Solutions_Hm2 - 1.20:1 Given mm = cf”* w:91“ 104 m(t 2...

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Unformatted text preview: 1.20. (:1) Given mm = cf” *» w.) :91“! .. 104 m(t) 2 6'1“ W931“) = (2’)” Since the system is linear, y o 1 -. ,, 1 ‘ mm = ye)” + W”) m» 111(1): QM“ + J“) , 1 Therefore, :1:1(t) = cos(2£) -—~) 311(t) = cos(3t) (b) We know that 1 ) 8"]9321 +e’e'72' $20.) 2 cos (2“ - -) 2 2 Using the linearity property, we may once again write 4 1f»)-u 1 ‘ . ‘ . 1 . = D .110) :: 5(6’7832' + €187”) ——-) y,(t) = i(e_1(r73‘ + HIE—’1'“) t mswt — 1) If (93 Therefore, 3:1(t) = cos(2(t-—1/2)) ——+ 3/;(t) = (:0s(3t w 1) 1.21. The signals are sketched in Figure $1.21, 13 ‘0 “2"” 1. 1 101+!) 7- Figure $1.21 1.22. The signals are sketched in Figure 81.22. 1.23. The even and odd parts are sketched in Figure 8123. 10 Figure 81.23 11 1.24. 1.25. 1.26. 1.27 Figure 51.24 (3) Periodic, period 2 27r/(4) = 1r/2. (b) Periodic, period = 21r/(1r) = 2. (c) 2:0.) =[1+ cos(4t — 21r/3)]/2. Periodic, period = 21r/(4) = 1r/2. (d) a:(t) = cos(47rt)/2. Periodic, period 2 21r/(41r) = 1/2. (e) :r(t) = [sin(47rt)u(t) -— sin(47rt)u(—t.)]/2. Not periodic. (f) Not periodic. (a) Periodic, period : 7. (b) Not, periodic. (c) Periodic, period = 8. (d) min] = (l/2)[cos(31m/4) + cos(1rn/4)] (8) Periodic, period = 16. . )3) Linear, stable. ,fb) Memoryless, linear, causal, stable. (0) Linear (d) Linear, causal, stable. (9) Time invariant, linear, causal, stable. (f) Linear, stable. /(,g) Time invariant, linear, causal. The even and odd parts are sketched in Figure 81.24. . Periodic, period = 8. 12 1.28. (a) Linear, stable. ,9?) Time invariant, linear, causal, stable I (c) anoryloss, linear, causal. (d) Linear, stalile. Xe) Linear, stable. ‘ if) Mernorylcss, linear“ causal, stahlc. ' Lg) Linear, stalilu 1.29. (3) Consider two inputs to the system such that unlit] 2) min] = Rehilnl} and min] 41> yflu] :1 'Rr{;r2[n]}. Now consider a third input 1,3[11] : :r.[n] t T2l"l~ The corresponding systcm output will lw, llxlnl = Rchtglnl} 2: Re{:r.[u] + :1:2[71.]} = 720mm + ‘Relzrrzlnl = yilnl+yzlnl Therefore, we may conclude that the systcm is additive. Let us now assume that the input-output rclationship is changcd to 31m] : 72.(-{:2~j"/4;r[71,]}. Also. consider two inputs to the system such that 21hr] 2) y|[n] : le{(jfi/4I1[72]} and q Izl’lt] 3—) yg[n] = Rc{c7"/4x2[n]}i Now consider a third input (1)3[7’11 = Illa] + :rzln]. The corresponding system output will he yxlnl —— ”Re{c]"/4m3[nl} cos(1ru/4)Re{mgln]} — sin(1m/4)Im{r;;[n]} + (:(_)s(7m/4)Rc{1'l [NH w sin(7m/4)I1n { .r. [11]} + cos(mz/4)Re{1'2[n]} »~ Slll(1l'7l/4)IHL{light}} : Re‘lcfl/qaflnl} + ’Rc{c’"/4:n2[n]} 111 [H] + y2lfl~l ll ll 'l‘hcrcforc, we may conclude that the systmn is additive. l3 1.30. (a) Invertilile. Inverse system: g(t) : 7(1. + 4). /(,h’) Non invertible. The signals a:(!.) and $10) 2 Mt) + 21: give the same output. /(/C) Non invertible. (H11) and 26In) give the same output. (d) Invertilile. Inverse system: y(t) : dx(t)/df. i/(e) lnvertible. Inverse system: 11hr.) : :rr[n + I] for n ;_> I) and yln} :2 .rlu] for 7:. <3: 1). (f) Non invertible. min] and “-IEIH-l give the same result. (4;) Invertilile. Inverse system: yln] = Ill —- n). (h) Invertihle. Inverse system: y(t) = :r(!.) + dar(!)/rlt. (i) Invertihle. Inverse system: yin] = xIn) -— (I/2):17[n — l]. (j) Non invertible. If '1:(t) is any constant, then y(t) = l). (1:) Non invertible. Mn) and 26in) result in y[n.] : U. (l) Invertihle. Inverse system: y(t) = .1:(t/'}.). (m)Non invertible. 1:1[11] : (fin) + (sin — 1] and 2:2[n] : 6hr] give yin] .-; Mn). (n) Invertihle. Inverse system: y[n] = :r[2n]. 1.31. (a) Note that 2112(t) 2: ml“) - 271(t —- 2). Therefore, using linearity we get. 1/2“) —.» mU) ,_ yr“. ~— 2). This is as shown in Figure 81.31. (b) Note that ;r:3(t) = 271(2) + 2:;(t + 1). Therefore, using linearity we get, ml!) .~» MU») 4 .711“ + I). This is as shown in Figure 81.31. 344:) mm Figure $1.31 1.32. All statements are true. (1) .r(t) periodic with period T; y] (t) periodic, period T/‘Z. (2) y‘(t) periodic period T; :12“) periodic, period 2T. (3) Mt) periodic, period T: 112(3) periodic, period 27‘. (4) in“) periodic, period T; 1(t) periodic, period 'I’/2. 1.33. (1) True. ;r[n] : :cln + N];yl[n] : yfln + N0]. i.e. periodic with N" - N/‘Z if N is even, and with period No 2 N if N is odd. 15 1.41. 1.42. 1.43. (a) y[n} = 2211:]. ’l“hureforc, the system is time invariant (b) yln] = (2n w Marlo]. This is not time—invariant because 3/[11 Nu} ¢ (2n - 1);r{n »»»»» No]. (C) 3/[11] : z[o]{l + (~-l)" + l + (wl)""‘} = ‘222[n]. 'l‘liereforo, the systum is timu invariant. (3) Consider two systems 51 and 52 connected in series. Assume that if :zrl (t) and 172(1) are the inputs to S], then yl(t) and y2(t) are the outputs, respectivvly. Also. EL‘RSUIHC that if yl (t) and 3120.) are the inputs to $2, then 21 (t) and 230.) arv tho outputs, rospevtively. Since S; is linear, we may write mm + 1mm 11+ mm + 01120). where a and b are constants. Since 5; is also linvar, wv may write (Ly1(t) + 01.12“.) :1) (121(1) + bz2(t), We may therefore conclude that (mm + (ix-1(a) 31:5)“ (mm + 6122(1) 'I‘herefore, the series combination of S] and $2 is linear. Since Si is time invariant, we may write 5'1 . . 11(t " TO) "*+ 3/1“ _., 10) and S. 3120-“ To) 74* 21(t - To)» Therefore, 53.52 , $1“ “ 7i!) ”4 31““ Iii). Therefore, the series combination of SI and .92 is time invariant. (b) False. Let y(t) 2: :I:(t) + l and 2(1) = y(t) ~~ I. These correspond to two nonlinear systems. If those systems are connected in scrir's. than 2(t) 7: ;r(f.) “'ilit‘ll is a linear system. (c) Let us name the output of system 1 as wln] anti the output of systt-m 2 as 211:]. Then, ii 1 MM z[211] :2 114211] + éiavl211~ I] t 31112“ , 2] l l : 1[1z]+ 5:5[11- l] + Z1112 -— 2] The overall system is linear and time—invariant. (a) We have mm 1+in 19 ...
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