GATE 2015

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

Q. 1 – Q. 25 carry one mark each. Q.1

⎧1 if a ≤ t ≤ b ⎩0 otherwise

The bilateral Laplace transform of a function f (t ) = ⎨ is (A)

Q.2

a −b s

The value of

(B)

e s ( a − b) s

5+ 20 2

4

Q.3

Q.4

(B) 5 −

e s ( a −b ) s

4 2 −10

(C) 1 − 5

Let ( ) = . If ( ) = ( ) for all equal to _______.



,

(D) 1 + 5

= 2,

= 4 and = 5, then should be

dy 1 + cos 2 y = is dx 1 − cos 2 x

The general solution of the differential equation (A) tan y − cot x = c ( c is a constant) (C) tan y + cot x = c ( c is a constant)

Q.5

(D)

for which all the eigen-values of the matrix given below are real is 10

(A) 5 + j

e − as − e − bs s

(C)

(B) tan x − cot y = c ( c is a constant) (D) tan x + cot y = c ( c is a constant)

of a periodic signal ( ) are The magnitude and phase of the complex Fourier series coefficients 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.

|ak|

3

3 2

2 1 -5 -4 -3 -2 -1 0

1

2

3 4

1

2

3 4

k

∠ ak

-5 -4 -3 -2 -1 0

k

−π

(A) ( ) ∈ (B) ( ) ∈ (C) ( ) ∈ ( − ) (D) the information given is not sufficient to draw any conclusion about ( )

EC

1/15

GATE 2015

Q.6

SET-2

The voltage (

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

) across the capacitor (in Volts) in the network shown is ______ . 80 V

40 V

~ 100 V, 50 Hz

Q.7

In the circuit shown, the average value of the voltage Vab (in Volts) in steady state condition is ________. 1 kΩ 5π sin(5000t)

Q.8

Q.9

EC

b

1 μF

1 mH

a

+

Vab

2 kΩ 5V

The 2-port admittance matrix of the circuit shown is given by

(A)

0.3 0.2 0.2 0.3

(B)

15 5

(C)

3.33 5

(D)

0.3 0.4 0.4 0.3

5 3.33

5 15

An n-type silicon sample is uniformly illuminated with light which generates 1020 electron-hole pairs per cm3 per second. The minority carrier lifetime in the sample is 1 µs. In the steady state, the hole concentration in the sample is approximately 10x, where x is an integer. The value of x is ___.

2/15

GATE 2015

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

Q.10

A piece of silicon is doped uniformly with phosphorous with a doping concentration of 1016/cm3. The expected value of mobility versus doping concentration for silicon assuming full dopant ionization is shown below. The charge of an electron is 1.6 ×10-19 C. The conductivity (in S cm −1 ) of the silicon sample at 300 K is _______.

Q.11

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

EC

(B) RC = 0.35 T

(C) RC ≈ T

(D) RC >> T

3/15

GATE 2015

Q.12

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

In the circuit shown, V0 = V0A for switch SW in position A and V0 = V0B for SW in position B. Assume that the opamp is ideal. The value of

V0 B is ___________. V0 A 1 kΩ

1 kΩ

5V

B

V0

1 kΩ

A

SW

1 kΩ 1 kΩ

1V

Q.13

In the bistable circuit shown, the ideal opamp has saturation levels of ± 5 V. The value of R1 (in kΩ) that gives a hysteresis width of 500 mV is ________. R2 = 20 kΩ R1 +

vin

Q.14

In the figure shown, the output respectively,

+ −

is required to be

(A) NOR, OR (C) NAND, OR Q.15

=

+ ̅ . The gates G1 and G2 must be,

(B) OR, NAND (D) AND, NAND

In an 8085 microprocessor, which one of the following instructions changes the content of the accumulator? (A) MOV B,M

EC

vout



(B) PCHL

(C) RNZ

(D) SBI BEH

4/15

GATE 2015

Q.16

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

A mod-n counter using a synchronous binary up-counter with synchronous clear input is shown in the figure. The value of n is _______. 4-Bit Binary Counter CLOCK

CLK CLEAR

Q.17

QA QB QC QD

QA QB QC QD

Let the signal ( ) = 0 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.18

Two causal discrete-time signals x[n] and y[ n ] are related as y[n] = If the z-transform of y[n] is

Q.19

)

, the value of x[2]

n m=0

x[m] .

is _______.

By performing cascading and/or summing/differencing operations using transfer function blocks G1 ( s ) and G2 ( s ) , one CANNOT realize a transfer function of the form (A) G1 ( s )G2 ( s )

⎛ 1 ⎞ + G2 ( s) ⎟ ⎝ G1 ( s) ⎠

(C) G1 ( s) ⎜

EC

(



(B)

G1 ( s) G2 ( s )

⎛ 1 ⎞ − G2 ( s) ⎟ ⎝ G1 ( s) ⎠

(D) G1 ( s ) ⎜

5/15

GATE 2015

Q.20

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

For the signal flow graph shown in the figure, the value of

C ( s) is R( s)

-H3

1 X1

X3 G1

R(s)

X2

1

X4 G3 X5 G4

G2

-H1

Q.21

-H2

(A)

G1G2G3G4 1 − G1G2 H1 − G3G4 H 2 − G2G3 H 3 + G1G2G3G4 H1 H 2

(B)

G1G2G3G4 1 + G1G2 H1 + G3G4 H 2 + G2G3 H 3 + G1G2G3G4 H1 H 2

(C)

1 1 + G1G2 H1 + G3G4 H 2 + G2G3 H 3 + G1G2G3G4 H1 H 2

(D)

1 1 − G1G2 H1 − G3G4 H 2 − G2G3 H 3 + G1G2G3G4 H1 H 2

A unity negative feedback system has an open-loop transfer function for the system to have a damping ratio of 0.25 is ________.

Q.22

Q.23

C(s)

( )=

( +10)

. The gain

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 31.8 dB, the number of levels in the quantizer is _______.

⎛ ⎝

The signal cos ⎜ 10π t +

π⎞

⎟ is ideally sampled at a sampling frequency of 15 Hz. The sampled 4⎠ π⎞ ⎛ sin(π t ) ⎞ ⎛ signal is passed through a filter with impulse response ⎜ ⎟ cos ⎜ 40π t − ⎟ . The filter 2⎠ ⎝ πt ⎠ ⎝

output is

15 π⎞ ⎛ cos ⎜ 40π t − ⎟ 2 4⎠ ⎝ 15 π⎞ ⎛ (C) cos ⎜10π t − ⎟ 2 4⎠ ⎝

(A)

EC

15 ⎛ sin(π t ) ⎞ π⎞ ⎛ ⎜ ⎟ cos ⎜10π t + ⎟ 2 ⎝ πt ⎠ 4⎠ ⎝ 15 ⎛ sin(π t ) ⎞ π⎞ ⎛ (D) ⎜ ⎟ cos ⎜ 40π t − ⎟ 2 ⎝ πt ⎠ 2⎠ ⎝ (B)

6/15

GATE 2015

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

Q.24

In a source free region in vacuum, if the electrostatic potential of constant must be _______.

Q.25

The electric field of a uniform plane electromagnetic wave is =

+ 4

exp[ (2 × 10

+

= 2

+

, the value

− 0.2 )].

The polarization of the wave is (A) right handed circular (C) left handed circular

(B) right handed elliptical (D) left handed elliptical

Q. 26 – Q. 55 carry two marks each. Q.26

Consider the differential equation

x(t ) for t > 0 is (A) 2 − e −0.2t Q.27

dx = 10 − 0.2 x with initial condition x(0) = 1 . The response dt

(B) 2 − e 0.2 t

The value of the integral

12 cos(2π t )

(C) 50 − 49e −0.2t

(D) 50 − 49e0.2t

sin(4π t ) dt is ___________. 4π t

Q.28

If

Q.29

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.30

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

denotes the counterclockwise unit circle, the value of the contour integral 1 { } 2 is _______.

(A) (C)

EC

1 2π LC 1 2π LC

1−

L R 2C

1

C L 2π LC 1 ⎛ 2 C ⎞ (D) ⎜1 − R ⎟ L⎠ 2π LC ⎝

(B)

1 − R2

7/15

GATE 2015

Q.31

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

In the circuit shown, the Norton equivalent resistance (in Ω ) across terminals a-b is _______. 2Ω a + -





4Ω Ι b

Q.32

In the circuit shown, the initial voltages across the capacitors C1 and C2 are 1 V and 3 V, respectively. The switch is closed at time = 0. The total energy dissipated (in Joules) in the resistor R until steady state is reached, is __________.

Q.33

A dc voltage of 10 V is applied across an n-type silicon bar having a rectangular cross-section and a length of 1 cm as shown in figure. The donor doping concentration ND and the mobility of electrons μn are 1016 cm-3 and 1000 cm2V-1s-1, respectively. The average time (in µs) taken by the electrons to move from one end of the bar to other end is ________. 10 V

n-Si

1 cm Q.34

EC

In a MOS capacitor with an oxide layer thickness of 10 nm, the maximum depletion layer thickness is 100 nm. The permittivities of the semiconductor and the oxide layer are εs and εox respectively. Assuming εs/εox = 3, the ratio of the maximum capacitance to the minimum capacitance of this MOS capacitor is ________.

8/15

GATE 2015

Q.35

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

The energy band diagram and the electron density profile n(x) in a semiconductor are shown in the figures. Assume that n(x) = 10 Given = 0.026 V, = 36 cm s , and

cm-3, with = 0.1 V/cm and x expressed in cm. = . The electron current density (in A/cm ) at

x = 0 is

(A) −4.4 × 10 Q.36

(C) 0

(B) −2.2 × 10

(D) 2.2 × 10

A function of Boolean variables X, Y and Z is expressed in terms of the min-terms as F(X, Y, Z) = Σ (1, 2, 5, 6, 7) Which one of the product of sums given below is equal to the function F(X, Y, Z)?

Q.37

(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 )

The figure shows a binary counter with synchronous clear input. With the decoding logic shown, the counter works as a Binary Counter Q3 Q2 CLK Q 1

Q0 CLR

(A) mod-2 counter

EC

(B) mod-4 counter

(C) mod-5 counter

(D) mod-6 counter

9/15

GATE 2015

Q.38

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

A 1-to-8 demultiplexer with data input Din , address inputs S0, S1, S2 (with S0 as the LSB) and Y0 to Y7 as the eight demultiplexed outputs, is to be designed using two 2-to-4 decoders (with enable input and address inputs A0 and A1) as shown in the figure. Din , S0, S1 and S2 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

Q.39

1E

1Y0 2-to-4 Decoder 1Y1 1A0 1Y2 1A1 1Y3

Y0 Y1 Y2 Y3

2E 2-to-4 2Y0 Decoder 2Y1 2A0 2Y2 2A1 2Y3

Y4 Y5 Y6 Y7

(A) S2, Din, S0, S1

(B) S1, Din, S0, S2

(C) Din, S0, S1, S2

(D) Din, S2, S0, S1

The diode in the circuit given below has the 4 kΩ resistor is _______.

= 0.7

but is ideal otherwise. The current (in mA) in

3kΩ

2kΩ D 1kΩ 1mA 4kΩ

Q.40

6kΩ

Assuming that the opamp in the circuit shown below is ideal, the output voltage is________.

(in volts)

2kΩ 12V

1kΩ

VO

1V

EC

-12V

10/15

GATE 2015

Q.41

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

For the voltage regulator circuit shown, the input voltage (Vin) is 20 V ± 20% and the regulated output voltage (Vout) is 10 V. Assume the opamp to be ideal. For a load RL drawing 200 mA, the maximum power dissipation in Q1 (in Watts) is ________. Q1

Vin Vref

4V

+

Vout R1

RL



R2 = 10 kΩ

Q.42

In the ac equivalent circuit shown, the two BJTs are biased in active region and have identical parameters with β >>1. 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 h (t ) of

the system is

1 3t 1 e u (−t ) + e−2t u (−t ) 5 5 1 3t 1 −2t (C) e u (−t ) − e u (t ) 5 5

(A)

EC

1 3t 1 e u (−t ) + e−2t u (−t ) 5 5 1 3t 1 −2t (D) − e u (−t ) − e u (t ) 5 5 (B) −

11/15

GATE 2015

Q.44

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

Consider two real sequences with time-origin marked by the bold value, [ ] = { , 2, 3, 0} , Let

Q.45

( )

( ).

[2] is ______.

Let ( ) = ( ) + (− ) with ( ) = Laplace transform of x(t ) is

X (s) = then the value of β is _______. Q.46

( )=

[ ] is derived by taking 4-point inverse DFT of

Another sequence The value of

[ ] and [ ], respectively.

( ) be 4-point DFTs of

( ) and

[ ] = { , 3, 2, 1}

( ), where ( ) is unit step function. If the bilateral

16 −4 < Re {s} < 4 ; s − 16 2

The state variable representation of a system is given as

⎡0 1 ⎤ x& = ⎢ ⎥x; ⎣0 −1⎦ y = [ 0 1] x

(0) = 1 0

The response ( ) is (A) sin(t )

Q.47

(D) 0

The output of a standard second-order system for a unit step input is given as

y (t ) = 1 − (A)

Q.48

(C) 1 − cos(t )

(B) 1 − et

2 −t π⎞ ⎛ e cos ⎜ 3t − ⎟ . The transfer function of the system is 6⎠ 3 ⎝

2 ( s + 2)( s + 3)

(B)

1 s + 2s + 1

(C)

2

3 s + 2s + 3 2

(D)

4 s + 2s + 4 2

The transfer function of a mass-spring-damper system is given by ( )=

1 +

+

The frequency response data for the system are given in the following table. ω in rad/s 0.01 0.1 0.2 1 2 3 5 10

| ( )| in dB -18.5 -18.5 -18.4 -16 -11.4 -21.5 -32.8 -45.3

arg ( ( -0.2 -1.3 -2.6 -16.9 -89.4 -151 -167 -174.5

)) in deg

The unit step response of the system approaches a steady state value of ________ . EC

12/15

GATE 2015

Q.49

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

A zero mean white Gaussian noise having power spectral density

N0 is passed through an LTI 2

filter whose impulse response h(t ) is shown in the figure. The variance of the filtered noise at

t = 4 is ℎ( )

0

2

1

3



(A)

3 2 A N0 2

3 2 A N0 4 1 2 (D) A N 0 2

(B)

(C) A 2 N 0

Q.50

{ X n }n=−∞ is an independent and identically distributed (i.i.d.) random process with equally n =∞ . The likely to be +1 or −1 . {Yn }n =−∞ is another random process obtained as = + 0.5 n =∞ autocorrelation function of {Yn }n =−∞ , denoted by RY[k], is n =∞

(A)

(B)

RY[k] 1

-3 -2 -1 0

1

2

3

1.25

0.5

0.5

-3 -2 -1 0

k

1

2

3

k

2

3

k

(D)

(C)

RY[k] 0.5

-2 -1 0

RY[k]

1.25 0.5

0.25

EC

RY[k]

0.25

0.25 1

2

k

-3 -2 -1 0

1.25

0.25 1

13/15

GATE 2015

Q.51

SET-2

Consider a binary, digital communication system which uses pulses g (t ) and − g (t ) 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)

1

1 0

(A)

(C)

Q.52

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

1

t (B)

g(t)

g(t)

1

1 0

1

t (D)

1

0

1

t

t

Let ∈ {0,1} and ∈ {0,1} be two independent binary random variables. If ( = 0) = and ( = 0) = , then ( + ≥ 1) is equal to (A) + (1 − )(1 − ) (C) (1 − )

Q.53

0

(B) (D) 1 −

An air-filled rectangular waveguide of internal dimensions a cm × b cm ( a > b ) has a cutoff frequency of 6 GHz for the dominant TE10 mode. For the same waveguide, if the cutoff frequency of the TM 11 mode is 15 GHz, the cutoff frequency of the TE01 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/2 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 60°, the distance d (in meters) between the antennas is _________. z

O d 60° x

EC

y OP >> d P

14/15

GATE 2015

Q.55

SET-2

ELECTRONICS AND COMMUNICATION ENGINEERING - EC

The electric field of a plane wave propagating in a lossless non-magnetic medium is given by the following expression ( , )=

5 cos(2 × 10

+

)+

3 cos 2 × 10

+



2



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

15/15

GATE 2015 Electronics and Communication Engineering Session 2 ...

Page 1 of 15. GATE 2015 SET-2 ELECTRONICS AND COMMUNICATION ENGINEERING - EC. EC 1/15. Q. 1 – Q. 25 carry one mark each. Q.1. The bilateral Laplace transform of a function. 1 if. ( ) 0 otherwise. atb. f t ⎧ ≤ ≤ = ⎨. ⎩. is. (A) a b. s. − (B) ( ) s ea b. s. − (C). as bs. e e. s. − − − (D). s( ) a b. e. s. −. Q.2 The value of ݔ ...

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