Question

each of the following systems, determine if the system could satisfy the additivity property, the scaling property and/or the time-invariance property:

Answer 1

Solution it e) y(t) + ay(t) = 2 X+) ci Additivity property- for 244) <,'t) + 2x,+) = 2 x44t) ----0 -- fg f(t) (+) + 22.(t) = 2X24) Fiy (x64) +8,4)) x++ ex suy 2 [26,1++ 2,)) - © ean. eaual to + een ③ Should be system to be stable can Orem » [Xi(+) + 2! ()] + 2[X164)+ 2,5+1] = 2( 3+2+ x2 (47) :: ,4)+mt) = x2ct) L₃(+) Latit Lat1= then 23't) + 28,4+) = 2 24) + 2x214) Bonast 2%24t) = 4,'(+)+ 2! (+ 2 [ x,'#7+ 4,14) a natured system is additive in (ii) fg scaling qalt) + ba X244) -7 [+) + 6 Lett) Xi'it) + 22,4) = 29 xq (t) for → a lt! ralt - La't) + 22,4)= 26 xat fg 6X₂ (4) TO a toy (t) by (t) - 1 y [ xo't) + 2X16+)] + [x34)+2x264).] [ 29% (H)] + [26 x(+1) a [2GH1} + b[zinelt?] ay, (+) + 6(4) it b LatH) as aytit by

b yast) followed. hence (iii) a Xift+ 6 424) = 9 %,(t) + systemis, scaling property time invarience property w Deley (to), alt-to) response - akt) + 2y(t) = 3-04(+-to) 2(t) — sestanse 4.' (t) + 2y() = 2x(+) Be gay viatay , 1 - Bothare some time Invariant system, 1. © ¢ft)= 16+). 46) (i) Additive property : 0464) - X16+ = X24t) — 42 (t) = 4(+) + &qlt) - 2,6+) = , (+). ult) oor-0 x2(+).utt) ---- [24(43+x35+)]ou(t) ---- eam eamorea OG (+) Ult) + x2(+)46+) X,CH) + 42() = 2,) + Lz(t) = [ 241) + x264)].4(+) Fromeno talt Ķit)t ,) = showing Additive property ; hence system is (ii) scaling - property:- pGct) -7 %, (t) = X(t) .44) ---0 K) - Watt) - mt). 464) -- a ZH) +84647 - Zy(t) = [9%4+6344) ). utt) ---* team b - quy, + 6.4alt le 9.34(+).lt) + b. x,t).ult. axeen

= [26,4) + 63426)].4(4) a 4, 4t by, (t) fromeano az,lt it by (t - Yalt) I hence system is showing scaling property (ii) Time in varience property: Response Mit to) ult) and s(t-to) Sponse &H) Delay (to) xH) Resanse, echo teen @ 361-44) Delay (to), 2(t-to).uft-to) ---☺ - system is Time- variant] (t)= t. ult) W Additive.it ...0 xH) — xult) - 4,ft)= L3H)= t. 4.(t) t. 42t) ----© 2414) 73621+ - zt) = t[ 4.46 )+ uglt)). ----☺ een ① t een > & olt)+ Lot) 4,(t) + + 4ylt) 2,1++ 4914) - t. [4,643+ 42(t) ] fromear Tito+ (+9 = 13(t) - - system is showing Additive property. ] (in scaling :- 2,64) = L21t1= Xz1+1 7.464) ---0 t.424) -...@ q. + 4,(t) + 6 t.426).

ах ек0 + by e "D 9. 4, (+) + b.yalt) - art. 4,6417 6.7.428) 9.4/tit bryz(t) = 36) - system is showing [scaling property ciii) time Invariance property: - ducti utt (to) Delay, ust-to) _Responses tuc responses Delay, to ut-to ... t-to) - 4(+-to) -- © too echo #ex" © - system is time a varient system/