Energy-efficiency resource allocation of very large multi-user MIMO systems

Energy-efficiency resource allocation of very largemulti-user MIMO systems

Ying Hu • Baofeng Ji • Yongming Huang •

Fei Yu • Luxi Yang

� Springer Science+Business Media New York 2014

Abstract With increasing demand in multimedia appli-

cations and high data rate services, energy consumption of

wireless devices has become a problem. At the user equip-

ment side, high-level energy consumption brings much

inconvenience, especially for mobile terminals that cannot

connect an external charger, due to an exponentially

increasing gap between the available and required battery

capacity. Motivated by this, in this paper we consider uplink

energy-efficient resource allocation in very large multi-user

MIMO systems. Specifically, both the number of antenna

arrays at BS and the transmit data rate at the user are adjusted

to maximize the energy efficiency, in which the power

consumption accounts for both transmit power and circuit

power. We proposed two algorithms. Algorithm1, we dem-

onstrate the existence of a unique globally optimal data rate

and the number of antenna arrays by exploiting the properties

of objective function, then we develop an iterative algorithm

to obtain this optimal solution. Algorithm2, we transform the

considered nonconvex optimization problem into a convex

optimization problem by exploiting the properties of frac-

tional programming, then we develop an efficient iterative

resource allocation algorithm to obtain this optimal solution.

Our simulation results did not only show that the the pro-

posed two algorithms converge to the solution within a small

number of iterations, but demonstrated also the perfor-

mances of the proposed two algorithms are close to the

optimum. Meanwhile, it also shows that with a given number

iterations the performance of proposed algorithm1 is supe-

rior to proposed algorithm2 under small pC. On the contrary,

the performance of proposed algorithm2 is superior to pro-

posed algorithm1 under large pC.

Keywords Energy efficiency � Multi-user � MIMO

1 Introduction

With increasing interest in multimedia applications and high

data rate services, energy consumption of wireless devices is

rapidly increasing. Reducing energy consumption at base

station side usually has a direct impact on operational

expenditure as well as CO2 emissions. At the user equipment

side, high-level energy consumption brings much inconve-

nience, especially for mobile terminals that are not able to

connect an external charger, due to an exponentially

increasing gap between the available and required battery

capacity [1]. Therefore, in addition to maximizing

throughput [2, 3], maximizing energy efficiency is becoming

increasingly important for wireless system design [4–15].

Additionally, MIMO technology has been a key technology

Y. Hu � B. Ji (&) � Y. Huang � F. Yu � L. Yang

School of Information Science and Engineering, Southeast

University, Nanjing, China

e-mail: [email protected]; [email protected]

Y. Hu

e-mail: [email protected]

Y. Huang


F. Yu


L. Yang


Y. Hu

Institute of Electronics and Information, Jiangsu University of

Science and Technology, Zhenjiang, China

B. Ji

Information Engineering College, Henan University of Science

and Technology (HAUST), Luoyang, China

123

Wireless Netw

DOI 10.1007/s11276-013-0674-x

for advanced wireless systems. Basically, more antennas the

transmitter/receiver are equipped with, more degrees of

freedom the propagation channel can provide, and the better

the performance in terms of data rate or link reliability [16–

18] is. In a multi-user scenario, multi-user MIMO systems

can provide a substantial gain in networks by allowing

multiple users to communication in the same frequency and

time slot [19–21]. Recently, there has been a great deal of

interest in MU-MIMO with very large antenna arrays at BS.

Very large arrays can substantially reduce intra-cell inter-

ference with simple signal processing, where ‘‘very large

MIMO’’ usually means the arrays comprising a hundred, or a

few hundreds, of antennas, simultaneously serving tens of

users [22]. It was also revealed in [22] that with a very large

antenna array, the effect of small-scale fading can be aver-

aged out. Whereas, it was pointed out in [23] that although

MIMO techniques have been shown to be effective in

improving capacity and spectral efficiency (SE) of wireless

systems, energy consumption also increases.

Recently energy-efficient design has emerged as a new

trend in wireless communications, where the energy con-

sumption usually account for circuit power in addition to the

transmitted power, e.g. [1, 4–7, 9–12, 14, 15]. In [1], the

authors address the energy-efficient resource allocation

problem in both downlink and uplink of OFDMA networks.

Optimal and low-complexity suboptimal algorithms are

developed to solve the QoS and priority/fainess issues. In [5,

6], the authors study uplink energy-efficient transmission in

single-cell OFDMA systems. The work in [7, 10] investigate

multi-cell interference-limited scenarios and develop a non-

cooperating game for energy-efficient power optimization.

Using throughput per Joule as a performance metric, link

adaptation and resource allocation techniques have been

studied in [9], which maximizes energy efficiency by adapting

both overall transmit power and its allocation, according to the

channel state information and the circuit power consumption.

In [11], we consider energy-efficient design of resource allo-

cation for a multi-user OFDMA and develop schemes of user

selection, rate allocation and power allocation under QoS

requirement to maximize the energy efficiency. In [14],

resource allocation for energy-efficient communication in an

OFDMA downlink network is studied. By exploiting the

properties of fractional programming, the non-convex opti-

mization problem in fractional form is transformed into an

equivalent optimization problem in subtractive form.

It is worth mentioning that the above work only consider

energy efficient resource in a single-antenna or fixed-beam

OFDM system, the energy efficient design concerning

antenna selection for very large MIMO system is still an open

system. Motivated by this, in this paper we consider uplink

energy-efficient resource allocation in very large multi-user

MIMO systems. Specifically, in our problem formulation the

number of antenna arrays at BS and the transmit data rate at

the user are jointly optimized to maximize the energy effi-

ciency, in which the power consumption accounts for both

transmit power and circuit power. We proposed two algo-

rithms. Algorithm1, we demonstrate the existence of a

unique globally optimal data rate and the number of antenna

arrays by exploiting the properties of objective function, then

we develop a iterative algorithm to obtain this optimal

solution. Because the convergence rate and accuracy of

direct algorithm depend on the value of step length, then we

develop the indirect algorithm. Algorithm2, we transform

the considered nonconvex optimization problem into a

convex optimization problem by exploiting the properties of

fractional programming, then we develop an efficient itera-

tive resource allocation algorithm to obtain this optimal

solution. Our simulation results did not only show that the the

proposed two algorithms converge fast to the optimal solu-

tion, but demonstrated also the performances of the proposed

two algorithms are close to the optimum.Meanwhile, it also

shows that with a given number iterations the performance of

proposed algorithm1 is superior to proposed algorithm2

under small pC. On the contrary, the performance of pro-

posed algorithm2 is superior to proposed algorithm1 under

large pC.

The remainder of this paper is organized as follow. In

Sect. 2, we introduce the very large multi-user MIMO

system model and formulate the optimization problem for

the uplink. In Sect. 3, we propose two iterative algorithms

to obtain the optimal solution. Then, we present numerical

results in Sect. 4. Finally, we conclude the paper in Sect. 5.

2 System model

We consider the uplink of a MU-MIMO system, consisting

of one BS equipped with an array of M antennas that

receive data from K single-antenna users. We assume that

the BS has perfect CSI and employs zero-forcing receiver

beamforming. Provided M C K ? 1, it is shown in [22]

that the achievable uplink rate for the kth user under

Rayleigh is lower bounded by

rk ¼ log2½1þ pkðM � KÞbk� ð1Þ

where bk denotes the large-scale channel factor for user

k. Without loss of generality, here we take the noise vari-

ance to be 1, to simplify notation. With this convention, pk

has the interpretation of normalized ‘‘transmit’’ SNR. Note

that in Eq. (1), the small-scale fading is removed, but the

effects of large-scale fading is remained. This may give

different users different SNRs. As a result, the optimal

transmit power for each user would depend only on othe

large-scale fading, so the introduction of such power con-

trol may bring the fairness between users near and far from

the BS. To issue with faireness versus throughput, which

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123

we would like to avoid here as this matter could easily

observe the main points of our analysis. Therefore, for

analytical tractability, we ignore the effect of the large-

scale fading here [22], i.e.

bk ¼ 1; k ¼ 1; 2; . . .K. Hence, the equation of rk can be

equivalent to

rk ¼ log2½1þ pkðM � KÞ� ð2Þ

Denote the data rate per user k as rk and the vector on all

users as

R ¼ ½r1; r2; . . .; rK �T ð3Þ

where ½��T is the transpose operator. Note that in Eq. (2), the

small-scale fading is removed and the effect of the large-

scale fading is ignored, so we consider

p1 ¼ p2 ¼ � � � ¼ pk ¼ � � � ¼ pK ¼ p

r1 ¼ r2 ¼ � � � ¼ rk ¼ � � � rK ¼ rð4Þ

Corresponding, the overall data rate is

RðrÞ ¼XK

k¼1

rk ¼ Kr ð5Þ

Denote the overall transmit power as PT(r, M) and

PTðr;MÞ ¼XK

k¼1

pk ¼ K2r � 1

M � Kð6Þ

Then the overall power consumption will be

Pðr;MÞ ¼ PTðr;MÞ þ PCðMÞ ð7Þ

We denote the circuit power as PC(M) = MpC, where pC

is the circuit power of each antenna. Here, the circuit

energy consumption includes the energy consumed by all

the circuit blocks along the signal path: analog to digital

converter (ADC), digital to analog converter (DAC),

freqency synthesizer, mixer, lower noise amplifier (LNA),

power amplifier, and baseband DSP [24]. In [7], it was

showed that transmit power is need to compensate path

loss.

The energy efficiency will be

Uðr;MÞ ¼ RðrÞPTðr;MÞ þ PCðMÞ

ð8Þ

Hence, the problem of maximizing the energy efficient

in the system can be expressed as

fM�; r�g ¼ argmax|fflfflfflffl{zfflfflfflffl}fM;rg

Uðr;MÞ ð9Þ

3 Energy-efficient resource allocation

The objective function is a ratio of two functions which is a

nonconvex function. As a result, we will develop the direct

and indirect approaches for energy-efficient resource allo-

cation. Algorithm1, we demonstrate the existence of a

unique globally optimal data rate and the number of antenna

arrays by exploiting the properties of objective function,

then we develop an iterative algorithm to obtain this optimal

solution. Algorithm2, we transform optimization problem

by exploiting the properties of fractional programming, then

we develop an efficient iterative resource allocation algo-

rithm to obtain this optimal solution.

3.1 Direct algorithm

In the following, we demonstrate that a unique globally

optimal data rate and the number of antenna arrays always

exists and give the necessary and sufficient conditions for a

data rate and the number of antenna arrays to be the unique

and globally optimum. In particular, we allow M to be a

positive real value instead of integer. Then we can take a

derivative with respect to M.

Lemma 1 The energy efficiency function U(r, M) is i.

strictly quasi-concave (furthermore, U(r, M) is either

strictly decreasing or first strictly increasing and then

strictly decreasing) w.r.t M for a fixed r(r = 0). ii. strictly

quasi-concave (furthermore, U(r, M) is first strictly

increasing and then strictly decreasing) w.r.t r for a fixed

M.

Proof see ‘‘Appendix 1’’.

Theorem 1 The energy efficiency function U(r, M)

i. there exists a unique globally optimal the number of

base station antennas M* for fixed r(r = 0), which can

be given by

(1) when

RðrÞ½Kð2r � 1Þ � pC� � 0; oUðr;MÞoMjM¼M� ¼ 0, i.e.

M� ¼ dK þffiffiffiffiffiffiffiffiffiffiffiffiffiK 2r��1

pC

qe.

(2) when R(r)[K(2r - 1) - pC] \ 0, M* = K ? 1

ii. there exists a unique globally optimal transmit rate r*

for a fixed M, where r* is given byoUðr;MÞ

orjr¼r� ¼ 0.

Theorem 1 provides the necessary and sufficient con-

ditions for a rate r* and the number of base station antennas

M* to be the unique and globally optimum one. However,

it is difficult to directly solve the joint optimization prob-

lems. Obviously, there exists an analytical solution of a

unique globally optimal the number of base station anten-

nas M* for fixed r(r = 0). Therefore, we develop an iter-

ative method to search the globally optimal r* and M* to

maximizing U(r, M). In Algorithm1, the U(r*, M*) is

obtained by exhaustive r*, therefore the energy efficiency

U(r, M) can converges to the optimal U(r*, M*).

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Algorithm 1 Description:

1 r* = r0, M* = M0, Umax = U0, initialize g[ 1

2 while U(r*, M*) [ Umax

3 do Umax �Uðr�;M�Þ4 if RðrÞ½Kð2r�1Þ � pC� � 0;M� ¼ dK þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiK 2r��1

pC

qe

5 else M* = K ? 1

6 r* = g* r*

7 Return Umax, r*, M*

3.2 Indirect algorithm

Obviously, the convergence rate and accuracy of direct

algorithm depend on the value of g, then we develop the

indirect algorithm. The fractional objective function can be

classified as a nonlinear fractional program [25]. We can

get the following theorem from theorem in [25].

Theorem 2 q� ¼ Rðr�ÞPT ðr�;M�ÞþPCðM�Þ ¼ max|{z}

fM;rg

RðrÞPT ðr;MÞþPCðMÞ if

and only if

max|{z}fM;rg

RðrÞ � q�½PTðr;MÞ þ PCðMÞ�

¼ Rðr�Þ � q�½PTðr�;M�Þ þ PCðM�Þ�¼ 0

ð10Þ

For R(r) C 0 and PT(r,M) ? PC(M) [ 0.

As a result, we can focus on the equivalent objective

function,

FðqÞ ¼ max|{z}fM;rg

RðrÞ � q½PTðr;MÞ þ PCðMÞ� ð11Þ

in the rest of the paper.

The problem above is now joint concave w.r.t all opti-

mization variables, c.f. ‘‘Appendix 2’’.

Using standard optimization technique, the rate

allocation

r� ¼ log2

K þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 � 4q2ðln2Þ2pCK

q

2q2ðln2Þ2pC

24

35 ð12Þ

and the number of base station antenna

M� ¼ d2r�qln2þ Ke ð13Þ

Therefore, we develop an iterative algorithm to search

the optimal r*, M* and q* to maximizing function f.

Algorithm 2 Description:

1 r* = r0, M* = M0, q* = 0, initialize e ¼ 0:01

2 while Rðr�Þ � q�½PTðr�;M�Þ þ PCðM�Þ�[ e

3 do q� � Rðr�Þ½PT ðr�;MÞþPCðM�Þ�

4 adopt formula above obtain rate allocation

5 adopt formula above obtain the number of base station

antennas

6 Return q*, r*, M*

Proof Please refer to ‘‘Appendix 3’’ for the proof of

convergence.

4 Simulations

In this section, we provide the simulation results to eval-

uate the energy efficiency, the overall transmit power, the

number of base station antenna M*, and the spectral effi-

ciency versus the number of user. Moreover, we provide

the simulation results to evaluate the energy-efficiency

versus the number of iterations. In [11, 12], it is found that

the value of pC plays an important role in increasing the

energy efficiency. In this paper, we assume a static circuit

power consumption pC = 1 mw, 10 mw, respectively.

Meanwhile, we assume an initialize optimal transmit rate

r0 = 2.0 bit/s/Hz, g = 1.002. The optimal algorithm is

obtained through exhaustive. The ‘‘proposed algorithm1

iteration 3’’ means that we adopt algorithm1 with a given

number iterations of 3, and the same to the proposed

algorithm2 iteration 3.

Figures 1 and 2 show the energy efficiency versus the

number of user under different pC value. As clearly seen,

the energy efficiency increases with the number of users.

Furthermore, it also shows that under the two situations,

the energy efficiency is larger under the small pC value.

Meanwhile, the energy efficiency increases quicker under

the small pC value and the energy efficiency increases

slower under the large pC value. Moreover, it also shows

that the performance of both the two proposed algorithms

are close to the optimal. On the other hand, the energy

efficiency of the proposed algorithms with a given num-

ber iterations under different pC value are different. For

instance, pC = 1 mw, the performance of proposed

algorithm1 is superior to proposed algorithm2. On the

contrary, pC = 10 mw, the performance of proposed

algorithm2 is superior to proposed algorithm1. This is

because under the small pC value, the energy efficiency is

larger, the convergence rate of optimal q* in proposed

algorithm2 is slow.

Figures 3 and 4 depict the optimal number of base sta-

tion antennas M* versus the number of users under dif-

ferent pC value. As expected, the number of base station

antennas increases with the number of users. This is

because M� ¼ dK þffiffiffiffiffiffiffiffiffiffiffiffiK 2r�1

pC

qe or M* = K ? 1. Mean-

while, the number of base station antennas is larger under

the small pC value. This is because the circuit power

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123

PC(M) = MpC. Furthermore, it also shows that the per-

formance of both the two proposed algorithms are close to

the optimal. On the other hand, the optimal number of base

station antennas of the proposed algorithms with a given

number iterations under different pC value are different.

For instance, pC = 1 mw, when the number of users is less

than 16, the performance of proposed algorithm2 is supe-

rior than proposed algorithm1, then the gap between the

proposed algorithm2 and optimal algorithm increases dra-

matically with the number of users. While the gap between

the proposed algorithm1 and optimal algorithm increases

slowly with the number of users. On the contrary, pC = 10

mw, the performance of proposed algorithm2 is superior to

proposed algorithm1, the gap between the performance of

proposed algorithm1 and optimal algorithm increases with

the number of users. Figures 3 and 4 also show when

serving tens of users, the optimal number of base station

antennas needs a hundred, or a few hundreds, especially

under the small pC value.

Figures 5 and 6 depict the overall transmit power

versus the number of users under different pC value.

Figures 7 and 8 depict the spectral efficiency versus the

number of users under different pC value. As clearly seen,

the trend of the overall transmit power and the spectral

efficiency versus the number of users are the same to the

optimal number of base station antennas M* versus the

number of users.

Figures 9 and 10 illustrate the energy efficiency versus

the number of iterations for K = 20 under different pC

value. pC = 1 mw, the number of iterations in the pro-

posed algorithms is set to 5. It can be observed that the

proposed algorithm1 converges 99 % of the optimal

10 15 20 25 30 35 4016

18

20

22

24

26

28

30

32

34

Number of Users

Ene

rgy

Effi

cien

cy(b

it/H

z/J)

optimal algorithm pc=10mw

proposed algorithm1 pc=10mw


proposed algorithm1 iterations 3 pc=10mw


Fig. 2 pC = 10 mw energy efficiency versus number of user

10 15 20 25 30 35 40150

200

250

300

350

400

450

500

550

600

650

Number of Users

Num

ber

of B

ase

Sta

tion

Ant

enna

s






Fig. 3 pC = 1 mw number of base station antennas versus number of

user

10 15 20 25 30 35 4060

80

100

120

140

160

180

Number of Users

Num

ber

of B

ase

Sta

tion

Ant

enna

s






Fig. 4 pC = 10 mw number of base station antennas versus number

of user

10 15 20 25 30 35 4050

60

70

80

90

100

110

120

Number of Users

Ene

rgy

Effi

cien

cy(b

it/H

z/J)




proposed algorithm1 iteration 3 pc=1mw

proposed algorithm2 iteration 3 pc=1mw

Fig. 1 pC = 1 mw energy efficiency versus number of user

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123

value in 2 iterations, while the proposed algorithm2

converges to the optimal value within 5 iterations. But,

the proposed algorithm1 still converges 99 % of the

optimal value within 5 iterations. pC = 10 mw, the

number of iterations in the proposed algorithms is set to

3. It can be observed that the proposed algorithm1

converges 99 % of the optimal value in 2 and 3 itera-

tions, while the proposed algorithm2 converges to the

optimal value within 3 iterations.In other words, the

maximum system energy efficiency can be achieved

within a few iterations. It also shows that with a given

number iterations the performance of proposed algo-

rithm1 is superior to proposed algorithm2 when pC = 1

mw. On the contrary, the performance of proposed

algorithm2 is superior to proposed algorithm1 when

pC = 10 mw.

10 15 20 25 30 35 400.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Number of Users

over

all t

rans

mit

pow

er(W

)






Fig. 6 pC = 10 mw overall transmit power versus number of user

10 15 20 25 30 35 4020

30

40

50

60

70

80

90

100

110

120

Number of Users

Spe

ctra

l Effi

cien

cy(b

it/s/

Hz)






Fig. 7 pC = 1 mw spectral-efficiency versus number of user

10 15 20 25 30 35 4020

30

40

50

60

70

80

90

100

110

120

Number of Users

Spe

ctra

l Effi

cien

cy(b

it/s/

Hz)






Fig. 8 pC = 10 mw spectral-efficiency versus number of user

1 1.5 2 2.5 3 3.5 4 4.5 530

35

40

45

50

55

60

65

70

75

80

Number of iterations

Ene

rgy

Effi

cien

cy(b

it/H

z/J)

proposed Algorithm1proposed Algorithm2maximum energy efficiency

Fig. 9 pC = 1 mw energy efficiency versus the number of iterations

10 15 20 25 30 35 40

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Number of Users

over

all t

rans

mit

pow

er(W

)






Fig. 5 pC = 1 mw overall transmit power versus number of user

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5 Conclusion

In this paper, we have investigated uplink energy-efficient

resource allocation in very large multi-user MIMO sys-

tems. Our goal is to jointly optimize rate allocation and the

number of antenna arrays at BS, such that the performance

measure in terms of throughput per Joule is maximized, in

which the power consumption accounts for both transmit

power and circuit power. We proposed two Algorithms.

Algorithm1, we demonstrate the existence of a unique

globally optimal data rate and the number of antenna arrays

by exploiting the properties of objective function, then we

develop a iterative algorithm to obtain this optimal solu-

tion. Algorithm2, we transform the considered nonconvex

optimization problem into a convex optimization problem

by exploiting the properties of fractional programming,

then we develop an efficient iterative resource allocation

algorithm to obtain this optimal solution. Our simulation

results did not only show that the the proposed two algo-

rithms converge to the solution within a small number of

iterations, but demonstrated also the performances of the

proposed two algorithms are close to the optimum.

Meanwhile, it also shows that with a given number itera-

tions the performance of proposed algorithm1 is superior to

proposed algorithm2 under small pC. On the contrary, the

performance of proposed algorithm2 is superior to pro-

posed algorithm1 under large pC. It is also found that we

can change the number of antenna arrays at BS to achieve

energy-efficient maximization.

Acknowledgments This work was supported by National Science

and Technology Major Project of China under Grant

2013ZX03003006-002, National Natural Science Foundation of China

under Grants 61271018, 61201176 and 61372101, Research Project of

Jiangsu Province under Grants BK20130019, BK2011597, and

BE2012167, Program for New Century Excellent Talents in University

under Grant NCET-11-0088.

Appendix 1

Proof of lemma 1

i. Denote the upper contour sets of U(r, M) as Sa =

{ M C K ? 1 | U(r,M) C a }.

According to proposition C.9 of [23], U(r, M) is strictly

quasi-concave if and only if Sa is strictly convex for any real

number a. When a B 0, no points exists on the contour

U(r, M) = a. When a[ 0, Sa is equivalent to Sa={ M C K?1

| a P_T(r,M) ? a PC(M) - R(r) B 0 }. Since PT(r, M) and

PT(M) are strictly convex in M, Sa is also strictly convex.

Hence, we have the strict quasiconcavity of U(r, M)

The partial derivative of U(r, M) with M is

oUðr;MÞoM

¼ �RðrÞ½P0Tðr;MÞ þ P0CðMÞ�

½PTðr;MÞ þ PCðMÞ�2

¼M /ðr;MÞ½PTðr;MÞ þ PCðMÞ�2

ð14Þ

where PT’ (r, M) is the first partial derivative of PT(r, M) with

respect of M, PC’ (M) is the first partial derivative of

PC(M) with respect of M. According to Lemma 1, if M*

exists such thatoUðr;MÞ

oMjM¼M� ¼ 0, it is unique, i.e. There is a

M* such that / (r,M) = 0, it is unique. In the following, we

investigate the conditions when M* exists.

The derivative of /(r, M) with respect of M is

/0 ðr;MÞ ¼ �RP00Tðr;MÞ\0. Where P00Tðr;MÞ is the second

partial derivative of PT(r, M) with respect of M. Hence, /(r, M) is strictly decreasing. According to the L’Hopital’s

rule, it is easy to show that

limM!1

/ðr;MÞ ¼ limM!1

f�RðrÞ½P0Tðr;MÞ þ P0

CðMÞ�g

¼ limM!1

f�RðrÞ½P0Tðr;MÞ þ P0CðMÞ�

MMg

¼ limM!1

½�RðrÞP00Tðr;MÞ1

M�\0 ð15Þ

Because of the M C K ? 1, where K is the number of

users, so

limM!Kþ1

/ðr;MÞ ¼ limM!Kþ1

f�RðrÞ½P0Tðr;MÞ þ P0

TðMÞ�g

¼ �RðrÞ½�Kð2r � 1Þ þ pC�¼ RðrÞ½Kð2r � 1Þ � pC� ð16Þ

1. when RðrÞ½Kð2r � 1Þ � pC� � 0; limM!Kþ1

/ðr;MÞ� 0.

We can see that M* exists and U(r, M) is first strictly

increasing and then strictly decreasing in M.

1 1.5 2 2.5 321

21.5

22

22.5

23

23.5

24

Number of iterations

Ene

rgy

Effi

cien

cy(b

it/H

z/J)

proposed Algorithm1proposed Algorithm2maximum energy efficiency

Fig. 10 pC = 10 mw energy efficiency versus the number of

iterations

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2. when RðrÞ½Kð2r � 1Þ � pC�\0; limM!Kþ1

/ðr;MÞ\0.

We can see that U(r, M) is always strictly decreasing

in M. Hence, U(r, M) is maximized at M = K ? 1.

ii. The quasi-concavity of U(r, M) with respect of r is the

same as the quasi-concavity of U(r, M) with respect of M.

The partial derivative of U(r, M) with r is

oUðr;MÞor

¼ R0 ðrÞ½PTðr;MÞ þ PCðMÞ� � RðrÞ �P

0Tðr;MÞ

½PTðr;MÞ þ PCðMÞ�2

¼M

uðr;MÞ½PTðr;MÞ þ PCðMÞ�2

ð17Þ

where �P0

Tðr;MÞ is the first partial derivative of

PT(r, M) with respect of r, R’(r) is the first partial deriva-

tive of R(r) with respect of r. According to Lemma 1, if r*

exists such thatoUðr;MÞ

orjr¼r� ¼ 0, it is unique, i.e. there is a

r* such that uðr�;MÞ ¼ 0, it is unique. In the following, we

investigate the conditions when r* exists.

The derivative of uðr;MÞ with respect of r is

u0 ðr;MÞ ¼ �RðrÞ�P00Tðr;MÞ\0 ð18Þ

where �P00Tðr;MÞ is the second partial derivative of

PT(r, M) with respect of r. Hence, uðr;MÞ is strictly

decreasing. According to the L’Hopital’s rule, it is easy to

show

limr!1

uðr;MÞ¼ limr!1fR0 ðrÞ½PTðr;MÞþPCðMÞ��RðrÞ�P0Tðr;MÞg

¼ limr!1

R0 ðrÞ½PTðr;MÞþPCðMÞ��RðrÞ�P0Tðr;MÞ

rr

� �

¼ limr!1

�RðrÞ�P00Tðr;MÞ1

r

� �\0

ð19Þ

Besides

limr!0

uðr;MÞ ¼ limr!0

R0 ðrÞ½PTðr;MÞ þ PCðMÞ�

� RðrÞ�P0Tðr;MÞ ¼ KPCðMÞ[ 0

We can see that r* exists and U(r, M) is first strictly

increasing and then strictly decreasing in r.

Appendix 2

Proof of the concavity of the problem

Without loss of generality, we define function

f ¼ RðrÞ � q½PTðr;MÞ þ PCðMÞ�

¼ Kr � qðK 2r � 1

M � KþMpCÞ

ð20Þ

The Hessian matrix is given by Hðf Þ ¼�qK

2rðln2Þ2M�K

qK 2r ln2

ðM�KÞ2

qK 2r ln2

ðM�KÞ2 �2qK 2r�1

ðM�KÞ3

0

@

1

A

So, f is jointly concave w.r.t r and M. Therefore, the

objective function is jointly concave w.r.t r and M.

Appendix 3

Proof of algorithm 2 convergence

We follow a similar approach as in [25] for proving to

the convergence of algorithm 2. We first introduce two

propositions.

Proposition 1 F(q) is strictly monotoinc decreasing, i.e.

Fðq00Þ\Fðq0 Þ if q0\q00

Proof let fr00;M00g maximize Fðq00Þ, then

Fðq00Þ ¼ max|{z}fM;rg

RðrÞ � q00½PTðr;MÞ þ PCðMÞ�

¼ Rðr00Þ � q00½PTðr00;M00Þ þ PCðM00Þ�\Rðr00Þ � q

0 ½PTðr00;M00Þ þ PCðM00Þ�� max|{z}fM;rg

RðrÞ � q0 ½PTðr;MÞ þ PCðMÞ�

¼ Fðq0 Þ

ð21Þ

Proposition 2 let r0, M0 be an arbitrary feasible solution

and q0 ¼ Rðr0 Þ

PT ðr0 ;M0 ÞPCðM0 Þ, then F(q’) [ 0.

Proof

Fðq0 Þ ¼ max|{z}fM;rg

RðrÞ � q0 ½PTðr;MÞ þ PCðMÞ�

�Rðr0 Þ � q0 ½PTðr

0;M

0 Þ þ PCðM0 Þ�

¼ 0

ð22Þ

We are now ready to prove the convergence of

algorithm 2.

a. First we shall prove that the energy efficiency q increases

in each iteration. Suppose qk = q* and qk?1 = q*

represent the energy efficiency of the considered system

in iteration k and k ? 1, respectively. Theorem 2 and

proposition 2 imply F(qk) [ 0. By definition we have

R(rk) = qk?1[PT(rk,Mk) ? PC(Mk)]. Hence

FðqkÞ ¼ RðrkÞ � qk½PTðrk;MkÞ þ PCðMkÞ�¼ qkþ1½PTðrk;MkÞ þ PCðMkÞ�� qk½PTðrk;MkÞ þ PCðMkÞ�¼ ðqkþ1 � qkÞ½PTðrk;MkÞ þ PCðMkÞ�[ 0 ð23Þ

We have qk?1 [ qk.

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b. Obviously, the energy efficiency q converges to the

optimal q* such that it satisfies the optimality condi-

tion in theorem 2, i.e. F(q*) = 0.

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Ying Hu received the B.S. and

M.S. degrees in 2003 and 2008,

and is currently working

towards Ph.D. degree at South-

east University, China. She also

held teaching/research positions

at the school of Jiangsu Uni-

versity of Science and Tech-

nology. Her major is

information and communication

engineering and her interests

mainly include signal process-

ing, cooperative communica-

tions, green communication.

Baofeng Ji received the B.S.

and M.S. degrees in 2006 and

2009, and is currently working

towards Ph.D. degree at South-

east University, China. He is

major in information and com-

munication engineering and

interested in signal processing,

cooperative communications,

WLAN.

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Yongming Huang received the

B.S. and M.S. degrees from

Nanjing University, China, in

2000 and 2003, respectively,

and the Ph.D. degree in electri-

cal engineering from Southeast

University, China, in 2007.

After graduation, he held

teaching/research positions at

the school of Information Sci-

ence and Engineering, Southeast

University, China, where he is

currently an Associate Profes-

sor. During 2008–2009, he was

visiting the Signal Processing

Laboratory, Electrical Engineering, Royal Institute of Technology

(KTH), Stockholm, Sweden. His current research interests include

MIMO communication systems, multiuser MIMO communications,

cooperative communications, and satellite mobile communications.

Fei Yu received the Ph.D.

degree in electrical engineering

from Southeast University in

2008, China. She held teaching/

research positions at the school

of Information Science and

Engineering, Southeast Univer-

sity, China, where she is cur-

rently a university lecturer. Her

current research interests

include MIMO communication

systems, multiuser MIMO

communications, interference

alignment and cooperative

communications.

Luxi Yang (M’96) received the

M.S. and Ph.D. degree in elec-

trical engineering from the

Southeast University, Nanjing,

China, in 1990 and 1993,

respectively. Since 1993, he has

been with the Department of

Radio Engineering, southeast

university, where he is currently

a professor of information sys-

tems ans communications, and

the Director of Digital Signal

Processing Division. His current

research interests include signal

processing for wireless com-

munications, MIMO communications, cooperative relaying systems,

and statistical processing. He is the author or co-author of two pub-

lished books and more than 100 journal papers, and holds 10 patents.

Prof.Yang received the first and second class prizes of science and

technology progress awards of the state education ministry of China

in 1998 and 2002. He is currently a member of Signal Processing

Committee of Chinese Institute of Electronics.

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Energy-efficiency resource allocation of very large multi-user MIMO systems