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April 2013, 20(2): 711www.sciencedirect.com/science/journal/10058885 http://jcupt.xsw.bupt.cn

The Journal of China

Universities of Posts and

Telecommunications

Resource allocation algorithm in LTE uplink SC-FDMA system fortime-varying channel with imperfect channel state information

XU Quan-sheng (), LI Xi, JI Hong, YAO Li-ping

Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract

In long term evolution (LTE) uplink single carrier frequency division multiple access (SC-FDMA) system, the restriction

that multiple resource blocks (RBs) allocated to a user should be adjacent, makes the resource allocation problem hard to

solve. Moreover, with the practical constraint that perfect channel state information (CSI) cannot be obtained intime-varying channel, the resource allocation problem will become more difficult. In this paper, an efficient resource

allocation algorithm is proposed in LTE uplink SC-FDMA system with imperfect CSI assumption. Firstly, the resource

allocation problem is formulated as a mixed integer programming problem. Then an efficient algorithm based on discrete

stochastic optimization is proposed to solve the problem. Finally, simulation results show that the proposed algorithm has

desirable system performance.

Keywords SC-FDMA, resource allocation, imperfect channel state information, discrete stochastic optimization, time-varying channel

1 Introduction

SC-FDMA has similar system structure and

performance with orthogonal frequency division multiple

access (OFDMA) but with many advantages such as low

peak-to-average-power ratio (PAPR) [1]. The characteristic

of low PAPR will greatly benefit mobile terminals in terms

of power efficiency. Therefore, localized mode SC-FDMA

is adopted for the uplink of long term evolution (LTE).

However, there are some restrictions in LTE uplink

SC-FDMA system [23], which will increase the difficulty

on resource allocation. (1) Exclusivity, i.e. a sub-channel

cannot be allocated to more than one user simultaneously,

unless multiplexing technique is adopted; (2) adjacency, i.e.

multiple sub-channels allocated to a user must be adjacent

to each other; (3) total power constraint, i.e. the total

power allocate to a user should be less than maximum

power level; (4) peak power constraint, i.e. the power on

every sub-channel should be less than peak power level;

(5) uniform power allocation, i.e. the power allocated to

Received date: 07-08-2012

Corresponding author: XU Quan-sheng, E-mail: xqs@bupt.edu.cn

DOI: 10.1016/S1005-8885(13)60020-5

multiple sub-channels of a user should be the same.

Although the resource allocation in SC-FDMA system is

still a combinatorial optimization problem like in OFDMA

system, the restriction of adjacency makes the problem

harder to solve [23]. Furthermore, when considering the

practical constraint that perfect CSI cannot be obtained in

time-varying channel, the problem will become

prohibitively difficult.

Some resource allocation algorithms that consider the

resource allocation constraints in localized mode

SC-FDMA system have been proposed. In Ref. [4], some

greedy sub-optimal algorithms are proposed. In Ref. [2],

an optimal algorithm is developed, where the resource

allocation problem is formulated as a binary-integer

program (BIP) problem. Considering computational

complexity of the BIP, with the help of canonical duality

theory, the authors in Ref. [3] propose a polynomial-

complexity resource allocation framework. Moreover,

some resource allocation algorithms that consider

particular situation are proposed in Refs. [56]. In Ref. [5],

the authors propose a framework for energy efficient

resource allocation with synchronous hybrid automatic

repeat request (HARQ) constraint. In Ref. [6], a resource

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8 The Journal of China Universities of Posts and Telecommunications 2013

allocation algorithm combined with user pairing scheme

are proposed. However, all aforementioned algorithms are

based on perfect CSI assumption, which is impractical,

especially in time-varying channel.

In this paper, an efficient resource allocation algorithmis proposed with imperfect CSI assumption, which can

satisfy all the resource allocation constraints in LTE uplink

SC-FDMA system. Firstly, we formulate the resource

allocation problem as a mixed integer programming

problem. Then an efficient algorithm based on discrete

stochastic optimization is proposed to solve the problem.

Finally, simulation results are presented to demonstrate the

performance of the algorithm.

2 System model and problem formulation

2.1 System model

Consider a single cell LTE situation. In LTE system, 12

adjacent sub-carriers are grouped into a sub-channel

named resource block (RB). And system bandwidth is

divided into N orthogonal RBs. Assume that there are K

users scatter in the cell andk

N with cardinalityk

N is

the set of RBs allocated to user k. When frequency domain

minimum mean square error (MMSE) equalizer is used at

receiver, the effective signal noise ratio (SNR) for user k

can be given as [3]1

112

, , , ,eff

1 , , , ,

11

12 1k

k n i k n i

k

n N ik k n i k n i

P

N P

=

= +

(1)

where , ,k n iP denotes the transmit power of user k on

sub-carrier i belonging to RB n; , ,k n i represents the SNR.

In constant parameter channel,, ,k n i

can be simply

expressed as2 2

, ,| |

k n ih . Where , ,k n ih denotes the

channel gain,2

is noise variance. However, in

time-varying channel the Doppler spread may cause serious

inter carrier interference (ICI), which will affect the SNR

calculation. The effect is approximated as [7]

( )

2

, ,

, , H 2 2

d s

| | 24,

2

k n i

k n i

hE

f T

=

(2)

where { } ( )1

H, 1 1E x y x y

+ ,

df is the Doppler

frequency ands

T is the symbol duration.

2.2 Problem formulation

In this paper, the objective of resource allocation is to

maximize weighted sum-rate subjecting to the resource

allocation constraints. Therefore the resource allocation

problem is formulated as

( )

( )

, ,

eff

Sum, 1

12max

, ,

1

peak

, ,

max lb 1

s.t. ,

, , ,12

,

k n i k

k

k

Kk

k kP N k

k k n i

n N i

k

k n i

k

k j

BNR

N

P P P k

PP P k n i

N

N N k j

=

=

= +

=

= =

I

(3)

where,max

kP and

peakP are the maximum transmit power

of user k and peak power constraint of per sub-carrier,

respectively; ( ) ( )1 2min , max ,k kn N n N = = { 1,kN n=

}1 2 21,... 1,n n n+ ;

k

denotes user weight chosen for

user k and defined as =1k k

r , wherek

r denotes

average rate of user k. The first and second constraints in

Eq. (3) represent the total power constraint, peak power

constraint, and uniform power allocation. The third

constraint indicates the constraints of exclusivity and

adjacency.

3 Resource allocation with imperfect CSI

In our situation the weighted sum-rate is a

non-decreasing function of power, which can be easilydemonstrated by ( ) ( )Sum , , 0k n iR P > . Therefore, the

power allocation is designed that user can transmit at their

maximum power without violating the peak power

constraint. Then the resource allocation problem is reduced

to RB allocation

( )effSum

1

max lb 1

s.t. ,

k

Kk

k kN

k

k j

BNR

N

N N k j

=

= +

=

I

(4)

where1

1max

peak

, ,12eff

max1 peak

, ,

min ,121

112

1 min ,12

k

kk n i

k

k

n N ik kk n i

k

P PN

N PP

N

=

= +

(5)

Let the RB allocation solution be { },k n K Nx =X , where

, =1k nx denotes RB n is allocated to user k; otherwise

,=0

k nx . When the system has N orthogonal RBs and K

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users ( N K ), there are totally1

1

1

C C !K

N KP

=

=

possible solutions [2]. The set of all possible solutions is

given as { }1 2, , ,... PX = X X X . Define [ ]H X as the

channel gain matrix of solution X. Then, the problem of

RB allocation can be reformulated as

[ ]( )* SumargmaxX X

R

=X H X (6)

where*

X denotes the optimal solution.

For imperfect channel state, only the noisy estimate can

be obtained. Therefore, suppose that at iteration l, the

evolved Node B (eNodeB) obtains the estimate of channel

gain [ ]lH , and selects a channel gain subset [ , ]lH X forsolution Xand then computes the relative noisy estimate of

the objective function ( )Sum , [ ] .r l H X If each

( )Sum , [ ]r l H X is an unbiased estimate of ( )Sum [ ]R H X ,

( )Sum , [ ]r l H X , 1,2,...l= is a sequence of i.i.d. random

variables [8]. Therefore, the problem of RB allocation

problem can be rewritten as

( ){ }* Sumarg max E , [ ]X X

r l

= X H X (7)

By the strong law of large numbers, the empirical average

( ) ( ) ( ){ }Sum _ Sum Sum1

[ ] 1 , [ ] , [ ]L

L

l

r L r l E r l =

= H X H X H X

almost surely as L . Therefore to solve Eq. (7) an

inefficient method is using exhaustive-search (ES)

algorithm. That is, we firstly compute the ( )Sum_ [ ]Lr H X

for each ,XX and then find ( )* Sum_max [ ]LX X

r

=X H X .

In ES algorithm, for each solution, L estimation

objectives should be computed. Thus,

1

1

1

C C !K

N KO L

=

operations should be conducted,

which causes enormous computational complexity. In this

paper, we propose a low computational complexity

algorithm based on discrete stochastic optimization [89],

which is named as proposed algorithm below. Assume[ ]1 1, ,..., P=E e e e , where pe denotes 1P vector with a

one in the p th situation and zeros elsewhere. At iteration

l, the 1P probability vector [[ ] [ ,1], [ ,2], ...,l l l =

][ , ],..., [ , ]l p l P is updated, where [ , ] [0,1]l p

represents the state occupation probabilities of p and

[ , ] 1p

l p = . Let ( )lX be the solution chosen at the

iteration l. For notational simplicity, define a

sequence { }[ ] ,lD where [ ] pl =D e if( )l

p=X X ,

1,2,...,p P .

The pseudo-code of the proposed algorithm is given as

1)Set 1l , and eNodeB selects a solution ( )1 XX ,and set ( )

1[1, ] 1 =X , [1, ] 0 =X for all ( )

1XX X .

2)For 1,2,...l= doa)Given ( )lX at iteration time l and choose another( ) ( )

/l l

XX X uniformly. Each user estimates CSI and

feedbacks it to eNodeB, then the eNodeB computes

( )( )Sum , [ ]lr l H X ,( )( )Sum , [ ]lr l H X .

b)If ( )( ) ( )( )Sum Sum, [ ] , [ ]llr l r l < H X H X , sets ( )1l+ =X ( )lX , otherwise sets ( ) ( )1l l+ =X X .

c)Then, the eNodeB updates all the solution occupationprobabilities as follow:

( )[ ] [ 1] [ ]l l l= + e D (8)

[ 1] [ ] [ ] [ ]l l l l + = + e (9)

{ }l[ 1] [ 1] [ ] [ ]Tl l v l l

+

+ = + + e J (10)

( )[ 1] 1 [ ] [ ] [ ]; [0] 0l l l l + = + =J J e J (11)

d)If ( ) ( )1 *[ 1, ] [ 1, ]l ll l ++ > +X X , the eNodeB sets( ) ( )1 1*l l+ +=X X , otherwise sets ( ) ( )

1*l l+ =X X .

e)Set 1l l + .End for.

3) Output the RB allocation solution of the last iteration.

In Eq. (10),l

v denotes the learning rate. The

expression of { }Y

+

denotes the projection of Yonto the

interval [ , ] + .

The proposed algorithm has low computational

complexity. In the algorithm, only one of the solutions is

chosen at each iteration. Then compute its objective

function, and compare it with the objective value of the

solution selected in previous iteration. Therefore, the

operations complexity will not more than ( )2O LK .

Furthermore, the algorithm also has the characteristic of

fast convergence. To prove the convergence, we firstly

present a sufficient condition based on Ref. [9].

Lemma 1 (Sufficient convergence conditions) The

proposed algorithm converges to the global maximum of

the objective function, if the independent observations

( )*Sum , [ ]r l H X , ( )Sum , [ ]r l H X , ( )Sum , [ ]r l H X meet thefollowing conditions

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( ) ( ){ }*Sum SumPr , [ ] , [ ]r l r l > >H X H X

( ) ( ){ }*Sum Sum Pr , [ ] , [ ]r l r l > H X H X (12)

( ) ( ){ }*Sum SumPr , [ ] , [ ]r l r l > >H X H X ( ) ( ){ }Sum Sum Pr , [ ] , [ ]r l r l > H X H X (13)

Then the sequence { }( )lX is a homogeneous irreducibleand aperiodic Markov chain with state space X , and it

spends more time on state*

X than other states [8].

Relying on the sufficient convergence conditions, the

convergence of the algorithm can be proved as following.

Theorem 1 (Global convergence). If the iteration is

sufficient, the algorithm converges to the global optimum.

Proof Suppose the mean and variance of the

objective function are ( ){ } ( )SumSum [ ], [ ] rE r l = H XH X and( ){ } ( )Sum

2

Sum [ ], [ ] ,

rV r l =

H XH X respectively. As in

Ref. [9], the estimated objective value can be

approximately regarded as the Gaussian distribution,

i.e., ( ) ( ) ( )( )Sum Sum2

Sum [ ] [ ], [ ] , .

r rr l N

H X H XH X Then if

( ) ( )Sum Sum, [ ] , [ ] ,p qr l r l > H X H X we can get

( ) ( )Sum Sum[ ] [ ]p qr r >

H X H X. Therefore, the condition in

Eq. (12) can be rewritten as

( ) ( ){ }Sum SumPr , [ ] , [ ] 0p qr l r l > >XH H X

( ) ( ){ }Sum Sum Pr , [ ] , [ ] 0q pr l r l >H X H X (14)The result of two Gaussian variables subtraction still

keeps the Gaussian property. Therefore, the inequality of

Eq. (14) is equivalent to

( ) ( ) ( ) ( )( ){ }

( ) ( ) ( ) ( )( ){ }Sum Sum Sum Sum

Sum Sum Sum Sum

2 2

[ ] [ ] [ ] [ ]

2 2

[ ] [ ] [ ] [ ]

Pr , 0

Pr , 0

p q p q

q p p q

r r r r

r r r r

N

N

+ > >

+ >

H X

H X

H X H X H X

H X H X H X

(15)

We know( ) ( ) ( ){Sum Sum Sum[ ] [ ] [ ]max , ,p p qr r r =H X H X H X

( )}Sum [ ]sr H X , which implies ( ) ( )( )Sum Sum[ ] [ ]p qr r >H X H X

( )( ( ) )Sum Sum[ ] [ ]q pr r H X H X . Therefore, condition in Eq. (12)is satisfied due to the same variance.

Similar to the condition in Eq. (12), the condition in

Eq. (13) can also be proved. Therefore, the convergence of

the algorithm is proved.

4 Simulation results and discussions

In this section, we study a system with 6N= RBs and

2.6 GHz carrier frequency. Assume that 4K= users

uniformly distribute in the cell. The path loss is100.8+20lg dBd , where d is the distance of users to the

eNodeB. The time-varying channel is modeled as that in

Ref. [8]. To simplify the analysis we assume that themax

kP

of different users is same, andmax

200 mWk

P = . The peak

power constraint of each sub-carrier is peak 10 mWP = .

The noise power spectrum density is 150 dBm Hz . Other

simulation parameters such as symbol duration are set

according to LTE standard. Furthermore, a resource

allocation algorithm [3] based on perfect CSI is also

simulated, which named as perfect CSI.Fig. 1 shows the convergence performance of different

algorithms in time-invariant channel. We can observe that as

the number of iterations increases the obtaining

weighted-sum rate of both exhaustive-search and proposed

algorithm converge to that of the perfect CSI. This is

because the estimated objective value is an unbiased

estimate of the objective function, as the number of

iterations increases, the performance of ES algorithm and

the proposed algorithm approximate to the optimal solution

by the strong law of large numbers. Furthermore, we can

also find that the proposed algorithm can achieve desirable

performance just like ES algorithm.

Fig. 1 The convergence performance of different algorithms

Fig. 2 shows the tracking channel capability of different

algorithms ( 600L= ). From the Fig. 2, we can find that

the weighted-sum rate is varying due to the time-varying

environment. And the tracking channel capability of both

proposed and ES algorithms become poor with increasing

of users speed. This can be explained that with the

increase of users speed, the channel changes more quickly

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and becomes more difficult to track. Furthermore, since

the lower computational complexity and similar

convergence performance, as is illustrated in Fig. 2, the

proposed algorithm has better capability of tracking

channel especially when users speed is high.

Fig. 2 The tracking channel capability of different algorithms

5 Conclusions

In this paper, we study the problem of resource

allocation in LTE uplink SC-FDMA system with imperfect

CSI assumption. Firstly, the resource allocation problem is

formulated as a mixed integer programming problem.

Then an efficient algorithm based on discrete stochastic

optimization is proposed to solve the problem, which has

been proved to be able to converge to the optimal solution.

Finally, simulation results show that the proposed

algorithm has desirable system performance.

Acknowledgements

This work was supported by Ministry of Industry and

Information Technology of the Peoples Republic of China

(2011ZX03001-007-03), the National Natural Science Foundation ofChina (61271182), and the National Natural Science Funds of China

for Young Scholar (61001115).

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