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

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

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Unformatted text preview: erty (dual of t he t ime differentiation): P rove t he following results, which are duals of each other: f (t) sin wat { =} -h[F(w - wa) - F(w - h[t(t + T ) - f(t - T)] { =} - jtf(t) + wo)] Using t he frequency-shifting property a nd T able 4.1, find t he inverse Fourier transform o fthe s pectra d epicted in Fig. P4.3-7. 4 .3-8 Using t he t ime convolution property, prove pairs 2, 4, 13 a nd 14 in Table 2.1 (assume >. < 0 in p air 2, >'1 a nd >'2 < 0 in pair 4, >'1 < 0 a nd .\2 > 0 in pair 13, a nd >'1 a nd >'2 > 0 i n p air 14). Hint: You will need p artial fraction expansion. For pair 2, you need t o a pply t he r esult in Eq. (1.23). ! F(W) ( b) Using t his p roperty a nd p air 1 (Table 4.1), determine t he Fourier transform of t e-atu(t). F(w) sin T w Using t he l atter r esult a nd Table 4.1, find t he Fourier transform of t he signal in Fig. P4.3-5. 4 .3-6 T he s ignals i n Fig. P4.3-6 are modulated signals with carrier cos lOt. F ind t he Fourier t ransforms o f these signals using t he a ppropriate properties of t he F ourier transform a nd T able 4.1. Sketch t he a mplitude a nd p hase spectra for p arts ( a) a nd ( b). { =} 4.4-1 For an LTIC s ystem with transfer function 1 H (s) = s + 1 find t he ( zero-state) response if t he i nput f (t) is ( a) e - 2t u(t) ( b) e -tu(t) ( c) e tu(-t) ( d) u (t) Hint: For p art ( d), you need t o a pply t he r esult in Eq. (1.23). 4 .3-7 4.4-2 A stable LTIC s ystem is specified by t he t ransfer function -1 H (w)=-.- JW - 2 314 4 C ontinuous-Time S ignal A nalysis: T he F ourier T ransform 315 P roblems specifies t he real p art, t he i maginary p art c annot b e specified independently. T he i maginary p art is p redetermined by t he real p art, a nd vice versa. This result also leads t o t he conclusion t hat t he m agnitude a nd angle of H (w) a re r elated provided all t he poles a nd zeros of H (w) lie in t he LHP. f,( t ) 4 .5-1 Consider a filter w ith t he t ransfer function H (w) = F ig. P 4.4-3 F ind t he i mpulse response of this system a nd show t hat t his is a n oncausal system. F ind t he ( zero-state) response of this system if t he i nput j (t) is ( a) e -tu(t) ( b)etu( - t). 4 .4-3 4 .4-4 4 .4-5 4 .4-6 Signals / I(t) = 104 rect(10 4 t) a nd h (t) = 6(t) are applied a t t he i npnts o f t he ideal lowpass filters Hl(W) = rect(40.~00") a nd H2(W) = rect(20.~") (Fig. P4.4-3). T he o utputs Y l(t) a nd Y2(t) o f these filters are multiplied to o btain t he signal y (t) = Y l(t)Y2(t). ( a) S ketch Fl(W) a nd F2(W). ( b) S ketch Hl(W) a nd H2(W), ( e) S ketch Yl(W) a nd Y2(W), ( d) F ind t he b andwidths o f Y l(t),Y2(t), a nd y (t). H int for p art ( d): Use t he convolution property a nd t he w idth property of convolution t o d etermine t he b andwidth o f Yl(t)Y2(t). A low pass system time constant is often defined as t he w idth of its u nit impulse response h (t) (see Sec. 2.7-2). An i nput pulse p(t) t o this system acts like an impulse of s trength e qual t o t he a rea of p(t) if t he w idth of p(t) is much smaller t han t he s ystem time c onstant. Assume p( t ) t o b e a lo...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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