (4.41). Using t his result a nd pairs 17 a nd 19 in Table 4.1, find t he F ourier transforms of t he signals shown in Fig. P4.3-4. 4 .3-5 3 6 5 --Q -4 -2 J 600- 246 ( b) 4 .3-9 A signal f (t) is b andlimited t o B Hz. Show t hat t he signal r (t) is bandlimited t o n B Hz. Hint: S tart w ith n = 2. Use frequency convolution p roperty a nd t he w idth property of convolution. -2 Hint: S ignals in Figs. b, c, a nd d c an b e expressed in the form f (t)[u(t) - u(t - a)]. f (t Doo- F ig. P 4.3-7 F ig. P 4.3-5 4 .3-4 J (a) F ig. P 4.3-4 -4 -3 F(oo) 4 .3-10 F ind t he Fourier transform of t he signal in Fig. P4.3-3a by t hree different methods: ( a) By direct integration using t he definition (4.8a). ( b) Using only pair 17 T able 4.1 a nd t he t ime-shifting property. ( c) Using t he t ime-differentiation a nd t ime-shifting properties, along with t he fact t hat 8(t) { =} 1. Hint: 1 - cos 2 x = 2 sin2 x . 4.3-11 ( a) Prove t he frequency differentiation prop...
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