GATE EC : EL ECTRONI CS & COM M UNI CATI ON ENGI NEERI NG. N o. of Qu est i on s : 65 Maxi mum Mar ks : 100 GENERAL APTITUDE

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1 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) GATE - 4 EC : EL ECTRONI CS & COM M UNI CATI ON ENGI NEERI NG N o. of Qu est i on s : 65 Mai mum Mar ks : GENERAL APTITUDE (Q. 5) Carry One Mark Each.. Choose the most appropriate phrase from the options given below to complete the following sentence. The aircraft take off as soon as its flight plan was filed. (a) is allowed to (b) will be allowed to (c) was allowed to (d) has been allowed to. Read the statements: All women are enterpreneurs. Some women are doctors. Which of the following conclusions can be logically inferred from the above statements? (a) All women are doctors (b) All doctors are enterpreneurs (c) All enterpreneurs are women (d) Some enterpreneurs are doctors. Choose the most appropriate word from the options given below to complete the following sentence. Many ancient cultures attributed disease to supernatural causes. However, modern science has largely helped such notions. (a) impel (c) propel (b) dispel (d) repel 4. The statistics of runs scored in a series by four batsmen are provided in the follwing table. Who is the most consistent batsman of these four? (a) K (c) M Batsman Average Standard deviation K L M N (b) L (d) N 5. What is the net number in the series? (Q. 6 ) Carry Two Marks Each. 6. Find the odd one from the following (a) W,E,K,O (c) F,N,T,X (b) I,Q,W,A (d) N,V,B,D 7. For submitting ta returns, all resident males with annual income below ` lakh should fill up Form P and all resident females with income below ` 8 lakh should fill up Form Q. All people with incomes above ` lakh should fill up Form R, ecept non residents with income above ` 5 lakhs, who should fill up Form S. All others should fill Form T, An eample of a person who should fill From T is (a) a resident male with annual income ` 9 lakh (b) a resident female with annual income ` 9 lakh (c) a non-resident male with annual income ` 6 lakh (d) a non-resident female with annual income ` 6 lakh 8. A train that is 8 metres long, travelling at a uniform speed, crosses a platform in 6 seconds and passes a man standing on the platform in seconds. What is the length of the platform in metres? 9. The eports and imports (in crores of `) of a country from to 7 are given in the following bar chart. If the trade deficit is defined as ecess of imports over eports, in which year is the trade deficit /5th of the eports? Eports Imports (a) 5 (b) 4 (c) 7 (d) 6

2 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC). You are given three coins: on has heads on both faces, the second has tails on both faces, and the third has a head on one face and a tail on the other. You choose a coin at random and toss it, and it comes up heads. The probability that the other face is tails is (a) /4 (b) / (c) / (d) / ELECTRONICS AND COMMUNICATION (Q. 5) Carry One Mark Each.. For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold? (a) (M T ) T M (b) (cm) T c(m) T (c) (M + N) T M T + N T (d) MN NM. In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is. C is a closed path in the z- plane given by z. The value of the integral (a) 4 ( + j) (b) 4 ( j) (c) 4 ( + j) (d) 4 ( j) C z z 4 j dz is z j 4. A real (4 4) matri A satisfies the equation A I, where I is the (4 4) identity matri. The positive eigen value of A is. 5. Let X, X and X be independent and identically distributed random variables with the uniform distribution on [, ]. The probability P {X is the largest} is 6. For maimum power transfer between two cascaded sections of an electrical network, the relationship between the output impedance Z of the first section to the input impedance Z of the second section is (a) Z Z (b) Z Z (c) Z Z * (d) Z Z * 7. Consider the configuration shown in the figure which is a protion of a larger electrical network i4 i R R i5 R i For R and currents i A, i 4 A, i 5, 4 A, which one of the following is TRUE? (a) i 6 5 A (b) i 4 A (c) Data is sufficient to conclude that the supposed currents are impossible (d) Data is insufficient to identify the currents i, i, and i 6 8. When the optical power incident on a photodiode is W and the responsivity is.8 A/W, the photocurrent generated (in A) is. 9. In the figure, assume that the forward voltage drops of the PN diode D and Schottky diode D are.7 V and. V, respectively. If ON denotes conducting state of the diode and OFF denotes non-conducting state of the diode, then in the circuit, V k D (a) both D and D are ON (b) D is ON and D is OFF (c) both D and D are OFF (d) D is OFF and D is ON. If fied positive charges are present in the gate oide of an n-channel enhancement type MOSFET, it will lead to (a) a decrease in the threshold voltage (b) channel length modulation (c) an increase in substrate leakage current (d) an increase in accumulation capacitance. A good current buffer has (a) low input impedance and low output impedance (b) low input impedance and high output impedance (c) high input impedance and low output impedance (d) high input impedance and high output impedance D i i6

3 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC). In the ac equivalent circuit shown in the figure, if i in is the input current and R F is very large, the type of feedback is J4 Q4 clk K4 J Q J Q clk clk K K J Q clk K J Q clk K small signal input iin M RD RF (a) voltage-voltage feedback (b) voltage-current feedback (c) current-voltage feedback (d) current-current feedback. In the low-pass filter shown in the figure, for a cut-off frequency of 5 khz, the value of R (in k) is. k C V i R + R M nf 4. In the following circuit employing pass transistor logic, all NMOS transistors are identical with a threshold voltage of V. Ignoring the body-effect, the output voltages at P, Q and R are, 5 V (a) 4V, V, V (c) 4V, 4V, 4V 5 V 5 V RD V o 5 V Vout P Q R (b) 5V, 5V, 5V (d) 5V, 4V, V 5. The Boolean epression (X + Y) (X + Y ) + (X Y)+X simplifies to (a) X (c) XY (b) Y (d) X+Y 6. Five JK flip-flops are cascaded to form the circuit shown in figure. Clock pulses at a frequency of MHz are applied as shown. The frequency (in khz) of the waveform at Q is. clock 7. A discrete-time signal [n] sin( n),n being an interger, is (a) periodic with period. (b) periodic with period. (c) periodic with period /. (d) not periodic. 8. Consider two real valued signals, (t) band-limited to [ 5 Hz, 5 Hz] and y(t) band-limited to [ khz, khz]. For z(t) (t). y(t), the Nyquist sampling frequency (in khz) is. 9. A continuous, linear time-invariant filter has an impulse reponse h(t) described by for t h( t) otherwise When a constant input of value 5 is applied to this filter, the steady state outout is.. The forward path transfer function of a unity negative feedback system is given by K G( s) ( s )( s ) The value of K which will place both the poles of the closed-loop system at the same location, is.. Consider the feedback system shown in the figure. The Nyquist plot of G (s) is also shown. Which one of the following conclusions is correct? + k Im G(j ) + G(s) (a) G(s) is an all-pass filter Re G(j ) (b) G(s) is a strictly proper transfer function (c) G(s) is a stable and minimum-phase transfer function (d) The closed-loop system is unstable for sufficiently large and positive k

4 4 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC). In a code-division multiple access (CDMA) system with N 8 chips, the maimum number of users who can be assigned mutually orthogonal signature sequences is. The capacity of a Binary Symmetric Channel (BSC) with cross-over probability.5 is 4. A two-port network has scattering parameters given by [S] s s s s. If the port- of the two- port is short circuited, the s parameter for the resultant one-port network is (a) (c) s s s s s s s s s s s s (b) (d) s s s s s s s s s s s s 5. The force on a point charge +q kept at a distance d from the suface of an infinite grounded metal plate in a medium of permittivity is (a) q (b) 6 d (c) (d) q 6 d q 4 d away from the plate towards the plate towards the plate (Q. 6 55) Carry Two Marks Eack. 6. The Taylor series epansion of sin + cos is (a) + (b) + (c) (d) For a function g(t), it is given that g(t) e jt dt e for any real value. If y(t) (a) (c) j g( ) d, then y(t) dt is (b) j (d) j 8. The volume under the surface z(, y) + y and above the triangle in the -y plane defined by { y and } is. 9. Consider the matri j 6 Which is obtained by reversing the order of the columns of the identity matri I 6. Let P I 6 + J 6, where is a non-negative real number. The value of for which det(p) is.. A Y-network has resistances of each in two of its arms, while the third arm has a resistance of. In the equivalent -network, the lowest value (in ) among the three resistances is.. A V rms source supplies power to two loads connected in parallel. The first load draws kw at.8 leading power factor and the second one draws kva at.8 lagging power factor. The comple power delivered by the source is (a) (8 + j.5) kva (c) ( + j.5) kva (b) 8 j.5) kva (d) ( j.5) kva. A periodic variable is shown in the figure as a function of time. The root-mean-square (rms) value of is. T/ T/. In the circuit shown in the figure, the value of capacitor C (in mf) needed to have critically damped response i(t) is. 4 i(t) 4 H C + Vo 4. A BJT is biased in forward active mode. Assume V BE.7 V, kt/q 5 mv and reverse saturation current Is - A. The transconductance of the BJT (in ma/v) is. t

5 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) 5 5. The doping concentrations on the p-side and n-side of a silicon diode are 6 cm and 7 cm, respectively. A forward bias of. V is applied to the diode. At T K, the intrinsic carrier concentration of silicon n i.5 cm and kt q 6 mv. The electron F V CC + V R k R C 5 k F v concentration at the edge of the depletion on the p-side is v i (a). 9 cm (b) 6 cm R k RS (c) 7 cm (d).5 6 cm 6. A depletion type N-channel MOSFET is biased in its linear region for use as a voltage controlled resistor. Assume threshold voltage V TH.5 V, V GS.V, V DS 5V, W/L, C OX 8 F/cm and n 8cm /V s. The value of the resistance of the voltage controlled resistor (in) is. 7. In the voltage regulator circuit shown in the figure, the op-amp is ideal. The BJT has V BE.7 V and, and the zener voltage is 4.7 V. For a regulated output of 9 V, the value of R (in ) is. V I V k V Z 4.7 V + V O 9 V k 8. In the circuit shown, the op-amp has finite input impedance, infinite voltage gain and zero input offset voltage. The output voltage V out is R R R E k C E mf 4. The output F in the digital logic circuit shown in the figure is X Y Z XOR XNOR (a) F X YZ + X Y Z (b) F X Y Z + X Y Z (c) F X Y Z + XYZ (d) F X Y Z + XYZ AND 4. Cosider the Boolean function, F(w,,y,z) wy + y + w yz + w y + y + y z. Which one of the following is the complete set of essential prime implicants? (a) w, y, z, z F (b) w, y, z R I I + Vout (c) y, y z (d) y, z, z 4. The digital logic shown in the figure satisfies the given state diagram when Q is connected to input A of the XOR gate. (a) I (R + R ) (b) I R (c) I R (d) I (R + R ) 9. For the amplifier shown in the figure, the BJT parameters are V BE.7 V,, and thermal voltage V T 5 mv. The voltage gain (v o /v i ) of the amplifier is. CLK D Q Q A S D Q Q S S S S S S S S

6 6 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) Suppose the XOR gate is replaced by an XNOR gate. Which one of the following options preserves the state diagram? (a) Input A is connected to Q (b) Input A is connected to Q (c) Input A is connected to Q and S is complemented (d) Input A is connected to Q n 4. Let [n] u( n) u( n ). The 9 Region of Convergence (ROC) of the z-transform of [n] (a) is z 9 (b) is n z (c) is z (d) does not eist Consider a discrete time periodic signal [n] sin n. Let ak be the comple Fourier 5 series coefficients of [n]. The cofficients {a k } are non-zero where k Bm ±, when m is any integer. The value of B is. 45. A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system. y ( t) 5 y( t) u( t) When y () and u(t) is a unit step function, y(t) is (a). +.8e 5t (b)..e 5t (c).8 +.e 5t (d).8.8e 5t 46. Consider the state space model of a system, as given below 4 u; y. The system is (a) controllable and observable (b) uncontrollable and observable (c) uncontrollable and unobservable (d) controllable and unobservable 47. The phase margin in degrees of G(s) calculated using the s. s s asymptotic Bode plot is. 48. For the following feedback system G(s). The %-settling time of the s s step response is required to be less than seconds. r + C(s) G(s) Which one of the following compensators C(s) achieves this? (a) s 5 (c) (s + 4) y. (b) 5 s (d) s 8 4 s 49. Let X be a real-valued random variable with E[X] and E[X ] denoting the mean value of X and X, respectively. The relation which always holds true is (a) (E[X]) > E[X ] (b) E[X ] > (E[X]) (c) E[X ] (E[X]) (d) E[X ] > (E[X]) 5. Consider a random process X(t) sin(t ), where the random phase is uniformly distributed in the interval [, ]. The autocorrelation E [X(t )X(t )] is (a) cos ((t + t )) (b) sin ((t t )) (c) sin ((t + t )) (d) cos ((t t )) 5. let Q( ) be the BER of a BPSK system over an AWGN channel with two-sided noise power spectral density N /. The parameter is a function of bit energy and noise power spectral density. A system with two independent and identical AWGN channels with noise power spectral density N / is shown in the figure. The BPSK demodulator receives the sum of outputs of both the channels. AWGN Channel / BPSK BPSK / Modulator Demodulator AWGN Channel If the BER of this system is Q(b ), then the value of b is.

7 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) 7 5. A fair coin is tossed repeatedly until a Head appears for the first time. Let L be the number of tosses to get this first Head. The entropy H(L) in bits is. 5. In spherical coordinates, let aˆ ˆ, a denote unit vectors the, directions. and E sin cos( t r) aˆ V / m r.65 H sin cos( t r) aˆ A / m r represent the electric and magnetic field components of the EM wave at large distances r from a dipole antenna, in free space. The average power (W) crossing hemispherical shell located at r km, / is 54. For a parallel plate transmission line, let v be the speed of propagation and Z be the characteristic impedance. Neglecting fringe effects, a reduction of the spacing between the plates by a factor of two results in (a) halving of v and no change in Z (b) no changes in v and halving of Z (c) no change in both v and Z (d) halving of both v and Z 55. The input impedance of a section of a lossless 8 transmission line of characteristic impedance 5 is found to be real when the other end is terminated by a load Z L ( R + jx). If X is, the value of R (in ) is ANSWERS GENERAL APTITUDE. (c). (d). (b) 4. (a) 5. (75 to 75) 6. (d) 7. (b) 8. (56 to 56) 9. (d). (b) ELECTRONICS AND COMMUNICATION. (d). (.65 to.68). (c) 4. (.99 to.) 5. (. to.4) 6. (c) 7. (a) 8. (7.99 to 8.) 9. (d). (a). (b). (b). (. to.6) 4. (c) 5. (a) 6. (6.4 to 6.6) 7. (d) 8. (.99 to.) 9. (44 to 46). (.4 to.6). (d). (7.99 to 8.). (. to.) 4. (b) 5. (c) 6. (a) 7. (b) 8. (86 to 866) 9. (.99 to.). (9.8 to 9.). (b). (.9 to.4). (9.99 to.) 4. (5.7 to 5.9) 5. (a) 6. (499 to 5) 7. (9 to 94) 8. (c) 9. ( 4 to ) 4. (a) 4. (d) 4. (d) 4. (c) 44. (9.99 to.) 45. (a) 46. (b) 47. (4 to 48) 48. (c) 49. (b) 5. (d) 5. (.4 to.4) 5. (.99 to.) 5. (55.4 to 55.6) 54. (b) 55. (9 to 4)

8 8 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) EXPLANATIONS GENERAL APTI TUDE 4. I f the standard deviation is less, there will be less deviation or batsman is more consistent. 5. The given series is, 5, 8, 7, 57,... The given series follows the following pattern 6. W, E, K, O, I, Q, W, A, F, N, T, X, N, V, B, D, Hence the odd one from the following group is N, V, B, D. 7. By reading the instructions, it is clear that person who fills form T is a resident female with annual income of ` 9 Lakh. 8. Let the length of platform.m and train length 8 m (given) According to Question m. 9. For 5, Now th 5 t rade deficit (9 7) cr or es crores of eport (7) crores 5 4 crores crores. Hence option (a) is wrong. For 4, Trade deficit (8 7) crores Now th 5 crores of eport (7) crores 5 4 crores crores. Hence option (b) is wrong. For 7, Now th 5 Trade deficit ( ) crores crores of eport () crores 5 crores crores. Hence option (c) is wrong. For 6 Now th 5 Trade deficit ( ) crores crores of eport () crores 5 crores Trade deficits Hence option (d) is correct.. The probaility that the other face is tail, is. EL ECTRONI CS AND COMM UNI CATI ON. M at ri mult iplication is not commut at ive in gener al.. Let the number of families in housing society be. family have single child. Total children Now remaining families have children. Total, sibling children Probability that a child picked at random has a sibling.667. Z j is a singularity lies inside C : Z By Cauchy s integral formula, C Z Z 4 j dz Z j j j Z j.[z Z 4 ] j[ 4 + j + 4j] 4[ + j]

9 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) 9 4. A real (4 4) matri A satisfies the equation A I where I is the (4 4) identity matri. The positive eigen value of A is. 5. X, X, X ar e i ndependent and i dent i cal l y dist r ibuted random var iables. So P{X is the largest} /. 6. Two cascaded sect ions Z Output impedance of first section Z I nput impedance of second section For maimum power transfer, upto st section is Z L Z * Z L Z Z * 7. Using KVL at all the three nodes. i 4 + i i i 6 + i i i 5 + i i By solving the equation, i 6 5A 8. Responsivity (R) IP P where I P photo current P I ncident power I P R P 8µA 9. Assume both the diode ON. Then circuit will be as per figure () I Now, I D.7 k 9. ma.7. ma I D I I D D is OFF and hence D ON.7 ma (Not Possible). I f fied positive charges are present in the gate oi de of an n -ch an n el en h an cemen t t ype M OSFETS it wi ll l ead t o a decr ease in t he threshold voltage.. I deal current Buffer has Z i Z. Output sample is voltage and is added at the input or current I t is voltage shunt negative feedback i.e., volt age-cur rent negative feedback. f 5KHz Cut off frequency (LPF) 5 KHz R C R.8 k Assume al NMOS are in saturation V DS (V GS V T ) For m, (5 Vp) (5 V p ) (5 Vp) > (4 Vp) Sat I D k(v GS V T ) I D K(4 V P )...() For m, I D K(5 V Q ) I D K(4 V Q )...() I D I D (4 V P ) (4 V Q ) V P V Q & V P + V Q 8 V P V Q 4V For m, I D K(5 V R ) Figur e () I D I D (4 V Q ) (4 V R ) V R V Q 4V V P V Q V R 4V

10 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) 5. f ( y)( y) ( y) y y ( y) y y ( y) ( y y) 6. f Q f Q input frequency modulus of counter MH z khz. 7. Time period of a discrete signal, w K N K N w K K N is a irrational number so signal is not periodic. 8. Multiplication in the t ime domain 9. Convolution in frequency domain (t) (t) X ( j)x ( j) Fundamental frequencies f, f ± f, f ± f... Nyquist rate 5 y(t) (t) * h(t) (t ) h (t ) y(t). Using r oot locus, Hz khz 5, d 45(steady state output) Break away point, K ( s )( s ) K ( s )( ) K ( s )( s ) dk ds s s.5 G(s) s.5 s s.5. For all pass system the pole zero pair must be symmetrical about imaginary ais with zero on the RHS and pole on the LHS of s-plane. This is not minimum phase system. Encirclement to the critical point (, ) N Open loop pole at RHS P N P z Thus system is unstable.. Spreading factor(sf) z (Close loop pole at RHS) chip rat e symbol rat e This if a single symbol is represented by a code of 8 chips Chip rate 8 symbol rate S.F (Spreading Factor) 8 symbol rat e 8 symbol rate Spread factor (or) process gain and determine to a cer t ain et ent t he upper limit of t he total number of uses supported simultaneously by a stat ion.. Channel capacity of BSC is 4. C P log P ( P) log ( P).5 log log.5 + [ log.5 ] I t is the case of channel with independent input and output, hence C b s a + s a b s a + s a X X b s s a b ; s a b s s a a By ver ificat ion Answer (b) sat isfies.

11 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) 5. Q Q F d d Q Q 4 R q q 4 ( d) 6 d 6. Taylor series epansion of sin + cos Taylor series is given by f() f ( ) + f ( ) ( ) + Here f() f() + f ()() + Now f() f + () f f ( ) ( )! ( ) ( )...! f () ( )......(i) 6 f() sin + cos Now f() cos + ( sin ) f() f() f() From equation (i), f() f() + f () + 7. Given, cos sin f() sin cos f () cos + sin f 6 () f () ( )... ( ) jwt g(t).e dt g(t)dt y(t) t g(z).dz... w.e (let G( j )) y(t) g(t) * u(t) [u(t) in unit step function] Y(j) G(j). U(j) Y(j) y(t).e Y(j) y(t) dt jt dt w.e ( ) j j j 8. The volume under the surface z(, y) + y and above the triangle in the y plane is given Volume Volume ( y) dy d y y 6 d d 6 () J 6 I 6

12 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) Now P I 6 + J 6 P P. X 9.9 y z X y ()() ()() ()() ()() ()() ()() ()() ()() ()() z i.e, lowest value among t hree r esistances is 9.9 Load I : P kw cos.8 S P jq j7.5 KVA Q P tan 7.5KVAR Load I I : S KVA cos.8 sin Q S. P () () (( )( )) + [( ) ( )] ( ) [ is non-negative real number] cos P S.8 P P 8kw Q 6 KVAR S P + jq 8 + j6 Compl e power del i ver ed by t he sour ce i s S I + S I I 8 j.5 KVA. RMS value T ( ) T f t dt where T is time period T T t dt () dt T T T / T / 4 T T t dt.48

13 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC). For ciritical damping, where For series circuit, L R C Q Q Quality factor Q R 4. g m L C C L R I C IC V T VBE VT IS e.7 I C.5 e I C ma g m ma V 5. Elect ron concent rat ion, 6. r DS n n N 4 mf. 4 i V in / VT A e (.5 ) 6 e. 9 / cm W ( C ) (V V ) n on L GS T. / 6mV r DS (.5) r DS 5 7. Given circuit is an op-amp series regulator, V is given by R V V R k 9V 4.7 R R 9. z 8. Given, Z i A L V i V (R //R ) I R KCL at inverting node R R V V V + R R R V R V R R I ( Z ) i...() V R R R R R I R R RR V I R 9. V BE.7V,, V T 5mV DC Analysis: V B k V k K V E.7.V I E. K.77mA I B.4 µa I C.6 ma 5 mv r e ma A V V R C V r ( )( R ) i e s 5K.98 () A V 7.76

14 4 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) 4. 5 y( s) sy( s ) s 4. Assume dummy variable K as a output of XOR gate K X Y XY XY F K.(K Z) (KZ + K.Z) K.KZ + K.K.Z) + K.Z ( K. K and K.K K) Put the value of K in above epression F (XY + XY) Z XYZ + XYZ 4z WX EPI z 4. D A S 4. (n) D Q S But D A S if A Q then D Q D Q EPJ z EPI y S S So input A is connected to Q 4( ) 4( n ) 9 Right side Signal n Left side signal y( s)[ s 5] y( s ) s s s( s 5) y(s) 4 5s 5( s 5) Applying inverse laplace transform 4 y(t) u t e u t 5 5 y(t) (. +.8 e 5t ) 46. From the given state model. A 5t ( ) ( ) B 4 c [ ] Controllable : Q c c [B AB A B] if Q c cont r ollable Q c Q c Uncontrollable Observable : Q C CA CA if Q obser vable Q Q 4 Observable The system is uncontrollable and observable. 47. G(S) ( s.)( s )( s ) ROC is z > 9 So ROC is z 9 dy y( t) dt u(t) y() sy( s) y() 5 y( s ) s ROC is z < G(S) G(S) s s. [ s].. [ s][ s][. s] By Approimation, G(S) [s ]

15 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) G(s) T(s) ( s 4) ( s )( s ) G(S) +G(S)+H(S) (S 4) (S )(S ) (S 4) Compare with S + W n S + w n W n.5 set t.8 sec. w n 49. E( ) [E()] can never be negative E[ ] [E()] Function of bit energy and noise P Count erllation diagr am of BPSK Channel is A WGN which implies noise sample as independent Let + n + n + n where n n + n SD N O 5. Given X( t) sin ( t ) f ( ) Now Bit error rate Q E is energy in E N O in uniformly distributed in the interval [, ] E[X(t ) X (t )] sin ( t ) sin( t ) f ( ) d sin( t )sin( t ).. d sin( ( t t ) ) d cos( ( t t )) d First integral will result into zero as we are integrating from to. Second integral result into cos[(t t )] E [X(t ) X(t )] cos ((t t )) E E 5. Bit error rate for BPSK Q Q NO N O Y E N O N is PSD of h O E 4E [as amplitudes are getting doubled] N N O O [independent and identical channel] Bit error rate Q 4E E Q N O N O 5. I n this problem random variable is L L can be,,... P{L } P{L } 4 P{L } 8 b or.44 H{L } log log log... / 4 / 4 8 / [Arithmatic geometric series summation]

16 6 SOLVED PAPER-4 (EL ECTRONI CS & COM M U NI CAT I ON EN GI NEER I NG - EC) j r 5. E sin e r.65 H Q sin e r jr * P avg E s H Q. ds (.65) s sin r sin d d r P aet 54. Z o (6.5)sin d d s /.5 sin d d P 55.5 w r Q 76 d log r.5 ( ) 55. Given, Z o 5 Z in Z in Z in s 8 Z JZ L o Zo Z o KZ L Z J5 5 5 JZ L L L L 5 5 JZ L 5 JZ L Z J5 5 JZ 5ZL 5ZL J(5 Z L ) 5 5 ZL Given, Z in Real So, I mg (Z in ) 5 Z L d distance between the two plates so, z o changes, if the spacing between the plates changes. V independent of spacing between the LC plat es Z L 5 R + X 5 R 5 X 5 R 4