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Sample Questions for MS/PhD Entrance Exams

Department of Electrical Engineering, IIT, Madras

GROUP - EE1 (Communications)

(For candidates interested in digital communications, wireless and telecom systems,

speech and image processing, computer networks, optical networks and microwave commu-nications).

1. For the signal x(t) shown below sketch and label x(2t 3)

t

x(t)

2 01 1

1

2

2

2. Given that X is a random variable with an uniform probability density function (pdf)

between 1 and +1, plot the pdf of Y = |X|. Plot the pdf ofZ = X 1.

3. Given the Fourier Transform G(f) (shown below) of a continuous time signal g(t),

answer the following:

(a) g(0) =?

(b) |g(t)|2dt = ?

1 2-1-2

1

f (in Hz)

G(f)

4. A signal f(t) = cos2(100t) + 3cos(80t) is sent through an ideal low pass filter with

cutoff at 60 Hz and sampled at 70 Hz to produce g(kT). What are the signal compo-

nents (give their frequency values) present in g(kT) for 0 f 60 Hz?

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5. The transfer function of a certain LTI system is given by

H(z) =

3

2 + z1

+

1

1 + 2z1

Shade the Region of Convergence (ROC) in the complex plane which will realise a

stable impulse response.

6. Classify the following as true or false. Give brief explanation for your conclusion.

(a) FIR filters require fewer computation then IIR filter for the same frequency re-

sponse.

(b) Both IIR and FIR filters are always stable.

(c) FIR filter always have linear phase response.

7. A signal x(t) of bandwidth W has a Fourier transform X(f) as shown in Figure

1. Sketch the Fourier transform of the sampled version of x(t) given by y(t) =

n=x(nT)(tnT) and the discrete-time Fourier transform, Z(ej), ofz[n] = x(nT).

Assume that 1/T > 2W.

W W

X(f)

f

1

Figure 1:

8. Choose the correct answer. The input to an LTI system is sin 20t+cos10t. The output

cannot be

(a) sin20t.

(b) cos20t + sin 10t.

(c) sin30t + cos 10t.

(d) cos20t.

9. Let {x(n)} be a real sequence defined for 0 n N1 with an N-point DFT {X(k)}.Let Y(k) = X(k), 0 k N 1. Determine {y(n)} (the N-point IDFT of{Y(k)})in terms of x(n) for 0 n N 1.

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10. The transfer function of a certain LTI system is given by

H(z) =3

2 + z1+

1

1 + 2z1.

Find the corresponding stable impulse response function {h(n)}.

11. Choose the correct answer with brief explanation. The system y(t) = 2 + x(t) withinput x(t) and output y(t) is

(a) non-linear and time-invariant.

(b) linear and time-variant.

(c) linear and time-invariant.

(d) non-linear and time-variant.

12. A signal x(t) = 100 cos(20000t) is ideally sampled with a sampling period of 50 sec.

Sketch possible sequences of the sampled output.

13. The transfer function of a certain LTI system is given by

H(z) =1 2z1

1 2z .

Find and plot the magnitude frequency response |H(ej)| versus for < < .

14. The impulse response of a linear time-invariant system is given as

h[n] = 2 n = 1,1.4 n = 2,2.

0 otherwise.

If the input to the above system is the sequence cos (n/4), determine the output

sequence.

15. Consider a satellite link and let Pu and Pd be the probability that a bit is in error

in uplink and downlink respectively. Compute the overall probability that a bit is in

error. Assume that the event that a bit is in error or not is independent in the uplink

and downlink.

16. In stop and wait ARQ (Automatic Repeat Request), if the probability of a successful

transmission of data frames (with out error) is p, then compute the mean number of

transmissions required to send a frame. Assume that acks or nacks are NEVER lost

and the occurances of errors in the data frames are independent.

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17. A coaxial cable consists of an inner conductor of radius a and an outer conductor of

inner radius 2a. Assume an infintely long cable with its axis along the z-axis. Current

I flows with uniform density in the +z direction in the inner conductor and returns

with uniform density in the z direction in the outer conductor.

(a) Use Gauss Law to calculate

|H

|. Draw a diagram clearly showing how you solve

for H.

(b) What is the direction of the H-field?

(c) How does | H| vary for a < r < 2a?

18. An electromagnetic wave Ei = E0[3

2x 1

2z] cos[6 109t 10(x +3z)] propagates

through a medium.

(a) What is the frequency of the wave? Specify units.

(b) What is the wavelength of the wave? Specify units.(c) What is the refractive index of the medium.

(d) What is the polarization of the wave? Draw a diagram, label the axis.

19. A coaxial cable has two layers between the conductors, with and as shown in Figure

2. A DC current I0 flows along the inner cable and returns along the outer conductor.

Determine the magnetic flux density B in the region between the conductors. Sketch

|B| vs. r.

1, 1

2

, 2

a

br0

Figure 2:

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