Q. 1 Q. 25 carry one mark each.

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1 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q. Q. 5 carry one mark each. Q. The bilateral Laplace transform of a function is if a t b f() t = otherwise (A) a b s (B) s e ( a b) s (C) e as e s bs (D) e s( a b) s Q. The value of for which all the eigen-values of the matrix given below are real is (A) 5+j (B) 5 (C) 5 (D) +5 Q.3 Let ( ) =. If ( ) = ( ) for all, =, = 4 and = 5, then should be equal to. Q.4 dy + cosy The general solution of the differential equation = dx cosx is (A) tan y cot x= c ( c is a constant) (B) tan x cot y = c ( c is a constant) (C) tan y+ cot x= c ( c is a constant) (D) tan x + cot y = c ( c is a constant) Q.5 The magnitude and phase of the complex Fourier series coefficients of a periodic signal ( ) are shown in the figure. Choose the correct statement from the four choices given. Notation: is the set of complex numbers, is the set of purely real numbers, and is the set of purely imaginary numbers. 3 a k k π a k 3 4 k (A) ( ) (B) ( ) (C) ( ) ( ) (D) the information given is not sufficient to draw any conclusion about ( ) EC /5

2 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.6 The voltage ( ) across the capacitor (in Volts) in the network shown is. 8 V 4 V ~ V, 5 Hz Q.7 In the circuit shown, the average value of the voltage V ab (in Volts) in steady state condition is. 5π sin(5t) kω μf mh kω b a V ab + 5 V Q.8 The -port admittance matrix of the circuit shown is given by.3. 5 (A) (B) (C) (D) Q.9 An n-type silicon sample is uniformly illuminated with light which generates electron-hole pairs per cm 3 per second. The minority carrier lifetime in the sample is µs. In the steady state, the hole concentration in the sample is approximately x, where x is an integer. The value of x is. EC /5

3 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q. A piece of silicon is doped uniformly with phosphorous with a doping concentration of 6 /cm 3. The expected value of mobility versus doping concentration for silicon assuming full dopant ionization is shown below. The charge of an electron is.6-9 C. The conductivity (ins cm ) of the silicon sample at 3 K is. Q. If the circuit shown has to function as a clamping circuit, then which one of the following conditions should be satisfied for the sinusoidal signal of period T? (A) RC << T (B) RC =.35 T (C) RC T (D) RC >> T EC 3/5

4 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q. In the circuit shown, V = V A for switch SW in position A and V = V B for SW in position B. V B Assume that the opamp is ideal. The value of is. V A kω 5 V kω B A kω V SW kω V kω Q.3 In the bistable circuit shown, the ideal opamp has saturation levels of ± 5 V. The value of R (in kω) that gives a hysteresis width of 5 mv is. R = kω v in + R + v out Q.4 In the figure shown, the output is required to be = +. The gates G and G must be, respectively, (A) NOR, OR (C) NAND, OR (B) OR, NAND (D) AND, NAND Q.5 In an 885 microprocessor, which one of the following instructions changes the content of the accumulator? (A) MOV B,M (B) PCHL (C) RNZ (D) SBI BEH EC 4/5

5 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.6 A mod-n counter using a synchronous binary up-counter with synchronous clear input is shown in the figure. The value of n is. CLOCK 4-Bit Binary Counter CLK CLEAR Q A Q B Q C Q D Q A Q B Q C Q D Q.7 Let the signal ( ) = outside the interval [, ], where and are finite. Furthermore, ( ) <. The region of convergence (RoC) of the signal s bilateral Laplace transform ( ) is (A) a parallel strip containing the Ω axis (B) a parallel strip not containing the Ω axis (C) the entire -plane (D) a half plane containing the Ω axis Q.8 Two causal discrete-time signals x[ n] and y[ n] are related as yn [ ] = xm [ ]. m= If the z-transform of y[n] is, the value of ( ) x [] is. n Q.9 By performing cascading and/or summing/differencing operations using transfer function blocks G () s and G () s, one CANNOT realize a transfer function of the form (A) G() s G() s (C) G() s + G() s G () s G () s (B) G () s (D) G() s G() s G () s EC 5/5

6 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q. For the signal flow graph shown in the figure, the value of Cs () R() s is -H 3 R(s) X X 3 X 4 C(s) G X G G 3 X 5 G 4 -H -H GGGG 3 4 (A) GGH GGH GGH + GGGGHH (B) (C) (D) GGGG GGH + GGH + GGH + GGGGHH GGH + GGH + GGH + GGGGHH GGH GGH GGH + GGGGHH Q. A unity negative feedback system has an open-loop transfer function ( ) =. The gain ( +) for the system to have a damping ratio of.5 is. Q. A sinusoidal signal of amplitude A is quantized by a uniform quantizer. Assume that the signal utilizes all the representation levels of the quantizer. If the signal to quantization noise ratio is 3.8 db, the number of levels in the quantizer is. Q.3 π The signal cos π t + is ideally sampled at a sampling frequency of 5 Hz. The sampled 4 sin( ) signal is passed through a filter with impulse response πt π cos 4 πt. The filter πt output is (A) 5 π cos 4π t (B) 5 sin( πt ) π cos π t + 4 πt 4 (C) 5 π cos π t (D) 5 sin( πt ) π cos 4π t 4 πt EC 6/5

7 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.4 In a source free region in vacuum, if the electrostatic potential = + +, the value of constant must be. Q.5 The electric field of a uniform plane electromagnetic wave is = + 4 exp[ (. )]. The polarization of the wave is (A) right handed circular (C) left handed circular (B) right handed elliptical (D) left handed elliptical Q. 6 Q. 55 carry two marks each. Q.6 dx Consider the differential equation =.x with initial condition () dt x =. The response x() t for t > is (A).t e (B).t e (C) 5 49.t e (D) 5 49e.t Q.7 The value of the integral cos( πt) sin(4 πt) 4π t dt is. Q.8 If denotes the counterclockwise unit circle, the value of the contour integral { } is. Q.9 Let the random variable X represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of X is. Q.3 An LC tank circuit consists of an ideal capacitor C connected in parallel with a coil of inductance L having an internal resistance R. The resonant frequency of the tank circuit is (A) C (B) R π LC π LC L (C) L C (D) R π LC RC π LC L EC 7/5

8 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.3 In the circuit shown, the Norton equivalent resistance (in Ω ) across terminals a-b is. Ω a Ι Ω 4Ω Ι b Q.3 In the circuit shown, the initial voltages across the capacitors C and C are V and 3 V, respectively. The switch is closed at time =. The total energy dissipated (in Joules) in the resistor R until steady state is reached, is. Q.33 A dc voltage of V is applied across an n-type silicon bar having a rectangular cross-section and a length of cm as shown in figure. The donor doping concentration N D and the mobility of electrons μ are 6 cm -3 and cm V - s -, respectively. The average time (in µs) taken by the electrons n to move from one end of the bar to other end is. V n-si cm Q.34 In a MOS capacitor with an oxide layer thickness of nm, the maximum depletion layer thickness is nm. The permittivities of the semiconductor and the oxide layer are ε s and ε ox respectively. Assuming ε s /ε ox = 3, the ratio of the maximum capacitance to the minimum capacitance of this MOS capacitor is. EC 8/5

9 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.35 The energy band diagram and the electron density profile n(x) in a semiconductor are shown in the figures. Assume that n(x) = cm -3, with =.V/cm and x expressed in cm. Given =.6 V, =36 cm s, and =. The electron current density (in A/cm ) at x= is (A) 4.4 (B). (C) (D). Q.36 A function of Boolean variables X, Y and Z is expressed in terms of the min-terms as F(X, Y, Z) = Σ (,, 5, 6, 7) Which one of the product of sums given below is equal to the function F(X, Y, Z)? (A) ( X + Y + Z ) ( X + Y + Z) ( X + Y + Z ) (B) ( X + Y + Z) ( X + Y + Z ) ( X + Y + Z) (C) ( X + Y + Z) ( X + Y + Z ) ( X + Y + Z) ( X + Y + Z ) ( X + Y + Z) (D) ( X + Y + Z ) ( X + Y + Z) ( X + Y + Z ) ( X + Y + Z) ( X + Y + Z ) Q.37 The figure shows a binary counter with synchronous clear input. With the decoding logic shown, the counter works as a Binary Counter Q 3 CLK CLR Q Q Q (A) mod- counter (B) mod-4 counter (C) mod-5 counter (D) mod-6 counter EC 9/5

10 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.38 A -to-8 demultiplexer with data input D in, address inputs S, S, S (with S as the LSB) and Y to Y 7 as the eight demultiplexed outputs, is to be designed using two -to-4 decoders (with enable input and address inputs A and A ) as shown in the figure. D in, S, S and S are to be connected to P, Q, R and S, but not necessarily in this order. The respective input connections to P, Q, R, and S terminals should be P Q R S E A A -to-4 Decoder Y Y Y Y 3 Y Y Y Y 3 E -to-4 Decoder A A Y Y Y Y 3 Y 4 Y 5 Y 6 Y 7 (A) S, D in, S, S (B) S, D in, S, S (C) D in, S, S, S (D) D in, S, S, S Q.39 The diode in the circuit given below has =.7 but is ideal otherwise. The current (in ma) in the 4 kω resistor is. kω D kω 3kΩ ma 4kΩ 6kΩ Q.4 Assuming that the opamp in the circuit shown below is ideal, the output voltage (in volts) is. kω kω V V O V -V EC /5

11 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.4 For the voltage regulator circuit shown, the input voltage (V in ) is V ± % and the regulated output voltage (V out ) is V. Assume the opamp to be ideal. For a load R L drawing ma, the maximum power dissipation in Q (in Watts) is. V in Q V out V ref 4 V + R R L R = kω Q.4 In the ac equivalent circuit shown, the two BJTs are biased in active region and have identical parameters with β >>. The open circuit small signal voltage gain is approximately. Q.43 Input x() t and output y() t of an LTI system are related by the differential equation y () t y () t 6 y() t = x() t. If the system is neither causal nor stable, the impulse response ht () of the system is 3t t 3t t (A) e u( t) + e u( t) (B) e u( t) + e u( t) t t 3t t (C) eu( t) e ut ( ) (D) e u( t) e u( t) EC /5

12 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.44 Consider two real sequences with time-origin marked by the bold value, [ ] = {,, 3, }, [ ] = {, 3,, } Let ( ) and ( ) be 4-point DFTs of [ ] and [ ], respectively. Another sequence [ ] is derived by taking 4-point inverse DFT of ( ) = ( ) ( ). The value of [] is. Q.45 Let ( ) = ( ) + ( ) with ( ) = ( ), where ( ) is unit step function. If the bilateral Laplace transform of x() t is 6 X ( s) = 4< Re { s} < 4; s 6 then the value of β is. Q.46 The state variable representation of a system is given as x& = x ; () = y = x [ ] The response ( ) is (A) sin( t ) t (B) e (C) cos( t) (D) Q.47 The output of a standard second-order system for a unit step input is given as t π yt () = e cos 3t. The transfer function of the system is (A) (B) (C) (D) ( s+ )( s+ 3) s + s+ s + s+ 3 s + s+ 4 Q.48 The transfer function of a mass-spring-damper system is given by ( ) = + + The frequency response data for the system are given in the following table. ω in rad/s ( ) in db arg ( ( )) in deg The unit step response of the system approaches a steady state value of. EC /5

13 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.49 N is passed through an LTI A zero mean white Gaussian noise having power spectral density filter whose impulse response ht () is shown in the figure. The variance of the filtered noise at t = 4 is h( ) 3 (A) (C) 3 AN (B) A N (D) 3 4 AN AN Q.5 { } n X n n = is an independent and identically distributed (i.i.d.) random process with = equally likely to be + or.{ Y } n = is another random process obtained as n n= = +.5. The autocorrelation function of { Y } n =, denoted by R n n= Y[k], is (A) R Y [k] (B) R Y [k] k (C) (D) R Y [k].5 R Y [k].5 3 k k k EC 3/5

14 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.5 Consider a binary, digital communication system which uses pulses gt () and gt () for transmitting bits over an AWGN channel. If the receiver uses a matched filter, which one of the following pulses will give the minimum probability of bit error? g(t) g(t) (A) t (B) t g(t) g(t) (C) t (D) t Q.5 Let {,} and {,} be two independent binary random variables. If ( =) = and ( =) =, then ( + ) is equal to (A) + ( )( ) (C) ( ) (B) (D) Q.53 An air-filled rectangular waveguide of internal dimensions a cm b cm ( a b) > has a cutoff frequency of 6 GHz for the dominant TE mode. For the same waveguide, if the cutoff frequency of the TM mode is 5 GHz, the cutoff frequency of the TE mode in GHz is. Q.54 Two half-wave dipole antennas placed as shown in the figure are excited with sinusoidally varying currents of frequency 3 MHz and phase shift of π/ between them (the element at the origin leads in phase). If the maximum radiated E-field at the point P in the x-y plane occurs at an azimuthal angle of 6, the distance d (in meters) between the antennas is. z d O 6 y OP >> d x P EC 4/5

15 GATE 5 SET- ELECTRONICS AND COMMUNICATION ENGINEERING - EC Q.55 The electric field of a plane wave propagating in a lossless non-magnetic medium is given by the following expression (, ) = 5 cos( + ) + 3cos + The type of the polarization is (A) Right Hand Circular. (C) Right Hand Elliptical. (B) Left Hand Elliptical. (D) Linear. END OF THE QUESTION PAPER EC 5/5

Q. 1 Q. 25 carry one mark each.

Q. 1 Q. 25 carry one mark each. Q. Q. 5 carry one mark each. Q. Consider a system of linear equations: x y 3z =, x 3y 4z =, and x 4y 6 z = k. The value of k for which the system has infinitely many solutions is. Q. A function 3 = is