Energy Efficiency of Large-Scale Multiple Antenna Systems with Transmit Antenna Selection

638 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 2, FEBRUARY 2014

Energy Efficiency of Large-Scale Multiple AntennaSystems with Transmit Antenna Selection

Hui Li, Lingyang Song, Senior Member, IEEE, and Merouane Debbah, Senior Member, IEEE

Abstract—In this paper, we perform transmit antenna selectionto improve the energy efficiency of large scale multiple antennasystems. We derive a good approximation of the distributionof the mutual information in this antenna selection system. Itshows that channel hardening phenomenon is still retained asfull complexity with antenna selection. Then, we use this closed-form expression to assess the energy efficiency performance.Specifically, we evaluate the performance of the energy efficiencyin two different cases: 1) the circuit power consumption iscomparable to or even dominates the transmit power, and 2)the circuit power can be ignored due to relatively much highertransmit power. The theoretical analysis indicates that thereexists an optimal number of selected antennas to maximizethe energy efficiency in the first case, whereas in the secondcase, the energy efficiency is maximized when all the availableantennas are used. Based on these conclusions, two simple butefficient antenna selection algorithms are proposed to obtain themaximum energy efficiency. All the analytical results are verifiedthrough computer simulations.

Index Terms—Large scale multiple antenna system, transmitantenna selection, channel hardening, energy efficiency, selectionalgorithms.

I. INTRODUCTION

IN the past years, the problem of energy efficiency hasmainly been studied for power-limited applications [1]–[2].

However, with growing energy demand and increasing energyprice, this problem has been noticed in the development offuture mobile cellular systems such as long term evolution-advanced (LTE-A) [3]. As is known to all, for a point-to-pointcommunication system, using multiple antennas can allowone to dramatically decrease the transmit power. Recently,there has been a great deal of interest in large scale multipleantenna systems, which typically means that wireless systemsare allowed to use much more antennas than in systems beingbuilt today, say, a hundred antennas or more [4]–[5]. Simplesignal detectors are proposed to alleviate the computationalcomplexity problem in multiple antenna systems [6]–[7]. In

Manuscript received June 5, 2013; revised November 8, 2013. The editorcoordinating the review of this paper and approving it for publication was T.Q. S. Quek.

This work was partially supported by the National 973 Project under grant2013CB336700, by the National Natural Science Foundation of China undergrant 61222104 and 61061130561, and the Ph.D. Programs Foundation of theMinistry of Education of China under grant number 20110001120129. Partof this work has been presented at the 2013 IEEE International Conferenceon Communications.

H. Li and L. Song are with the State Key Laboratory of Advanced OpticalCommunication Systems and Networks, School of Electronics Engineeringand Computer Science, Peking University, Beijing, China, 100871 (e-mail:{hui.li, lingyang.song}@pku.edu.cn).

M. Debbah is with SUPELEC, Gif-sur-Yvette, France (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCOMM.2014.011414.130498

[5], the authors show that when the number of Base Station(BS) antennas M grows without bound, it can reduce thetransmit power of each user proportionally to 1/M if the BShas perfect channel state information (CSI).

However, one of the disadvantages of employing multipleantennas is the associated complexity which results fromemploying a separate radio frequency (RF) chain for everyemployed antenna. This also gives rise to a significant increasein the implementation cost [8]. For example, the hardwarecomplexity of optimal signal detection grows exponentiallywith the number of used antennas [9]. Most of the energyefficient communication techniques typically focus on min-imizing the transmit power only, which is reasonable whenthe transmit power is large enough and the number of usedRF chains is small. But, when the transmit power is relativelysmall, especially in large scale multiple antenna system wherethe circuit power consumption can be comparable to or evendominates the transmit power [10], it would be worthwhile toinvestigate whether large scale multiple antenna systems canoutperform the systems with less antennas in energy efficiency.

With these problems stated above, antenna selection can beintroduced as a means to alleviate this hardware complexity,while still retaining the diversity advantages [11]. Antennaselection in multiple antenna systems has been evaluated interms of capacity, outage probability and so on [12]–[13].Also, due to the computational burden required to selectthe best set of antennas among all available antennas, fastalgorithms have been proposed [14]–[15]. To the best of ourknowledge, there are very few papers that analyze the behaviorof antenna selection in large scale multiple antenna systems.In [16], the authors explore the behavior of MIMO systemsin the limit of large number of antennas and a phenomenoncalled channel hardening is observed, i.e., the variance of themutual information shrinks as the number of antennas grows.The same phenomenon has been observed in [17] with onlyselecting the best one transmit antenna. However, for a largescale multiple antenna system that a certain number of transmitantennas has to be selected due to lower power RF constraint,few results of the approximation of the distribution of themutual information have been given yet.

In this paper, we focus on transmit antenna selectionwith a large number of available antennas at the transmitter.Under such a scenario, for the first time we derive a goodapproximation of the distribution of the mutual informationwith selecting any number of antennas. In our system, withlarge number of available antennas at the transmitter, it showsthat our results include the asymptotic distribution of themutual information in [16] as a special case when the number

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LI et al.: ENERGY EFFICIENCY OF LARGE-SCALE MULTIPLE ANTENNA SYSTEMS WITH TRANSMIT ANTENNA SELECTION 639

of selected antennas equals to the number of total availabletransmit antennas. Besides, we model the power consumptionas the addition of the transmit power and the circuit powerconsumption, and analyze the performance of the energyefficiency. We find that when the circuit power consumptionis comparable to or dominates the transmit power, an optimalnumber of selected antennas can be obtained, and thus, lessor more used antennas can decrease the energy efficiency.Based on these conclusions, we propose two antenna selectionalgorithms to improve the energy efficiency.

The rest of this paper is organized as follows. Section IIpresents the system model and then the channel hardeningphenomenon in antenna selection system with large numberof available antennas is proved. In Section III, the issues ofenergy efficiency are discussed and we propose two selectionschemes. Simulation results are given in Section IV. Section Vsummarizes the paper. The proofs are given in the Appendixes.

II. SYSTEM MODEL AND CHANNEL HARDENING

PHENOMENON

A. System Model

We consider the downlink of a point-to-point MIMO sys-tem, which is typical of a cellular network when one useris scheduled for a certain time slot or frequency. The BS isequipped with N antennas and the user has M antennas. Weassume that N is large and N � M . Our goal is to obtainthe closed-form expression for the energy efficiency and tryto find ways to improve it. To start with, we assume the useris equipped with only one antenna, which is common for acellphone. The case of multiple antennas at the receiver willbe discussed later.

Under these assumptions, the received signal can be writtenas

y=hTx+n, (1)

where the N × 1 complex vector x is the transmitted signalvector, the vector h = [h1, h2, . . . , hN ]T , the element hi is thefading coefficient between the i-th transmitter and the receiver,which is modeled as independent identically distributed (i.i.d.)complex Gaussian random variables with zero mean andvariance 0.5 per real dimension, n is additive white zero-meanGaussian noise. Without loss of generality and to minimizenotations, we take the noise variance to be one.

It is impractical to use all the antennas to achieve high ratedue to the expensive RF chains for every employed antenna.Thus, with limited number of RF chains, transmit antenna se-lection is performed to choose the best L antennas from all theavailable N ones. We consider the case where the receiver usesall the available antennas while the transmitter uses antennaselection. Note that full complexity is the special case whenall transmit antennas are selected. We assume perfect CSIat the transmitter. In a Time-Division-Duplex(TDD)system,the channel is reciprocal and the BS estimates the downlinkchannels by using uplink received pilots. Typically, in asufficiently slowly changing environment, the L RF chains canbe cycled through N antennas during the training bits. In otherwords, RF chains are connected to the first L antennas duringthe first part of the training sequence, then to the second L

antennas during the next part, and so on. At the end of trainingsequence, we pick the best L antennas. Similar methods canbe found in [18]–[19].

There is a total power constraint of ρ across the transmitantennas. According to [20], by transmit beamforming withCSI, the achievable rate of transmit antenna selection in thissystem Isel is given by

Isel= log2 (1+ρL∑

i=1

|hi|2) bits/s/Hz, (2)

where |h1|2 > |h2|2 > · · ·> |hN |2.

B. Channel Hardening Phenomenon

According to the results in [16], the approximation of thedistribution of the mutual information with full complexitycan be written as

Ifull ∼ N(log2 (1+Nρ) ,

(log2 e)2

N

). (3)

Therefore, in a large scale multiple antenna system, as thenumber of antennas grows, the channel quickly “hardens”, inthe sense that the mutual information fluctuation decreasesrapidly relative to its mean. This form of channel hardeningis generally welcome for voice and other traffic that is sensitiveto channel fluctuations and delay. The implementation of thischannel hardening result for data and voice services, schedul-ing and rate feedback are also discussed in [16]. It is statedthat the number of bits needed for rate feedback decreasesas the number of transmit or receive antennas increases. Theoutage probability can also be reduced as a result of the stabletransmission rate achieved by large number of antennas. Dueto these advantages stated above, we are interested in seeingif there is a similar channel hardening phenomenon in suchan antenna selection system. We have the following results.

Theorem 1: With transmit antenna selection, for large Nand 1 ≤ L ≤ N , a good approximation of the distribution ofthe mutual information is given by

Isel∼FN(log2

(1+

(1 + ln

N

L

)ρL

),(log2 e)

2ρ2L

(2− L

N

)(1+(1 + ln N

L

)ρL)2).

(4)

Proof: Please see Appendix A.Discussion: As can be seen from Theorem 1, when N →

∞, the numerator of the variance approaches to a constant2 (log2 e)

2 ρ2L, and the denominator of the variance ap-proaches to infinity, thus the variance approaches to zero.So we can conclude that the same channel hardening phe-nomenon with full complexity also exists in such an an-tenna selection system. With large N , the mutual informationconverges to the mean, the tails of the distribution can beextremely small. We can use the normal distribution to ap-proximate to the folded normal distribution. By letting L = N ,we can achieve the following result:

Isel ∼ N(log2 (1+ρN) ,

(log2 e)2ρ2

N(

1N+ρ

)2). (5)


With sufficiently large N , our result in Eq. (5) is almostthe same as Eq. (3). Therefore, the asymptotic distribution ofthe mutual information derived in [16] is a special case ofthe generalized distribution with antenna selection derived inthis paper when L = N . Although the asymptotic distributionof Isel in Eq. (4) is obtained with the assumption that N islarge, we find it also works well when N is “not so large”.The results are verified through simulations.

III. IMPROVE ENERGY EFFICIENCY WITH TRANSMIT

ANTENNA SELECTION

A. Analysis of Energy Efficiency

The energy efficiency is defined as the spectral efficiencydivided by the total power consumed, which is first introducedin [23]. It can be written as

η =E [I]

Ptotal, (6)

where E denotes the expectation, I is the mutual information,Ptotal is the total power consumption which is the addition ofthe transmit power ρ and the circuit power consumption Pc

in our model, i.e., Ptotal = ρ+ Pc.As is known to all, for every employed antenna, a separate

radio frequency chain is need. We use the circuit powerconsumption model in [10]. It is given by

Pc ≈Nt (PDAC + Pmix + Pfilt) + 2Psyn

+Nr (PLNA + Pmix + PIFA + Pfilr + PADC) ,(7)

where Nt and Nr are the numbers of transmitter and receiverantennas, respectively, PDAC , Pmix, Pfilt, Psyn, PLNA,PIFA, Pfilr , and PADC are the power consumption valuesfor the DAC, the mixer, the active filters at the transmitterside, the frequency synthesizer, the low noise amplifier, theintermediate frequency amplifier, the active filters at the re-ceiver side, and the ADC, respectively. For more details aboutthe parameters of these variables, please see [10] and thereferences therein.

To simplify notation, we denote that P1 = 2Psyn+PLNA+Pmix+PIFA+Pfilr+PADC and P2 = PDAC+Pmix+Pfilt,so Pc = P1 + LP2. Obviously, P1 > P2. With large N , thetails of the distribution of I is extremely small, the mean of Ican be approximated by the mean of its corresponding normaldistribution. Thus, the closed-form expression of the energyefficiency in Eq. (6) becomes

η =log2

[1 +

(1 + ln N

L

)ρL]

ρ+ P1 + LP2. (8)

Most of the studies about antenna selection focus on theperformance with small and fixed number of selected antennas.But in large scale multiple antenna systems, there are largenumber of available antennas. Thus, the range to be selectedextends widely. Next, we analyze the effects of the transmitpower and the number of selected antennas on the energyefficiency. For our power consumption model stated above,we get the following statement.

Theorem 2: If the total power consumption is modeledas the addition of the transmit power and the circuit powerconsumption, the energy efficiency η in Eq. (8) increases first

and then decreases as the transmit power ρ or the number ofselected antennas L increases.

Proof: Please see Appendix B.Discussion: From Theorem 2 we can see that too small or

large transmit power is not fit for energy efficient systems.When the circuit power consumption is comparable to thetransmit power, using more antennas than the optimal valueL∗ that maximizes the energy efficiency can reduce theenergy efficiency. In this scenario, the increase of the mutualinformation is not sufficient to compensate for the increase ofthe total power consumption.

Thus, antenna selection could be an alternative to improvethe energy efficiency by selecting the optimal number ofantennas that maximizes the energy efficiency.

However, two special cases need to be explained. The firstcase is that the circuit power consumption totally dominatesthe transmit power, i.e., the transmit power consumption ismuch smaller than the circuit power consumption, then wehave the following conclusion.

Corollary 1: If the circuit power consumption totally dom-inates the transmit power, the conclusions in Theorem 2 stillhold true.

It has been proved in Appendix B that q(N) ≤ 0 and q(1) >0 in Eq. (32) and Eq. (33), respectively. As the value of L isalways an integer from 1 to N , two special cases should beconsidered, which are listed below:⎧⎨

⎩q (N) < 0q (N − 1) > 0r (N − 1) < r(N)

(9)

and ⎧⎨⎩

q (1) > 0q (2) < 0r (1) > r(2)

(10)

When Eq. (9) is satisfied, the energy efficiency increasesmonotonically as L increases from one to N . However, whenEq. (10) is satisfied, the energy efficiency decreases monoton-ically as L increases from one to N . If the transmit power ismuch smaller than the maximum circuit power consumption,i.e., ρ � P1 +NP2, it is necessary to analyze the behaviorsof q(L) in Eq. (30) when L takes the values of two and N−1.The same method in Appendix B can also be applied to thiscase. It is easy to demonstrate that the conclusions in Theorem2 still hold true. We omit the proof here because of the limitedspace.

However, in the other special case where the transmit powerdominates the circuit power, i.e., the transmit power is muchlarger than the circuit power consumption, the optimal value ofL can be N . From a theoretical perspective, Eq. (9) can be sat-isfied with massive transmit power. Thus, the energy efficiencyis an increasing function of the number of selected antennas L.It is also easy to explain from a practical perspective analysis:if the transmit power is large enough, an increase number ofantennas can improve the mutual information significantly buthas very little contribution to the total power consumption.The transmit power is so large that even all the antennas areused, the circuit power consumption is still not comparable tothe transmit power. An extreme case of this scenario is theone when the circuit power consumption equals to zero.


TABLE IALGORITHM 1: SSA

Algorithm 1: Improve Energy Efficiency by Using Sequential SearchAlgorithm

Given the transmit power ρ and the number of available RF chains Rch.

Initialize U1.

while |U1| < Rch

Add the best antenna from UC1 into U1, forming U ′

1.

if η(U ′1

) ≤ η (U1)

break;

else

set U1 = U ′1.

end if

end while

U∗ = U1.

Next, we give a brief analysis of this extreme case relatedto the energy efficiency by letting P1 = P2 = 0 in Eq. (8).Then we get the following conclusion.

Corollary 2: If only the transmit power is considered asthe total power consumption, the energy efficiency increasesmonotonically as the transmit power decreases or the numberof selected antennas increases.

Proof: please see Appendix C.Discussion: In this case, the energy efficiency is maxi-

mized at ρ = 0 and minimized at ρ = ∞, which can becalculated as lim

ρ→0η =

(1 + ln N

L

)L log2 e and lim

ρ→∞ η = 0.

But we must realize that if ρ = 0, the mutual information Ialso equals to zero, which is meaningless. As is studied in[5], a trade-off appears between the spectral efficiency andthe energy efficiency. However, another way to improve theenergy efficiency is to take advantage of the other propertyof Corollary 2, i.e., increasing the number of actually usedantennas, which not only improves the energy efficiency butalso enhances the spectral efficiency when using this form ofenergy consumption model.

From all the analysis above, we find that when the transmitpower is large, we need to use all the antennas to maximizethe energy efficiency. But with moderate or small transmitpower, more used antennas can reduce the energy efficiency. Inconclusion, antenna selection is needed in large scale multipleantenna system to maximize the energy efficiency by selectingthe optimal number of antennas.

B. Selection Algorithm

Most of the previous works related to antenna selectionassume that the number of available RF chains is fixed,and the problem is focused on how to select the optimalantenna subset efficiently. As is stated in section I, manyfast algorithms have been proposed to reduce the complexity[14]–[15]. In the last subsection, we find that for a fixedtransmit power, it is not necessarily optimal to employ allthe available RF chains. Based on the above conclusions, wepropose two simple but efficient selection algorithms whichjointly considers the number of employed antennas and theantenna subset. We assume perfect CSI at the transmitterthroughout the implementation of both algorithms.

TABLE IIALGORITHM 2: BSA

Algorithm 2: Improve Energy Efficiency by Using Binary SearchAlgorithm

Given the transmit power ρ and the number of available RF chains Rch.

Initialize low value, high value, and calculate mid value.

while (high value − low value) > 1

if η(mid value+ 1) > η(mid value)

set low value = mid value+ 1;

elseif η(mid value+ 1) < η(mid value)

set high value = mid value;

else

break;

end if

end while

if high value− low value = 1

η∗ = max{η(low value),η(high value)};

else

η∗ = η(mid value);

end if

1) Sequential Search Algorithm (SSA): The first algorithmuses a sequential search algorithm to obtain the optimalantenna subset. U1 is denoted as the current optimal subsetafter every loop. We initialize U1 by adding the best antennainto it. In the following, when we refer to “the best antenna”,we mean the antenna with the best channel condition. Next,U1 is updated by trying to add one more antenna throughcomparison. If the number of elements in U1(i.e., |U1|) islarger than or equals to Rch, i.e., the number of available RFchains, the algorithm terminates. This is due to the considera-tion of the RF chains constraint. With the RF chains constraintsatisfied, at every loop, we choose the best antenna from thecomplementary set of U1(i.e., UC

1 ) to form a temporary newset U ′

1. If the energy efficiency with U ′1, η (U ′

1), is smallerthan or equals to the energy efficiency with U1(i.e., η (U1)),the algorithm terminates, otherwise, set U1 = U ′

1. This can beillustrated by the conclusions we obtained in Theorem 2, i.e.,the energy efficiency increases first and then decreases as thenumber of selected antennas increases. When the algorithmterminates, U1 is the optimal antenna subset. The algorithmuses at most O(Rch − 1) comparisons to find the optimalsubset. The algorithm is briefly summarized in Table I.

2) Binary Search Algorithm (BSA): The traditional BSA isused to find the position of a specified value within a sortedarray. This algorithm requires far fewer comparisons than alinear search especially for large arrays. It takes logarithmictime, but it imposes the requirement that the list be sorted.Although, the energy efficiency with different number ofselected antennas is not sorted monotonously, we can decidewhich part the maximum value locates in according to themiddle value. In the following, we will use the binary searchalgorithm with modification to find the maximum value.

First, we initialize the three variables: the lower boundof the used antenna, the upper bound of the used an-tennas, and the midpoint of the lower bound and upperbound, which are denoted as low value, high value, andmid value respectively. The initial value of low value and


high value are 1 and Rch respectively. mid value is calcu-lated as mid value = (low value+ high value)/2. Theenergy efficiency achieved by using L antennas is denotedas η(L). At every loop, we compare η(mid value) withη(mid value + 1), and decide which subset the maximumvalue locates in. If η(mid value + 1) > η(mid value), themaximum value is in the upper subset, change low value =mid value+1 to search the upper subset; if η(mid value+1) < η(mid value), the maximum value is in the lowersubset, change high value = mid value to search the lowersubset; otherwise, the maximum value is found by selectingmid value antennas. At the end of every loop, mid valueis updated with new low value and high value. To avoidlocking up, we terminate the algorithm when the equationhigh value−low value = 1 is satisfied. Then, the maximumvalue is either η(low value) or η(high value). We denote L∗

as the optimal number of selected antennas corresponding tothe maximum energy efficiency. The optimal antenna subsetis achieved by selecting the best L∗ antennas. The algorithmwill never use more than O(log2Rch + 1) comparisons tofind the maximum value. We know that the complexity ofthe traditional BSA is O(log2Rch). The one more comparisonin our modified BSA is due to the property that we need tocompare the two consecutive values in the array. The algorithmis summarized in Table II.

C. Discussion for the Energy Efficiency with M > 1

According to Eq. (6), in order to analyze the energy effi-ciency, we have to obtain the closed-form expression for themutual information of MIMO with transmit antenna selection.For a MIMO system with N transmit antennas and M receiveantennas, M > 1, the received signal is given by

y = Hx+ n, (11)

where y, x, n are the received signal, the transmitted signaland the zero-mean additive noise vectors, respectively, H isthe M×N channel matrix, the element is the same as definedin Section II.

For simplicity, we assume that H is unknown at thetransmitter. Thus, to maximize the mutual information, thetotal power ρ is equally allocated to all transmit antennas.The capacity of the MIMO channel in Eq. (11) is given by[20]

Ifull = log(det(IM +

ρ

NHH†

)), (12)

where (·)† indicates the matrix conjugate-transpose operation.Transmit antenna selection is performed to select the best

L antennas from all the available N ones, where L > M , sothe energy efficiency in Eq. (6) becomes

η=E

[log(det(IM +

ρ

LHH

†))]/Ptotal, (13)

where H is created by adding L columns from H .By using the Sherman-Morisson formula for determinants,

the authors of [24] found that

det(IM +

ρ

LHH

†)=

L∏i=1

(1 +

ρ

Lh†i Pihi

), (14)

0 20 40 60 80 1003

4

5

6

7

8

9

10

Number of Selected Antennas L

Mea

n of

the

Mut

ual I

nfor

mat

ion

N=100,ρ = 0dB,simulated

N=100,ρ = 0dB,analytical



2 4 6 8 102

3

4

5

6

7





Fig. 1. The effect of the number of selected antennas on the mean of themutual information.

where hi is a certain column selected at the ith step, Pi =

I− Hi−1

(I + H†

i−1Hi−1ρ/L)−1

H†i−1ρ/L is a projection

matrix of rank M − i + 1. Please refer to [15] for moreinformation.

For large ρ and large N , it can be achieved that [24]

det(HH

†) ∼L∏

i=1

χ22(M−i+1),N−i+1, (15)

where χ22p,n is a random variable which is the maximum of

n independent χ22p random variables.

Eq. (15) is only the approximation with large ρ, but it is stilldifficult to obtain the closed-form expression for det

(HH

†).

Therefore, the closed-form expression for Eq. (13) is too com-plicated to achieve. Although, the proposed antenna selectionscheme, i.e., jointly considers the number of selected antennasand the antenna subset, is based on the conclusions drawn fromMISO system, the numerical simulation results indicate thatit can also be applied to the MIMO system to improve theenergy efficiency. The simulation results will be discussed inSection IV.

IV. SIMULATION RESULTS

The comparisons between the simulation and our derivedresults of the mean and variance in section II are plotted inFig. 1 and Fig. 2. From the figures we can see that withdifferent N and ρ, they all match perfectly. It can be seen fromFig. 1 that 90% of the mutual information with full complexitycan be achieved with only a quarter of the antennas selected.In Fig. 2, we find that with large N , the variance can be verysmall. Moreover, for a fixed ρ, when the number of selectedantennas L is very small, the channel hardens at a lower rate.But for a large range of L, channel hardens at a high ratealmost the same as the full complexity. Fig. 1 and Fig. 2show that the asymptotic distribution in Eq. (4) is also veryaccurate for even small N . These simulations are consistentwith our derived result in Eq. (4) and the relevant discussions.

The effects of the transmit power and the number ofactually used antennas on the energy efficiency are plotted


0 20 40 60 80 1000.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09


Var

ianc

e of

the

Mut

ual I

nfor

mat

ion





2 4 6 8 10

0.2

0.25

0.3

0.35



N=10,ρ=10dB,simulated

N=10,ρ=10dB,analytical

Fig. 2. The effect of the number of selected antennas on the variance of themutual information.

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Transmit Power ρ

Ene

rgy

Effi

cien

cy

L=2L=4L=8L=20

43 44 45 46 47

4

5

6

7

8

x 10−4

−48 −46 −44

1

1.5

2

2.5

x 10−3

Fig. 3. The effect of the transmit power on the energy efficiency with thecircuit power consumption considered, where N=200.

and compared respectively in Fig. 3 – Fig. 6. In Fig. 3, theenergy efficiency approaches to zero when the transmit poweris either too small or too large. We can see that when thecircuit power consumption is comparable to or even dominatesthe transmit power, appropriate more antennas could improvethe energy efficiency, but too many used antennas can addtoo much additional circuit complexity and therefore reducethe energy efficiency. However, when the transmit power ismuch larger than the circuit power consumption, the energyefficiency increases monotonically as the number of selectedantennas increases. The performance of the extreme case withthe transmit power consumption considered only is displayedin Fig. 4, which is consistent with the discussions in Corollary2.

In order to have a better explanation of this problem, theeffects of the number of selected antennas stated in Theorem2 and Corollary 2 are plotted and compared in Fig. 5–Fig. 6individually. In Fig. 5, the figure in the small box shows thatwhen the circuit power consumption is comparable to thetransmit power, too many used antennas decrease the energyefficiency. However, if the transmit power is so large that

−60 −40 −20 0 20 40 600

10

20

30

40

50

60

70

80

90

100

Transmit Power ρ

Ene

rgy

Effi

cien

cy

L=2L=4L=8L=20

Fig. 4. The effect of transmit power on energy efficiency without circuitpower consumption, where N=200.

0 20 40 60 80 100 120 140 160 180 2001.2

1.4

1.6

1.8

2

2.2

2.4

2.6x 10

−4


Ene

rgy

Effi

cien

cy

ρ =50 dB

ρ =52 dB

0 50 100 150 2000.4

0.5

0.6

0.7

0.8

0.9


Ene

rgy

Effi

cien

cy

ρ =10 dB

ρ =12 dB

Fig. 5. The effect of the number of selected antennas on energy efficiencywith moderate and large transmit power, where N=200.

the circuit power can be neglected, the energy efficiency ismaximized when all the available antennas are used. Theeffect of the number of selected antennas when the circuitpower dominates the transmit power is plotted in the smallbox of Fig. 6. For comparison, the special case when thetransmit power dominates the circuit power in Corollary 2is also demonstrated in Fig. 6. All the simulations meet thediscussions stated above, which confirms our analysis.

In Fig. 7, we investigate the convergence of both selectionalgorithms. We can see that both algorithms converge to themaximum energy efficiency after a number of comparisons.In reference to the rate of convergence, in some cases, SSAoutperforms BSA, in another case, BSA can outperform SSA.According to the property of the energy efficiency we obtainedabove, we know that with different transmit power, we needto select different number of antennas to maximize the energyefficiency. SSA is much more sensitive to the optimal numberof selected antennas than BSA. To illustrate the performanceof rate of convergence, we run the algorithms over 10000 timesto obtain the average number of comparisons. In Fig. 8, we cansee that BSA converges fast regardless of the position of the


0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300


Ene

rgy

Effi

cien

cy

ρ = −50 dB

ρ = −52 dB

0 50 100 150 2000

0.2

0.4

0.6

0.8

1x 10

−3


Ene

rgy

Effi

cien

cy

ρ = −50 dBρ = −52 dB

Fig. 6. The effect of the number of selected antennas on energy efficiencywhen the circuit power dominates the transmit power (in the small box) andthe extreme case in Corollary 2, where N=200.

0 10 20 30 40 500

2

4

6

8

10

12

Number of Comparisons

Ene

rgy

Effi

cien

cy

ρ = −20 dB,Sequential Search

ρ = 0 dB,Sequential Search

ρ = 10 dB,Sequential Search

ρ = −20 dB,Binary Search

ρ = 0 dB,Binary Search

ρ = 10 dB,Binary Search

0 5 10

0.4

0.5

0.6

0.7

0 10 200.4

0.6

0.8

Fig. 7. The energy efficiency versus the number of comparisons, whereN=100, Rch=50.

optimal value. Moreover, the average number of comparisonwith Rch = 50 is only a little larger than Rch = 20.BSA has a good performance in large arrays. However, thecomplexity of SSA is proportional to the optimal number ofselected antennas. If the optimal number of selected antennasis larger than Rch, Rch antennas are used. In some cases, thecomplexity of SSA can be intolerable. In conclusion, BSAis a simple and efficient algorithm to solve the problem inour model. The proposed algorithm by joint selecting thenumber of antennas and antenna subset is compared with thetraditional method in Fig. 9. The traditional method with bluedashed line is curved by always using all the available RFchains regardless of the transmit power. We can observe thatour algorithm can improve the energy efficiency significantly.According to the previous analysis, with high transmit power,the optimal choice is to use all the available RF chains.Therefore, the curve of our algorithm has some overlap withthe traditional one at high transmit power, which is consistentwith the analysis we derived.

In Fig. 10, the proposed method(PM) drawn from the

−40 −30 −20 −10 0 10 20 30 400

5

10

15

20

25

30

35

40

45

50

Transmit Power ρ /dB

Ave

rage

Num

ber

of C

ompa

rison

s

Rch

= 20,Sequential Search

Rch

= 50,Sequential Search

Rch

= 20,Binary Search

Rch

= 50,Binary Search

Fig. 8. The average number of comparisons versus the transmit power ρ,where N=100.

−40 −30 −20 −10 0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5

4

Transmit Power/dB

Ene

rgy

Effi

cien

cy

Proposed MethodTraditional Method

Fig. 9. The energy efficiency versus transmit power ρ, where N=100,Rch=50.

MISO system is compared with the traditional method(TM)in a MIMO system with M = 8. What’s more, in largescale MIMO systems, it is difficult to place antennas farapart with so many antennas, but the spatial correlation mayexist between antennas due to the large antenna number. Theproblem of spatial correlation has been examined in recentpapers [25] – [26]. For simplicity, we assume that the effect ofspatial correlation in large scale MIMO system is the same astraditional systems. It is not a trivial work to derive the closed-form expression for energy efficiency with spatial correlation,but we can benefit from the numerical results. In Fig. 10,the exponential correlation matrix model in [27] is used toinvestigate the effect of spatial correlation. The correlationcoefficient of neighboring antennas r is 0.7 in the simulation.From the figure, we can see that the conclusions derived inthis paper can also be applied to correlated channels(CC) inMIMO Systems. The correlation between antennas results inthe reduction of the energy efficiency. However, the proposedmethod still outperforms the traditional method.


−40 −30 −20 −10 0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

SNR/dB

Ene

rgy

Effi

cien

cy

PM with CCPM with UCCTM with CCTM with UCC

Fig. 10. The energy efficiency in MIMO system with correlated anduncorrelated channel, where N = 100, M = 8, Rch = 50.

V. CONCLUSIONS

In this paper, a good approximation of the distribution ofthe mutual information is achieved with antenna selection forthe first time, from which a channel hardening phenomenoncan be observed. Based on these results, we obtained aclosed-form expression for the energy efficiency. We findthat if the circuit power is much larger than or comparableto the transmit power, using too many extra antennas canreduce the energy efficiency. However, if the transmit powertotally dominates the circuit power consumption, the energyefficiency increases monotonically as the number of selectedantennas increases. The extreme case of this scenario hasalso been demonstrated. The proposed selection algorithms,which adaptively selects the number of used antennas, canimprove the energy efficiency significantly. As stated above,we can conclude that antenna selection can be a good choice toimprove the energy efficiency in large scale multiple antennasystems.

APPENDIX A

According to [21], let x1 > x2 > · · · > xN denote theordered random variables, as N → ∞ and 1 ≤ L ≤ N ,

the distribution of the trimmed sumL∑

i=1

xi is asymptotically

normal. Reference [22] gives the mean and variance when xi

is a chi-square random variable with two degrees of freedom.Thus, we can get the following conclusion:

L∑i=1

|hi|2 ∼ N(L

(1 + ln

N

L

), L

(2− L

N

)). (16)

Note that the left side hand in Eq. (16) is approximatedas a random variable of normal distribution which can takenegative values. However, the left hand side in Eq. (16) ispositive for sure. This is due to the fact that the normaldistribution is just an approximation. The approximation isaccurate around the mean but not very precise in the tailsof the distribution. However, in large scale MIMO systemswith large N , the mean of the normal distribution increaseswhile the variance almost remains constant. Therefore, the

probability of taking negative values becomes smaller and theapproximation gets more accurate.

We have the following derivations:

Isel = log2

∣∣∣∣∣1 + ρ

L∑i=1

|hi|2∣∣∣∣∣

= log2

∣∣∣∣(1 +

(1 + ln

N

L

)ρL

)x

∣∣∣∣= log2

(1 +

(1 + ln

N

L

)ρL

)+ log2 |x| ,

(17)

where |·| denotes the absolute value, and x is given by

x = 1 +

ρ

(L∑

i=1

|hi|2 − L(1 + ln N

L

))1 +

(1 + ln N

L

)ρL

. (18)

According to Eq. (16), it is easy to obtain that

x ∼ N(1,

ρ2L(2− L

N

)(1 +

(1 + ln N

L

)ρL)2). (19)

Given a normally distributed random variable x with meanμ and variance σ2, the random variable y = |x| has a foldednormal distribution, i.e., FN (

μ, σ2), the probability density

function of which is given by

f (y) =1√2πσ

e−(y−μ)2

2σ2 +1√2πσ

e−(y+μ)2

2σ2 , y > 0, (20)

where μ = 1, σ2 =ρ2L(2− L

N )(1+(1+ln N

L )ρL)2 .

Then, the equation in Eq. (17) becomes

Isel = log2

(1 +

(1 + ln

N

L

)ρL

)+ log2 (1 + (y − 1))

(21)

The mean of (y − 1) is zero, and the variance of (y − 1) is

σ2 =ρ2L(2− L

N )(1+(1+ln N

L )ρL)2 <

(2− LN )

(1+ln NL )

2L

. Note that σ2 → 0 is

almost surely for large N . Then

Isel = log2

(1+

(1 + ln

N

L

)ρL

)+(y − 1)log2e+O

((y − 1)

2),

(22)

O((y − 1)

2)

is asymptotically negligible. Thus, the distribu-tion of Isel is given by

Isel∼FN(log2

(1 +

(1 + ln

N

L

)ρL

),(log2e)

2ρ2L(2− L

N

)(1 +(1 + ln N

L

)ρL)2).

(23)

APPENDIX B

A. The effect of transmit power ρ

First, we denote the energy efficiency as a function of ρ

f (ρ) =log2

[1 +

(1 + ln N

L

)ρL]

ρ+ P1 + LP2. (24)

Then, take the first derivative with respect to ρof f (ρ), we find that the denominator of f ′ (ρ),ln 2

[1 +

(1 + ln N

L

)ρL](ρ+ P1 + LP2)

2, is surely a


positive value, so we can just concentrate on the numeratorand let it be

g (ρ) =

(1 + ln

N

L

)L (ρ+ P1 + LP2)

−[1 +

(1 + ln

N

L

)ρL

]ln

[1 +

(1 + ln

N

L

)ρL

].

(25)

Next, take the first order derivative of g (ρ) and it isobserved obviously that

g′ (ρ) = −(1 + ln

N

L

)L ln

[1 +

(1 + ln

N

L

)ρL

]< 0.

(26)

So, g (ρ) is a decreasing function of ρ, thus we haveg (∞) ≤ g (ρ) ≤ g (0). As

limρ→0

g (ρ) =

(1 + ln

N

L

)L (P1 + LP2) > 0, (27)

and

limρ→∞ g (ρ) = lim

ρ→∞

[1 +

(1 + ln

N

L

)ρL

]

×(1− ln

[1 +

(1 + ln

N

L

)ρL

])

− (ρ+ P1 + LP2)

ρ< 0,

(28)

the positive f ′ (ρ) at small ρ becomes negative as the transmitpower increases. Thus, we conclude that the energy efficiencyincreases first and then decreases as the transmit powerincreases with lim

ρ→0η = lim

ρ→∞ η = 0.

B. The effect of the number of selected antennas L

In order to evaluate the performance of the energy efficiencywith reference to the number of selected antennas, we denotethe energy efficiency as a function of L

r(L) =log2

[1 +

(1 + ln N

L

)ρL]

ρ+ P1 + LP2. (29)

As in the last part, take the first derivative but with respectto L and ignore the positive denominator, the numerator iswritten as

q (L) = (ρ+ P1 + LP2) ρ lnN

L

− P2

[1 +

(1 + ln

N

L

)ρL

]ln

[1 +

(1 + ln

N

L

)ρL

].

(30)

Take the first derivative of q(L), we achieve

q′ (L) =− (ρ+ P1 + LP2) ρ

L

− P2ρ lnN

Lln

[1 +

(1 + ln

N

L

)ρL

],

(31)

which is negative for sure.So, q(L) is a decreasing function of L, thus q (N) ≤

q (L) ≤ q (1). The lower bound can be obtained easily as

q (N) = −P2 (1 +Nρ) ln (1 +Nρ) < 0, (32)

whereas the upper bound

q (1) = (ρ+ P1 + P2) ρ lnN

− P2 [1 + (1 + lnN) ρ] ln [1 + (1 + lnN) ρ](33)

is hard to decide.However, if we take the second derivative of q(1) with

respect to ρ, we will find that

q′′1 (ρ) > 2 lnN − P2 (1 + lnN)2. (34)

As denoted in the previous part, P2 could be a very smallvalue much less than one. Even with very large N , say amagnitude of thousands, the value lnN is still less than ten.Therefore, the expression in Eq. (34) is positive. So q′1 (ρ) ≥q′1 (0) > 0, that is to say, q(1) is an increasing function ofρ. Besides, we also have lim

ρ→0q (1) = 0, hence q (1) > 0.

The conclusion that the energy efficiency increases first andthen decreases as the number of selected antennas increasesis proved.

APPENDIX C

By letting P1 = P2 = 0, the energy efficiency in Eq. (8)becomes

η =log2

[1 +

(1 + ln N

L

)ρL]

ρ. (35)

The process is almost the same as the demonstration inAppendix C, but some special cases may happen in thisscenario. The expression in Eq. (27) becomes

limρ→0

g (ρ) = 0. (36)

As g(ρ) is a decreasing function of ρ, thus g (ρ) ≤ 0. Asa result, the energy efficiency is a decreasing function of thetransmit power.

In addition, Eq. (30) will be given as

q (L) = ρ2 lnN

L, (37)

which is surely positive.Thus, the energy efficiency increases monotonically as the

number of selected antennas increases.

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Hui Li received the B.S. degree in telecommunica-tion engineer from Xidian University, Xi’an, China,in 2011. She is currently pursuing her Master’sdegree at Peking University with a major in wirelesscommunication and signal processing. Her researchinterests mainly include large scale MIMO, vehicu-lar network, and millimeter wave communication.

Lingyang Song received his PhD from the Uni-versity of York, UK, in 2007, where he receivedthe K. M. Stott Prize for excellent research. Heworked as a postdoctoral research fellow at theUniversity of Oslo, Norway, and Harvard University,until rejoining Philips Research UK in March 2008.In May 2009, he joined the School of ElectronicsEngineering and Computer Science, Peking Univer-sity, China, as a full professor. His main researchinterests include MIMO, OFDM, cooperative com-munications, cognitive radio, physical layer security,

game theory, and wireless ad hoc/sensor networks. He is co-inventor of anumber of patents (standard contributions), and author or co-author of over100 journal and conference papers. He received the best paper award inIEEE International Conference on Wireless Communications, Networking andMobile Computing (WiCOM 2007), the best paper award in the First IEEEInternational Conference on Communications in China (ICCC 2012), the beststudent paper award in the7th International Conference on Communicationsand Networking in China (ChinaCom2012), and the best paper award in IEEEWireless Communication and Networking Conference (WCNC2012).

Dr. Song is an Associate Editor of IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS since 2012. He is the recipient of 2012 IEEE Asia Pacific(AP) Young Researcher Award, and 2012 National Science Foundation ofChina (NSFC) Outstanding Young Investigator Award. He is a senior memberof IEEE.

Merouane Debbah entered the Ecole NormaleSuperieure de Cachan (France) in 1996 where hereceived his M.Sc and Ph.D. degrees respectively.He worked for Motorola Labs (Saclay, France) from1999-2002 and the Vienna Research Center forTelecommunications (Vienna, Austria) until 2003.He then joined the Mobile Communications de-partment of the Institut Eurecom (Sophia Antipo-lis, France) as an Assistant Professor until 2007.He is now a Full Professor at Supelec (Gif-sur-Yvette, France), holder of the Alcatel-Lucent Chair

on Flexible Radio and a recipient of the ERC grant MORE (AdvancedMathematical Tools for Complex Network Engineering). His research interestsare in information theory, signal processing and wireless communications. Heis a senior area editor for IEEE TRANSACTIONS ON SIGNAL PROCESSINGand an Associate Editor in Chief of the journal Random Matrix: Theory andApplications. Merouane Debbah is the recipient of the “Mario Boella” awardin 2005, the 2007 General Symposium IEEE GLOBECOM best paper award,the Wi-Opt 2009 best paper award, the 2010 Newcom++ best paper award, theWUN CogCom Best Paper 2012 and 2013 Award as well as the Valuetools2007, Valuetools 2008, Valuetools 2012 and CrownCom2009 best studentpaper awards. He is a WWRF fellow and a member of the academic senateof Paris-Saclay. In 2011, he received the IEEE Glavieux Prize Award.

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Energy Efficiency of Large-Scale Multiple Antenna Systems with Transmit Antenna Selection