Energy and Spectral Efficiencies Tradeoff in Multiple Access InterferenceAware Networks


 Norah Hodge
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1 XXX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES  SBrT, 36 DE SETEMBRO DE, BRASÍLIA, DF Energy and Spectral Efficiencies Tradeoff in Multiple Access InterferenceAware Networs Álvaro R. C. Souza, Taufi Abrão, Fábio R. Durand, Lucas H. Sampaio, Paul Jean E. Jeszensy Abstract This wor analyzes the power allocation problem in DS/CDMA systems under the energy efficiency (EE) metric, maximizing the transmitted information per energy unit. As the spectral efficiency (SE) is one of the most important performance metrics and is maximized with infinite power, the tradeoff between these two metrics (EESE tradeoff) are investigated and characterized in relation to multipleaccess interference (MAI) power level, resulting that the optimum operating point is the maxee point and the EE limitation imposed by MAI. To corroborate these results, we develop two algorithms for power allocation based on distributed instantaneous SINR level. Keywords Energy efficiency, spectral efficiency, DS/CDMA, distributed power control. I. INTRODUCTION Due to the explosive demand of mobile services worldwide and to battery lifetime limitations, resource allocation techniques in wireless networs had become a great concern and a challenge to many researchers in the last decades. In general, the problems aims to maximize system throughput given the maximum power or minimize power consumption given a minimum rate criterium. Besides this scenario, the increasing importance of rational use of scarce resources (specially energy), core of Green Communications approach, brings the metric of Energy Efficiency (EE), which aim to maximize the number of transmitted bits per energy unit, measured in bits per Joule. The problem of maximizing EE is mainly investigated in wireless systems for codedivision multiple access (CDMA) [], multicarrier direct sequence CDMA (MCDSCDMA) [] and orthogonal frequencydivision multiple access (OFDMA) [3], [4]. The strategy presented in those papers is that all users transmit at the maximum achievable EE, constrained by the maximum power available. As a consequence, some users can allocate the maximum power without transmitting at the optimum EE, increasing the generated interference. As CDMA systems are limited by interference, specially when using singleuser detections techniques, putting some of these nonoptimum users in outage can result in interference reduction and greater EE. On the other hand, since rate and spectral efficiency (SE) remain as fundamental metrics for modern wireless systems, it becomes necessary to investigate the gap between energy and spectral efficienciesbased allocation processes, determining the tradeoff among these two metrics and the optimum operating point. This issue is an important opened topic in Green Communications area []. Previously, Álvaro R. C. Souza and Taufi Abrão, Dept. of Electrical Engineering and Computer Science. State University of Londrina, LondrinaPR, Brazil. Fábio R. Durand, Dept. of Electrical Engineering, UTFPR  Campo Mourão PR. Lucas H. Sampaio and Paul Jean E. Jeszensy, Dept. of Telecomm. and Control Engineering, Escola Politécnica of the University of São Paulo. São PauloSP, Brazil. s: {alvarorcsouza, the EESE tradeoff has been investigated in [6] considering OFDMA systems and some simplifications in the system model, given the difficulty in the analytical characterization. This wor proposes to analyze the existing EESE tradeoff for DSCDMA systems under matchedfilter (MF) detection, determining the optimum EESE tradeoff, as well as to the impact of multipleaccess interference (MAI) on this tradeoff. In order to corroborate the conclusions of this analysis, we develop two algorithms for power allocation using game theory (specially noncooperative games), since the overall networ EE deps on the behavior of all users. II. NETWORK SYSTEM MODEL For analysis simplicity, we have assumed an uplin direct sequence code division multiple access (DS/CDMA) networ. However, the extension for multicell multicarrier multiple access systems is straightforward. The received signal by the base station can be described as K y = p h b s +η, () = where p is the transmitted power for the th user, h is the channel gain between the th user and the base station, constant during the chip period, s is the th user spreading code, b is the modulated symbol and η is the thermal noise vector, assumed to be AWGN, zeromean and covariance matrix given by σ I N, where N is the processing gain. The uplin K channel gain vector, considering path loss, shadowing and fading effects, between users and the base station, is given by h = [h,,h K ], with h i = h i e hi and assumed to be static over the optimization window. The signaltointerferenceplusnoise ratio (SINR) is defined by the received signal power to the sum of interfering power plus bacground noise, measured after demodulation. Adopting a matched filter (MF) receiver and random spreading sequence, the SINR output for th user is given by: γ = p h K p j h j s j s T +σ j j= = p h I +σ, () where I represents the MAI power level and s s T =. A. QoS Requirements In order to guarantee the quality of service (QoS), a minimum data rate R,min must be provided for each user by the system networ service, being an important requirement to be warranted. So, in general, data rate for the th user is assumed to be a function of SINR γ. To model it, we use a modified version of Shannon capacity equation: [7], given by: r = C gap = wlog (+θ γ ), [bit/s], (3)
2 XXX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES  SBrT, 36 DE SETEMBRO DE, BRASÍLIA, DF where w is the system bandwidth and θ is the gap describing the limitations and imperfections in real communication systems, approaching to real data rates [8], given by. θ = log( BER ), with θ ];], (4) where BER is the maximum tolerable bit error rate by the th user [9]. So, the SE is readily obtained from (3): ζ = log (+θ γ ), [bit/(s Hz)]. () From (3), the minimum data rate for the th lin, R,min, which is able to guarantee the QoS, can be easily mapped into the minimum SINR R,min rc γ,min =, =,...,K. (6) θ III. PROBLEM FORMULATION In a MAI limited communication system, the th user selfishly (non cooperative approach) allocates his own transmit power p in order to maximize his own energy efficiency function, given by []: L f(γ ) ξ = r M p +p c [ ] bit, =,...,K (7) Joule where M is the number of bits in each transmitted data pacet; L is the number of information bits contained in each data pacet, p c is the circuit power consumption, and f(γ ) is the efficiency function, which approximates the probability of errorfree pacet reception. Given a noncoded communication, it can be approximated by [] f(γ ) = ( e γ ) M, (8) which is widely accepted for BPSK and QPSK modulation. It is worth noting that both transmission power and circuit power consumptions are very important factors for energyefficient communications. While p is used for reliable data transmission, circuit power represents average energy consumption of device electronics [4]. Besides, note that ξ is measured in [ bit Joule], which represents the number of successful bit transmissions that can be made for each energyunit drained from the battery and effectively used for transmission. In a more general context, we can define the global energy efficiency function as the ratio of the total achievable capacity over the total power transmission consumption: K = ξ = l [ ] r f(γ ) bit, (9) Joule P Tot where P Tot = K = (p +p c ), and l = ( L M A. Distributed Noncooperative EE Power Optimization Game The networ energy efficiency deps on the behavior of all users; so, the power control can be properly modeled as a noncooperative game []. In this context the noncooperative power control game is defined as G = [K, {A }, {u }], () ) where K = {,,...,K} is the set of players (users), {A } = [, ] is the strategy set for the th user, with being the maximum allowed power for transmission, and the utility function {u } is performed by (7). Consider the power allocation for the th user, p, and denote the respective power vector of other users as p = [p,p,...,p,p +,...,p K ]. () Hence, given the power allocation of all interfering users,p, the bestresponse of the power allocation for the th user can be expressed as p best = f (p ) = argmax p u (p,p ), () where f (p ) is the th bestresponse function. Finally, the distributed energyefficiency problem with power constraint under noncooperative game perspective may be posed by: f(γ ) arg max ξ = arg max l r (3) p p p +p c s.t. p which solution consists in adopting the bestresponse strategy for user. Indeed, the bestresponse strategy consists in obtain the best user utility function (EE) individually for each user, as posed by (). B. Best SINR Response Since the EE utility function (7) deps on p and γ and both of them are related, we can define p as a function of γ using (), obtaining p = γ I +σ h = γ Ĩ, (4) where Ĩ is the multiple access interference plus noise normalized by the channel gain. Applying (4) into (7) and taing it s first derivative ( ξ γ = ), we can determine the point(s) γ that maximizes the th user s EE. So, after some simplifications, the solution of ξ γ = for a fixed value of Ĩ is given by Me γ log (+θ γ )+ θ ( e γ ) (+θ γ )ln = () Ĩ ( e γ ) (γ Ĩ +p c ) log (+θ γ ). In order to guarantee that () has only one maximizer, we introduce the concept of strict quasiconcavity, defined as [3]: Definition (Strict quasiconcavity): A function z, that maps a convex set of ndimensional vectors D into a real number is strict quasiconcave if for any x,x D,x x, z(λx +( λ)x ) > min{z(x ),z(x )}, λ [,]. (6) With this definition, the strict quasiconcavity of u is summarized by Lemma : Lemma (Strict quasiconcavity of u ): The utility function u (p,p ) is strict quasiconcave in p This result is very important in the proof of existence and uniqueness of the system equilibrium. However, due to space limitation all proofs are not developed herein.
3 XXX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES  SBrT, 36 DE SETEMBRO DE, BRASÍLIA, DF IV. INCREASING INTERFERENCE EFFECT AND NASH EQUILIBRIUM ON EESE TRADEOFF In this section we present the tradeoff between noncooperative energyefficient and spectralefficient power control schemes. This tradeoff is determined by the MAI level, which is responsible by the gap among the maximal EE and the optimum SE (only attainable with infinity power allocation). In realistic interferenceaware systems the increasing number of active users brings to an increment on system capacity; therefore, the SE of the system grows accordingly. We define the gap between the maxee and the optse as Λ = ζ(γ,se) ζ(γ,ee), [bit/(s Hz)] (7) and then characterized its reduction when the interference level increases. To quantify this effect, we have defined the coupling networ parameter β = h / h j, : interest; j : interfering users, where is the operator temporal average. We consider a ring cell geometry, where the th interest user is located at the interior radius (d) and the interfering (K ) users are located at the exterior radius (d interf ), varying the number of interfering users and exterior radius size (to affect the β factor). Hence, in Fig., the maxee and the optse are presented and parameterized in terms of the system loading and d i interf,i. It is clear the gap reduction between the maxee and the asymptoticse when the interference level increases. Even increasing the maximum power (increasing the maximum SE achievable), we observe the same behavior, besides the EE reduction given the MAI level increasing (approximately two magnitude orders). Table I shows the system parameters used in this hypothetic simulation scenario. Given the fact that in real communication DS/CDMA systems the interest is in high system loading which implies in higher MAI, we observe that the optimal SINR for the EESE tradeoff is the SINR that maximizes EE (γ,ee ). As we can see from Fig., when the number of users is increased, the gap between the SE at the maxee point and optse is reduced and even become null. We can see from Fig..d, in some cases the MAI power level is so high that maes impossible to some users achieve the maxee point without using a higher power than the maximum power allowed. In that cases, these users allocate maximum power to achieve the bigger EE they can, since the utility function is strictly increasing in the interval [, p ]. This approach increases the MAI power level, which could reduce system s EE; however, if some of these nonoptimal users are put in outage, we can increase the energy efficiency as a consequence of the MAI level reduction. When circuit power consumption is much smaller than the transmitted power (p c << p ), an interesting result emerges: the optimum SINR obtained from the EE optimization problem in () is the same for any MAI level, while the asymptotic SINR necessary to the SE maximization still related to the interference power level, Ĩ. Hence, under this hypothesis, the best SINR for maxee criterium deps only on the system parameters, such as maxber (QoS), modulation level, coding and pacet coding size. It is worth noting that when the MAI a) b). x 8 EE SE trade off x Number of active users d int = m.. Λ (K = 9) Λ (K = 6) Λ (K = 3) = dbm x 6 EE SE trade off x Number of active users d int = m Λ (K = 9) Λ (K = 6) Λ (K = 3) = dbw x 8 EE SE trade off x Interfering users distance (K = 6) = 8m) Λ (d int = m) = m) 3 4 c) d) 8 x EE SE trade off x Interfering users distance (K = 6) = m) = m) = 8m) = dbm = m = m = 8m = m = m = 8m = dbw = m = m = 8m = m = m = 8m Fig.. EESE tradeoffs considering different interfering scenarios and maximum power. a) and b) Fixed d interf = m, β =., K [3,6,9]; c) and d) Fixed K = 6, d interf [,,8]m, β [.,.,.63]... Power (mw) Power (W) Power (mw) Power (W)
4 XXX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES  SBrT, 36 DE SETEMBRO DE, BRASÍLIA, DF increases, the transmitted power becomes higher and indeed the condition p c << p holds, and the optimum SINR value ts to converge. TABLE I EESE TRADEOFF ANALYSIS Parameters Adopted Values DS/CDMA Noise Power P n = 9 [dbm] Processing Gain N = Max. power per user = [dbm], [dbw] Interest user distance d = [m] Interfering users distance d interf = [8,,] [m] Pacet size M = 8 [bits] Data bits L = [bits] Maximum BER BER = 3 Circuit Power p c = 7 [dbm] Bandwidth w = 6 [Hz] Channel Gain Path loss d Fading coefficients Rayleigh distribution, mean over samples Verhulst PCA Convergence factor α =. Number of iterations N it = V. PROPOSED EESE ALGORITHM The proposed algorithm to implement the optimal EESE tradeoff solution is described in Algorithm and is based on Verhulst power control algorithm [4]. On the other hand, in order to avoid users outage, in which users are not able to achieve the optimal SINR in terms of EE (due to constraint), but are able to maintain the minimum data rate, R,min, an alternative algorithm is proposed in Algorithm 3. Both algorithms uses the same structure, that corresponds to the basic algorithm defined in the literature [], [], [] and presented in Algorithm. Algorithm Basic EE Power Allocation Algorithm [], [] Initialization: i, N it, p [] = σ, for i = : N it for = : K: Evaluate Ĩ (via SINR measurement); Find γ, by solving (); Find p iteratively using Verhulst Algorithm [4]. Output: p Algorithm EESE with Verhulst Optimum Power Allocation Initialization: K out = {} Execute Algorithm, obtaining p ; Compute achieved SINR and optimum SINR (γ) for each user; Compute K out, where K out if γ < γ; if {K out} choose user with worst channel gain in K out (jth user); set γj = ; Restart. Output: p (EESE tradeoff solution with maxee) Existence and uniqueness of the achieved equilibriums. Given that the equilibrium is defined by p = (p,...,p,...,p K ), the Nash equilibrium can be defined as: Definition (Nash Equilibrium): An equilibrium is said to be a Nash equilibrium if and only if any user cannot improve their response by changing unilaterally the optimum value achieved []. In the context of the energyefficiency problem: u (p,p ) u (p,p ), p p. (8) Algorithm 3 EESE R,min and Verhulst Power Allocation Initialization: K out = {} Execute Algorithm, obtaining p ; Compute achieved SINR (γ ) and rate (R ) for each user; Compute the optimum SINR (γ ) for each user; Compute K out, where K out if γ < γ and R < R,min ; if {K out} Choose user with worst channel gain in K out (jth user); Set γ j = ; Restart. Output: p (EESE tradeoff solution with R,min ) The conditions for the existence of the Nash equilibrium in the EE SE tradeoff noncooperative game, implicit in Algorithms and 3, are summarized by Theorems and : Theorem : The system achieves at least one equilibrium p for Algorithm, and p,...,p,...,p K p are defined by the following conditions: ) If p and (u (p,p )) =, then p = p ) Else, p = Theorem : The system achieves at least one equilibrium p for Algorithm, with p p obeying the conditions: ) If p and (u (p,p )) =, then p = p (u ) If p =, (p,p )) and R R,min, then p = 3) Else, p = The first condition in the Theorem occurs when the user has sufficient power to achieve the optimum SINR point; the second one occurs when the user cannot achieve the optimum point (the outage scenario). The first condition in the Theorem is the same condition of the first condition of Theorem ; however, when the user cannot achieve the maximum efficiency but is able to achieve a minimum rate criterion, in the second condition, th user set his transmit power to the maximum available. Finally, when the two criteria fails, the user must set his Tx power to zero. The uniqueness of the Nash equilibrium for both noncooperative games is summarized in Lemma. Lemma : When the equilibrium p is achieved without removing any user, this Nash equilibrium is unique. When is needed to remove any user, multiple equilibriums could exist, but for our adopted criterion, the equilibrium is also unique. VI. NUMERICAL RESULTS In this section, some changes are introduced in the system parameters in Table I. Analysis hereafter assumes a ring geometry, with internal radius r int = m and external radius r ext = m, with K mobile users uniformly distributed on U[r int, r ext ], and the base station in the center of the ring. The processing gain was assumed N = 63; maximal transmitted power per user was = dbm, while
5 XXX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES  SBrT, 36 DE SETEMBRO DE, BRASÍLIA, DF circuit power was fixed p c = 7 dbm per user; number of mobile terminals K {; }. For all users, same QoS have been adopted, i.e., maximal BER = 3 and R min = [bps]. Fading is modeled as Rayleigh distribution (module), simulated by a complex Gaussian random process, with zero mean and variance given by d j. In order to analyze the average networ behavior, the average results have been taen over realizations with random positions, channel and spreading sequences. We assume that the mobile transmitter has perfect channel state information (CSI) available, but the measurement of other mobile users CSI can only be made through the quantized bits transmitted by the base station. Figs. and 3 present four metrics to analyze the two proposed algorithms: sum of rates of all users, R, sum of power consumption, including the circuit power, P, overall energy efficiency and the outage probability. The first three metrics also bring the comparison with the classical approach []. From Fig. one can conclude that the Algorithm 3 obtains the best result for sum rate maximization, mainly when the system loading increases, since only a user is dropped when the rate achieved by this user is lower than the minimum rate R min. As a consequence, the power consumption (rightside axis) is increased, because any user that doesn t achieve the optimum SINR tries to achieve it using maximum power allowed, increasing remarably the interference level. R (bps) Fig.. x Sum Rate x Sum Power 7 3 SR (Alg. ) SR (Alg. 3) SR (Alg. Lit) SP (Alg. ) SP (Alg. 3) SP (Alg. Lit) # of active users (K) P (mw).6 x R (bps) Sum rate versus Sum Power for the proposed algorithms. Fig. 3.a indicates that the rate improvement obtained by the Algorithm 3 and literature s approach reduces the system s energy efficiency, once that there are users transmitting with nonoptimal powers. The bestresponse in terms of energy efficiency is achieved by Algorithm, but incurs in more users in outage (Fig. 3.b). Although there is a marginal powerrate tradeoff difference among the two proposed algorithms, both are more energyefficient than the common literature approach. VII. CONCLUSIONS In this wor we have analyzed the distributed energy efficiency (EE) cost function taing in perspective the two conflicting metrics, throughput maximization and the power consumption minimization. We have found that SINR on the maxee equilibrium deps on the MAI power level when considering circuit power, mainly when MAI level is low. As the interference increases, the EESE tradeoff gap is reduced; the optimum SINR is that maximizes EE. Finally, we have shown that removing nonoptimal EE users allows better energyefficiency, at the cost of higher outage probability. Fig Energy Efficiency x System Loading (N = 63) Algorithm Algorithm 3 Algorithm (Lit) a) System Loading (L) Percentage of users in outage b) Percentage of users in outage x System Loading (N = 63) Algorithm Algorithm # of active users (K) a) Energy Efficiency and b) outage probability. REFERENCES [] S. Buzzi and H. Poor, Joint receiver and transmitter optimization for energyefficient cdma communications, Selected Areas in Communications, IEEE Journal on, vol. 6, no. 3, pp , Apr. 8. [] F. Meshati, H. V. Poor, and S. C. Schwartz, Energyefficient resource allocation in wireless networs: An overview of gametheoretic approaches, IEEE Signal Process. Mag., vol. 4, pp. 8 68, May 7. [3] G. Miao, N. Himayat, G. Li, and S. Talwa, Lowcomplexity energyefficient ofdma, in IEEE International Conference on Communications, 9. ICC 9., Dresden, Germany, Jun. 9, pp.. [4] G. Miao, N. Himayat, and Y. Li, Energyefficient lin adaptation in frequencyselective channels, IEEE Trans. Commun., vol. 8, no., pp. 4 4, Feb.. [] Y. Chen, S. Zhang, S. Xu, and G. Li, Fundamental tradeoffs on green wireless networs, IEEE Commun. Mag., vol. 49, no. 6, pp. 3 37, Jun.. [6] G. Miao, N. Himayat, G. Li, and S. Talwar, Distributed interferenceaware energyefficient power optimization, IEEE Trans. Wireless Commun., vol., no. 4, pp , Apr.. [7] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, vol. 7, pp and 63 66, 948. [8] D. Tse and P. Viswanath, Fundamentals of Wireless Communications. England: Cambridge University Press, Feb.. [9] A. J. Goldsmith and S. G. Chua, Variablerate variablepower mqam for fading channels, IEEE Trans. Commun., vol. 4, no., pp. 8 3, Oct [] D. J. Goodman and N. B. Mandayan, Power control for wireless communication, IEEE Pers. Comm. Mag., vol. 7, no. 4, pp. 48 4, Apr.. [] F. Meshati, H. Poor, S. Schwartz, and N. Mandayam, An energyefficient approach to power control and receiver design in wireless data networs, IEEE Trans. Commun., vol. 3, no., pp , Nov.. [] D. Fudenberg and J. Tirole, Game Theory. USA: MIT Press, 99. [3] V. Rodriguez, An analytical foundation for resource management in wireless communication, in GLOBECOM 3, vol.. San Francisco, USA: IEEE, Dec. 3, pp Vol.. [4] T. J. Gross, T. Abrão, and P. J. E. Jeszensy, Distributed power control algorithm for multiple access systems based on verhulst model, AEU  Intern. Jour. Elect. & Commun., vol. 6, no. 4, pp ,.
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