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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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15

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.

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ii)