PANIMALAR ENGINEERING COLLEGE

Transcription

1 PANIMALAR ENGINEERING COLLEGE DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGG QUESTION BANK (7-8) II YEAR THIRD SEMESTER ECE INDEX PAGE S.NO SUBJECT CODE / NAME PAGE NO.. MA65- TRANSFORMS & PARTIAL DIFFERENTIAL EQUATIONS.-.7. EC6- OOPS AND DATA STRUCTURE.-.. EC6- DIGITAL ELECTRONICS EC6- SIGNALS AND SYSTEMS EC64- ELECTRONIC CIRCUITS EE65- ELECTRICAL ENGINEERING & INSTRUMENTATION

2 BLOOM S TAXONOMY LEVELS (BTL) LEVEL - REMEMBERING(R) LEVEL - UNDERSTANDING (U) LEVEL - APPLYING (A) LEVEL 4 - ANALYZING (AZ) LEVEL 5 - EVALUATING (E) LEVEL 6 - CREATING(C)

3 PANIMALAR ENGINEERING COLLEGE MA65 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS SOLVED TWO MARKS & PART - B QUESTIONS (FOR THIRD SEMESTER).

4 UNIT-I PARTIAL DIFFERENTIAL EQUATIONS PART-A. Form a partial differetial equatio by elimiatig the arbitrary costats from a b y. [N/D] [C] Solutio: Give () a b y Differetiatig () partially with respect to, we have a ==> p a ==> a p Differetiatig () partially with respect to y b y ==> q b y usig () ad () i (), we have p q y y (), we have q ==> b () y y ==> p q y p q y, which is the required partial differetial equatio.. Form a partial differetial equatio by elimiatig the arbitrary costats from a b y. [M/J4] [C] Solutio: Give () a b y Differetiatig () partially with respect to, we have a ==> p a ==> Differetiatig () partially with respect to b y y ==> q b y ==> usig () ad () i (), we have p q y p a () y, we have y ==> p q y q b () y p q y, which is the required partial differetial equatio.. Form the partial differetial equatio by elimiatig the arbitrary costats a ad b from log a a y. [A/M5] [C] b.

5 log a a y Solutio: Give b..() Differetiatig () partially with respect to, we have a a ==> a p a Differetiatig () partially with respect to y, we have a a a q a a Dividig () by (), we have a y () a p a p a a q a q a a Substitutig the value of a i (), we have q p p q q q p p p p pq q p q p q q p q which is the required partial differetial equatio. 4. Form the partial differetial equatio by elimiatig a a a y Solutio: Give b.[n/d5] [C] a a y b Differetiatig () partially with respect to, we have a p a.() Differetiatig () partially with respect to y, we have a y y Usig () i (), we have partial differetial equatio. q a y a p q y q p () p q q p ad b from () q q p 4 p y q which is the required y 4 y.

6 5. Form the PDE by elimiatig the arbitrary costats a, b from the relatio a a y 4 b. [A/M5] [C] Solutio: Give 4 a a y b () Differetiatig () partially with respect to, we have 4 a a y b 4 a p a y b Differetiatig () partially with respect to, we have 4 a a y ba y Dividig () by (), we have p q 4 a From (), we have a y b y a q a a y b 4 q a....(4) a p p.(5) 4 Usig (5) i (), we have a a 4 p 6 a p a a p 4 4 Usig (4) i (6), we have p which is the required partial differetial equatio..() () q p q p p p q p 6. Form the partial differetial equatio from a y b r Solutio: Give a y b r () Differetiatig () partially with respect to, we have a ==> a p ==> a p a p ==> a p () Differetiatig () partially with respect to y y, we have y b ==> y b q ==> y b q b q y ==> y b q () Usig () ad () i (), we have p q r ==> p q r p q r, which is the required partial differetial equatio.. [M/J] [C].4

7 7. Form the partial differetial equatio by elimiatig a ad b from a y b [A/M]. [C] Solutio: Give a y b () Differetiatig () partially with respect to, we have p y b ==> p y b ==> y b () Differetiatig () partially with respect to y, we have y a y usig () ad () i (), we have ==> q y a ==>. q a () y q p y 4 y p q, which is the required partial differetial equatio. 8. Form the partial differetial equatios of all plaes passig through the origi.[m/j6] [C] Solutio: Let the equatio of the plae be a b y c d () where a, c ad d are costats. Sice plae () passes through the origi, we have a b c d ==> d substitutig d i (), we have a b y c () Differetiatig () partially with respect to, we have a c ==> a c p ==> a c p () Differetiatig () partially with respect to y, we have b c ==> b c q ==> b c q (4) y usig () ad (4) i (), we have c p cqy c ==> c p c q y c p q y p yq c ==> p y q, which is the required partial differetial equatio. 9. Costruct the partial differetial equatio of all spheres whose cetres lie o the -ais, by the elimiatio of arbitrary costats. [N/D, N/D5]. [A] Solutio: Let the cetre of the sphere be,, c a poit o the -ais ad be its radius. Hece, its equatio is y c r,b r.5

8 c r y () Differetiatig () partially with respect to, we have c ==> c p ==> c p c p ==> c p ==> Differetiatig () partially with respect to y c () p y, we have y c ==> y cq ==> y cq y cq ==> cq y From () ad (), we have y ==> p q q ==> y p y c () q q p y, which is the required partial differetial equatio.. Fid the PDE of all spheres whose ceters lie o the -ais. [N/D6] [A] Solutio: Let the cetre of the sphere be a,, a poit o the radius. Hece, its equatio is a y r a y r () Differetiatig () partially with respect to, we have a ==> a p a p ==> a p ==> a p () Differetiatig () partially with respect to y ==> q y y, we have y ==> y q y q ==> y q () Usig () ad () i (), we have p q r ==> p q r p q r, which is the required partial differetial equatio. ais ad r be its. By elimiatig the arbitrary costats, form the partial differetial equatio from the relatio a y b. [A/M]. [C] Solutio: Give ay b ().6

9 Differetiatig () partially with respect to, we have y p b ==> p y b ==> y b () Differetiatig () partially with respect to y a y usig () ad () i (), we have ==> q y a.7 y, we have ==> q p y 4 y p q, which is the required partial differetial equatio. q a () y. Form partial differetial equatio by elimiatig the arbitrary fuctio from f y Solutio: Give f y () Differetiatig () partially with respect to, we have f y y ==> p y f y () Differetiatig () partially with respect to y, we have y f y Equatios p q ==> q f y implies y f y ==> f y p y q () p y ==> p y q q, which is the required partial differetial equatio...form the partial differetial equatio by elimiatig f y y. [C] Solutio: Give f y y () Differetiatig () partially with respect to, we have f y p f y ==> p f y Differetiatig () partially with respect to y, we have f y y q y f y y q () ==> y f y () f. [C] from the relatio

10 Equatios p f q y f y p q implies y y ==> p q y ==> y p y q y, which is the required partial differetial equatio. 4. Elimiate the arbitrary fuctio from y f equatio. [N/D, N/D4]. [C] Solutio: Give y f () Differetiatig () partially with respect to, we have y y y y f ==> p f () Differetiatig () partially with respect to, we have y f y y ==> q f f y ad form the partial differetial () y y f p Equatios implies p y ==> ==> p y q q y q f p y q, which is the required partial differetial equatio. 5. Fid the partial differetial equatio by elimiatig the arbitrary fuctio from y,.(or) Form the partial differetial equatio by elimiatig the arbitrary fuctio from y f. [N/D,M/J]. [C] Solutio: The give relatio is of the form u, v Hece the required pde is of the form P p Q q R Where P P u v y u v y ==> P where u y ad v.8

11 .9 v u v u Q y Q ==> y Q v y u y v u R y R ==> R Therefore, the required equatio is q y p ==> q y p q y p, which is the required partial differetial equatio. 6. Form the partial differetial equatio by elimiatig the arbitrary fuctio from, y. [N/D4] [C] Solutio: The give relatio is of the form, v u where y u ad v Hece the required pde is of the form R Q q p P Where y v u v y u P y P ==> y P v u v u Q Q ==> Q v y u y v u R y R ==> R Therefore, the required equatio is q p y q p y, which is the required partial differetial equatio. 7. Form the partial differetial equatio by elimiatig the arbitrary fuctios from, y y f. [M/J6] [C] Solutio: The give relatio is of the form, v u f where y u ad y v Hece the required pde is of the form R Q q p P

12 Where P u v y u v y y P ==> P y Q u v u v y Q ==> Q R u v y u v y y y ==> R y y R Therefore, the required equatio is y p q y y p q y, which is the required partial differetial equatio. 8. Fid the geeral solutio of Solutio: Give. [U] () Itegratig () with respect to, we have f y () Oce agai itegratig with respect to, we have f y y, which is the geeral solutio. 9. Solve si y. [A] Solutio: Give si y () Itegratig () with respect to, we have si y f y () Agai itegratig () with respect to, we have si y f yy.. Fid the complete itegral of p q. [N/D4] [U] Solutio: Give p q This is of the form F p, q.

13 Hece, the complete itegral is a b y c where a b (ie) b a a a y. Therefore, the complete solutio is c. Fid the complete itegral of p q. [M/J6] [U] Solutio: Give p q This is of the form F p, q Hece, the complete itegral is a b y c where b a (ie) a Therefore, the complete solutio is c. b a b a a y.. Fid the complete solutio of the partial differetial equatio p q. [M/J6] [U] Solutio: Give p q This is of the form F p, q Hece, the complete itegral is a b y c where a b (ie) b a Therefore, the complete solutio is. a a yc. Fid the complete itegral of p q p q.[m/j] [U] Solutio: Give p q p q This is of the form p, q F. Hece the complete itegral is a b y c Therefore, the complete solutio is where a b ab b a b a aba a b a a b a a a y c. a 4. Fid the complete solutio of the partial differetial equatio p q 4 p q. [U] Solutio: Give p q 4 p q. This is of the form p, q F. Hece the complete itegral is a b y c where a b 4ab

14 b b b 4ab a 4a 4a 6a a 4a 4a a b b a a a Therefore, the complete solutio is a a y c. 5. Fid the complete itegral of the partial differetial equatio p yq p y q Solutio: Give p p q q y p q y p q This is of the form p q y f p, q Hece, the complete solutio is a b y a b. y 6. Fid the complete itegral of p q. [N/D6] [U] p q q p Solutio: Give p q pq y p q () q p p q y pq pq This is of the form p q y f p, q Hece, the complete solutio is a b y ab ab. [U] 7. Fid the complete solutio of q p. [A/M5] [U] Solutio: Give q p () This of the form F, p, q Assume Substitutig But q a d q a i (), we have a p implies p d q d y a d d a d y Itegratig o both sides, we have. p a

15 8. Solve p q y Solutio: Give 9. Solve a log a y c.[n/d5] [A] p p q y q y This of the form F p Gy, q But p p a d y q a which is the complete itegral., (separable equatio) ad p d q d y q y d a d d y a Itegratig o both sides, we have a Differetiatig y a y c () which is the complete itegral a which is absurd partially with respect to c, we have Hece there is o sigular solutio. Substitutig c f a i (), we have a y a f a () Differetiatig () partially with respect to a, we have Elimiatig p y f a () a a q y Solutio: Give Here betwee the equatios () ad () we get the geeral solutio. p. [N/D4] [A] q y This is of Lagrage s type. P Q The subsidiary equatios are y d P d y Q d d y y R d R d.

16 d d y y d y y d d y d y y d d y Itegratig, we have y y a b a y b y Hece the solutio is,. y y D.[U]. Fid the geeral solutio of 4 D D 9 D Solutio: Auiliary equatio is 4m m9 Hece the solutio is. Solve D D D m m m, m y y.. [N/D9]. [A] Solutio: The auiliary equatio is m m m m m,, Hece the solutio is y y y. Solve D D.. [A/M,N/D].[A] Solutio: The auiliary equatio is m m,, Hece the solutio is y y y Solve D D. [ M/J4] [A] 4 Solutio: The auiliary equatio is m m m m,, i, i.4

17 Hece the solutio is y y y i y i 4. Solve D 7 D D 6 D 4.. [M/J]. [A] Solutio: The auiliary equatio is m 7 m6. m m 6 m, 6 Hece the solutio is y y 6. D D D e. [N/D].[U] 5. Fid the particular itegral of y Solutio: P. I D D e D D D y e y e y D D e y e! y e y. 6. Solve D 4D D 4D D. [A/M5][A] Solutio: Auiliary equatio is m 4m 4m m m m 4m4 mm m, m, m Hece the solutio is y y y y y y. D D DD.[N/D5] [A] D D DD 7. Solve Solutio: Give Here m, c, m, c Hece the solutio is e y e y. 8. Solve D D D Solutio: Give D D D. [N/D].[A] Here m, c, m, c Hece the solutio is e ye y. 9. Solve. [N/D].[A] y Solutio: Give equatio ca be writte i the operator form as.5

18 Here D D D D D D D D D D D m, c, m, Hece the solutio is ye y. c PART-B. Form the partial differetial equatio of the family of plaes that are at costat distace k from the origi. [A/M,N/D6]. [C]. Form the partial differetial equatio by elimiatig the arbitrary fuctio from the relatio g y, y.[c]. Form the partial differetial equatio by elimiatig the arbitrary fuctio y, y f. [M/J6][C] 4. Form the PDE by elimiatig the arbitrary fuctio y, a b y c. [N/D,M/J].[C] 5. Form the PDE by elimiatig the arbitrary fuctio from the relatio y f log y. [M/J4][C] 6. Form the partial differetial equatio by elimiatig the arbitrary fuctios f ad g i f y g y. [C] 7. Form the partial differetial equatio by elimiatig the arbitrary fuctios f ad g i f y y g. [N/D][C] 8. Form a partial differetial equatio by elimiatig arbitrary fuctios from f y g y. [C] 9. Form the PDE by elimiatig the arbitrary fuctios ad ' f ' '' from the relatio y f y. [N/D6][C]. Form the partial differetial equatio by elimiatig the arbitrary fuctios f ad from f y y. [C]. Form the partial differetial equatio by elimiatig arbitrary fuctios f ad from f ct ct [A/M].[C]. Form the PDE by elimiatig the arbitrary fuctios f, f from the relatio f t f t. [A/M5][C]. Solve a give that whe, asi y ad. [N/D4] [A] y 4. Solve the equatio p q p q. [A] 5. Fid the sigular itegral of p q y pq.[u] 6. Solve g from p q y p q ad fid the complete ad sigular solutios. [N/D9,A/M5].[A].6

19 7. Fid the sigular solutio of p qy p q 6. [U] 8. Solve p qy p q. [N/D,M/J,N/D,N/D5,M/J6].[A] 9. Fid the sigular solutio of p q y p q. [N/D4] [U]. Fid the sigular itegral of the partial differetial equatio p q y p [N/D].[U] q. Fid the sigular solutio of the equatio p q y p.[u] p. Fid the sigular itegral of p q y pq.[u]. Solve p q.[a] 4. Solve p q.[a] 5. Solve pq q 6. Solve 9p q 4 7. Solve. [A]. [N/D4][A] p q q [A/M].[A] 8. Solve 9. Solve p p y q. [A] [A/M].[A] q y p q. Solve y. [A/M5][A]. Solve p y q. Solve p q y. Solve p q y.[a] 4. Solve p q.[m/j4, N/D5][A]. [A] 5. Obtai the complete solutio of 6. Solve p.[a] q y y. [N/D5][A] p y q 7. Solve p y q. [A] 8. Solve y p q y.[a] 9. Solve y py q y y [N/D9].[A] 4. Solve y p yq.[a] 4. Fid the geeral solutio of 4 y p4 q y 4. Solve y p y q y[n/d, N/D4]. [A] 4. Solve y p y q y.[a] 44. Solve y p y q y y p y q y. [A] 45. Solve 46. Solve y y p y q y 47. Solve p yq y 48. Solve y p q y [N/D].[A].[A]. [A/M5,M/J6][A]. [U] [A/M,A/M5, M/J6].[A].[A] p q q.7

20 49. Solve y p y q.[n/d] [A] 5. Fid the geeral solutio of 5. Solve p y q y y p. [A/M5][A] 5. Solve y p y q y 5. Solve y p y q y q y.8.[a].[u].[m/j, N/D4, N/D5][A] 54. Solve the Lagrage s equatio y p y q y 55. Solve y p y q y.[a] 56. Solve the Lagrage s equatio y p y q y 57. Solve the equatio y p y q 58. Solve the partial differetial equatio 59. Fid the geeral solutio of. [M/J6,N/D6][A]. [N/D,M/J].[A].[A] m y p l q ly m [A/M].[A] y p q y.[u] p y q. [M/J4][A] 6. Solve the Lagrage s equatio y 6. Fid the geeral solutio of y y py q y 4 D D 4 D e. [A] y 6. Solve D D D D y 6. Solve D si h y e 64. Solve D 5D D D 5si y. [A] 65. Solve D 4 D D5D si y. [A] 66. Solve D 7 D D 6 D si ( y) 67. Solve D D D D D D cos y 68. Solve D D D 4 D cos y.[a] 69. Solve D D D 4 D D 4 D cos y D D D D si cos 7. Solve y.[a] 7. Solve 4. [N/D9].[A] [M/J].[A].[A] D 7 D D 6 D cos y.[a] D 7 D D 6 D e y cos y 7. Solve 5 y 7. Solve D D D D e si 4 y y 74. Solve D 4 D D5D e si y 75. Solve D 7 D D 6 D si y. [A]. [N/D,M/J].[A]. [A]. [N/D4] [A] y 76. Solve.[A] D 7 DD 6D si y e. [M/J6][A] D D D D si 5 y. [N/D4][A] 77. Solve y 78. Solve 4 D 4 D D D e si.[a] y 79. Solve D D D D D D e cos y y y 8. Solve e 4si y y 8. Solve D 4 D D D e.. [A] [A]. [A].[N/D5][U]

21 D D D y e [N/D].[A] 6 D D D y e. [A] D D y e. [N/D4] [A] 4 y 8. Solve D y 8. Solve D y 84. Solve D D D6 D y e.[a] y 85. Solve D 86. Solve D 6 D D 5D e si h y y y 87. Solve si h y y y D D D y e. [A] y 88. Solve D 89. Solve D D D e y 9. Solve D D D 4 D si y. [A]. [A] [N/D,M/J6]. [A]. [A] D D D cos y y e y 9. Solve D 9. Solve D 7 D D 6 D cos y 9. Solve D D D 4 D cos y y 94. Fid the geeral solutio of D D y 95. Solve D D e si y y 96. Solve D D e si y.[a] y 97. Solve D DDD 4 e y. [A] [A/M]. [A].[A] [N/D]. [A]. [N/D5] [A]. [A/M5] [A] 98. Fid the geeral solutio of D DD D cos y si y 99. Solve D DD 6D ycos. [N/D5] [A] [M/J,M/J4,N/D6].(OR) Solve r s6t ycos. [A/M5] [A] D 5D D6 D ysi.[a]. Solve D D D D e. [A] y. Solve D D D DD e [N/D]. [A] y. Solve D D D D D D e.[a] y. Solve D y 4. Solve D D D D D D e 4.[A] 5. Solve y D D D D DD e y 6. Solve D D D D e 4 [N/D]. [A] 7. Solve D D D D D D si y e.[a] [A/M, N/D5,M/J6]. [A] [A] y 8. Solve D D D D e y 9. Solve D D D D D D y si y D D D D y. Solve 7 D D D D 6 DD y. Solve [N/D9]. [A] e [N/D, M/J]. [A] [A/M]. [A].9

22 UNIT-II FOURIER SERIES PART-A. State Dirichlet s coditios for a give fuctio to epad i Fourier series. [N/D 9, A/M,N/D,M/J,M/J4,N/D4,A/M5,N/D6]. [R] Solutio: A fuctio series of the form (i) (ii) (iii) f f f f defied i c c l ca be epaded as a ifiite trigoometric a a cos b si provided l l c, c l is sigle-valued ad fiite i is cotiuous or piecewise cotiuous with fiite umber of fiite discotiuities i c, c l. has o or fiite umber of maima or miima i c, c l.. State Euler s formula for Fourier co-efficiets of a fuctio defied i c, c l.[r] Solutio: If a fuctio f defied i c, c l ca be epaded as the ifiite a trigoometric series a cos b si, the l l c l a f d l c c l a f cos d l l c c l b f si d l l c Formulas give above for a ad b are called Euler s formula for Fourier co-efficiets. f ta possess a Fourier series epasio?[u] Solutio: No, f ta ifiite discotiuity. (ie) Dirichlet s coditio is ot satisfied.. Does does ot possess a Fourier epasio. Because f ta has a 4. If f is discotiuous at a Solutio:, what does its Fourier series represet at that poit?[u].

23 If a is a iterior poit of discotiuity of f i c, c l, the the Fourier series of at coverges to lim f a h f a h h (ie) [ Sum of Fourier series of ] = lim f a h f a h. h f a 5. Fid the value of Solutio: Here a a l 6. If the fuctio f a i Fourier series epasio of implies l f e i l f l f d e d e e series epasio of the fuctio Solutio: a f i the iterval f. [N/D5] [U],. [N/D] [U], the fid the costat term of the Fourier d d a The costat term of the Fourier series epasio. 7. Let f be defied i, the value of f.[u] Solutio: f lim f h h by f h cos ; f ad f f. Fid cos ; cos lim h h h lim cos h cos h h lim lim cos h h h h 8. If 4 si h cos lim. h cos Solutio: i, the fid. [M/J6] [U].

24 Give 4 cos 4 cos cos cos The poit is the poit of discotiuity (right etreme poit) 4 cos cos cos Therefore, cos i, the deduce that the value of.[n/d4][u] Solutio: cos 4 9. If 4 Give The poit is the left etreme poit of discotiuity 4 cos cos cos f f 4 cos cos cos

25 Therefore, f, with period. If the Fourier series of the fuctio f is give by si si si 4 si..., the fid the sum of the series [A/M5] [U] 5 7 Solutio: Give Substitute Hece, The poit f si si si 4 si....() 4 is the poit of cotiuity. i (), we have si f si si si If the Fourier series epasio of the fuctio f k, is k, the fid the sum of the series.... [N/D5][U] 5 7 Solutio: Give f f f 4k,,5,,5 4k si si,, 5 4k si si si 5 si () The poit is the poit of cotiuity. Substitute i (), we have 4k si,.

26 4k f si si 4k k si si... 7 Hece, Give the epressio for the Fourier series co-efficiet for the fuctio defied i. [A/M].[U] Solutio: The Fourier series co-efficiet b for the fuctio defied i is, give by b f si d.. Fid the costat term i the Fourier series correspodig to f cos iterval,. [N/D,M/J].[U] Solutio: f cos Give f cos f cos cos f Therefore, is a eve fuctio. f The costat term i the Fourier series is a cos d cos d d cos d a where b f si f, epressed i the Therefore, the costat term a is..4

27 4. State TRUE or FALSE: Fourier series of period for the fuctio cotais oly sie terms. Justify your aswer. [M/J6] [U], Solutio: f cos i the iterval Fourier series of period for the fuctio f cos i the iterval oly sie terms is TRUE. Sice f cos f cos is a odd fuctio. f cos f The costat term i the Fourier series is a cos d cos d d cos d a where si, cotais Therefore, the costat term a is. 5. If f is a odd fuctio defied i l, l, what are the values of Solutio: Give f is a odd fuctio, the values of a a. a ad a? [U] 6. Obtai the first term of the Fourier series for the fuctio Solutio: Here l implies l Give f f f Therefore a l l f is a eve fuctio. f d d Therefore the first term of Fourier series is a f, [N/D9]. [U]..5

28 7. Fid the value of b i the Fourier series epasio of Solutio: i, Give f i, Let Therefore, is a eve fuctio. Hece, b f. f i,. [M/J6] [U] i, 8. Fid the value of the Fourier series of f Solutio: Give f i c, i, c lim. h i i Value of f f h f h lim h b c,, c 9. Fid the Fourier costats for si i,. [U] Solutio: Give f si f si si si f Therefore, is a eve fuctio. Hece, b. f b at the poit of discotiuity [M/J6] [U]. Fid the co-efficiet of the Fourier series for the fuctio f si i,. [N/D]. [U] Solutio: Give f si f si si si f Therefore, f is a eve fuctio. Hece, b.. If f is epressed as a Fourier series i the iterval,, to which value this? [U] series coverges at..6

29 Solutio: Fourier series of f coverges at is f f. Fid the half rage sie series epasio of l Solutio: Here Fourier sie series is give by f b si l f b si l l l f, i. [N/D] [U] where b f si d si d cos f si d cos 4 4 si si Therefore,,5,,5. Epad f as a half rage sie series i the iterval, Solutio: Here l Fourier sie series is give by f b si l f where b b si. 4, if, if i s. [M/J4,A/M5,N/D5,N/D6] [U] cos si si f d d cos 4 4, if, if Therefore f si si,,5 4,,5 i s odd is eve. odd is eve.7

30 4. To which value, the half rage sie series correspodig to, 5 coverges at 5? [U] Solutio: We defie F as ; 5 F ; 5 Sum of Therefore, at 5 5 f 5 f Fourier Series 5 5 5, the series coverges to ero. 5. To which value, the half rage sie series correspodig to, Solutio: We defie coverges at F F as Sum of Therefore, at ; ;? [U] f f Fourier Series 4 4, the series coverges to ero. 6. Defie root mea square value of a fuctio [R] Solutio: The root mea square value of a fuctio give by y b b a a f d. 7. Fid the root mea square value of f l Solutio: by f epressed i the iterval f epressed i the iterval f over the iterval a, b. [M/J, N/D]. f over the iterval b a, is i l.[n/d5] [A] The root mea square value of a fuctio y l l l l f l i l is give l 4 f d l d l l d l.8

31 .9 l l l l l l = l l l l l l l l. 8. Fid the root mea square value of the fuctio f i the iterval l,. [N/D].[A] Solutio: The root mea square value of f i l, is give by l l l l d l y l l. 9. Fid the root mea square value of f i l,. [N/D].[A] Solutio: The root mea square value of f i l, is give by l l l l d l d l y l l l.. Write dow Parseval s formula o Fourier coefficiets. [N/D4][R] Solutio: If f y ca be epaded as Fourier series of the form si cos l b l a a i l,, the the root-mea square value y of f y i l, is give by 4 a b a y where l d f l y.. If cos a a f is the Fourier cosie series of f i. State the correspodig Parseval s idetity. [R] Solutio: Parseval s idetity is give by 4 a a d f 4 a a d f.

32 . State Parseval s idetity for the half rage cosie epasio of Solutio: Parseval s idetity for the half rage cosie epasio of give by y 4 a a f d a a 4 f d a a. 4 f f i. [R], i is,. If the Fourier series correspodig to a a cos b si a of a b. [A] 4 Solutio: By Parseval s idetity, we have. f i the iterval a, a, b,, without fidig the values of, fid the value a a b y 4 f d d a, a, b 4. Without fidig the values of, the Fourier co-efficiet of Fourier series, for the fuctio f a i the iterval, fid the value of a b. [A/M].[A] Solutio: By Parseval s idetity, we have a 5 4 a b y f d d What do you mea by Harmoic Aalysis?[M/J][R] Solutio: is

33 f Whe a fuctio is give by its umerical values at q equally spaced poits, the co-efficiets i the Fourier series represetig f ca be obtaied by umerical itegratio. PART-B for ) Fid the Fourier series epasio of f ( ). Also deduce that for.... [N/D].[U] 5 8 si ; ) Epad f ( ) as a Fourier series of periodicity ad hece evaluate ;....[U] ) Epad f as a Fourier series i the iterval. Deduce the sum of the series... [A/M]. [U] f si i. [N/D4] [U] 4) Determie the Fourier series for the fuctio 5) Fid the Fourier series of.[u] 6 f i, 6) Fid the Fourier series of f, ad periodic with period l. Hece deduce that i, of periodicity. [M/J]. [U] l ; l 7) Obtai Fourier series for f () of period ad defied as follows f ( ). ; l l Hece deduce that... ad....[u] ) Fid the Fourier series for f i the iterval. [A/M]. [U] 9) Fid the Fourier series of periodicity for f i. [N/D, N/D4]. [U] ; ) Fid the Fourier series for the fuctio f ( ). Deduce that ;....[U] 5 8 i ) Fid the Fourier series for f.[u] 6 i 6 ( ) ; ) Obtai the Fourier series for the fuctio f ( ).[U] ( ) ;.

34 ) Fid the Fourier series epasio of... to.[n/d]. [U] 5 8 4) Fid the Fourier series of [U] 5) Obtai the Fourier series for, f. Also deduce, f i. Hece deduce the value of f ( ) i (, ). [N/D4]. Deduce that.... [U] 6 6) Epad the fuctio f si as a Fourier series i the iterval.[n/d ].[U] 7) Fid the Fourier series of i. [M/J6] [U] f ( ) f, ad deduce 8) Obtai the Fourier series to represet the fuctio 8. [M/J]. [U] 9) Fid the Fourier series for f ( ) cos i the iterval (, ). [M/J6] [U] ) Fid the Fourier series of f si ) Fid the Fourier series of ) Fid the Fourier series of f i of periodicity. [A/M5] [U] f e i, f e i,.[u].[u] ) Epad ( ) as Fourier series i (, ).[U], 4) Fid a Fourier series to represet.[u] 5) Fid the Fourier series of. [N/D]. [U] 6 6) Fid the Fourier series for f 5... to. [M/J6] [U] 7) Fid the Fourier series of the fuctio [U] f i, of periodicity. Hece deduce that,. Hece deduce the sum of the series,, f ( ) ad hece evaluate si,.

35 . 8) Determie the Fourier series for the fuctio f,, ) (. Hece deduce that [A] 9) Fid the Fourier series of f,, ) (. Hece deduce that [A/M, N/D]. [A] ) Fid the Fourier series epasio of the periodic fuctio f of the period defied by f,,. Hece deduce that 8.[U] ) Fid the Fourier series epasio the followig periodic fuctio of period 4 ; ; f. Hece deduce that [N/D5] [U] ) Obtai the Fourier series of the periodic fuctio defied by f,,.deduce that [N/D9]. [U] ) Fid the Fourier series epasio of the periodic fuctio ) ( f of period l defied by l l l l f ; ; ) (. Deduce that 8 ) (.[U] 4) Epad f,, as a full rage Fourier series i the iterval,. Hece deduce that [M/J4] [U] 5) Fid the Fourier series of f, L L.[U] 6) Fid the Fourier series epasio of f i the iterval,. [N/D]. [U] 7) Fid the Fourier series of ; ; ) ( f.[u] 8) Fid the Fourier series of periodicity for,,, ) ( for for f. Hece show that the sum of the series [U] 9) Fid the Fourier series of the fuctio,, k f. Hece fid [U]

36 4) Fid the half rage sie series of f ( ) cos i.[u] 4) Epad f si as a cosie series i ad show that [N/D]. [A] 4) Obtai the Fourier epasio of as a cosie series i. Hece show that.... [N/D5] [A] ) Obtai the half rage cosie series for i. [N/D,]. [U] 44) Fid the half rage sie series for f ( ) ( ) i the iterval.[u] 45) Fid the half rage sie series of si f ( ) 46) Fid the half-rage sie series of f. [A/M5] [U] f ( ) (, ) (, ) i (, ).[U] 47) Fid the half rage sie series of f l i,l (, ). [N/D] [U] 48) Fid the half rage cosie series of f l, l.[u],,. Hece deduce the sum of the series, l k, 49) Obtai the Fourier cosie series of f. [M/J] [U] l k l, l l ; 5) Obtai the Fourier sie ad cosie series of f ( ).[A/M]. [U] l l ; l 5) Fid the half rage sie series for f si a i, l. [A/M5] [U] ; 5) Epad f ( ) as a series of cosies i the iterval,. [N/D6] [U] ; 5) Obtai the Fourier cosie series of, ad hece show that....[m/j] [A] 6 54) Fid the half rage cosie series epasio of 55) Obtai the half rage cosie series for ( ).[A] 8 i. [N/D4] [U] f ( ) ( ) i the iterval. Deduce that.4

37 56) Fid the half-rage sie series of of the series f 4 i the iterval. Hece deduce the value.... [M/J4] [U] 5 7 f cos. [M/J6] [U], 57) Fid the half rage sie series of i 58) Epad f as a Fourier series i square value of f f i the iterval 59) Epad of f L L i the iterval. [N/D9]. [U] as a Fourier series i.[u], 6) Fid the Fourier series of period of the series for the fuctio.... [N/D] [U] 5 6) Fid the Fourier series of periodicity for f,4 ad usig this series fid the root mea ad usig this series fid the R.M.S value ;(, ) f ( ) ad hece fid the sum ;(, ), i. Hece show that [M/J,A/M5,N/D5,N/D6] [A] 9 l ;(, ) 6) Fid the Fourier series epasio of period l for the fuctio f ( ). Hece deduce l l,(, l) the sum of the series.[a] 4 ( ) 4 6) By usig cosie series for f ( ) i, show that.... [N/D4] [A] 64) Fid the half rage cosie series of f ( ) ( ) i the iterval (, ). Hece fid the sum of the series....[a] ) Fid the half rage Fourier cosie series of f i the iterval,. Hece fid the sum of the series.... [M/J,N/D5]. [A] ) Fid the half rage cosie series for the fuctio f ( ) ( ) i. Deduce that [A/M]. [A] 9 67) Fid the half rage Fourier Cosie series epasio for the fuctio deduce the sum of the series.... [M/J6] [A] f i l. Hece.5

38 68) Fid the half rage sie series of i. Deduce the sum of....[a] 69) Obtai the Fourier cosie series epasio of i. Hece deduce the value of... to. [N/D4] [A] ) Fid the Fourier series of period as far as the first harmoic to represet the fuctio defied by the followig table: [A] y f f ( ) a f (, l) 4 y ) Fid the first two harmoics i the Fourier series of y f which is defied i the followig table [A], 6 y ) Obtai a Fourier series upto the secod harmoic from the data [A] 4 5 y ) Fid the first two harmoics of the Fourier series of y f ( ) from the data :[A] y ) Fid the Fourier series up to third harmoics to represet the fuctio give by the followig discrete data: [A/M,N/D,M/J4,N/D4, A/M5, M/J6][A] 4 5 y ) Obtai the costat term ad the first harmoic i the Fourier series epasio for f () is give i the followig table.6

39 f () Hece fid f ().[A] 76) Fid the first fudametal harmoic of the Fourier series of f () give by the followig table: [A/M,N/D,M/J,N/D,N/D6].[A] 4 5 f () ) Determie the first two harmoics of the Fourier series [N/D5][A] 4 5 y ) The followig table gives the variatios of a periodic curret over a period T T T T 5T 6 6 T T f Fid the fudametal ad first harmoics of f () to epress f () i a Fourier series i the form a f a cos b si where. [N/D9, ]. [U] T 79) The followig table gives the variatios of a periodic curret over a period. t secs T 6 T A amps By harmoic aalysis, show that there is a direct curret part of. 75 amps i the variable Curret. Also obtai the amplitude of the first harmoic. [A/M5] [A] 8) Fid the costat term ad the first two harmoics of the Fourier cosie series of y f [U] 6 T T 5T f () 5 7 T.7

40 8) Obtai the first three coefficiets i the Fourier cosie series for, where is give i the followig table [N/D5,M/J6][A] 4 5 y y y ) Fid the comple form of Fourier series of f e i 8) Show that the comple form of fourier series for the fuctio si h i f f is e i f. [N/D4] [A]. [A/M5,M/J6] [U] f e whe ad 84) Fid the comple form of the Fourier series of the fuctio f e whe ad f f. [U] 85) If a is ot a iteger, fid the comple Fourier series of f cos a i a 86) Fid the comple form of Fourier series of costat. Hece deduce that a f e i the iterval,,.[a/m,n/d5,n/d5][u] asi ha, 87) Fid the comple form of Fourier series of.[u] 88) Epad f si as a comple form of Fourier series i, 89) Fid the comple form of the Fourier series of f e other tha a iteger cos si l a 9) Epad f e l. [M/J4] [U].[M/J][U] where a ad is a real, l l. Deduce that whe is costat l i e l. [A/M5] [U], as a comple form Fourier series. [N/D6] [U] UNIT-III APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS PART-A u u ) Solve y by method of separatio of variables. [N/D5][A] y u u Solutio: Give y.() y Let the solutio of () be u XY y.() u u X Y ad X Y () y Usig () i (), we have X Y y X Y X Y y X Y.8

41 X Y y X Y L.H.S is a fuctio of aloe ad R.H.S is a fuctio of y aloe. They are equal for all values of ad. This is possible if each is a costat. X Y y k X Y X X k k (4) X X Y Y k y k.(5) Y Y y Itegratig (4) with respect to ad (5) with respect to, we have log X k log log A log Y k log y log B log X log X y k log log A log Y log y log B log k A k y k log Y log By k k X A k k X a Y b y Hece the solutio of () is u a b y ab y. ) Classify the followig partial differetial equatio: [AZ] u u (a) y (b) u u u y. y y u u Solutio: Give y Here A, B, C B 4 AC 4 4 Therefore, the give pde is hyperbolic. (b) u u u y y y Here A, B, C B 4 AC 4 Therefore, the give pde is hyperbolic. k ) Classify the partial differetial equatio u uy y.[az] k Y k By k.9

42 Solutio: If If If Here A, B, C 4 B 4 AC 4, B 4 AC 4, therefore the give pde is elliptic, B 4 AC, therefore the give pde is parabolic., B 4 AC 4, therefore the give pde is hyperbolic. u u u u u 4) Classify the differetial equatio 4 6 u.[az] y y y Solutio: Here,, B 4 AC A Therefore, the give pde is elliptic. B 4 C ) Classify the followig partial differetial equatio Solutio: Here A 4, B, C B 4 AC 44 Therefore, the give pde is parabolic. u 6) Classify the partial differetial equatio u f y Solutio: Here A, B, C B 4 AC 4 Therefore, the give pde is hyperbolic. 7) Classify the partial differetial equatio y, y y y y yy y u u 4. [N/D9]. [AZ] t. [M/J6] [AZ]. [A/M5] [AZ] Solutio: Give y y y Here B A y B y yy C y y 4 y 4 y y y 4 4 y 4 AC y If y, the B 4 AC. Hece the give PDE lies o a circle. If y, the B 4 AC. Hece the give PDE lies iside the circle. If y, the B 4 AC. Hece the give PDE lies outside the circle. y y 8) I the wave equatio c t Solutio:, what does c stad for? [N/D]. [U] y.4

43 The costat a i the wave equatio ut t a u stads for Tesio Mass (i.e) a 9) What are the possible solutios of oe dimesioal wave equatio? [N/D9,M/J4,N/D4].[R] Solutio: The possible solutios of oe dimesioal wave equatio are (i) y, t A e B e C e at D e at (ii) y, t A cos B si C cos at D si at (iii) y, t A B C t D. ) State the assumptios i derivig the oe dimesioal wave equatio y t t y. [A/M5,N/D6][R] Solutio: The assumptios ivolved i derivig oe dimesioal wave equatio are (i) The motio takes place etirely i oe plae. This plae is chose as the y plae. (ii) I this plae, each particle of the strig moves i a directio perpedicular to the equilibrium positio of the strig. (iii)the tesio T caused by stretchig the strig before fiig it at the ed poits is costat at all times at all poits of the deflected strig. (iv) The tesio T is very large compared with the weight of the strig ad hece the gravitatioal force may be eglected. ) A tightly stretched strig with fied ed poits ad l give by y y is iitially i a positio, si. If it is released from rest i this positio, write the l boudary coditios. [A/M]. [U] Solutio: The boudary coditios are i y, t ii yl, t y ( iii ) t, ( iv) y, y si, l. l ) Write the oe dimesioal heat equatio.[r] u u Solutio: The oe dimesioal heat equatio is where t ) Write the partial differetial equatio goverig oe dimesioal heat coductio.[r] Solutio: The partial differetial equatio goverig oe dimesioal heat u u coductio is give by. t k c. T M..4

44 4) I the oe dimesioal heat equatio ut c u. What is Solutio: I the equatio (ie) c k c u c, t u c stads for k c c?[m/j][u] is called the diffusivity cm \sec of the substace. u u 5) How may coditios are required to solve.[u] t u u Solutio: Three coditios are required to solve. t 6) State ay two laws which are assumed to derive oe dimesioal heat equatio. [M/J4][R] Solutio: The laws which are assumed to derive oe dimesioal heat equatio are (i) Heat flows from a higher to lower temperature. (ii) The amout of heat required to produce a give temperature chage i a body is proportioal to the mass of the body ad to the temperature chage. This costat of (iii) proportioality is kow as the specific heat of the coductig material. The rate at which heat flows through ay area is proportioal to the area ad to the temperature gradiet ormal to the area. This costat of proportioality is kow as the thermal coductivity k of the material. 7) State Fourier law of heat coductio.[r] Solutio: The rate at which heat flows through ay area is proportioal to the area ad to the temperature gradiet ormal to the area. This costat of proportioality is kow as the thermal coductivity k of the material. It is kow as Fourier law of heat coductio. 8) State the three possible solutios of the oe dimesioal heat flow (usteady state) equatio. [N/D,N/D4,M/J6,N/D6].[R] Solutio: The various possible solutios of oe dimesioal heat equatio are t i u, t A e B e e t ii u, t A cos B si e iii u, t A B. 9) Defie steady state coditio o heat flow. [N/D][U] Solutio: The state i which the temperature at ay poit does ot deped o t, but oly o is called steady state. ) What is the basic differece betwee the solutios of oe dimesioal wave equatio ad.4 c

45 oe dimesioal heat equatio. [M/J][A] Solutio: The correct solutio of oe dimesioal wave equatio is of periodic i ature. But the solutio of heat flow equatio is ot i periodic ature. ) The eds log have the temperature at state prevails. Fid the steady state temperature. [N/D4] [U] A ad B of a rod cm Solutio: The steady state equatio of oe dimesioal heat flow is C d u d ad 8 C..() util steady u The solutio of () is b Here l a..() The boudary coditios are ( i ) u ( ii ) ul 8 Applyig coditio (i) i (), we have u a b ==> ==> Substitutig b i (), we have u a () Applyig coditio (ii) i () ad substitutig l, we have b b u l al ==> 8 a ==> a 5 ==> 5 Substitutig a i (), we have u. ) A isulated rod of legth 6 cm has its eds at A ad maitaied at c ad 8 c respectively. Fid the steady state solutio of the rod. [N/D,M/J].[U] Solutio: d u The steady state equatio of oe dimesioal heat flow is..() d The solutio of () is u a b..() Here l 6.4 B a The boudary coditios are ( i ) u ( ii ) ul 8 Applyig coditio (i) i (), we have u a b ==> b ==> b Substitutig b i (), we have u a () Applyig coditio (ii) i (), we have u l al ==> 8 a 6 ==> 6a 6 ==> a Substitutig a i (), we have u. 5

46 ) Write dow the goverig equatio of two dimesioal steady state heat coductio. [R] u u Solutio: is the goverig equatio of two dimesioal steady state heat y coductio. 4) Write the steady state heat flow equatio i two dimesio i Cartesia equatio ad polar form. [M/J].[R] u u Solutio: The Cartesia equatio of two dimesioal heat flow is. y u u u The polar form of two dimesioal heat flow is r r. r r 5) Write dow the three possible solutios of Laplace equatio i two dimesios. [A/M, N/D, A/M,A/M5,N/D5][R] Solutio: The two solutios of the Laplace equatio obtaied by the method of separatio of variables are ( i) u, y A B C e y De y cos si ( ii) u, y A e B e C cos y D si y. iii u, y A B C y D. 6) A plate is bouded by the lies, y, l ad y l edge coicidig with -ais is kept at The other two edges are kept at two dimesioal heat flow equatio. [N/D, N/D][U] Solutio: The boudary coditios are i u, c, l ii u, y 5 c, y l iii u, l c, l iv u l, y c, y l c.44. Its faces are isulated. The c. The edge coicidig with y-ais is kept at 5 c. Write the boudary coditios that are eeded for solvig 7) Write dow the partial differetial equatio that represets steady state heat flow i two dimesios ad ame the variables ivolved. [M/J][R] Solutio: The partial differetial equatio that represets steady state heat flow i two dimesio u u is. y 8) Write dow the two dimesioal heat equatio both i trasiet ad steady states. [M/J][R].

47 Solutio: The two dimesioal heat equatio i trasiet state is u u two dimesioal heat equatio i steady state is. y u u u ad the t y PART-B ) A tightly stretched strig has its eds fied ad. At time, the strig is give a shape defied by f ( ) k ( l ), where k is a costat, ad the released from rest. Fid the displacemet of ay poit of the strig at ay time. [U] ) A uiform strig is stretched ad fasteed to two poits apart. Motio is started by displacig the strig ito the form of the curve y k( l ) ad the releasig it from this positio at time. Fid the displacemet of the poit of the strig at a distace from oe ed at time. [A/M,N/D,A/M5,N/D5]. [U] ) A tightly stretched strig with ed poits is iitially i a positio give by y, k L. If it is released from this positio, fid the displacemet y, t at ay poit of the strig.[u] 4) A tightly stretched strig with fied ed poits ad is iitially i a positio give by y( y. It is released from rest from this positio. Fid the displacemet at ay l time. [N/D]. [U] 5) A tightly stretched strig of legth has its eds fasteed at ad. The midpoit of the strig is the take to a height epressio for the displacemet of the strig at ay subsequet time. [A/M]. [A] l 6) A strig of legth has its eds, fied. The poit where is draw aside a y y small distace h, the displacemet y (, t) satisfies a. Fid y (, t) at ay time t t.[a] 7) A tightly stretched strig of legth has its eds fasteed at, l. The midpoit of the strig is the take to height 'b' ad the released from rest i that positio. Fid the lateral displacemet of a poit of the strig at time 't' from the istat of release. [N/D].[A] 8) A strig of legth l, fasteed at both eds. Motio is started by displacig the strig ito the form y k ( l ) ad the releasig it from this positio at time t. Fid the displacemet of the poit of the strig at a distace from oe ed at time 't '.[U] 9) A uiform elastic strig of legth 6cms is subjected to a costat tesio of Kg. If the eds t ad l t 'l' L l t 't' l l ' l' l h ad the released from rest i that positio. Obtai a l fied ad the iitial displacemet y, 6, 6 fid the displacemet fuctio y, t. [A/M5][U], while the iitial velocity is ero, t.45

48 ) A strig is stretched ad fasteed to poits at a distace 'l ' apart. Motio is started by displacig the strig i the form y a si, l, from which it is released at time t. Fid l the displacemet at ay time t. [M/J4][U] ) A tightly stretched strig with fied ed poits ad l is iitially displaced to the form si cos ad the released. Fid the displacemet of the strig at ay distace l l from oe ed at ay time t. [M/J6][U] ) A tightly stretched strig with fied ed poits ad is iitially at rest i its equilibrium positio. If it is set vibratig by givig each poit a velocity. Fid the displacemet of the strig at ay time.[m/j,n/d4][u] ) A tightly stretched strig with fied ed poits is iitially at rest i its equilibrium positio. If it is set vibratig givig each poit a iitial velocity l, fid the displacemet. [N/D9].[U] 4) A tightly stretched strig with fied ed poits is iitially at rest i its equilibrium positio. If it is set vibratig givig each poit a velocity, fid the displacemet at ay time ad at ay distace from ed 5) A tightly stretched strig of legth is iitially at rest i its equilibrium positio ad each of its poits is give the velocity v vo si, fid the displacemet of the strig at ay subsequet l time. [N/D, N/D4].[U] 6) If a strig of legth l is iitially at rest i its equilibrium positio ad each of its poits is give a l c; velocity v such that v, fid the displacemet at ay time t.[u] l c( l ); l 7) A elastic strig of legth fied at both eds is disturbed from its positio at equilibrium positio by impartig to each poit a iitial velocity of magitude k(l ). Fid the displacemet fuctio y (, t).[u] 8) Fid the displacemet of a strig stretched betwee two fied poits at a distace of apart whe the strig is iitially at rest i equilibrium positio ad poits of the strig are give iitial i, l l velocities v where v, beig the distace measured from oe ed. l i l,l l [M/J6][U] y l l ad l l ad.[u] k( l ) l.46

49 9) A tightly stretched strig with fied ed poits ad l is iitially at rest i its equilibrium k l i l positio. If it is vibratig by givig to each of its poits a velocity v. k l l i l l Fid the displacemet of the strig at ay distace from oe ed at ay time t. [N/D5][U] ) A tightly stretched strig of legth positio. If it is set vibratig by givig each poit a velocity y t, v si cos, l l where l. Fid the displacemet of the strig at a poit, at a distace from oe ed at ay istat 't'. [N/D6][U] u u ) Fid the solutio to the equatio a that satisfies the coditios, t u l, t, for t ad u 'l' with fied ed poits is iitially at rest i its equilibrium l,,. [N/D,A/M5][U] l l, l u t u, is ) Solve the problem of heat coductio i a rod give that the temperature fuctio t u u subjected to the coditio,, l, t (i) u is fiite as t t u (ii) for ad l, t (iii) u l fort, l. [A/M5][U] L ) A isulated rod of legth has its eds ad ad C respectively, util steady state coditios prevail. If B is suddely reduced to ad that at A is maitaied at, fid the temperature at a distace from A at time t.[a] 4) A rod cm log, has its eds A ad B kept at c ad respectively, util steady state coditios prevail. The temperature at each ed is the suddely reduced to ad kept so. Fid the resultig temperature fuctio U, t takig 5) A rod of legth cm has its eds A ad B kept at c ad respectively util steady state coditios prevail. If the temperature of A is suddely raised to 4 c while that the other ed B is reduced to 6 c, fid the temperature distributio at ay poit i the rod.[a] 6) A metal bar cm log with isulated sides, has its eds A ad B kept at c ad 4 c respectively util steady state coditios prevail. The temperature at A is the suddely raised to 5 c ad at the same istat that at B is lowered to c. Fid the subsequet temperature at ay poit at the bar at ay time.[a] 7) The eds A ad B of a rod lcm log have their temperatures kept at c ad 8 c, util steady state coditios prevail. The temperature of the ed B is suddely reduced to 6 c ad that of A is icreased to 4 c. Fid the temperature distributio i the rod after time t.[a] 8) The eds A ad B of a rod lcm log have the temperatures 4 c ad 9 c util steady state prevails. The temperature at A is suddely raised to 9 c ad at the same time that at B is C A B maitaied at 8 c at A. [N/D9].[A] 8 c C C c.47

50 lowered to 4 c. Fid the temperature distributio i the rod at time t. Also show that the temperature at the midpoit of the rod remais ualtered for all time, regardless of the material of the rod.[a] 9) A bar of cm log, with isulated sides has its eds A ad B maitaied at temperatures ad respectively, util steady-state coditios prevail. The temperature at A is suddely raised to ad at B is lowered to. Fid the temperature distributio i the bar thereafter.[n/d5] [A] ) The eds ad log have their temperature kept at ad respectively, util steady state coditio prevails. The temperature of the ed is the suddely reduced to ad kept so, while that of the ed is kept at. Fid the subsequet temperature distributio i the rod. [M/J].[A] ) Fid the steady state temperature at ay poit of a square plate if two adjacet edges are kept at ad the others at. A rectagular plate is bouded by the lies, y, a, y b. Its surfaces are isulated. The temperature alog are kept at ad the others at. Fid the steady state temperature at ay poit of the plate.[a] ) A square plate is bouded by the lies,, ad. Its surfaces are isulated c c 9 c A C B of a rod 4 ad the temperature alog C c 4 cm u (, t) c C 6 c a y A y b c c ad B y 5 c 8 c y b is kept at C, while the temperature alog other three edges are at. Fid the steady state temperature at ay poit i the plate.[n/d4][a] ) The boudary value problem goverig the steady state temperature distributio i a flat thi u u square plate is give by, a, y a, y u (,), u (, a ) 4si a, a ad u (, y), u ( a, y), y a. Fid the steady state temperature distributio i the plate. [U] 4) Fid the steady state temperature distributio i a rectagular plate of sides a ad b isulated at the lateral surface ad satisfyig the boudary coditios u (, y) u( a, y) for y b u (, b) ad u(,) ( a ) for a. [N/D].[U] 5) A rectagular plate of sides a ad b has its faces isulated ad the edges ad ad the edge is kept at temperature k ( y b ). Fid the steady state temperature distributio i the plate. [U] 6) A square plate is bouded by the lies, y, ad y. Its faces are isulated. The temperature alog the upper horiotal edge is give by u,, while the other three edges are kept at C. Fid the steady state temperature distributio i the plate. [N/D,N/D,N/D4,N/D6].[U] 7) A square plate is bouded by the lies, y, ad y. Its faces are isulated. The temperature alog the upper horiotal edge is give by u,, while the other edges are kept at. Fid the temperature distributio i the plate.[a] are kept at C C a y y b ad.48

51 8) A rectagular plate is bouded by the lies, a, y ad y b ad the edge temperatures 5 are u, si 8si, u, y, u, b ad u a, y. Fid the steady a a state temperature distributio at ay poit of the plate. [M/J6][U] 9) A ifiitely log plate i the form of a area is eclosed betwee the lies y, y for positive values of. The temperature is ero alog the edges y, y ad the edge at ifiity. If the edge is kept at temperature ky, fid the steady state temperature distributio i the plate.[a] 4) A rectagular plate with isulated surface is cm so log compared to its width that it may be cosidered ifiite legth. If the temperature alog short edge is give u(,) 8si whe, while the two log edges ad as well as the other short edge are kept at fid the steady state temperature fuctio.[m/j4][a] 4) A rectagular plate with isulated surface is cm wide ad so log compared to its width that it may be cosidered ifiite i legth without itroducig appreciable error. The temperature at, 5 short edge y is give by u ad all the other three edges are kept at ( ),5. Fid the steady state temperature at ay poit i the plate.[a/m,m/j,n/d5][a] 4) A rectagular plate with isulated surfaces is cm wide ad so log compared to its width that it may be cosidered ifiite i legth without itroducig a appreciable error. If the y, for y temperature of the short edge is give by u ad the two log y, for y edges as well as the other short edge are kept at. Fid the steady state temperature distributio i the plate. [A/M].[A] 4) A rectagular plate with isulated surfaces is wide ad so log compared to its width that it may be cosidered as a ifiite plate. If the temperature alog short edge is u, si, 8, while two log edges 8 as well as the other short edge are kept at C. Fid the steady state temperature at ay poit of the plate.[a] 44) A log rectagular plate with isulated surface is wide. If the temperature alog oe short edge ( y ) is u, k l degrees, for l, while the other two log edges ad l as well as the other short edge are kept at C, fid the steady state temperature fuctio u (, y). [M/J,M/J6]. [A] c c, u, y 8cm l cm u(, y) c ad y 8 y UNIT-IV FOURIER TRANSOFRMS PART-A ) State the Fourier itegral theorem. [M/J4,A/M5,M/J6][R].49

52 Solutio: If fiite iterval i f f is piecewise cotiuous, has piecewise cotiuous derivatives i every ad absolutely itegrable i, the, is t f t e dt ds, or equivaletly f f t cos st dt ds. ) Write the Fourier trasform pair. [N/D, N/D].[R] Solutio: The Fourier trasform pair is F f s F F f f Fs Fs e i s e d i s ds ad. ) Defie self reciprocal with respect to Fourier trasform. [N/D][R] Solutio: If is the Fourier trasform of is said to be self reciprocal f uder Fourier trasform. s f, the 4) Prove that Fourier trasform is liear. [N/D5][A] F a f b g af Solutio: We have to prove that f bfg i By defiitio, we have Fa f b g a f b ge s d F F F i a f b g a f e s d b g i a f b g a f e s d b g a f b g af f bfg. 5) Fid the Fourier trasform of f Solutio: F F f f b e i e i s d i s k f e i k e i s d e k a f e i s e i s d d, a b. [N/D9][U], a ad b b a d.5

53 i s k e e i s k i s k b i s k i a e s k 6) If F s is the Fourier trasform of f, write the formula for the Fourier trasform of a f cos i terms of F.[OR] State ad prove modulatio theorem o Fourier trasforms.[n/d4][a] Solutio: By defiitio, F f i s f e d F F f cos a f f cos a f cos a e e i a f e i a e i s d f f Fs a F s a i s e e d i a i a e e i s i s i s a i s a e d f e ( ie ) F f a Fs a Fs a cos. 7) State the shiftig property o Fourier trasform.[r] Solutio: If F s a is the Fourier trasform of a s f. ( ie ) F f a e i Fs. f F s e i a, the s 8) If is the Fourier trasform of f f.[m/j6][a] Solutio: F f f Ff Ff F s e is F f. d d d, obtai the Fourier trasform of will be the Fourier trasform of is is is se Fs Fse e Fscos hs Fscos hs 9) What is the Fourier trasform of f a if the Fourier trasform of f is F s i s a F f a e F s. [N/D4,A/M5] [A/M, M/J, N/D].Prove that Solutio: By defiitio, F f f F f a f a e e i s i s d d. [A]..5

54 F Put a t t a d dt s t a f a f t e i e i a s f t e i s t whe, t whe, t dt dt e ias f t e i s t f e i a s e dt i s d e ias Fs. ) State the Fourier trasforms of the derivatives of a fuctio.[r] Solutio: If the Fourier trasform of is, the the Fourier trasform of the derivatives f f of a fuctio (ie) Ff i s Fs, if as ) If Ff Fs, the give the value of f.. (OR) State ad prove the chage of scale property of Fourier trasform. [A/M,M/J,N/D5,N/D6].[A] i s Solutio: By defiitio, F f f e d F F whe F F f F s a f a f a a f a f t whe put, s F a a a, a t t a dt d a f a f t e i s put t a dt a e i s a t t a dt d a a () e i s whe whe t a dt a d, t, t a whe whe f f t t e e, t, t s i t a s i t a dt dt.5

55 a From () ad (), we have F f a t e s i t a s a f a F. dt s F a a () i a ) If Ff Fs, the fid F e f. [N/D4][A] Solutio: By defiitio, F f f e i s d a i a f e i f F e e i s a i i s a f f e d F s a F e d ) Write dow the Fourier cosie trasform pair.[r] Solutio: Fourier cosie trasform pair is F C 4) If s f F s f C cos s d F C is the Fourier cosie trasform of s f a is F a C a. [A/M].[A] Solutio: F f a F s f a C put C a a t t a dt d a f t f cos s (ie) F f a F C a C f s a ad f F Fs F s C, prove that Fourier cosie trasform of cos s d whe, t whe, t t a cos s a dt a a d f t cos s a t dt C cos s ds..5

56 s for s a 5) Give that F S f Solutio: F C d ds a f F f S, hece fid f d ds s a s s s s.54 a F C. [M/J6][U] s a s a s. 6) Fid the Fourier cosie trasform of Solutio: F e e cos s d C 7) Fid the Fourier sie trasform of Solutio: s a s a s a f a e, a a a e a a F S.[N/D5][U] a a. a s s a put si s s d d s. [N/D9,M/J4,A/M5,N/D6].[U] s d d si s s si d, a s whe, whe, a cos s s si s s si d s. 8) Fid the Fourier sie trasform of a. [N/D, M/J].[U] Solutio: F S a a e e si s d e e a a asi s s s 9) Fid the Fourier sie trasform of a s s s. s a e. [ M/J].[U] cos s

57 Solutio: F S e e si s d si s s cos s e s s s. 9 s s 9 e ) Fid the Fourier sie trasform of Solutio: F s Fs e f f e si f si s d s d 4s 4. [A/M5][U] e s si s scos s 4s 4s s 4s 4 s ) State Covolutio theorem o Fourier trasforms. [N/D].[R] Solutio: If ad are the Fourier trasforms of ad (ie) F f g Ff. Fg F s G s trasform of the covolutio of f g. ) State Parseval s idetity o Fourier trasform. [N/D].[R] Solutio: If f ad g is the product of their Fourier trasforms. f respectively, the the Fourier Fs is the Fourier trasform of, the f d Fs ds ) Solve the itegral equatio f cos d e Solutio: Give f cos d e.[u]. Multiplyig both sides by f, we have cos d e.55

58 f F e f e F C e e d C cos cos d. e cos si ) Fid the Fourier itegral represetatio of PART-B f defied as f e for for for. ) Epress the fuctio si cos d ad ) Fid the Fourier trasform of si d si [N/D, M/J ].[U], f as a Fourier itegral. Hece evaluate, d. [U] 4) Fid the Fourier trasform of si d.[n/d5] [U] ; f ( ). Hece prove that ; otherwise e a, if a 5) Fid the Fourier trasform of f ( ) defied by si of d usig Parseval s idetity prove that. Deduce that d if a.[u] ( a ) 4a, a f ( ) ad hece fid the value, a si t t dt. [A/M,M/J,A/M5,M/J6].[U].56

59 6) Fid the Fourier trasform of a f ( ) si t t cos t dt. [N/D,N/D5]. [U] t 4 7) Fid the Fourier trasform of f cos si i cos d ii 8) If is the Fourier trasform of f (a).[a] f () a ad ; ; a. Hece deduce that a, a. Hece evaluate, a cos si d.[u], fid the Fourier trasform of f () 9) Fid the Fourier trasform of the fuctio defied by ; f ( ) ; f ( a) ad. Hece prove that si s s cos s s cos ds ad si s s cos s s 6 ds. [U] 5 s [A/M,N/D,M/J6] ; if 4 ) Fid the Fourier trasform of f ( ). Hece deduce that si ; if t dt t ad si t t dt. [N/D, N/D, N/D4,N/D5,M/J6,N/D6].[U] ) Fid the Fourier trasform of give by si t t dt ad 4 f f si t dt. [N/D9,A/M5].[U] t ) Fid the Fourier trasform of e,. Hece show that the Fourier trasform. [N/D4,N/D6] [U] s ) If F f ( ) F( s). Prove that F f ( a) F a a. [A] a a a 4) Fid the Fourier trasform of e, a () cos t dt a t a a () F e i e a a s s a a for a. Hece show that for a e ad hece deduce that, here F stads for Fourier trasform. is self reciprocal uder [M/J4,N/D4][U].57

60 5) Show that the Fourier trasform of e Fourier trasform of f e i 6) Show that the trasform of e 7) Fid the Fourier trasform of f 8) Fid the Fourier trasform of [A/M5][U], s e is e s. [M/J6][U]. [A/M,N/D,M/J].[OR]Fid the is by fidig the Fourier trasform of,. [N/D4,A/M5][U]. [M/J4][U] e ad hece deduce that 9) Verify the covolutio theorem for Fourier trasform if f g e ) Fid the Fourier sie itegral represetatio of the fuctio f e ) If F C s ad G C s are the Fourier cosie trasform of prove that f gd F sg s C C ds.[a] f ad e a a cos t dt e t si g.[n/d5][a].[n/d5][u] respectively, the d d ) Prove that FC f FS f ad FS f FC f. [A/M5][A] ds ds si ; ) Fid the Fourier sie trasform of f ( ).[U] ; cos 8 4) Fid the Fourier cosie trasform of. Deduce that d e ad 6 8 si d e 6 8.[U] e 4 5) Fid the Fourier sie trasform of e.[u] 6) Fid the Fourier cosie trasform of f ( ) si cos cos d.[u] 6 7) Fid the Fourier sie ad cosie trasforms of f 8) Prove that f ;. Hece prove that ; otherwise si, a. [A/M].[U], a e is self reciprocal uder the Fourier cosie trasform.[u] a 9) Fid the Fourier sie ad cosie trasform of e, a ad hece deduce the iversio formula. [N/D,N/D4].[U]..58

61 ) Fid the Fourier sie trasform of ssi s d s e a s ) Prove that a. [U] f a e where a ad hece deduce that is self reciprocal uder Fourier sie ad cosie trasforms. [N/D9].[A] ) Fid the Fourier sie trasform of si m d e, m m.[u] e a e a, a ad hece show that ) Fid the Fourier sie trasform of ad hece evaluate Fourier cosie trasforms of ad e a si a. [N/D].[A] 4) Fid the Fourier sie ad cosie trasform of ad hece fid the Fourier sie e a trasform of e ad Fourier cosie trasform of 5) Fid the Fourier sie ad cosie trasform of ad hece fid the Fourier sie trasform of ad Fourier cosie trasform of. [M/J6][A] a a 6) Fid the Fourier sie ad cosie trasform of [N/D, A/M].[U] e e a.[a] ; f ( ) ;. ; 7) Fid the Fourier cosie trasform of. [N/D9].[U] a e 8) Fid the ifiite Fourier sie trasform of ad hece deduce the ifiite Fourier sie trasform of. [N/D6][U] a 9) Fid the Fourier cosie trasform of the fuctio f, 4) Fid the Fourier cosie trasform of a e.[u] a e e e b. [N/D5][U]. Hece evaluate the Fourier sie trasform of 4) Fid the Fourier cosie trasform of a e, a. Hece show that the fuctio reciprocal. [N/D].[U] 4) Fid the Fourier cosie trasform of. [A/M5][U] e is self 4) Fid Fourier sie ad cosie trasform of ad hece prove that uder Fourier sie ad cosie trasforms. [M/J].[U] is self reciprocal.59

62 44) Fid the Fourier sie trasform of e a, a 45) Fid Fourier sie trasform of f () defied as 46) Prove that e F e a. Hece fid.[u] si ; a f ( ).[U] ; a is self reciprocal uder Fourier cosie trasform.[u] e 47) Fid the Fourier cosie ad sie trasform of itegrals cos s a s ds ad ssi s ds. [M/J6][U] a s f a for, a. Hece deduce the e a s 48) Fid the fuctio whose Fourier sie trasform is a. [N/D] [A] s a 49) Fid Fourier sie trasform ad Fourier cosie trasform of e, a. Hece evaluate ( a ) d ad d ( a )( b.[a] ) e a,, a 5) Fid the Fourier sie trasforms of g e b,, b f. Hece evaluate ad d.[a] ( a )( b ) 5) State ad prove covolutio theorem for Fourier trasforms. [N/D, M/J].[R] 5) Derive the Parseval s idetity for Fourier trasforms. [N/D, M/J].[R] a 5) By fidig the Fourier cosie trasform of f ( ) e ( a ) ad usig Parseval s idetity for cosie trasform evaluate d ( a ).[M/J,N/D][A] 54) Verify Parseval s theorem of Fourier trasform for the fuctio 55) Evaluate 56) Evaluate d a b d 4 5 usig Fourier cosie trasforms of ; f ( ).[A] e ; a e ad [N/D, A/M, N/D4,N/D5,M/J6].[A] usig trasform methods. [N/D9].[A] 57) Usig Parseval s idetity evaluate the followig itegrals () () a d, where a. [M/J4] [A] b e. a d.6

63 58) Solve for 59) Solve for f () f from the itegral equatio from the itegral equatio f cos d e. [N/D4,N/D6][A], s f si s d, s., s [M/J4,A/M5,N/D5] [A] UNIT-V Z-TRANSFORMS AND DIFFERENCE EQUATIONS PART-A ) Defie Z-trasform of the sequece. [R] Solutio: Let be a sequece defied for,,,......, the the two-sided Z- trasform f of the sequece Z f is defied as f F f f, where is a comple variable. If f is a casual sequece, the the -trasform reduces to oe-sided Z-trasform ad its defiitio is f F f Z. ) State the fial value theorem i Z-trasform.[N/D5][R] Solutio: If f t F lim f t lim F ) Fid Z. t Z, the.[m/j, N/D4][U] Solutio: Z Z...6

64 4) If Z, the fid Z Solutio: Z Z 5) Fid Z isi Solutio: Z cos isi. [M/J6][U] Z Z Z Zcos isi cos. [M/J6][U] cos si Z cos i Zsi i. cos cos 6) State dampig rule related to Z-trasform ad the fid Solutio: Dampig rule: If Z f F, the Za f Z f Z a Z a. Z a a a a. Z a a a a. [M/J6][A] a a a a a a. 7) Fid Z.[N/D][U] Solutio: Z log log log Z log. log.6

65 .6 8) Fid the Z-trasform of.[a/m5,n/d5][u] Solutio:.... Z log log log log. 9) Fid! a Z i Z-trasform. [N/D 9, M/J ].[U] Solutio:!! a a Z!! a a a e a a a......!!!. ) Fid the Z-trasform of!. [N/D,M/J6][U] Solutio:!! Z!! e...!!!. ) Fid the Z-trasform of.[u] Solutio: Z... ) Fid the Z-trasform of a. [A/M,N/D].[U]

66 Solutio: Z a a a a a a a a a... a ) Fid the Z-trasform of.[u] Solutio: Z Z 4) Fid Z e i a t Z usig Z-trasform.[A] Solutio: i at i at Z e Z e. Z Z Z i a T e i at e at e i. i a T e 5) Fid Ze t t si. [N/D5][A] si T e t Solutio: Ze t Z si t 6) If F t Z e si t e T T e si T T e cost 4 4 Solutio: Give F 4 4 By iitial value theorem, we have f lim F lim.64 si T cost, fid f. [N/D9].[A] 4 4 T e

67 lim lim ) Fid the iverse Z-trasform of Solutio: Let X X. is a pole of order R where R Re s X lim. [N/D] [U] is the sum of residue of d! d. X lim d d lim 8) State covolutio theorem of Z-trasform. [M/J4,A/M5,A/M5,N/D6][R] Solutio: If w is the covolutio of two sequeces ad, the Z w W Z. Z y. 9) Form a differece equatio by elimiatig arbitrary costat from Solutio: Give u A u A u A u u Elimiatig A u A. A u 4 A u (). () from ad, we have 4 4u u u u y. u A. [C] [N/D,N/D5]..65

68 ) Form the differece equatio geerated by Solutio: Give y y a b a b () y ab y y ab a b () ab y ab a4b () Elimiatig a ad b from (), () ad (), we have y y y y 4 y 4 y 4 y y y y y y 4. [A/M].[C] ) Solve y y give y Solutio: Give y y. [N/D][A] Takig Z-trasform o both sides of the above equatio, we have y Z y y Y Y Y Y Y Y implies Z y y Z.. ) Defie uit step sequece. Write its Z-trasform.[R] Solutio: The uit step sequece give by Z u. 6) Fid the Z-trasform of u is defied as u. [M/J4] [A] d d Solutio: Z Z. Z d d for for. Its Z-trasform is.66

69 ) Prove that Z a f f. [N/D4] (OR) If X a X a Z. [A/M5] [E] a Solutio: By defiitio, Z a f a f f f a Z, the show that f 8) If Z f f, the prove that Z f f Z Solutio: f f Z put u u a., u ad, a. [N/D4][E] u u u u f f u f u f u f f u u u PART-B ) State ad prove the secod shiftig theorem i Z-trasform.[M/J][R] ) State ad prove Iitial ad fial value theorem i Z-trasform.[N/D4][R] ) If Z f F, fid Z f k ad Z f k. [N/D][U] 4) Fid the Z-trasforms of the sequeces f ad g.[a] 5) Fid Z. [M/J].[U] 6) Prove that Z log.[e] 7) Fid the Z-trasform of.[u] 8) Fid the Z-trasform of.[n/d][u] ( )( ) 9) Fid Z. [N/D5][U] ) Fid the Z-trasform of for. [N/D4,N/D6][U]..67

70 ) Fid, if f Z f ) Fid [U] Z cos ad hece deduce Z cos.[m/j][u] ) Fid the Z-trasforms of cos cos ad. [M/J6][U] 4) Fid the Z-trasform of ad a cos.[u] 5) Fid the Z-trasform of ad.[u] 6) Fid the Z-trasform of ad hece fid Z-trasform of.[u] 7) Fid the Z-trasform of ad. 8) Fid the Z-trasforms of si ad cos. [N/D ].[A] 4 4 9) Fid Z a si. [A/M].[U] ) Fid the Z-trasforms of a cos ad e at si bt. [A/M].[U] ) Fid the Z - trasform of cos ad si. Hece deduce the Z -trasforms of cos ad a si. [N/D].[U] ) Fid the Z-trasform of r cos ad e t cos bt. [M/J4][U] Z si. [N/D4][U] ) Fid a cos a Z cos ad 4) Fid the Z-trasform of cos t ad t 5) Fid () Z ad () Z e t 6) Fid Z r si, Z a a cos t e t. [N/D6][U]. [N/D4][U].[A/M5][U] 4 a. [A/M5][U] 7) Fid Zr cos ad Z 8) Fid Z usig partial fractio.[a] ( ) ( ) ( ) 9) Fid Z by usig the method of partial fractio.[a] ( )( ) ) Fid Z by the method of partial fractios.[a] ( )( 4) ) Fid the iverse Z-trasform of. [N/D9].[U] ) Fid the iverse Z-trasform of 4. [N/D9].[U] cos

71 ) Fid the iverse Z-trasform of 4) Fid Z ad 5) Fid Z.[U] Z.[U] 6) Fid the iverse Z-trasform of 7) Evaluate Z 5 for 8) Fid the iverse Z-trasform of 9) Usig residue method, fid Z 5. [N/D].[U] ( ) ( ) 4) Usig comple residue theorem evaluate.[a/m].[u], usig partial fractio. [N/D4][A] by residue method. [N/D].[U]. [A/M5,M/J6][A] Z 9. [A/M5,N/D6][A] 4) Usig the iversio itegral method (Residue Theorem), fid the iverse Z-trasform of U 4.[N/D5][A] 4) State the iitial value theorem. Use it to fid u, u, u ad u, where Z 5 4 U. 4 [N/D5,M/J6][A] 4) If U, fid the value of u, u ad u.[n/d5][a] 44) State ad prove covolutio theorem o Z-trasformatio. Fid Z ( a)( b) [M/J,M/J4,A/M5,N/D5,N/D6][R] 45) Usig covolutio theorem evaluate iverse Z-trasform of. ( )( ) [A/M,N/D].[A].69

72 46) Usig covolutio theorem fid Z. [N/D].[A] ( a) 47) Usig covolutio theorem evaluate Z. [M/J6][A] a 48) Usig covolutio theorem fid the iverse Z-trasform of ) Fid the iverse Z-trasform of by covolutio theorem. [M/J,N/D4] [A] ( )(4 ) 5) Usig covolutio theorem, fid Z 4.7. [N/D 9,N/D5].[A] 5) By covolutio theorem, prove that the iverse Z-trasform of a b b a b a.[a] 5) Usig covolutio theorem, fid the iverse Z-trasform of 4.[A] is.[a/m][a] 5) Usig covolutio theorem, fid Z. [A/M5,M/J6] [A] 4 54) Usig Z-trasform method solve y y give that y y. [M/J6][A] 55) Usig Z-trasform solve differece equatio y ( ) 4y( ) 4y( ) give that y ( ), y ( ).[A] 56) Usig Z-trasform solve y ( ) y( ) 4y( ), give that y ( ) ad y ( ). [A] 57) Solve the differece equatio y ( ) y( ) y( ) give that y ( ) 4, y ( ) ad y ( ) 8, by the method of Z-trasform. [A/M,N/D,N/D4].[A] 58) Solve 59) Solve y give y y 6y 9y, usig Z-trasform. [N/D9,N/D,M/J6,N/D6].[A] y 4y y with y ad y, usig Z-trasform. [N/D,N/D5].[A] 6) Solve u 4u u with u ad u. [N/D,A/M5,N/D5].[A] 6) Solve the differece equatio y y y, give y y usig Z-trasform.[N/D][A] 6) Solve by Z-trasform u u u with u ad u. [A/M].[A] 6) Usig Z-trasform, solve y 4y 5y 4 8 give that y ad y 5.[A] 64) Solve the differece equatio y 4 y 4 y, give y ad y, by

73 the method of Z-trasform.[A] 65) Solve the differece equatio y k yk, [M/J][A] 66) Usig Z-trasform solve y y y, y( ), y ad y( ) usig Z-trasform. y.[m/j, M/J4] [A] 67) Solve y y. usig Z-trasform. [M/J].[A] 68) Solve u u u 4, give that u, u. [M/J4][A] 69) Usig Z-trasforms solve u u u give that u, u. [N/D4,A/M5][A] y A B 7) Form the differece equatio whose solutio is.[n/d][c] 7) Derive a differece equatio by elimiatig the costats from 7) Derive the differece equatio from y A B y A B. [M/J6][C]. [A/M].[C] 7) Form the differece equatio of secod order by elimiatig the arbitrary costats A ad B from y A B. [N/D].[C] 74) Form the differece equatio from the relatio y a b. [N/D]. [C] ASSIGNMENT UNIT-I PARTIAL DIFFERENTIAL EQUATIONS. Form the PDE by elimiatig the arbitrary fuctio from the relatio y f log y.[c] q. Fid the sigular solutio of the equatio p q y p.[u] p y y p y q y.[a]. Solve y y si h y. [A] 4. Solve y 5. Solve D D D D D D y si y. [A] UNIT-II FOURIER SERIES. Obtai the Fourier series for.... [U] 6 f ( ) i (, ). Deduce that.7

74 . Fid the half rage sie series of f ( ) cos i (, ). Hece deduce the value of the series [U] 4. Fid the Fourier series of period as far as the first harmoic to represet the fuctio defied by the followig table: [A] [U]. Fid the half-rage sie series of f 4 i the iterval y f,4 y Fid the comple form of the Fourier series of e, l l costat other tha a iteger cos si l l l. Deduce that whe is i e l. [U] UNIT-III APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS. A uiform elastic strig of legth 6cms is subjected to a costat tesio of the eds fied ad the iitial displacemet y, 6, 6 velocity is ero, fid the displacemet fuctio y, t. [U]. A tightly stretched strig with fied ed poits lcm ad l Kg. If, while the iitial is iitially displaced to the form si cos ad the released. Fid the displacemet of the strig at l l ay distace from oe ed at ay time t.[u]. The eds A ad B of a rod log have their temperatures kept at ad, util steady state coditios prevail. The temperature of the ed B is suddely reduced to 6 c ad that of A is icreased to 4 c. Fid the temperature distributio i the rod after time t.[a] 4. A square plate is bouded by the lies, y, ad y. Its faces are isulated. The temperature alog the upper horiotal edge is give by u,, while the other edges are kept at C. Fid the temperature distributio i the plate.[a] 5. A rectagular plate with isulated surfaces is 8 cm wide ad so log compared to its width that it may be cosidered as a ifiite plate. If the temperature alog short edge y is u, si, 8 8 other short edge are kept at plate.[a], while two log edges c 8 c ad 8 as well as the C. Fid the steady state temperature at ay poit of the.7

75 . Fid the Fourier cosie trasform of si cos cos d.[u] 6. Fid the Fourier cosie trasform of UNIT-IV FOURIER TRANSFORMS f ( ) e.[u]. Fid the Fourier trasform of the fuctio defied by ;. Hece prove that ; otherwise ; f ( ) ; that si s s cos s s cos ds ad si s s cos s s 6 s f [U] 4. Fid the Fourier trasform of 5. Fid the Fourier sie trasform of si ; f ( ).[U] ; UNIT-V Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Hece prove ds. [U] 5. Fid Z.[U]. Fid the Z-trasforms of si ad 4. Fid the iverse Z-trasform of 4 cos. [A] 4. [U] 4. Usig covolutio theorem, fid the iverse Z-trasform of 4. [A] 5. Usig Z-trasform, solve y 4y 5y 4 8 give that y ad y 5.[A].7

76 PANIMALAR ENGINEERING COLLEGE DEPARTMENTOFELECTRONICS&COMMUNICATION ENGINEERING EC6 OBJECTORIENTEDPROGRAMMING ad DATASTRUCTURES SOLVED TWOMARKS&PART-B QUESTIONS(FOR THIRD SEMESTER ECE)

77 EC6 - OBJECT ORIENTED PROGRAMMING AND DATA STRUCTURES UNIT-I DATA ABSTRACTION AND OVERLOADING PART - A. What is the output of the followig program, if it is correct? Otherwise idicate the mistake: [May6] (E) it l=; void mai [ ] {it l=; {it l=; cout<<l<<::l; }} The program is i correct due to syta error. Withi the mai fuctio, there is o eed of aother opeig braces i the it l=; ad also closig braces.. Differece betwee Class ad structure? [April -, Dec-] (R) Class is the ADT where as structure is user defied datatype. Class eeds access specifier such as private, public & private where as structure members ca be accessed by public by default & do t eed ay access specifiers. Class is oops where structure is borrowed from traditioal structured [pop] cocept.. What is abstract Class? [Nov-9] (R) A abstract class is a class that is desiged to be specifically used as a base class. A abstract class cotais at least oe pure virtual fuctio. You declare a pure virtual fuctio by usig a pure specifier [= ] i the declaratio of a virtual member fuctio i the class declaratio. 4. List out the advatages of ew operator over malloc[] [Dec-] (R) It automatically computes the sie of the data object. It automatically returs the correct poiter type It is possible to iitialie the objects while creatig the memory space. It ca be overloaded. 5. What are the basic cocepts of OOS? [ April -][Nov 9] (R) Objects. Classes. Data abstractio ad Ecapsulatio. Iheritace. Polymorphism. Dyamic bidig. Message Passig 6. What is the differece betwee local variable ad data member? [Nov-] (U) A data member belogs to a object of a class whereas local variable belogs to its curret scope. A local variable is declared withi the body of a fuctio ad ca be used oly from the poit at which it is declared to the immediately followig closig brace. A data member is declared i a class defiitio, but ot i the body of ay of the class member fuctios. Data members are accessible to all member fuctio of the class..

78 7. What is the fuctio parameter? Differece betwee parameter ad Argumet. [Nov-] (R) A fuctio parameter is a variable declared i the prototype or declaratio of a fuctio: void foo[it ]; // prototype -- is a parameter void foo[it ] // declaratio -- is a parameter { } A argumet is the value that is passed to the fuctio i place of a parameter. 8. What is data hidig? [April -,Nov-] (R) The isulatio of data from direct access by the program is called as data hidig or iformatio bidig. The data is ot accessible to the outside world ad oly those fuctios, which are wrapped i the class, ca access it. 9. What are the advatages of Default Argumets? [Nov-] (U) The fuctio assigs a default value to the parameter which does ot have a matchig argumet i the fuctio call. They are useful i situatios where some argumets always have the same value. e.g., float amt [float P, float, float r =.5];. What are abstract classes? [Nov 9, Apr ] (U) Classes cotaiig at least oe pure virtual fuctio become abstract classes. Classes iheritig abstract classes must redefie the pure virtual fuctios; otherwise the derived classes also will become abstract. Abstract classes caot be istatiated.. Defie abstractio ad Ecapsulatio [Apr ] (U) Data Abstractio Abstractio refers to the act of represetig the essetial features without icludig the backgroud details or eplaatios. Data Ecapsulatio The wrappig up of data ad fuctios ito a sigle uit is kow as data ecapsulatio.. What is the Need for Static Members [April ] (U) Class members ca be declared usig the storage class specifier static i the class member list. Oly oe copy of the static member is shared by all objects of a class i a program. Whe you declare a object of a class havig a static member, the static member is ot part of the class object.. Defie Polymorphism. [Apr ] (U) Polymorphism is aother importat oops cocept. Polymorphism meas the ability to take more tha oe form. For eample, a operatio may ehibit differet behavior i differet istaces. Behavior depeds upo the types of data used i the operatio. 4. What do you mea by pure virtual fuctios? [Dec8] (U) A pure virtual member fuctio is a member fuctio that the base class forces derived classes to provide. Ay class cotaiig ay pure virtual fuctio caot be used to create object of its ow type. 5. What is fuctio Prototype? [DEC ] (U) A fuctio prototype or fuctio iterface i C, Perl, PHP or C++ is a declaratio of a fuctio that omits the fuctio body but does specify the fuctio's retur type, ame ad argumet types. While a fuctio defiitio specifies what a fuctio does, a fuctio prototype ca be thought of as specifyig its iterface. 6. List out four Storage Classes i C++ [Nov 8] (AZ).

79 Storage classes are used to specify the lifetime ad scope of variables. How storage is allocated for variables ad how variable is treated by complier depeds o these storage classes. These are basically divided ito 5 differet types :. Global variables. Local variables. Register variables 4. Static variables 5. Eter variables 7. What is a idetifier? (R) Idetifiers are ames for various programmig elemets i c++ program. such as variables, arrays, fuctio, structures, uio, labels ect., A idetifier ca be Composed oly of uppercase, lower case letter, uderscore ad digits, but should start oly with a alphabet or a uderscore. 8. What is a keyword? (R) Keywords are word whose meaigs have bee already defied i the c compiler. They are also called as reserved words. (eg) mai(), if, else, else, if, scaf, pritf, switch, for, goto, while ect., 9. List out the beefits of oops. (A) Ca create ew programs faster because we ca reuse code Easier to create ew data types Easier memory maagemet Programs should be less bug-proe, as it uses a stricter syta ad type checkig. `Data hidig', the usage of data by oe program part while other program parts caot access the data. List out the applicatio of oops. (A) Cliet server computig Simulatio such as flight simulatios. Object-orieted database applicatios. Artificial itelligece ad epert system Computer aided desig ad maufacturig systems.. Defie data hidig. (AZ) The purpose of the eceptio hadlig mechaism is to provide a meas to detect ad report a eceptioal circumstaceǁ so that appropriate actio ca be take.. What is the use of scope resolutio operator? (U) I C, the global versio of the variable caot be accessed from withi the ier block. C++ resolves this problem by itroducig a ew operator :: called the scope resolutio operator. It is used to ucover a hidde variable. Syta: :: variable ame. Whe will you make a fuctio ilie? (E) Whe the fuctio defiitio is small, we ca make that fuctio a ilie fuctio ad we ca maily go for ilie fuctio to elimiate the cost of calls to small fuctios. 4. What is overloadig? (U) Overloadig refers to the use of the same thig for differet purposes. There are types of overloadig: Fuctio overloadig Operator overloadig.4

80 5. What is the differece betwee ormal fuctio ad a recursive fuctio? (U) A recursive fuctio is a fuctio, which call it whereas a ormal fuctio does ot. Recursive fuctio ca t be directly ivoked by mai fuctio 6. What are objects? How are they created? (U) Objects are basic ru-time etities i a object-orieted programmig system. The class variables are kow as objects. Objects are created by usig the syta: classame obj,obj,,obj; (or) durig defiitio of the class: class classame { }obj,obj,,obj; 7. List some of the special properties of costructor fuctio. (E) They should be declared i the public sectio. They are ivoked automatically whe the objects are created. They do ot have retur types, ot eve void ad caot retur values. Costructors caot be virtual. Like other C++ fuctios, they ca have default argumets 8. Describe the importace of destructor. (E) A destructor destroys the objects that have bee created by a costructor upo eit from the program or block to release memory space for future use. It is a member fuctio whose ame is the same as the class ame but is preceded by a tilde. Syta: ~classame(){ } 9. What do you mea by fried fuctios? (U) C++ allows some commo fuctios to be made friedly with ay umber of classes, thereby allowig the fuctio to have access to the private data of thse classes. Such a fuctio eed ot be a member of ay of these classes. Such commo fuctios are called fried fuctios.. What are member fuctios? (U) Fuctios that are declared withi the class defiitio are referred as member fuctio.. Defie dyamic bidig.[ NOV/DEC 5] (U) Dyamic bidig meas that the code associated with a give procedure call is ot kow util the time of the call at ru-time..write ay four properties of costructor.[dec ) Costructors should be declared i the public sectio. They are ivoked automatically whe the objects are created. They do ot have retur types They caot be iherited. (U).List ay four Operators that caot be overloaded.(dec ) (DEC 9) (DEC ) ( E) Class member access operator (.,.*) Scope resolutio operator (::) Sie operator ( sieof ) Coditioal operator (?:) 4.What is a Destructor? (DEC )[ NOV/DEC 5] (U) A destructor is used to destroy the objects that have bee created by a costructor. It is a special member fuctio whose ame is same as the class ad is preceded by a tilde ~ symbol. Whe a object goes out from object creatio, automatically destructor will be eecuted..5

81 Eample: class File { public: ~File(); //destructor declaratio }; File::~File() { close(); // destructor defiitio } 5.What is the Need for iitialiatio of object usig Costructor? (DEC ) (U) If we fails to create a costructor for a class, the the compiler will create a costructor by default i the ame of class ame without havig ay argumets at the time of compilatio ad provides the iitial values to its data members. So we have to iitialie the objects usig costructor. 6. Give a eample for a Copy Costructor (JUNE ) (C) #iclude<iostream> #iclude<coio.h> usig amespace std; class Eample { //Variable Declaratio it a,b; public: //Costructor with Argumet Eample(it,it y) { // Assig Values I Costructor a=; b=y; cout<<"\im Costructor"; } void Display() { cout<<"\values :"<<a<<"\t"<<b; } }; it mai() { Eample Object(,); //Copy Costructor Eample Object=Object; // Costructor ivoked. Object.Display(); Object.Display(); // Wait For Output Scree getch(); retur ; } 7. What is the Need for Destructors? (Jue ) (C) Destructor is used to destroy a object. By destroyig the object memory occupied by the object is released. 8. Eplai the fuctios of Default Costructor(MAY ) (E) The mai fuctio of the costructor is, if the programmer fails to create a costructor for a class, the the compiler will create a costructor by default i the ame of class ame without havig ay argumets at the time of compilatio ad provides the iitial values to its data members. Sice it is created by the compiler by default, the o argumet costructor is called as default costructor. 9.What is the eed for Overloadig a operator(may ) To defie a ew relatio task to a operator, we must specify what it meas i relatio to the class to which the operator is applied. This is doe with the help of a special fuctio called operator fuctio. It allows the developer to program usig otatio closer to the target domai ad allow user types to look like types built ito the laguage. (E).6

82 The ability to tell the compiler how to perform a certai operatio whe its correspodig operator is used o oe or more variables. 4.What is the fuctio of get ad put fuctio (MAY ) Ci.get(ch) reads a character from ci ad stores what is read i ch. Cout.put(ch) reads a character ad writes to cout. (U) 4. What is referece variable?(april/may 5) (R) c++ refereces allow you to create a secod ame for the a variable that you ca use to read or modify the origial data stored i that variable. 4.What is fried fuctio??(april/may 5) (R) A fried fuctio of a class is defied outside that class' scope but it has the right to access all private ad protected members of the class. Eve though the prototypes for fried fuctios appear i the class defiitio, frieds are ot member fuctios. PART - B ) Classify the eeds for object orieted paradigm? (U) ) Eplai i detail of object orieted cocepts. (U) ) Eplai the characteristics of OOPS i detail. (U) 4) List the features of object orieted programmig. (A) 5) Eplai the structure of C++ program. (U) 6) Eplai various cotrol statemets used i C++. (U) 7) Eplai Do-While with a eample. (U) 8) Compare ilie fuctios of C++ with ordiary fuctios ad with macros. (U) 9) Elaborate dyamic bidig? How is it achieved? (C) ) Criticie short otes o call ad Retur Referece. (E) ) Distiguish New ad Delete Operators. With ad eample. (AZ) ) Develop a C++ program to implemet dyamic memory allocatio. (A) ) Origiate the cocept of ADT with some illustrative eample. (C) 4)Eplai i detail about data hidig. (U) 5) Eplai i detail about class, objects, methods ad messages. (U) 6) Eplai fried fuctio with a eample. (U) 7) Illustrate the cocept of fuctio overloadig to fid the maimum of two umbers. (U) 8) Justify about operator overloadig with a eample. (E) 9)Evaluate the list of rules for overloadig operators with oe eample. (E) PART- C ) Develop a C++ program that will ask for a temperature i Fahreheit ad display.(c) ) Desig calculator usig fuctio overloadig. (C) ) Develop a C++ program to implemet C=A+B, C=A-B ad C-A*B where A, B ad C are objects cotaiig a it value (vector). (C).7

83 4)Develop a program to cocateate two strigs usig + operator overloadig. (C) 5) Develop a program usig operator overloadig to add two time values i the format HH:MM:SS to the resultig time alog with roudig off whe 4 hrs is reached. A time class is created ad operator + is overloaded to add the two time class objects. (C) 6) Develop a program to perform multiplicatio usig a iteger ad object. Use fried fuctio. (C).8

84 Uit II INHERITANCE AND POLYMORPHISM PART - A. What are templates? (AUC DEC 9) (U) Template is oe of the features added to C++. It is ew cocepts which eable us to defie geeric classes ad fuctios ad thus provides support for geeric programmig. A template ca be used to create a family of classes or fuctios.. Illustrate the eceptio hadlig mechaism. (AUC DEC 9) (E) C++ eceptio hadlig mechaism is basically upo three keywords, amely, try, throw,ad catch. try is used to preface a block of statemets which may geerate eceptios. throw - Whe a eceptio is detected, it is throw usig a throw statemet i the try block. Catch- A catch block defied by the keyword catch catches the eceptio throw by the throw statemet i the try block ad hadles its appropriately.. What happes whe a raised eceptio is ot caught by catch block? (AUC MAY )(AZ) If the type of object throw matches the arg type i the catch statemet, the catch block is eecuted for hadlig the eceptio. If they do ot match the program is aborted with the help of abort() fuctio which is ivoked by default. Whe o eceptio is detected ad throw, the cotrol goes to the statemet immediately after the catch block. That is the catch block is skipped. 4. What is a template? What are their advatages(auc DEC /JUN /DEC ) (U) A template ca be used to create a family of classes or fuctios. A template is defied with a parameter that would be replaced by a specified data type at the time of actual use of the class or fuctios. Supportig differet data types i a sigle framework. The template are sometimes called parameteried classes or fuctios. 5. How is a eceptio hadled i C++? (AUC DEC ) (A) Eceptios are ru time aomalies or uusual coditios that a program may ecouter while eecutig.. Fid the problem (Hit the eceptio). Iform that a error has occurred.( Throw the eceptio). Receive the error iformatio. ( Catch the eceptio) 4. Take corrective actios.( Hadle the eceptio) 6. What is fuctio template? Eplai. (AUC MAY ) (A) Like class template, we ca also defie fuctio templates that could be used to create a family of fuctios with differet argumet types. The geeral format of a fuctio template is: template<class T> returtypefuctioame ( argumets of type T) { // //.Body of fuctio with type T wherever appropriate //.. } 7. List five commo eamples of eceptios. (AUC MAY ) (AZ) Divisio by ero Access to a array outside of its bouds.9

85 Ruig out of memory or disk space. 8. What is class template? (AUC DEC ) (A) Templates allows to defie geeric classes. It is a simple process to create a geeric class usig a template with a aoymous type. The geeral format of class template is: template<class T> classclassame { // class member specificatio. //. with aoymous type T wherever appropriate //,.. } 9. What are basic keywords of eceptio hadlig mechaism? (AUC DEC ) (R) C++ eceptio hadlig mechaism is basically built upo three keywords try throw catch. What are the c++ operators that caot be overloaded? (U) Sie operator (sieof) Scope resolutio operator (::) member access operators(.,.*) Coditioal operator (?:). What is a virtual base class? (U) Whe a class is declared as virtual c++ takes care to see that oly copy of that class is iherited, regardless of how may iheritace paths eist betwee the virtual base class ad a derived class.. What is the differece betwee base class ad derived class? (U) The biggest differece betwee the base class ad the derived class is that the derived class cotais the data members of both the base ad its ow data members. The other differece is based o the visibility modes of the data members..what are the rules goverig the declaratio of a class of multiple iheritace? (U) More tha oe class ame should be specified after the : symbol. Visibility modes must be take care of. If several iheritace paths are employed for a sigle derived class the base class must be appropriately declared. 4. Metio the types of iheritace. (R).Sigle iheritace.. Multiple iheritace.. Hierarchical iheritace. 4. Multilevel iheritace. 5. Hybrid iheritace. 5. Defie dyamic bidig. (APRIL MAY ) (U) Dyamic bidig meas that the code associated with a give procedure call is ot kow util the time of the call at ru-time. 6. What do you mea by pure virtual fuctios? (U) A pure virtual fuctio is a fuctio declared i a base class that has o defiitio relative to the base class. I such cases, the compiler requires each derived class to either defie the fuctio or redeclare.

86 it as a pure virtual fuctio. A class cotaiig pure virtual fuctios caot be used to declare ay objects of its ow. 7. Metio the key words used i eceptio hadlig. (U) The keywords used i eceptio hadlig are throw try catch 8. List the ios format fuctio. (U) The ios format fuctios are as follows: width() precisio() fill() setf() usetf() 9. List the maipulators. (U) The maipulators are: setw() setprecisio() setfill() setiosflags() resetiosflags(). Defie fill fuctios. (R) The fill( ) fuctio ca be used to fill the uused positios of the field by ay desired character rather tha by white spaces (by default). It is used i the followig form: cout.fill(ch); where ch represets the character which is used for fillig the uused positios..give the syta of eceptio hadlig mechaism. (E) The syta of eceptio hadlig mechaism is as follows: try { throw eceptio } catch(type argumets) { } What is overloadig(april/may 5) (U) Overloadig is the cocept of Compile time Polymorphism. It does ot eed iheritace. Method ca have differet data types. Two fuctio havig same ame ad retur type, but with differet type ad/or umber of argumets is called as overloadig.. Why there is eed for operator overloadig? (April/May 5) (U) Most fudametal data types have pre-defied operators associated with them. For eample, the C++ data type it, together with the operators +, -, *, ad /, provides a implemetatio of the.

87 mathematical cocepts of a iteger. To make a user-defied data type as atural as a fudametal data type, the user-defied data type must be associated with the appropriate set of operators. It deals with overloadig of operators to make Abstract Data Types(ADTS) more atural, ad closer to fudametal data types. 4. Differetiate Private ad protected members of a class. [NOV/DEC 5] (U) Public variables, are variables that are visible to all classes. Private variables, are variables that are visible oly to the class to which they belog. Protected variables, are variables that are visible oly to the class to which they belog, ad ay subclasses. PART - B ) Discuss the characteristics of costructors ad destructors. (C) ) What are costructors? Eplai the cocept of destructor with a eample. (U) ) Eplai the costructors ad destructors i detail with a eample program. (U) 4) Eplai data ecapsulatio ad iheritace i detail. (U) 5) Eplai with eamples, the types of iheritace i C++. (E) 6)Eamie how ca you pass parameters to the costructors of base classes i multiple iheritaces? (AZ) 7) What is iheritace? List out the advatages of iheritace. (AZ) 8) Eplai the order i which the costructors are called whe a object of derived class is created. (U) 9) Eplai about implemetatio of ru time polymorphism i C++ with a eample. (U) ) What is a abstract class? What is dyamic bidig? How is it achieved? (U) ) What is the differece betwee a virtual fuctio ad a pure virtual fuctio? Give eample of each. (U) ) What is multiple iheritaces? Discuss the syta ad rules of multiple iheritaces i C++. (U) ) Eplai about dyamic method dispatch with a eample (U) PART C )Develop a program to illustrate multiple costructors ad default argumet for a sigle class. (C) )Develop a C++ program to defie overload costructor to perform strig iitialiatio, strig copy ad strig destructio. (C) )Write a C++ program to illustrate the cocept of hierarchical iheritace. (C) 4)Write a C++ program to implemet multilevel iheritace (C) 5)Write a C++ program to implemet multiple iheritace (C) 6)Write a C++ program to geerate user defied eceptio user iputs odd umbers.(c) 7)State the rules for virtual fuctios. Write a C++ program to declare a virtual fuctio &demostrate (C).

88 UNIT III LINEAR DATA STRUCTURES PART A. Write dow the defiitio o f data structures? NOV DEC A data structure is a mathematical or logical way of orgaiig data i the memory that cosider ot oly the items stored but also the relatioship to each other ad also it is characteried by accessig fuctios.. Biary Heap NOV DEC, APRIL MAY 9 The implemetatio we will use is kow as a biary heap. Its use is so commo for priority queue implemetatios that whe the word heap is used without a qualifier Structure Property A heap is a biary tree that is completely filled, with the possible eceptio of the bottom level, which is filled from left to right. Such a tree is kow as a complete biary tree. Defie Algorithm? Algorithm is a solutio to a problem idepedet of programmig laguage. It cosist of set of fiite steps which, whe carried out for a give set of iputs, produce the correspodig output ad termiate i a fiite time. 4. What are the features of a efficiet algorithm? Free of ambiguity Efficiet i eecutio time Cocise ad compact Completeess Defiiteess Fiiteess 5. List dow ay four applicatios of data structures? Compiler desig Operatig System Database Maagemet system Network aalysis 6. What is meat by a abstract data type (ADT)? A ADT is a set of operatio. A useful tool for specifyig the logical properties of a datatype is the abstract data type.adt refers to the basic mathematical cocept that defies the datatype. Eg.Objects such as list, set ad graph alog their operatios ca be viewed as ADT's. 7. What are the operatios of ADT? Uio, Itersectio, sie, complemet ad fid are the various operatios of ADT. 8. What is meat by list ADT? List ADT is a sequetial storage structure. Geeral list of the form a, a, a.., a ad the sie of the list is ''. Ay elemet i the list at the positio I is defied to be ai, ai+ the successor of ai ad ai- is the predecessor of ai. 9. What are the two parts of ADT? Value defiitio Operator defiitio. What is a Sequece? A sequece is simply a ordered set of elemets.a sequece S is sometimes writte as the eumeratio of its elemets,such as S =If S cotais elemets,the legth of S is.. Defie le(s),first(s),last(s),ilseq? le(s) is the legth of the sequece S. first(s) returs the value of the first elemet of S last(s) returs the value of the last elemet of S ilseq :Sequece of legth is ilseq.ie., cotais o elemet..

89 . What are the two basic operatios that access a array? Etractio: Etractio operatio is a fuctio that accepts a array, a,a ide,i,ad returs a elemet of the array. Storig: Storig operatio accepts a array, a,a ide i, ad a elemet.. Defie Structure? A Structure is a group of items i which each item is idetified by its ow idetifier,each of which is kow as a member of the structure. 4. Defie Uio? Uio is collectio of Structures,which permits a variable to be iterpreted i several differet ways. 5. Defie Automatic ad Eteral variables? Automatic variables are variables that are allocated storage whe the fuctio is ivoked. Eteral variables are variables that are declared outside ay fuctio ad are allocated storage at the poit at which they are first ecoutered for the remeider of the program s eecutio. 6. What is a Stack? NOV DEC 8 A Stack is a ordered collectio of items ito which ew items may be iserted ad from which items may be deleted at oe ed, called the top of the stack. The other ame of stack is Last-i -First-out list. 7. What are the two operatios of Stack? _ PUSH _ POP 8. What is a Queue? APR/MAY 5 A Queue is a ordered collectio of items from which items may be deleted at frot ed ad iserted at rear ed 9. What is a Priority Queue? NOV DEC Priority queue is a data structure i which the itrisic orderig of the elemets does determie the results of its basic operatios. Ascedig ad Descedig priority queue are the two types of Priority queue.. What is a liked list? Liked list is a kid of series of data structures, which are ot ecessarily adjacet i memory. Each structure cotai the elemet ad a poiter to a record cotaiig its successor.. What is a doubly liked list? I a simple liked list, there will be oe poiter amed as 'NEXT POINTER' to poit the et elemet, where as i a doubly liked list, there will be two poiters oe to poit the et elemet ad the other to poit the previous elemet locatio.. Defie double circularly liked list? I a doubly liked list, if the last ode or poiter of the list, poit to the first elemet of the list,the it is a circularly liked list.. What is a circular queue? The queue, which wraps aroud upo reachig the ed of the array is called as circular queue. 4. Defie ma ad mi heap? A heap i which the paret has a larger key tha the child's is called a ma heap. A heap i which the paret has a smaller key tha the child is called a mi heap 5.What is ADT? (April/May 5) A data type ca be cosidered abstract whe it is defied i terms of operatios o it, ad its implemetatio is hidde. is a mathematical model for data types where a data type is defied by its behavior (sematics) from the poit of view of a user of the data, specifically i terms of possible values, possible operatios o data of this type, ad the behavior of these operatios..4

90 PART B. Eplai the isertio ad deletio operatio i sigly liked list. [Comprehesio]. Eplai array based implemetatio of list with a eample program. [Comprehesio]. Give sigly liked list whose first ode is poited to by the poiter variable C formulate a algorithm to delete the first occurrece of X from the list ad Isert the elemet X after the positio P i the list..[ Comprehesio] 4. Write the ADT operatio for isertio ad deletio routie i stack. [Sythesis] 5. Eplai the process of postfi, prefi, ifi epressio evaluatio with a eample. [Comprehesio] 6. Give a procedure to covert a ifi epressio a+b*c+(d*e+f)*g to postfi otatio.[ Comprehesio] 7. Write a routie to isert a elemet i a liked list. [Sythesis] 8. Eplai the process of coversio from ifi epressio to postfi usig stack. [Comprehesio] 9. Write the ADT operatio for isertio ad deletio routie i liked lists & Queue. [Sythesis] PART C. Eplai about Liked list, its Types, isertio ad deletio routies with suitable eample. [Comprehesio]. Eplai the process of postfi, prefi, ifi epressio evaluatio with a eample. [Comprehesio]. What is a queue? Write a algorithm to implemet queue.[ Kowledge] 4. Eplai the implemetatio stack usig liked list. [Comprehesio].5

91 UNIT IV NON- LINEAR DATA STRUCTURES PART A. Defie o-liear data structure? Data structure which is capable of epressig more comple relatioship tha that of physical adjacecy is called o-liear data structure.. Defie tree? A tree is a data structure, which represets hierarchical relatioship betwee idividual Data items.. Defie child ad paret of a tree. The root of each subtree is said to be a child of r ad r is the paret of each subtree root. 4. Defie leaf? I a directed tree ay ode which has out degree o is called a termial ode or a leaf. 5. What is a Biary tree? NOV/DEC 5 A Biary tree is a fiite set of elemets that is either empty or is partitioed ito three disjoit subsets. The first subset cotais a sigle elemet called the root of the tree. The other two subsets are themselves biary trees called the left ad right sub trees. 6. What are the applicatios of biary tree? Biary tree is used i data processig. a. File ide schemes b. Hierarchical database maagemet system 7. What is meat by traversig? Traversig a tree meas processig it i such a way, that each ode is visited oly oce. 8. What are the differet types of traversig? The differet types of traversig are a.pre-order traversal-yields prefi form of epressio. b. I-order traversal-yields ifi form of epressio. c. Post-order traversal-yields postfi form of epressio. 9. What are the two methods of biary tree implemetatio? Two methods to implemet a biary tree are, a. Liear represetatio. b. Liked represetatio. Defie Graph? A graph G cosist of a oempty set V which is a set of odes of the graph, a set E which is the set of edges of the graph, ad a mappig from the set for edge E to a set of pairs of elemets of V.It ca also be represeted as G=(V, E).. Defie adjacet odes? Ay two odes which are coected by a edge i a graph are called adjacet odes. For Eample, if ad edge ÎE is associated with a pair of odes (u,v) where u, v Î V, the we say that the edge coects the odes u ad v..name the differet ways of represetig a graph? a. Adjacecy matri b. Adjacecy list. What are the two traversal strategies used i traversig a graph? a. Breadth first search b. Depth first search 4. What is a acyclic graph? A simple diagram which does ot have ay cycles is called a acyclic graph..6

92 5. Give some eample of NP complete problems. Hamiltoia circuit. Travellig salesme problems 6. WhatAVL treares?the algorithms used to fid the miimum spaig tree? A AVL Prim treeis algorithmbiarysearch tree with a balacig coditio.for every ode i the tree the height of the left ad right subtrees ca differ at most by.the height of Kruskal s algorithmof empty tree is defied to be -.It esures that the depth of the tree is O(log N) 7.What is topological sort? A topological sort is a orderig of vertices i a directed acyclic graph,such that if there is a path from vi the vj appears after vi i the orderig. 8.What is sigle source shortest path problem? Give as a iput a weighted graph, G=(V,E) ad a distiguished verte, s fid the shortest weighted path from s to every other verte i G. 9.Metio some shortest path problems Uweighted shortest paths Dijikstra s algorithm All-pairs shortest paths. Defie complete biary tree. It is a complete biary tree oly if all levels, ecept possibly the last level have the maimum umber of odes maimum.a complete biary tree of height h has betwee h ad h+ ode.defie biary search tree. Why it is preferred rather tha the sorted liear array ad liked list? Biary search tree is a biary tree i which key values of the left sub trees are lesser tha the root value ad the key values of the right sub tree are always greater tha the root value. I liear array or liked list the values are arraged i the form of icreasig or decreasig order. If we wat to access ay elemet meas, we have to traverse the etire list. But if we use BST, the elemet to be accessed is greater or smaller tha the root elemet meas we ca traverse either the right or left sub tree ad ca access the elemet irrespective of searchig the etire tree.. various implemetatios of trees. Liear implemetatio Liked list implemetatio. List out the applicatio of trees. Ordered tree Cǁ represetatio of tree 4 Show the maimum umber of odes i a biary tree of height H is H+. Cosider H = No. of odes i a full biary tree = H+ = + = 4 = 5 odes We kow that a full biary tree with height h= has maimum 5 odes. Hece proved 5. A + (B-C)*D+(E*F), if the above arithmetic epressio is represeted usig a biary tree, Fid the umber of o-leaf odes i the tree. Epressio tree is a biary tree i which o leaf odes are operators ad the leaf odes are operads. I the above eample, we have 5 operators. Therefore the umber of o-leaf odes i the tree is Write a algorithm to declare odes of a tree structure. (AUC APR / MAY) Struct tree ode { it elemet; Struct tree *left; Struct tree *right; } 7.How Graph is represeted? (April/May 5) Graph Data Structure. Verte Each ode of the graph is represeted as a verte. A graph is a.7

93 pictorial represetatio of a set of objects where some pairs of objects are coected by liks. The itercoected objects are represeted by poits termed as vertices, ad the liks that coect the vertices are called edges.formally, a graph is a pair of sets (V, E), where V is the set of vertices ad E is the set of edges, coectig the pairs of vertices. Take a look at the followig graph I the above graph, V = {a, b, c, d, e} E = {ab, ac, bd, cd, de} 8.What is a Tree? (April/May 5) Tree represets odes coected by edges. I computer sciece, a tree is a widely used abstract data type (ADT) or data structure implemetig this ADT that simulates a hierarchical tree structure, with a root value ad subtrees of childre with a paret ode, represeted as a set of liked odes. 9. Whe a graph is said to be coected? NOV/DEC 5 A graph is coected whe there is a path betwee every pair of vertices. I a coected graph, there are o ureachable vertices. A graph that is ot coected is discoected. A graph with just oe verte is coected. A edgeless graph with two or more vertices is discoected.. How array elemets are sorted usig merge sort? NOV/DEC 5 A merge sort works as follows:. Divide the usorted list ito sublists, each cotaiig elemet (a list of elemet is cosidered sorted).. Repeatedly merge sublists to produce ew sorted sublists util there is oly sublist remaiig. This will be the sorted list PART B. Describe the isertio deletio with eample. [Comprehesio]. Draw the biary search tree for the followig iput list 6, 5,75,5,5,66,,44. Trace a algorithm to delete the odes 5, 75, 44 from the tree. [Aalysis].8

94 . Eplai the operatios of isertio of odes ito ad deletio of odes from, a biary search tree with code. [Comprehesio] 4. Eplai the two applicatios of trees with a eat eample. [Comprehesio] 5. How do you isert a elemet i a biary search tree? [ Kowledge] 6. What are the graph traversal methods? Eplai it with eample. [ Kowledge] 7. Give a algorithm to fid miimum Spaig tree, eplai it with suitable eample.[ Comprehesio] 8. Eplai coected compoet. [Comprehesio] PART C. Eplai the operatios of isertio of odes ito ad deletio of odes from, a biary search tree with code. [Comprehesio]. Write a algorithm for fidig miimum spaig tree ad eplai applicatio, illustrate the algorithm with typical data of yours ow eample. [Sythesis]. Give the aalysis of isertio ad deletio operatios of odes i biary search tree.[ Comprehesio] 4. Eplai the algorithm for depth first search ad breadth first search with the followig graph. [Comprehesio].9

95 UNIT V SORTING AND SEARCHING PART A. What is maheap? Apr/May, If we wat the elemets i the more typical icreasig sorted order, we ca chage the orderig property so that the paret has a larger key tha the child. it is called ma heap.. What is divide ad coquer strategy? I divide ad coquer strategy the give problem is divided ito smaller problems ad solved recursively. The coquerig phase cosists of patchig together the aswers. Divide ad coquer is a very powerful use of recursio that we will see may times.. Differetiate betwee merge sort ad quick sort? Apr/May, Mergesort. Divide ad coquer strategy. Partitio by positio Quicksort. Divide ad coquer strategy. Partitio by value 4. Metio some methods for choosig the pivot elemet i quicksort?. Choosig first elemet. Geerate radom umber. Media of three 5. What are the three cases that arise durig the left to right sca i quicksort? I ad j cross each other I ad j do ot cross each other I ad j poits the same positio 6. What is the eed of eteral sortig? Eteral sortig is required where the iput is too large to fit ito memory. So eteral sortig is ecessary where the program is too large. It is a basic eteral sortig i which there are two iputs ad two outputs tapes. 7. What is replacemet selectio? We read as may records as possible ad sort them. Writig the result to some tapes. This seems like the best approach possible util oe realies that as soo as the first record is writte to a output tape the memory it used becomes available for aother record. If the et record o the iput tape is larger tha the record we have just output the it ca be icluded i the item. Usig this we ca give algorithm. This is called replacemet selectio. 8. What is sortig? APR/MAY 5 Sortig is the process of arragig the give items i a logical order. Sortig is a eample where the aalysis ca be precisely performed. 9. What is mergesort? The mergesort algorithm is a classic divide ad coquer strategy. The problem is divided ito two arrays ad merged ito sigle array. What are the p roperties ivolved i heapsort? Apr/May,. Structure property. Heap order property. D efie articulatio poits. If a graph is ot bicoected, the vertices whose removal would discoect the graph are kow as articulatio poits. What is time compleity? APR/MAY 5.

96 The time compleity of a algorithm quatifies the amout of time take by a algorithm to ru as a fuctio of the legth of the strig represetig the iput. PART B. Write the sequece of sortig o,, 4,7,5,9,,6,5 usig Isertio sort. [Sythesis]. Eplai the operatio ad implemetatio of eteral sortig. [Comprehesio]. Write dow the merge sort algorithm ad give its worst case, best case ad average case aalysis. [Sythesis] 4. Eplai the Quick sort algorithm with eample. [Comprehesio] 5. Discuss i detail about quick sort algorithm. [Sythesis] 6. Eplai liear search algorithm with a eample. [Comprehesio] 7. Discuss the differetiatio of liear search algorithm with biary search algorithm. [Sythesis] PART C. Write quick sort algorithm ad eplai. [Sythesis]. Eplai liear search & biary search algorithm i detail. [Comprehesio]. Eplai the operatio ad implemetatio of merge sort. [Comprehesio] 4. Write the quick sort algorithm for the followig list of umbers. 9,77,6,99,55,88,66. [Sythesis].

97 PANIMALAR ENGINEERING COLLEGE DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6 DIGITAL ELECTRONICS SOLVED TWO MARKS AND PART B &C QUESTIONS.

98 UNIT I - MINIMIZATION TECHNIQUES AND LOGIC GATES PART- A ) What are the basic properties of Boolea algebra? (R) The basic properties of Boolea algebra are commutative property, associative Property ad distributive property. ) State the associative property of Boolea algebra. (R) The associative property of Boolea algebra states that the OR ig of several variables results i the same regardless of the groupig of the variables. The associative property is stated as follows: A+(B+C)=(A+B)+C. ) State the commutative property of Boolea algebra. (R) The commutative property states that the order i which the variables are ORed makes o differece. The commutative property is: A+B=B+A 4) State the distributive property of Boolea algebra. (R) The distributive property states that AND ig several variables ad OR ig the result with a sigle variable is equivalet to OR ig the sigle variable with each of the several variables ad the AND ig the sums. The distributive property is: A+BC= (A+B) (A+C) 5) State the absorptio law of Boolea algebra. (R) The absorptio law of Boolea algebra is give by X+XY=X, X(X+Y) =X. 6) Simplify the followig usig De Morga's theorem [((AB)'C)'' D]' (AZ) [((AB)'C)'' D]' = ((AB)'C)'' + D' [(AB)' = A' + B'] = (AB)' C + D' = (A' + B ) C + D' 7) State De Morga's theorem. (R) De Morga suggested two theorems that form importat part of Boolea algebra. They are) the complemet of a product is equal to the sum of the complemets. (AB)' = A' + B' ) The complemet of a sum term is equal to the product of the complemets. (A + B)' = A'B' 8) Simplify A.A'C (AZ) A.A'C =.C [A.A' = ] = 9) Simplify A(A + B) (AZ) A(A + B) = AA + AB = A( + B) [ + B = ] = A..

99 ) Simplify A'B'C' + A'BC' + A'BC (AZ) A'B'C' + A'BC' + A'BC = A'C'(B' + B) + A'B'C = A'C' + A'BC [A + A' = ] = A'(C' + BC) = A'(C' + B) [A + A'B = A + B] ) Simplify AB + (AC)' + AB C (AB + C) (AZ) AB + (AC)' + AB C (AB + C) = AB + (AC)' + AAB'BC + AB'CC = AB + (AC)' + AB'CC [A.A' = ] = AB + (AC)' + AB'C [A.A = ] = AB + A' + C' =AB'C [(AB)' = A' + B'] = A' + B + C' + AB'C [A + AB' = A + B] = A' + B'C + B + C' [A + A'B = A + B] = A' + B + C' + B'C =A' + B + C' + B' =A' + C' + = [A + =] ) Simplify the followig epressio Y = (A + B) (A + C' )(B' + C' ) Y = (A + B)(A + C' )(B' + C' ) (AZ) = (AA' + AC +A'B +BC) (B' + C') [A.A' = ] = (AC + A'B + BC) (B' + C ) = AB'C + ACC' + A'BB' + A'BC' + BB'C + BCC' = AB'C + A'BC' ) Show that (X + Y' + XY)( X + Y')(X'Y) = (U) (X + Y' + XY) (X + Y')(X'Y) = (X + Y' + X)(X + Y' )(X' + Y) [A + A'B = A + B] = (X + Y ) (X + Y ) (X'Y) [A + A = ] = (X + Y ) (X'Y) [A.A = ] = X.X' + Y'.X'.Y = [A.A' = ] 4) Prove that ABC + ABC' + AB'C + A'BC = AB + AC + BC (E) ABC + ABC' + AB'C + A'BC=AB(C + C') + AB'C + A'BC.

100 =AB + AB'C + A'BC =A(B + B'C) + A'BC =A(B + C) + A'BC =AB + AC + A'BC =B(A + C) + AC =AB + BC + AC =AB + AC +BC...Proved 5) Covert the give epressio i caoical SOP form Y = AC + AB + BC (C) Y = AC + AB + BC =AC (B + B ) + AB (C + C ) + (A + A') BC =ABC + ABC' + AB'C + AB'C' + ABC + ABC' + ABC =ABC + ABC' +AB'C + AB'C' [A + A =] 6) Defie duality property. (R) Duality property states that every algebraic epressio deducible from the postulates of Boolea algebra remais valid if the operators ad idetity elemets are iterchaged. If the dual of a algebraic epressio is desired, we simply iterchage OR ad AND operators ad replace 's by 's ad 's by 's. 7) Fid the complemet of the fuctios F = 'y' + 'y' ad F = (y'' + y). (R) By applyig De-Morga's theorem. F' = ('y' + 'y')' = ('y')'('y')' = ( + y' + )( + y +') F' = [(y'' + y)]' = ' + (y'' + y)' = ' + (y'')'(y)' = ' + (y + ) (y' + ') 8) Simplify the followig epressio. (AZ) Y = (A + B) (A = C) (B + C) = (A A + A C + A B + B C) (B + C) = (A C + A B + B C) (B + C) = A B C + A C C + A B B + A B C + B B C + B C C = A B C.4

101 9) What are the methods adopted to reduce Boolea fuctio? (R) i) Karaugh map ii) Tabular method or Quie Mc-Cluskey method iii) Variable etered map techique. ) State the limitatios of karaugh map. (R) i) Geerally it is limited to si variable map (i.e.) more the si variable ivolvig epressio are ot reduced. ii) The map method is restricted i its capability sice they are useful for Simplifyig oly Boolea epressio represeted i stadard form. ) What is a karaugh map? (R) A karaugh map or k map is a pictorial form of truth table, i which the map diagram is made up of squares, with each squares represetig oe miterm of the fuctio. ) Fid the miterms of the logical epressio Y = A'B'C' + A'B'C + A'BC + ABC' (R) Y = A'B'C' + A'B'C + A'BC + ABC' =m + m +m +m6 =Σm (,,, 6) ) Write the materms correspodig to the logical epressio. (R) Y = (A + B + C' )(A + B' + C')(A' + B' + C) = (A + B + C' )(A + B' + C')(A' + B' + C) =M.M.M6 = M (,, 6) 4) What are called do t care coditios? (R) I some logic circuits certai iput coditios ever occur, therefore the Correspodig output ever appears. I such cases the output level is ot defied, it ca be either high or low. These output levels are idicated by X or d i the truth tables ad are called do t care coditios or icompletely specified fuctios. 5) What is a prime implicat? (R) A prime implicat is a product term obtaied by combiig the maimum possible umber of adjacet squares i the map. 6) What is a essetial implicat? (R) If a mi term is covered by oly oe prime implicat, the prime implicat is said to be essetial..5

102 7) What is a Logic gate? (R) Logic gates are the basic elemets that make up a digital system. The electroic gate is a circuit that is able to operate o a umber of biary iputs i order to perform a particular logical fuctio. 8) Give the classificatio of logic families. (R) Bipolar i) Saturated. TTL. ECL. I I C 4. RTL ii) No Saturated. Schottky TTL. DTL Uipolar. PMOS. NMOS. CMOS 9) What are the basic digital logic gates? (R) The three basic logic gates are AND gate OR gate NOT gate. ) Which gates are called as the uiversal gates? What are its advatages? (R) The NAND ad NOR gates are called as the uiversal gates. These gates are used to perform ay type of logic applicatio. ) Classify the logic family by operatio? (R) The Bipolar logic family is classified ito Saturated logic Usaturated logic. The RTL, DTL, TTL, I L, HTL logic comes uder the saturated logic family. The Schottky TTL, ad ECL logic comes uder the usaturated logic family..6

103 ) Metio the importat characteristics of digital IC s? (R) Fa out Power dissipatio Propagatio Delay Noise Margi Fa I Operatig temperature Power supply requiremets. ) Defie Fa-out? (R) Fa out specifies the umber of stadard loads that the output of the gate ca drive without impairmet of its ormal operatio. 4) Defie power dissipatio? (R) Power dissipatio is measure of power cosumed by the gate whe fully drive by all its iputs. 5) What is propagatio delay? (R) Propagatio delay is the average trasitio delay time for the sigal to propagate from iput to output whe the sigals chage i value. It is epressed i s. 6) Defie oise margi? (R) It is the maimum oise voltage added to a iput sigal of a digital circuit that does ot cause a udesirable chage i the circuit output. It is epressed i volts. 7) Defie fa i? (R) Fa i is the umber of iputs coected to the gate without ay degradatio i the voltage level. 8) What is High Threshold Logic? (R) Some digital circuits operate i eviromets, which produce very high oise sigals. For operatio i such surroudigs there is available a type of DTL gate which possesses a high threshold to oise immuity. This type of gate is called HTL logic or High Threshold Logic. 9) What are the types of TTL logic? (R). Ope collector output. Totem-Pole Output. Tri-state output. 4) Metio the characteristics of MOS trasistor? (R).7

104 . The - chael MOS coducts whe its gate- to- source voltage is positive.. The p- chael MOS coducts whe its gate- to- source voltage is egative. Either type of device is tured of if its gate- to- source voltage is ero. 4) How schottky trasistors are formed ad state its use? (R) A schottky diode is formed by the combiatio of metal ad semicoductor. The presece of schottky diode betwee the base ad the collector prevets the trasistor from goig ito saturatio. The resultig trasistor is called as schottky trasistor. The use of schottky trasistor i TTL decreases the propagatio delay without a sacrifice of power dissipatio. 4) Why totem pole outputs caot be coected together. (R) Totem pole outputs caot be coected together because such a coectio might produce ecessive curret ad may result i damage to the devices. 4) State advatages ad disadvatages of TTL (R) Adv: Easily compatible with other ICs Low output impedace Disadv: Wired output capability is possible oly with tristate ad ope collector types Special circuits i Circuit layout ad system desig are required. 44) What happes to output whe a tristate circuit is selected for high impedace. (R) Output is discoected from rest of the circuits by iteral circuitry. 45) Implemet the Boolea Epressio for EX OR gate usig NAND Gates. (R) 46) Defie Miterm & Materm. (R) A miterm of variables is a product (logical AND) of the variables i which each appears eactly oce i true or complemeted form. A materm of variables is a sum (Logical OR) of the variables i which each appears eactly oce i true or complemeted form. 47) List out the advatages ad disadvatages of Quie-Mc Cluskey method? (R) The advatages are, i) This is suitable whe the umber of variables eceed four. ii). Digital computers ca be used to obtai the solutio fast. iii)essetial prime implicats, which are ot evidet i K-map, ca be clearly see i the fial results..8

105 The disadvatages are, i) Legthy procedure tha K-map. ii) Requires several groupig ad steps as compared to K-map. iii) It is much slower. iv) No visual idetificatio of reductio process. v) The Quie Mc Cluskey method is essetially a computer reductio method 48) What is the sigificace of high impedace state i tri-state gates? (R) I digital electroics three-state, tri-state, or -state logic allows a output port to assume a high impedace state i additio to the ad logic levels, effectively removig the output from the circuit. This allows multiple circuits to share the same output lie or lies (such as a bus which caot liste to more tha oe device at a time). 49) What is totem pole output? (R) Output stage cosists of pull up trasistors, diode resistor ad pull dow trasistor. I this output of two gates caot be tied together. Whose operatig speed is also high. 5) Draw the logic diagram of OR gate usig uiversal gates. (R) PART-B & C ) Eplai the operatio of CMOS NAND ad NOR Gate with the circuits ad truth Table (6) [Nov/Dec ] (U) ) Simplify the followig Boolea fuctio usig 4 variable map F(w,,y,) = Σ (,,,,,,4,5) (8) [Nov/Dec ](AZ) F(w,,y,) = Σ (,,,4,5,6,8,9,,,4) () [Apr/May ] ) Draw a NAND logic diagram that implemets the complemets of the followig fuctio. F(A,B,C,D) = Σ (,,,,4,8,9,) (8) [Nov/Dec ](A) 4) Usig QM method simplify the Boolea epressio. f(v,w,,y,) = Σ (,,4,5,6,7,,5,9) (6) [Nov/Dec ](AZ). П M(,,4,,,5) + П d(5,7,8) ad verify the result usig K-map method. [Apr/May ]. F(A,B,C,D) = Σ M(,,,6,7,8,,,) [Apr/May ].9

106 4. f(a,b,c,d) = Σ m(,,4,5,9,,) + Σd(6,8) ad realie usig NAND gates.(6) [Nov/Dec ] 5.F= Σ (,,,8,,,4,5) (8) [May/Jue ] 6.F (A,B,C,D) = Σm (,,,4,5,7,,,4,5) 5) Epress the Boolea fuctio i POS form ad SOP form (C) i. F = (A + B )(B + C ) (4) [Apr/May ] ii. F= A = B C (6) [Apr/May ] 6) Implemet the followig fuctio usig NOR gates. (AZ) (8) [Apr/May ] Output = whe the iputs are Σm(,,,,4) = whe the iputs are Σm(5,6,7) 7) Discuss the geeral characteristics of TTL ad CMOS logic families.(8) [Apr/May ]U 8) Draw the schematic ad eplai the operatio of a CMOS iverter. Also eplai its characteristics. (8) [Apr/May ] U 9) Draw a TTL circuit with totem pole output ad eplai its workig.(8) [Nov/Dec ] U ) Simplify the followig Boolea epressio F = y + y +y +y.(az)(8) [May/Jue ] ) Differetiate betwee Mi term ad Ma term. U (4) [May/Jue ] )Usig K-map simplify the followig epressios ad implemet usig basic gates.. F = Σ (,,4,6). F = Σ (,,7,,5) + d(,,5) AZ [May/Jue ]() )Simplify the give Boolea fuctio usig tabulatio method F=Σm(,,,5,7,9,,,,5) AZ [Nov/Dec ](6) 4) Draw the circuit diagram of a two iput TTL NAND gate with tri-state output ad eplai its actio,clearly showig logic ad voltage levels.(6) [Nov/Dec ] U 5) Prove that (+).(. +) ( +.) = (8) E 6) Simplify usig K-map to obtai a miimum POS epressio: AZ (A +B +C+D)(A+B +C+D)(A+B+C+D )(A+B+C +D ) (A +B+C +D )(A+B+C +D) 7)Reduce the followig equatio usig Quie McClucky method of miimiatio F (A,B,C,D) = Σm (,,,4,5,7,,,4,5) AZ (6) 8) Usig a K-Map, Fid the MSP from of F= Σ (, 4, 8,,, 7,, 5) +d (5).

107 Usig a K-Map, Fid the MSP form of F= Σ (-, -5) + d (7, ) U 9)Simplify the followig usig the Quie McClusky miimiatio techique D = f(a,b,c,d) = Σ(,,,,6,7,8,9,4,5).Does Quie McClusky take care of do t care coditios? I the above problem, will you cosider ay do t care coditios? Justify your aswer AZ )List also the prime implicats ad essetial prime implicats for the above case (8) R ) Determie the MSP ad MPS focus of F= Σ (,,6,8,,,4,5) (8) E ) State ad Prove Demorga s theorem (8) [May/Jue ] R ) Determie the MSP form of the Switchig fuctio E F( a,b,c,d) =Σ (,,4,6,8)+Σd(,,,,4,5) 4) Fid the Mi term epasio of f(a,b,c,d) =a (b +d)+acd (8) R 5)Fid a Mi SOP ad Mi POS for f = b c d + bcd + acd + a b c + a bc d (6) R 6) Fid a epressio for the followig fuctio usig Quie McCluscky method F= Σ (,,,5,7,9,,,4,6,8,4,6,8,) (6) R 7)Fid the MSP represetatio for F(A,B,C,D,E) = Σm(,4,6,,,,4,6)+ Σd (,,6,7) usig K- Map method, Draw the circuit of the miimal epressio usig oly NAND gates (6) U 8)i. Show that if all the gates i a two NAND gates the output fuctio does ot chage. level AND-OR gate etworks are replaced by (8) U ii) Why does a good logic desiger miimie the use of NOT gates? (8) U 9)Simplify the Boolea fuctio F(A,B,C,D) = Σ m (,,7,,5) + Σd (,,5).if do t care coditios are ot take care, What is the simplified Boolea fuctio.what are your commets o it? Implemet both circuits (6) AZ ) i.implemet Y = (A+C) (A+D ) ( A+B+C ) usig NOR gates oly (8) AZ ii) Fid a etwork of AND ad OR gate to realie f(a,b,c,d) = Σ m (,5,6,,,4) (8) ) i) What is the advatages of usig tabulatio method? (8) E ii) Determie the prime implicats of the followig fuctio usig tabulatio method F(W,X,Y,Z) = Σ (,4,6,7,8,9,,,5) (8 ) ) Eplai the operatio of I/P TTL NAND Gate with required diagram ad truth table. (6) U )Covert the followig fuctio ito product of Ma terms. F(A,B,C) = (A+B ) (B+C) (A+C ). (4) [Nov/Dec 4]C.

108 4) Usig Quie McClusky method simplify the give fuctio. F(A,B,C,D) = Σ(,,,5,7,9,,,4) AZ() Nov/Dec 4] 5)Draw the multiplicad two iput NAND circuit for the followig epressio: F = (AB +CD )E + BC(A+B). (4) U 6) Draw ad eplai Tri state TTL iverter circuit diagram ad eplai its operatio.() U ASSISGNMENT. Defie aalog, discrete time ad digital sigals. R. What are some of the advatages digital systems compared to aalog systems? R. Covert the followig umbers as required i each case. 4 = ( ) 5.65 = ( ) 6. = ( ) ABCD 6=( ) 5C.8 6=( ) R 4. Eplai the differece betwee positive logic ad egative logic. U 5. What are uiversal gates? Why are they called so? R 6. Use Boolea Algebra to show that A BC +AB C +AB C+ABC + ABC = A+BC AZ 7. Usig -variable K - Map simplify the Boolea fuctio give by AZ F (a, b, c) = Σm (,, 4, 5, 6, 7) 8. Usig 4-variable K-Map simplify the Boolea fuctio give by AZ F (w,, y, ) = Σm (,,,,,, 4, 5) 9. Usig K Map simplify i the product-of-sum form the fuctio give by AZ F(A,B,C,D} = ΠM (, 6,, ). I all digital systems ad circuits two logic levels are defied, HIGH ad LOW, sometimes called ad. Takig the eample of two voltage levels, i positive logic, HIGH level is deoted by higher voltage ad LOW level is deoted by lower voltage, e.g., 5 volts ad volts. I egative logic, o the other had, HIGH level is deoted by lower voltage ad LOW level by higher voltage, e.g., volts ad 5 volts. I other words, the two are dual of each other. For eample, a AND gate i positive logic will represet a OR gate i egative logic. C. NAND ad NOR gates are called uiversal gates. Normally ay logic circuit will have AND, OR, INVERT gates. These three gates are required to implemet ay logic fuctio. Idividually either NAND or NOR gate ca implemet AND, OR, INVERT gates ad hece either oe of them (NAND or NOR) ca implemet ay logic fuctio. C.

109 UNIT II - COMBINATIONAL CIRCUITS PART-A ) Defie combiatioal logic. (R) Whe logic gates are coected together to produce a specified output for certai specified combiatios of iput variables, with o storage ivolved, the resultig circuit is called combiatioal logic. ) Eplai the desig procedure for combiatioal circuits. (R) The problem defiitio Determie the umber of available iput variables & required O/P variables. Assigig letter symbols to I/O variables Obtai simplified Boolea epressio for each O/P. Obtai the logic diagram. ) Defie Half adder ad full adder. (R) The logic circuit that performs the additio of two bits is a half adder. The circuit that performs the additio of three bits is a full adder. 4) Defie Decoder? (R) A decoder is a multiple - iput multiple output logic circuit that coverts coded iputs ito coded outputs where the iput ad output codes are differet. 5) What is biary decoder? (R) A decoder is a combiatioal circuit that coverts biary iformatio from iput lies to a maimum of out puts lies. 6) Defie Ecoder? (R) A ecoder has iput lies ad output lies. I ecoder the output lies geerate the biary code correspodig to the iput value. 7) What is priority Ecoder? (R) A priority ecoder is a ecoder circuit that icludes the priority fuctio. I priority ecoder, if or more iputs are equal to at the same time, the iput havig the highest priority will take precedece. 8) Defie multipleer? (R) Multipleer is a digital switch. If allows digital iformatio from several sources to be routed oto a sigle output lie. 9) What do you mea by comparator? (R) A comparator is a special combiatioal circuit desiged primarily to compare the relative magitude of two biary umbers..

110 ) Give the applicatios of De-multipleer. (R) De-multipleers are used i digital processig, telecommuicatios, istrumetatio ad computer architecture etc ) Give other ame for Multipleer ad Demultipleer. (R) Multipleer is a data selector, Demultipleer is a data distributer. ) List out the applicatios of multipleer? (R). It ca be used to realie a Boolea fuctio. It ca be used i commuicatio systems e.g., time divisio multipleig.. Data routig 4. Logic fuctio geerator 5. Cotrol sequecer 6. Parallel-to-serial coverter ) What is multipleer? (R) A multipleer (or mu) is a device that selects oe of several aalog or digital iput sigals ad forwards the selected iput ito a sigle lie. A multipleer of iputs has select lies, which are used to select which iput lie to sed to the output 4) Distiguish betwee a decoder ad a demultipleer. (U) A demu simply selects a output lie, othig more. It's a glorified switch. A decoder takes iputs, ad uses those iputs to determie which of the output lies is high. This is the differece, A decoder is desiged to simply keep oe lie high. A demu is desiged to set oe output equal to the iput (whether it be high, low, or a chagig sigal). 5) Write a epressio for borrow ad differece i a full subtractor circuit. (R) D = y + y + y + y B = + y + y 6) Draw the circuit diagram for 4 bit Odd parity geerator. (R) 7) Write a Epressio for borrow ad differece i a half subtractor circuit. (R) D = y + y B = y 8) Relate carry geerate, carry propagate, Sum ad carry out of a carry look ahead adder.(az) Pi = Ai Bi Gi = Ai Bi Si = Pi Ci Ci+ = Gi + PiCi.4

111 9) Distiguish betwee decoder ad ecoder. (AZ) Decoder Ecoder Oe of the iput lies is activated correspodig to the biary iput The iput lies geerate the biary code, correspodig to the iput value Iput of the decoder is a ecoded Iput of the ecoder is a decoded iformatio preseted as iput iformatio preseted as iputs producig possible outputs producig outputs. The iput code geerally has a fewer bits tha the output code. The iput code geerally has more bits tha the output code ) Metio the uses of decoders. (R). Decoders are used i couter system.. They are used i aalog to digital coverter.. Decoder outputs ca be used to drive a display system. 4. Used i code coverters. ) Give the applicatios of Demultipleer. (R). It fids its applicatio i Data trasmissio system with error detectio.. Oe simple applicatio is biary to Decimal decoder ) What will be the maimum umber of outputs for a decoder with a 6-bit data word? (R) For a 6-bit data word the maimum umber of output for a decoder is, 6 = outputs. ) Metio the uses of decoders. (R). Decoders are used i couter system.. They are used i aalog to digital coverter.. Decoder outputs ca be used to drive a display system. 4. Used i code coverters. 4) What is BCD adder? (R) A BCD adder is a circuit that adds two BCD digits i parallel ad produces a sum digit also i BCD. PART-B & C. Implemet the switchig fuctio F= Σ (,,,4,7) usig a 4 iput MUX ad eplai (6) AZ. Eplai how will build a 64 iput MUX usig ie 8 iputmuxs (6) U. Implemet the followig fuctio usig suitable Multiplier F(A,B,C,D) = Σ (,,,4,,4,5) (8) [Apr/May ]AZ.5

112 F(A,B,C,D) = Σ (,,,4,8,9,5) (b) [No/Dec 4] 4. Eplai how will build a 6 iput MUX usig oly 4 iput MUXs (6) U 5. Eplai the operatio of4 to lie decoder with ecessary logic diagram (6)U 6. Desig full adder ad full sub tractor. (8) C 7. Desig a 4 bit magitude comparator to compare two 4 bit umber (6) [May/Jue ] C 8. Desig a bit magitude comparator ad eplai its operatio i detail. (6) [Nov/Dec ] C 9. Costruct a combiatioal circuit to covert give biary coded decimal umber ito a Ecess code for eample whe the iput to the gate is the the circuit should geerate output as (6) C.Usig a sigle 748, draw the logic diagram of a 4 bit adder/sub tractor (8) U.Realie a Biary to BCD coversio circuit startig from its truth table (8) C.Desig a combiatioal circuit which accepts bit biary umber ad coverts its equivalet ecess code (6) C.Eplai carry look ahead adder. (8)[Apr/May ] U 4. Desig a combiatioal circuit of -digit BCD adder. (8) [Apr/May ] [May/Jue ] [Nov/Dec ] [Nov/Dec ].C 5.Draw ad eplai the block diagram of a 4 bit serial adder to add the cotets of two registers.()[ Apr/May ] U 6. Desig ad implemet coversio circuit for Biary to Gray code.(8)[apr/may ]C 7.Multiply () by () usig additio ad shiftig operatio also draw block diagram of the 4 bit serial adder to add the cotets of two registers.(8) [Apr/May ]U 8.Desig a full adder usig two half adder ad a OR gate.(6) [Apr/May ] C 9.Draw the logic diagram of a bit biary multiplier ad eplai its operatio.(8) [Apr/May ] U.Desig a combiatioal circuit that produces the product of biary umber A= (A, A) B =(B,B,B). () [Nov/Dec ]C. Desig 4 bit parallel adder/ subtractor ad draw the logic diagram.(8) [Nov/Dec ] C.Desig a 4bit BCD to ecess - code coverter. Draw the logic diagram.(6) [Nov/Dec ]. C.Costruct a four bit eve parity geerator circuit usig gates.(4) [Nov/Dec 4] C.6

113 UNIT III SEQUENTIAL CIRCUITS PART A. Defie Flip flop. (R) The basic uit for storage is flip flop. A flip-flop maitais its output state either at or util directed by a iput sigal to chage its state.. What are the differet types of flip-flop? (R) There are various types of flip flops. Some of them are metioed below they are, i. RS flip-flop ii. SR flip-flop iii. D flip-flop iv. JK flip-flop v. T flip-flop.. What is the operatio of RS flip-flop? (R) Whe R iput is low ad S iput is high the Q output of flip-flop is set. Whe R iput is high ad S iput is low the Q output of flip-flop is reset. Whe both the iputs R ad S are low the output does ot chage. Whe both the iputs R ad S are high the output is upredictable. 4. What is the operatio of D flip-flop? (R) I D flip-flop durig the occurrece of clock pulse if D=, the output Q is set ad if D=, the output is reset. 5. What is the operatio of JK flip-flop? (R) Whe K iput is low ad J iput is high the Q output of flip-flop is set. Whe K iput is high ad J iput is low the Q output of flip-flop is reset. Whe both the iputs K ad J are low the output does ot chage Whe both the iputs K ad J are high it is possible to set or reset the flipflop (ie) the output toggle o the et positive clock edge. 6. What is the operatio of T flip-flop? (R) T flip-flop is also kow as Toggle flip-flop. Whe T= there is o chage i the output. Whe T= the output switch to the complemet state (ie) the output toggles. 7. Defie race aroud coditio. (R) I JK flip-flop output is fed back to the iput. Therefore chage i the output results chage i the iput. Due to this i the positive half of the clock pulse if both J ad K are high the output toggles cotiuously. This coditio is called race aroud coditio. 8. What is edge-triggered flip-flop? (R).7

114 The problem of race aroud coditio ca solved by edge triggerig flip flop. The term edge triggerig meas that the flip-flop chages state either at the positive edge or egative edge of the clock pulse ad it is sesitive to its iputs oly at this trasitio of the clock 9. What is a master-slave flip-flop? (R) A master-slave flip-flop cosists of two flip-flops where oe circuit serves as a master ad the other as a slave.. Defie rise time. (R) The time required to chage the voltage level from % to 9% is kow as rise time(tr).. Defie fall time. (R) The time required to chage the voltage level from 9% to % is kow as fall time(tf).. Defie skew ad clock skew. (R) The phase shift betwee the rectagular clock waveforms is referred to as skew ad the time delay betwee the two clock pulses is called clock skew.. Defie setup time. (R) The setup time is the miimum time required to maitai a costat voltage levels at the ecitatio iputs of the flip-flop device prior to the triggerig edge of the clock pulse i order for the levels to be reliably clocked ito the flip flop. It is deoted as setup. 4. Defie hold time. (R) The hold time is the miimum time for which the voltage levels at the ecitatio iputs must remai costat after the triggerig edge of the clock pulse i order for the levels to be reliably clocked ito the flip flop. It is deoted as thold. 5. Defie propagatio delay. (R) A propagatio delay is the time required to chage the output after the applicatio of the iput. 6. Defie registers. (R) A register is a group of flip-flops flip-flop ca store oe bit iformatio. So a -bit register has a group of flip-flops ad is capable of storig ay biary iformatio/umber cotaiig -bits. 7. Defie shift registers. (R) The biary iformatio i a register ca be moved from stage to stage withi the register or ito or out of the register upo applicatio of clock pulses. This type of bit movemet or shiftig is essetial for certai arithmetic ad logic operatios used i microprocessors. This gives rise to group of registers called shift registers. 8. What are the differet types of shift registers? (R).8

115 There are five types. They are, _ i. Serial I Serial Out Shift ii. Serial I Parallel Out iii. Shift Register Parallel I iv. Serial Out Shift Register v. Parallel I Parallel out Shift vi. Bidirectioal Shift Register. 9. Eplai the flip-flop ecitatio tables for RS FF. (R) RS flip-flop I RS flip-flop there are four possible trasitios from the preset state to the et state. They are, i. trasitio: This ca happe either whe R=S= or whe R= ad S=. ii. trasitio: This ca happe oly whe S= ad R=. iii. trasitio: This ca happe oly whe S= ad R=. iv. trasitio: This ca happe either whe S= ad R= or S= ad R=.. Eplai the flip-flop ecitatio tables for JK flip-flop. (R) I JK flip-flop also there are four possible trasitios from preset state to et state. They are, i. trasitio: This ca happe whe J= ad K= or K=. ii. trasitio: This ca happe either whe J= ad K= or whe J=K=. iii. trasitio: This ca happe either whe J= ad K= or whe J=K=. iv. trasitio: This ca happe whe K= ad J= or J=.. Defie sequetial circuit? (R) I sequetial circuits the output variables depedet ot oly o the preset iput variables but they also deped up o the past history of these iput variables.. Give the compariso betwee combiatioal circuits ad sequetial circuits. U Combiatioal circuits Sequetial circuits Memory uit is ot required Parallel adder is a combiatioal circuit Memory uity is required Serial adder is a sequetial circuit.9

116 . Defie sychroous sequetial circuit. (R) I sychroous sequetial circuits, sigals ca affect the memory elemets oly at discrete istat of time. 4. Defie Asychroous sequetial circuit? (R) I asychroous sequetial circuits chage i iput sigals ca affect memory elemet at ay istat of time. 5. Give the compariso betwee sychroous & Asychroous sequetial circuits? (R) Sychroous sequetial circuits Asychroous sequetial circuits. Memory elemets are clocked flip-flops Memory elemets are either ulocked flip -flops or time delay elemets. Easier to desig More difficult to desig. 6. Draw the logic diagram for SR latch usig two NOR gates. (U) 7. The followig wave forms are applied to the iputs of SR latch. Determie the Q waveform Assume iitially Q =. (U) Here the latch iput has to be pulsed mometarily to cause a chage i the latch Output state ad the output will remai i that ew state eve after the iput pulse is over. 8. Give the compariso betwee sychroous & Asychroous couters. (U) Asychroous couters I this type of couter flip-flops are coected i such a way that output of Sychroous couters I this type there is o coectio betwee output of first flip-flop ad st flip-flop drives the clock for the et clock iput of the et flip - flop flipflop All the flip-flops are Not clocked simultaeously simultaeously.

117 9. The t pd for each flip-flop is 5 s. Determie the maimum operatig frequecy for MOD ripple couter.(e) f ma (ripple) = 5 5 s = 4 MHZ. What is a ripple couter? (R) A register that goes through a prescribed sequece of states upo the applicatio of iput pulse called a couter. A flip flop output trasitio serves as a source for triggerig other flip flops is called ripple couter.. Defie state diagram. (R) State diagram is a graphical represetatio of state table. A state is represeted by a circle, ad the trasitios betwee states are idicated by direct lie coectig the circles. This lie is labeled by two biary umbers (Iput /Output) separated by a slash.. What is the use of state diagram? (R) It gives a pictorial view of state trasitios ad is the form more suitable for huma iterpretatio of the circuit s operatio.. What is state table? (R) The time sequece of iputs, outputs,ad flip flop states ca be eumerated i a state table. It cosists preset state, iput, et state ad output sectios 4. What is a state equatio? (R) The behavior of a clocked sequetial circuit ca be described algebraically by meas of state equatios. It specifies the et state as a fuctio of the preset state ad iputs. 5. What is the classificatio of sequetial circuits? (R) Sychroous sequetial circuit ad Asychroous sequetial circuit 6. Metio ay two differeces betwee the edge triggerig ad level triggerig. (R) I level trigger mode, the iput sigal is sampled whe the clock sigal is either high or low whereas i edge trigger mode the iput sigal is sampled at risig or at the fallig edge, level triggerig is sesitive to glitches whereas edge trigger is o sesitive 7. Write dow the characteristic equatio of JK Flip flop. (R) Q(t+) = JQ + K Q 8. Realie T-FF from JK FF. (R) J ad K are havig a complemet iput (with each other) meas that is called Toggle (T) Flip flop. J = : K = 9. Draw the logic diagram of T flip flop usig JK flip flop. (U).

118 4. How do you elimiate the race aroud coditio i a JK Flip flop? (R) Race aroud coditio i a JK FF is elimiated by usig either edge triggered flip flop or master slave flip flop. 4. Draw the state table ad ecitatio table of T Flip flop. (R) PART-B & C ) Eplai the operatio of JK ad clocked JK flip-flops with suitable diagrams (6)U ) Draw the state diagram of a JK flip- flop ad D flip flop (6) U ) Desig ad eplai the workig of a sychroous mod couter (6) C 4) Desig ad eplai the workig of a sychroous mod 7 couter (6) C 5) Desig a sychroous couter with states,,,,,. Usig JK FF (8) C 6) Usig SR flip flops, desig a parallel couter which couts i the sequece (8)C,,,,,,. 7) Usig JK flip flops, desig a parallel couter which couts i the sequece (8)C,,,,,,. 8) Draw ad eplai Master-Slave JK flip-flop (8) [Nov/Dec 9] U 9) Draw as asychroous 4 bit up-dow couter ad eplai its workig (6) U ) Usig D flip flop,desig a sychroous couter which couts i the sequece (6) C,,,,,,,, )Desig a biary couter usig T flip flops to cout i the followig sequeces (i),,,,,,,, (ii),,,,, (6) C )Desig a bit sychroous couter usig JK Flip flop (8) [Nov/Dec 4] C )Draw ad eplai the operatio of four bit Johso couter (8) U 4)How will you covert a D flipflop ito JK flipflop? (8) [Nov/Dec 9] U.

119 5)Eplai i detail the operatio of a 4 bit biary ripple couter. (6) U 6)Desig a sequece detector which detect the sequece usig D flip flops (oe bit over lappig.(6) [Nov/Dec ] C 7) Desig a 4 bit bi-directioal shift register. (8) [Nov/Dec ] C 8) Eplai the differece betwee a state table, a characteristic table ad a ecitatio table,(4) [Nov/Dec 4] AZ 9) Desig a bit Johso couter ad eplai its operatio.(8) [N ov/dec ] C ) Eplai i detail the operatio of a 4 bit biary ripple couter.(6) [Nov/Dec 9] U )Costruct a clocked JK flip flop which is triggered at the positive edge of the clock pulse from a clocked SR flip flop cosistig of NOR gates (6) [Apr/May ] C )What is meat by Uiversal Shift Register? Eplai the priciple of operatio of 4 bit uiversal shift register.(6) [Apr/May ] [Nov/Dec ] U )How will you covert a D flipflop ito T flipflop? (6) [Nov/Dec 9] U 4)Desig a serial biary adder (8) [Apr/May ] C 5) Eplai the operatio of a BCD ripple couter with JK flip flops(8) [Apr/May ] U 6)Desig ad draw the output waveform of UP/DOWN couter usig JK FF.(6) [May/Ju ] C 7) Eplai the desig steps of Mod couter.(8) [May/Ju ] C 8) Eplai the operatio of shift ad rig couters.(8) [May/Ju ] U 9) Draw the logic diagram of master slave SR flip flop ad eplai the workig with truth table.() [Nov/Dec ]U )Desig a D flip flop usig JK FF ad eplai with its truth table.(6) [Nov/Dec ]C )Draw the logic diagram of 4 bit biary UP?DOWN sychroous couter ad eplai with its truth table. Also draw the timig diagram.(6) [Nov/Dec ]U )Provide the characteristic table, characteristic equatio ad ecitatio table of D flip flop ad JK flip flop.(6) [Nov/Dec ]C )With a eat state diagram ad logic diagram, desig ad eplai the sequece of states of BCD couter.(6) [Nov/Dec ]C 4)Desig a bit rig couter ad fid the mod of the desiged couter (6) [Nov/Dec ]C.

120 5) Desig a clocked sequetial machie usig T FF for the followig state diagram.(use straight biary assigmet)c ASSIGNMENT. Implemet the followig Boolea fuctio usig a sigle 4 to Multipleer. F (A,B,C,D) = Σ m (,,, 4, 6, 9,, 4 ) AZ. Implemet a Full Subtractor usig a to 8 Decoder. AZ. If Ai ad Bi are the two bits to be added i the ith stage of a multi-bit adder, Gi = Ai. Bi ad Pi = Ai EXOR Bi These parameters are used to predict Co of each stage if A ad B of the stage is kow usig the Ci of the first stage. The equatio Ci+ = Gi + Pi Ci is used to predict the carry out of ith stage if the carry i of that stage is kow, which i tur ca be predicted recursively from the first stage. U 4. Desig a circuit to geerate odd parity if the data is represeted with 4 bits.. Desig a -bit full adder with two half adders ad miimum umber of additioal gates. C 5. A 4-bit Serial-i, Parallel-out left shift-shift Register is loaded with a iitial bit patter. Write the sequece of bit patters as the clock is applied to the Shift Register if bits b ad b are coected to the serial iput through a EXOR gate. b) After how may clock cycles will the iitial patter retur? U 6. A uiversal shift register ca be used a) as a delay elemet to delay a serial bit stream by a fied umber of clock cycles, b) as serial to parallel coverter of a it stream, c) as a parallel to serial coverter of a bit stream. d) All of the above. U.4

121 UNIT- IV MEMORY DEVICES PART -A. Eplai ROM.(U) A read oly memory (ROM) is a device that icludes both the decoder ad the OR gates withi a sigle IC package. It cosists of iput lies ad m output lies. Each bit combiatio of the iput variables is called a address. Each bit combiatio that comes out of the output lies is called a word. The umber of distict addresses possible with iput variables is.. What are the types of ROM?.PROM.EPROM.EEPROM. Eplai PROM. (U) PROM (Programmable Read Oly Memory) It allows user to store data or program. PROMs use the fuses with material like ichrome ad polycrystallie. The user ca blow these fuses by passig aroud to 5 ma of curret for the period 5 to μs.the blowig of fuses is called programmig of ROM. The PROMs are oe time programmable. Oce programmed, the iformatio is stored permaet. 4. Eplai EPROM. (U) EPROM(Erasable Programmable Read Oly Memory) EPROM use MOS circuitry. They store s ad s as a packet of charge i a buried layer of the IC chip. We ca erase the stored data i the EPROMs by eposig the chip to ultraviolet light via its quart widow for 5 to miutes. It is ot possible to erase selective iformatio. The chip ca be reprogrammed. 5. Eplai EEPROM. (U) EEPROM(Electrically Erasable Programmable Read Oly Memory) EEPROM also use MOS circuitry. Data is stored as charge or o charge o a isulated layer or a isulated floatig gate i the device. EEPROM allows selective erasig at the register level rather tha erasig all the iformatio sice the iformatio ca be chaged by usig electrical sigals. 6. What is RAM?(R) Radom Access Memory. Read ad write operatios ca be carried out. 7. Defie ROM. (R) A read oly memory is a device that icludes both the decoder ad the OR gates withi a sigle IC package..5

122 8. Defie address ad word. (R) I a ROM, each bit combiatio of the iput variable is called o address. Each bit combiatio that comes out of the output lies is called a word. 9. What are the types of ROM. (R). Masked ROM.. Programmable Read oly Memory. Erasable Programmable Read oly memory. 4. Electrically Erasable Programmable Read oly Memory.. What is programmable logic array? How it differs from ROM? (R) I some cases the umber of do t care coditios is ecessive, it is more ecoomical to use a secod type of LSI compoet called a PLA. A PLA is similar to a ROM i cocept; however it does ot provide full decodig of the variables ad does ot geerates all the miterms as i the ROM.. What is mask - programmable? (R) With a mask programmable PLA, the user must submit a PLA program table to the maufacturer.. What is field programmable logic array? (R) The secod type of PLA is called a field programmable logic array. The user by meas of certai recommeded procedures ca program the EPLA.. List the major differeces betwee PLA ad PAL. (R) PLA: Both AND ad OR arrays are programmable ad Comple Costlier tha PAL PAL: AND arrays are programmable OR arrays are fied Cheaper ad Simpler 4. Defie PLD. (R) Programmable Logic Devices cosist of a large array of AND gates ad OR gates that ca be programmed to achieve specific logic fuctios. 5. Give the classificatio of PLDs. (R) PLDs are classified as PROM(Programmable Read Oly Memory), Programmable Logic Array(PLA), Programmable Array Logic (PAL), ad Geeric Array Logic(GAL) 6. Defie PROM. (R) PROM is Programmable Read Oly Memory. It cosists of a set of fied AND gates coected to a decoder ad a programmable OR array. 7. Defie PLA (R) PLA is Programmable Logic Array(PLA). The PLA is a PLD that cosists of a programmable AND array ad a programmable OR array. 8. Defie PAL (R).6

123 PAL is Programmable Array Logic. PAL cosists of a programmable AND array ad a fied OR array with output logic. 9. What is CPLD? (R) CPLDs are Comple Programmable Logic Devices. They are larger versios of PLDs with a cetralied iteral itercoect matri used to coect the device macro cells together.. How may words ca a 68 memory ca store? (R) A 68 memory ca store 6,84 words of eight bits each.. Why RAMs are called as Volatile? (R) RAMs are called as Volatile memories because RAMs lose stored data whe the power is tured OFF.. Defie Static RAM ad dyamic RAM. (R) Static RAM use flip flops as storage elemets ad therefore store data idefiitely as log as dc power is applied. Dyamic RAMs use capacitors as storage elemets ad caot retai data very log without capacitors beig recharged by a process called refreshig.. List the basic types of DRAMs. (R) Asychroous DRAM FPM DRAM, EDO DRAM, BEDO DRAM, SDRAM, RDRAM 4. List the two categories of RAMs. (R) Static RAM ad dyamic RAM 5. List the two types of SRAM. (R) No-volatile SRAM, Asychroous SRAM 6. Defie Cache memory. (R) It is a relatively small, high-speed memory that ca store the most recetly used istructios or data from larger but slower mai memory. 7. Give the feature of flash memory. (R) The ideal memory has high storage capacity, o-volatility; i-system read ad write capability, comparatively fast operatio. The traditioal memory techologies such as ROM, PROM, EEPROM idividually ehibits oe of these characteristics, but o sigle techology has all of them ecept the flash memory. 8. List the three major operatios i a flash memory. (R) Programmig, Read ad Erase operatio. 9. List basic types of programmable logic devices. (R). Read oly memory. Programmable logic Array. Programmable Array Logic. What is mask - programmable? (R).7

124 With a mask programmable PLA, the user must submit a PLA program table to the maufacturer.. List the major differeces betwee PLA ad PAL. (R) PLA has Programmable AND Gate liked with Programmable OR Gate. PAL is a combiatioal PLD that was developed to overcome certai disadvatage of PLA. PLA shows loger delay due to additioal fusible liks which results from usig two programmable array ad icrease circuit compleity. Thus, PAL is used which is less comple ad fast to implemet. PAL cosists of programmable AND liked with fied OR.. Defie FPGA. (R) FPGA is a field programmable gate array, which is the et geeratio i the programmable logic devices. The word field refers to the ability of the gate arrays to be program for a specific fuctio by the ed user. The word array idicates a series of colums ad roes of gates that ca be programmed.. Compariso betwee SRAM & DRAM. (U) SRAM DRAM SRAM is static Iteral latches are used to store the iformatio Stored iformatio is valid till the power is applied to the uit. So No eed of refreshmet. SRAM is more epesive Storage capacity is less SRAM is commoly used i cache memory DRAM is dyamic capacitors store the biary iformatio i the form electrical charges which is provided iside the chip by MOS trasistors. It eeds cotiuous refreshig. DRAM is cheap. Storage capacity is more Cheaper DRAM is used i mai memory 4. Compariso betwee PROM, PAL, PLA. (U) PROM cosists of set of AND gates coected to a decoder ad a programmable OR array. I PAL AND array is programmable ad OR array is fied. But PLA, Both array are programmable. 5. What is PAL? How does it differ from PLA? (U) PAL cosists of programmable AND liked with fied OR. PAL is a combiatioal PLD that was developed to overcome certai disadvatage of PLA. PLA shows loger delay due to additioal fusible liks which results from usig two programmable array ad icrease circuit compleity. Thus, PAL is used which is less comple ad fast to implemet..8

125 6. What is access time ad cycle time of a memory? (R) Access time of memory is the time required to select a word ad read it. Cycle time of memory is the time required to complete a write operatio. 7. What is the differece betwee PAL ad PLA? (R) PLA is a kid of programmable logic device used to implemet combiatioal logic circuit. It has Programmable AND Gate liked with Programmable OR Gate. 8. Give the differece betwee RAM ad ROM. (U) Radom Access Memory Read ad write operatio are performed Read Oly Memory -- It performs oly the read operatio 9. Metio few applicatios of PLA ad PAL. (R) It is used as mu ad demu, used for implemet half adder & full adder, by program it. 4. How the memories are classified? (R) Memories are classified as Radom access memory ad Read oly memory. RAM are further classified as Static RAM ad Dyamic RAM. ROM are further classified as PROM, EPROM, EEPROM PART-B & C. Draw a RAM cell ad eplai its workig. (8) U. Write short otes o (i) RAM (ii) Types of ROM s (8) U. List the PLA programmable for BCD to Ecess --code covertor circuits ad show its implemetatio for ay two output fuctios (6) U 4. Geerate the followig Boolea fuctios with PAL with 4iputs ad 4outputs Y = A BC D+A BCD +A BCD+ABD Y = A BCD +A BCD+ABCD Y = A BC +A BC+AB C+ABC Y = ABCD (6)C 5. Implemet the followig fuctios usig PLA. (6) AZ F=Σm(,,4,6); F=Σm(,,6,7) F=Σm(,6) 6. Implemet the give fuctios usig PROM ad PAL (6) AZ F=Σm(,,,5,7,9); F=Σm(,,4,7,8,,)) Implemet the give fuctios usig PLA F=Σm(,,,4,6,7); F=Σm(,,5,7); F=Σm(,,,6) (6)U.9

126 7. Draw the block diagram of a PLA device ad briefly eplai each block. (8) U 8. Write short ote o Field Programmable Gate Array (FPGA) (8) U 9. Desig a 6 bit ROM array ad eplai the operatio (8) C.Write short ote o Field Programmable Gate Array (FPGA) (8) U.Implemet the followig Boolea fuctios with a PLA (6) [Nov/Dec 9]. AZ a. F ( A, B,C ) = Â (,,, 4 ) b. F ( A, B,C ) = Â (,5, 6, 7 ) c. F ( A, B,C ) = Â (,,5, 7 )..Desig a combiatioal circuit usig a ROM. The circuit accepts a three bit umber ad outputs a biary umber equal to the square of the iput umber. (6) [Nov/Dec 9]. C. We epad the word sie of a RAM by combiig two or more RAM chips. For istace, we ca use two 8 memory chips where the umber represets the umber of words ad 8 represets the umber of bits per word, to obtai 6 RAM. I this case the umber of words remais the same but legth of each word will two bytes log. Draw a block diagram to show how we ca use two 6 4 memory chips to obtai a 6 8 RAM. (8) [Apr/May ].C 4.Eplai the priciple of operatio of Bipolar SRAM cell. (8)[Apr/May ].U 5.A combiatioal circuit is defied as the fuctios F = A B C + A B C + A B C F = A B C + A B C + A B C Implemet the digital circuit with a PLA havig iputs, product terms, ad outputs. (8) [Apr/May ].AZ 6. Draw a PLA circuit to implemet the fuctios F = A B + A C + A B C ; F = ( AC + A B + BC ). (8) [Nov/Dec ].AZ 7.Write a ote o FPGA. (8) [Nov/Dec ]. U 8.Desig a combiatioal circuit usig a ROM. The circuit accepts a -bit umber ad geerates a output biary umber equal to the square of the iput umber.() [Apr/May ]. C 9. Briefly eplai the EPROM ad EEPROM techology. (6) [Apr/May ]. U. Implemet the followig fuctios usig iput, 4 product term ad output PLA.

127 F = A B + AC + A B C F = (AC + B C ) (8) [Apr/May ].AZ. With logic diagram, eplai the basic macrocell. (8) [Apr/May ]. U. Draw the basic block diagram of PLA device ad eplai each block. List out its applicatio. Implemet a combiatioal circuit usig PLA by takig a suitable Boolea fuctio. (6) [Nov/Dec ]. U.Eplai the operatios of static ad dyamic MOS RAM cell with ecessary diagrams. () [Nov/Dec ]. U 4.What are advatages of FPGA? (4) R 5.Discuss the classificatio of ROM ad ROM memories. (8) [May/Ju ]. AZ 6.Eplai Memory decodig ad Memory epasio of Digital System. (8) [May/Ju ].U 7.Write short otes o : (i) FPGA ad its applicatio (8) (ii) Programme Logic Array (8) [May/Ju ]. U 8.Implemet a -bit up/dow couter usig PAL devices. (8) [Nov/Dec ]. AZ 9.Implemet biary to Gray code coverter usig PROM devices. (8) [Nov/Dec ]. AZ.Write short otes o : (i) Memory decodig (8) (ii) Memory epasio (8) [Nov/Dec ]. U.Usig eight 648 ROM chips with a eable iput ad a decoder, costruct a 58 ROM. () [Nov/Dec 4] AZ.Compare ad cotrast PAL ad PLA (4). [Nov/Dec 4]. U.

128 UNIT V - SYNCHRONOUS AND AYNCHRONOUS SEQUENTIAL CIRCUITS PART-A ) What is the classificatio of sequetial circuits? (R) i. Sychroous sequetial circuit ii. ASychroous sequetial circuit ) Give the compariso betwee sychroous & Asychroous sequetial circuits? (U) S.No Sychroous sequetial circuits Asychroous sequetial circuits Memory elemets are clocked flip- Memory elemets are either uclocked flops flip-flops or time delay elemets. The chage i iput sigals ca The chage i iput sigals ca affect memory elemet at ay istat of time. activatio of clock sigal. The maimum operatig speed of Because of the absece of clock, it ca clock depeds o time delays operate faster tha sychroous ivolved. circuits. 4 Easier to desig More difficult to desig ) Defie Asychroous sequetial circuit? (R) Asychroous sequetial circuit output depeds o the iput ad preset state. It does ot use clock pulse, the chage of iteral states deped o the chage i iput variables. The memory elemets i asychroous sequetial circuit are time delay elemets. 4) What is fudametal mode? (R) I fudametal mode circuit, all of the iput sigals are cosidered to be levels. Fudametal mode operatio assumes that the iput sigals will be chaged oly whe the circuit is i a stable state ad that oly oe variable ca chage at a give time. 5) What are the problems ivolved i asychroous circuits? (R) Timig problem ivolved i the feedback path 9) What are secodary variables? (R) Preset state variables i asychroous sequetial circuits are called secodary variables.

129 ) What are ecitatio variables? (R) Net state variables i asychroous sequetial circuits. ) What is fudametal mode sequetial circuit? (R) -iput variables chages if the circuit is stable. -iputs are levels, ot pulses. -oly oe iput ca chage at a give time. ) What are pulse mode circuit? (R) -iputs are pulses. -width of pulses are log for circuit to respod to the iput. -pulse width must ot be so log that it is still preset after the ew state is reached. ) What are the sigificace of state assigmet? (R) I sychroous circuits-state assigmets are made with the objective of circuit reductio. Asychroous circuits-its objective is to avoid critical races. 4) Whe the race coditio occurs? (R) Two or more biary state variables chage their value i respose to the chage i i/p variable. 5) What is o critical race? (R) -fial stable state does ot deped o the order i which the state variable chages. -race coditio is ot harmful. 6) What is critical race? (R) -fial stable state depeds o the order i which the state variable chages. -race coditio is harmful. 7) Whe does a cycle occur? (R) I asychroous circuit makes a trasitio through a series of ustable state. 8) What are the differet techiques used i state assigmet? (R) Shared row state assigmet. Ad Oe hot state assigmet. 9) What are the steps for the desig of asychroous sequetial circuit? (R) i. Costructio of primitive flow table. ii. Reductio of flow table. iii. State assigmet is made. iv. Realiatio of primitive flow table. ) What is haard? (R).

130 I a uwated switchig trasiets occurs i the sequetial circuit is called haards. ) What is static haard? (R) -output goes mometarily whe it should remai at. ) What is static haard? (R) -output goes mometarily whe it should remai at. ) What is dyamic haard? (R) -output chages or more times whe it chages from to or to. 4) What is the cause for essetial haards? (R) Uequal delays alog or more path from same iput itroduce a essetial haard. 5) What is flow table? (R) I a state table of a sychroous sequetial etwork is called flow table 6) What is SM chart? (R) Chart which describes the behavior of a state machie is called SM Chart. It is used i hardware desig of digital systems. 7) What are the advatages of SM chart? (R) i. Easy to uderstad the operatio. ii. Easy to covert to several equivalet forms. 8) What is primitive flow chart? (R) Oly oe stable state per row. 9) What is combiatioal circuit? (R) Output depeds o the give iput. It has o storage elemet. ) What is state equivalece theorem? (R) Two states SA ad SB, are equivalet if ad oly if for every possible iput X sequece, the outputs are the same ad the et states are equivalet i.e., if SA (t + ) = SB (t + ) ad ZA = ZB the SA = SB. ) What do you mea by distiguishig sequeces? (R) Two states, SA ad SB of sequetial machie are distiguishable if ad oly if their eist at least oe fiite iput sequece which, whe applied to sequetial machie causes differet output sequeces depedig o whether SA or SB is the iitial state. ) Defie merger graph. (R).4

131 The merger graph is defied as follows. It cotais the same umber of vertices as the state table cotais states. A lie draw betwee the two state vertices idicates each compatible state pair. It two states are icompatible o coectig lie is draw. ) Eplai the procedure for state miimiatio. (U). Partitio the states ito subsets such that all states i the same subsets are - equivalet.. Partitio the states ito subsets such that all states i the same subsets are - equivalet.. Partitio the states ito subsets such that all states i the same subsets are - equivalet. 4) Defie state table. (R) For the desig of sequetial couters we have to relate preset states ad et states. The table, which represets the relatioship betwee preset states ad et states, is called state table. 5) What are the steps for the desig of asychroous sequetial circuit? (R). Costructio of a primitive flow table from the problem statemet.. Primitive flow table is reduced by elimiatig redudat states usig the state reductio.. State assigmet is made 4. The primitive flow table is realied usig appropriate logic elemets. 6) Defie primitive flow table. (R) It is defied as a flow table which has eactly oe stable state for each row i the table. The desig process begis with the costructio of primitive flow table. 7) What are the types of asychroous circuits? (R). Fudametal mode circuits. Pulse mode circuits. 8) What are races? (R) Whe or more biary state variables chage their value i respose to a chage i a iput variable, race coditio occurs i a asychroous sequetial circuit. I case of uequal delays, a race coditio may cause the state variables to chage i a upredictable maer. 9) Defie o critical race. If the fial stable state that the circuit reaches does ot deped o the order i which the state variable chages, the race coditio is ot harmful ad it is called a o critical race. 4) Defie critical race? Give a Eample. (R).5

132 If the fial stable state depeds o the order i which the state variable chages, the race coditio is harmful ad it is called a critical race. 4) What is a cycle? (R) A cycle occurs whe a asychroous circuit makes a trasitio through a series of ustable states. If a cycle does ot cotai a stable state, the circuit will go from oe ustable to stable to aother, util the iputs are chaged. 4) Defie flow table i asychroous sequetial circuit. (R) I asychroous sequetial circuit state table is kow as flow table because of the behaviour of the asychroous sequetial circuit. The stage chages occur i idepedet of a clock, based o the logic propagatio delay, ad cause the states to.flow. from oe to aother. 4) Write short ote o shared row state assigmet. (R) Races ca be avoided by makig a proper biary assigmet to the state variables. Here, the state variables are assiged with biary umbers i such a way that oly oe state variable ca chage at ay oe state variable ca chage at ay oe time whe a state trasitio occurs. To accomplish this, it is ecessary that states betwee which trasitios occur be give adjacet assigmets. Two biary are said to be adjacet if they differ i oly oe variable. 44) Write short ote o oe hot state assigmet. (R) The oe hot state assigmet is aother method for fidig a race free state assigmet. I this method, oly oe variable is active or hot for each row i the origial flow table, ie, it requires oe state variable for each row of the flow table. Additioal row are itroduced to provide sigle variable chages betwee iteral state trasitios. 45) Compare the ASM chart with a covetioal flow chart. (U) A covetioal flow chart describes the procedural steps ad decisio paths of a algorithm i a sequetial maer, without takig ito cosideratio their time relatioship. The ASM chart describes the sequece of evets,as well as timig relatioship betwee the states of a sequetial cotroller ad the evets that occur whilke goig from oe state to the et. 46) Draw the block diagram for Moore model. (U) 47) What are haard free digital circuits? (R).6

133 Haard i combiatioal circuits ca be removed by coverig ay two miterms that may produce a haard with a product term commo to both. The removal of haards are require the additio of redudat gates to the circuit. 48) Differetiate Moore machie from Mealy machie.(u) I Mealy model, the output is a fuctio of both preset state ad the iput. I Moore model, the output is a fuctio of oly the preset state. 49) What are the basic buildig blocks of a Algorithmic state machie chart. (R) The basic buildig blocks of ASM chart are State bo, decisio bo ad coditioal bo. 5) What is meat by haard ad how it could be avoided? (R) I a uwated switchig trasiets occurs i the sequetial circuit is called haards. I t could be avoided by addig etra gate. 5) What is state Table? (R) The time sequece of iputs, outputs,ad flip flop states ca be eumerated i a state table. It cosists preset state, iput, et state ad output sectios 5) Differetiate fudametal mode ad pulse mode asychroous sequetial circuits.(u) fudametal mode circuit All of the iput sigals are cosidered to be levels. Fudametal mode operatio assumes that the iput sigals will be chaged oly whe the circuit is i a stable state Oly oe variable ca chage at a give time. pulse mode circuits The iputs are pulses rather tha levels I this mode of operatio the width of the iput pulses is critical to the circuit operatio The iput pulse must be log eough for the circuit to respod to the iput but it must ot be so log as to be preset eve after ew state is reached. The state of the circuit may make aother trasitio..7

134 PART B & C ) What is the objective of state assigmet i asychroous circuit? (6) R ) Realiatio for the followig Boolea fuctio f(a,b,c,d) = Σm(,,6,7,8,,) (6)R ) Summarie the desig procedure for asychroous sequetial circuit Discuss o Haards ad races.u 4) Develop the state diagram ad primitive flow table for a logic system that has iputs, Ad y ad a output. Ad reduce primitive flow table. The behavior of the circuit is stated as follows. Iitially =y=. Wheever = ad y = the =, wheever = ad y = the =.Whe =y= or =y= o chage i to remais i the previous state. The logic system has edge triggered iputs without havig a clock the logic system chages state o the risig edges of the iputs. Static iput values are ot to have ay effect i chagig the Z output. (6) C 5) A pulse mode asychroous machie has two iputs. It produces a output wheever two cosecutive pulses occur o oe iput lie oly. The output remais at util a pulse has occurred o the other iput lie. Draw the state table for the machie (6) C 6) Costruct the state diagram ad primitive flow table for a asychroous etwork that has two iputs ad oe output. The iput sequece XX =,, causes the output to has two iputs ad oe output. C 7) Discuss o the differet types of Haards that occurs i asychroous sequetial circuits. (6) U 8) Write short ote o races ad cycles that occur i fudametal mode circuits. (6)U 9) Desig a asychroous sequetial circuit with two iputs X ad Y ad with oe output Z. Wheever Y is,(6) iput X is trasferred to Z. Whe Y is,the output does ot chage for ay chage i X. (6)C ) Draw the ASM chart for the followig state diagram.(8) [Apr-May ]R.8

135 ) Desig the followig sequetial circuit usig D Flip flop ad logic gates. (8)C )What is a Haard? What are the types of haards? Check whether the followig circuit cotais a haard or ot Y= + If the haard is preset, demostrate its removal. (6)AZ )Desig a three bit biary couter usig T flipflops. (6) [Nov/Dec 9].C 4) Desig a egative-edge triggered T flipflop. (6) [Nov/Dec 9]. C 5)What are called as essetial haards? How does the haard occur i sequetial circuits? How ca the same be elimiated usig SR latches? Give a eample. (6) [Apr/May ]. U 6)Differetiate critical races from o-critical races. (6) [Nov/Dec ].U 7)Eplai the steps ivolved i the reductio of state table. () [Nov/Dec ]. U 8)What is a Haard? What are the types of haards? Check whether the followig circuit cotais a haard or ot Y = +. If the haard is preset, demostrate its removal. (6) [Apr/May ].U 9)Write short otes o shared row state assigmet with a eample. (8) [Nov/Dec ]. U )Eplai the method to elimiate static haard i a asychroous circuit with a eample. () [Nov/Dec ]. U ) Write short otes o VERILOG. (6) [Nov/Dec ] U )A sequetial circuit has three D flip flops. A, B ad C ad oe iput. It is desired by the followig flip flop iput fuctios D A = ( B C + B C ) + ( B C + B C ) D B = A, D C = B Derive the state table for the circuit ad draw two state diagrams for =, ad other for =. (8) [Nov/Dec ]C )With a eample, eplai the use of algorithmic state machies. (8) [May/Ju ]. U 4)Describe the method to aalye the behavior of a sychroous sequetial circuit. (8) [May/Ju ]. AZ.9

136 5)Write short otes o: (i) Icompletely specified state machies (8) (ii) Haard free switchig circuits (8) [May/Ju ]. U 6)Desig the followig circuits usig verilog (i) 4 to multipleer (8) (ii) bit up/dow couter. (8) [Nov/Dec ]. C 7)Write short otes o races ad haards that occur i asychroous circuits. Discuss a method used for race free assigmet with eample. U 8)For the state diagram show i fig (), desig a sychroous sequetial circuit usig JK flip-flop. C 9)Desig a asychroous sequetial circuit with iputs A a B ad a output Y. Iitially ad at ay time if both the iputs are, the output, Y is equal to. Whe A or B becomes, Y becomes. Whe the other iput also becomes, Y becomes. The output stays at util the circuit goes back to iitial state. C ASSIGNMENT. List the PLA programmig table for followig Boolea fuctios. A a) W = A+BC+BD b) X = B C + B D + BC D c) Y = CD + C D d) Z = D. What is the sie specificatio of PLA that could implemet oe digit BCD adder? Give the umber of iputs, outputs, ad product terms required. R. Costruct the ROM table to implemet the followig fuctios usig 4 4 ROM assumig all iputs ad outputs are active high. AZ a) Y (a,b,c,d) = Σm(,,,5,7,8,,4,5) b) Y (a,b,c,d) = Σm(,,4,5,6,7,8,,) c) Y (a,b,c,d) = Σm(,,,,4,6,8,9,,) 4. How may 6 ROMs with eable will be required to costruct a 86 ROM? What type of decoder will you eed for this desig?. You are asked to build a digital lock which opes ONLY with the combiatio. Desig a

137 sychroous digital system that the electroic system of this lock would comprise, usig: a) D flip-flops b) J-K flip-flops 5. Desig a sychroous -bit Up/Dow couter havig Gray code sequece.. How may 568 RAM chips are eeded to provide a memory capacity of 496 bytes? a. How may address lies must be used to access 496 bytes? How may of these lies are coected to the address iputs of all chips? b. How may lies must be decoded for the chip-select iputs? C 6. Implemet the followig fuctio with 8 multipleer. F(A,B,C,D) = Σ(,,,4,8,9,5) with A,B,C coected to S, S, S respectively. AZ 7. Fid the Boolea fuctio that a 8 multipleer implemet with A,B,C coected to select lies S,S,S respectively, if I=, I=D, I=, I=D, I4=I5=D, I6=, I7=. AZ 8. Specify the o of address lies, o of data lies ad total memory sie i bytes for followig chips specificatios. a. K 6 b. 64K. U 9. Obtai the ecitatio table of the JK' flip-flop, i.e. A JK flip-flop with a iverter betwee eteral iput K' ad iteral iput K. U. A set-domiate flip-flop has set ad reset iputs. It differs from a covetioal RS flip-flop i that a attempt to simultaeously set ad reset results i settig the flip-flop. Obtai the ecitatio table of such a flip-flop? U. Desig a biary couter havig the followig repeated biary sequece. Use JK flip-flops.,,... C. How may flip-flops will be complemeted i a -bit biary ripple couter to reach the et cout after the followig cout:. A.4

138 PANIMALAR ENGINEERING COLLEGE DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING EC6 SIGNALS & SYSTEMS SOLVED TWO MARKS & PART - B QUESTIONS (FOR THIRD SEMESTER ECE) 4.

139 EC6 SIGNALS AND SYSTEMS L T P C 4 OBJECTIVES: To uderstad the basic properties of sigal & systems ad the various methods of classificatio To lear Laplace Trasform &Fourier trasform ad their properties To kow Z trasform & DTFT ad their properties To characterie LTI systems i the Time domai ad various Trasform domais UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS 9 Cotiuous time sigals (CT sigals) - Discrete time sigals (DT sigals) - Step, Ramp, Pulse, Impulse, Siusoidal, Epoetial, Classificatio of CT ad DT sigals - Periodic & Aperiodic sigals, Determiistic & Radom sigals, Eergy & Power sigals - CT systems ad DT systems- Classificatio of systems Static & Dyamic, Liear & Noliear, Time-variat & Time-ivariat, Causal & Nocausal, Stable & Ustable. UNIT II ANALYSIS OF CONTINUOUS TIME SIGNALS 9 Fourier series aalysis-spectrum of Cotiuous Time (CT) sigals- Fourier ad Laplace Trasforms i CT Sigal Aalysis - Properties. UNIT III LINEAR TIME INVARIANT- CONTINUOUS TIME SYSTEMS 9 Differetial Equatio-Block diagram represetatio-impulse respose, covolutio itegrals-fourier ad Laplace trasforms i Aalysis of CT systems. UNIT IV ANALYSIS OF DISCRETE TIME SIGNALS 9 Basebad Samplig - DTFT Properties of DTFT - Z Trasform Properties of Z Trasform. UNIT V LINEAR TIME INVARIANT-DISCRETE TIME SYSTEMS 9 Differece Equatios-Block diagram represetatio-impulse respose - Covolutio sum- Discrete Fourier ad Z Trasform aalysis of Recursive & No-Recursive systems TOTAL (L: 45+T:5): 6 PERIODS OUTCOMES: Upo the completio of the course, studets will be able to: Aalye the properties of sigals & systems. Apply Laplace trasform, Fourier trasform, Z trasform ad DTFT i sigal aalysis. Aalye cotiuous time LTI systems usig Fourier ad Laplace Trasforms. Aalye discrete time LTI systems usig Z trasform ad DTFT. TEXT BOOK:. Alla V.Oppeheim, S.Wilsky ad S.H.Nawab, Sigals ad Systems, Pearso, 7. REFERENCES:. B. P. Lathi, Priciples of Liear Systems ad Sigals, Secod Editio, Oford, 9.. R.E.Zeimer,W.H.Trater ad R.D.Fai, Sigals & Systems - Cotiuous ad Discrete, Pearso, 7.. Joh Ala Stuller, A Itroductio to Sigals ad Systems, Thomso, 7. 4.

140 4. M.J.Roberts, Sigals & Systems Aalysis usig Trasform Methods & MATLAB, Tata McGraw Hill, 7. FORMULAS Fiite summatio N N + N + - a a - Â a = or, a < = - a a Ifiite summatio 7. a < 7(a). Â a a = where = - a a < 8. a < 9. a <. Eergy of cotiuous time sigal, E=. Eergy of discrete time sigal, E=. Power of cotiuous time sigal, P=. Power of discrete time sigal, P= Note: Eergy sigal: E= fiite, P= Power sigal: E=, P=fiite 4. Periodic Sigal: 4.

141 CT sigal (t+t) = (t) DT sigal (+N) = () 5. Eve ad Odd Sigal: CT sigal: Eve sigal (-t) = (t) Odd Sigal (-t) = -(t) DT sigal: Eve sigal (-) = () Odd Sigal (-) = -() 6. Eve ad Odd Compoet: CT sigal: Eve compoet e (t) = ½[(t) + (-t)] odd compoet o (t) = ½[(t) - (-t)] DT sigal: Eve compoet e () = ½[() + -()] odd compoet o () = ½[() - (-)] 7. Stability of the system: CT System h ( ) < DT System 8. Coditio for Causality of sigal: CT sigal: (t)=, t< i.e.,t= egative values DT sigal: ()=, < i.e.,= egative values 9. Coditio for Causality of system: CT system: h(t)=, t< i.e.,t= egative values DT system: h()=, < i.e.,= egative values. Time domai operatio:. time reversal. Time shiftig. Time scalig. Fourier series Ú - h ( t) dt < Â - =. Si =-si. Si =si. Si =cos 4. Cos =-Cos 5. Cos =Cos 6. Cos =-Si 7. ta =ta 8. ta =ta 9. Si( )=, Cos( )=(-) Si Cos, Cos Si, ta Cot, Sec Cosec, Cosec Sec 4.4

142 . SiC+SiD=Si( )Cos( ). SiC-SiD=Cos( )Si( ). CosC+CosD=Cos( )Cos( ). CosC-CosD=Si( )Si( ) 4. SiACosB= 5. SiASiB= 6. CosACosB= 7. CosASiB= 8. Trigometric series: a = a= b= (t)= a + 9. Cosie represetatio: A =a = a= b= A= Or 4.5

143 (t)= A+ A --> amplitude coefficiet of FS. Phase coefficiet of FS.. Epoetial FS: C = C = X(t)=. Eistace of Fourier series: The coditio uder which periodic sigal ca be represeted by Fourier series are Dirichlet coditio a. The fuctio (t) have oly fiite umber of maima ad miima. b. The fuctio (t) have a fiite umber of discotiuities. c. The fuctio (t) is absolutely itegrable over oe period Fourier Trasform. F[] Poit of discotiuity: (t d ) =. F[e ]. F[e ] 4. F[sg(t)] 5. F[u(t)] 6. F[ 7. F[ 8. F[(t-t)] e X(j 9. F[(t)e ] X(j( )). F[ ] (j X(j ) 4.6

144 . F[t(t)] j. F[e u(t)]. F[e u(-t)] 4. F[e u(-t)] 5. F[e ]= 6. F[e ] e 7. F[e ] e 8. F[ Laplace Trasform. X(S) = (bilateral o-causal system). X(S) = (uilateral causal system). L[e u(t)] ROC: s>-a 4. L[e u(-t)] ROC: s<-a 5. L[ 6. L(u(t)]= 7. L[r(t)]= 8. L[cos( )]= 9. L[si( )]=. L[Cosh(at)]=. L[sih(at)]= Z-Trasform. [ All Z plae 4.7

145 . [ Z,. u() or 4. Z(-u(--)] or 5. u() or 6. a u() or 7. (-a u(--)] or 8. (a u()] 9. (-a u(--)]. [e ] or. a u(-m) -m ( ). (+)a u(). a u(-) 4. a u() UNIT-I CLASSIFICATION OF SIGNALS AND SYSTEMS PART-A. Defie Sigal. (R) A sigal is defied as physical Quatity that varies with time, space or ay other idepedet variables which cotai some iformatio. Eg: Radio sigal, TV sigal, Telephoe sigal etc.. Give some eamples of CT ad DT sigals. (R) CT eample : AC waveform, ECG,Temperature recorded over a iterval of time etc. DT eample : Amout deposited i a bak per moth.. Defie determiistic ad radom sigals. (R) A determiistic sigal is oe which ca be completely represeted by Mathematical equatio at ay time.i a determiistic sigal there is o ucertaity with respect to its value at ay time. Eg : (t)= coswt A radom sigal is oe which caot be represeted by ay mathematical equatio. 4.8

146 Eg: Noise geerated i electroic compoets, trasmissio chaels, cables etc. 4. Defie periodic ad Aperiodic sigals. (R) A sigal is said to be periodic sigal if it repeated itself every fudametal period. Aperiodic sigals do ot repeat at regular itervals. A CT sigal which satisfies the equatio (t)=(t+t) is said to be periodic ad a DT sigal which satisfies the equatio ()=(+N)is said to be periodic 5. Compare Eergy ad Power sigals. (AZ) S.No Power sigal Eergy sigal The ormalied average Total ormalied eergy is power is fiite ad o-ero fiite ad o- ero. periodic sigals are power sigals Aperiodic sigals are eergy sigals. 6. Defie step fuctio ad delta fuctio. (May/Jue 9) (R) CT uit step fuctio, u(t) = {, for t, for t< DT uit step fuctio, u[] = {, for, for < CT delta fuctio, (t)=, for t= =, for t π DT delta fuctio, [] =, for = =, for π 7. Show that the comple epoetial sigal (t)= e is periodic ad that fudametal period is. (May/Jue 9) (A) A Cotiuous time sigal (t) is said to be periodic with period T if there is a positive oero value of T for which (t+t)=(t) for all t. Therefore give (t) will be periodic if Proof: e = e e = e e we kow e = If T=m (Or) T=m, where m= positive iteger. the e = e = e 4.9

147 Thus the comple epoetial sigal (t)= e is is periodic with fudametal period 8. Fid average power of the sigal u[]-u[-n]. (A) P= = = = 9. What is the total eergy of the discrete time sigal [] which takes the value of uity at =-,,. (Nov/Dec ) (A) Eergy of the sigal is give by E= = = [-] + [] + [] = ++ =. Classify the followig sigals as, a. Periodic or o periodic ad b. Eergy or power sigal i) e, > ii) e (Nov/Dec 4) (AZ) i) Periodicity i)e ii) e w=pf. ii) Eergy or power is Aperiodic sice it is real epoetial sigal. is periodic sigal sice it is Comple epoetial sigal with i) E= = This sigal is either eergy or power. ii) e is power sigal. P=T = T = T = Power is fiite, therefore power sigal. 4.

148 . Fid the Fudametal period of the sigal []= (Nov/Dec 4) (A) N= m = m = / (ratioal umber, so periodic) Fudametal period of the sigal []=/ sec. Is diode liear device? Give your reaso. (Nov/Dec 4) (U) Diode is o liear device sice it operates oly whe forward biased. For egative bias, diode does ot coduct.. Fid the eve part ad odd part of (t)= e (Nov/Dec 8) (A) Let (t)= cost +j si t (-t)= cos(-t) +j si(-t) =cos t j si t e (t)= /{(t)+(-t)}=/{ cos t +j si t+ cos t j si t}=cos t (t)= /{(t)-(-t)}=/{ cos t +j si t- cos t +j si t}=j si t 4. Is the system y(t)=y(t-)+ty(t-) time ivariat? (Nov/Dec ) (AZ) Respose for delayed iput, y(t,t )= y(t--t )+ty(t--t ) Delayed respose, y(t-t )= y(t-t -)+(t-t )y(t-t -) y(t,t ) π y(t-t ), This is time variat system. 5. Check whether the system havig iput- output relatio y(t)= is liear Time ivariat or ot. (April/May 4) (U) This is a itegratio of iput. Itegratio is always idepedet of time shift. Hece this is time ivariat system. 6. Check whether the system is dyamic or ot. Y[]=[]+[+] (U) The output depeds o the preset iput ad future iput. Hece the system is dyamic. 7. State coditio for causal system ad o-causal system.(april/may 8) (R) Causal system: A causal system is oe for which the output at ay time t depeds o the preset ad past iputs but ot future iputs. These systems are also kow as o-aticipative systems No-causal system: A o-causal system is oe whose output depeds o future values. 8. Determie whether the sigal is periodic or ot.(april/may 8) (A) N takes smallest positive iteger value whe m=, N= ad sequece is periodic. 9. Is the sigal periodic (Nov/Dec 8) (A) 4.

149 ( N= For ay value of m, N is ot a iteger. It is aperiodic sequece.. Defie liear time ivariat system. (April/May ) (R) The time shift i iput sigal results i a idetical time shift i output sigal. For CT (t-t ) = y(t-t ) ad For DT (- ) = y(- ). Whe is a system said to be memoryless? Give a eample.(april/may ) (R) A system is said to be memeoryless or static if its output for each value of idepedet variable at a give time is depedet oly o the iput at that same time (depeds oly o preset value) Eample: Resistor is a memoryless system with the iput (t) take as curret ad output y(t) take as voltage y(t)=r(t). PART B. Check for liearity, time ivariace, causality ad stability. () y()=( ) () y()=(-) (Nov/Dec & 8) (AZ). What is (-) ad (-)? (Nov/Dec ) (E). Compare eergy sigal ad power sigal. (Nov/Dec ) (AZ) 4. Determie whether (t)= rect (t/)cos t is eergy sigal or power sigal. (Nov/Dec ) (AZ) 5. Derive the relatioship betwee uit step ad delta fuctio. (Nov/Dec ) (U) 6. Distiguish betwee the followig: i. Cotiious time sigal ad discrete time sigal. ii. Uit bstep ad uit ramp fuctios. iii. Periodic ad aperiodic sigal. iv. Determiistic ad radom sigals. (April/May ) (U) 7. Fid whether the followig sigals are periodic or ot i. (t)=cos(t+)-si(4t-) ii. (t)=cos4t+sit (April/May ) (A) 8. Fid the summatio (April/May ) (E) 9. Eplai the properties of uit impulse fuctio. (April/May ) (U). Fid the fudametal period T of the cotiuous time sigal 4.

150 X(t)= cos( ) (April/May ) (A). Check the followig for liearity, time ivariace, causality ad stability for () y()=()+ (+) () y()=log[()] ()y()= (April/May ) (AZ). Check whether the followig are periodic:. ()=si. ()=e (April/May ) (A). Describe the basic properties of systems with eamples. (April/May ) (U) 4. Sketch the followig sigal (April/May ) (E) i. (t)=r(t) ii. (t)=r(-t+) iii. (t)= -r(t) where r(t) is a ramp sigal. 5. A discrete time system is give as y()=y (-)=(). A bouded iput of ()= () is applied to the system. Assume that the system is iitially relaed. Check whether system is stable or ustable. (May/Jue ) (AZ) 6. Determie the value of power ad eergy for each of the sigals () ()=, () = (April/May 8) (A) 7. Let (t)= be a periodic sigal with fudametal period T= ad fourier coefficiets a k. i. determie the value of a. ii. determie the fourier series represetatio of. iii. use the result of part(ii) ad the differetiatio property of cotiuous time fourier series to help determie the fourier series coefficiets of (t). (April/May 8) (E) 8. Check whether the system is liear, time ivariat, causal, memory less ad stable y(t)=(t-)+(-t) ad y(t)= (April/May 8) (AZ) 4.

151 9. Determie the eergy ad power of the followig sigals: () (t)=t.u(t) () ()= (Nov/Dec 8) (A). Determie whether the sigal is periodic or ot () (t)=si (t) () ()=cos(/4) (Nov/Dec 8) (A). Evaluate the itegral () (Nov/Dec 8) (E). Write the mathematical equatio ad eplai the differet classificatio of CT ad DT sigals. (April/May ) (R). Determie eergy ad power of the sigals () ()= () (t)=5cos( t)+si5 t, (April/May ) (A) 4. A discrete time sigal is give by ()= Fid y()=(+) (April/May ) (A) 5. Eplai all classificatio of DT sigals with eamples for each category. (Nov/Dec ) (U) 6. Fid out whether the followig systems are i. Liear or o-liear ii. Causal or o- Causal iii. Fied or time-variet iv. Dyamic or istatataeous. () y()=()+ () (Nov/Dec ) (A) 7. Describe through eamples, the classificatio of cotiuous time sigals. PART - C (April/May ) (U). Sketch the followig sigals (t)= [u(t-)+u(t-4)] 4.4

152 (t)= (t -4) [ u(t -)-u(t -4)] (April / May 5) (AZ). (i). Check if (t) = 4cos( πt + ) + cos( 4πt) is periodic. (ii). For the system y() = log [()], check for liearity, causality, time ivariace ad stability (April/May 5) (AZ). Fid whether the followig sigals are periodic or aperiodic. If periodicfid the fudametal period ad fudametal frequecy X ()=si( πt)+ cos(πt) X ()=si( )cos( ) (May /Jue 6) (A) 4. Fid whether the followig sigals are power or eergy sigals. Determie power ad eergy of the sigals (May /Jue 6) (A) G(t)=5 cos(7 πt + )+si(9 πt + ) G()=(.5) u() 5. Fid whether the followig systems are time variat or fied. Also fid whether the systems are liear or oliear (May /Jue 6) (A) y()= a ()+b (-). 6. Determie whether the system is liear. Time variat, Causal ad memory less y(t)= (Nov /Dec 6) (AZ) 7. Sketch the followig sigals. (Nov /Dec 6) (AZ) (i)u(-t+) (ii)r(-t+) (iii)δ[+]+δ[]-δ[-]+δ[-] (iv) u[+] u[-+] Where u(t), r(t), δ[],u[] represet cotiuous time uit step, cotiuous time ramp, discrete time impulse ad discrete time step fuctios respectively. UNIT-II ANALYSIS OF CT SIGNAL PART - A. Fid the iverse Fourier trasform of X( )= ( ) (A) We kow, F [ ( )]=p 4.5

153 F [ ( )]=. Write the Fourier trasform pair of (t). (Nov/Dec ) (R) (t) X( ) X( )= (t)=. What are the differeces betwee Fourier series ad Fourier Trasform? (Nov/Dec 4) (U) S.No Fourier Series Fourier Trasform Fourier Series is calculated for Fourier Trasform is calculated for o periodic sigals. periodic as well as periodic sigals. Epad the sigal i time domai. Represets the sigal i frequecy domai. Three types of Fourier series such as trigoometric, polar ad comple epoetial. Fourier trasform has o such types. 4. A sigal (t)=cosp ft is passed through a device whose iput-output is related by y(t)= (t). What are the frequecy compoets i the output? (Nov/Dec 4) (AZ) y(t)= (t) =( cosp ft) = =/ + /cos Thus the frequecy preset i the output is f. 5. Defie Parsevel s relatio for cotiuous time periodic sigals. ( Nov/Dec 6) (R) It states that the total average power of the periodic sigal (t), is equal to the sum of the average powers of its phasor compoets. i.e., P=. 6. Fid the Laplace Trasform for the sigal (t)= - te u(t) ( Nov/Dec 6) (A) We kow that L[e u(t)] = By differetiatio with time property, L[t e u(t)] = L[-t e u(t)] =, Re(s)>-. 7. Fid the S-domai trasfer fuctio, if the poles are located at P = -+j, P =--j ad a ero at s=.5. (May/Jue 7) (A) H(S)= 4.6

154 = = 8. Write the coditio to be satisfied for the eistece of Fourier Trasform of aperiodic eergy sigal. (May/Jue 8) (R) Fourier trasform does ot eist for all aperiodic fuctios. The coditios for (t) to have Fourier Trasform are i. (t) is absolutely itegrable over (-, ), that is ii. (t) has fiite umber of discotiuities ad a fiite umber of maima miima i every fiite time iterval. 9. List out the four properties of Laplace trasform used i sigal aalysis. (May/Jue 8) ad (R) Time shiftig, (t-t ) e X(s) Time reversal, (-t) Frequecy shiftig, (t)e X(-s) X(s+a) Differetiatio i S-domai, (-t) (t).. List the Dirichlet coditios. (April/May 8) (R) i. The fuctio (t) should be withi the iterval T. ii. The fuctio (t) should have fiite umber of maima ad miima i the iterval T. iii. The fuctio (t) should have at the most fiite umber of discotiuities i the iterval T. iv. The fuctio should be absolutely itegrable. i.e., <. Fid Laplace trasform ad ROC of (t)= t e u(t). (April/May 8) (A) L[t e u(t)] = ; ROC : Re(s)>-a. What is the Laplace Trasform of (t)= e si t u(t). (Nov/Dec 4) (A) L[e si t u(t).]= 4. Fid Fourier Trasform of e at u(-t). (May/Jue 9) (A) We kow FT[e u(t)]= By the property of Time reversal, (-t)=x(-j ) 4.7

155 i.e., replace, t = -t ad j = -j FT[e u(-t)]= 5. State Rayleigh s eergy theorem. (R) Rayleigh s eergy theorem states that the eergy of the sigal may be writte i frequecy domai as superpositio of eergies due to idividual spectral frequecies of the sigal. 6. What are the methods for evaluatig iverse Laplace trasform. (R) The two methods for evaluatig iverse laplace trasform are. By Partial fractio epasio method.. By covolutio itegral. 7. Fid the LT for sigal (t)=u(t-) (April/may ) (A) Time shiftig property :LT{ (t-t )}= X(S)=u(t) ad t = 8. Fid FT of e at u(-t) (April/May ) (A) (t)=e at u(-t) is time reversal of sigal e -at u(t) Time reversal property of FT[(-t)]= 9. Defie Laplace Trasform (April/May ) (R) The Laplace trasform of a sigal (t) is defied as, where s is comple frequecy deoted by s=.. Determie the FT of sigal (t)=e -at u(t) (April/May ) (A) 4.8

156 .. Fid LT of uit ramp fuctio (Nov/Dec 8) (A) (t)=r(t)=t, for t =(t ) =-(- ) =.. State iitial value theorem ad fial value theorem of Laplace Trasform. (U) (April/May ) Iitial Value Theorem: The Iitial Value allows to calculate () directly from trasform X(S) without the eed for fidig iverse trasform of X(S). (t)= SX(S) Fial value theorem: The Fial value of (t) as t teds to ifiity may also be foud directly from Laplace Trasform X(S). (t)= SX(S) 4. What is the Fourier trasform of (t)= (April/May ) (U) PART B F - [ ]= F - [ ]= F[]=. Obtai the Fourier trasform of rectagular pulse of duratio T ad amplitude A. (8 mark, Nov/Dec ) (A). Fid out the iverse laplace trasform of F(S)=. (8 mark, Nov/Dec ). Fid the epoetial Fourier series of a impulse trai. Plot its magitude ad phase 4.9 (A)

157 Spectrum. ( mark, Nov/Dec ) (AZ) 4. What are the two types of Fourier series represetatio? Give the relevat mathematical represetatio. (4 mark, Nov/Dec ) (U) 5. Fid the trigoometric Fourier series for the periodic sigal (t) show i the figure give below: ( mark, April/May ) (A) 6. Eplai the Fourier spectrum of a periodic sigal (t). (6 mark, April/May ) (A) 7. Fid the Laplace trasform of the sigal (t)=e -at u(t)+e -bt u(-t). (8 mark, April/May ) (A) 8. Fid the Fourier trasform of (t)=e for - t = otherwise (8 mark, April/May ) (A) 9. Distiguish betwee Fourier series aalysis ad Fourier trasforms (4 mark, April/May ) (U). Obtai the Fourier series of the followig half wave rectified sie wave. ( mark, April/May ) (A). Fid the Laplace Trasform of the followig:. (t)=u(t-). (t)= (8 mark, April/May ) (A). Fid the step respose of RL circuit usig Laplace trasform.( mark,may ) (AZ). Determie the Laplace trasform of (t)=e cos (8 mark, May/Jue ) (A) 4. Fid the comple epoetial Fourier series coefficiet of the sigal (t)=si p t+cos4 t (8 mark, May/Jue ) (A) 5. Fid the trigoometric Fourier series of full wave rectifier. From the result calculate the coefficiets of epoetial Fourier series. (6 mark, Nov/Dec ) (A) 6. State ad prove parsevel s power theorem for cotiuous time sigal. (8 mark, Nov/Dec ) (U) 4.

158 7. Fid the Fourier trasform of the sigal (t)=e -at u(t), a> ad calculate magitude ad phase spectrum. (6 mark, Nov/Dec ) (AZ) PART - C. Determie the Fourier series epasio for a periodic ramp sigal with uit amplitude ad a period T. (April/May 5) (A). Fid the Fourier Trasform of (t) = te -at u(t). (April/May 5) (A). If (t) => X(ω), the usig time shiftig property show that (t+t) + (t-t) => X(ω) cos(ωt) (April/May 5) (A) 4. Fid the Iverse Laplace trasform of X(s) = (April/May 5) (A) 5. Obtai the Fourier series coefficiet ad plot the spectrum for the give waveform (May/Jue 6) (A) 6. From basic formula, determie the Fourier trasform of the give sigals. Obtai the magitude ad phase spectra of the give sigals. (t) = te -at u(t) a> (t) = e -a t a> (May/Jue 6) (A) 7. State ad prove Rayleigh s eergy Theorem. (May/Jue6) (U) 8. Fid the Fourier trasform of the sigal (t) = cosω t u(t) (Nov/Dec6) (A) 9. State ad Prove the multiplicatio ad covolutio property of Fourier trasform. (Nov/Dec6) (A) UNIT - III LINEAR TIME INVARIENT-CONTINUOUS TIME SYSTEMS PART - A. Give four steps to compute covolutio itegral. (Oct/Nov ) (R). Foldig : Oe of the sigal is first folded at t=.. Shiftig:The folded sigal is shifted right or left depedig upo time at which output is to be calculated.. Multiplicatio: The shifted sigal is multiplied to other sigal. 4. Itegratio: The multiplied sigals are itegrated ti get the covolutio output.. State the properties of covolutio. (May/Jue 9)(R). Commutative property : (t)*h(t)=h(t)*(t). Associative property : [(t)*h (t)]*h (t)=(t)* [h (t)*h (t)]. Distributive property: [(t)*h (t)]+[(t)*h (t)]=(t)* [h (t)+h (t)] 4.

159 . What is the relatioship betwee iput ad output of a LTI system? (R) y(t)= i.e., it is equal to covolutio. 4. What is the output of a system whose impulse respose h(t)=e -at for a delta iput? (Nov/Dec 5) (A) Give (t)= (t) We kow y(t) = Accordig to the property of covolutio with impulse respose is equal to the sigal itself. i.e, (t)* (t)=(t) y(t) =(t)*h(t) = (t)* e at = e at 5. What is the overall impulse respose of h (t) ad h (t) whe they are i (a) series (b)parallel. (Nov/Dec 5) (U) List out differet ways of iter coectig ay two system (April/May ) System coected i series: h (t) h (t) y(t) Overall impulse respose of the system : h(t)= h (t)*h (t) System coected i parallel: Overall impulse respose of the system : h(t)= h (t)+h (t) 6. How Laplace trasform aalysis will help i the aalysis of cotiuous time LTI systems? (May/Jue 6) (U) Laplace trasform ca aalysis both stable ad ustable system. 7. Fid the Fourier Trasform of the impulse respose. (May/Jue 6) (A) X(j )= h (t) h (t) y(t) = = =e = 8. Fid the Laplace trasform of the sigal u(t). (May/Jue 6) (A) 4.

160 X(s)= = We kow u(t)=, for t =, for t< X(s)= =[e -st /(-s)] =-(-/s) =/s 9. Fid the impulse respose h(t) for the system described by the differece equatio +5y(t) = (t)+ (AZ) sy(s)+5y(s) = X(s)+S(s) Y(s)(s+5) = X(s)(s+/) H(S)=Y(s)/X(s) = By partial fractio H(s) = = A = (s+5) S= - 5 = - 4/9 B = (s+/) S= - / = 4/9 H(s) = Take iverse LT we get h(t) = - 4/9 e 5t +4/9 e - (½)t. Defie atural respose (R) Natural respose is the respose of the system with ero iput. It depeds o the iitial state of the system. It is deoted by y (t).. Defie forced respose ad complete respose. (R) The Forced respose is the respose of the system due to iput aloe whe the iitial state of the system is ero. It is deoted by y f (t). The complete respose of a LTI-CT system is obtaied by addig the atural respose ad forced respose. y(t)= y (t)+ y f (t).. Fid the iitial ad fial value, give that X(s)= (May/Jue 7) (A) Iitial Value theorem : () = s 4. [sx(s)] ()= s [s ] = s [ ] =

161 Fial value theorem : ( ) = s [sx(s)] = s [s ] =. Draw the direct form-ii realiatio of the system described by the differetial equatio (April/May 5) (U) From the equatio a =5, a =4 ad b = (t) + y(t) + -a = -5 b = -a = Defie Eige fuctio ad Eige value. (R) I the equatio give below, y(t)=h(s)e st H(s) is called Eige value ad e st is called Eige fuctio. 5. Defie Causality ad stability usig poles. (R) For a system to be stable ad causal, all the poles must be located i the left half of the s plae. 6. What are the differet forms of realiatio (April/May ) (R). Direct form I realiatio. Direct form II realiatio. Cascade form realiatio 4. Parallel form. 7. What is the coditio for a system to be causal? (April/May ) (U) A LTI cotiuous system is causal if ad oly if its impulse respose is ero for egative values of t. h(t)=, for t<. 8. State ad write covolutio itegral formula. (Nov/Dec 8) (R) The output of system ca be obtaied usig covolutio itegral if the iput to the system (t ad impulse respose of system h(t) are kow. The covolutio itegral is give by y(t)=(t)*h(t) =. 9. Give differetial equatio.fid the frequecy respose of system. (Nov/Dec 8) (U) 4.4

162 Frequecy respose: PART- B. Solve the differetial equatio with y( - )= ad ad (t)=u(t).fid the system trasfer fuctio, frequecy respose ad impulse respose.(nov/dec ) (AZ). The system is described by iput output relatio, Fid the system trasfer fuctio, frequecy respose ad impulse respose. (Nov/Dec ) (AZ). Draw the direct form I ad II implemetatio of system described by (Nov/Dec ) (AZ) 5. Eplai the steps to compute the covolutio itegral. ( April/May ) (A) 6. Fid the covolutio of the followig sigal. (April/May ) (A) (t)=e -t u(t) ad h(t)=u(t+) 7. Usig Laplace trasform, fid the impulse respose of a LTI system described by the differetial equatio (April/May ) (A) 8. Eplai the properties of covolutio itegral. (April/May ) (U) 9. Realie the followig i direct form II (April/May ) (AZ). Obtai the covolutio of the sigals (t)=e -at u(t), (t)=e -bt u(t) usig Fourier Trasform. (April/May ) (A). The iput ad output of a causal CTI system are related by the differetial equatio.fid impulse respose of the system.. Compute ad plot the covolutio y(t) of the give sigals i. (t)=u(t-)-u(t-5), h(t)=e -t u(t) 4.5 (April/May ) (AZ)

163 ii. (t)=u(t), h(t)=e -t u(t) (May/Jue &April/May ) (A). The LTI system is characteried by impulse respose fuctio give by H(S)=, ROC: Re> -.Determie the output of a system whe it is ecited by the iput (t)= -e -t u(-t)-e -t u(t). (May/Jue ) (AZ) 4. Usig Laplace trasform, fid the output respose of the system described by the differetial equatio., iput sigal=(t)=e -at u(t), iitial coditio: y( + )=,. (Nov/Dec ) (AZ) 5. Fid the covolutio of h(t) ad (t) usig graphical method.(nov/dec ) (A) h(t)=t, <t<t (t)=u(t), <t<t 6. Cosider a CT-LTI system fid the system fuctio ad determie if the impulse respose h(t) for the system is causal, the system is stable ad the system is either causal or stable. (Nov/Dec 8& April/May ) (AZ) 7. Determie the voltage y(t) i the circuit show i figure for a applied voltage (t)=e - t u(t). the voltage across the capacitor at time t= is 5V. April/May ) (AZ) 8. Solve the followig differetial equatio usig laplace trasform (April/May ) 9. Fid the impulse ad step respose of the followig system H(S)= (AZ) (April/May ) (AZ). Cosider the system show i figure with RC=. Fid frequecy respose of the output across the capacitor ad sketch H(ω) as a fuctio ω. (April/May 8) (AZ). Fid the trasfer fuctio ad impulse respose of the system. (April/May 8) (A). Give (t)=e -at u(t) ad h(t)=u(t-) covolve ad fid the respose y(t). (Nov/Dec 8) (A) 4.6

164 . Give (t)=e -t u(t) ad h(t)=e -t u(t) usig the properties of cotiuous time fourier trasform fid the respose y(t). (Nov/Dec 8) (A) 4. Defie covolutio itegral ad derive its equatio. (Nov/Dec ) (U) 5. A stable LTI system is characteried by the differetial equatio,fid the frequecy respose ad impulse respose usig Fourier trasform. (Nov/Dec ) (A) 6. Draw direct form, cascade ad parallel form of a system with system fuctio H(S)= (Nov/Dec ) (AZ). Solve the differetial equatio (D +D+)y(t)=D(t)usig the iput (t)=e -t ad with iitial coditios are y( + )= ad y ( + )= (APRIL-MAY 5) (A). Draw the block diagram represetatio for H(s) = (APRIL-MAY 5) (AZ). For a LTI system with H(s) = fid the differetial equatio. Fid the system output y(t) to the output (t)= e -t u(t) (APRIL-MAY 5) (AZ) 4. Usig graphical method covolve (t)= e -t u(t) with h(t)= u(t+). (APRIL-MAY 5) (A) PART - C. Usig graphical covolutio, fid the respose of the system whose impulse respose is h(t)= e -t u(t) for a iput (t)= (MAY-JUNE 6) (A). Realie the followig is idirect form II. A LTI system is defied by the differetial equatio (MAY-JUNE 6) (AZ) with y( - )= ad ad (t)=u(t).fid the system respose y(t). (MAY-JUNE 6) (AZ) 4.7

165 4. Determie the frequecy respose ad impulse respose for the system described by the followig differetial equatio. Assume ero iitial coditios. (MAY-JUNE6) (AZ) 5.Covolute the followig sigals (t)= e -t u(t) h(t)= u(t+). (NOV-DEC 6) (A) 6. A system is described by the differetial equatio. Fid the trasfer fuctio ad output sigal y(t)for (t)=δ(t) (NOV-DEC 6) (AZ) UNIT IV ANALYSIS OF DISCRETE TIME SYSTEMS PART - A. Obtai Z- trasform of []={,,,4} (Nov/Dec 5) (A) The sigal is give i sequece. We kow by power series X()={..(-),(-),(),(),()..} X() = = =()+() - +() - +() - = State samplig theorem. (Nov/Dec 7) (R) A bad limited sigal (t) with X(j )= for > is uiquely determied determied from its samples (T), if the samplig frequecy f s f m, i.e., samplig frequecy must be at least twice the highest frequecy f m,which uder samplig theorm must be atleast twice the highest frequecy preset i the sigal.. Write the properties of Regio of covergece of the -trasform. (Nov/Dec 7) (R) i. The ROC is a rig or disk i the -plae cetered at the origi. ii. The ROC caot cotai ay poles. iii. If [] is a causal sequece the ROC is the etire -plae ecept at = iv. The ROC of a LTI stable system cotais the uit circle. 4. State ad prove time shiftig property of DTFT. (Nov/Dec 7) (U) Time shiftig X(t) X(j ) The, (t-t ) e X(j ) F[(t-t )] = t-t = p ad dt = dp 4.8

166 F[(t-t )] = = F[(t-t )]= X(j ) 5. Write DTFT pair. (May/Jue 7) (U) X(t) X(j ) (t) = for all X(j )=, for all t 6. List ay two properties of Z-trasform. ( May/Jue 7) (U). Liearity: {a []+b []} {ax []+bx []}. Time shiftig: [-k] Z k X[] 7. Defie Parsevels relatio for discrete time periodic sigals. (May/Jue 6) (R) The Parsevel s relatio for discrete time periodic sigals is give by 8. Defie Zero paddig. (R) The method of appedig ero i the give sequece is called as Zero paddig. 9. State the covolutio property of Z trasforms. (R) The covolutio property states that the covolutio of two sequeces i time domai is equivalet to multiplicatio of their Z trasforms.. Obtai the iverse trasform of X()=/-a, > a (A) Give X()= By time shiftig property, X()=a. u(-). List the methods of obtaiig iverse Z trasform. (R) Iverse trasform ca be obtaied by usig. Partial fractio epasio.. Cotour itegratio. Power series epasio 4. Covolutio.. Fid the - trasform for []= u[-]. (May/Jue 7) (A) We kow ZT[ u[]]= ROC: > a Usig time shiftig property of -trsform we have ZT{[-k]}= The ZT{ u[-]} = = ROC : > a. State the coditio for eistece of DTFT? (R) 4.9

167 The coditios are If ()is absolutely summable the If () is ot absolutely summable the it should have fiite eergy for DTFT to eit. 4. Write ay two properties of Z-trasform (April/May 8) (R) Time shiftig property: If X(Z)=ZT{()} with the iitial coditio for () are ero, the ZT{(m)}=,where m is a positive or egative iteger. Scalig property: If X(Z)=ZT{()}, ROC:r < <r the ZT{a ()}=X(a Z), ROC: r < < r 5. What is the Z- trasform ad associated ROC of the sigal ()=u(- )? (Nov/Dec 8) (A) By time shiftig property ZT{(-m)}= ZT{u(- )}= 6. Defie aliasig effect. (April/May ) (R) The super impositio of high frequecy compoet o the low frequecy is kow as frequecy aliasig. Because of aliasig the spectrum X(j ) is o loger recoverable from spectrum of X S (j ). To avoid aliasig the samplig frequecy must be greater tha twice the highest frequecy preset i the sigal. 7. Defie oe sided Z-trasform. (April/May ) (R) Oe sided Z-trasform:Right had sequece is oe for which ()= for all <, where is positive or egative but fiite. If is greater tha or equal to ero,resultig sequece is causal or positive time sequece. For such type of sequece the ROC is etire Z-plae ecept at Z=. Left had sequece is oe for which ()= for all, where is positive or egative but fiite. If, the resultig sequece is aticausal sequece. For such type of sequece the ROC is etire Z-plae ecept at Z= 8. Fid DTFT OF u() (Nov/Dec ) (A) X(e )= = = = 9. What is the mai coditio to be satisfied to avoid aliasig? (Nov/Dec ) (U) To avoid aliasig the samplig frequecy must be greater tha twice the highest frequecy preset i the sigal. 4.

168 If f s = the.. If X(e ) is the FT of (), Fid the FT of * (-) i terms of X(e ). (Nov/Dec 8) (A) By time reversal property: (-) X(e ) ad by cojugate property: (-) X (e ). What is the relatioship betwee Z-Trasform ad DTFT? (Nov/Dec 8) (U) The Z-Trasform of sequece () is give by X(Z)=ZT{()} = ----() Where Z=r e ----() Sub () i () we get X(re )= ----() The Fourier Trasform of () is give by X(e )= ----(4) Equatio () ad (4) are idetical whe r=.i this Z-plae this correspods to the locus of poits o the uit circle.hece H(e ) is equal to H(Z) evaluated alog the uit circle. X(e )= PART B. State ad prove the property of Z-trasform. (Nov/Dec ) (U). Determie the Z-trasform of i. ()= ii. u(), for. (Nov/Dec ) (A). State ad prove samplig theorem. (Nov/Dec ) (U) 4. Fid the iverse Z trasform of the followig: (Nov/Dec ) (A) 4.

169 i. X(Z)= ii. X(Z)= 5. Fid the Fourier trasform of ()=A, =, (April/May )(A) 6. Eplai ay four properties of DTFT. (April/May )(U) 7. Fid Z-trasform of the give sigal () ad fid ROC. ()=[si ]u() (April/May )(A) 8. Describe the samplig operatio ad eplai how aliasig error ca be preveted. (April/May ) (U) 9. What is aliasig? Eplai with a eample. (April/May ) (R). State ad prove the followig properties of DTFT: (April/May ) (U) Covolutio property Time shiftig Time reversal Frequecy shiftig.. Obtai iverse Z trasform of X(Z)= for ROC: (April/May ) (A). Determie the Z-trasform ad sketch the pole ero plot with the ROC for each of the followig sigals. (May/Jue ) (A). ()=(.5) u()-(/) u(). X()=(/) u()+(/) u(-).. Fid iverse Z-trasform of the (May/Jue ) (A) 4. Epress the Fourier trasforms of the followig sigals i terms of X(e j ). ()=(-) 4.

170 . ()=(-) (). (May/Jue ) (A) 5. Prove samplig theorem ad eplai how the origial sigal ca be recostructed from the sampled versio. (Nov/Dec ) (AZ) 6. Fid DTFT for sigal ()=u(-). (Nov/Dec ) (A) 7. Determie the Z trasform for the sequece ()=4 cos u(--).sketch the pole-ero plot ad idicate the ROC. (April/May 8) (A) 8. Fid iverse Z trasform of X(Z)=, > (April/May 8) (A) 9. Suppose that the algebraic epressio for the Z trasform of () is How may differet regios of covergece could correspod to X(Z)? (April/May 8) (AZ). Fid the Nyquist s frequecy ad Nyquist s rate for each of the followig sigals:. (t)=5 cos(5πt). (t)=5sect(t/) (Nov/Dec 8) (A). Fid iverse Z trasform of X(Z)= (Nov/Dec 8) (A). Determie the iitial ad fial values for the sigal () with the trasfer fuctio ()= (Nov/Dec 8) (A). Cosider a aalog sigal (t)= cos πt i. determie the miimum samplig rate to avoid aliasig ii. If samplig rate F S =H, what is the DT sigal after samplig? iii. fid the DT sigal give is sampled at 75 H. iv. What is the frequecy <F<F S / of a siusoidal that yields samples idetical to those obtaied i part iii. (April/May ) (AZ) 4. Determie the Z trasform, the ROC ad the locatio of poles ad eros of X(Z) the sigal ()= (April/May ) (A) 4.

171 5. Fid the iverse Z Trasform of X(Z)= with ROC >/ usig a power series epasio. (April/May ) (A) 6. Eplai the recostructio of the origial sigal from the sampled sigal. (April/May ) (U) 7. Fid the discrete time sigal of X(e -jw )= (April/May 8) (A) 8. Determie the Z trasform ad the ROC for the followig sequece (April/May 8) i. ()=(-) u(--) ii. ()=(/) +(-)(/) (A) 9. State ad prove parsevel s relatio for discrete aperiodic sigals.(nov/dec 8) (U) PART - C. A cotiuous time siusoid cos(πft+θ) is sampled at a rate f s = H. Determie the resultig sigal samples if the iput sigal frequecy f is 4H, 6H ad H respectively. (April/May 5) (A). Prove the followig DTFT properties (i) () => j (ii) () e jωc => X(Ω Ωc) (April/May 5) (A). Fid the DTFT of () = (/) - u(-) (April/May 5) (A) 4. Usig suitable trasform properties fid X() if () = (-)(/) - u(-) (April/May 5) (A) 5. Fid the Z-trasform of () = a <a< (April/May 5) (A) 6. State ad prove samplig theorem. (May/Jue6) (U) 7. What is Aliasig? Eplai the steps to be take to avoid aliasig. (May/Jue6)(U) 8. State ad Prove the followig theorems (i) Covolutio theorem of DTFT (ii) Iitial value theorem of -trasform. (May/Jue6) (U) 9. Discuss the effects of udersamplig a sigal usig ecessary diagram. (Nov/Dec6) (U). Fi the Z-trasform of () = a u() = b u(--) ad specify its ROC. (Nov/Dec6) (A). State ad Prove the time shiftig property ad time reversal property of -trasform. (Nov/Dec6) (A) 4.4

172 UNIT V LINEAR TIME INVARIENT DISCRETE TIME SYSTEMS PART-A. Determie the trasfer fuctio of the system described by y[]=ay[-]+[] (Nov/Dec 5) (U) Y()=a - Y()+X() Y()-a - Y()=X() Y()(-a - )=X() Trasfer fuctio, H()=Y()/X() =/(-a - ). Fid the covolutio sum for []={,,,} ad h[]={,,,} (May/Jue 6) (A) [] = h[] = []*h[] = {,,5,6,5,,}. What is the differece betwee the spectrums of the CT sigal ad the spectrum of the correspodig sampled sigal? (Nov/Dec 4) (U) i. The spectrums of CT sigal ad sampled sigal are related as, X ( ) = ii. This meas spectrum of sampled sigal is periodic repetitio of spectrum of CT sigal. iii. It repeats at samplig frequecy ad amplitude is also multiplied by. 4. Differetiate betwee atural respose ad forced respose. (May/Jue 7) (U) Natural respose: It is the respose of the system with ero iput.it depeds o the iitial state of the system. Forced respose: It is the respose of the system due to iput aloe whe the iitial state of the system is ero. 5. What is the respose of a LSI system with impulse h[]= []+ [-] for the iput []={,,}? (April/May 5) (A) h[]= []+ [-] ca be represeted as h[]={,} The respose of a LSI system is give as y[]={,4,7,6} 4.5

173 6.Is the output sequece of a LTI system fiite or ifiite whe the iput [] is fiite? Justify your aswer. (Oct/Nov ) (AZ) If the impulse respose of the system is ifiite, the output sequece is ifiite eve though iput is fiite. For eample cosider, Let iput [] = [] fiite legth Impulse respose, h[] = u[] Ifiite legth Output sequece, y[]=h[]*[] = u[]* [] = u[] 7. List the steps ivolved i fidig covolutio sum? (April/May 9) (R). Foldig. Shiftig. Multiplicatio 4. Summatio 8. List the properties of covolutio? (April/May 9) (R). Commutative property of covolutio () * h() = h() * () = y(). Associative property of covolutio [ () * h()] * h() = () * [h() * h()]. Distributive property of covolutio () * [h() + h()] = () * h() + () * h() 9. Determie the rage for which the LTI system with impulse respose h()=a u() is stable. (Nov/Dec ) (AZ) The impulse respose is stable whe H(Z) pole is H(Z)=, where the pole is at =a ad hece for system to be stable the a value should be a <.. Fid system fuctio for the give differece equatio y()=.5y(-)+() (May/Jue ) (A) Y(Z)=.5 - Y(Z)+y(-)+X(Z) where y(-)= =.5 - Y(Z)+X(Z) Y(Z)-.5 - Y(Z)=X(Z) Y(Z)(-.5 - )=X(Z) H(Z)= =.. Compare FIR ad IIR filter realiatio (U) FIR: It stads for fiite impulse respose ad it is o-recursive type. The preset output sample depeds o preset iput samples ad previous iput samples IIR: It stads for ifite impulse respose.preset output sample depeds o preset iput ad past iput samples ad output samples. It is recursive type.. State the coditio for a DT-LTI system to be causal ad Stable. (April/May 8) (U) Causal system: A LTI system is causal if ad oly if, h() = for <.This is the sufficiet ad ecessary coditio for causality of the system. 4.6

174 Stable system:the bouded iput () produces bouded output y() i the LTI system oly if, Whe this coditio is satisfied,the system will be stable.. Defie o recursive ad recursive systems? (April/May 8) (R) No-recursive system: Whe the output y() of the system depeds upo preset ad past iputs the it is called o-recursive system. Recursive system: Whe the output y() of the system depeds upo preset ad past iputs as well as past outputs, the it is called recursive system. 4. What is the advatage of direct form II over direct form I structure? (U) The direct form II structure has reduced memory requiremet compared to direct form I structure. 5. A causal DT system is BIBO stable oly if its trasfer fuctio has. (U) A causal DT system is stable if poles of its trasfer fuctio lie withi the uit circle. 6. Determie the rage of values of the parameter a for which liear time ivariat system with impulse respose h()=a u() is stable? (U) H()=, There is oe pole at =a. The system is stable, if all its poles.i.e., withi the uit circle. Hece a < for stability. 7. What are the drawbacks of trasfer fuctio aalysis method? (R) i. Trasfer fuctio is defied uder ero iitial coditios. ii. Trasfer fuctio approach ca be applied oly to liear time ivariat systems. iii. It does ot give ay idea about the iteral state of the system. iv. It caot be applied to multiple iput multiple output systems. v. It is comparatively difficult to perform trasfer fuctio aalysis o computers. PART-B. The system fuctio of the LTI system is give as. Specify the ROC of ad determie for the followig coditio. Stable system ad Casual system. (Nov/Dec ) (A). Obtai the direct form II structure for. (Nov/Dec ) (AZ). A discrete time casual system has a trasfer fuctio 4.7

175 i) Determie the differece equatio of the system. ii) Show pole ero diagram. iii) Fid the impulse respose (Nov/Dec ) (AZ) 4. Fid the impulse respose of the discrete time system described by the differece equatio. (April/May ) (AZ) 5. Discuss the block diagram represetatio for LTI discrete time systems. 6. Obtai the cascade realiatio of (April/May ) (U). (April/May ) (AZ) 7. Obtai the relatioship betwee DTFT ad Z trasforms. (April/May ) (U) 8.Fid the impulse respose of the differece equatio. (May/Jue ) (A) 9.Draw the direct form II block diagram represetatio for the system fuctio. (May/Jue ) (AZ). Fid the iput which produces output whe passed through the system havig. (May/Jue ) (AZ). Determie the system fuctio ad impulse respose of the casual LTI system defied by the differece equatio, Usig Z-trasform, determie if. (Nov/Dec ) (A). Determie the trasfer fuctio ad impulse respose for the causal LTI system described by the differece equatio usig Z trasform. (Nov/Dec ) (A) 4.8

176 . Draw the direct form I ad direct form II implemetatio of the dsyatem described by differece equatio 4. Give y()+4y(-)+4y(-)=(). Fid the impulse respose h(). (Nov/Dec ) (AZ) (April/May ) (A) 5. Fid the Z trasform of ()=(/) u()+( /) u(--) (April/May 8) (A) 6. Covolve () = α u(), < α < h() = u() (April/May 8) (A) 7. Fid Covolutio sum of ()= (/) u() ad h() = δ () + δ (-) + (/) u(). (April/May 8) (A) 8. Fid Trasfer fuctio of system y() = () + / (-) + (-), for ( ) = u(). Plot the pole - ero locatios i Z Plae.(April/May 8) (AZ) 9. Determie system fuctio ad impulse respose of Causal LTI system defied by differetial equatio: (April/May ) (A). Fid the trasposed direct form II realiatio for the system described by differece equatio y()=()-/(-)-/4y(-)-/4y(-) ad write dow the set of equatios correspodig to this realiatio. (April/May ) (A). Discuss the block diagram represetatio for LTI DT system. (U). Compute Covolutio of two sequeces show. Plot y() Vs (A) () h(). Fid overall respose of the system show with : (Nov/Dec 8) (AZ) h()= δ(), h() = (-)u() & h() = δ() + u(-) + δ(-) h() - h() + h() 4. Derive trasfer fuctio of FIR & IIR system. (Nov/Dec 8) (U) 5. Realie the give DT system i Cascade & Parallel form: (Nov/Dec 8) (AZ) 4.9

177 6. Whe a iput () = δ (-) is applied to a causal LTI system, whose output is foud to be y() = (-/) + 8(/4). Fid the impulse respose h() of the system. (AZ) PART C. Determie the impulse respose ad step respose of y() + y(-) y(-) = (- ) + (-). (April/May 5) (A). Fid the covolutio sum betwee () = {,4,,} ad h( = {,,,}. (April/May 5) (A). A causal system has () = δ()+ /4 δ(-) /8 δ(-) ad y() = δ ( )- /4 δ(- ). Fid the impulse respose ad output if () = (/) u(). (April/May 5) (A) 4. Compare recursive ad o recursive systems. (April/May 5) (U) 5. Realie the followig systems i cascade form (May/Jue6) (A) 6. Covolve () = {,,,} ad h() = {,-,-,4) (May/Jue6) (A) 7. A system is govered by a liear costat coefficiet differece equatio y() =.7 y(-). y(-) + () (-). Fid the output respose of the system y() for a iput () = u(). (May/Jue6) (A) 8. Covolve the followig sigals () = u() u(-) h() = (.5) u() (Nov/Dec6) (A) 9. Determie whether the give system is stable by fidig H() ad plottig the poleero diagram y() = y(-) -.8y(-) +() +.8(-) (Nov/Dec6) (A) ADDITIONAL UNIVERSITY QUESTIONS. Defie power sigal (April/May 5) (R). How the impulse respose of a discrete system is useful i determiig its stability ad causality? (April/May 5) (U). Fid the Fourier co-efficiets of the sigal 4.4 (April/May 5) (A) 4. Draw the spectrum of a CT rectagular pulse (April/May 5) (U) 5. Give (t)=δ(t). Fid X(s) ad X(ω). (April/May 5) (A) 6. State the covolutio itegral (April/May 5) (R) 7. Determie the Nyquist samplig rate for (t)=si(πt)+si (πt). (April/May 5)(A) 8. List the methods used for fidig the iverse Z trasform. (April/May 5) (R) 9. Name the basic buildig blocks used i LTIDT system block diagram

178 (April/May 5)(U). Write the th order differece equatio. (April/May 5) (R). Sketch the followig sigals : Rect ( ; 5 ramp (.t) (April/May 6) (U). Give g()= e --, Write out ad simplify the fuctios (i)g(-) (ii) g ( ) (April/May 6) (A). What is the iverse fourier trasform of (i) e -jπft (ii)δ(f-f ) (April/May 6)(A) 4. Give the laplace trasform of (t)= e -t u(t)- e -t u(t) with ROC. (April/May 6) (A) 4.4

179 PANIMALAR ENGINEERING COLLEGE DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING EC 64 ELECTRONIC CIRCUITS - I SOLVED TWO MARKS, PART - B, PART-C QUESTIONS ASSIGNMENT QUESTIONS (FOR THIRD SEMESTER ECE 7-8) 5.