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Cairo University
Faculty of Engineering
Electronics and Communication
Department
Multiuser Detection
Final Exam
2014
Time: 2 hours
This exam consists of five problems with a total number of 55
points. The maximum
grade is 50 points.
You are allowed to have a single-sided cheat sheet during the
exam period. Put your
cheat sheet inside your answer booklet at the end of the
exam.
Do NOT use any external paper sheets for your answers; there is
enough room for your
answers in this booklet.
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1. (13 points) Consider a downlink channel with two users. All
terminals have one antenna. The channels from the base station to
user-1 and user-2 are denoted by and , respectively. The base
station schedules the user that has the strongest channel
magnitude. Assume the two channels are i.i.d. symmetric
complex Gaussian random variables with variance one.
i. (2 points) Does this scheduling technique provide fair
network access to the users? Justify your answer (state the
reason).
ii. (2 points) What might be the limitation when using such
scheduling technique in practical cellular networks?
iii. (2 points) State some other scheduling techniques and
discuss their advantages and disadvantages.
iv. (2 points) Would the system benefit from a multi-user
diversity gain? Justify your answer.
v. (2 points) Write an expression for the outage probability of
the scheduled user as a function of the channel.
vi. (3 points) Using the outage probability of part-v, evaluate
the outage-diversity.
Hint: if | | and | |
are two independent identically distributed
exponential random variables with density function , then the
cumulative distributive function (CDF) of a random variable ( ) is
( )
( ) ( ) . You may also use the approximation:
( ) ( )
( )
( ) for small values of .
i)
ii)
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iii)
iv)
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v)
vi)
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2. (7 points) Consider a cyclic-delay diversity scheme shown in
Figure-1. The modulated symbols
{ ( ) are transmitted on M antennas after applying a cyclic
shift and a cyclic-prefix (CP)
insertion. Specifically, the modulator outputs a block of
symbols, each block has N symbols. Each
block is cyclically shifted by symbols, where is a symbol-delay
unit. A cyclic-prefix is
inserted for each shifted block{ ( )} to obtain the block { ( )
.The length of the CP equals the
maximum delay spread of the channel. The channel is assumed
quasi-static flat fading generated by
complex Gaussian process.
i. (2 points) Assume N=2, M=2, and that there is a total power
constraint P over the antennas.
Suggest (draw and explain) a single-antenna receiver to detect
the symbols { ( ) .
ii. (3 points) Assume a zero-forcing equalizer is used. Write an
expression for the equalizer and compute the symbol error
probability.
Hint: Recall that cyclic-prefix can be used to diagonalize the
channel. The diagonal channel
elements have the same distribution as the original channel
coefficients when N=M.
iii. (2 points) Now assume a minimum mean square error (MMSE)
equalizer is used. Write an
expression for the equalizer and compute the symbol error
probability.
i)
Figure-1: Transmitter side for a cyclic-delay diversity (CDD)
scheme. The symbol denotes the cyclic delay for antenna i, and CP
denotes insertion of a cyclic extension
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ii)
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iii)
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3. (15 points) Consider an AWGN uplink channel with one base
station and two single-antenna users.
The base station is equipped with three antennas. The channels
from user-1 and user-2 to the base
station are denoted by the vectors and , respectively. Assume
that the noise variance is
and that user transmits with power per channel use, where .
i. (3 points) Write an expression for the maximum achievable
sum-rate of this channel (this rate
will be called SDMA rate throughout this problem)
ii. (3 points) Assume orthogonal transmission (with a parameter
for resource sharing). Write an
expression for the achievable sum-rate
iii. (3 points) What is the degree-of-freedom achieved by the
SDMA scheme and the orthogonal
scheme?
iv. (2 points) Evaluate the sum-rates obtained in the SDMA
scheme and the orthogonal scheme
assuming that , and that the channels and are given by
[
] [
]
v. (2 points) Compare the sum-rates obtained in part (iv), and
verify which system (if any) gives a
higher rate. Justify your result.
vi. (2 points) For the given values in part (iv), suggest a
communication scheme (i.e. transmission
and reception techniques) that can achieve the SDMA
sum-rate.
Hint: Note the special structure (relationship) of and in part
(iv).
i)
ii)
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iii)
iv)
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v)
vi)
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4. (15 points) A base station, with two antennas, uses
zero-forcing precoding for downlink transmission.
The base station serves two single-antenna users per
time-frequency grid. The zero-forcing precoder,
denoted by , can be found by solving the optimization
problem
( )
( )
where is the power transmitted for stream-k (dedicated for
user-k), and is the total transmitted
power constraint.
i. (4 points) Show that the power is given by (
( ) )
where the constant b is
calculated from the equation ( ( ) ) , and ( )
denotes the ( )
ii. (4 points) The channel from the base station to user-k is
given by . Let the channel
[
] denote the overall channel matrix such that [
], where and are the
power levels obtained from part-i. Find and for the channel
matrix [
], and a
given total power unit of power.
iii. (3 points) The SVD of the channel matrix is given by:
[
] [
] . Assume that the base station knows the channels
( ) and that each user knows its own channel (there is no
cooperation between the
users).
a. If the base station precodes the data with a precoding matrix
and then applies a water-
filling algorithm, find the power allocations obtained from the
water-filling algorithm.
b. Describe the receiving scheme.
iv. (4 points) Find the rates achieved by the schemes used in
part-ii and part-iii assuming unit noise
variance and that capacity-achieving codes are used.
i)
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ii)
iii)
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iv)
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5. (5 points) A base station, with three antennas, uses
zero-forcing successive interference cancellation
(ZF-SIC) for uplink reception. The base station serves two
users; the first one has two antennas and the
second one has one antenna. The channel between the users and
the base station is denoted by
. The first two columns of correspond to the channel between the
first user and the base station,
whereas the third column corresponds to the channel between the
second user and the base station. The
first user transmits two independent streams on his antennas
(the first stream on the first antenna and
the second stream on the second antenna), and the second user
transmits one stream on his single
antenna. The channel (the overall channel observed at the base
station) is assumed constant for all
transmissions and known by the base station. Assume user-1
transmits with a total power
(equally distributed on his two streams), and user-2 transmits
with power . The matrices and
are given by
[
] [
]
i. (3 points) Let the vector denote the i-th row-vector of the
matrix . Using the vectors
{ design the ZF-SIC utilized at the base station (draw and
explain the different stages).
ii. (2 points) Assuming unit noise variance and that
capacity-achieving codes are used, calculate the achievable
sum-rate of this uplink channel.
i)
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ii)