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1 _ —_— a { | © He Step Function and Associatecl Waveform | . bi wit) ja defined a4 The anik atop fonction aw x § SSSSPAVAAAAAAAAAAAAY A VOVvv VEU eEULLEUGUEEGEECeCoCessscEe ~ ~ amplinde 1 and pero 2T tor E20. At) Using Aime ahifting property, — ' woe Con wArte 3 T = Ath) = ul) -2(t-D+ 24 (4-27) - 2 ult-3T) 4+... Another w b t aero a frenction. where arguement Lecomer negotire cyclically. wo: fm) ae Another Way je by defining Aignum func. sgn[ fo} = 1 Fflt>° =o flt)=o --1 f(t) 0 Zt —> é €5 % & 2 & t 4 &, 5b function san he atfined Aewrdtically o Foving property ) Slo) = d(t)=0 fo to L) The arta undy Sb) ia O+ f sce at = fs ot A 3 ily <) Entire arco uncer Slt) 0 cm 0- oo dl) = = J at = [ S(Qolt =o Oo+ Mirivative of A Abp function oG Aeght A gelds an impulse function ASCE-T) wh tHe oliscontinur’g wo ak T. PUVVVVWOVEUHUELUUEELEGEEUESESEEEYS £y £4 4) wing this, any oliscont uty shouts Bowe ar impulae frurctior Sit-T) at the pocak of dideontinun &y 4 =T. . Dig continurty Aecght A is given a0 Ae £CTH- £1) Le $10 Le olefined as 4) fr 6ST ao £ lt), tue wothout Denrvative of f> fib = fin+A slt-T) Conoioer A Aquare woe flt)= Ault-a)-Ault-b po, LI) = A dlt-a) — A SC4-6) 1 F ~A lrmacdy a function shou in kgure. 4 #08), flys 2 Alt) + 2u¢t-9) a, #lO= 2 ult) +2 6le9 o #0) f £) sle-1) t= $07) . ~o flt) slt-T) <0 for €#T 5 % T wee vaed fom -2 fo +0, ther %2 woul obtain Ontin'ly of Plt). Thin operation io aims [at bo Aeannhng of Abe function. f(t) ft) oO t t t > pprrrmor mo mdsdeshdQXXQQXXAAQar' — EE a ee es eee e) @ The Laplece Largo ect) The Kaphace Tram afew cs ae fon tan fio) Ets) be deiner ao LL 404) ] ae XT & (| = Fts) = Spe oy < fue Comper fee puineny yauable 82 7+ 7G. — Note: The integratim Lint io tek an O- te . Alcommoodlet, Aho pov A) of oliswons nhnty Cimpute) wn 40 ak &-0. 4. Step, donpolre eke. << | se muler sor she furction fa porsrm Captnce Wroil 4 A mnt fey the conoltiow | Livin {lili - - Ym f |fce)| ov’ lk < @ fer real, prorbie Tamme o- Nx@: A function 40 have Fourser Aranafem, i mate obey the lowes tors tion - < [ l£lee <2” . L anf hin or a Atep ‘oa wae me Oe astr Lronrgam, but < eit, parser Laphect — : ot i, a COnvengine peter A gundeon ~ Noe e ° peowe Laphok Anonatoun . he t* will nok OH ° t ; ee Onverse Mranrform L rea] « & too fi x [pio a ~ ov -jo &m Where a je a Mak pewrove quantity that so pectin Rin. par convirgenck acer C. ~ VUE EUEEESESEEESEEES vvve evve £4 £[ ¢U0))- Ele), tA 4l- 4*) t at |~ a fe! FCs) = | i) ee ot - = ; (=) = O- ,. (6) at ) Sg ft flf)- 2 u . 9 e + -B-a)t Fid= f o** yu ee dk= J on oe ~ eat |7 oa : (6-9) = £-0. @® Propertico of Laplace Damages OA finite sum 63) cpe4o]- 2 zie | a) t i ej. tf - — ZL [Aine } zj yar Ade = ! | = HS 2) | gt w Arte S% POVVVIVVEUEEEUSEESEEECEESEECESETCE gringrating by pote. we get “ t a t <[ J fod] « |< [ swat o- 0° J -At dt t zle 4 t Sinn 2° > 0 ao kro, Se Rowe f 4cae | t on t=o x fliaddz] 2 FW [fs dar] = Fe 4 & Le flt)= ult). fi alode = £ ud o- Lf tute] - luce) _ = a \ ° (iv) Differentiation Ay 4) Difgeerttiation hy 4° 1m compli froquency domain > PP PPPBsRrS Vw VG UVF OUECELECECECECEECECECCCEVCECCOOOOC EH Defra o fun chon fil) vhih in Aowe eo 4) Ayer tuve 0 oT. = alt) - alt-% 4 as C4) a LCqcj= f- fe = a posod As 41) ie pecodic func on AEE” - = tf t4_ pe have fil - Fia- J [= | £4 Find tae snveue dnowafow Of Fia)= WS __ (4+4) hw L tiati. property « -ak We . a wa ut) Lei). & (e . = (a sAunwt + w coowt) & Note thot 9 yin wt] oe =O- we Aave Maung 1 Evaluation @ Vreo of Laplore Jranrfoms 0 gale &4. 5- J oe an st dt. Faard g 0 Replace gr? by @t* phen the wntegratd becomes the Laplace Cronapom OG den ©. nw stfs 2 — ZL [sw ¢ J 42s Replacing 3-2, we Hoe $+ Z G: Le gs fre ol ue. Note thak thie in an been fancies jen feet ot hegre, YYYEFCGEGCLECCECCEEy I ) yeu” ) a Y 7 YS © Func snverse A vancgrmn of X(A). L [xt] > xo. x3)= Ray + Ke e-) Gr) (4 +2) Solving tor Ky, Ky And ke grrr Bie enol lution ja the snverse of x(a, -t -2t - z 4 elt) = ze -£ +48 Lf, / Kecl Aarne £3: binen Aho Act 0f simultaneous oAcfpeoti'at Lquationd an) +4) 4 y tt) +79) = sult) eile) + ut) +4) +39) = 5 SLE) fend K0Q and lt). " 1 Co-) = y Co-) =0. 3 A Dramaporming the Ack of equations, we Ace 2 (442) x04) + 4D YA - (4) XA) +4 @4+y9IA = © alee) (any) xa ; [ " (+) (asd | | XD * [A 7 Lx] = [@] x - ae D= 2542) CA43) ~-) (44D = gC Ar*eset+h)- [a7 gat) = Ay 2ZKREE [ety 2 Pe er (542245) - (+) 2a+2) , K,= + a —— sitll naam Fotuing fer X(8), X(d)2 -GA - 308 415 Ee 2 Vaing partial factions ACS 4244 5) Xt) = Ki Kae+ Ka = 24s (A, 24+ s) 5X8) > Ke 3. Azo D> X(R)-3_. = -83-36 4 A423 45 X= 2 gs4d + 1402) “ (A+ *4 a The inveroe zvancfour Lo [xesy] 4d xlt) = 3- (ec° Coo 2t - ae * bin 2t) uct) Scmblarty, 4 = (141 et cos2t +307 © sen at) uw) O Partial faction parr? — (a) Simple root (2) Complex conju gat pots &: Fls= NOS) p35 4 38 +? =¢- — -_ : a a4 2A+2 () Mulkiple yoots . Sines lege, of NUS) io Ai ghey phan Aayoe of DA), we ciwole NCA) by De). F(ae= (At) - A Aeezats ; an we note the dlenoninatey “O* be L (544-2442) 2 Ary) +! FA: B+ e (A+) -) OO it) +t = -t : Lt [Fw] - 4l)= Silt+ MO +2 (ain t - cost) —parppp pp yyy >>> >> od >> eoRRRBRVEDY ee — ei ee for He simple root b--2, we fae Ke Vs. ke GaAs) 2 7 (A%2 sts) - Fo, the complex conjugete Avot, (S*424+5 = (A+ i+ 2j) C3+ I. 2)) HX ttajps Ks jh, we Aave Ks -1 and Br 2 (4742445) s (4-4-jR—) (4-a +jp) 2 2 ee (oi+2)) +3 ** = 1-2j-4 +3 2J + —_. a A> (1 42j) 2 (-1+3)) -2 + 4j i-2y i = 2 a r 4 e whee g- 2+k Je d han Z So knverce Monofom xan Le utter ae £*[ed]= 2 os sn [-ftan'e +E) 2¢] os = ae* + Gos (2t — tad! 2) ‘ Lt He poots of FCA) olenominator he +jp) wa K-jB - Kc Theepere, Flad= hy + ____— “ (s- a cia)) . Me sn (Bt +4) rrp ppp ppp pp > ppd > vn e es dod &% iw > wh ey } u VUULULLOUUCUUCELCECCECECLECEYUYEY wuvvov We not. hat me’? . Meas p + J MAing = -26 4 j2A = 2jk, ) Multiple Asoo: Parkiel fraction involves Aepeatect fy: Method 4? Lee Fy, NO) _— U-45)" De® Wsth ruthiple rots of cages not aod, we Aare Kar N,C Fis)e Key Ky Be be 5 Oa) Ba SA ee) 4 she Hie AomaincnS Aired whee N,W)/D,ds) ja te bod Ko. Ki, Kaos Mer cupreviom The prettenr Define Fs) (6-4) FLA). Thus, lade ko t Ki (aay te © Feet sehen Ris) indicates nemrosning Lene thor Ko= (4-46)" Fis) Iho: Note: We know . ra Mefe appl, Are sore equation AP Wher we fue woul tna up ~f n (4-8). + RA) C-4o) n-2 EUW: K+ 2h AAdtoe + Kpa (n-@ Ae) + —_ ‘ ~- 1 as EC Therefore, K,< Z i) = Ao L bane K,-t A Fw) Simibarty, o6e Arawe ay fe _ Dn genmral, we hae y~ 1 Ah £9) js 42,.,04- vo yl det Ae As deze CR)= Anz ACA+I)" Ze Ols)= Keo 4 b+ to 4 A uy er Bt 4 
