Download mecce 101 analytical foundations for communication engineering

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M.TECH DEGREE EXAMINATION
Model Question Paper
Specialization: Communication Engineering
First Semester
MECCE 101 ANALYTICAL FOUNDATIONS FOR COMMUNICATION
ENGINEERING
(Regular-2013 Admissions)
Time: Three Hours
Maximum: 100 marks
Each question carries 25 marks.
1. a) Prove that a non-empty subset W of V is a subspace of V if and only if for each pair
of vectors α, β in W and each scalar c in F the vector c α + β is again in W.
(5)
b) Let V and W be vector spaces over the field F and T be a linear transformation from V
onto W, if V is finite dimensional then, prove that
Rank (T) + Nullity (T) = dim V
(8)
c) Are the vectors
α1 = (1, 1, 2, 4)
α2 = (2, -1, -5, 2)
α3 = (1, -1, -4, 0)
α4 = (2, 1, 1, 6)
linearly independent in R4 ?
(12)
OR
2. a) Consider the bases B = {(1, 2), (3, -1)} and B’= {(1, 0), (0, 1)} of R2. If u is a vector
having co-ordinate vector uB =
, find uB’ .
(5)
b) From the non orthogonal a, b, c find orthonormal vectors q1, q2, q3
a=
,
b=
,
c=
.
(8)
c) Find the Eigen values and Eigen vectors of
A=
and write two different diagonalising matrices.
(12)
3. a) Explain the notion of ergodicity and stationarity of a random process
(5)
b) Prove that if a WSS process is input to a linear system the output is also WSS.
(8)
c) A WSS random process X(t) with ACF RX (τ) =0.5 e- 2|τ| is fed as an input to an LTI system
with impulse response h(t) = e-t u(t). If Y (t) is the system output find its autocorrelation. (12)
OR
4. a) Define a Gaussian process, White Noise process and a Wiener process
(5)
b) X1 and X2 are two jointly Gaussian variables with X1 = X2 = 0 and σ1 = σ2 = σ. The
correlation coefficient between X1 and X2 is ρ. Write the joint PDF of X1 and X2 in i) the
matrix form and ii) the expanded form
(8)
c) A random process X (t) = A Cos (2 π fc t + Ѳ); where Ѳ is a uniformly distributed random
variable over the interval (-π, π). Determine the power spectral density of the random process
and the average power of X (t).
(12)
5. a) State and prove the CK equation for a DTMC
(5)
b) Suppose that whether or not it rains today depends on previous weather conditions
through the last two days. Specifically, suppose that if it has rained for the past two days, then
it will rain tomorrow with probability 0.7; if it rained today but not yesterday, then it will rain
tomorrow with probability 0.5; if it rained yesterday but not today, then it will rain tomorrow
with probability 0.4; if it has not rained in the past two days, then it will rain tomorrow with
probability 0.2. Convert this into a suitable Markov chain and obtain its Transition
probability matrix. Now, given that it rained on Monday and Tuesday, what is the probability
that it will rain on Thursday?
(8)
c) Define a Continuous Time Markov chain and obtain its limiting distribution. Compare it
with that of a DTMC
(12)
OR
6. a) Define a Poisson process and state its important properties.
(5)
b) With respect to classification of states in a Markov chain, differentiate between a
recurrent and transient state. Prove that recurrence is a class property.
(5)
c) Customers arrive at a bank according to a Poisson process with rate λ = 10 per hour.
What is the probability that (i) there are exactly 2 arrivals in the first 15 minutes (ii) there are
at least 2 arrivals in the first 20 minutes (iii) Probability of ( at least one arrival in the first 15
minutes / there are exactly 2 arrivals in the first 15 minutes)
(8)
d) Define a Birth Death process and obtain a steady state solution for the BD process
(7)
7. a) Define an M/M/1 queue and obtain its steady state balance equation
b) Customers arrive at an ISD Telephone exchange according to a Poisson process with
average inter arrival time of 18 minutes. The length of a telephone call is assumed to be
exponential with a mean of 7 minutes. Assume that only one customer can use the telephone
at a time. (i) What is the probability that a person arriving at the booth has to wait in the
queue? (ii) What is the average number of customers in the system? (iii) What is the average
waiting time of a customer in the system?
(8)
c) Derive Erlang’s B formula and find out the average waiting time and number of customers
in the system. In what way is the M/M/m/m system different from this?
(12)
OR
8. a) State and prove Little’s theorem
(5)
b) Define the PK formula for an M/G/1 Queue. Find out the number of customers served
during a busy period in an M/G/1 queue.
(8)
c) Derive the Blocking probability of an M/M/m/m queue
(12)