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sol_7 - We see that G(jw is periodic with a period of 8...

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Unformatted text preview: We see that G'(jw) is periodic with a period of 8. Using the multiplication property, we know that we = 2—17,- [fr {4} * Gem] 7rt If we denote FT {$793} by A(jw), then Xlflu) = U/2w)pfljw)*8n E: 6(w~—8k) k=—oo = 4 i AW — 8k) k=—oo X ( jw) may thus be viewed as a replication of 4A(jw) periodic. Using Table 4.2, we obtain A(jw):{1, IWISI every 8 rad / see. This is obviously 0, otherwise Therefore, we may specify X ( jw) over one period as . _ 4, iwlSl X(’”)*{0, l<lw$4 4.17. (a) From Table 4.1, we know that a real and odd signal and odd Fourier transform X (jw). Let us now consider the purely imaginary and odd signal jx(t). Using linearity, we obtain the Fourier transform of this signal to be jX(jw). The function jX(jw) will clearly be real and odd. Therefore, the given statement is false. signal :1:(t) has a purely imaginary (b) An odd Fourier transform corresponds to an odd signal, while an even Fourier transform corresponds to an even signal. The convolution of an even Fourier transform with an Odd Fourier may be viewed in the time domain as a multiplication of odd signal. Such a multiplication will always result in an odd time signal. transform of this odd signal will always be odd. Therefore, the given an even and The Fourier statement is true. 8 Using Table 4.2, we see that the rect ”transform X1010) ~~We may Write angular pulse 321 (t) shown in Figure 84.18 has a Fourier = sin(3w)/w. Using the convolution property of the Fourier transform, . 2 mm = m1(t) * me) 313 Xzew) = X1(jw)X1<jw) = ( 81115.3 a”) 8"" signal $2(t) is shown in Figure 84.18. Using the shifting property, we also note that . 2 émit + 1) (—F—Ta éej‘“ (sm(3w)) LI.) 139 and ~ 2 gm _ 1) a; law (SN-M) . Adding the two above equations, we obtain h(t) = w - 2 gm“ + 1) + ~12-m2(t—— 1) <55) cos(w) (Sln(3w)) ‘ The signal h(t) is as shown in Figure 84.18. We note that h(t) has the given Fourier transform H ( jw). x (0 ma 1 5/; l/ —-3 O 3 t ‘6 0 6 t Mt ) 5/4 —; —s 0 5 a t Figure $4.18 Mathematically h(t) may be expressed as g, |t| < 1 .151 + Q, 1 S it] S 5 W) = {a 3 0, otherwise 4.19. We know that Y(' ) . .7“) H w = . - (J ) X(Jw) Since it is given that y(t) := e’3tu(t) — e‘4tu(t), we can compute Y(jw) to be Since, H(jw) = 1/(3 +jw), we have X(J'w) = 28:; =1/(4 +y‘w) Taking the inverse Fourier transform of X ( jw), we have m(t) = e‘4tu(t). 4.20. From the answer to Problem 3.20, we know that the frequency response of the circuit is 1 ~w2+jw+1' Breaking this up into partial fractions, we may write H (.7141) = H(jw)~—1[ ‘1 + “1 ] N5 %—‘/7§j+jw é+§j+jw USing the Fourier transform pairs provided in Table 4.2, we obtain the Fourier transform of H ( jw) to be 1 _1 fi~ _r__\/.'§' hi =—~—— —e( 2+2J)‘++e( 2 23)” ut () NJ ] () Simplifying, Mi) 2 —2—3e“§‘sin(—~t)u(t). (a) The given signal is 1 - l - 6“”t cos(w0t)u(t) = fiew‘tewot’ufl) + 56"“te_3“’°l11.(t). Therefore, 1 1 X ' = —- ——————————————. ‘5’”) 2m — jwo + jw) 2(a — jwo + M (b) The given signal is x(t) = e”3" sin(2t)u(t) + e3tsin(2t)u(—t). We have _ . , . l/2j 1/23" :, t = 3‘. 21 2‘. fix — — . . M) e “I“ M) 10“) 3—j2+jw 3+32+jw Also 3 , . , . 1 2' 1 2‘ 12(1) = ersmewuet) = —x1(—t) «53> now) = —X1<—Jw) = :fiE—mfi Therefore, ‘ . . 3' 3.7' XW) = X1(]w + X2(Jw) = 3173,2175)? ” W' 141 (m) We have __= —t FT _. h (t) 2t Mt) (#4 Hg(yw) _../'(1 +310)?- Therefore, 7r YUw) ’ Xiawwmw) = 3M» 4 1) , 5(a) ~ 1)] we obtain 2. Note that Mt) = In“ .. 1), where 4.3 is the impulse re Therefore, h(t) i d is in the range ' ing the shift property, . (a) We have . X100): “6’ 12 It is clear that This result is equivaie (b) We have Xzijw) = 71. Xfl‘aflaw ~ ‘7 k=0 Therefore, ' ' Tuummw This implies that 1 ygm == 5 sin(3t — 1). iy the sinusoid with frequen at on (c) We have 6]“), {col < 4 0, otherwise X300) : { new) = X3(jw)H(jw) = awe—1"“- This implies that sin(4t) 7rt ' 313(15): 153(t'"1)= We may have obtained the same result by noting that X3(jw) lies entirely in the passband of H(jw). (d) X4(jw) is as shown in Figure 84.32. “.le ~¢+ ° + t» Figure $4.32 Therefore, 4 HOW = X4(jw)H(jw) = X4(jw)6”’w. This implies that ‘ __ 2 We may have obtained the same result by noting that X4(jw) lies entirely in the passband of H(jw). 33- (a) Taking the Fourier transform of both sides of the given differential equation, we obtain Y(jw) 2 X(jw) ~w2+2jw+8. H (N) = Using partial fraction expansion, we obtain 1 1 jw+2 jw+4' HOW) = 151 e inverse Fourier transform, Taking th Mt) = e‘QtuU) w e'4tu(t). (b) For the given signal $(t), we have X( m) - #3., 3 (2 +21»)? Therefore, . . 2 1 /. ff 2. Y(jw) = X(jw)H(le = (W2 + 2,,» + 8) (2 + Jw) pansion, we obtain Using partial fraction ex . 1/4 1/2 1 1/4 Y = __.._——— -— / ._.————-—~ _. . (9w) jw+2 (jw+2)2 + (jw+2)3 jw+4 Taking the inverse Fourier transform, 1 1. ”Wm — atea—”mm + {la—mum — Ze"4‘u(t). 1 1N) = 718 ansform of both sides of the given differential equation, we obtain (c) Taking the Fourier tr Using partial fraction expansion, we obtain —— 2 — 2+ 2 2" H(jw)=2+_._‘—/—'-——f\-/‘Jf+jw fl «3/3. 3L0 __ — 2+l —' 2;2\/§ Taking the inverse Fourier transform, h(t) = 25m « f (1 + 2 j‘)e—< WWW) — flu — 2j)e’(1"j)"/‘/§u(t). 4.34. (a) We have Y(jw _ jw + 4 X(jw) 6~w2 +5jw‘ Fourier transform, we obtain d taking the inverse d2 t d t d,t 3:2)“; sighting: 294—41170). Cross-multiplying an (b) We have . 2 “ 1 Taking the inverse Fourier t (c) We have 1 1 X ' = — M. (W) 4 +jw (4 +jw)2 Therefore, 1 (4 +jw)(2 + jw)’ Finding the partial fraction expansion of Y(jw) form, YUW) = X(jw)H(jw) = and taking the inverse Fourier trans- . 1 _ 1 _ y(t) 2 ie 2tu(t) — 5e 4tu(t). 4.35. (a) From the given information, «m mew): = ——~——— = 1. fifi$§ Also, LU <1H(jw) = ~tan‘1 E — tan“1 2 = —2 tan’1 —. a a (1. Also, 2 H(jw) = —1 + a +“jw => h(t) = ~6(t) + ale-aims. (b) Ifa = 1, we haveH(jw)[ = 1, <IH(jw) = —2tan”1 w. Therefore, y(t) = cos(-\% — g) —- cos(t ~ 32:) + cosh/gt ~ 233:) 4.36. (a) The frequency response is . _me_ m+m> ”W * qu) - m (b) Finding the partial fraction expansion of answer in part (a) and t aking its inverse Fourier transform, we obtain h(t) = [0(0) [6—4t + 8—2t] u(t). (c) We have Y(jw) _ (9 + 3jw) qu) * 8 +6jw —w2' Cross-multiplying and taking the inverse Fourier transform, we obtain 2 x ' gagawigaaymflddi” +9241»- 153 ...
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