performance analysis of device-to-device communications underlaying cellular networks

Noname manuscript No.(will be inserted by the editor)

Performance Analysis of Device-to-Device CommunicationsUnderlaying Cellular Networks

Wenjun Wu Wei Xiang Yingkai Zhang Kan Zheng Wenbo Wang

Received: date / Accepted: date

Abstract Device-to-device (D2D) communications un-derlaying cellular networks are considered to be promis-ing communication modes to improve network radio re-

source eciency and provide higher transmission datarates to devices close to each other. However, when D2Dcommunications reuse cellular resources, the resulting

interference will cause signicant performance loss tocellular users. In this paper, the spacial distributionof D2D communication users is modeled as a homoge-

neous spatial poisson point process (SPPP). With thisassumption, the closed-form expressions of the cumu-lative distribution functions (CDF) of the uplink inter-

ference power from the D2D communications and thesignal power from the serving cellular users to the basestation (BS) are derived, respectively. The approximate

CDF of the uplink signal-to-interference-ratio (SIR) ofcellular users is also given in our analysis. With theseanalytical results, one can readily obtain the outage

probability and the CDF of the achievable data rates ofthe cellular users in a straightforward manner withoutexhaustive simulations. More attractively, the analyti-

cal results can be used to help design the constraints onthe congurations of D2D communications consideringthe minimum requirements of cellular users. Simulation

results validate our analysis. Application examples ofthe analytical results are also given in this paper.

W. Wu Y. Zhang K. Zheng W. WangWireless Signal Processing and Network Lab, Key Labora-tory of Universal Wireless Communications, Ministry of Ed-ucation, Beijing University of Posts & Telecommunications,Beijing, China.Tel.: 86-10-6228-2245-5E-mail: [email protected]

W. XiangFaculty of Engineering and Surveying, University of SouthernQueensland, Toowoomba, QLD 4350, Australia.

Keywords Device-to-device interference analysis spacial poisson point processes

1 Introduction

Communications by means of wireless networks havebecome indispensable in modern social lifes. Increas-

ingly large volumes of data are envisaged to be trans-mitted over wireless networks in the foreseeable future.To meet the demands of such data explosion, wireless

networks tend to support increasingly higher data ratesfor all the users. New techniques such as relay, pico celland femtocell have received a lot of attentions in both

academia and industry. However, no matter where thesource and destination in communications are locatedand what kind of new node they access to, they have

to transmit data through a centralized controller suchas the base station (BS). This will cause inecient re-source utilization when the source and destination are

close to each other.

In this article, the BS controlled device-to-device(D2D) communications are considered. With this com-

munication mode, the source and destination only re-ceive necessary control signaling from the BS, and trans-mit data directly to each other when their locations are

in close proximity, as illustrated in Fig. 1. This commu-nication mode can avoid unnecessary data transmissionto the BS and thus improve on the eciency of resource

utilization. When the number of communication devicesis large, such as the devices of the Internet of Things(IoT), enabling such a communication mode is espe-

cially imperative. It is proved that enabling the D2Dcommunication mode can provide a higher sum ratecompared with the pure cellular communications [1].

In comparison with other short distance communica-

2 Wenjun Wu et al.

Fig. 1 D2D communications underlaying wireless cellularnetworks.

tion modes, the BS controlled D2D communications are

transparent to users. Moreover, it is more reliable sincethey communicate via an accredited network.

However, the D2D communication mode poses con-siderable new challenges, e.g., the signaling of D2D com-

munications, the interference problem between D2D com-munication mode and cellular mode, and the resourceallocation of D2D communications. In [2], the session

control of D2D communications is considered. Relatedfunctional blocks to enable D2D communications arealso addressed in [3]. The mode selection schemes, which

decide when to enable the D2D communication modeare discussed to maximize the system utility [4]. In [5]and [6], the resource sharing mode selection is discussed

when the D2D communication mode is selected. Tomanage interference, power control is considered. WhenD2D communications reuse the cellular resources, the

mechanisms that limit the maximumD2D transmit powerare proposed in [1] and [2]. In [7], several practicalpower control methods for D2D communications in an

OFDMA FDD system are investigated. Eective re-source allocation including time-frequency resource andpower allocation is also helpful to manage interference.

Reserving resources for D2D communications is a straight-forward way to allocate resources, but not eective interms of resource eciency [2]. Dynamic resource allo-

cation is more eective, the instantaneous interferenceenvironment can be considered when allocating [8]-[11].The performance of cellular networks with underlaying

D2D communication links is analyzed in [12][13] in asimple isolated cell where one cellular user and two D2Dusers share the available radio resources. Although it is

helpful to design some interference avoidance schemes,

it is not enough to give statistical characteristics on the

general interference scenario of D2D communications.

This paper analyzes the performance of cellular usersin an OFDMA wireless cellular network with under-laying D2D communications. The interference range isconsidered to be two-layer cells around the observing

one. Each cell serves only one cellular user on a sub-channel, which can be located inside the coverage areaof the BS uniformly. Meanwhile, the number of D2D

communications on a subchannel depends on the den-sity of D2D communication users. Since the locationof D2D communication users is random, the homoge-

neous spacial poisson point processes (SPPP) [14] isused in our analysis. The statistical distribution of theinterference from D2D communications and the signal

from cellular users to the BS are derived, respectively.An approximate statistical distribution of the signal-to-interference-ratio (SIR) for cellular users in uplink

transmission is also given. With these analytical results,the outage probability of cellular users and an estimateddistribution of the achievable transmission data rates

are given. The analysis also help to determine the con-straints on the congurations of D2D communications,such as density and the maximum transmit power.

The remainder of the paper is organized as follows.

In Section II, the system model is introduced. In SectionIII, the statistical distribution of the cellular user per-formance is derived. The applications of the analytical

results are given in Section IV. The validation of the an-alytical results and their application examples are givenin Section V with numerical simulation results. At last,

Section VI concludes this paper.

2 System model

We consider an OFDMA wireless cellular network with

underlaying D2D communications. On the downlink,D2D users receive control signaling information fromthe cellular BS. Since user devices are always half-duplex,

they can not transmit and receive simultaneously. Asa result, the D2D communication users transmit sig-nals to each other on the cellular uplink interval. A

schematic of such a two-tier network is illustrated inFig. 1, where the dashed line represents the transmis-sion of control signaling information on the downlink.

There are generally two basic frequency resource alloca-tion methods between the cellular communication modeand the D2D communication mode in such a two-tier

network. One is the fully reuse method, whereas theother is termed orthogonal allocation. We focus on thefully reuse scheme to analyze the most serious interfer-

ence scenario in this paper.

Performance Analysis of Device-to-Device Communications Underlaying Cellular Networks 3

Denote by AR2 and HA the coverage area ofthe cellular network and one hexagonal macro cell, re-spectively. The BS is located in the center of each cell.Assume the use of omni-antenna by the BS for simplic-

ity of analysis. Users adopting the traditional cellularcommunication mode are distributed randomly in A ,forming a homogeneous SPPP denoted by C with in-

tensity C . Thus, C is also the density of the cellu-lar users, and the average number of them in each cellis NC = C kH k, where kH k represents the area ofone cell. The underlaying D2D communication pairs arealso randomly distributed in A . We set one user of eachpair as a dominator of the communication. All these

dominant users form a homogeneous SPPP denoted byD with intensity D. Similarly, D is also the densityof these dominant users and the average number of D2D

communication pairs in each cell is ND = D kH k.In practice, the distance between the BS and users

should be greater than a certain threshold Rc;min. How-ever, this threshold is considered to be very small (closeto zero) in our analysis for theoretic tractability. There

are still some restrictions on distance Rd between thetwo D2D communication users. Assume Rd follows anuniform distribution. The probability density function

(PDF) of Rd is given by

pRd (r) =1

Rd;max Rd;min =1

ard; (1)

where Rd;max and Rd;min are the maximum and min-

imum distance between the two D2D communicationusers, respectively. Similarly, Rd;min is also consideredto be very small in our analysis.

For brevity of exposition, the propagation model is

considered to be a simple distance-dependent path loss,with an extra log-normal distributed random factor ac-counting for shadow fading, which is similar with that

adopted in [15]. The received power from transmitter yto receiver x deployed on a tower is

Pr;x;y = Pt;yjy xj10(my=10)10(mx;y=10); (2)where Pt;y is the transmit power at y, jy xj indicatesthe distance between x and y, and is the path-lossfading coecient (typically = 4). my and mx;y aretwo independent zero-mean Gaussian random variableswith an identical standard deviation of s. Obviously,10(my=10) and 10(mx;y=10) are log-normal distributed,

with the former representing the shadow fading inducedby the local environment of y and the latter represent-ing the shadow fading induced by the propagation en-

vironment between x and y. In practice, coecients and often take the value of 2 = 2 = 1=2. This prop-agation model is in line with the fact that the received

powers of two separate high located receivers x1 and x2

from the same location y are correlated random vari-

ables, since they both share the same local environmentof y. However, for the receiver x located at a particulartower, the received powers from two dierent locations

y1 and y2 are assumed to be independent since there arehardly any scatters in the local environment of x, andthe local environments of y1 and y2 are independent.

This means mx;y1 and mx;y2 are independent randomvariables.

When the transmitter and receiver are both userdevices, their surroundings contribute to shadow fading

in general. The propagation model should be modiedas

Pr;x;y = Pt;yjy xj10(my=10)10(mx=10); (3)where mx is the shadow fading induced by the local

environment of x.

To provide a basic understanding on the perfor-mance of D2D communications underlaying a cellularnetwork, we assume that the network does not em-

ploy any power control and interference mitigation tech-niques. The BS allocates frequency resources to regularcellular users and D2D communication pairs randomly.

In an OFDMA network, there is always only one cellularuser on each subchannel in a cell, denoted by N 0C = 1.The average number of subchannels assigned to each

cellular user can be given by

Nsc;c =NscNC

; (4)

where Nsc is the total number of subchannels in use inthe OFDMA network. Assume the average number ofsubchannels assigned to each D2D communication pair

is denoted by Nsc;d. The density of D2D communica-tion pairs on each subchannel is harmoniously diluted.Denote by 0D the sets of D2D communication pairs onone subchannel. The average number of pairs per-celland the density of 0D are given by

N 0D =NDNsc;dNsc

and 0D =DNsc;dNsc

: (5)

3 Performance analysis

For fully reuse resource allocation method, the interfer-ence received by the BS from D2D users will give rise

to signicant outage of cellular uplink transmission. Inthe following, we primarily analyze the statistical char-acteristics of the interference induced by D2D commu-

nications to the BS. Then, the constraints on the con-gurations of D2D users are given with respects to theminimum performance requirement of cellular users.

The interference received by the BS is comprised

of two components, i.e., that from cellular users served

4 Wenjun Wu et al.

by the neighboring BSs and that from the surrounding

D2D users. Since each cell serves only one cellular useron each subchannel irrespective of the resource alloca-tion method employed, the statistical characteristics of

the cellular interference is not changed in comparisonwith the traditional cellular network.

On a candidate subchannel, the set of interfering

D2D users is 0D with density 0D as dened in (5).

The interference from D2D users can be expressed as

PId2c=Xd20D

Pr;c;d; (6)

where c is the location of the candidate BS which is setas the origin of the two-dimensional plane. Pr;c;d is thereceived power dened by (2) since the antennas of the

BS are always deployed on a tower. Assuming all theD2D users transmit at the same power level of Ptd andsubstituting (2) into (6), a more detailed expression of

PId2c is given by

PId2c=Xd20D

Ptdjdj10(md=10)10(mc;d=10): (7)

For two dierent points d1 and d2 of 0D, md1 and md2

are independent. mc;d1 and mc;d2 are also independentsince the environments along the two paths are dier-

ent. As a result, 10(md=10)10(mc;d=10) is a log-normaldistributed random variable, which is independent fordierent points of 0D. According to Appendix A withQD = 10

(mD=10)10(mc;D=10), the characteristic func-tion of PId2c can be given by

PId2c(!)=exp

0DPtd

12 e

acd16

1

2

e

j4 !

12

=exp

r

2(1j)

0DpPtde

acd16

p!

; (8)

where acd = 22s

2 and = (ln 10) =10. The PDF and

CDF of PId2c are given by

fPId2c(pI) =

20DpPtd p

32I e

acd16 e

30D

2Ptde

acd8

4pI (9)

and

FPId2c(pI) = erfc

320D

pPtd e

acd16

2ppI

!; (10)

respectively.

According to (2), the received signal power of the

BS from its serving user located at s is

PSc2c=Ptcjs cj10(ms=10)10(ms;c=10)=Ptcr

c Qcs; (11)

where Ptc is the transmit power of the cellular user andrc is the distance between the BS and the user. Qcsis the comprehensive shadow fading comprised of the

shadow fading induced by the surrounding of the user

and the environment of the transmission path. Obvi-ously, Qcs is log-normal distributed with variance s

2

in dB whose PDF is given by

fQcs(qcs) =1

qcspacs

e(ln qcs)

2

acs ; (12)

where acs = 22s

2.

Since the number of cellular users transmitting onthe candidate subchannel is xed to one, the distribu-tion of the user served on this subchannel is no longer

an SPPP. In fact, it is uniformly distributed in the cov-erage area of the BS. Thus, the distance between thecellular user and its serving BS has the following PDF

pRc (rc) =2rc

R2c;max R2c;min=

2rcarc

; (13)

where Rc;max and Rc;min are the maximum and mini-mum distances between the BS and the users, respec-tively. Since Rc;min is very small, we assume it is zero

for convenience of analysis, and thus arc = R2c;max. To

simplify expressions, we dene yr = rc and

fYr (yr) = 2

arcy

21r : (14)

The CDF of PSc2c can be calculated as (15).

FPSc2c(pS)

=

Z pSPtcR

c;max

0

fQcs (qcs)

Z pSPtcqcs

Rc;max

fYr (yr) dyr dqcs

=

Z pSPtcR

c;max

0

e(ln qcs)

2

acs

qcspacs

1arc

pSPtc

2

q 2cs dqcs

Z pS

PtcRc;max

0

e(ln qcs)

2

acs

qcspacs

R

2c;max

arc

!dqcs

=1

2f1 + sign (A3) erf [abs (A3)]g

12arc

pS2Ptc

2 eacs2 f1+sign (A4) erf [abs (A4)]g ;(15)

where A3=ln

pS

PtcRc;max

pacs

and A4=A3 +pacs .

Since the CDFs of PId2c and PSc2c both are very

complicated according to the previous analysis, it isdicult to nd a closed-form expression for the sys-tem performance. As the received interference of the BS

from the neighboring cellular users is always the sameas that in the conventional cellular network, the outageof cellular users can be well controlled via controlling

the interference from D2D users. It is easy to sketch theCDFs of PId2c and PSc2c , and the gap between them isvery clear. Designing the gap will be employed as one

option in the subsequent simulation section to aid the


design of D2D transmission. However this method can-

not be used to analyze other performance metrics suchas the achievable rate and the exact outage probability.

A more feasible method is analyzing the distribution

of the SIR of cellular users approximately. The receivedSIR denoted by Tc is given by

Tc =PSc2c

PIc2c + PId2c; (16)

where PIc2c is the received interference at the observ-ing BS from cellular users served by neighbors. Whenthe density and transmit power of D2D pairs are large,

PIc2c is relatively small compared with the interferencefrom D2D users. In this paper, this kind of interferenceis neglected. However, it is noted that this will cause

inaccuracy especially when the transmit power and thedensity of D2D communication devices is low. A moreaccurate analysis will be undertaken in further study.

Using Td2c to denote an approximation of Tc which isgiven by

Td2c =PSc2cPId2c

: (17)

To make the analysis process clear, we rst ignore theshadowing fading factor in PSc2c , and denote by T

0d2c

this interim SIR. We can obtain

T 0d2c =P 0Sc2cPId2c

; (18)

where P 0Sc2c = Ptcrc with

fP 0Sc2c(pS) = 2

arcPtc

2 pS21: (19)

The CDF of T 0d2c can be calculated as follows.

FT 0d2c(t0)

=

Z 1PtcR

c;maxt0

fPId2c (pI)

"Z pIt0PtcRc;max

fP 0Sc2c(pS) dpS

#dpI

= 12arc

pP 2tc A5

A52

1+ 4t0

2

1

2 2

1

2 2;A26

+R2c;maxarc

erf (A6) (20)

where A5 = 320D

pPtde

acd16 and A6 =

A5pt0

2pPtcRc;max

.

Since Td2c=T0d2cQcs, the CDF of Td2c is given by

FTd2c (t) =

Z 10

fQcs(qcs)FT 0d2c

t

qcs

dqcs: (21)

4 Application of the analysis results

With the distribution of the SIR of cellular users, the

system performance can be analytically determined with-out running time consuming simulations. We can alsosuggest on system design without exhaustive simula-

tions.

4.1 Achievable Data Rate

When fast fading is not considered, the geometry-basedachievable data rate of the cellular user on one subchan-nel can be calculated as a function of the SIR, which is

given by

CC;1=Bsubclog2 (1 + Tc)

Bsubclog2 (1 + T 0d2c) : (22)where CC;1 is the achievable data rate of the cellularuser on one subchannel, and Bsubc is the bandwidth ofthe subchannel. Considering the average number of sub-

channels assigned to each cellular user, the achievabledata rate of cellular users can be expressed as

CC = Nsc;cCC;1 Nsc;cBsubclog2 (1 + T 0d2c) : (23)According to the geometry-based static SIR, the CDFof CC is given by

FCC (c) FT 0d2c2

cNsc;cBsubc 1

: (24)

Since the logarithm function is a convex function, the

achievable data rate calculated with this method is largerthan its actual value aected by fast fading. However,when fast fading is considered, opportunistic schedul-

ing will provide additional gains, which requires furtherinvestigation.

4.2 Outage Probability

With the CDF function at hand, it is easy to obtain theoutage probability. Assume that the outage SIR thresh-

old of the cellular users is Tout according to the mod-ulation threshold of the control signal. If the transmitpower Ptc of the cellular users, the transmit power Ptdand the density 0D of the D2D users are given, theoutage probability pout of the cellular users is

pout = FT 0d2c(Tout) : (25)

4.3 Constraints on the Congurations of D2DCommunications

In this section, the constraints on the congurationssuch as the density and the transmit power are investi-

gated. With the consideration of the outage probabilityof the cellular user, the density and the transmit powershould be below a certain threshold. As can be observed

from (20), FT 0d2c() can be expressed as a function of0DpPtd

denoted by 0. When the transmit power

of cellular user Ptc is xed, there will be a range ofFT 0d2c(t

0; 0) with dierent values of 0. Supposing thatthe outage SIR threshold of the cellular users is Tout,

6 Wenjun Wu et al.

and the desired outage probability of the cellular user

is below Pout, we can calculate FT 0d2c(Tout; 0) until the

condition FT 0d2c(Tout; 0) Pout is satised. The process

of nding the proper 0 is given below.

{ Step 1 : Assign an initial value to 0.{ Step 2 : Calculate FT 0d2c(Tout;

0).{ Step 3 : if FT 0d2c(Tout;

0) > Pout, set 0 = 0and return to Step 2. if FT 0d2c(Tout;

0) < Pout,stop calculation, and record the current value of 0.

where < 1 denotes the step coecient of 0. When

we obtain the value of 0, we can determine density 0Dand transmit power Ptd of the D2D users according tothe system requirement, such as the receiver sensitivity

of D2D devices and the number of the D2D pairs in acell. This can be expressed as Ptd Ptd;min and 0D 0D;min.

5 Numerical results

In this section, we will rst validate our analytical re-sults via simulations for a wide range of scenarios with

dierent system parameters. Then, we give some exam-ples of using these results to guide the design of thesystem.

5.1 Validation

To validate the analytical results of the CDF of the re-

ceived interference power, the received signal power andthe SIR of the cellular uplink transmission, an OFDMAnetwork with a bandwidth of 10 MHz and Nsc = 50 is

considered. Assume NC = 10, and ND = 100. The av-erage numbers of subchannels allocated to each cellularuser and each D2D communication pair are Nsc;c = 5

and Nsc;d = 5, respectively. To observe the robustnessof the analytical results, a group of parameters are usedas shown in Table 1. Since we only analyze the statisti-

cal characteristics of the geometry-based performancein the OFDMA networks, the simulations are based onsnapshots of the network.

In our simulations, we also assume the antenna gains

of the BS and the user are GB = 14dB and GU = 0dB,respectively. Comparative results between the simula-tions and the analysis are shown in Figs. 2-4. To es-

timate the error of our analytical results, the meansquared error (MSE) of the CDF is calculated in termsof the value of the probability when the abscissa is the

same. The results are shown in Table 2.

The MSEs of FPId2c and FPSc2c in Fig. 2 and Fig.3 are relatively stable. Since we have considered shad-

owing in the analysis of the CDFs of these tow kinds

Table 1 Parameters for simulation on a subchannel

CellularParameters

ISD(m)

Rc;min(m)

s(dB)

N 0C(=cell)

Ptc(dBm=subchannel)

Scenario I 500 0.05 4 1 3

Scenario II 500 0.05 4 1 3

Scenario III 500 0.05 8 1 3

Scenario IV 500 0.05 8 1 3

Scenario V 500 10 4 1 3

Scenario VI 500 10 4 1 3

D2DParameters

Rd;max(m)

Rd;min(m)

s(dB)

N 0D(=cell)

Ptd(dBm=subchannel)

Scenario I 10 0.05 4 10 3

Scenario II 10 0.05 4 10 -15

Scenario III 10 0.05 8 10 3

Scenario IV 10 0.05 8 10 -15

Scenario V 10 0.05 4 10 3

Scenario VI 10 0.05 4 10 -15

of power, increasing the standard deviation will not in-crease the error. This can be observed from the MSEs of

Scenarios group I /III and Scenarios group II/IV. Sincewe assume the minimum distance between the users andthe BS is zero in the analysis, the error increases with

the increase of the actual minimum distance. From Ta-ble 2 we can see the MSE of Scenario V is larger thanthose of Scenarios I and III, and the MSE of Scenario VI

is larger than those of Scenarios II and IV too. How-ever, the MSEs are still relatively small, and there isno distinct gap between the simulation and analytical

results in Figs. 2 and 3.

When observing the results in Fig. 4, the MSE be-tween the simulation and the analytical CDFs of FTcchanges in dierent scenarios, since we use FT 0d2c to ap-proach FTc in the analysis. In Scenario I, the standarddeviation of shadowing is low and the transmit power

of the D2D users is relatively large. Ignoring the shad-owing of the received signal and the interference fromneighboring cellular users causes little loss of accuracy.

With the increase of the standard deviation, the MSEincreases very fast. We can see that the MSEs in Scenar-ios III and IV are larger than those in Scenarios I and

II, respectively. Meanwhile, when the transmit powerof D2D users is low, the proportion of the interferencefrom the neighboring cellular users will increase. The

approach employed by our analysis will result in moreerrors, and the MSEs in Scenarios II, IV and VI arelarger than those in Scenarios I, III and V, respectively.

Comparing Scenarios V and VI to Scenarios I and II, aninteresting phenomenon can be observed that the MSEdecreases when the actual minimum distance increases.

This is contrary to the previous results on FPId2c and


-80 -60 -40 -20 0

0.0

0.2

0.4

0.6

0.8

1.0

Scenario I

Scenario II

Scenario III

Scenario IV

Scenario V

Scenario VI

Solid : Simulation results

Dash : Analytical results

CDF

MeNB receiving interference from D2D links, dBm/subchannel

Fig. 2 CDF validation of the BS receiving interference fromD2D communications.

-100 -80 -60 -40 -20

0.0

0.2

0.4

0.6

0.8

1.0

Scenario I

Scenario II

Scenario III

Scenario IV

Scenario V

Scenario VI



CDF

MeNB receiving signal from cellular users, dBm/subchannel

Fig. 3 CDF validation of the BS receiving signal power fromserving cellular users.

FPSc2c , due to that counteraction among serval factors

may reduce the error. In a word, ignoring the shadowingof the received signal and the interference from neigh-boring cellular users will cause a distinct gap between

the simulation and analytical results when the standarddeviation of shadowing is large and the transmit powerof D2D users is low. However, this is still acceptable

and our analytical results are valid in most cases.

Table 2 MSE of the analytical CDF

XXXXXXXXScenarioCDF

FPId2c FPSc2c FTc

Scenario I 9.5149E-05 7.1099E-06 8.4503E-05

Scenario II 1.1169E-04 1.7757E-06 4.1965E-04

Scenario III 9.0064E-05 6.3717E-06 5.8865E-04

Scenario IV 1.0366E-04 1.2540E-06 9.4950E-04

Scenario V 2.2416E-04 1.0520E-05 6.7498E-05

Scenario VI 1.5682E-04 6.4085E-06 4.4839E-04

-80 -60 -40 -20 0 20 40 60

0.0

0.2

0.4

0.6

0.8

1.0

CDF

Uplink SIR of cellular users, dB

Scenario I

Scenario II

Scenario III

Scenario IV

Scenario V

Scenario VI



T

out

Fig. 4 CDF validation of the uplink SIR for cellular users.

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Scenario I

Scenario II

Scenario III

Scenario IV

Scenario V

Scenario VI



CDF

Achievable data rate, Mbps

Fig. 5 CDF of the uplink achievable data rate of cellularusers.

5.2 Achievable Data Rate and Outage Probability

Utilizing the analytical FTc to obtain the achievabledata rate and outage probability is straightforward. Us-

ing the parameters in Table 2 and assuming that thebandwidth of the subchannel is Bsubc=180 kHz, theachievable data rate is shown in Fig. 5. The error in

the analytical achievable data rate is similar to thatof the analytical SIR. The corresponding outage prob-ability is also given in Table 3. Since the interference

from neighboring cellular users is ignored in the anal-ysis, the analytical outage probability is lower than itsactual value when the transmit power of the D2D links

is low. When one uses the analytical results to guidethe system design, these errors need to be taken intoconsideration.

5.3 Constraints on the Congurations of D2DCommunications

We rst use a graphic method to obtain the constraints

on the density and transmit power of D2D communica-

8 Wenjun Wu et al.

Table 3 outage probability

Scenario Simulation outage Analytical outage

Scenario I 0.8628 0.8719

Scenario II 0.3567 0.2977

Scenario III 0.8678 0.9068

Scenario IV 0.4869 0.3927

Scenario V 0.8587 0.8719

Scenario VI 0.3453 0.2977

Note: Simulated outage is read from the solid curve in Fig.4 and the analytical outage is calculated as FT 0

d2c(Tout)

assuming Tout=7:44 dB according to the required SNRof PUCCH in [16].

tions. We assume the acceptable outage probability ofcellular users is below 10%. Before we use this methodto design the system, we should nd a reasonable gap

between the CDF of the BS receiving signal power fromits serving users and the interference from D2D com-munications. Once the desired gap is identied, we can

use it in the system design when needed.

The cellular parameters are xed which are the sameas in Scenario II in Table 1. A range of values for 0

given in Table 4 are used to sketch the analytical CDFsof the BS receiving interference from D2D communica-tions shown in Fig. 6. Accordingly, simulation results

with the same parameters are also given in Fig. 7. Ascan be seen from Figs. 6 and 7, we can obtain a graph ofthe gap versus the outage probability as shown in Fig.

8. With these statistics and under conservative estima-tion, we suggest a gap of 15 dB between the analyticalCDF of the signal power from cellular users and the in-

terference power from D2D communications to the BS.

Table 4 Parametric values of the intensity and transmitpower for D2D communications

0105

pmw=m2

01 =4:1214

02 =3:1703

03 =2:4387

04 =1:8759

N 0D (=cell) 10 10 5 5

Ptd(dBm=subchannel)

14:9897 17:2686 13:5268 15:8057

0105

pmw=m2

05 =1:4430

06 =1:1100

07 =0:8539

08 =0:6568

N 0D (=cell) 5 5 5 5

Ptd(dBm=subchannel)

18:0846 20:3634 22:6423 24:9212

Note: considering antenna gains in simulation, 0 =0D

pPtdGB .

In the sequel, we give an example of using the graphic

method to assist in the design of the density and trans-

-100 -80 -60 -40

0.0

0.2

0.4

0.6

0.8

1.0

CDF

BS receiving power, dBm/subchannel

signal from serving user

Dash: interference from D2D

'

1

'

2

'

3

'

4

'

5

'

6

'

7

'

8

Fig. 6 CDFs of PId2c with various D2D communication den-sities and transmit powers.

-40 -20 0 20 40

0.0

0.2

0.4

0.6

0.8

1.0

CDF

Simulation uplink SIR of cellular users, dB

'

1

'

2

'

3

'

4

'

5

'

6

'

7

'

8

-8 -7

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-7.44

Fig. 7 CDFs of the uplink SIR for cellular users with variousD2D communication densities and transmit powers.

-2 0 2 4 6 8 10 12 14

5

10

15

20

25

30

35

40

Outage Probability (%)

Gap between BS receiving signal power from serving cellular

users and interference power from D2D links, dBm

Fig. 8 Analytical gap vs. actual cellular outage probability.

mit power of D2D communications. The cellular param-eters are given in Table 5 and xed. A proper value of

0 will be chosen to satisfy the suggested gap in the pre-vious study. Using (15) and (10), we can draw a groupof curves with dierent values of 0 as illustrated inFig. 9. When 0 = 4:0554 106pmw=m2, the gap can


-100 -80 -60 -40

0.0

0.2

0.4

0.6

0.8

1.0

CDF

BS receiving power, dBm/subchannel

signal from serving user

interference from D2D

-97.17

-81.91

-97.5 -97.0 -96.5

0.20

-82.5 -82.0 -81.5

0.20

Fig. 9 Theoretical gap of the system design example.

meet the suggested requirement of 15 dB. We design thedensity and transmit power of D2D communication asN 0D = 3=cell and Ptd = 24:6726 (dBm=subchannel),respectively. The minimum requirement of the refer-

ence sensitivity power level of user equipment is chosento be -94 dBm according to the specication in [17].Since the maximum distance between the D2D users

is Rd;max = 10m, the minimum transmit power of theD2D users is about -54 dBm. As a result, our designedtransmit power is much larger than the minimum trans-

mit power that is feasible.

We carry out system-level simulations according tothe designed parameters. Results shown in Fig. 10 demon-strate that the outage probability of cellular users is

8:6161%. Since this value is less than 10%, our methodis thus validated.

Table 5 Parameters for the system design example

ISD(m)

Rc;min(m)

Rd;max(m)

Rd;min(m)

s(dB)

N 0C(=cell)

Ptc(dBm=subchannel)

500 0.05 10 0.05 4 1 0

At last, a more convenient method using the ap-

proximate CDF of the uplink SIR for cellular users de-scribed in Section IV.C is adopted to design the densityand transmit power of D2D communications. The cellu-

lar parameters are the same as in the previous examplegiven in Table 5 with = 0:9. Since the errors re-sulted from ignoring the interference from neighboring

cellular users increase when the density and transmitpower of D2D communications are low, 5% is set asthe target outage probability in our analytical com-

putation, which is half of the actually required out-age probability. We choose N 0D = 5=cell and Ptd =14:21 (dBm=subchannel) as the initial values, whichindicates 0 = 2:2547 105pmw=m2. When 0 =

-60 -40 -20 0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

CDF


Outage Probability = 8.6161%

-7.75 -7.50 -7.25

0.05

0.06

0.07

0.08

0.09

0.10

-7.44

Fig. 10 Simulation SIR of the system design example usinggraphic method.

-60 -40 -20 0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

CDF


Outage Probability = 8.7449%

-7.75 -7.50 -7.25

0.05

0.06

0.07

0.08

0.09

0.10

-7.44

Fig. 11 Simulation SIR of the system design example usinganalytical approximate method.

4:6422106pmw=m2, the approximate outage proba-bility is about 4:9707% that meets our design target. Wedesign the density and transmit power of D2D commu-nication asN 0D = 3=cell and Ptd = 23:4985 (dBm=subchannel).Simulation results are shown in Fig. 11 indicating thatthe outage probability of the cellular users is 8:7449%which is lower than 10%. This method is validated too.

6 Conclusion

In this paper, the interference induced by D2D com-munications to cellular users is analyzed when the twocommunication modes reuse the same radio resources.

The closed-form CDFs of both the BS receiving inter-ference from D2D communications and the signal powerfrom its serving cellular user are derived given random

user location and shadow fading. An approximate CDFof the uplink SIR of cellular users is derived in lieu of theactual one for convenience of analysis. Based on the an-

alytical results, the outage probability and the CDF of

10 Wenjun Wu et al.

the achievable data rate is obtained. Simulation results

demonstrate the accuracy of the derived closed-formCDFs, and the error of the approximate uplink SIR forcellular users is acceptable in most cases. The analyt-

ical results provided here are also helpful for networkdesign. The D2D communication conguration calcu-lation is given as an example in this paper. Since the

distance between the source and destination is short,the transmit power of D2D communication devices canbe extremely low, and the density of D2D devices can

be relatively large compared to the users adopting thecellular communication mode. The results also conrmthe use of this new communication mode to support

emerging service such as the IoT.

Appendices: Derivation of the Statistical Char-

acteristics of the Sum of Received Power

Assume Y is an SPPP on A with intensity , wewill analyze the statistical characteristics of the sum

of received power at an arbitrary point x in the two-dimensional plane comprised of all the points in Y .

Since shadow fading is a random variable, it is nec-essary to form a marked SPPP. Dene by QY 2 R+as the random marking of Y . For any two dierentpoints y1 and y2, Qy1 and Qy2 are independent. Thedistribution of QY is the same for all the points in Y ,

and 10log10(QY ) N (0; 2), where 2 is the variance.The probability density function (PDF) is

fQY (q) =1

qpay

e (ln q)2ay : (26)

where ay = 222 and = (ln 10) =10.

The pair (Y;QY ) can then be regarded as a ran-dom point Y in the product space C = A R+. Thetotality of points Y forms a random countable sub-set Y = f(Y;QY ) ;Y 2 Y g of C. According to theMarking Theorem [14], Y is still a SPPP on C, withmean measure given by

(C)=ZZ

(y;Qy)2C

(dy) p (dQy)=

ZZ(y;Qy)2C

dyfQY (q) dq: (27)

Dene the received power from a transmitter y to the

particular receiver x as

Pr;x;y=g (y;Qy)=PtryQy: (28)

where ry is the distance between y and x. The sum of

the received power is

P=Xy2Y

g (y;Qy): (29)

Using Campbell's Theorem [14], the characteristic func-

tion of P is

P (!) = Eej!P

=exp

264 ZZ(y;Qy)2C

ej!g(y;Qy) 1

(dy;dQy)

375=exp

8>:EQy264 Zry2(0;1)

ej!Ptry

Qy 12ry dry

3759>=>; : (30)

Dene u= Ptry

Qy that falls under the range of (1; 0)since is always negative. With this variable substitu-tion, (30) can be rewritten as

P(!)

=exp

8>:EQy264 Zu2(1;0)

ej!u 1 1

PtQy

2 2

u

21du

3759>=>;

=expPt

2A1A2: (31)

where

A1 =

Zu2(1;0)

ej!u 1 2

u

21du; (32)

and

A2 = EQy

hQy 2i: (33)

We rst simplify A1 as follows.

A1=

Z 10+

ej!u 1 du 2

=!2

hj cos

2+

sin

2+

i

1 +

2

=! 2 e j

1 +

2

: (34)

where (x) =R10

tx1etdt.In order to simplify A2, the average kth power of the

log-normal distributed random variable QY is derivedas follows.

Eqk=

Zm2R+

qkfQY (q) dq=1pay

Z +10

qk1qln qay dq: (35)

Dening v= ln q v 2 (1;+1), we have

Eqk=

1pay

Z +11

e v2ay +kvdv = e

k2ay4 : (36)

Substituting (36) in to (33), A2 can be simplied as

A2 = EQy

hQy 2i= e

ay

2 : (37)


As a result, the characteristic function is given by

P(!)=exp

Pt 2 e

ay

2

1 +

2

ej !

2

=exp

Ae j ! 2

; (38)

where A= Pt

2 eay

2 1 + 2

.

The laws of probability with characteristic functions

given by (38) are the stable laws of exponent 2 withthe restriction of 0 < 2 < 1 [19]. Similar to (22)in [18], we give the PDF of the sum of the received

power as

fP(p;)=1

p

1Xk=1

1 2k

k!

A

p2

ksin k

1 +

2

:(39)

Following the inverse Gaussian probability law for =4, we can have a density given by a closed-form ex-pression. The characteristic function is

P(!)=exp

Pt 12 e

ay16

1

2

e

j4 !

12

=exp

r

2(1 j)

pPte

ay16

p!

: (40)

The PDF and CDF of P are given by

fP(p) =

2pPte

ay16 p

32 e

32Pte

ay8

4p (41)

and

FP(p) = erfc

32pPte

ay16

2pp

!; (42)

respectively, where erfc(x) = 1 2p

R x0et

2

dt.

Acknowledgments

This work was supported in part by National Basic Re-search Program of China (973 Program): 2012CB316005and Program for New Century Excellent Talents (NCET)

in University.

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