Power Control for D2D Underlay Cellular Networks With ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 2017

Power Control for D2D Underlay Cellular Networks With Channel Uncertainty Amen Memmi, Student Member, IEEE, Zouheir Rezki, Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE

Abstract— Device-to-device (D2D) communications underlying the cellular infrastructure are a technology that have been proposed recently as a promising solution to enhance cellular network capabilities. It improves spectrum utilization, overall throughput, and energy efficiency while enabling new peerto-peer and location-based applications and services. However, interference is the major challenge, since the same resources are shared by both systems. Therefore, interference management techniques are required to keep the interference under control. In this paper, in order to mitigate interference, we consider centralized and distributed power control algorithms in a onecell random network model. Existing results on D2D underlay networks assume perfect channel state information (CSI). This assumption is usually unrealistic in practice due to the dynamic nature of wireless channels. Thus, it is of great interest to study and evaluate achievable performances under channel uncertainty. Differently from previous works, we are assuming that the CSI may be imperfect and include estimation errors. In the centralized approach, we derive the optimal powers that maximize the coverage probability and the rate of the cellular user while scheduling as many D2D links as possible. These powers are computed at the base station (BS) and then delivered to the users, and hence the name “centralized". For the distributed method, the ON–OFF power control and the truncated channel inversion are proposed. Expressions of coverage probabilities are established in the function of D2D links intensity, pathloss exponent, and estimation error variance. Results show the important influence of CSI error on achievable performances and thus how crucial it is to consider it while designing networks and evaluating performances. Index Terms— Device-to-device communications, imperfect channel state information, Poisson point process, power control.

I. I NTRODUCTION

T

O MEET the ever increasing data demand, researchers from industry and academia are seeking for new paradigms to revolutionize the traditional communication methods of cellular networks. Device-to-Device (D2D) communications appear to be of those promising paradigms expected to become

Manuscript received January 7, 2016; revised June 1, 2016 and October 13, 2016; accepted December 9, 2016. Date of publication December 26, 2016; date of current version February 9, 2017. The associate editor coordinating the review of this paper and approving it for publication was T. M. Lok. A. Memmi is with the Institut National de la Recherche Scientifique, Quebec City, QC G1K 9A9, Canada. Z. Rezki is with the University of Idaho, Moscow 83844, ID, USA. M.-S. Alouini is with the King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2016.2645210

a key feature supported by next-generation cellular networks. Propositions in this field mainly focused on sharing all the cellular spectrum between the two systems (cellular & D2D) in what is called D2D communications underlay cellular networks. This is for the purpose of maximizing the spectrum efficiency. However, this is challenging since D2D links may generate significant interference to the host network. Here comes the importance of interference management techniques such as power control (PC). PC is a simple, yet effective approach broadly used in current wireless networks to mitigate interference. In this work, we propose PC methods under imperfect CSI and analyze their performances in a random network model. A. Related Works Persuaded of the promising future and the expected gains offered by D2D communications underlay cellular networks, research efforts have been invested to analyze and optimize its operation. Several works focused on its main challenge: interference mitigation. Particularly, there has been considerable interest in PC techniques for D2D underlay cellular networks. Authors of [1], propose a centralized and dynamic PC mechanism in order to reduce interference generated by D2D communications while improving the cellular performance in Downlink (DL). The algorithm has two phases. First, BSs assign resources to D2D communications by reusing the same resources allocated to cellular users. Then, PC is applied for D2D communications to decrease interference they create on cellular users: This is done through the BS adjusting the transmit power of the D2D transmitter based on the channel information. However, it has been shown in [2] that uplink resources are preferable to downlink ones for D2D transmissions. This is mainly due to performance issues: In addition to regulation and hardware concerns, using the uplink would improve spectrum utilization since uplink resources are often less used than the downlink ones. Further, it minimizes D2D interference as it is handled by BSs which have more computational power than terminal users. This explains why most of literature considers sharing the cellular uplink with D2D communications. A simple PC scheme was proposed in [3] to regulate the transmit power of D2D users and protect the existing cellular links in a single-cell scenario and deterministic network model. The proposed algorithm fixes SINR constraints to tolerate quality degradation of cellular links until threshold levels are reached. A PC method for D2D communications was proposed in [4] to maximize the

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MEMMI et al.: POWER CONTROL FOR D2D UNDERLAY CELLULAR NETWORKS WITH CHANNEL UNCERTAINTY

network sum rate, also for a deterministic model of networks. Essentially, early works on PC for D2D [1], [3]–[7], mainly developed and evaluated PC strategies in deterministic D2D link deployment scenarios. For a more realistic representation of networks, random network models were then proposed. For example, spectrum sharing between “ad hoc” and “cellular networks” taking randomness into consideration was studied in [8] and [9]. In [10], authors consider that a cellular user needs to share uplink resources with multiple D2D links whose locations are random and modeled via stochastic geometry. Two forms of PC were proposed: centralized (managed by the BS) and distributed. In the centralized case, using perfect global Channel State Information (CSI), BS decides power profile based on optimizing the cellular signal to interference plus noise ratio under power and quality constraints for other users. In the distributed case, transmit powers of D2D users are set based on the knowledge of direct link information and the minimum channel gain threshold that is fixed and known by all users. In [11], stochastic geometry has been used more thoughtfully for modeling the random aspect of the network: BSs, cellular and D2D users were represented by Poisson Point Processes (PPPs). All these previous works considered perfect knowledge of the CSI, but in reality, obtaining the accurate CSI is usually difficult and causes high overhead. Recent works for D2D communications started to consider this issue. In [12], assuming partial CSI, authors suggest resources allocation and investigate the signaling overhead and performance tradeoff in D2D communications with channel uncertainty. The same assumption was considered in [13] to analyze achievable rate and outage probability of a time-division hybrid D2D-infrastructure underlay the cellular uplink. In practice, the CSI, especially that between D2D pairs, is often imperfect and the difficulty to acquire perfect knowledge is mainly due to channel estimation errors, mobility and feedback delay. We refer here to a close field, cognitive radio, where this issue has been elaborated further when dealing with the link between primary and secondary users. Authors of [14]–[16] proposed channel models to this imperfect CSI and analyzed the impact of channel estimation quality on the ergodic capacity and the performances of cognitive networks.In this paper, we analyze performance of some PC algorithms considering the impact of imperfect CSI in a random network model. B. Contribution of the Paper We propose some PC algorithms and analyze their performances in D2D underlay cellular networks. We take into account the estimation error that may affect the CSI. For this purpose, we consider a single cellular cell that underlay several D2D links. Specifically, a cellular user intends to communicate with the BS in uplink while several D2D communications are established using the same cellular spectrum. We take randomness of locations into consideration by modeling D2D transmitters’ positions through a spatial homogeneous PPP, using stochastic geometry theory. Differently from most of the previous works, we take into account the difficulty to acquire perfect channel knowledge and consider noisy CSI. In this

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framework, we analyze performance of a centralized and two distributed PC algorithms considering the impact of imperfect CSI in a random network model and compare it to the perfect case treated in [10] to show how the estimation error affects the PC schemes’ performances. • In the centralized approach, based on global noisy CSI, the BS handles the design of transmit power profile of all users in order to maximize the signal-to-interference-plusnoise ratio (SINR), and thus the rate, of the cellular link while satisfying quality of service (QoS) requirements (SINR constraints) for D2D users. The centralized PC allows a significant improvement of the overall performance of the network since it protects the cellular while supporting some D2D links. However, as we show in the sequel, this PC method is sensitive to CSI estimation error. • Coverage probabilities in the decentralized case are expressed in function of the estimation error α. The perfect case is then captured as a particular one which suggests that our framework may be regarded as a generalization of previous works, e.g., [10], [11]. • For the distributed approach, an on-off PC as well as a truncated channel inversion PC are proposed. Such approaches have the benefit of reducing the high CSI feedback overhead required for centralized PC and only use local CSI about the direct link between the transmitter and its corresponding receiver. Coverage probabilities and transmit powers are then established and analyzed for each of the proposed PCs. For distributed PC, even if the cellular communication is less reliable, there is still an overall network gain, in terms of sum rate, provided by D2D links. This gain is more noticeable in the high target SINR. Again, the estimation error quickly degrades the performance and reduces this gain. C. Outline of the Paper The paper is organized as follows. Section II describes the system model and provides SINR expressions and network performances. In section III, the centralized algorithm is presented. Distributed coverage probabilities are then computed in section IV. Section V treats the on-off and truncated channel inversion PCs while selected numerical results and performance analysis are presented in section VI. Finally, section VII concludes the paper. II. S YSTEM M ODEL AND R ELATED BACKGROUND A. Network Model In this paper, we are considering a single-cell D2D underlay cellular network where a BS is in the center of a circular coverage area with radius R, as shown in Fig. 1. We assume that one cellular uplink user intends to communicate with the BS while several other D2D pairs are communicating using the same spectrum. The cellular user is uniformly located in this region i.e., the probability distribution function (pdf) of its distance to the BS, d0,0 , is given by: fd0,0 (d) =

2d , R2

0 ≤ d ≤ R.

(1)

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assume that the channel is not known perfectly at the transmitter but only imperfect CSI is available. Specifically, a minimum mean square error (MMSE) estimation is used to obtain ˆ The fading channel model is expressed as: the estimate h. √ √ (4) h k,l = 1 − α hˆ k,l + α h˜ k,l ,

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

Further, to take the network randomness into account, we assume that locations of the D2D transmitters are distributed according to a homogeneous PPP,  with intensity λ. Thus, the expected number K of D2D transmitters, a Poisson random variable, is E [K ] = K¯ = λπ R 2 . Distances dk,l between two random nodes k and l in a circular cell with radius R are distributed according to the following pdf [18]: ⎛ ⎞    2 d d 2d 2 d ⎠, − 1− fdk,l (d) = 2 ⎝ cos−1 R π 2R πR 4R 2 0 ≤ d ≤ 2R.

(2)

We assume also that all users have a single antenna each. Each of them has a maximum transmit power constraint such that p0 ≤ Pmc and pk ≤ Pmd , where Pmc and Pmd are maximum powers for the cellular and D2D users, respectively. B. Radio Channel Model We consider a general power-law pathloss model in which the signal power decays at the rate d −δ with the propagation distance d, where δ ≥ 2 is the pathloss exponent. The channel (distance independent) fading h is also considered. For a particular realization of the PPP , the received signals at the BS and the D2D receiver k are written as: K g0,l sl + n 0 y0 = g0,0 s0 + l=1 K yk = gk,k sk + gk,0 s0 + l=1,l =k gk,l sl + n k ,

(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, • n k and n 0 denote the additive noise at D2D receiver k and the BS. They have a complex normal distribution with zero mean and σ 2 as variance, i.e., n k ∼ CN (0,σ 2 ), • gk,l and g0,l represent the total channel gains from D2D transmitter l to receiver k and the one fromδ D2D − transmitter l to the BS, respectively: gk,l = h k,l dk,l2 . To mitigate interference, perfect CSI is necessary. However, due to the dynamic nature of wireless channels, perfect CSI is unfeasible. Thus, for a more realistic representation, we

where h k,l is the the fading channel gain, hˆ k,l represents the estimate of h k,l and h˜ k,l refers to the estimation error independent of hˆ k,l and whose entries are assumed to be CN (0,1). The parameter α is the estimation error variance and has a fixed value between 0 and 1. We also specify that the estimate hˆ k,l is known by both the transmitter and the receiver. We assume it is CN (0,1) so that |hˆ k,l |2 is exponentially distributed with unit mean. We consider an SINR capture model. That is, a communication is established and the message can be successfully decoded at the receiver if and only if the SINR at the receiver is greater than a certain threshold β. Obviously, this requires specific encoding and decoding strategies that we are not discussing here for brevity. C. SINR and Performance Yardsticks Due to the channel model along with the random nodes locations, the transmit powers and the SINRs experienced by the receivers are also random. Transmit powers depend on the PC policy and are discussed in sections III and IV. To define our performance yardsticks, we need first to establish SINRs’ expressions as function of transmit powers, SINR threshold and estimation error variance. We start by expressing the expected received signal power at the BS. Lemma 1: The conditionally expected received power at the BS when only an estimate hˆ 0 = [hˆ 0,0 , ..., hˆ 0,K ] and the distribution of estimation error h˜ k,l (∼ CN (0, 1)) are available, is as follows: 

−δ −δ p0 + αd0,0 p0 E y0 y0∗ | hˆ 0 = (1 − α)|hˆ 0,0 |2 d0,0

K −δ + k=1 (1 − α)|hˆ 0,k |2 + α d0,k pk + σ 2 . (5) Proof: Please see Appendix A. Note that the case α = 0 corresponds the to perfect channel knowledge equal to

and then an−δexpected Kreceived2 power −δ E y0 y0∗ |h = |h 0,0 |2 d0,0 p0 + k=1 |h 0,k | d0,k pk + σ 2 . In case α = 1, there is no CSI and the receiver must

decode non-coherently. The expected power would be E y0 y0∗ = K −δ −δ d0,0 p0 + k=1 d0,k pk + σ 2 , which depends solely on the pathloss. Now, using Lemma 1, the conditionally expected cellular SINR in the proposed network model with imperfect CSI is: SINRc (K , p, α)

−δ p0 (1 − α)|hˆ 0,0 |2 d0,0

−δ K −δ αd0,0 p0 + k=1 (1 − α)|hˆ 0,k |2 + α d0,k pk + σ 2 u 0,0 p0 = . (6) K v 0,0 p0 + m=1 w0,m pm + σ 2

−δ −δ where u k,l = (1 − α)|hˆ k,l |2 dk,l , v k,l = αdk,l ; wk,l = u k,l + v k,l and p = [ p0 , p1 , ... p K −1, p K ] is the vector of transmit

=

MEMMI et al.: POWER CONTROL FOR D2D UNDERLAY CELLULAR NETWORKS WITH CHANNEL UNCERTAINTY

powers i.e., pi denotes the transmit power of transmitter i . The interpretation of SINR in (6) is that the receiver decodes the interference and the noisy received signal as a background additive white Gaussian noise (AWGN). Similarly, SINR expression of the D2D link k could be computed exactly like the cellular uplink case: u k,k pk . SINRk (K , p, α) = K v k,k pk + wk,0 p0 + m=1 wk,m pm + σ 2 (7) We note that (6) and (7) represent the SINR as seen by the transmitter and receiver, considering interference as noise, and ˜ As a conseaveraging out the uncertainty in the channel (h). ˆ quence, the achievable rate for a given h is log(1 + SINR). We recall here that there is no outage formulation therein. Based on the SINR, we are using two metrics which are the coverage probability and the achievable sum rate to evaluate performances. Precisely, the PC algorithms aim to maximize those quantities while maintaining a minimum level of QoS (SINR threshold). We define the coverage probabilities for cellular and D2D users and the sum rate of D2D links as follows:

c = E P(SINRc (K , p, α) ≥ β0 ) , (8) Pcov

D Pcov = E P(SINRk (K , p, α) ≥ βk ) , (9)

 K R D2D = E (10) k=1 log (1 + SINRk (K , p, α)) , where β0 and βk represent the minimum SINR value for reliable uplink and D2D connections, respectively. The expectations above are with respect to locations of different users and their transmit powers. III. C ENTRALIZED P OWER C ONTROL The centralized power control is proposed when global imperfect CSI (4) is available at the BS. Its objective is to find optimal transmit power profile pi , i ∈ [0, K ], that maximizes the SINR, and thus the achievable rate, of the cellular link. This is done while a required QoS is guaranteed for cellular and D2D links. The problem can be formulated as follows: max

p0 , p1 ,..., pk

SINRc (K , p, α) SINRk (K , p, α) ≥ βk , k = 1, . . . , K 0 ≤ p0 ≤ Pmc 0 ≤ pk ≤ Pmd

(11)

We can write this compactly in a vector form: Gu p max p Gi p + σ 2 subject to (I − Q) p ≥ d 0 ≤ p ≤ pmax

(12)

where G u = [u 0,0 , 0, ..., 0]T ; G i = [v 0,0 , w0,1 , w0,2 , ..., 2 2 w0,K ]T ; d = [ u 0,0σ−ββ00v 0,0 , ..., u K ,Kσ−ββKKv K ,K ]T ; pmax = and

Q

Problem (12) can be solved using standard optimization tools since the objective function is linear-fractional and hence quasi-convex (or it can even be transformed into linear problem [20]) and the constraint set is convex [19]. Hence, the optimal solution exists if the feasible set is non empty. A necessary and sufficient condition for that is provided in [19]: the spectral radius of the matrix Q should be less than one i.e., ρ( Q) ≤ 1. Due to randomness of transmitters’ locations, the condition may be unsatisfied and the PC problem unfeasible. To remedy this without doing an exhaustive complex search, a simple algorithm was proposed in [10]: It consists in shutting down the k t h D2D transmitter that creates the maximum sum of interference power to all other receivers. This means eliminating the k t h row and column of Q with the highest euclidean norm ||.||2 and then reducing the size of Q. The same process is repeated until satisfying ρ( Q) ≤ 1. To summarize, the algorithm assumes that all nodes feedback the estimated CSI and required threshold to the BS. The latter computes all users transmit powers according to the optimal solution of (12). For that purpose, it can reduce the number of D2D links as much as needed to assure feasibility. The BS sends then the optimal power policy to all users. IV. C OVERAGE P ROBABILITY A NALYSIS FOR D ISTRIBUTED P OWER C ONTROL In this section, we compute cellular and D2D coverage probabilities in the distributed scenario. We present then the optimal cellular PC under the peak and average transmit power constraints. Since we are utilizing stochastic geometry, we assume that the transmit powers of D2D transmitters are independent and identically distributed (i.i.d.). The coverage probability expressions provided here are valid for both proposed distributed PC methods in the next section (or any distributed ones where the user decides its power based solely on its own channel knowledge without considering other transmitters). A. Cellular Link Coverage Probability

subject to SINRc (K , p, α) ≥ β0

[P  mc , Pmd , ..., Pmd ] wk,l βk u k,k −βk v k,k k  = l 0 k =l

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is

defined

by

Q k,l

=

Let Id denote the conditional average interference K created (1 − by all D2D users on the uplink cellular user: Id = k=1

−δ α)|hˆ 0,k |2 + α d0,k pk . To compute the coverage probability for the cellular user in the proposed network configuration, we first establish the Laplace transform L(·) of Id in the following lemma: Lemma 2: The Laplace transform of Id is given by:    2 2 2 LId (t) = exp −λπt δ E pk pkδ (1 − α) δ (α, δ) (13) α

α )(1 − 2δ ), (a) is the where (α, δ) = e 1−α (1 + 2δ , 1−α Gamma function ([21, Ch. 1]) at the point a and (a, x) is the incomplete Gamma function defined as (a, x) =  ∞ upper a−1 e −t dt. t x Proof: Please see Appendix B. Now, using Lemma 2, we provide an analytical expression for the uplink coverage probability.

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Proposition 1: The coverage probability of the cellular user in the proposed network and channel model is given by:

2 αβ c (14) Pcov (β, α) = e− 1−α E X e−c1 X −c2 X δ , ⎧ −1 δ p X = d0,0 ⎪ 0 ⎪ ⎨ −1 c1 = βσ 2 (1 − α)  2 wher e : 2 ⎪ ⎪ ⎩ c2 = λπβ δ E p p δ (α, δ) k

k

Proof: Please see Appendix C. Proposition 1 shows how network parameters affect the celc decreases when λ and lular link coverage probability: Pcov 2δ E pk pk increase since this implies more interference generated by D2D links. Estimation error affects all its terms and thus reduces the coverage probability. To get a closed c and compare it to simulation results, we further form of Pcov assume that δ = 4 and that D2D transmitters are sending the √ signal √ using power Pmd with probability 0.5 (thus E[ pk ] = 0.5 Pmd ). A closed form expression is given by the following corollary: Corollary 1: Assuming that δ = 4, p0 = Pmd and √ √ E[ pk ] = 0.5 Pmd , the cellular coverage probability is given by: c (β, α) Pcov

=e

α −β 1−α

 √  α 1 1−α 3 α 1 − ex p − λ2 π R 2 β PPmd ( )e ( , ) 2 2 1−α mc . √  Pmd 1 α λ 2 1−α ( 3 , α ) π R β ( )e 2 Pmc 2 2 1−α (15)

Proof: c Pcov α

= e−β 1−α =e

α −β 1−α

 

R

−c2 √rP

e

−c2 √rP

mc

f (r )dr

0 R

2 mc

2r dr R2 2

=e

α

= e−β 1−α

Fig. 3. Cellular coverage probability performance versus the threshold β for different values of α and λ = 0.00002 (plain line) and λ = 0.00005 (dashed line).

2

e

0 α −β 1−α

Fig. 2. Cellular coverage probability (analytical and simulation) versus the threshold β for various values of α and λ.

1 − ex p(−c2 √RP ) mc

2

c2 √RP

mc  α √  1 1−α 3 α 1 − ex p − λ2 π R 2 β PPmd ( )e ( , ) 2 2 1−α mc . √  Pmd 1 α λ 3 α 2 1−α ( , π R β ( )e ) 2 Pmc 2 2 1−α

Corollary 1 shows that cellular coverage performance is affected by 4 factors: D2D links number (λπ R 2 ), D2D-cellular power ratio ( PPmd ), SINR threshold β and the estimation error mc variance α. Note that when α = 0, corresponding to perfect c reduces to: channel knowledge, Pcov

 √  1 3 1 − ex p − λ2 π R 2 β PPmd ( )( ) 2 2 mc c  Pcov = , (16) √ P λ 1 3 md 2 β 2πR Pmc ( 2 )( 2 ) which represents an upper bound on the achievable coverage. Figure 2 depicts the analytical expression of the coverage probability (15) alongside the results from corresponding Monte Carlo simulations for the entire range of β and some

values of λ and α: The results are well matched which validate the analytical expression. Figure 3 provides more details on coverage probability performance for the entire range of β, for α ∈ {0, 0.1, 0.5, 0.9} and λ ∈ {0.00002, 0.00005}. These two values of λ correspond to an average number of D2D links K¯ ∈ {15, 39}. The latter curve emphasizes again that imperfect CSI has a negative impact on the performance especially for high target SINR where coverage probability becomes virtually zero for values of α starting from 0.5: the cellular user can no more be covered for threshold values greater than 6 dB and -3 dB when α = 0.5 and α = 0.9, respectively.

B. Optimal Power Strategy for the Cellular User When Uplink Distance is Known Conditioning on d0,0 , the coverage probability for c (p ) = a given transmit power p0 , reduces to Pcov 0 αβ

δ −1

2 δ

e− 1−α e−c1 d p0 −c2 d p0 . Under the average and peak power constraints of the uplink transmission power p0 , and following the same methodology as [10], the optimal strategy for the uplink cellular user would be the on-off PC. The proof is similar to [10, Th. 2] and hence is omitted for brevity. The cellular transmit power p˜ 0 that maximizes the coverage 2

MEMMI et al.: POWER CONTROL FOR D2D UNDERLAY CELLULAR NETWORKS WITH CHANNEL UNCERTAINTY

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probability for a given uplink distance d, is given by:

2 δ 2 2 2 λπβ δ E pk pkδ (α, δ) d 2 . p˜ 0 (d) = (17) δ Thus, under the average and peak power constraints (Pac ≤ Pmc ), the cellular user will be transmitting with power p0∗ (d) = max(min( p˜ 0 (d), Pmc ), Pac ) with probability pP∗ ac (d) and remaining silent (i.e., p0 = 0) with probability 1 −

0

Pac p0∗ (d)

.

In interference limited regime and using the optimal binary PC, the cellular user coverage probability, for an uplink distance d, reduces to:  

αβ −2 c c = Pcov (d, β, α) = e− 1−α E p0 exp −c2 d 2 p0 δ Pcov =

  Pavg,c − αβ 2 1−α exp −c d 2 p ∗ (d)− δ . e 2 0 p0∗ (d)

(18)

The cellular user coverage probability becomes as follows:   2  ⎧ 2 αβ 2 Pac − 1−α d 2 −δ ⎪ δ ¯ δ Ep ⎪ e exp − K β ] P p

(α, δ)[ mc , ⎪ Pmc k k R ⎪ ⎪ ⎪ ⎪ ⎪ if p˜ 0 (d) ≥ Pmc ⎪ ⎪ ⎪ α −β 1−α ⎪ δ ⎪ Pac e exp(− 2 ) ⎨ , if Pac < p˜ 0 (d) < Pmc   δ c Pcov =  2 2 δ δ 2 d δ ⎪ ¯ K 2β R ⎪ δ E pk pk (α,δ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if p˜ 0 (d) ≤ Pac ⎪  2 ⎪

 ⎪ ⎪ − αβ 2 −2 ⎪ ⎩e 1−α exp − K¯ β δ E pk pkδ (α, δ)[ dR ]2 Pac δ (19) C. D2D Users Coverage Probability Proposition 2: The coverage probability of a D2D user k, k = 1..K , in the proposed network and channel model is given by:

2 αβ δ D Pcov (20) (β) = e− 1−α EY e−c3 Y −c4 Y , ⎧ δ p −1 Y = dk,k ⎪ ⎪

k ⎪

−δ  ⎨ β c3 = 1−α p0 σ 2 + (1 − α)|hˆ k,0 |2 + α dk,0 wher e :  2 ⎪ 2 ⎪ ⎪ ⎩ c4 = λπβ δ E pk pkδ (α, δ) Proof: Similar to Proposition 1. Note that in interference limited regime (σ 2 = 0), for fixed distance between D2D pairs (dk,k = d) and constant power D reduces to: for cellular user ( p0 = Pmc ), Pcov   2  2 − 2δ D 2 δ δ Pcov (β, α) = exp −λπβ E pk pk pk (α, δ)d   αβ β Pmc ×e− 1−α Edk,0 exp − 1 − α pk 

  d δ 2 ˆ × (1 − α)|h k,0 | + α dk,0  2   2 αβ 2 − = e− 1−α exp −λπβ δ E pk pkδ pk δ (α, δ)d 2  2   αβ 2 −2 = e− 1−α exp −λπβ δ E pk pkδ pk δ (α, δ)d 2

Fig. 4. D2D coverage probability versus the threshold β in Example 2 configuration: Analytical approximated expression and simulation results for different values of α and λ.

× Edk,0

× Edk,0



1

1 + β Ppkmc  



d dk,0

δ



   d δ βα Pmc ) exp −( . (1 − α) pk dk,0

(21)

The last equality follows from the fact that |hˆ k,0 |2 ∼ exp(1) 1 with a moment generating function equal to 1−t for t < 1. Note that the coverage probability established in (21) is still averaged with respect to dk,0

. To obtain closed form, 1

and we can approximate it using E 1+1 μ μ2/δ

− δμ dk,0



δ dk,0

2/δ − μ 2 E[dk,0 ]

1+

E[dk,0 ]2

e which are obtained from numerical E e observations [10]. Knowing that the first moment of dk,0 , E[dk,0 ] = 128R 45π [18], we have:   2 αβ D 2 ¯ δ − λπβ (α, δ)d Pcov (β, α) exp − 1−α   αβ p0 2/δ d2 exp −[ (1−α) 2 pk ] (128R/(45π)) × β p0 2/δ d2 1 + [ pk ] (128R/(45π)) 2   2 αβ − Ps β δ

exp − 1−α   αβ 2/δ exp −κ( 1−α ) . (22) × 1 + κβ 2/δ d 0 ) δ ( 128R/(45π) )2 , λ¯ = λPs and = where κ = ( Ppmd λπ (α, δ)d 2 . For p0 = Pmc , Ps = 0.5 and δ = 4, simulation results are shown in Fig. 4. It compares results obtained from the analytical approximated expression (22) to those of a Monte Carlo simulation for different values of λ and α: The agreement is accurate and this match validates the legitimacy and accuracy of the expression and the approximation. We want to provide relevant references for the choice of parameter 2

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settings used throughout the simulation in the paper. For the transmit power, looking into the LTE technical specification in [[22], Table 6.2.2 −1], we can find that the maximum transmit power for a cellular user equipment is defined as 23 dBm with a tolerance of 3 dBm. In our case we took Pmc = 100 mW or 200 mW, i.e., 20 or 23 dBm , which does lie in this interval and meet the specifications. D2D communications are supposed to be of short range compared to the cellular link that is why we chose Pmd = 0, 1 mW . For the path loss exponent δ, “measurements taken in urban and suburban areas usually find a path loss exponent close to 4” [23]. Several models based on the path loss exponent use the value 4 while describing the urban and suburban environments such that the Clutter Factor Models [23] and others consider values close to 4 like 3.8 in the NLOS model [24]. Secondly, we want to admit that this is only an approximation and there is no claim that it is universally good for all parameters. For a certain range of parameters, it may be loose. However, for the range of those we used, which are typical for such a scenario, the approximation is quite accurate. V. D ISTRIBUTED P OWER C ONTROL A LGORITHMS In this section, we propose two distributed PC algorithms for D2D users: on-off and truncated channel inversion PCs. For the first one, we optimize the on-off threshold to maximize the D2D sum rate. For the second algorithm, we perform transmit power analysis and establish D2D coverage probability expression. A. On-off Power Control On-off PC is a simple yet effective method to interference mitigation when there is no coordination between nodes. Each transmitter has to decide individually whether to transmit or not using solely the direct channel condition to his receiver. The link is activated ( pk = Pmd ), only  if the channel quality is good enough, i.e., E gk,k |hˆ k,k = −δ (1 − α)|hˆ k,k |2 dk,k ≥ gmin (Ps ) where gmin is a fixed quality threshold known by all users. The probability of activation is as follows:

−δ (23) ≥ gmin Ps = P (1 − α)|hˆ k,k |2 dk,k The average D2D transmit power would be pk = Ps Pmd . Therefore, gmin (Ps ) plays a decisive role in determining the overall (sum rate) performance of the D2D links: When gmin is small, the system allows more D2D users to be active. However this will also increase inter D2D interference. On the other hand, when gmin is large, less D2D users are active, but there are suffering less inter D2D interference. The effect of the on-off threshold on the D2D sum rate is illustrated by Fig.5: We see that for big values of the threshold, the D2D sum rate gets lower since less D2D links are activated. We notice also that for very small values of gmin , the sum rate decreases also, especially for bigger λ, and this is due to, as stated previously, to the increasing inter-D2D interference when activated D2D links gets higher. This shows the existence of an optimal value of the on-off threshold, which we establish in the following.

Fig. 5. of λ.

D2D sum rate versus the on-off threshold gmin for different values

1) Sum Rate of D2D Links: We assume an interference limited regime (σ 2 = 0), on-off PC for D2D links and constant power for the cellular uplink user (Pmc ). It is recalled that D2D communications are done using pk = Pmd with probability ¯ R2 = Ps , so the number of active D2D links is Nact = λπ 2 λPs π R . The achievable D2D sum rate would be:  K  

D R =E log (1 + SIRk ) = Nact E log(1 + SIRk ) k=1

= λPs π R 2 × R D2D

(24)

where R D2D is the average rate of a single D2D link. Using expression (22), the rate R D2D can be expressed then as follows:  ∞ ∂ R D2D = − log(1 + β) (P[SIRk ≥ β]) dβ ∂β 0  ∞ D Pcov (β) dβ = 1+β 0 αβ 2/δ  ∞ 2 αβ 1 e−κ( 1−α ) sβ δ e− 1−α − P

! " 1 + κβ 2/δ dβ. (25) 1+β 0  ! " A B

We can note that the approximated expression in (25) is determined by two factors: (1) the Laplace transform of the total interference power created by all D2D active links (A) and (2) the approximated effect of the uplink interference (B), both affected by the estimation error α. 2) Optimizing the On-off Threshold: We can optimize the D2D on-off threshold by maximizing the average transmission rate defined as: TrD (β) = λPs π R 2 log(1 + β) P[SIRk ≥ β]

(26)

The optimization problem becomes: max

TrD (β)

subject to

0 ≤ Ps ≤ 1

Ps

(27)

Using the first order optimality condition, the maximizer is the following: 2 opt Ps = min( −1 β − δ , 1). (28)

MEMMI et al.: POWER CONTROL FOR D2D UNDERLAY CELLULAR NETWORKS WITH CHANNEL UNCERTAINTY

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TABLE I AVERAGE S UM R ATE P ERFORMANCE OF D2D L INKS

Given Ps in (23) and since |h k,k |2 ∼ exp(1), we obtain the optimal threshold: −δ ln(1/Ps ). gmin = (1 − α)dk,k opt

opt

opt

(29)

opt

Likewise Ps , gmin has two expressions depending on β. Now integrating TrD with respect to β provides the sum rate of D2D links. Proposition 3: The average D2D sum rate is given by: 

δ

− 2

D

R (β)

αx

+

2

2 2/δ x δ

2

(1 + κ x δ )(1 + x)

0



α

λπ R 2 e− 1−α exp(− β δ )e−κ( 1−α )

∞ −δ 2

R 2 −δ − ( dk,k ) x e 2

Γ (1 +

Fig. 6. Relative error between simulation and approximate results for the average sum rate performance of D2D links for different values of λ and α.

dx

α(x+1) α 2/δ 2/δ x −1 1−α −κ( 1−α )

2 α 2 2/δ) δ , 1−α )Γ (1− δ )(1+x)(1+κ x

×d x. (30) Proof: Please see Appendix D. To validate this approximated expression (AR), we compare it to the average D2D sum rate obtained through Monte Carlo simulations (SR). Table I presents results of comparison with different values of λ and α using the following parameters: Pmc = 100 mW, Pmd = 0.1 mW, dk,k = 50 m, R = 500 m and δ = 4. The relative error between approximate values and the ones obtained from Monte Carlo simulation are plotted for different values of the estimation error α and the D2D link intensity λ. The relative error is defined here as | S R−AR S R |. As can be seen in Fig. 6, error is around 0.1 and thus the approximate expression of the sum-rate is accurate up to 14% precision.

to the truncated channel inversion PC, the D2D proximity 1 is reduced to Rd = ( Pρmd ) δ . That is, the pdf of the D2D 0 link distance is given by fdk,k (d) =

≤ d ≤ Rd . ˜ The intensity of D2D links reduces also to λ = μλ where 2 d )2 = ( ρρmin ) δ , μ is the power coverage probability μ = ( RRmax 0 Due to the assumed PC along with the random locations of D2D users, the transmit powers and the SINRs experienced by the receivers are random also. First, we characterize the transmit power via its pdf and its ηt h moment (η ≥ 0). Then, we characterize the SINR and the coverage probability. 1) Transmit Power Analysis: For the D2D transmitter to be active, two conditions are required: a maximum power constraint and a minimum quality requirement. Thus, the transmit δ  power pk can be written as pk = ρ0 dk,k −δ {(1−α)|hˆ k,k |2 dk,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 users with cutoff threshold ρ0 and quality threshold T , the pdf of the transmit power of a D2D user is given by:

B. Truncated Channel Inversion Under the maximum transmit power (Pmd ) constraint, the D2D transmitters use a truncated channel inversion PC: The transmit power compensates the path-loss to keep the average signal power at the receiver equal to a certain threshold ρ0 . The latter should be higher than the receiver sensitivity ρmin , (ρ0 ≥ ρmin [25, Ch. 4]). Therefore, a connection is established only if the transmit power required for the path-loss inversion is less than or equal to Pmd . Otherwise, the D2D user does not transmit and remains idle due to insufficient transmit power. Another condition for the D2D transmitter to be active is that −δ ≥T the link quality is good, in the sense that (1−α)|hˆ k,k |2 dk,k where T is a nonnegative quality threshold that is fixed and known by all users. We assume that each D2D transmitter has its unique receiver uniformly distributed within its proximity Rmax . It is deter1 md δ mined by Pmd and ρmin , such as Rmax = ( ρPmin ) . But, due

2d 2 ,0 Rd2

f pk (x) =

2 δ

T 2

x δ −1 e 2

T − (1−α)ρ

0

x

T Pmd ρ0δ (1 − α) δ γ ( 2δ , (1−α)ρ ) 0 2

.

(31)

where γ (·, ·) is the lower incomplete Gamma function. Furthermore, the transmit power moments can be obtained as: η

E[ pk ] =

η

T Pmd ) ρ0 (1 − α)η γ ( 2δ + η, (1−α)ρ 0

. (32) T Pmd T η γ ( 2δ , (1−α)ρ ) 0 Proof: Please see Appendix E. 2) SINR and Coverage Probability: We assume now constant uplink cellular power p0 = Pmc . Proposition 5: The coverage probability of D2D users is: D Pcov,tr (β, α) = e

β (ρ0 α+σ 2 ) 0 (1−α)

−ρ

L Id (

β β )L Ic ( ), (1 − α)ρ0 (1 − α)ρ0

(33)

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where: β ) (1 − α)ρ0 2/δ T Pmax,d

 ρ (1 − α)2/δ γ ( 4δ , (1−α)ρ ) −1 2δ 0 0 ˜ = exp − λπ(βρ0 )

(α, δ) T Pmax,d T 2/δ γ ( 2δ , (1−α)ρ ) 0

L Id (

β ) (1 − α)ρ0

 βα −δ −δ = E exp(−βρ0−1 |hˆ k,k |2 dk,0 dk,0 p0 ) p0 − 1−α # $−1 (βp0 )2/δ

1 + 2/δ ρ0 (128R/(45π))2   − (45π)2 (αβp0 )2/δ × exp ((1 − α)ρ0 )2/δ (128R)2

L Ic (

Fig. 7. Cellular coverage probability versus the threshold β in centralized PC for different values of estimation error α.

and Ic denotes the conditional average interference created by the cellular user on a D2D

−δuser k and can be expressed as: p0 . Ic = (1 − α)|hˆ k,0 |2 + α dk,0 Proof: Please see Appendix F. To validate our analysis, we compare the analytical result of D with that obtained through Monte Carlo simulation. It Pcov,tr can be seen in Fig. 11 that resuts do match which shows that the obtained model captures well the coverage probability. VI. N UMERICAL R ESULTS In this section, we provide numerical results for the D2D underlay cellular system. We validate the analytical expressions through Monte Carlo simulations. Through results, we show the performance gain of using different proposed PC schemes (compared to the no PC case) in terms of coverage probability and show how the estimation error affects these performances.

Fig. 8. D2D coverage probability versus the threshold β in centralized PC for different values of estimation error α.

A. Optimal Centralized Power Control In the simulation setup for the centralized PC, we consider a BS positioned in the center of the coverage area (with radius R = 500 m) where a cellular user is uniformly located. The D2D transmitters are distributed according to a PPP with density λ = 0.00002 which corresponds to an average number of D2D links K¯ = 15. Each D2D receiver is located at a fixed distance d = 50 m from its transmitter. We assume that the maximum transmit powers for cellular and D2D communications are Pmc = 0.1 W and Pmd = 0.1 mW , respectively. This is due to the fact that D2D links are supposed to be of a short range, compared to cellular ones. The effect of estimation error on coverage probabilities is illustrated by Fig. 7 and Fig. 8. As shown in the former, the centralized PC is quite sensitive to estimation error and coverage chances get lower when the estimation error variance increases. For example, for a target SINR of 3 dB, the cellular coverage drops from 95% in the perfect case to 78% and 14% when α is equal to 0.1 and 0.5, respectively. The cellular can not get covered anymore when α gets close to 1. As shown by Fig. 8, D2D coverage is also negatively affected and decreases when α increases. This shows that the central controller (BS) can not get good decisions when the CSI is very noisy and the

Fig. 9. Cellular coverage probability versus the threshold β for both on-off PC and no PC for K=39.

overall performance gets worse when error variance increases: imperfect CSI and estimation error should be considered when designing PC algorithms. B. Suboptimal Decentralized Power Control 1) On-off Power Control: Now we evaluate performances for the on-off PC for D2D users and for the cellular and compare them to those obtained when no PC is applied. Simulation parameters are similar to those in the centralized part. Figure 9 and Figure 10 provide comparison for the onoff PC with no PC for cellular and D2D links, respectively. In the first figure K=39 and in the second both cases of K=15

MEMMI et al.: POWER CONTROL FOR D2D UNDERLAY CELLULAR NETWORKS WITH CHANNEL UNCERTAINTY

Fig. 10. D2D coverage versus the threshold β for on-off PC and no PC for K = 15 (dashed line) and K = 39 (plain line).

D2D coverage probability versus the threshold β using the truncated channel inversion PC for K=15.

Fig. 11.

and K=39 are displayed. It can be seen in these figures that the on-off PC yields performance gains for both cellular and D2D links compared to that of no PC case when the target SINR is larger than 3 d B (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 for β ≤ 3 d B, since the optimal opt activation probability (28) Ps = min( 1 2 , 1) = 1; while β δ

it is activated when β ≥ 3 d B. For example, for K=39 at β = 9d B, performance gain for the cellular user is about 50% and 15% for D2D users. This gains is considerably affected by the estimation error: it is reduced to 30% when α = 0.1 and there is almost no gain when α increases more. This shows that the on-off PC is sensitive to CSI estimation error, since it relies on this information to make decision, the more this information is noisy the more performance tends to the no PC case. 2) Truncated Channel Inversion Power Control: Now for the truncated channel inversion model, we first validate our model by simulations and then present some numerical results for the following parameters: We set the maximum cellular and D2D transmit powers to Pmc = 0.2 W and Pmd = 0.1 mW respectively, the cutoff threshold ρ0 = −80 d Bm, the receiver sensitivity ρmin = −100 d Bm the quality threshold T = −50 d Bm, the path-loss exponent δ = 4 and the SINR threshold β ranging from −18 to 18 dB. The following figure provides D2D coverage probability when the truncated channel inversion PC is applied for D2D links (λ = 0.00002) while

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Fig. 12. D2D coverage probability versus the PC cutoff threshold ρ0 : Effect of ρ0 on SINR and power coverages.

the cellular user transmit with constant power Pmd . Figure 11 shows that truncated channel inversion PC provides the same overall response for D2D links: their coverage probability decreases when SINR requirements increase (due to power constraint) and the estimation error reduces efficiency and degrades considerably the coverage probability. Results are close to those obtained for the on-off PC, but the advantage here is the ability to adapt signal power to the channel with an additional parameter ρ0 whose effect is shown in Fig. 12. The latter shows that increasing the PC cutoff threshold increases the SINR coverage probability for D2D links: Increasing the cutoff threshold ρ0 increases the power of the useful signal and increases the SINR coverage probability. However, this comes at the expense of decreased power coverage probability since increasing ρ0 requires a higher transmit power to invert the channel which increases the power outage. So, ρ0 introduces an important trade-off between power constraints and SINR coverage. VII. C ONCLUSION In the present paper, we proposed a single cell model for a D2D underlay cellular system: random aspect of the network was expressed based on stochastic geometry and imperfect CSI was taken into account. Results for the centralized PC proved that it does improve the network performance since it continues to protect the cellular communication while supporting additional underlay D2D links. We observed also that channel estimation error degrades the algorithm performance and reduces considerably its efficiency. The distributed approach, on the other hand, has the merit of relying only on local CSI to decide the transmit power by the user itself. Results showed also that it presents overall network gain in terms of sum rate but it fails to guarantee reliable cellular communication whose coverage decreases noticeably compared to the centralized case. Again here, imperfect CSI and misinformation lead to the degradation of performance especially for high target SINR where coverage becomes virtually equal to zero. This shows that estimation error is a key parameter that should be taken into account during network design. Our approach in this paper was to understand first the effect of CSI error in a relatively simpler, but not straightforward, setting, leaving the general case to future work: The framework generalization could cover the several cellular users scenario

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and eventually include out-of-cell interference and multiple antennas into the analysis. A PPENDIX A P ROOF OF L EMMA 1 The conditionally expected received power at the BS could be expressed as follows: 



¯ 2 + |I |2 + |n 0 |2 + 2Re(S I ∗ ) E y0 y0∗ | hˆ 0 = E |S|2 + | S| ¯ ∗0 ) + 2Re(Sn ∗0 ) + 2Re(I n ∗0 ) + 2Re( Sn  + 2Re(S S¯ ∗ ) + 2Re(I S¯ ∗ )| hˆ 0 , √ √ where S = 1 − α hˆ 0,0 d0,0 s0 ; S¯ = α h˜ 0,0 d0,0 s0 and K −δ I = k=1 h 0,k d0,k2 sk . Below the expectation of each of the terms: 

(1) E |S|2 | hˆ 0 − 2δ

− 2δ

−δ p0 . = (1 − α)|hˆ 0,0 |2 d0,0 

¯ 2 | h˜ 0 (2) E | S|

−δ −δ = α E |h˜ 0,0 |2 d0,0 p0 = αd0,0 p0 .

 (3) E |I |2 | hˆ 0

K 

 −δ ˆ E |h 0,k sk |2 d0,k | h0 k=1 K −1 

K 

& %

−δ −δ d0,i2 d0,k2 Re E h 0,k sk si∗ h ∗0,i | hˆ 0

k=1 i=k+1

=

K 

E

%

−δ −δ (1 − α)|hˆ 0,k |2 d0,k + α|h˜ 0,k |2 d0,k

k=1

 ' & −δ + 2 α(1 − α)d0,k Re(hˆ 0,k h˜ ∗0,k ) |sk |2 | hˆ 0 +2

K −1 

K 

−δ

−δ

d0,i2 d0,k2

k=1 i=k+1

% & × Re E[h 0,k ] E[sk ] E[si∗ ] E[h ∗0,i | hˆ 0 ]  !" !" =0

=

=0

K  k=1

−δ −δ (1 − α)|hˆ 0,k |2 d0,k |sk |2 + α|sk |2 E |h˜ 0,k |2 d0,k  ! "

' −δ + 2 α(1 − α)hˆ 0,k d0,k Re(E [h˜ ∗0,k ]) ! "  =0

=

K  k=1

−δ (1 − α)|hˆ 0,k |2 + α d0,k pk ,

A PPENDIX B P ROOF OF LEMMA 2 The Laplace transform of Id is expressed as follows: LId (t) = E[e−t Id ]

K 



−δ = E, pk ,hˆ 0,k exp(−t pk ) (1 − α)|hˆ 0,k |2 + α d0,k

 = exp λ

k=1

+2

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

(1)

K 

 −δ =E | h 0,k dk,02 sk |2 | hˆ 0

=

The third equality is obtained since we assume, without loss of generality, )that si and h are independent ( E(h 0,k si ) = E(h 0,k ) E(si ) and that si are with zero mean {E(si ) = 0} while the fourth equality follows from the distriˆ bution of h.

 (4) E |n 0 |2 | hˆ 0 = σ 2 .

=1

+∞  2π

k=1

  −δ ˆ 2 E pk ,hˆ 0,k e−t [(1−α)|h 0,k | +α] pk r − 1

0 0  × dφr dr  +∞   

−δ ˆ 2 E pk ,hˆ 0,k 1−e−t [(1−α)|h 0,k | +α] pk r r dr = exp − 2πλ 0  

2 2 2 (2) δ = exp − λπt E pk ( pkδ ) Ehˆ 0,k [(1 − α)|hˆ 0,k |2 + α] δ 2  × Γ (1 − ) δ

2 α 2 2 α 2 (3) ) = exp − λπt δ E pk ( pkδ )(1 − α) δ e 1−α Γ (1 + , δ 1−α  2 × Γ (1 − ) δ Equality (1) follows from Campbell’s theorem for PPPs (See p.78 [17]). Equalities (2) and (3) are detailed below:  +∞    −δ ˆ 2 (2) I = E pk ,hˆ 0,k 1 − e−t [(1−α)|h 0,k | +α] pk r r dr 0 +∞    −δ E pk ,hˆ 0,k 1 − e−t H pk r r dr = 0   % 2& 2 1 +∞

1 − E(e−y ) E (t H pk ) δ y −1− δ d y =− δ 0 2 2 1 2 = − t δ E pk ( pkδ ) Eh (H δ ) δ +∞ * δ

2 × − 1 − E((1 − e−y )y δ ) 0 2  +∞ + δ −y − 2δ − E e y dy 2 0 2 2 2 1 2 = t δ E pk ( pkδ ) Eh (H δ )(1 − ) 2 δ 2 2 1 2 2 = t δ E pk ( pkδ ) Eh ([(1 − α)|hˆ 0,k |2 + α] δ )(1 − ) 2 δ (3) We define Y as Y := (1 − α)h + α where h ∼ exp(1) (and thus, fh (x) = e−x (x≥0)). We deduce then that 1 − y−α e 1−α (y≥α) Y follows this distribution: f Y (y) = 1−α

MEMMI et al.: POWER CONTROL FOR D2D UNDERLAY CELLULAR NETWORKS WITH CHANNEL UNCERTAINTY

The ( 2δ )t h moment of Y is then: α  +∞  +∞ y 2 2 2 e 1−α δ δ y f Y (y)d y = y δ e− 1−α d y Eh (Y ) = 1−α α α  +∞ α 2 2 = e 1−α (1 − α) δ u δ e−u du

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

α 1−α

=e

α 1−α

Ps

subject to 0 ≤ Ps ≤ 1

2 α (1 − α) (1 + , ). δ 1−α 2 δ

2 ∂ TrD = 0 ⇔ 1 − β δ Ps = 0 ∂ Ps

c Pcov (β, α)

= P(SINRc ≥ β)

−δ   (1 − α)|hˆ 0,0 |2 d0,0 p0 =P ≥ β

K −δ −δ (1 − α)|hˆ 0,k |2 + α d0,k αd0,0 p0 + k=1 pk + σ 2  βα + = P |hˆ 0,0 |2 ≥ 1−α K δ

  βd0,0

−δ + (1− α)|hˆ 0,k |2 + α d0,k pk + σ 2 (1− α) p0 k=1

δ σ 2β

d0,0 αβ ) exp(− = E exp(− ) 1−α (1 − α) p0 K δ β  d0,0

−δ  (1 − α)|hˆ 0,k |2 + α d0,k pk × exp(− (1 − α) p0

=e

=e =e

αβ − 1−α

αβ − 1−α

EX

k=1

*

EX EX

exp(−

δ σ 2β d0,0

αβ

EX

δ β d0,0

(1 − α) p0 (1 − α) p0 K + 

−δ (1 − α)|hˆ 0,k |2 + α d0,k × pk )

*

#

exp −

* exp(−



$

δ σ 2β d0,0

(1 − α) p0 δ σ 2β d0,0 α 1−α

× (1 − α) e 2 e−c1 X −c2 X δ .

# LId

δ β d0,0

$

+

opt

So we have Ps

α 2 + 2 )(1− )) (1 + , δ 1−α δ

where  is the PPP representing the locations of D2D transmitter, Ex,y [.] denotes the expectation with respect to the random variables x and y, (ii) follows from the independence between , pk , and hˆ , and (iii) is obtained by using the Laplace transform expression established in Lemma 2.

1 2

β δ

, 1) and since |h k,k |2 ∼ exp(1), opt

opt

gmin = (1 − α)

ln(1/Ps ) . δ dk,k

(36)

It follows that the approximated transmission capacity can be δ re-expressed, depending on β compared to − 2 as: ⎧ αβ 2 log (1+β) −κ( α )2/δ β 2/δ 2 ⎪ 1−α λπ R 2 e− 1−α exp(− β δ ) 1+κβ ⎪ 2/δ e ⎪ ⎪ ⎪ ⎪ δ ⎪ ⎨ if β ≤ − 2 D Tr (β)

α(β+1) α )2/δ β 2/δ −1 −κ( 1−α − 2 ⎪ ⎪ ⎪ e 21−αα log2 (1 + β)( dRk,k )2 β − δ 2 ⎪ 2/δ ) Γ (1+ , )Γ (1− )(1+κβ ⎪ δ 1−α δ ⎪ ⎪ δ ⎩ if β > − 2 . (37) Integrating TrD with respect to β provides the sum rate of D2D links. A PPENDIX E P ROOF OF P ROPOSITION 4

2 δ

) × exp(−λπt δ E pk ( pk )

= min(

(35)

we obtain the optimal threshold:

(1 − α) p0 2

(1 − α) p0 2 δ

= e− 1−α

) E pk ,hˆ 0,k exp(−

k=1

(34)

The objective function is not concave but we still can get optimal Ps that maximizes the average transmission rate by using the first order optimality condition (since the objective function has a unique optimum point):

A PPENDIX C P ROOF OF P ROPOSITION 1

αβ − 1−α

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δ denote the unconditional transmit power Let X d = ρ0 dk,k required to invert the channel for the D2D link. We have 2d 2 fdk,k (d) = 2 , 0 ≤ d ≤ Rd and we easily prove that: Rd

2x δ −1 2

f X d (x) =

2 δ

δρ0 Rd2

2x δ −1 2

=

2 δ

δ Pmd

, 0 ≤ d ≤ Pmd

(38)

Now the D2D transmit power can be expressed as: δ pk = ρ0 dk,k 1{(1−α)|hˆ k,k |2 d −δ ≥T } k,k

A PPENDIX D P ROOF OF P ROPOSITION 3 The average transmission rate TrD is defined as: TcD (β) = λPs π R 2 log(1 + β) P[SIRk ≥ β] αx 2 log(1 + β)

λPs π R 2 e− 1−α exp(− Ps β δ ) 1 + κβ 2/δ α

× e−κ( 1−α )

2/δ β 2/δ

.

=

δ {ρ0 dk,k

if

δ ρ0 dk,k

≤

= {X d if X d ≤ X 0 }

(1 − α)ρ0 ˆ 2 |h k,k | } T

0 ˆ where X 0 = (1−α)ρ |h k,k |2 . We can now express the pdf of T pk as follows:  ∞ f X d |X 0 (x|y) P(X d ≤ y)f X 0 (y) f pk (x) = dy P(X d ≤ X 0 ) x

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 2017 T

− (1−α) y T (1−α)ρ0 e

Since |hˆ k,k |2 ∼ exp(1), we have f X 0 (y) =

and

δ T δ T dk,k dk,k ) = E[exp(− )] P(X d ≤ X 0 ) = P(|hˆ k,k |2 ≤ (1 − α) (1 − α)  Rd  Rd rδ T rδ T 2r = e− (1−α) fdk,k (r )dr = e− (1−α) 2 dr Rd 0 0 T Rdδ 1−α



2 1−α 2 = ( )δ δ T Rdδ

y δ −1 e−y dr 2

0

2 1 − α 2 2 T Rdδ ( ) )δ γ( , δ T Rdδ δ 1−α

=

We get back to the pdf of pk :  f pk (x) =

∞ x

2x δ −1 δ T Rdδ 2 1 T ( )δ 2 2 1 − α γ ( 2 , T Rdδ ) (1 − α)ρ0 δ δ Pmd δ 1−α 2

T

× e− (1−α) y d y =

=

T 2 δ



x δ −1

2 δ

2

T Pmd ρ0 (1 − α) γ ( 2δ , (1−α)ρ ) 0 2 δ

T

2 δ

x δ −1 e 2

2

T − (1−α)ρ

0

∞ T (1−α)ρ0

e−z dz x

x

T Pmd ρ0δ (1 − α) δ γ ( 2δ , (1−α)ρ ) 0 2

The ηt h moment is obtained by computing

 Pmd 0

x η f pk (x) d x.

A PPENDIX F P ROOF OF P ROPOSITION 5

Pcov (β, α) = P (SINRd ≥ β)  β αβ = P |hˆ k,k |2 ≥ + 1−α ρ0 (1 − α) ⎤ ⎡ K   −δ −δ 2⎦ Wk, j dk, p + W d p + σ ×⎣ j k,0 0 j k,0 j =1

=e

β(ρ α+σ 2 ) − ρ 0(1−α) 0

⎛

⎞ K  β −δ × E exp ⎝− Wk, j dk, j p j ⎠ (1 − α)ρ0 j =1   β −δ × exp − Wk,0 dk,0 p0 (1 − α)ρ0 β β − β (ρ α+σ 2 ) = e ρ0 (1−α) 0 L Id ( )L I ( ) (1− α)ρ0 c (1− α)ρ0

L Id is the Laplace transform of the interference Id created by the rest of D2D users and can be deduced from Lemma 2

β 2 at (1−α)ρ . Then, E pk [ pkδ ] is replaced by its expression 0 established in Proposition 4.

L Id

TP

2/δ

= exp − λ˜ π(βρ0−1 ) δ

2

max,d ρ0 (1 − α)2/δ γ ( 4δ , (1−α)ρ ) 0

TP

max,d T 2/δ γ ( 2δ , (1−α)ρ ) 0



(α, δ)

L Ic is the Laplace transform of the interference created by the cellular user: L Ic



dk,k δ  β p0 (1 − α)|hˆ k,0 |2 + α ( = Edk,0 exp{− ) } 1 − α pk dk,0

βp0 ˆ 2 dk,k δ βα p0 dk,k δ  ) ( |h k,0 | ( ) } exp{−( ) } = Edk,0 exp{− pk dk,0 1− α pk dk,0



1 βα p0 dk,k δ  E exp{−( ) ( )} = Edk,0 d k,0 d 1 − α pk dk,0 1 + β pp0k ( dk,k )δ k,0 $ # 2 αβp0 2/δ dk,k 1 exp −[ ]

2 128R 2 dk,k (1 − α) pk β p0 2/δ ( (45π) ) 1 + [ pk ] (128R/(45π))2 # # $−1 $ (βp0 )2/δ − (αβp0 )2/δ = 1+ 2/δ exp 128R 2 ((1− α)ρ0 )2/δ ( (45π) ) ρ0 (128R/(45π))2 We obtain now the result stated in Proposition 5. R EFERENCES

[1] J. Gu, S. J. Bae, B.-G. Choi, and M. Y. Chung, “Dynamic power control mechanism for interference coordination of device-to-device communication in cellular networks,” in Proc. 3rd Int. Conf. Ubiquitous Future Netw. (ICUFN), Jun. 2011, pp. 71–75. [2] X. Lin, J. G. Andrews, A. Ghosh, and R. Ratasuk, “An overview of 3GPP device-to-device proximity services,” IEEE Commun. Mag., vol. 52, no. 4, pp. 40–48, Apr. 2014. [3] C.-H. Yu, O. Tirkkonen, K. Doppler, and C. Ribeiro, “On the performance of device-to-device underlay communication with simple power control,” in Proc. IEEE Veh. Technol. Conf. (VTC-Spring), Apr. 2009, pp. 1–5. [4] C.-H. Yu, K. Doppler, C. B. Ribeiro, and O. Tirkkonen, “Resource sharing optimization for device-to-device communication underlaying cellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2752–2763, Aug. 2011. [5] G. Fodor et al., “Design aspects of network assisted device-to-device communications,” IEEE Commun. Mag., vol. 50, no. 3, pp. 170–177, Mar. 2012. [6] K. Doppler, M. Rinne, C. Wijting, C. B. Ribeiro, and K. Hugl, “Deviceto-device communication as an underlay to LTE-advanced networks,” IEEE Commun. Mag., vol. 47, no. 12, pp. 42–49, Dec. 2009. [7] C. H. Yu, O. Trikkonen, K. Doppler, and C. Ribeiro, “Power optimization of device-to-device communication underlaying cellular communication,” in Proc. IEEE ICC, Dresden, Germany, Jun. 2009, pp. 1–5. [8] K. Huang, V. K. Lau, and Y. Chen, “Spectrum sharing between cellular and mobile ad hoc networks: Transmission-capacity trade-off,” IEEE J. Sel. Areas Commun., vol. 27, no. 7, pp. 1256–1267, Sep. 2009. [9] J. Lee, J. G. Andrews, and D. Hong, “Spectrum-sharing transmission capacity,” IEEE Trans. Wireless Commun., vol. 10, no. 9, pp. 3053–3063, Sep. 2011. [10] N. Lee, X. Lin, J. G. Andrews, and R. W. Heath, “Power control for D2D underlaid cellular networks: Modeling, algorithms, and analysis,” IEEE J. Sel. Areas Commun., vol. 33, no. 1, pp. 1–13, Jan. 2015. [11] H. ElSawy, E. Hossain, and M.-S. Alouini, “Analytical modeling of mode selection and power control for underlay D2D communication in cellular networks,” IEEE Trans. Commun., vol. 62, no. 11, pp. 4147–4161, Nov. 2014. [12] D. Feng, L. Lu, Y. Yuan-Wu, Y. Li, G. Feng, and S. Li, “QoS-Aware resource allocation for device-to-device communications with channel uncertainty,” IEEE Trans. Veh. Technol., vol. 65, no. 8, pp. 6051–6062, Aug. 2015. [13] A. Abu Al Haija and M. Vu, “Spectral efficiency and outage performance for hybrid D2D-infrastructure uplink cooperation,” IEEE Trans. Wireless Commun., vol. 14, no. 3, pp. 1183–1198, Mar. 2015. [14] Z. Rezki and M.-S. Alouini, “Ergodic capacity of cognitive radio under imperfect channel-state information,” IEEE Trans. Veh. Technol., vol. 61, no. 5, pp. 2108–2119, May 2012. [15] V. N. Q. Bao, D. Q. Trung, and C. Tellambura, “On the performance of cognitive underlay multihop networks with imperfect channel state information,” in Proc. IEEE ATC, Oct. 2013, pp. 125–130.

MEMMI et al.: POWER CONTROL FOR D2D UNDERLAY CELLULAR NETWORKS WITH CHANNEL UNCERTAINTY

[16] B. Prasad, S. D. Roy, and S. Kundu, “Outage and SEP of secondary user with imperfect channel estimation and primary user interference,” in Proc. IEEE Int. Conf. Electron. Comput. Commun. Technol. (CONECCT), Jan. 2014, pp. 1–6. [17] M. Haenggi, Stochastic Geometry for Wireless Networks. Cambridge, U.K.: Cambridge Univ. Press, 2013. [18] D. Moltchanov, “Distance distributions in random networks,” Ad Hoc Netw., vol. 10, no. 6, pp. 1146–1166, Aug. 2012. [19] G. J. Foschini and Z. Miljanic, “A simple distributed autonomous power control algorithm and its convergence,” IEEE Trans. Veh. Technol., vol. 42, no. 4, pp. 641–646, Nov. 1993. [20] L. Vandenberghe, Linear-Fractional Optimization, document EE236A, 2013. [21] G. Andrews, R. Askey, and R. Roy, Special Functions. Cambridge, U.K.: Cambridge Univ. Press, 1999. [22] ETSI (European Telecommunications Standards Institute), 3GPP Version 12.7.0 Release 12, document TS 36.101, 2015. [23] S. R. Saunders and A. Aragón-Zavala, Antennas and Propagation for Wireless Communication Systems, 2nd ed. Wiley, p. 165. [24] Propagation Data and Prediction Methods for the Planning of ShortRange Outdoor Radiocommunication Systems and Radio Local Area Networks in the Frequency Range 300 MHz to 100 GHz, document ITU-R Rec. P.1411-3, International Telecommunication Union, Geneva, Switzerland, 2005 [25] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005. Amen Memmi (S’16) was born in Ksar Hellal, Tunisia. He received the Diplôme National d’Ingénieur degree in signals and systems from the École Polytechnique de Tunisie in 2015. He is currently pursuing the M.Sc. degree with the Institut National de la Recherche Scientifique-Énergie, Matériaux, et Télécommunications, Université du Québec, Montréal, QC, Canada. His research interests include wireless communications and performance analysis for next generation cellular networks.

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Zouheir Rezki (S’01–M’08–SM’13) was born in Casablanca, Morocco. He received the Diplôme d’Ingénieur degree from the École Nationale de l’Industrie Minérale, Rabat, Morocco, in 1994, the M.Eng. degree from the École de Technologie Supérieure, Montreal, QC, Canada, in 2003, and the Ph.D. degree from the École Polytechnique, Montreal, QC, in 2008, all in electrical engineering. From 2008 to 2009, he was a Post-Doctoral Research Fellow with the Data Communications Group, Department of Electrical and Computer Engineering, The University of British Columbia. He has been a Senior Research Scientist with the King Abdullah University of Science and Technology, Saudi Arabia, until 2016. He joined the University of Idaho as a Faculty Member in 2016. His research interests include performance limits of communication systems, physical-layer security, cognitive and sensor networks, and low-complexity detection algorithms.

Mohamed-Slim Alouini (S’94–M’98–SM’03–F’09) was born in Tunis, Tunisia. He received the Ph.D. degree in electrical engineering from the California Institute of Technology, Pasadena, CA, USA, in 1998. He served as a Faculty Member with the University of Minnesota, Minneapolis, MN, USA, the Texas A&M University at Qatar, Education City, Doha, Qatar. He has been a Professor of Electrical Engineering with the King Abdullah University of Science and Technology, Thuwal, Saudi Arabia, since 2009. His current research interests include the modeling, design and performance analysis of wireless communication systems.