Power Control for D2D Underlay Cellular Networkswith Imperfect CSI

Amen Memmi, Zouheir Rezki and Mohamed-Slim AlouiniComputer, Electrical, and Mathematical Sciences and Engineering (CEMSE) Division

King Abdullah University of Science and Technology (KAUST), Saudi Arabiaamen.memmi@gmail.com, zouheir.rezki,slim.alouini@kaust.edu.sa

Abstract—Device-to-Device (D2D) communications underlyingthe cellular infrastructure is a technology that has recentlybeen proposed as a promising solution to enhance cellularnetwork capabilities: It improves spectrum utilization, overallthroughput and energy efficiency while enabling new peer-to-peer and location-based applications and services. However,interference is the major challenge since the same resourcesare shared by both systems. Therefore, interference managementtechniques are required to keep the interference under control.In this work, in order to mitigate interference, we considercentralized and distributed power control algorithms in a one-cell random network model. Differently from previous works, weare assuming that the channel state information (CSI) may beimperfect and include estimation errors. We evaluate how thisuncertainty impacts performances. In the centralized approach,we derive the optimal powers that maximize the coverageprobability and the rate of the cellular user while schedulingas many D2D links as possible. These powers are computed atthe base station (BS) and then delivered to the users, and hencethe name ”centralized”. For the distributed method, the on-offpower control and the truncated channel inversion are proposed.Expressions of coverage probabilities are established in functionof D2D links intensity, pathloss exponent and estimation errorvariance.

Index Terms—Power control, device-to-device communications,Poisson point process, imperfect channel state information.

I. Introduction

To meet the ever increasing data demand, researchers fromindustry and academia are seeking for new paradigms to revo-lutionize the traditional communication methods of cellularnetworks. Device-to-Device (D2D) communications appearto be of those promising paradigms expected to become akey feature supported by next-generation cellular networks.Propositions in this field mainly focused on sharing all thecellular spectrum between the two systems (cellular & D2D)in what is called D2D communications underlaying cellularnetworks. This is for the purpose of maximizing the spectrumefficiency. However, this is challenging since D2D links maygenerate significant interference to the host network. Herecomes the importance of interference management techniquessuch as power control (PC). PC is a simple, yet effectiveapproach broadly used in current wireless networks to mitigateinterference. In this work, we propose PC methods underimperfect CSI and analyze their performances in the proposedrandom network model.

A. Related Works

Several works focused on interference mitigation. Particu-larly, there has been considerable interest in PC techniquesfor D2D underlaid cellular networks. First, it has been shownin [1] that uplink resources are preferable to downlink onesfor D2D transmissions. This is mainly due to performanceissues. This explains why most of literature considers sharingthe cellular uplink with D2D communication. A simple PCscheme was proposed in [2] to regulate the transmit power ofD2D-capable users and protect the existing cellular links ina single-cell scenario and deterministic network model. Theproposed algorithm fixes SINR constraints to tolerate qualitydegradation of cellular links until threshold levels are reached.A PC method for D2D communications was proposed in [3]to maximize the network sum rate, also for a deterministicmodel of network. Essentially, early works on PC for D2D,mainly developed and evaluated PC strategies in deterministicD2D link deployment scenario.

For a more realistic representation of networks, randomnetwork models were then proposed. For example, spectrumsharing between ”ad hoc” and ”cellular networks” takingrandomness into consideration was studied in [4] and [5]. In[6], authors consider that a cellular user needs to share uplinkresources with multiple D2D links whose locations are randomand modeled via stochastic geometry. Two forms of PC wereproposed: centralized (managed by the BS) and distributed.In the centralized case, using perfect global Channel StateInformation (CSI), BS decides power profile based on opti-mizing a criteria. In the distributed case, transmit powers ofD2D-capable users are set based on the knowledge of directlink information and the minimum channel gain that is fixedand known by all users. In [7], stochastic geometry has beenused more thoughtfully for modeling the random aspect ofthe network: BSs, cellular and D2D users were representedby Poisson Point Processes (PPPs). All these previous worksconsidered perfect knowledge of the CSI, but in reality, CSIestimation error are very likely to happen. In this work,we analyze performance of some PC algorithms consideringimperfect CSI in a random network model.

B. Contribution of the Paper

In this work, we propose and analyze performance of somePC algorithms in D2D underlay cellular network. We take intoaccount the estimation error that may affect the CSI. For this

Figure 1: A single-cell D2D underlay cellular system: onecellular user establishes uplink with the BS while five activeD2D links are established in a circular disk with radius R.

purpose, we consider a single cellular cell underlaying D2Dlinks. Specifically, a cellular user intends to communicate withthe BS in uplink while several D2D communications are estab-lished using the same cellular spectrum. We take randomnessof locations into consideration by modeling D2D transmitters’positions through a spatial homogeneous PPP, using stochasticgeometry theory. In this framework, a centralized and twodistributed PC algorithms are proposed.• In the centralized approach, based on global noisy CSI,the BS handles the design of transmit power profile of allusers in order to maximize the signal-to-interference-plus-noise ratio (SINR), and thus the rate, of the cellular link whilesatisfying QoS requirements (SINR constraints) for D2D users.The centralized PC allows a significant improvement of theoverall performance of the network since it protects the cellularwhile supporting some D2D links. However, this PC methodis sensitive to CSI estimation error.• Distributed coverage probabilities are expressed in functionof the estimation error α. The perfect case is then capturedas a particular one which suggest that our framework may beregarded as a generalization of previous works, e.g., [6,7].• For the distributed approach, an on-off PC as well as a trun-cated channel inversion PCs are proposed. Such approacheshave the benefit of reducing the high CSI feedback overheadrequired for centralized PC and using only local CSI aboutthe direct link between the transmitter and its correspondingreceiver. Coverage probabilities and transmit powers are thenestablished and analyzed. For distributed PC, even if thecellular communication is less reliable, there is still an overallnetwork gain provided by D2D links. This gain is more notablein the high target SINR. Again, the estimation error quicklydegrades the performance and reduces this gain.

The paper is organized as follows. Section II describes thesystem model and provides SINR expressions and networkmetrics. In section III, the centralized algorithm is presented.Distributed coverage probabilities are then computed in sectionIV along with on-off and truncated channel inversion PCs.Selected numerical results and performance analysis are pre-sented in section V. Finally, section VI concludes the paper.

II. SystemModel and Related BackgroundA. Network Model

In this paper, we are considering a single-cell D2D underlaycellular network where a BS is in the center of a circular

coverage area with radius R, as shown in Fig. 1. We assumethat one cellular uplink user intends to communicate with theBS while several other D2D pairs are communicating usingthe same spectrum. The cellular user is uniformly located inthis region i.e., the probability distribution function (pdf) ofits distance to the BS, d0,0, is given by:

fd0,0(d) =

d2

R2 , 0 ≤ d ≤ R. (1)

Further, to take the network randomness into account, weassume that locations of the D2D transmitters are distributedaccording to a homogeneous PPP, Φ with intensity λ. Thus,the expected number K of D2D transmitters, a Poisson randomvariable, is E (K) = λπR2. Distances dk,l between two randomnodes k and l in a circular cell with radius R are distributedaccording to the following pdf [9]:

fdk,l(d) =

2dR2

2π

cos−1(

d2R

)−

dπR

√1 −

d2

4R2

, 0 ≤ d ≤ 2R.

(2)We assume also that all users have a single antenna each.Each of them has a maximum transmit power constraint suchthat p0 ≤ Pmax,c and pk ≤ Pmax,d, where Pmax,c and Pmax,d aremaximum powers for the cellular and D2D users, respectively.

B. Radio Channel Model

We consider a general power-law pathloss model in whichthe signal power decays at the rate d−δ with the propagationdistance d, where δ ≥ 2 is the pathloss exponent. The channel(distance independent) fading h is also considered. For aparticular realization of the PPP Φ, the received signals atthe BS and the D2D receiver k are written as:

y0 = g0,0s0 +∑K

l=1 g0,lsl + n0

yk = gk,k sk + gk,0s0 +∑K

l=1,l,k gk,lsl + nk,(3)

where:• the subscript 0 is used to reference the cellular uplink

user and the other k, k , 0 are for the D2D links,• yk and y0 represent the received signal at D2D receiver k

and the BS, respectively,• sk and s0 denote the signal sent by D2D transmitter k and

the uplink user, respectively,• nk and n0 denote the additive noise at D2D receiver k

and the BS. They have a complex normal distributionwith zero mean and σ2 as variance: nk ∼ CN(0,σ2),

• gk,l and g0,l represent the total channel gains from D2Dtransmitter l to receiver k and the one from D2D trans-mitter l to the BS, respectively: gk,l = hk,ld

− δ2

k,l .To mitigate interference, perfect CSI is necessary. However,

due to the dynamic nature of wireless channels, perfect CSIis unfeasible. Thus, for a more realistic representation, we as-sume that the channel is not known perfectly at the transmitterbut only imperfect CSI is available. Specifically, a minimummean square error (MMSE) estimation is used to obtain theestimate h. The fading channel model is expressed as:

hk,l =√

1 − α hk,l +√α hk,l, (4)

where hk,l is the the fading channel gain, hk,l represents theestimate of hk,l and hk,l refers to the estimation error indepen-dent of hk,l and whose entries are assumed to be CN(0,1). Theparameter α is the estimation error variance and has a fixedvalue between 0 and 1. We also specify that the estimate hk,l

is known by both the transmitter and the receiver. We assumeit is CN(0,1) so that |hk,l|

2 would be exponentially distributedwith unit mean. We consider an SINR capture model. Thatis, a communication is established and the message can besuccessfully decoded at the receiver if and only if the SINRat the receiver is greater than a certain threshold β. Obviously,this requires specific encoding and decoding strategies that weare not discussing here for brevity.

C. SINR and Performance YardsticksDue to the channel model along with the random nodes

locations, the transmit powers and the SINRs experienced bythe receivers are also random. Transmit powers depend onthe PC approach and are discussed in sections III and IV. Todefine our performance yardsticks, we need first to establishSINRs’ expressions as function of transmit powers, SINRthreshold and estimation error variance. We start by expressingthe expected received signal power at the BS.

Lemma 1: The conditionally expected received power atthe BS when only an estimate h = [h0,0, ..., h0,K] and thedistribution of estimation error hk,l (∼ CN(0, 1)) are available,is as follows:

E[y0y∗0 |h

]= (1 − α)|h0,0|

2d−δ0,0 p0 + αd−δ0,0 p0

+∑K

k=1[(1 − α)|h0,k |

2 + α]d−δ0,k pk + σ2.

(5)

Proof: Please see Appendix A.Note that the case α = 0 corresponds the to perfect channelknowledge and then an exact expected power E

[y0y∗0

]=

|h0,0|2d−δ0,0 p0 +

∑Kk=1 |h0,k |

2d−δ0,k pk + σ2. In case α = 1, thereis no CSI and the receiver must decode non-coherently. Theexpected power would be E

[y0y∗0

]= d−δ0,0 p0 +

∑Kk=1 d−δ0,k pk +σ2,

which depends no more on h but solely on α and the pathloss.Now, using Lemma 1, the conditionally expected cellularSINR in the proposed network model with imperfect CSI is:

SINRc(K, p, α) =(1 − α)|h0,0|

2d−δ0,0 p0

αd−δ0,0 p0 +∑K

k=1[(1 − α)|h0,k |

2 + α]d−δ0,k pk + σ2

=u0,0 p0

v0,0 p0 +∑K

m=1 w0,m pm + σ2.

(6)where uk,l =

[(1−α)|hk,l|

2]d−δk,l , vk,l = αd−δk,l ; wk,l = uk,l + vk,l andp = [p0, p1, ...pK−1, pK] is the vector of transmit powers i.e., pi

denotes the transmit power of transmitter i. The interpretationof SINR in (6) is that the receiver decodes the interferenceand the noisy received signal as a background additive whiteGaussian noise (AWGN). The rate achieved by the cellularuser is given by Rc(K,p, α) = log

[1+SINRc

]. Similarly, SINR

expression of the D2D link k could be computed exactly likethe cellular uplink case:

SINRk(K, p, α) =uk,k pk

vk,k pk + w0,0 p0 +∑K

m=1 wk,m pm + σ2. (7)

Based on the SINR, we are using two metrics which arethe coverage probability and the sum rate to evaluate perfor-mances. Precisely, the PC algorithms aim to maximize thosequantities while maintaining a minimum level of QoS (SINRthreshold). We define the coverage probabilities for cellularand D2D users and the sum rate of D2D links as follows:

Pccov = E

[P(SINRc(K, p, α) ≥ β0)

], (8)

PDcov = E

[P(SINRkK, p, α) ≥ βk)

], (9)

RD2D = E[∑K

k=1 log (1 + SINRk(K, p, α))], (10)

where β0 and βk represent the minimum SINR value forreliable uplink and D2D connections, respectively. The expec-tations above are with respect to locations of different usersand their transmit powers.

III. Centralized Power Control

The centralized power control is proposed when globalimperfect CSI (4) is available at the BS. Its objective is to findoptimal transmit power profile pi , i ∈ [0,K], that maximizesthe SINR, and thus the achievable rate, of the cellular link.This is done while a required QoS is guaranteed for cellularand D2D links. The problem can be formulated as follows:

maxp0,p1,...,pk

SINRc(K, p, α)

subject to SINRc(K, p, α) ≥ β0

SINRk(K, p, α) ≥ βk, k = 1, . . . ,K0 ≤ p0 ≤ Pmax,c

0 ≤ pk ≤ Pmax,d

(11)

We can write this compactly in a vector form:

maxp

Gu p

Gi p + σ2

subject to (I −Q)p ≥ d

0 ≤ p ≤ pmax

(12)

where Gu = [u0,0, 0, ..., 0]T ; Gi = [v0,0,w0,1,w0,2, ...,w0,K]T

d = [ σ2β0u0,0−β0v0,0

, ..., σ2βKuK,K−βK vK,K

]T ; pmax = [Pmax,...,Pmax ]and Q

is defined by Qk,l =wk,lβk

uk,k−βkvk,kif k , l and Qk,k = 0 for

k, l ∈ [0,K]2.Problem (12) can be solved using standard optimization

tools since the objective function is linear-fractional andhence quasi-convex (or it can even be transformed into linearproblem [11]) and the constraint set is convex [10]. Hence,the optimal solution exists if the feasible set is non empty. Anecessary and sufficient condition for that is provided in [10]:the spectral radius of the matrix Q should be less than one i.e.,ρ(Q) ≤ 1. Due to randomness of transmitters’ locations, thecondition may be unsatisfied and the PC problem unfeasible.To remedy this without doing an exhaustive complex search,a simple algorithm was proposed in [6]: It consists in shuttingdown the kth D2D transmitter that creates the maximumsum of interference power to all other receivers. This meanseliminating the kth row and column of Q with the highest

euclidean norm ||.||2 and then reducing the size of Q. Thesame process is repeated until satisfying ρ(Q) ≤ 1.

To summarize, the algorithm assumes that all nodes feed-back the estimated CSI and required threshold to the BS. Thelatter computes all users transmit powers according to theoptimal solution of (12). For that purpose, it can reduce thenumber of D2D links as much as needed to assure feasibility.The BS sends then the optimal power policy to all users.

IV. Distributed Power ControlIn this section, we compute cellular and D2D coverage

probabilities in the distributed scenario. We present then theoptimal cellular PC under the peak and average transmitpower constraint. For D2D users, we propose two distributedPC algorithms: on-off and truncated channel inversion PCs.Since we are utilizing stochastic geometry, we assume that thetransmit powers of D2D transmitters are i.i.d. The coverageprobability expressions provided in this section are valid forboth proposed distributed PC methods (or any distributed oneswhere the user decides its power based solely on its channelknowledge without considering other transmitters).

A. Coverage Probabilities

1) Cellular Link Coverage Probability: To compute thecoverage probability for the cellular user in the proposednetwork configuration, we first establish the Laplace transformL(·) of the interference, Id, created by D2D users on uplinkcellular user in the following lemma:

Lemma 2: The Laplace transform of Id is given by:

LId (t) = exp[−λπt

2δ Epk

[p

2δ

k

](1 − α)

2δ Ω(α, δ)

](13)

where Ω(α, δ) = eα

1−α Γ(1 + 2δ, α

1−α )Γ(1 − 2δ), Γ(a) is the gamma

function at the point a and Γ(a, x) is the upper incompletegamma function defined as Γ(a, x) =

∫ ∞x ta−1e−y dy

Proof: Please see Appendix B.Now, using the previous lemma, we can provide an analyt-

ical formula for the uplink coverage probability.Proposition 1: The coverage probability of the cellular user

in the proposed network and channel model is given by:

Pccov(β, α) = e−

αβ1−α EX

[e−c1X−c2X

2δ], (14)

where :

X = dδ0,0 p−1

0c1 = βσ2(1 − α)−1

c2 = λπβ2δ Epk

[p

2δ

k

]Ω(α, δ)

Proof: Please see Appendix C.Proposition 1 shows how network parameters affect the cel-lular link coverage probability: Pc

cov decreases when λ andEpk

[p

2δ

k]

increase since it means more interference generatedby D2D links. Estimation error affects all its terms and thusreduces the coverage considerably as α increases. Note thatin interference limited regime (σ2 = 0), for fixed distancebetween D2D pair (dk,k = d) and constant power for cellularand D2D users (p0 = Pmax,c, pk = Pmax,d), Pc

cov reduces to:

Pccov(β, α) = exp

(−λπβ

2δ Epk

[p

2δ

k

]p− 2δ

k Ω(α, δ)d2k,k

)

× e−αβ

1−α Edk,0

exp

− β

1 − αp0

pk

[(1 − α)|hk,0|

2 + α] (dk,k

dk,0

)δ

= e−αβ

1−α exp(−λπβ

2δ Ω(α, δ)d2

k,k

)Edk,0

[ 1

1 + β p0pk

( dk,k

dk,0

)δ ]× Edk,0

exp

−(βα

1 − α)

p0

pk

(dk,k

dk,0

)δ . (15)

The last equality follows from the fact that |hk,0|2 ∼ exp(1)

with a moment generating function equal to 11−t for t < 1.

2) Optimal Power Strategy for the Cellular User whenUplink Distance is Known: Conditioning on d0,0, the cov-erage probability for a given transmit power p0, reduces to

Pccov(p0) = e−

αβ1−α e−c1dδp−1

0 −c2d2 p2δ

0 . Under the average and peakpower constraints of the uplink transmission power p0, andfollowing the same methodology as [6], the optimal strategyfor the uplink cellular user would be the on-off PC. Theproof is similar to Theorem 2 in [6]. The cellular transmitpower p0 that maximizes the coverage probability for a given

uplink distance d, is: p0(d) =(

2δλπβ

2δ Epk

[p

2δ

k

]Ω(α, δ)d2

) δ2.

Thus, under the average and peak power constraints (Pavg,c ≤

Pmax,c), the cellular user will be transmitting with powerp?0 = max(min( p0, Pmax,c), Pavg,c) with probability Pc

cov =Pavg,c

p∗0and remaining silent (i.e., p0 = 0) otherwise. In interferencelimited regime and using the optimal binary PC, the cellularuser coverage probability, for an uplink distance d, reduces to:

Pccov = Pc

cov(d, β, α) = e−αβ

1−α Ep0

[exp

(−c2d2 p

− 2δ

0

) ]=

Pavg,c

p?0 (d)e−

αβ1−α exp

(−c2d2 p?0 (d)−

2δ

).

(16)

So the cellular user coverage probability becomes as follows:

Pccov =

Pavg,c

Pmax,ce−

αβ1−α exp

[−E[K]β

2δ Epk

[p

2δ

k

]Ω(α, δ)[ d

R ]2P− 2δ

max,c

],

if p0(d) ≥ Pmax,cPavg,ce−β

α1−α exp(− δ

2 )[2δ Epk

[p

2δ

k

]Ω(α,δ)

] δ2

E[K]δ2 β[ d

R ]δ, if Pavg,c < p0(d) < Pmax,c

if p0(d) ≤ Pavg,c

e−αβ

1−α exp[− E[K]β

2δ Epk

[p

2δ

k

]Ω(α, δ)[ d

R ]2P− 2δ

avg,c

](17)

3) D2D Users Coverage Probability:Proposition 2: The coverage probability of a D2D user in

the proposed network and channel model is given by:

PDcov(β) = e−

αβ1−α EY

[e−c3Y−c4Y

2δ], (18)

where :

Y = dδk,k p−1

k

c3 =β

1−α

[σ2 +

[(1 − α)|hk,0|

2 + α]d−δk,0 p0

]c4 = λπβ

2δ Epk

[p

2δ

k

]Ω(α, δ)

Proof: Similar to Proposition 1.

B. On-off Power Control

On-off power control is a simple yet effective method tointerference mitigation when there is no coordination betweennodes. Each transmitter has to decide individually whether totransmit or not using solely the direct channel condition tohis receiver. The link is activated i.e., pk = Pmax,d, only ifthe channel quality is good enough: the useful part of theconditional expected power (1−α)|hk,k |

2d−δk,k ≥ βmin where βmin

is a fixed quality threshold known by all users. The probabilityof activation is Ps = P

[(1 − α)|hk,k |

2d−δk,k ≥ βmin]

and theaverage transmit power of D2D link would be pk = Ps Pmax,d.Therefore, βmin(Ps) plays a decisive role in determining theoverall (sum rate) performance of the D2D links: When βis small, the systems allows more D2D users to be active.However this will also increase inter D2D interference. Onthe other hand, when β is large, less D2D users are active, butthere are suffering less inter D2D interference.

1) Sum Rate of D2D Links: We assume an interferencelimited regime (σ2 = 0), on-off PC for D2D links and constantpower for the cellular uplink user (Pmax,c). It is recalled thatD2D communications are done using pk = Pmax,d with proba-bility Ps, so the number of active D2D links is Nact = λPsπR2.We suppose also Gaussian signal transmission from all activelinks, so that the interference distribution becomes Gaussiantoo. The achievable sum rate of all D2D links would be:

RD = E

K∑k=1

log (1 + SIRk)

= Nact E[log(1 + SIRk)

]= λPsπR2 × RD2D

(19)

where RD2D is the average rate of a single D2D link. Now thecoverage probability established in (15) is still averaged withrespect to dk,0. To obtain closed form, we can approximate

it using E[

11+

µ

dδk,0

]' 1

1+µ2/δ

E[dk,0]2

and E[e−

µ

dδk,0]' e

−µ2/δ

E[dk,0]2 which

are obtained from numerical observations [6]. Knowing thatthe first moment of dk,0, E[dk,0] = 128R

45π [9], the approximatedexpression would be as follows:

PDcov(β, α) ' e−

αβ1−α exp(−λπβ

2δ Ω(α, δ)d2

k,k)

×1

1 + [ βp0pk

]2/δ d2k,k

(128R/(45π))2

e−[ αβp0(1−α)pk

]2/δd2k,k

(128R/(45π))2 .(20)

The rate RD2D can be expressed then as follows:

RD2D =

∫ ∞

0log(1 + β)

∂

∂β

[P[SIRk ≥ β]

]dβ =

∫ ∞

0

PDcov(β)

1 + βdβ

'

∫ ∞

0

11 + β

e−αx

1−α−ΛPsβ2δ︸ ︷︷ ︸

A

11 + κβ2/δ e−κ(

αβ1−α )2/δ

︸ ︷︷ ︸B

dβ, (21)

where κ = ( Pmax,c

Pmax,d)

2δ ( dk,k

128R/(45π) )2 and Λ = λπΩ(α, δ)d2

k,k. We cannote that the approximated expression in (21) is determined bytwo factors: (1) the Laplace transform of the total interferencepower created by all D2D active links (A) and (2) the approx-imated effect of the uplink interference (B), both affected bythe estimation error α.

2) Optimizing the On-off Threshold: We can optimize theD2D on-off threshold by maximizing the transmission rate[6] defined as T D

r (β) = λPsπR2 log(1 + β) ∂∂β

(P[SIRk ≥ β]).The maximizer is Popt

s = min(Λ−1β−2δ , 1). Given the activation

probability expression and since |hk,k |2 ∼ exp(1), we obtain the

optimal threshold βoptmin = (1 − α)d−δk,k ln(1/Popt

s ). Like Popts , βopt

minhas two expressions depending on β. Now integrating T D

r withrespect to β provides the sum rate of D2D links.

Proposition 3: The average D2D sum rate is given by:

RD(β) '∫ Λ

− δ2

0

λπR2e−αx

1−α exp(−Λβ2δ )e−κ(

α1−α )2/δx

2δ

(1 + κx2δ )(1 + x)

dx (22)

+

∫ ∞

Λ− δ2

e−α(x+1)

1−α −κ(α

1−α )2/δx2/δ−1

Γ(1 + 2δ, α

1−α )Γ(1 − 2δ)(1 + x)(1 + κx2/δ)

(R

dk,k)2x−

2δ dx.

Proof: Please see Appendix D.C. Truncated Channel Inversion

Under the maximum transmit power (Pmax,d) constraint, theD2D transmitters use a truncated channel inversion PC: Thetransmit power compensates the path-loss to keep the averagesignal power at the receiver equal to a certain threshold ρ0.The latter should be higher than the the receiver sensitivityρmin, (ρ0 ≥ ρmin [ [12], chapter 4]). Therefore, a connectionis established only if the transmit power required for thepath-loss inversion is less than or equal to Pmax,d. Otherwise,the D2D user does not transmit and remains idle due to theinsufficient transmit power. Another condition for the D2Dtransmitter to be active is that the link quality is good, in thesense that (1 − α)|hk,k |

2d−δk,k ≥ T where T is a nonnegativequality threshold that is fixed and known by all users.

We assume that each D2D transmitter has its unique re-ceiver uniformly distributed within its proximity Rmax. It isdetermined by Pmax,d and ρmin, such as Rmax = ( Pmax,d

ρmin)

1δ . But,

due to the truncated channel inversion PC, the D2D proximityis reduced to Rd = ( Pmax,d

ρ0)

1δ . That is, the pdf of the D2D link

distance is given by fdk,k(d) = d2

R2d, 0 ≤ d ≤ Rd. The intensity of

D2D links reduces also to λ = ηλ where µ = ( RdRmax

)12 = ( ρmin

ρ0)

2δ ,

µ is the power coverage probabilityDue to the assumed PC along with the random locations of

D2D users, the transmit powers and the SINRs experiencedby the receivers are random also. First, we characterize thetransmit power via its pdf and its ηth moment (η ≥ 0). Then,we characterize the SINR and the coverage probability.

1) Transmit Power Analysis: For the D2D transmitter tobe active, two conditions are required: a maximum powerconstraint and a minimum quality requirement. Thus, thetransmit power pk can written as pk = ρ0dδk,k1(1−α)|hk,k |

2d−δk,k≥T ,

and can be characterized by the following proposition:Proposition 4: In the proposed network model and using

the truncated channel inversion power control for D2D userswith cutoff threshold ρ0 and quality threshold T , the pdf ofthe transmit power of a D2D user is given by:

fpk (x) =T

2δ x

2δ−1 e

−T

(1−α)ρ0x

ρ2δ

0 (1 − α)2δ γ( 2

δ,TPmax,d

(1−α)ρ0). (23)

where γ(·, ·) is the lower incomplete gamma function. Further-more, the transmit power moments can be obtained as:

E[pηk] =ρη0 (1 − α)η γ( 2

δ+ η,

TPmax,d

(1−α)ρ0)

T η γ( 2δ,TPmax,d

(1−α)ρ0)

. (24)

Proof: Please see Appendix E.

D. SINR and Coverage Probability

We assume now that the uplink cellular uses constant powerp0 = Pmax,c.

Proposition 5: The coverage probability of D2D users is:

PDcov,tr(β, α) = e−

βρ0(1−α) (ρ0α+σ2)

LId (β

(1 − α)ρ0)LIc (

β

(1 − α)ρ0),

(25)

where :

LId ( β(1−α)ρ0

) = exp[− λπ(βρ−1

0 )2δ

×ρ2/δ

0 (1 − α)2/δγ( 4δ,TPmax,d

(1−α)ρ0)

T 2/δ γ( 2δ,TPmax,d

(1−α)ρ0)

Ω(α, δ)]

LIc (β

(1−α)ρ0) = E

[exp(−βρ−1

0 |hk,k |2d−δk,0 p0)

× exp(−βα(1 − α)−1d−δk,0 p0)]

'

(1 +

(βp0)2/δ

ρ2/δ0 (128R/(45π))2

)−1exp

(− (αβp0)2/δ

((1−α)ρ0)2/δ(128R/(45π))2

)Proof: Please see Appendix F.

To validate our analysis, we compare the analytical resultof PD

cov,tr with that obtained through Monte Carlo simulation.It can be seen in Fig.6 that resuts do match which shows thatthe obtained model captures well the coverage probability.

V. Numerical Results

In the simulation setup for the centralized PC, we considera BS positioned in the center of the coverage area (with radiusR=500m) where a cellular user is uniformly located. The D2Dtransmitters are distributed according to a PPP with densityλ = 0.00002 to which corresponds mean number of D2D linksE[K] = 15. Each D2D receiver is located at a fixed distanced=50m from its transmitter. We assume that the maximumtransmit powers for cellular and D2D communications arePmax,c = 0.1W and Pmax,d = 0.1mW, respectively. This isdue to the fact that D2D links are supposed to be of a shortrange, compared to cellular ones. The effect of estimationerror on coverage probabilities is illustrated by Fig.2 andFig.3. As shown in Fig.2, the centralized PC is quite sensitiveto estimation error and coverage chances gets lower whenthe estimation error variance increases. For example, for atarget SINR of 3 dB, the cellular coverage drops from 95%in the perfect case to 78% and 14% when α is equal to0.1 and 0.5, respectively. The cellular can not get coveredanymore when α gets close to 1. As shown by Fig. 5, D2Dcoverage is also negatively affected and decreases when αincreases. This shows that the central controller (BS) cannot get good decisions when the CSI is very noisy and theoverall performance gets worse when error variance increases:imperfect CSI and estimation error should be considered when

β (dB)-15 -10 -5 0 5 10 15

Cel

lula

r co

vera

ge p

roba

bilit

y

0

0.2

0.4

0.6

0.8

1

α=0α=0.1α=0.5α=0.9

Figure 2: Cellular coverage probability versus the thresholdβ in centralized PC for different values of estimation error α

β (dB)-15 -10 -5 0 5 10 15

Cov

erag

e pr

obab

ility

of D

2D u

sers

0

0.2

0.4

0.6

0.8

1

α=0α=0.1α=0.5α=0.9

Figure 3: D2D coverage probability versus the threshold β incentralized PC for different values of estimation error α

designing PC algorithms. Now we evaluate performances forthe on-off PC for D2D users and for the cellular and comparethem to those obtained when no PC is applied. Simulationparameters are similar to those in the centralized part. Fig.4and Fig.5 provide comparison for the on-off PC with no PC forcellular and D2D links, respectively. In the first figure K=39and in the second both cases of K=15 and K=39 are displayed.It can be seen in these figures that the on-off PC yieldsperformance gains for both cellular and D2D links comparedto that of no PC case when the target SINR is larger than 3dB (for the K=39 case) when perfect CSI is available (α = 0).This is because the on-off PC acts like the no PC case forβ ≤ 3dB, since the optimal activation probability (in IV-B2)Popt

s = min( 1

Λβ2δ, 1) = 1; while it is activated when β ≥ 3dB.

For example, for K=39 at β = 9dB, performance gain for thecellular user is about 50% and 15% for D2D users. This gainsis considerably affected by the estimation error: it is reducedto 30% when α = 0.1 and there is almost no gain when αincreases more. This shows that the on-off PC is sensitiveto CSI estimation error, since it relies on this information tomake decision, the more this information is noisy the moreperformance tends to the no PC case.

Now for the truncated channel inversion model, we firstvalidate our model by simulations and then present somenumerical results for the following parameters: We set the

β (dB)-15 -10 -5 0 5 10 15

Cel

lula

r co

vera

ge p

roba

bilit

y

0

0.2

0.4

0.6

0.8

1α=0 with On-Off PCα=0 with no PCα=0.1 with On-Off PCα=0.1 with no PCα=0.5 with On-Off PCα=0.5 with no PC

Figure 4: Cellular coverage probability versus the thresholdβ for both on-off PC and no PC for K=39

β (dB)-15 -10 -5 0 5 10 15

D2D

cov

erag

e pr

obab

ility

0

0.2

0.4

0.6

0.8

1α=0 with on-off PCα=0 with no PCα=0.1 with on-off PCα=0.1 with no PCα=0.5 with on-off PCα=0.5 with no PC

Figure 5: D2D coverage versus the threshold β for on-off PCand no PC for K = 15 (dashed line) and K = 39 (plain line)

maximum cellular and D2D transmit powers to Pmax,c = 0.2 Wand Pmax,d = 0.1 mW respectively, the cutoff threshold ρ0 =

−80 dBm, the receiver sensitivity ρmin = −100 dBm the qualitythreshold T = −50 dBm, the path-loss exponent δ = 4 and theSINR threshold β ranging from -18 to 18 dB. The followingfigure provides D2D coverage probability when the truncatedchannel inversion PC is applied for D2D links (λ = 0.00002)while the cellular user transmit with constant power Pmax,d.Fig. 6 shows that truncated channel inversion PC providesthe same overall response for D2D links: their coverageprobability decreases when SINR requirements increase (dueto power constraint) and the estimation error reduces efficiencyand degrades considerably the coverage probability. Resultsare close to those obtained for the on-off PC, but the advantagehere is the ability to adapt signal power to the channel with anadditional parameter ρ0 which introduces an important trade-off between power constraints and SINR coverage.

VI. Conclusion

In the present paper, we proposed a single cell model for aD2D underlay cellular system: random aspect of the networkwas expressed based on stochastic geometry and imperfectCSI was taken into account. Results for the centralized PCproved that it does improve the network performance sinceit continues to protect the cellular communication while sup-porting additional underlaid D2D links. We observed also thatchannel estimation error degrades the algorithm performance

β (dB)-15 -10 -5 0 5 10 15

D2D

cov

erag

e pr

obab

ility

0

0.2

0.4

0.6

0.8

1α=0α=0.1α=0.5α=0.9Simulation

Figure 6: D2D coverage probability versus the threshold βusing the truncated channel inversion PC for K=15

and reduces considerably its efficiency. The distributed ap-proach, on the other hand, has the merit of relying only onlocal CSI to decide the transmit power by the user itself.Results showed also that it presents overall network gains butit fails to guarantee reliable cellular communication whosecoverage decreases noticeably compared to the centralizedcase. Again here, imperfect CSI and misinformation lead tothe degradation of performance especially for high target SINRwhere coverage becomes virtually equal to zero. This showsthat estimation error is a key parameter that should be takeninto account during network design.

Appendix AProof of Lemma 1

E[y0y∗0 |h

]= E

[|S |2 + |S |2 + |I|2 + |n0|

2 + 2Re(S I∗) + 2Re(S n∗0)

+ 2Re(In∗0) + 2Re(S n∗0) + 2Re(S S ∗) + 2Re(IS ∗) |h],

where S =√

1 − α ˆh0,0d−δ2

0,0 s0; S =√α ˜h0,0d−

δ2

0,0 s0 and I =∑Kk=1 h0,kd−

δ2

0,k sk. Below the expectation of each of the terms:(1) E

[|S |2 |h

]=

[(1 − α)|h0,0|

2]d−δ0,0 p0.

(2) E[|S |2 |h

]=

[αE |h0,0|

2]d−δ0,0 p0 = αd−δ0,0 p0.

(3) E[|I|2 |h

]= E

[|

K∑k=1

h0,kd−δ2

k,0 sk |2 |h

]=

K∑k=1

E[|h0,k sk |

2d−δ0,k |h]

+ 2K−1∑k=1

K∑i=k+1

d−δ2

0,i d−δ2

0,k Re(E

[h0,k sk s∗i h∗0,i|h

])=

K∑k=1

E[(

(1 − α)|h0,k |2d−δ0,k + α|h0,k |

2d−δ0,k

+ 2√α(1 − α)d−δ0,kRe(h0,kh∗0,k)

)|sk |

2 |h]

+ 2K−1∑k=1

K∑i=k+1

d−δ2

0,i d−δ2

0,k Re(E[h0,k] E[sk]︸︷︷︸

= 0

E[s∗i ]︸︷︷︸= 0

E[h∗0,i|h])

=

K∑k=1

(1 − α)|h0,k |2d−δ0,k |sk |

2 + α|sk |2 E |h0,k |

2︸ ︷︷ ︸= 1

d−δ0,k

+ 2√α(1 − α)h0,kd−δ0,kRe(E [h∗0,k]); Re(E [h∗0,k]) = 0

=

K∑k=1

[(1 − α)|h0,k |

2 + α]d−δ0,k pk,

since we assume, without loss of generality, that si and h areindependent and that si are with zero mean.(4) E

[|n0|

2 |h]

= σ2.The rest of terms are equal to zero each and the sum givesthe result stated in (5).

Appendix BProof of lemma 2

LId (t) = E[e−tId ]

= EΦ,pk ,h0,k

[exp(−t

K∑k=1

[(1 − α)|h0,k |

2 + α]d−δ0,k pk)

](1)= exp

[λ

∫ +∞

0

∫ 2π

0Epk ,h0,k

(e−t[(1−α)|h0,k |

2+α]pkr−δ − 1)dφrdr

]= exp

[− 2πλ

∫ +∞

0Epk ,h0,k

(1 − e−t[(1−α)|h0,k |

2+α]pkr−δ)rdr

](2)= exp

[− λπt

2δ Epk (p

2δ

k ) Eh0,k

([(1 − α)|h0,k |

2 + α]2δ

)Γ(1 −

2δ

)]

(3)= exp

[− λπt

2δ Epk (p

2δ

k )(1 − α)2δ e

α1−αΓ(1 +

2δ,

α

1 − α)Γ(1 −

2δ

)]

where (1) follows from Campbell’s theorem for PPPs (See p.78[8]). Equalities (2) and (3) are detailed below.

(2) I =

∫ +∞

0Epk ,h0,k

(1 − e−t[(1−α)|h0,k |

2+α]pkr−δ)rdr

]=

∫ +∞

0Epk ,h0,k

(1 − e−tHpkr−δ

)rdr

]= −

1δ

∫ +∞

0

[1 − E(e−y) E

((tHpk)

2δ)]

y−1− 2δ dy

= −1δ

t2δ Epk (p

2δ

k ) Eh(H2δ )−δ

2

[1 − E((1 − e−y)y

2δ )]+∞

0

−δ

2E

∫ +∞

0e−yy−

2δ dy

=

12

t2δ Epk (p

2δ

k ) Eh(H2δ )Γ(1 −

2δ

)

=12

t2δ Epk (p

2δ

k ) Eh([(1 − α)|h0,k |2 + α]

2δ )Γ(1 −

2δ

)

(3) h ∼ E(1) : fh(x) = e−x 1(x≥0)

Y := (1 − α)h + α⇒ fY (y) =1

1 − αe−

y−α1−α 1(y≥α)

Eh(Y2δ ) =

∫ +∞

α

y2δ fY (y)dy =

eα

1−α

1 − α

∫ +∞

α

y2δ e−

y1−α dy

= eα

1−α (1 − α)2δ

∫ +∞

α1−α

u2δ e−udu

= eα

1−α (1 − α)2δ Γ(1 +

2δ,

α

1 − α).

Appendix CProof of Proposition 1

Pccov(β, α) = P(SINRc ≥ β)

= P( [

(1 − α)|h0,0|2]d−δ0,0 p0

αd−δ0,0 p0 +∑K

k=1[(1 − α)|h0,k |

2 + α]d−δ0,k pk + σ2

≥ β)

= P(|h0,0|

2 ≥βα

1 − α+

+βdδ0,0

(1 − α)p0

[ K∑k=1

[(1 − α)|h0,k |

2 + α]d−δ0,k pk + σ2

])= E

[exp(−

αβ

1 − α) exp(−

dδ0,0σ2β

(1 − α)p0)

× exp(−dδ0,0β

(1 − α)p0

K∑k=1

[(1 − α)|h0,k |

2 + α]d−δ0,k pk

]= e−

αβ1−α EX

exp(−

dδ0,0σ2β

(1 − α)p0) Epk ,h0,k

[exp(−

dδ0,0β

(1 − α)p0

×

K∑k=1

[(1 − α)|h0,k |

2 + α]d−δ0,k pk)

]= e−

αβ1−α EX

exp

− dδ0,0σ2β

(1 − α)p0

LId

dδ0,0β

(1 − α)p0

= e−

αβ1−α EX

e−

dδ0,0σ2β

(1−α)p0 e−λπt2δ Epk (p

2δ

k )(1−α)2δ e

α1−α Γ(1+ 2

δ ,α

1−α )Γ(1− 2δ )

= e−αβ

1−α EX

[e−c1X−c2X

2δ].

Appendix DProof of Proposition 3

The transmission rate T Dr is defined as:

T Dc (β) = λPsπR2 log(1 + β)

∂

∂β(P[SIRk ≥ β])

' λPsπR2e−αx

1−α exp(−ΛPsβ2δ )

log(1 + β)1 + κβ2/δ e−κ(

α1−α )2/δβ2/δ

.

Rather than optimizing over the sum rate, we do it over thetransmission capacity (whose integral with respect to β is thesum rate). Thus, the optimization problem becomes:

maxPs

T Dr (β)

subject to 0 ≤ Ps ≤ 1(26)

The objective function is not concave but we still can getoptimal Ps that maximizes the transmission capacity by usingthe first order optimality condition (since the objective functionhas a unique optimum point):

∂T Dr

∂Ps= 0⇔ 1 − Λβ

2δ Ps = 0 (27)

So we have Popts = min( 1

Λβ2δ, 1) and since |hk,k |

2 ∼ exp(1), we

obtain the optimal threshold:

βoptmin = (1 − α)

ln(1/Popts )

dδk,k. (28)

It follows that the approximated transmission capacity can bere-expressed, depending on β compared to Λ−

δ2 as:

T Dr (β) '

λπR2e−

αβ1−α exp(−Λβ

2δ ) log2(1+β)

1+κβ2/δ e−κ(α

1−α )2/δβ2/δ

if β ≤ Λ−δ2

e−α(β+1)

1−α −κ( α1−α )2/δβ2/δ−1

Γ(1+ 2δ ,

α1−α )Γ(1− 2

δ )(1+κβ2/δ)log2(1 + β)( R

dk,k)2β−

2δ

if β > Λ−δ2 .

(29)

Integrating T Dr with respect to β provides the sum rate of D2D

links.

Appendix EProof of Proposition 4

Let Xd = ρ0dδk,k denote the unconditional transmit powerrequired to invert the channel for the D2D link. We have

fdk,k(d) =

d2

R2d

, 0 ≤ d ≤ Rd and we easily prove that

fXd(x) = 2x

2δ −1

δρ2δ

0 R2d

= 2x2δ −1

δP2δ

max,d

, 0 ≤ d ≤ Pmax,d.

Now the transmit power pk = ρ0dδk,k1(1−α)|hk,k |2d−δk,k≥T

=

ρ0dδk,k if ρ0dδk,k ≤(1−α)ρ0T|hk,k |

2 = Xd if Xd ≤ X0 whereX0 =

(1−α)ρ0T|hk,k |

2.We can now express the pdf of pk as follows:

fpk(x) =

∫ ∞

x

fXd |X0(x|y) P(Xd ≤ y)fX0

(y)P(Xd ≤ X0)

dy

Since |hk,k |2 ∼ E(1), we have fX0

(y) = T

(1−α)ρ0e−

T(1−α) y and

P(Xd ≤ X0) = P(|hk,k |2 ≤

dδk,kT

(1 − α)) = E[exp(−

dδk,kT

(1 − α))]

=

∫ Rd

0e−

rδT(1−α) fdk,k

(r)dr =

∫ Rd

0e−

rδT(1−α)

2rR2

d

dr

=2δ

(1 − αTRδ

d

)2δ

∫ TRδd1−α

0y

2δ−1e−ydr =

2δ

(1 − αTRδ

d

)2δ δ(

2δ,TRδ

d

1 − α)

We get back to the pdf of pk:

fpk(x) =

∫ ∞

x

2x2δ−1

δP2δ

max,d

δ

2(TRδ

d

1 − α)

2γ

1

γ( 2δ,TRδd1−α )

T

(1 − α)ρ0e−

T(1−α) y dy

=T

2δ x

2δ−1

ρ2δ

0 (1 − α)2δ γ( 2

δ,TPmax,d

(1−α)ρ0)

∫ ∞

T(1−α)ρ0

xe−z dz

=T

2δ x

2δ−1 e

−T

(1−α)ρ0x

ρ2δ

0 (1 − α)2δ γ( 2

δ,TPmax,d

(1−α)ρ0)

The ηth moment is obtained by computing∫ Pmax,d

0 xηfpk(x) dx.

Appendix FProof of Proposition 5

Pcov(β, α) = P (SINRd ≥ β)

= P(|hk,k |

2 ≥αβ

1 − α+

β

ρ0(1 − α)

×

K∑j=1

Wk, jd−δk, j p j + Wk,0d−δk,0 p0 + σ2

)= e−

β(ρ0α+σ2)ρ0(1−α) E

[exp

− β

(1 − α)ρ0

K∑j=1

Wk, jd−δk, j p j

× exp

(−

β

(1 − α)ρ0Wk,0d−δk,0 p0

) ]= e−

βρ0(1−α) (ρ0α+σ2)

LId (β

(1 − α)ρ0) LIc (

β

(1 − α)ρ0)

LId is the Laplace transform of the interference Id created bythe rest of D2D users and can be deduced from Lemma 2 at

β(1−α)ρ0

. Then, Epk [p2δ

k ] is replaced by its expression establishedin Proposition 4. We obtain the result stated in Proposition 5.

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