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Parameter Optimization for Energy Efﬁcient Indoor

Massive MIMO Small Cell Networks

Chen Chen, Yan Jiang, Jiliang Zhang, Xiaoli Chu, Jie Zhang

Department of Electronic and Electrical Engineering, The University of Shefﬁeld, Shefﬁeld, S1 4ET

United Kingdom, c.chen2@shefﬁeld.ac.uk

Abstract—To better characterize indoor small cell networks

(SCN), we consider the blockages caused by interior walls and

employ the bounded path loss model to derive the expression

for energy efﬁciency (EE) of a downlink massive multiple-input

multiple-output (MIMO) SCN. Our EE expression demonstrates

that a higher penetration loss of interior walls leads to a higher

EE. For the purpose of maximizing EE, we propose a novel

genetic algorithm (GA) based scheme to jointly optimize the

number of antennas per base station (BS), the number of users

per cell, and the transmission power per antenna. Numerical

results show that our proposed scheme can achieve almost the

identical EE as the optimal greedy search algorithm, while

signiﬁcantly reducing the computational time.

Index Terms—Indoor, energy efﬁciency, massive MIMO, small

cell networks, genetic algorithm

I. INTRODUCTION

To address the requirement of thousand fold network capac-

ity increase without further increasing the power consumption,

energy efﬁciency (EE) has been one of the most important

considerations for the ﬁfth generation (5G) cellular network

[1]. Two potential techniques to improve EE are massive

MIMO and small cell networks (SCN) [2].

Massive MIMO systems can provide dramatical throughput

gains and reduction in transmission power [2]. In [3], closed-

form expressions were derived for the number of antennas per

base station (BS), the number of users per cell, and the emitted

power in a single-cell scenario with perfect CSI. The multi-

cell scenario under imperfect CSI was studied through Monte

Carlo simulations. Recently, in [5], the authors showed that

EE can be improved through adapting the number of antennas

used at a BS according to the daily load variation. However,

none of these studies has considered the effect of BS density.

SCN saves power for each BS by reducing the transmission

distance to an associated user. In [6], it was shown that EE can

be improved by increasing the BS density or the number of

antennas only when the circuit power consumption of the BS

is less than a certain threshold. Recent work [10] showed that

the network coverage probability of a SCN was overestimated

under the simpliﬁed unbounded channel model which violates

the conservation of energy. Nevertheless, the EE of the SCN

under the bounded model, especially the indoor SCN equipped

with massive MIMO, has not been studied yet.

The indoor SCN is largely characterized by the blockages of

interior walls [7]. In [8], the walls were distributed following

a Poisson point process (PPP), and the numerical results

revealed that a higher penetration loss of interior walls can

provide a higher network coverage probability and an higher

area spectral efﬁciency.

In this paper, we analyze the EE of the indoor massive

MIMO SCN considering the blockages of interior walls. To

maximize EE, we propose a novel genetic algorithm based

scheme. The contributions of this paper are summarized as

follows:

•Taking into account of the penetration loss of interior

walls, we derive the average user rate and EE in an

indoor massive MIMO multi-cell scenario as functions

of the number of antennas per BS, the number of users

per cell, the transmission power per antenna, and the

number of cells. We apply the bounded channel model

to better capture the path loss of short distances. Our

analytical results show that the network densiﬁcation may

degrade the EE when the BS density is sufﬁciently large.

Furthermore, we ﬁnd that a higher penetration loss of

interior walls can provide a higher EE.

•We propose a novel, low-complexity genetic algorithm

based scheme to maximize the EE, subject to the quality

of service (QoS) constraints. Our scheme jointly opti-

mizes the number of antennas per BS, the number of users

per cell and the transmission power per antenna. The

performance of our scheme is evaluated in comparison

with the optimal greedy search algorithm.

Fig. 1. Illustration of an indoor multi-cell scenario in a 30 m×30 m area,

where the cell under study is located in the centre. Each cell covers a room

and the bold lines denote the walls. The number of cells in each column is

λ.

II. SY ST EM MO DE L

A. Network Model

As shown in Fig. 1, we consider the downlink of an

indoor massive MIMO SCN in a 30m×30m area consisting

of C=λ2square cells, which are denoted by the set

C={1,2,3, ..., C}. We assume that each room has a BS,

so each cell corresponds to a room. The bold lines denote

the walls with the same penetration loss z(0 < z < 1). For

each cell, the BS is equipped with Mantennas to serve K

uniformly distributed single-antenna users. We assume that

the system is completely symmetric where all the cells have

the same parameters: M,Kand the transmission power per

antenna p. A time-division duplex (TDD) protocol is employed

with perfect synchronization. The channels are assumed to be

static within a time-frequency block U=BCTC, where BC

is the coherence bandwidth and TCis the coherence time. For

analytical tractability, we assume perfect CSI. Imperfect CSI

will be studied in our future work.

B. Channel Model

Since the distances between users and the BS are much

larger than the space between antennas, the large-scale fading

is considered to be the same for the channels from a user to

all BS antennas. To better characterize the power attenuation

with small transmit distances, we model the large-scale fading

using a bounded path loss model [10], [11]:

l(d) = β0

(d2+ 1)α/2,(1)

where dis the transmit distance, αis the path-loss exponent,

and the constant β0is the channel attenuation at unit reference

distance.

We assume that the space between antennas in the BS is

adequate so that the correlation between the channels from s

BS to a single-antenna user can be omitted. The channel model

between the BS to a speciﬁc user kis denoted by the vector

hk= [hk,1, hk,2, ..., hk,M ]T∈CM×1, where hk,j denotes

the channel from the jth antenna to the kth user. Rayleigh

distribution with unit variance is employed to model small-

scale fading, i.e., hk∼ CN (0M, l(dk)IM),dkis the distance

between the BS and user k,0Mand IMare the M-dimensional

null and unit column vectors.

C. User Rate Model

For simplicity, we assume that the BS can obtain perfect

CSI from uplink pilots and use zero-forcing (ZF) precoding

for downlink transmission. The ZF matrix is:

W=H(HHH)−1,(2)

where H= [h1,h2, ..., hK]is the channel vector for all the

users in a cell.

The cell cin the centre is our cell under study, and all the

other cells fare interference cells, f6=cand f∈ C. The

desired signal received by user kin cell cis:

yck =hH

cck

K

X

i=1

wcisci +X

f6=c

zfchH

fck

K

X

i=1

wfisf i +nck,(3)

where hcck and hfck are the channels from the serving BS

in cell cand interfering BS in cell fto user k,wci and sci

are the ZF vector and the transmitted signal from the serving

BS in cell cto user iin cell c, respectively, wfi and sf i are

the ZF vector and the transmitted signal from the BS in cell

fto user iin cell f,zfc denotes the penetration loss of walls

between cell cand cell f,nck is the additive noise.

The tractable lower bound of average user rate in cell cis

given in [5]:

Rc=B1−τK

Ulog2 1+ pM

K(M−K)

σ2Gcc +pM Pf6=cGcf !,

(4)

where Bis the channel bandwidth, σ2is the variance of

additive white Gaussian noise, τis the pilot reuse factor,

1−τK

Udenotes the overhead used for channel estimation, p

is the average transmission power per antenna, so pM

Kdenotes

average transmission power per user, Gcc is the average inverse

channel attenuation, while Gcf is the average ratio of the

channel attenuation between interference cell fand serving

cell c. For our model, Gcc can be calculated as:

Gcc =Zr

−rZr

−r

(x2+y2+ 1)α/2

4r2β0

dxdy, (5)

where ris the radius of the cell.

Pf6=cGcf is the function of λand the expression can be

found in Appendix A. As aforementioned, Rcis a function of

M,K,pand λ.

D. Power Consumption Model

The power consumption consists of two parts: power am-

pliﬁer (PA) power consumption PPA and circuit power con-

sumption PCP in Watt (Joule/second). So the total power

consumption for BS cis:

Pc=PPA,c +PCP,c,(6)

where PPA,c =BMp

η, and ηdenotes the PA efﬁciency [12].

The circuit power consumption is the sum of both analog

and digital parts as [3]:

PCP,c =PFIX +PTC +PCE +PZF +PC/D+PBH,(7)

where PFIX is the ﬁxed power consumption for signal control

and load-independent power, PTC is the power consumption

of transceiver chains, the typical value can be computed as

PTC =MPBS +PSYN, where PBS is the power required for

circuit components in each antenna and PSYN is the power

needed for local oscillator, PCE is the channel estimation

power, which is given by:

PCE =B

U

2τMK2

LBS

,(8)

where LBS is the computational efﬁciency for BS processing.

During downlink transmission, channel coding and modula-

tion are employed before the BS transmits signals to its served

Kusers. The power for this operation is proportional to the

user rate, i.e.,

PC/D=KRc(PCOD +PDEC ),(9)

where PCOD (Watt per bit/s) and PDEC (Watt per bit/s) are

the coding and decoding power.

PZF is the power consumption for ZF precoding, this costs:

PZF =B1−τK

U2MK

LBS

+B

UK3

3LBS

+3MK2+MK

LBS ,

(10)

where the ﬁrst term is for matrix-vector multiplication and

the second one is used for the computation of ZF precoding

matrix W.

The last part of power consumption is for backhaul, which

is proportional to the average user rate, it can be computed as:

PBH =KRcPBT ,(11)

where PBT is backhaul trafﬁc power.

The expressions above allow us to rewrite PCP,c as:

PCP,c =AKRc+

3

X

i=0

CiKi+M

2

X

i=0

DiKi,(12)

where A=PCOD +PDEC +PBT,C0=PFIX +PSYN ,C1= 0,

C2= 0,C3=B

3ULBS ,D0=PBS ,D1=B

LBS (2 + 1

U),D2=

3B

ULBS .

III. ENERGY EFFIC IE NC Y OPTIMIZATION

A. Problem Formulation

We compute EE as the ratio between the sum average rate

of the users (in bit/second) and total power consumption (in

Watt) of BS c, i.e.,

EE(M, K, p) = KRc

Pc

,(13)

where EE(M, K, p)is a function with parameters M,Kand

p. Since from Fig. 4, the EE monotonically decreases with the

BS density λ, we will not optimize λ.

Then we can formulate our optimization problem as:

arg max

M,K,p

EE(M, K, p),(14)

subject to constraints:

C1 : M∈Z+, K ∈Z+

C2:1≤K≤Kmax

C3 : K < M ≤Mmax

C4:0≤p≤pmax

C5 : Rc≥Rmin

C6 : pM ≤Pmax

where, in C5,Rmin is the threshold for user rate, which guar-

antees the QoS. In C6,Pmax is the maximum transmission

power for BS and the total power can not go beyond this.

With the expressions in (4) and (6), it is intractable to derive

closed-form expressions for each parameter. In addition, from

Fig. 2, it is observed that EE is a quasi-concave function for

each parameter when the other two parameters are ﬁxed. These

make the optimization of all the parameters difﬁcult. To obtain

the near optimal parameter conﬁguration that maximizes EE,

we propose a novel genetic algorithm based scheme.

B. Proposed Scheme

Genetic algorithm (GA) is a kind of evolutionary algorithm,

which is especially suitable for solving complex optimization

problems with many parameters that are intractable to derive

the analytical expressions [13].

GA mimics the process of evolution theory. The probable

solutions are called individuals and a population of selected

individuals are called chromosomes. At ﬁrst, a random popula-

tion of individuals are selected. In each generation, the current

population will go through crossover and mutation to generate

a new population, the worst individuals will be replaced by

better individuals in order to obtain better ﬁtness.

We take use of GA to jointly optimize M,Kand p

for maximizing EE, considering the constraints in (14). The

detailed steps are given in Algorithm 1.

Algorithm1 :Joint parameter optimization

1) Input: Input Mmax,Kmax ,pmax, power constraint

Pmax, rate threshold Rmin and maximum number of

generations Imax.

2) Initialize: Generate chromosomes by combining four

parameters M,Kand p. Each chromosome has three

points. Select Nchromosomes to form initial popula-

tion, check constraints. Set iteration counter i= 1.

3) Fitness: Compute the ﬁtness of each chromosome ac-

cording to the EE function.

4) Sorting: Sort the chromosomes in initial population in

terms of ﬁtness.

5) Iteration: Repeat the next steps until convergence.

a) Crossover: Decide whether to apply crossover

according to pre-deﬁned crossover probability ρc.

If applied, choose two parent chromosomes with

higher ﬁtness ranking, then randomly select an ex-

change point and exchange the information. Check

constraints.

b) Mutation: For each chromosome in current pop-

ulation, decide whether to apply mutation to a

chromosome according to mutation probability ρm.

If applied, choose a point and change its value

randomly. Check constraints.

c) Accepting: Compute ﬁtness for current population

and sort the values. The chromosomes with lower

ﬁtness will be replaced by those having higher

ranking.

d) Check termination: If i=Imax, select the best

chromosome in current population, terminate the

algorithm. Otherwise, continue the iteration.

41 50 60 70 80 90 100 110 120

Number of Antennas

0

10

20

30

40

50

60

70

Energy Efficiency [Mbit/Joule]

z = -20 dB

z = -10 dB

z = 0 dB

(a)

1 10 20 30 40 50 60 70 79

Number of Users

0

10

20

30

40

50

60

70

Energy Efficiency [Mbit/Joule]

z = -20 dB

z = -10 dB

z = 0 dB

(b)

0 0.005 0.01 0.015 0.02

Transmission Power Per Antenna (W)

0

10

20

30

40

50

60

Energy Efficiency [Mbit/Joule]

z = -20 dB

z = -10 dB

z = 0 dB

0246

10-4

56

58

1 2

10-4

34

36

5 10 15

10-5

12.5

13

13.5

14

(c)

Fig. 2. Energy efﬁciency versus: (a) The number of antennas per BS. (b) The number of users per cell. (c) Transmission power per antenna.

41 50 60 70 80 90 100 110 120

Number of Antennas

0

50

100

150

200

Average User Rate [Mbit/s]

z = -20 dB

z = -10 dB

z = 0 dB

(a)

1 10 20 30 40 50 60 70 79

Number of Users

0

50

100

150

200

250

300

Average User Rate [Mbit/s]

z = -20 dB

z = -10 dB

z = 0 dB

(b)

0 0.2 0.4 0.6 0.8 1

Transmission Power Per Antenna (W) 10-3

0

50

100

150

200

Average User Rate [Mbit/s]

z = -20 dB

z = -10 dB

z = 0 dB

(c)

Fig. 3. Average user rate versus: (a) The number of antennas per BS. (b) The number of users per cell. (c) Transmission power per antenna.

In Algorithm 1, the constraints in (14) are checked for

all the chromosomes in initialization and for new generated

chromosomes every time crossover and mutation are applied.

If the constraints are not satisﬁed, the chromosome will repeat

mutation operation until it satisﬁes all the constraints.

IV. SIMULATION RESULTS

This section presents simulation results to validate our pro-

posed scheme. The simulations are performed on a laptop with

Intel i7 CPU and 16G memory. The corresponding simulation

parameters are summarized in Table I [3].

In Fig. 2 and Fig. 3, we analyze the inﬂuence of M,Kand

prespectively when the other two parameters are ﬁxed. We

set λ= 3 and assume an initial combination of M,Kand

pto be 80, 40 and 0.002 W. From Fig. 2, it is observed that

EE is a quasi-concave function of M,Kand p, respectively.

While in Fig. 3, the average user rate increases monotonically

with M,pand decreases monotonically with K. In addition,

it is obvious that higher penetration loss of the walls can lead

to both higher EE and higher average user rate.

In Fig. 4, we show the inﬂuence of network densiﬁcation.

The BS density can be computed as λ2/900 BSs/m2. Different

from [2], we observe that under the bounded path loss model,

the system EE will decrease with the increasing BS density

when the BS density is sufﬁciently large.

TABLE I

SIMULATION PARAMETERS

Parameter Value

Transmission bandwidth: B20 MHz

Channel coherence bandwidth: BC180 kHz

Channel coherence time: TC10 ms

Total noise power: Bσ2−96 dBm

Path loss at reference distance: βL

010−3.85

Path loss exponent: α3

Pilot length: 1

Computational efﬁciency at BS: LBS 12.8Gﬂops/W

PA efﬁciency at BS: η0.39

Fixed power consumption: PFIX 18 W

Power for local oscillator: PSYN 2W

Power for circuit components at BS: PBS 1W

Power required for coding: PCOD 0.1W/(Gbit/s)

Power required for decoding: PDEC 0.8W/(Gbit/s)

Power required for backhaul: PBT 0.25 W/(Gbit/s)

Next, we compare our proposed joint optimization scheme

with the greedy search algorithm in Fig. 5. We set λ= 3 so

that the BS density is 0.01 BSs/m2. For the constraints in (14),

we assume Kmax = 80,Mmax = 120,pmax = 0.001 W, the

user rate threshold Rmin = 30 Mbit/s, maximum power for

the BS Pmax = 0.1W. For GA, the number of generations is

50, the size of population is 100, ρc= 0.8and ρm= 0.04.

After the algorithm converges, we get the optimal EE: 14.48

Mbit/Joule, 38.78 Mbit/Joule and 66.5 Mbit/Joule when zis

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Density (BSs/m2)

5

10

15

20

25

30

35

40

45

50

55

60

Energy Efficiency [Mbit/Joule]

z = -20 dB

z = -10 dB

z = 0 dB

Fig. 4. Energy efﬁciency versus BS density.

5 10 15 20 25 30 35 40 45 50

Generations

10

20

30

40

50

60

70

Energy Efficiency [Mbit/Joule]

Proposed scheme, z = -20 dB

Proposed scheme, z = -10 dB

Proposed scheme, z = 0 dB

Greedy search, z = -20 dB

Greedy search, z = -10 dB

Greedy search, z = 0 dB

Fig. 5. Comparison of our proposed scheme and the greedy search algorithm.

0dB, −10 dB and −20 dB, respectively. For z= 0 dB, the

optimal parameters are: M= 108,K= 46,p= 0.1mW. For

z=−10 dB, M= 89,K= 57,p= 0.3mW and for z=

−20 dB, M= 81,K= 61,p= 0.8mW. Therefore, higher

penetration of walls can provide higher EE with less antennas

and higher transmission power while serving more users. As

shown in Fig. 5, our proposed scheme can obtain nearly the

same EE with greedy search. The average computational time

for our proposed scheme is 21.89 s while for greedy search is

1288.83 s. These demonstrate that our proposed scheme is a

nearly optimal algorithm with lower time complexity.

V. CONCLUSIONS

In this paper, we formulate the network energy efﬁciency of

an indoor Massive MIMO network with blockages caused by

interior walls. We employ the bounded path loss model and

show that the network will suffer from lower EE with higher

BS density. Furthermore, we maximize the EE by jointly

optimizing the number of antennas per BS, the number of users

per cell and the transmission power per antenna, taking into

account the user rate and total transmission power constraints.

To solve this joint optimization problem, we propose a low-

complexity GA based scheme to obtain the optimal EE. The

results show that the penetration loss of walls is beneﬁcial for

both EE and average user rate. In the future, we will extend our

research to millimeter wave massive MIMO indoor scenarios

while considering the impact of pilot contamination.

APPENDIX A

DERIVATION OF INTE R- CE LL INTERFERENCE

The average ratio of the channel attenuation between inter-

ference cell fand serving cell cdepends on the value of λ,λ

is an odd number and λ≥3. Deﬁne two odd numbers iand

j,i= 1,3...λ −2,j=−1,1...λ −2. Deﬁne two functions

f(x, y) = x2+y2+ 1,g(x, y) = [x−(i+ 1)r]2+ [y−(j+

1)r]2+ 1, then Pf6=cGcf can be computed as follows:

X

f6=c

Gcf =4

λ−2

X

i=1

λ−2

X

j=−1Rr

−rRr

−r(f(x, y))α/2zi+j+2

2dxdy

Rr

−rRr

−r(g(x, y))α/2dxdy,

(15)

ACKNOWLEDGMENT

This work was funded by the European Union’s Horizon

2020 research and innovation programme under grant agree-

ment No. 766231.

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