ESE-2018 PRELIMS TEST SERIES Date: 12 th November, 2017 ANSWERS. 61. (d) 121. (c) 2. (a) 62. (a) 122. (b) 3. (b) 63. (a) 123. (c) 4. (b) 64.

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1 ESE-8 PRELIMS TEST SERIES Da: h Novmbr, 7 ANSWERS. (d). (a) 6. (d) 9. (d). (c). (a). (c) 6. (a) 9. (d). (b). (b). (a) 6. (a) 9. (a). (c). (b). (b) 6. (a) 9. (b). (a) 5. (d) 5. (c) 65. (b) 95. (a) 5. (c) 6. (d) 6. (c) 66. (a) 96. (b) 6. (d) 7. (c) 7. (d) 67. (b) 97. (a) 7. (b) 8. (b) 8. (b) 68. (b) 98. (a) 8. (a) 9. (c) 9. (b) 69. (d) 99. (b) 9. (c). (d). (a) 7. (c). (c). (a). (b). (c) 7. (b). (c). (c). (c). (b) 7. (b). (d). (a). (a). (c) 7. (b). (d). (b). (b). (b) 7. (b). (d). (b) 5. (a) 5. (b) 75. (b) 5. (b) 5. (c) 6. (c) 6. (b) 76. (c) 6. (d) 6. (b) 7. (b) 7. (d) 77. (a) 7. (c) 7. (c) 8. (a) 8. (b) 78. (c) 8. (b) 8. (b) 9. (d) 9. (a) 79. (c) 9. (c) 9. (c). (c) 5. (c) 8. (b). (a). (c). (a) 5. (a) 8. (b). (b). (c). (d) 5. (b) 8. (a). (b). (b). (c) 5. (a) 8. (b). (b). (b). (b) 5. (b) 8. (c). (a). (c) 5. (a) 55. (b) 85. (c) 5. (c) 5. (d) 6. (a) 56. (a) 86. (d) 6. (b) 6. (c) 7. (a) 57. (d) 87. (b) 7. (a) 7. (b) 8. (b) 58. (b) 88. (b) 8. (b) 8. (c) 9. (c) 59. (d) 89. (c) 9. (c) 9. (a). (b) 6. (c) 9. (c) (c) 5 (c)

2 () (Ts - 8)- Novmbr 7. (d). (a). (b) VLF ( o Hz) SONAR VHF ( o MHz) Air Traffic Corol SHF ( o GHz) Salli commuicaio MF ( o Hz) Mariim radio (A) o Hz (LF) Navigaio (B) o MHz (HF) Tlgraph (C). o GHz (UHF) Wird lvisio (D) o GHz (EHF) Radar Hr,. (b) 5. (d) 6. (d) B 6 Hz SNR Chal capaciy C B log ( + SNR)bis/sc C 6 log ( + )bis/sc 6 log bis/sc 6 bis/sc Daa ra ( 6 bis/sc) > chal capaciy Daa rasmid will hav rrors. Hr, chal capaciy C 56 bis/sc Now, C B log ( + SNR) bis/sc (SNR) db db log SNR SNR B C log ( SNR) 56 log 8.5 Hz All h hr sams ar corrc. Modulaio is usd for frqucy raslaio, Muliplxig rducig aa high ad covrs wid-bad sigal o arrow bad sigal. Wh modulaio idx i AM is grar ha, h vlop is disord ad dos o coai acual mssag. 7. (c) 8. (b) 9. (c). (d). (b). (c). (a). (b) For a SSB-SC, h badwidh of modulad sigal is qual o B.W. of mssag sigal Opraio of muliplyig a sigal wih siusoidal sigal is calld Mixig or Hrodyig m() A cos For o vlop disorio c A m()cos c c c K m(), ls h modulad sigal dos o coai acual mssag sigal. If f c < Mssag sigal badwidh, h modulad sigal will coai ovrlappig sidbads. I-phas compo dpds iry o mssag sigal Quadraur compo irfr wih h I-phas compo o rduc or limia powr i o of h sidbads dpdig upo how i is dfid. Quadraur compo of modulad wav is a filrd vrsio of mssag sigal ad is usd o limia o of h sidbads hrby modifyig h spcral of modulad wav. s() A m()cosf c c a A cosf y() A m()cosf A cosf c c c c y() A [ m()]cosf c AM sigal (Ka ) V max V, V mi V Modulaio idx c c c y() Vmax Vmi.5 V V max mi Hr, V max.5 V ad V mi.5 V Modulaio idx Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

3 (Ts - 8)- Novmbr 7 () 5. (a) 6. (c) 7. (b) V V max max Hr, f c + f m 55 f c f m 5 f c V V mi mi.5 (.5).5 (.5) Hz Am Modulaio Idx.8 A 5 c (raio of mssag sigal volag of carrir sigal volag) Toal powr P LSB A c m R 5 (.8).65 W P USB m A c 8R c A Toal powr P T P T 8 6 Modulaio fficicy A c m R 6 W 8. (a) P SB P T.% m P T PC 9. (d) P T P C.89P T (.5) PC 89% is carrir powr ad hc % is sidbad powr. m Toal powr i AM P T PC As m icrass P T icrass P C. (c). (a) A c R m P T Pc P c P c idpd of m 5. A c R A c. 8.6V Pa ampliud of carrir afr modulaio A c ( + m) V max V V max A c ( + m) V V mi A c ( m) V V max + V mi A c + A c m + A c A c m A c V max V mi 7V Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

4 () (Ts - 8)- Novmbr 7. (d) I C 8A ad I T A m I T IC m 8 Giv o K Effciv ois mpraur of h rcivr T (F ) To F log T ( ) K Th ovrall ffciv ois mpraur. (c). (b) m.565 m.5.6 s() [.cos( ).8 si( )]cos( ) m. ad m.8 N modulaio idx m 8 m m (.) + (.8).8 Modulaio Efficicy m m % m.8 s() [. si( ).cos( )]cos( ) Hr, m., m. 5 9 m m m...5 P T P T Also, 5. (a) m A c m PC R (.5) 5W f Hz ad f Hz m m Badwidh Highs frqucy compo of mssag sigal 5 Hz Hz DSB wih carrir or AM wav ca b grad usig: (i) (ii) 6. (a) Squar law Modulaor Swichig Modulaor 5 T IN T a +T + K Ovrall ois figur 7. (a) 8. (b) 9. (c). (b) F T T IN o F db log db For a bad-pass sysm, miimum samplig frqucy f s whr K f H K fh fh f L fh 8 fh f L.5 f s MHz 8 I Mid-ris quaizr, ipu valu bw ad is mappd o a oupu valu of.5. For X() o b saioary µ x () mus b idpd of i.. x () E[X()] E[P] si + E[Q] cos mus b idpd of which is possibl oly if x () E[P] E[Q]. Givh() u() h(f) F T [h()] jf Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

5 (Ts - 8)- Novmbr 7 (5). (a). (c) so,s y (f) S f hf N f x Auocorrlaio fucio R y Ivrs F.T. [S y (f)] y R F N N f For a uiformaly disribud radom variabl W hav variac E X EX Ma µ x E[X] b a x f x dx a b x dx b a ad E X b So E X EX a x x dx b a b a b a, hr b 5 ad a So 5.75 b a Lmpl-ziv codig schm dos dpd upo sourc saisics, hc i is uivrsal codig schm.. (b) h(x) log dx log dx log log Muual iformaio of a BSC is I(X, Y) H (Y) + p log p + ( p) log ( p) which is maximum wh H(Y) is maximum. Maximum H(Y) is achivd wh ach oupu has probabiliy of.5 ad H(Y) max So chal capaciy of BSC C s 5. (c) 6. (c) 7. (d) max I (X, Y) + p log p + ( p) P(x) log ( p) I a bloc cod d mi, h miimum disac bw ay wo cod words of h cod is ( + ). cod r of a (,, L) liar covoluio cod rpas afr (L + ) h sag. Fro a raisd-cosi spcrum Badwidh B W R b WhrW T Giv,R b 8 Kbps b. (a) Diffrial ropy of a coiuous radom variabl is giv as h(x) Hr f x (x) f x(x)log dx f x(x), K x K, ohrwis 8. (b) W 8 KHz Badwidh KHz Giv Log i log P P I ca b wri as i LogPi i logpi i Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

6 (6) (Ts - 8)- Novmbr 7 Muliplyig by Pi ad summig ovr i yilds m m m P log P P P log P...() i i i i i i il i i m m m i i i i P log P Pi log P P H X i i i 9. b. a Thus q. () rducs o H(x) L H X Giv () c()cosc s()si c vlop E() () ad c variabl s c s () ar idpd radom So E() has rayligh dsiy Th oupu of rcifir is giv by x( ), x( ) -x, x( ) y( ) Th probabiliy dsiy fucio of guassia procss is x f ( ) x ma x f ( ) x x f y( ) y x x y( ), x( ) y( ), x( )< dx fx x y fx x y dy dx dy fy y y y / dx dy y /. c W hav s C Blog N s Blog B N B Wh B lim B C lim B B x. b powr spcral dsiy of ois s Blog B s B s lim log s B lim x log x log. Hc lim C s. B Erophy H P log log log log P i i i 9 7 log 6 log 9 log log. bis Mssag Prob m y y m y y y m y y y 9 9 m y y 9 9 m y y m y 6 7 m y 7 7 y -ary cod Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Avrag lgh Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

7 (Ts - 8)- Novmbr 7 (7) L 7 Pi Li i bis. c. b 5. b Efficicy H % L. %. % Giv BW of sigal f m 8Hz Nyquis ra f s,mi f m 6 Hz Samplig ra.5 Nyquis ra fs.5 6 Hz If umbr of bis i a word ar h umbr of quaizaio lvls Iformaio ra f s bps Sic fucio is dfid as sic (x) si Hc f() x x si si Nyquis ra is drmid by highs badwidh/frqucy compo. For sic () highs fq 5 Hz For sic () badwidh Hz Hc Nquis ra Hz Prs SNR db Dsird SNR db Hc, a icrm of db Sic icrasig by o bi rsuls i 6dB icras i SNR. So, o hav db icrm i SNR, w d o icras by Dsird + 6. b Fracioal icras i rasmissio badwidh % % I PCM, h B.W. rquirm f m whr o. of bis 7. d 8. b 9. a Quaizaio lvl icrass from o 6 mas o bis o bis BW fm BW fm (BW ) doubld Giv f m KHz f s KHz (Nyquis ra) L 5 bis Bi Ra KHz 5 bps For a raisd-cosi puls, daa ra is [B Bad widh 5KHz] B R b T 5 56bps.5 For a raisd cosi pulss B Hr w hav T b. R b 6 H T b B KHz.Mbps T 6 b 5 scod B T b z Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

8 (8) (Ts - 8)- Novmbr 7 5. c 5. (a) 5. (b) B T b 5 6 (Idal low pass filr) S For a PCM sysm giv Nq db or or L q db,mi S N L L or L Drawig h TTL om pol NAND ga as show blow ad applyig ipus A, B, A B E R B C E Q ON V CC R C B OFF Q D Q R Q ON o/p y C OFF For NAND ga o/p y for ipus A, B. I ordr o hav o/p y, Q should b i OFF sa. This will happ bcaus whvr Q ON h bas of Q is o forward biasd hc. Q is OFF. Wh Q is OFF Q bcoms ON as bas of Q is forward biasd as Q is ON curr flows hrough h capacior C ad gs mos chargd o V CC hrfor boh sids of h diod hav volag ad diod D gs rvrs biasd ad hc Q bcoms OFF. Sam I is ru bcaus i cas of rippl 5. (a) 5. (b) 55. (b) addr (or) paralll addr hr is propagaio dlay i carry from o flip-flop o ohr. Sam II is fals bcaus his propagaio dlay i carry is rducd by usig carry loo ahad addr. MUX-Srial o paralll covrsio DEMUX - Paralll o srial covrsio FULL ADDER - 9 NAND gas ar usd. Carry loo ahad addr - Rducio i carry propagaio dlay. W ow ha from absorpio law x xy x y x xy x y Proof : 56. (a) x xy x xx y [From disribuio law] Hc provd. () (x + y) ( x x ), By Complimaio law (x + y) So absorpio law is usd i sp (). From h giv circui diagram For o/p Y 5. Hc LED 5 will b forward biasd ad glow. For o/p Y. Hc LED will b forwards biasd ad glow. For hr is o valid oupu hc o of LEDs will glow. ( 9) +9 For gaiv Sig Now, s complm form of 9 is producd by aig s complm of +9 s complm : 's complm of +9 s complm : + s complm of 9 is. Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

9 (Ts - 8)- Novmbr 7 (9) 57. (d) Numbr of Boola xprssios havig variabls Hr, 58. (b) 59. (d) h 8 56 F A B. A B Applyig D Morga s horm Th, F. (A) A.B AB A.B. A.B AA.BB A BA C AA BA AC BC (B) BA AC BCA A BA BCA AC BCA AB C AC B AB AC A B A C AA BA AC BC (C) (D) 6. (c) Sac : A B AC BC A AC BC A C BC A BC AB AC BC AB AC BCA A AB ABC AC ABC AB C AC B AB AC A BA CB C A BA CB CA A B ACA AC AC AB 6. (d) I is a R/w mmory rsrvd for sorig iformaio mporarily. EI (Eabl irrups) MVI A, 8 H A -8 H SOD SDE R 7.5 MSE M 7.5 M 6.5 M 5.5 Srial R 7.5 is o availabl oupu Daa ON as No availabl i is Srial daa abl Hc all h irrups RST 7.5 RST 6.5 ad RST 5.5 ar abld. 6. (a) Idxd addrssig mod of addrssig is vry usful for arrays. This addrssig is o prs i microprocssor 885. Th 5 yps of addrssig mods prs i 885 ar. (i) 6. (a) Immdia Addrssig Mod: A immdia is rasfrrd dircly o h rgisr.eg: - MVI A, H (H is copid io h rgisr A) MVI B,H(H is copid io h rgisr B). (ii) Rgisr Addrssig Mod: Daa is copid from o rgisr o aohr rgisr.eg: MOV B, A (h co of A is copid io h rgisr B) MOV A, C (h co of C is copid io h rgisr A). (iii) Dirc Addrssig Mod: Daa is dircly copid from h giv addrss o h rgisr.eg: LDA H (Th co a h locaio H is copid o h rgisr A). (iv) Idirc Addrssig Mod: Th daa is rasfrrd from h addrss poid by h daa i a rgisr o ohr rgisr. Eg: MOV A, M (daa is rasfrrd from h mmory locaio poid by h rgisr o h accumulaor). (v) Implid Addrssig Mod: This mod dos rquir ay oprad. Th daa is spcifid by opcod islf.eg: RAL CMP. ALU prforms arihmaic ad logic opraios. Thr is o sorig lm i Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

10 () (Ts - 8)- Novmbr 7 ALU. Th oupu dpd o h curr ipu o ALU. I has irly combiaioal circuiry. 7. (b) A B A AB A B 6. (a) I mmory mappd I/O MEMW or MEMR sigals ar usd as corol sigals. B AB A B F 65. (b) 66. (a) 67. (b) 68. (b) 69. (d) B C 7. (c) Daa bus is a bidircioal bus. Th daa flows i boh h dircio bw MPU ad mmory ad priphral dvic. Addrss bus is a uidircioal bus. For xpadig 6K 8 o K 8, h rquird 6K 8 RAMs ar, 6K 8 8 6K Sofwar Irrups : RST;whr,,,,, 5, 6,7. i.. RST : RST, RST, RST, RST, RST 5, RST 6 ad RST 7. Hardwar Irrup : TRAP, RST7.5, RST 6.5, RST 5.5 ad INTR Poi P is suc a C B Th, Z W f A P X BC Y BC A Z A W f A X BC, Y X BC BC. A. A Th oal o. of ar (b) 7. (b) 7. (b) 75. (b) F A BA B AB AB A B A B EX-OR ga Th oupu of NAND ga is ABC A B C Y A A B C C A B C Which is h oupu of opio (b) Y ABC A B C Sic, A A ad, A A So, A A A A A A CD AB F BD CD ABCD d Giv is a sychroous cour, o. of FF Truh Tabl : CLK Q Q Thus mod- cour. Hc K. d d Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

11 (Ts - 8)- Novmbr 7 () 76. (c) 77. (a) 78. (c) 79. (c) CLK bi modulo-6 rippl cour No. of FFs Miimum im priod of o cloc puls 5 sc. sc. Maximum cloc frqucy 6 5MHz Cas : Wh Q D Q T Q + (Rquird for D FF) Cas : Wh Q D Q T Q So, T D Q DQ + DQ (i.. Q ) (i.. Q ) (i.. Q ) (i.. Q ) i.. rquird ga is EXOR ga. S R 8. (b) Th maximum cloc frqucy Hz MHz Hz pd(ff) 8. (b) 8. (a) 8. (b) Frqucy of oupu 8 T 8sc. f 8 CLK QA QB QC QD QB QD 5 6 Afr 6 cloc pulss, h oupu is rpad. Th oupu of EX-OR ga is h ipu o D flip-flop D Q I Q I As I is s high D Q Q Q I D flip-flop Q + D Q + Q Q Th umbr of flip-flops rquird o ma a modulo rippl cour log log () f max pd Th oupu of dividr 6 6 Hz Th Johso cour will ma h oupu 8. (c) pd f max 6 sc. frqucy Hz 8 Th Schmi riggr dos chag h frqucy Y AB A B AB A(B B) B AB AB AB B Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

12 () (Ts - 8)- Novmbr 7 B AB B 85. (c) AB.x.x.x (d) 87. (b) 88. (b) 89. (c) x x x + x 5 x 7, 7.5 x 7 ax x ca o 7.5. A B Bulb OFF ON ON OFF Th abov ruh abl is sam as ha of EX-OR ga Y AB AB. f W WZ ZXY W( Z) ZXY f W ZXY Th abov xprssio of f shows ha ga No. is rduda. Toal 5 NAND gas ar rquird o implm X Y 9. (c) 9. (d) F Y XZ. From h ruh abl i is ifrd ha h oupu is high wh boh h ipus ar sam ad i is low wh h ipus ar diffr V V av 5V ; ; T / ; T / T / ; T / T T T 5 5 T 5T 5 T T V av Oupu of ga, Y x x T/ F XY XYZ XY(Z Z) XYZ XYZ XYZ XYZ Oupu of ga, Y x x + x Oupu of ga, Y (x x + x ) x x x x + x x Oupu of ga, Y x x x + x x + x Oupu of ga 5, Y 5 (x x x + x x + x ) x 5 x x x x 5 + x x x 5 + x x 5 Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

13 (Ts - 8)- Novmbr 7 () 9. (d) Hc f x x x... x + x x x 5... x x + x FFFF 5AB A5E A5F i.. Y High oly of A B i.. ihr A, B or, A, B So, opio (b) is corrc. 95. (a) So, F s complm of (5AB) 6 (A5F) 6 9. (a) 9. (b) Giv, X (X) (.) 6 Sic, 6 7 () 6 To ur off LED, oupu of NOR ga should b high. Oupu of NOR ga, Y AB AB AB AB AB A B A B A B AA BA AB BB AB AB A B 96. (b) 97. (a) A K - map. : Q J Q K Q Hc afr h arrival of h cloc dg Q ad Q. For NOR lach B C D Prvious sa ivalid sa Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

14 () (Ts - 8)- Novmbr (a). (c) If propagaio dlay for o FF FF h, h for -bi cour; propagaio dlay for Sychroous cour, FF.sc. Rippl cour, FF.sc. i.. opio (c). (d) 99. (b) (Q, Q ) is,,,,,,,.... (c) [SOLN] Truh abl of J-K FF : J K Q Q Q.(d) Truh Tabl of D-FF : D Q Truh Tabl for h abov circui, D CLK D Q Q Q Q Q Q Q i.. opio (d) y() x() * h() or y() h( ) X ( ) d... Igraio x( ) im shifig i.. Q + JKQ JK JKQ A A B Q A A B A A B Q ABQ AB ABQ B AQ AQ AB B A Q AB.(d) h( ) x( ) muliplicaio Giv y[] x[] x[ ]...() y[] A [] [ ] [Puig h valu of x() i q ()] y[] bcaus [ ] ; Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org [] ; ad Sic y[] dpd o pas valus so i is o mmorylss.

15 (Ts - 8)- Novmbr 7 (5) 5.(b) Sic y[] for all valus of so i is o ivribl. L x () cos( )[u( ) u( )] () & x () u() () & y() x () * x () y() x ( ) x ( ) d 6. (d) H H 7.(c) () Puig h valus from q. () ad q. () x ( ) cos( )[u( ) u( )] () x ( ) u( ) x ( ) for < ad > + y() for < < So from q. () y() y() x ( ) cos( )d si( ) si( ) ohrwis (5) I is causal bcaus h oupu dos o appar bfor h ipu. I is o causal bcaus h oupu appars a, o im ui bfor h dlayd ipu a +. Sic o pol li i righ half of s pla so fial valu horm is o applicabl. Giv impuls rspos is causal h ROC Img 8. (b) 9. (c) H(s) s (s ) (s ) Usig parial fracio H(s) A B s (s ) By solvig, w g A ad B H(s) s (s ) By aig ivrs laplac rasform, w g h() u() H( ) + h( ) For causal h() for < For saic rasfr fucio of h LTI, sysm should b idpd of frqucy ad aur of impuls rspos should b impuls a origi which is valid for opio (c) oly. S () (7) si() s () xp ( 7) si ( ) No priodic Ral S () cos + cos + cos 5,, 5 Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

16 (6) (Ts - 8)- Novmbr 7 T, T, T (a) a T, T, T 5 x() ( ) d (hr a ) T T. (a). (b) raioal Similarly, is priodic T T, T T ar also raioal. So S () S j8 is also priodic wih frqucy 8 rad / sc Giv y[] u[] u[] y[] x [] * x [] y[] x [] * x [ ] y[] u[] u[ ] u[]...() u[ ]...() from () ad () y[]... u[] im u[] for y[] ( + ) y[] (+) u[] Giv, x() ( ) d.(b) if h oly ( ) is dfid For, ( ) x () ( a)d x (a) ( a) x() x() x() x() ( )d ( )d FT cos X(j ) [ ( ) ( )] F.T h() H( j ) y() x() * h() F.T 5 () j y() Y( j ) H(j )X(j ) By puig h valus from quaio () ad quaio () Y(j ) 5 [ ( ) ( )] j ( ) ( ) ( ) ( ) 5 j j x() ( ) x( ) ( ) ( ) ( ) Y(j ) x() [( ) d ( ) ( ) 5 j j Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

17 (Ts - 8)- Novmbr 7 (7). (b) ( ) ( ) + 5 ( ( ) ( )] + 5 j( ( ) j( ( ) j ( ) ( ) IFT Y( j ) y() si() Taig fourir rasform of giv diffrial quaio (j ) Y(j ) + (j ) Y(j ) + Y(j ) (j ) X(j ) + X(j ) Y(j ) [(j ) + (j ) + ] X(j )[(j ) + ] Y(j ) X(j ) j (j ) (j ) Th frqucy rspos is ( j ) H(j ) (j ) (j ) H(j ) A B (j ) (j ) j (j )(j ) O solvig valu of A / ad B, w g H(j ) j j Taig ivrs fourir rasform of H(j ) y[] x[] h[ ] () puig h x[] ad h[] i q. (), w g u[ ] u[ ] u[ ] for 5. (c) & u[ ] for Sic h[] u[] h[] will b o zro for So rag of is { o } y[] Giv sigal x() si / ( ) d (a) si / ( ) d x() ( /a) a hr a si / ( ) d h() u() u() si / ( ) d.(a) Giv y[] x[] * h[] x() ( a) x(a) ( a) hr a si / ( )d x() Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

18 (8) (Ts - 8)- Novmbr 7 6. (b) x() x() si / L y() x () * x () () x () u() o comparig by giv x () u( ) y() So, y() x ( ) x ( )d [from q. () [ u( ) u( )d from q. () u( ) x ( ) x ( ) & x ( ) u( ) u( ( ( )) + If + < or < h y() for So, y() () u( ) d (.) y() 7. (a) 8. (b) 9.(c) 9. Giv y[] x[] * h[] So y[] x[]h[ ] Puig h valus of x[] ad h[], w g a u[] u[ ] Sic u[] for u[ ] for So, y[] a y[] a + a + a + a Bcaus x[], h[] is valid for bcaus u[] for y[] + a + a + a y[] y[] a a a a u[] Cojuga symmric x[] (x[]) CS x[] x * [ ] x[] [ 5j + j ] x*[ ] [ j 5j] x() + x* [ ] [ 5j 5j] x[] x * [ ] [.5j j.5] If x() is ral h a mus b cojuga symmric Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

19 (Ts - 8)- Novmbr 7 (9).(c) a As a * a * a ; for all valus of j ( j) a a * ( ) Sic his is o ru i his cas so x() is o ral. If x() is v, h x() x( ) Sic his is ru for his cas, so x() is v. a +(a ) a a j j W ow X(j ) ( ) x() j j d () a By puig h valu of x(), w g X(j ) X(j ) j u( ) d ( j ) u( ) d Sic u( ) for, so. (b) W ow x() Pu j X( ) d x() X( ) d from giv figur of x() i is clar ha x() so X( ) d.56 Giv x() si x() si x() Sa( )...() si Sa() L x() A Sa()...() o comparig q () ad (), w g A A Sa() x() x( ) FT A /K X(j ) ( j ) d Puig valu of x() from q () X(j ) ( j ) ( j ) Sa () FT x( ) / X j j j. (c) Sa ( ) FT Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

20 () (Ts - 8)- Novmbr 7 x( ) h(). (c) usig Prsrval Egry horm E E E Giv x( ) d () d X(j ) ( ) ( ) ( 5) Taig ivrs Fourir rasform x() j Sigal x() has wo complx xpoials whos fudamal frqucis ar 5 rd/ sc.ad rad/sc. Ths wo complx xpoials ar o harmoically rlad. So sigal x() is o priodic. Cosidr y() x() * h() Y(j ) X(j )H(j ) h() u() u( ) h() L h () j5 h () h() h ( ) Y(j ). (a) ' Sa ( ) h () H ( j ) H(j ) j j Sa( ) si j si ( ) ( ) ( 5) Sic wh, h H(j ) If h, H(j ) Y(j ) [ ( ) ( 5)] y() j5 y() is a complx xpoial summd wih a cosa ad y() is priodic Sic w ow A Sa() A A A L si() si() si y () x () () A A A X (j ) Y (j ) Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

21 (Ts - 8)- Novmbr 7 () x () y () X (j ) * Y (j ) from q. () Usig parsval s rlaio x() d T (a ) T si L x () x (j ) si * x () X'(j ) x () from q. () dx (j ) j d si d(x( j )) j d si j/ x(j ) j/ () j / X(j ) j / ohrwis T Sic x() d (a ) x() d (giv) (a ) + (a ) a (a ) a a 6. (d) j Two possibl sigals which saisfy h giv iformaio x() j j si( ) x().5 L x () j( /) j( /) (c) Sic x() is ral ad odd Fourir sris coffici a is purly imagiary ad odd hrfor a a ad a Sic a, > oly a ad a xis / X () / / +/ x() ca b rprsd i rm of x () ad x () as Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

22 () (Ts - 8)- Novmbr 7 x() x (.5) x (.5) C j / () 7. (b) x (.5).Sa x (.5) Sa x() X( j ).5j j.5.sa Sa si si si si si si Giv x() + j j j.5j j.5 j.5 + j.5 j j j( /) j( /) Sic w ow x() j C...() j x() j C C + j C + C j + C...() o comparig q () ad q. () C + C j / j / 8. (a) 9. (c) cos / cos / Th rgy of h raisd puls is E [x()] Puig h valu of x() from giv i h qusio / / / (cos ( ) ) d / cos cos d cos() cos ( ) d O solvig h igraio ad puig h uppr ad lowr limis w s E E Giv: x() x() L a 5 x() 5 5 a a Sic w ow 5 FT X( j ) C j / () x() a a u() u( ) a a Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

23 (Ts - 8)- Novmbr 7 () Usig dualiy propry, Pu i lf sid rm ad muliply by Pu i righ sid rm w g y() u() O comparig wih giv q of y(), w g a a a FT u( )); a a ( u( ) + A B C D A + B + C + D. (c). (a) x() a a a X(j ) a 5,. a( ) 5 (.) X(j.).5 Giv y() x() * h() a ; a Taig Fourir rasform boh sid w g Y( j ) X(j ). H(j )...() Taig Fourir rasform of x(), w g X( j ) as, X(j ) ( j )...() Taig Fourir rasform of h(), w g H j as, H j j Puig h valu of H(j ) ad q () from q () ad () w g...() X( j ) i Y j j ( j ) ( j ) ( j ) (By usig parial fracio) Now aig ivrs Fourir rasform of y(), w g. (a) L g() dx() d Taig Fourir rasform of giv sigal G(j ) jx(j ) si(.) j si ( ) ( ) j si cos x( ) j j j j Taig Fourir rasform of h(), w g h(j ) a j Taig Fourir rasform of x() yilds X(j ) Sic b j y() x() * h() Taig Fourir rasform boh sid Y(j ) X(j )H(j ) Puig h valu of q. () ad () Y(j ) (a j ) b j Usig parial fracio H(j ) ad () () () X(j ) from () Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

24 () (Ts - 8)- Novmbr 7 Y(j ) A B a j b j (5) Solvig for q. () ad q. (5), w obai b purly imagiary ad odd. Thrfor, a ad a a a a. (b) A B b a Y(j ) (b a) a j b j Taig ivrs Fourir rasform y() Giv a b u() u() (b a) F.T h() H(j ) x() cos X(j ) y() h() * x() ( j ) () F.T ( ) ( ) Y(j ) X(j ) H(j ) x() ( ) x( ) ( ) () Puig h valus from q. () ad q. () Y(j ) ( ) ( ) ( j( ) ( j) ( ) ( ) ( j) ( j) j ( ) j ( ) IFT j{ ( ) ( )} si a a Fially a j a j a j So, a + a + a + a 6j j(a + a + a + a ) j( 6j) 6 5. (c) H(s) H(s) s s 7s s s I.L.T. u(); R{s} s I.L.T. u( ); R {s} s 6. (b) H() u() + u( ) W ow x(s) x() s d By puig h valu of x(), w g x(s) s u() u() d O solvig ad simplifyig, w g x(s) s s s s s To drmi ROC :.(b) Sic h Fourir sris cofficis rpa a vry N, w hav a a 5, a a 6 ad a a 7 Furhr mor, sic h sigal is ral ad odd, h Fourir sris cofficis a will u() u() ; ROC > s s ; > for which boh rms covrg is > Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

25 (Ts - 8)- Novmbr 7 (5) 7. (c) Giv x() b x() b u() + b u( )...() 8. (b) b u() LT ; b s b b u( ) LT ; b s b So, Laplac rasform of x() is X(s) Giv b ; b b s b...()...() Y(z) x (z)...() x(z) IZT x[] u[] aig z rasform of x[] x(z) z x (z) z ( z) Y(Z) ( z ) y(z) z(z ) z (z ) x(z) z z [from q ()] (By addig z ad subracig z i umaraor) z z z (z ). (c). (c) x(z) 6z + 8z x[] + x[ ] z + x[ ] z O comparig, w g x[] x[ ] 6 x[ ] 8 So x[] + x[ ] + x[ ] 5 For a fii lgh sigal, h ROC is h ir pla. Thrfor, hr ca b o pols i h fii z-pla for a fii lgh sigal. Sic h sigal is absoluly summabl, h ROC mus iclud ui circl for lf sidd sigal < z < dos o iclud ui circl, so sigal is righ sidd sigal ad sabl bcaus i iclud ui circl. Giv x[] b, < b < x[] b u[] + b u[ ] b u[] Ad b ZT b z u[ ] ZT, z b b z, 9. (c) y(z) z z ( z ) Taig ivrs z-rasform of y(z) y[] u() + u() ( + ) ( + ) Usig log divisio z b So x(z), bz b z o x(z) b z b (z b) (z b ) b z b ; Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

26 (6) (Ts - 8)- Novmbr 7. (b) b z b X(s) (s )(s ) pol locaios (s+) (s+) s, Possibl ROC () ROC > () < ROC < () ROC < Img Ral () Sigal will b lf sidd sigal, sic j axis is o icludd i ROC so sysm is o sabl W ow: L.T u() ; s L.T u() ; s L.T u( ) ; s L.T u( ) Usig parial fracio for quaio () X(s) X(s) A B s s s s o solvig A, B ad () Taig ivrs Laplac rasform for quaio (). (b) x() u( ) u( ) x() ( ) u ( ) X(z) log( + az ), z > a W ow X(z) x[]z diffriaig boh sid wr z w g dx(z) dz x[]z Muliplyig by z boh sid dx(z) z. dz dx(z) z. dz x[]z x[] From q. () ad q. () az az I.Z.T x[] Z.T ( a) u(), az Z.T a a( a) u[], az z a () () z a () Combiig im shifig propry o q. () yilds a( a) u[ ] Z.T z a x[] a( a) u[ ] a[ a] u[ ] x[] a, ( ) u[ ] x[] ( ) u() x[] az az, Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

27 (Ts - 8)- Novmbr 7 (7). (c) Z.T x[] X(z) ROC R x [] Z.T x[] X(z) ROC R/ T s 5 µ sc So samplig frqucy s will b f s T 5 s Sic m() 6 Hz cos( ) 5. (d) Z.T x [] x[] X[8z] ROC R/8 8 Sic x [] is absoluly summabl h R icluds ui circl ad X(z) has a pol z, w may coclud ha R is dfialy ousid h circl wih radius. Sic x [] is o absoluly summabl h R 8 dos o iclud h ui circl i is clar ha his is o h cas. R icluds h ui circl, R. mi R R So, R > R 8 R 8 R 8 R < 8 dos o iclud ui circl R max. So, < R < 8 wo sidd sigal as R has pol a so R mus b mor ha i.. ru for Boh sidd sigal Giv samplig im priod cos( m ) m rad / sc. [ m 6. (c) frqucy of bad limid sigal] Frqucy compos prs afr samplig f F s ± f m ( igr) wh f Hz wh f ± Hz, 8 Hz f c 5 Hz (cu off frqucy of LPF) So frqucy compo prs i h rag of 5 Hz will pass hrough LPF h (f ) LPF 8 KHz ad KHz m() m () * m () aig Fourir rasform boh sid M(j ) M (j ). M (j ) h lows frqucy compo from ( & ) will prs i m(j ) as a highs frqucy compo m() m (). m () aig fourir rasform boh sid M (j * M (j ) M(j ) h highs frqucy compo will b m() m () + m () M(j ) M (j ) + M (j ) Max (, ) will b prs as highs frqucy compo i M(j ) So x () cos() cos(,) Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

28 (8) (Ts - 8)- Novmbr 7 Nyquis ra s pla xcp {,, } rad / sc. A S X(s) x () si() si( ) X(s) d ( s ) ds ds ds s 8 rad/sc. x () si() cos() si cos ', ' rad / sc. x () si() * cos() si( ) * cos( ), 7. (b) [mi(, )] mi, 6 rad / sc. From giv fucio of x(), i ca b wri as x() u() u( ) L.T u() s L.T u( ) s s By aig Laplac rasform of x(), w g X(s) s s Sic h giv sigal is a fii duraio sigal so rgio of covrgc will b whol 8. (c) s X(s) s X(s) X(s) so (idrmia) ROC whol s pla xclud ( ) Th pols of z-rasform obaid from characrisic q z j, z j, z, z Basd o hs pol locaios, w may choos from h followig rgios of covrgc (i) z (ii) z (iii) z 9. (a) Characrisic quaio 5 z z z 8 If x() x () x () h x(j ) x (j ) * x (j ) whr * do covoluio highs frqucy compo i b. x(j ) will Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org

29 (Ts - 8)- Novmbr 7 (9) 5. (c) x() Sa () Sa (8) LT For Sa () 8 LT For Sa (8) 8 So Nyquis ra ( ) [ 8 ] [ ] 6 As iiial codiio y[ ] is giv h us uilral z-rasform (U.Z.T) U.ZT y[ ] z Y(z) y[ ] So by usig giv q. (aig z rasform) z Y(z) Y[ ] + Y(z) X(z) () as zro ipu rspos x[], X(z) So, q. () bcom as Y(z) y[ ] z z (z) () Taig ivrs uilral z-rasform of q. () y[] u[] Rgd. offic : F-6, (Lowr Basm), Kawaria Sarai, Nw Dlhi-6 Pho : -656 Mobil : 89955, ifo@ismasrpublicaios.com, ifo@ismasr.org