Fractional Power Control for Decentralized Wireless Networks

Transcription

1 Fractiona Power Contro for Decentraized Wireess Networks Nihar Jinda, Steven Weber, Jeffrey G. Andrews Abstract arxiv: v2 [cs.it] 28 Apr 2008 We consider a new approach to power contro in decentraized wireess networks, termed fractiona power contro FPC). Transmission power is chosen as the current channe quaity raised to an exponent s, where s is a constant between 0 and. The choices s = and s = 0 correspond to the famiiar cases of channe inversion and constant power transmission, respectivey. Choosing s 0, ) aows a intermediate poicies between these two extremes to be evauated, and we see that usuay neither extreme is idea. We derive cosed-form approximations for the outage probabiity reative to a target SINR in a decentraized ad hoc or unicensed) network as we as for the resuting transmission capacity, which is the number of users/m 2 that can achieve this SINR on average. Using these approximations, which are quite accurate over typica system parameter vaues, we prove that using an exponent of s = 2 minimizes the outage probabiity, meaning that the inverse square root of the channe strength is a sensibe transmit power scaing for networks with a reativey ow density of interferers. We aso show numericay that this choice of s is robust to a wide range of variations in the network parameters. Intuitivey, s = 2 baances between heping disadvantaged users whie making sure they do not food the network with interference. I. INTRODUCTION Power contro is a fundamenta adaptation mechanism in wireess networks, and is used to at east some extent in virtuay a terrestria wireess systems. For a singe user fading channe in which the objective is to maximize expected rate, it is optima to increase transmission power and rate) as a function of the instantaneous channe quaity according to the we-known waterfiing poicy [2]. On the other hand, if the objective is to consistenty achieve a target rate or ), then the power shoud be adjusted so that this target eve is exacty met. Such an objective is phiosophicay the opposite of waterfiing, since power is inversey reated to the instantaneous channe quaity: we ca this channe inversion. Athough suboptima from an information theory point of view, some channe inversion is used in many modern wireess systems to adapt to the extreme dynamic range often > 50 db due to path oss differences as we as mutipath fading) that those systems experience, to provide a baseine user experience over a ong-term time-scae. A. Background and Motivation for Fractiona Power Contro In a muti-user network in which users mutuay interfere, power contro can be used to adjust transmit power eves so that a users simutaneousy can achieve their target SINR eves. The Foschini-Mijanic agorithm is an iterative, distributed power contro method that performs this task assuming that each receiver tracks its instantaneous SINR and feeds back power adjustments to its transmitter [3]. Considerabe work has deepy expored the properties of these agorithms, incuding deveoping a framework that describes a power contro probems of this type [4], as we as studying the feasibiity and impementation of such agorithms [5], [6], incuding with varying channes [7]; see the recent monographs [8][9] for exceent surveys of the vast body of iterature. This body of work, whie in many respects quite genera, has been primariy focused on the ceuar wireess communications architecture, particuary in which a users have a common receiver i.e., the upink). More recenty, there has been considerabe interest in power contro for decentraized wireess networks, such as unicensed spectrum access and ad hoc networks [0], [], [2], [3], [4], [5]. A key distinguishing trait of a decentraized network is that users transmit to distinct receivers in the same geographic area, which causes the power contro properties to change consideraby. The contact author N. Jinda nihar@umn.edu) is with the University of Minnesota, S. Weber is with Drexe University, J. Andrews is with the University of Texas at Austin. This research was supported by NSF grant no Jinda), no Weber), nos and Andrews), and the DARPA IT-MANET program, grant no. W9NF a authors). An eary, shorter version of this work appeared at Aerton 2007 []. Manuscript date: Apri 28, 2008.

2 2 In this paper, we expore the optima power contro poicy for a muti-user decentraized wireess network with mutuay interfering users and a common target SINR. We do not consider iterative agorithms and their convergence. Rather, motivated by the poor performance of channe inversion in decentraized networks [6], we deveop a new transmit power poicy caed fractiona power contro, which is neither channe inversion nor fixed transmit power, but rather a trade-off between them. Motivated by a recent Motoroa proposa [7] for fairness in ceuar networks, we consider a poicy where if H is the channe power between the transmitter and receiver, a transmission power of H s is used, where s is chosen in [0, ]. Ceary, s = 0 impies constant transmit power, whereas s = is channe inversion. The natura question then is: what is an appropriate choice of s? We presume that s is decided offine and that a users in the network utiize the same s. B. Technica Approach We consider a spatiay distributed decentraized) network, representing either a wireess ad hoc network or unicensed spectrum usage by many nodes e.g., Wi-Fi or spectrum sharing systems). We consider a network that has the foowing key characteristics. Each transmitter communicates with a singe receiver that is a distance d meters away. Channe attenuation is determined by path oss with exponent α) and a fat) fading vaue H. Each transmitter knows the channe power to its intended receiver, but has no knowedge about other transmissions. A muti-user interference is treated as noise. Transmitters do not schedue their transmissions based on their channe conditions or the activities of other nodes. Transmitter node ocations are modeed by a homogeneous spatia 2-D) Poisson process. These modeing assumptions are made to simpify the anaysis, but in genera reasonaby mode a decentraized wireess network with random transmitter ocations, and imited feedback mechanisms. In particuar, the above assumptions refer to the situation where a connection has been estabished between a transmitter and receiver, in which case the channe power can be earned quicky either through reciprocity or a few bits of feedback. It is not however as easy to earn the interference eve since it may change suddeny as interferers turn on and off or physicay move and reciprocity does not hep). The fixed transmit distance assumption is admittedy somewhat artificia, but is significanty easier to hande anayticay, and has been shown to preserve the integrity of concusions even with random transmit distances. For exampe, [6], [8] prove that picking the source-destination distance d from an arbitrary random distribution reduces the transmission capacity by a constant factor of E[d 2 ]/E[d]) 2. Therefore, athough fixed distance d can be considered best-case as far as the numerica vaue of transmission capacity, this constant factor wi not change fractiona power contro s reative effect on the transmission capacity, which is the subject of this paper. C. Contributions and Organization The contributions of the paper are the suggestion of fractiona power contro for wireess networks and the derivation of the optimum power contro exponent s = 2. The exponent s = 2 is shown to be optima for an approximation to the outage probabiity/transmission that is vaid for reativey ow density networks that are primariy interference-imited i.e., the effect of therma noise is not overy arge); if the reative density or the effect of noise is arge, then our numerica resuts show that no power contro s = 0) is generay preferred. In the reativey arge parameter space where our primary approximation is vaid, fractiona power contro with the choice s = 2 is shown to greaty increase the transmission capacity of a -hop ad hoc network for sma path oss exponents as α 2), with more modest gains for higher attenuation channes. The resuts open a number of possibe avenues for future work in the area of power contro, and considering the prevaence of power contro in practice, carry severa design impications. The remainder of the paper is organized as foows. Section II provides background materia on the system mode, and key prior resuts on transmission capacity that are utiized in this paper. Section III hods the main resuts, namey Theorem 3 which gives the outage probabiity and transmission capacity achieved by fractiona power contro, and Theorem 4 which determines the optimum power contro exponent s for the outage probabiity approximation. Section IV provides numerica pots that expore the numericay computed optima s, which provides insight on

3 3 how to choose s in a rea wireess network. Section V suggests possibe extensions and appications of fractiona power contro, whie Section VI concudes the paper. A. System Mode II. PRELIMINARIES We consider a set of transmitting nodes at an arbitrary snapshot in time with ocations specified by a homogeneous Poisson point process PPP), Πλ), of intensity λ on the infinite two-dimensiona pane, R 2. We consider a reference transmitter-receiver pair, where the reference receiver, assigned index 0, is ocated without oss of generaity, at the origin. Let X i denote the distance of the i-th transmitting node to the reference receiver. Each transmitter has an associated receiver that is assumed to be ocated a fixed distance d meters away. Let H i0 denote the random) distance independent fading coefficient for the channe separating transmitter i and the reference receiver at the origin; et H ii denote the random) distance independent fading coefficient for the channe separating transmitter i from its intended receiver. We assume that a the H ij are i.i.d. incuding i = j), which impies that no sourcedestination S-D) pair has both a transmitter and receiver that are very cose ess than a waveength) to one another, which is reasonabe. Received power is modeed by the product of transmission power, pathoss with exponent α > 2), and a fading coefficient. Therefore, the random) SINR at the reference receiver is: P 0 H 00 d α SINR 0 = i Πλ) P ih i0 Xi α + η, ) where η is the noise power. Reca our assumption that transmitters have knowedge of the channe condition, H ii, connecting it with its intended receiver. By expoiting this knowedge, the transmission power, P i, may depend upon the channe, H ii. If Gaussian signaing is used, the corresponding achievabe rate per unit bandwidth) is og 2 + SINR 0 ). The Poisson mode requires that nodes decide to transmit independenty, which corresponds in the above mode to sotted ALOHA [9]. A good scheduing agorithm by definition introduces correation into the set of transmitting nodes, which is therefore not we modeed by a homogeneous PPP. We discuss the impications of scheduing ater in the paper. B. Transmission Capacity In the outage-based transmission capacity framework, an outage occurs whenever the SINR fas beow a prescribed threshod, or equivaenty whenever the instantaneous mutua information fas beow og 2 + ). Therefore, the system-wide outage probabiity is qλ) = PSINR 0 < ) 2) Because 2) is computed over the distribution of transmitter positions as we as the iid fading coefficients and consequenty transmission powers), it corresponds to fading that occurs on a time-scae that is comparabe or sower than the packet duration if 2) is to correspond roughy to the packet error rate). The outage probabiity is ceary a continuous increasing function of the intensity λ. Define λɛ) as the maximum intensity of attempted transmissions such that the outage probabiity is no arger than ɛ, i.e., λɛ) is the unique soution of qλ) = ɛ. The transmission capacity is then defined as cɛ) = λɛ) ɛ)b, which is the maximum density of successfu transmissions times the spectra efficiency b of each transmission. In other words, transmission capacity is area spectra efficiency subject to an outage constraint. For the sake of carity, we define the constants δ = 2/α < and = pd α η. Now consider a path-oss ony environment H i0 = for a i) with constant transmission power P i = p for a i). The main resut of [8] is given in the foowing theorem. Theorem [8]): Pure pathoss. Consider a network where the SINR at the reference receiver is given by 2) with H i0 = and P i = p for a i. Then the foowing expressions give bounds on the outage probabiity and transmission attempt intensity for λ, ɛ sma: { q p λ) q p λ) = exp λπd 2 ) } δ, 3) λ p ɛ) λ p u ɛ) = og ɛ) πd 2 ) δ. 4)

4 4 Here p denotes pathoss. The transmission attempt intensity upper bound, λ p u ɛ), is obtained by soving q p λ) = ɛ for λ. These bounds are shown to be approximations for sma λ, ɛ respectivey, which is the usua regime of interest. Note aso that og ɛ) = ɛ + Oɛ 2 ), which impies that transmission density is approximatey inear with the desired outage eve, ɛ, for sma outages. The foowing coroary iustrates the simpification of the above resuts when the noise may be ignored. Coroary : When η = 0 the expressions in Theorem simpify to: q p λ) q p λ) = exp { λπd 2 δ}, 5) λ p ɛ) λ p u ɛ) = og ɛ) πd 2 δ. 6) III. FRACTIONAL POWER CONTROL The goa of the paper is to determine the effect that fractiona power contro has on the outage probabiity ower bound in 3) and hence the transmission capacity upper bound in 4). We first review the key prior resut that we wi use, then derive the maximum transmission densities λ for different power contro poicies. We concude the section by finding the optima power contro exponent s. A. Transmission capacity under constant power and channe inversion In this subsection we restrict our attention to two we-known power contro strategies: constant transmit power or no power contro) and channe inversion. Under constant power, P i = p for a i for some common power eve p. Under channe inversion, P i = p E[H ] H ii for a i. This means that the received signa power is P i H ii d α = p E[H ] d α, which is constant for a i. That is, channe inversion compensates for the random channe fuctuations between each transmitter and its intended receiver. Moreover, the expected transmission power is E[P i ] = p, so that the constant power and channe inversion schemes use the same expected power. We woud ike to emphasize the distribution of H is arbitrary and can be adapted in principe to any reevant fading or compound shadowing-fading mode. For some possibe distributions such as Rayeigh fading, i.e. H exp)), the vaue E[H ] may be undefined, stricty speaking. In practice, the transmit power is finite and so P i = p E[H ] H ii is finite. The vaue E[H ] is simpy a normaizing factor and can be interpreted mathematicay to mean that H minh, δ) for an arbitrariy sma δ. Such a definition woud not affect the resuts in the paper. A main resut of [6] extended to incude therma noise is given in the foowing theorem, with a genera proof that wi appy to a three cases of interest: constant power, channe inversion and fractiona power contro. Note that cp and ci are used to denote constant power and channe inversion, respectivey. Theorem 2: Constant power. Consider a network where the SINR at the reference receiver is given by 2) with P i = p for a i. Then the foowing expressions give good approximations of the outage probabiity and transmission attempt intensity for λ, ɛ sma. q cp λ) q cp λ) = P H 00 q cp λ) = P H 00 λ cp ɛ) λ cp ɛ) = og ɛ P H 00 ) [ { E exp λπd 2 E[H δ H00 ] ) { exp λπd 2 E[H δ ]E ) πd 2 E[H δ ] E [ H00 ) } ] δ H00 ) ]} δ H00 ) ] δ H00. 7) [ H00 Channe inversion. Consider the same network with P i = E[H ] H ii for a i. Then the foowing expressions give tight bounds on the outage probabiity and transmission attempt intensity for λ, ɛ sma: ) δ } q ci λ) q { λπd ci λ) = exp 2 E[H δ ]E[H δ ] E[H ] 8) λ ci ɛ) λ ci u ɛ) = og ɛ) ) δ πd 2 E[H δ ]E[H δ ] E[H ]. 9) p

5 5 Proof: The SINR at the reference receiver for a generic power vector {P i } is SINR 0 = P 0 H 00 d α i Πλ) P ih i0 X α i + η, 0) and the corresponding outage probabiity is ) P 0 H 00 d α qλ) = PSINR 0 < ) = P i Πλ) P ih i0 Xi α + η <. ) Rearranging yieds: qλ) = P i Πλ) P i H i0 X α i P 0H 00 d α η. 2) Note that outage is certain when P 0 H 00 < ηd α. Conditioning on P 0 H 00 and using f ) to denote the density of P 0 H 00 yieds: qλ) = P P 0 H 00 ηd α ) + P i H i0 Xi α p 0h 00 d α η P0 H 00 = p 0 h 00 fp 0 h 00 )dp 0 h 00 ). ηd α P i Πλ) 3) Reca the generic ower bound from [6]: if Πλ) = {X i, Z i )} is a homogeneous marked Poisson point process with points {X i } of intensity λ and iid marks {Z i } independent of the {X i }, then P i Πλ) Z i X α i > y exp Appying here with Z i = P i H i0 and y = p0h00 d η: α { qλ) P P 0 H 00 ηd α ) + exp πλe[p i H i0 ) δ ] ηd α { = exp πλe[p i H i0 ) δ p0 h 00 ] ηd d α α [ { = P P 0 H 00 ηd α ) E exp { πλe[z δ ]y δ}, 4) p0 h 00 d α ) } δ η fp 0 h 00 )dp 0 h 00 ) λπd 2 E[P i H i0 ) δ P0 H 00 ] ) }) δ η fp 0 h 00 )dp 0 h 00 ) η ) } ] δ P0 d α H 00 ηd α. 5) The imation for this quantity is: { qλ) P P 0 H 00 ηd α ) exp λπd 2 E[P i H i0 ) δ ]E [ P0 H 00 η ) ]} δ P0 d α H 00 ηd α. 6) For constant power we substitute P i H i0 = ph i0 for a i) into 5) and 6) and manipuate to get the expressions for q cp λ) and q cp λ) in 7). To obtain λ cp ɛ), we sove q cp λ) = ɛ for λ. For channe inversion, P 0 H 00 = p E[H ] whie for i 0 we have P i H i0 = p H i0 E[H ] H ii. Pugging into 5) and using the fact that H ii and H i0 are i.i.d. yieds 8), and 9) is simpy the inverse of 8). Note that channe inversion ony makes sense when E[H ] = pd α ηe[h ], the effective interference-free after taking into account the power cost of inversion, is arger than the SINR threshod. The vaidity of the outage ower bound/density upper bound as we as of the Jensen s approximation are evauated in the numerica and simuation resuts in Section IV. When the therma noise can be ignored, these resuts simpify to the expressions given in the foowing coroary:

6 6 Coroary 2: When η = 0 the expressions in Theorem 2 simpify to: [ { [ q cp λ) q cp λ) = E exp λπd 2 δ E H δ] }] H00 δ { [ q cp λ) = exp λπd 2 δ E H δ] [ E H δ]}, [ q ci λ) q { λπd ci λ) = exp 2 δ E H δ] [ E H δ]}, λ cp ɛ) λ cp ɛ) = og ɛ) πd 2 δ E [H δ ] E [H δ ], λ ci ɛ) λ ci u ɛ) = og ɛ) πd 2 δ E [H δ ] E [H δ ]. 7) Note that these expressions match Theorem 3 and Coroary 3 of the SIR-anaysis performed in [6]. In the absence of noise the constant power outage probabiity approximation equas the channe inversion outage probabiity ower bound: q cp λ) = q ci λ). As a resut, the constant power transmission attempt intensity approximation equas the channe inversion transmission attempt intensity upper bound: λ cp ɛ) = λ ci u ɛ). Comparing λ cp ɛ) = λ ci u ɛ) in 7) with λ p u ɛ) in 6) it is evident that the impact of fading on the transmission capacity is measured by the oss factor, L cp = L ci, defined as L cp = L ci = E [H δ ] E [H δ <. 8) ] The inequaity is obtained by appying Jensen s inequaity to the convex function /x and the random variabe H δ. If constant power is used, the E[H δ ] term is due to fading of the desired signa whie the E[H δ ] term is due to fading of the interfering inks. Fading of the interfering signa has a positive effect whie fading of the desired signa has a negative effect. If channe inversion is performed the E[H δ ] term is due to each interfering transmitter using power proportiona to Hii. When the path oss exponent, α, is cose to 2 then δ = 2/α is cose to one, so the term E[H δ ] is neary equa to the expectation of the inverse of the fading, which can be extremey arge for severe fading distributions such as Rayeigh. As a ess severe exampe, α = 3, the oss factor for Rayeigh fading is L cp = L ci = 0.4. B. Transmission capacity under fractiona power contro In this section we generaize the resuts of Theorem 2 by introducing fractiona power contro FPC) with p parameter s [0, ]. Under FPC the transmission power is set to P i = E[H s ] H s ii for each i. The received power at receiver i is then P i H ii d α = p E[H s ] H s ii d α, which depends upon i aside from s =. The expected transmission power is p, ensuring a fair comparison with the resuts in Theorems and 2. Note that constant power corresponds to s = 0 and channe inversion corresponds to s =. The foowing theorem gives good approximations on the outage probabiity and maximum aowabe transmission intensity under FPC. Theorem 3: Fractiona power contro. Consider a network where the SINR at the reference receiver is given by 2) with P i = p E[H s ] H s ii for a i, for some s [0, ]. Then the foowing expressions give good approximations of the outage probabiity and maximum transmission attempt intensity for λ, ɛ sma q fpc λ) q fpc λ) = P H 00 κs)) [ { q fpc E exp λ) = P H 00 κs)) { exp λ fpc ɛ) λ fpc ɛ) = og λπd 2 E[H sδ ]E[H δ ] λπd 2 E[H sδ ]E[H δ ]E ɛ P H 00 κs)) ) πd 2 H s 00 [ H s 00 E[H sδ ]E[H δ ] [ H s 00 E ) δ } E[H s ] H00 κs)] ) δ E[H s ] H00 κs)]} ) δ E[H s ] H00 κs)])

7 7 ) s where κs) = E[H s ]. p Proof: Under FPC, the transmit power for each user is constructed as P i = E[H s ] H s ii. Substituting this vaue into the proof for Theorem 2 immediatey gives the expression for q fpc λ). Again, the transmission attempt intensity approximation is obtained by soving q λ) = ɛ for λ. As with Theorem 2, the approximation q fpc λ) q fpc λ) is accurate when the exponentia term in q fpc λ) is approximatey inear in its argument and thus Jensen s is tight. In other words, this approximation utiizes the fact that e x is neary inear for sma x. Looking at the expression for q fpc λ) we see that this reasonabe when the reative density λπd 2 is sma. If this is not true then the approximation q fpc λ) is not sufficienty accurate, as wi be further seen in the numerica resuts presented in Section IV. The FPC transmission attempt intensity approximation, λ fpc ɛ), is obtained by soving q fpc λ) = ɛ for λ. The foowing coroary iustrates the simpification of the above resuts when the noise may be ignored. Coroary 3: When η = 0 the expressions in Theorem 3 simpify to: [ { [ q fpc λ) q fpc λ) = E exp λπd 2 δ E H δ] [ E H sδ] }] H s)δ 00 { [ q fpc λ) = exp λπd 2 δ E H δ] [ E H sδ] [ E H s)δ]}, λ fpc ɛ) λ fpc ɛ) = og ɛ) πd 2 δ E [H δ ] E [H sδ ] E [ H s)δ]. 9) The oss factor for FPC, L fpc, is the reduction in the transmission capacity approximation reative to the pure pathoss case: L fpc s) = E [H δ ] E [H sδ ] E [ H s)δ]. 20) Ceary, the oss factor L fpc for FPC depends on the design choice of the exponent s. C. Optima Fractiona Power Contro Exponent Fractiona power contro represents a baance between the extremes of no power contro and channe inversion. The mathematica effect of fractiona power contro is to repace the E[H δ ] term with E[H sδ ]E[H s)δ ]. This is because the signa fading is softened by the power contro exponent s so that it resuts in a eading term of H s) rather than H ) in the numerator of the SINR expression, and utimatey to the E[H s)δ ] term. The interference power is aso softened by the fractiona power contro and eads to the E[H sδ ] term. The key question of course ies in determining the optima power contro exponent. Athough it does not seem possibe to derive an anaytica expression for the exponent that minimizes the genera expression for q fpc λ) given in Theorem 3, we can find the exponent that minimizes the outage probabiity approximation in the case of no noise. Theorem 4: In the absence of noise η = 0), the fractiona power contro outage probabiity approximation, q fpc λ), is minimized for s = 2. Hence, the fractiona power contro transmission attempt intensity approximation, λ fpc ɛ) is aso maximized for s = 2. Proof: Because the outage probabiity/transmission density approximations depend on the exponent s ony through the quantity E [ H sδ] E [ H s)δ], it is sufficient to show that E [ H sδ] E [ H s)δ] is minimized at s = 2. To do this, we use the foowing genera resut, which we prove in the Appendix. For any non-negative random variabe X, the function hs) = E [ X s] E [ X s ], 2) is convex in s for s R with a unique minimum at s = 2. Appying this resut to random variabe X = Hδ gives the desired resut. The theorem shows that transmission density is maximized, or equivaenty, outage probabiity is minimized, by baancing the positive and negative effects of power contro, which are reduction of signa fading and increasing interference, respectivey. Using an exponent greater than 2 over-compensates for signa fading and eads to interference eves that are too high, whie using an exponent smaer than 2 eads to sma interference eves but an under-compensation for signa fading. Note that because the key expression E [ H sδ] E [ H s)δ] is convex, the oss reative to using s = 2 increases monotonicay both as s 0 and s.

8 8 One can certainy envision fractiona power contro schemes that go even further. For exampe, s > corresponds to super channe inversion, in which bad channes take resources from good channes even more so than in norma channe inversion. Not surprisingy, this is not a wise poicy. Less obviousy, s < 0 corresponds to what is sometimes caed greedy optimization, in which good channes are given more resources at the further expense of poor channes. Waterfiing is an exampe of a greedy optimization procedure. But, since E [ H sδ] E [ H s)δ] monotonicay increases as s decreases, it is cear that greedy power aocations of any type are worse than even constant transmit power under the SINR-target set up. The numerica resuts in the next section show that FPC is very beneficia reative to constant transmit power or channe inversion. However, fading has a deeterious effect reative to no fading even if the optima exponent is used. To see this, note that x 2 is a convex function and therefore Jensen s yieds E[X 2 ] E[X]) 2 for any non-negative random variabe X. Appying this to X = H δ we get E [ H δ 2 ]) 2 E[H δ ] ), which impies L fpc /2) = [ ]) 2. E [H δ ] E H δ 2 Therefore, fractiona PC cannot fuy overcome fading, but it is definitey a better power contro poicy than constant power transmission or traditiona power contro channe inversion). IV. NUMERICAL RESULTS AND DISCUSSION In this section, the impications of fractiona power contro are iustrated through numerica pots and anaytica discussion. The tightness of the bounds wi be considered as a function of the system parameters, and the choice of a robust FPC exponent s wi be proposed. As defaut parameters, the simuations assume α = 3, = 0 db), d = 0m, = pd α η = db), λ = users m 2. 22) Furthermore, Rayeigh fading is assumed for the numerica resuts. A. Effect of Fading The benefit of fractiona power contro can be quicky iustrated in Rayeigh fading, in which case the channe power H is exponentiay distributed and the moment generating function is therefore E[H t ] = Γ + t), 23) where Γ ) is the standard gamma function. If fractiona power contro is used, the transmission capacity oss due to fading is L fpc = E [H δ ] E [H sδ ] E [ H s)δ] = Γ + δ) Γ sδ) Γ s)δ) In Fig. this oss factor L) is potted as a function of s for path oss exponents α = {2., 3, 4}. Notice that for each vaue of α the maximum takes pace at s = 2, and that the cost of not using fractiona power contro is highest for sma path oss exponents because Γ + x) goes to infinity quite steepy as x. This pot impies that in severe fading channes, the gain from FPC can be quite significant. It shoud be noted that the expression in 24) is for the case of no therma noise η = 0). In this case the power cost of FPC competey vanishes, because the same power normaization by E[H s ]) is performed by each transmitting node and therefore this normaization cances in the SIR expression. On the other hand, this power cost does not vanish if the noise is stricty positive and can potentiay be quite significant, particuary if is not arge. A simpe appication of Jensen s shows that the power normaization factor E[H s ] is an increasing function of the exponent s for any distribution on H. For the particuar case of Rayeigh fading this normaization factor is Γ s) which makes it prohibitivey expensive to choose s very cose to one; indeed, the choice s = requires infinite power and thus is not feasibe. On the other hand, note that Γ.5) is approximatey 2.5 db and thus the cost of a moderate exponent is not so arge. When the interference-free is reasonaby arge, this normaization factor is reativey negigibe and the effect of FPC is we approximated by 24). 24)

9 9 B. Tightness of Bounds There are two principe approximations made in attaining the expressions for outage probabiity and transmission capacity in Theorem 3. First, the inequaity is due to considering ony dominant interferers; that is, an interferer whose channe to the desired receiver is strong enough to cause outage even without any other interferers present. This is a ower bound on outage since it ignores non-dominant interferers, but nevertheess has been seen to be quite accurate in our prior work [8], [20], [6]. Second, Jensen s inequaity is used to bound E[expX)] expe[x]) in the opposite direction, so this resuts in an approximation to the outage probabiity rather than a ower bound; numerica resuts confirm that this approximation is in fact not a ower bound in genera. Therefore, we consider the three reevant quantities: ) the actua outage probabiity q fpc λ), which is determined via Monte-Caro simuation and does not depend on any bounds or approximations, 2) a numerica computation of the outage probabiity ower bound q fpc λ), and 3) the approximation to the outage probabiity q fpc λ) reached by appying Jensen s inequaity to q fpc λ). Note that because of the two opposing bounds one ower and one upper), we cannot say a priori that method 2) wi produce more accurate expressions than method 3). The tightness of the bounds is expored in Figs Consider first Fig. 2 for the defaut parameters given above. We can see that the ower bound and the imation both reasonaby approximate the simuation resuts, and the approximation winds up serving as a ower bound as we. The Jensen s approximation is very accurate for arge vaues of s i.e., coser to channe inversion), and whie ooser for smaer vaues of s, this error actuay moves the Jensen s approximation coser to the actua simuated) outage probabiity. The Jensen s approximation approaches the ower bound as s because the random variabe H s)δ approaches a constant, where Jensen s inequaity triviay hods with equaity see, e.g., 9)). Changing the path oss exponent α, the, the target SINR, or the density λ can have a significant effect on the bounds, as we wi see. With the important exception of high density networks, the approximations are seen to be reasonaby accurate for reasonabe parameter vaues. Path oss. In Fig. 3, the bounds are given for α = 2.2 and α = 5, which correspond to much weaker and much stronger attenuation than the more ikey) defaut case of α = 3. For weaker attenuation, we can see that the ower bound hods the right shape but is ess accurate, whie the Jensen s approximation becomes very oose when the FPC exponent s is sma. For path oss exponents near 2, the dominant interferer approximation is weakened because the attenuation of non-dominant interferers is ess drastic. On the other hand, both the ower bound and Jensen s approximation are very accurate in strong attenuation environments as seen in the α = 5 pot. This is because the dominant interferer approximation is very reasonabe in such cases.. The behavior of the bounds aso varies as the background noise eve changes, as shown in Fig. 4. When the is 0 db, the bounds are quite tight. However, the behavior of outage probabiity as a function of s is quite different from the defaut case in Fig. 2: outage probabiity decreases sowy as s is increased, and a rather sharp jump is seen as s approaches one. When the interference-free is ony moderatey arger than the target SINR in this case there is a 0 db difference between and ), a significant portion of outages occur because the signa power is so sma that the interference-free received fas beow the target ; this probabiity is captured by the P H 00 κs)) terms in Theorem 3. On the other hand, if is much arger than the target, outages are amost aways due to a combination of signa fading and arge interference power rather than to signa fading aone i.e., P H 00 κs)) is insignificant compared to the tota outage probabiity). When outages caused purey by signa fading are significant, the dependence on the exponent s is significanty reduced. Furthermore, the power cost of FPC becomes much more significant when the gap between and is reduced; this expains the sharp increase in outage as s approaches one. When = 30 db, the behavior is quite simiar to the 20 db case because at this point the gap between and is so arge that therma noise can effectivey be negected. Target SINR. A defaut SINR of = was chosen, which corresponds roughy to a spectra efficiency of bps/hz with strong coding, and ies between the ow and high SINR regimes. Exporing an order of magnitude above and beow the defaut in Fig. 5, we see that for = 0. the bounds are highy accurate, and show that s = 2 is a good choice. For this choice of parameters there is a 30 db gap between and and thus therma noise is essentiay negigibe. On the other hand, if = 0 the bounds are sti reasonabe, but the outage behavior is very simiar to the earier case where = 0 db and = 0 db because there is again ony a 0 db gap between and. Despite the quaitative and quantitative differences for ow and high target SINR from the defaut vaues, it is interesting to note that in both cases s = 2 is sti a robust choice for the FPC exponent. Density. The defaut vaue of λ = corresponds to a somewhat ow density network because the expected

10 0 distance to the nearest interferer is approximatey 50 m, whie the TX-RX distance is d = 0 m. In Fig. 6 we expore a density an order of magnitude ower and higher than the defaut vaue. When the network is even sparser, the bounds are extremey accurate and we see that s = 2 is a near-optima choice. However, the behavior with s is very different in a dense network where λ =.00 and the nearest interferer is approximatey 7 m away. In such a network we see that the nearest neighbor bound is quite oose because a substantia fraction of outages are caused by the summation of non-dominant interferers, as intuitivey expected for a dense network. Athough the bound is oose, it does capture the fact that outage increases with the exponent s. On the other hand, the Jensen approximation is oose and does not correcty capture the reationship between s and outage. The approximation is based on the fact that the function e x is approximatey inear for sma x. The quantity x is proportiona to πλd 2, which is arge when the network is dense reative to TX-RX distance d, and thus this approximation is not vaid for reativey dense networks. C. Choosing the FPC exponent s Determining the optimum choice of FPC exponent s is a key interest of this paper. As seen in Sect. III-C, s = 2 is optima for the Jensen s approximation and with no noise, both of which are questionabe assumptions in many regimes of interest. In Figs. 7 0, we pot the truy optima choice of s for the defaut parameters, whie varying α,,, and λ, respectivey. That is, the vaue of s that minimizes the true outage probabiity is determined for each set of parameters. The FPC exponents s ) and s u ) are aso potted, which provide % error beow and above the optimum outage probabiity. For the pots, we et = and = 0. The key findings are: ) In the pathoss α) pot, s = 2 is a very robust choice for a attenuation regimes; 2) For, s = 2 is ony robust at high, and at ow constant transmit power is preferabe; 3) For target SINR, s = 2 is robust at ow and moderate SINR targets i.e. ow to moderate data rates), but for high SINR targets constant transmit power is preferred; 4) For density λ, s = 2 is robust at ow densities, but constant transmit power is preferred at high densities. The expanation for findings 2) and 3) is due to the dependence of outage behavior on the difference between and. As seen earier, therma noise is essentiay negigibe when this gap is arger than approximatey 20 db. As a resut, it is reasonabe that the exponent shown to be optima for noise-free networks s = 2 ) woud be near-optima for networks with very ow eves of therma noise. On the other hand, outage probabiity behaves quite differenty when is ony sighty arger than. In this case, power is very vauabe and it is not worth incurring the normaization cost of FPC and thus very sma FPC exponents are optima. Intuitivey, achieving high data rates in moderate or moderate data rates in ow are difficut objectives in a decentraized network. The ow case is somewhat anomaous, since the is cose to the target SINR, so amost no interference can be toerated. Simiary, to meet a high SINR constraint in a random network of reasonabe density, the outage probabiity must be quite high, so this too may not be particuary meaningfu. To expain 4), reca that the Jensen-based approximation to outage probabiity is not accurate for dense networks and the pot shows that constant power s = 0) is preferred at high densities. Fractiona power contro softens signa fading at the expense of more harmfu interference power, and this turns out to be a good tradeoff in reativey sparse networks. In dense networks, however, there generay are a arge number of nearby interferers and as a resut the benefit of reducing the effect of signa fading by increasing exponent s) is overwhemed by the cost of more harmfu interference power. Note that this is consistent with resuts on channe inversion s = ) in [6], where s = 0 and s = are seen to be essentiay equivaent at ow densities as expected by the imation) but inversion is inferior at high densities. V. POSSIBLE AREAS FOR FUTURE STUDY Given the historicay very high eve of interest in the subject of power contro for wireess systems, this new approach for power contro opens many new questions. It appears that FPC has potentia for many appications due Based on the figure it may appear that choosing s < 0, which means users with good channes transmit with additiona power, outperforms constant power transmission. However, numerica resuts not shown here) indicate that this provides a benefit ony at extremey high densities for which outage probabiity is unreasonaby arge. Intuitivey, a user with a poor channe in a dense network is extremey unikey to be abe to successfuy communicate and goba performance is improved by having such a user not even attempt to transmit, as done in the threshod-based poicy studied in [6].

11 to its inherent simpicity, requirement for ony simpe pairwise feedback, and possibe a priori design of the FPC parameter s. Some areas that we recommend for future study incude the foowing. How does FPC perform in ceuar systems?. Ceuar systems in this case are harder to anayze than ad hoc networks, because the base stations receivers) are ocated on a reguar grid and thus the tractabiity of the spatia Poisson mode cannot be expoited. On the other hand, FPC may be even more hepfu in centraized systems. Note that some numerica resuts for ceuar systems are given in reference [7], but no anaysis is provided. Can FPC be optimized for spectra efficiency?. In this paper we have focused on outage reative to an SINR constraint as being the metric. Other metrics can be considered, for exampe maximizing the average spectra efficiency, i.e. max E[og 2 + SINR)], which coud potentiay resut in optima exponents s < 0, which is conceptuay simiar to waterfiing. What is the effect of scheduing on FPC? If scheduing is used, then how shoud power eves between a transmitter and receiver be set? Wi s = 2 sti be optima? Wi the gain be increased or reduced? We conjecture that the gain from FPC wi be smaer but non-zero for most any sensibe scheduing poicy, as the effect of interference inversion is softened. Can FPC be used to improve iterative power contro? At each step of the Foschini-Mijanic agorithm as we as most of its variants), transmitters adjust their power in a manner simiar to channe-inversion, i.e., each transmitter fuy compensates for the current SINR. Whie this works we when the target SINR s are feasibe, it does not necessariy work we when it is not possibe to satisfy a users SINR requirements. In such a setting, it may be preferabe to perform partia compensation for the current SINR eve during each iteration. For exampe, if a ink with a 0 db target is currenty experiencing an SINR of 0 db, rather than increasing its transmit power by 0 db to fuy compensate for this gap as in the Foschini-Mijanic agorithm), an FPC-motivated iterative poicy might ony boost power by 5 db e.g., adjust power in inear units according to the square root of the gap). VI. CONCLUSIONS This paper has appied fractiona power contro as a genera approach to pairwise power contro in decentraized e.g. ad hoc or spectrum sharing) networks. Using two approximations, we have shown that a fractiona power contro exponent of s = 2 is optima in terms of outage probabiity and transmission capacity, in contrast to constant transmit power s = 0) or channe inversion s = ) in networks with a reativey ow density of transmitters and ow noise eves. This impies that there is an optima baance between compensating for fades in the desired signa and ampifying interference. We saw that a gain on the order of 50% or arger reative to no power contro or channe inversion) might be typica for fractiona power contro in a typica wireess channe. APPENDIX We prove that for any non-negative random variabe X, the function hs) = E [ X s] E [ X s ], 25) is convex in s for s R with a unique minimum at s = 2. In order to show hs) is convex, we show h is og-convex and use the fact that a og-convex function is convex. We define and reca Höder s inequaity: Hs) = og hs) = og E [ X s] E [ X s ]), 26) E[XY ] E[X p ]) p E[Y q ]) q, p + =. 27) q The function Hs) is convex if Hλs + λ)s 2 ) λhs ) + λ)hs 2 ) for a s, s 2 and a λ [0, ]. Using Höder s with p = λ and q = λ we have: [ Hλs + λ)s 2 ) = og E X λs+ λ)s2)] E [X λs+ λ)s2) ]) ] = og E [X λs X λ)s2 E [X λs ) X λ)s2 )]) og E [ ] λ X s E [X s 2 ] λ E [ X s ] λ [ E X s ] ) 2 λ = λ og E [ ] [ X s E X s ]) + λ) og E [X s2 ] E [ X s2 ]) = λhs ) + λ)hs 2 ). 28)

12 2 This impies Hs) is convex, which further impies convexity of hs). The derivative of h is h s) = E [ X s] E [ X s og X ] E [ X s ] E [ X s og X ], 29) and it can easiy be seen that s = 2 is the unique minimizer satisfying h s) = 0. REFERENCES [] N. Jinda, S. Weber, and J. G. Andrews, Fractiona power contro for decentraized wireess networks, in Proc., Aerton Conf. on Comm., Contro, and Computing, Monticeo, IL, Sept. 2007, avaiabe at [2] A. Godsmith and P. Varaiya, Capacity of fading channes with channe side information, IEEE Trans. on Info. Theory, pp , Nov [3] G. J. Foschini and Z. Mijanic, A simpe distributed autonomous power contro agorithm and its convergence, IEEE Trans. on Veh. Technoogy, vo. 42, no. 8, pp , Nov [4] R. D. Yates, A framework for upink power contro in ceuar radio systems, IEEE Journa on Se. Areas in Communications, vo. 3, no. 7, pp , Sept [5] N. Bambos, S. Chen, and G. J. Pottie, Radio ink admission agorithms for wireess networks with power contro and active ink quaity protection, in Proc., IEEE INFOCOM, Boston, MA, Apr. 995, pp [6] J. Herdtner and E. Chong, Anaysis of a cass of distributed asynchronous power contro agorithms for ceuar wireess systems, IEEE Journa on Se. Areas in Communications, vo. 8, no. 3, Mar [7] J. F. Chamberand and V. V. Veeravai, Decentraized dynamic power contro for ceuar CDMA systems, IEEE Trans. on Wireess Communications, vo. 2, no. 3, pp , May [8] M. Schubert and H. Boche, QoS-Based Resource Aocation and Transceiver Optimization. NOW: Foundations and Trends in Communications and Information Theory, [9] M. Chiang, P. Hande, T. Lan, and C. W. Tan, Power Contro in Ceuar Networks. NOW: Foundations and Trends in Networking, To appear [0] T. EBatt and A. Ephremides, Joint scheduing and power contro for wireess ad hoc networks, in Proc., IEEE INFOCOM, June 2002, pp [] R. Cruz and A. V. Santhanam, Optima routing, ink scheduing and power contro in mutihop wireess networks, in Proc., IEEE INFOCOM, Apr. 2003, pp [2] M. Haenggi, The impact of power ampifier characteristics on routing in random wireess networks, in Proc., IEEE Gobecom, San Francisco, CA, Dec. 2003, pp [3] S. Agarwa, S. V. Krishnamurthy, R. H. Katz, and S. K. Dao, Distributed power contro in ad-hoc wireess networks, in Proc., IEEE PIMRC, Oct. 200, pp [4] V. Kawadia and P. R. Kumar, Power contro and custering in ad hoc networks, in Proc., IEEE INFOCOM, [5] M. Chiang, Baancing transport and physica ayers in wireess mutihop networks: Jointy optima congestion contro and power contro, IEEE Journa on Se. Areas in Communications, vo. 23, no., pp. 04 6, Jan [6] S. Weber, J. G. Andrews, and N. Jinda, The effect of fading, channe inversion, and threshod scheduing on ad hoc networks, IEEE Trans. on Info. Theory, vo. 53, no., pp , Nov [7] W. Xiao, R. Ratasuk, A. Ghosh, R. Love, Y. Sun, and R. Nory, Upink power contro, interference coordination and resource aocation for 3GPP E-UTRA, in Proc., IEEE Veh. Technoogy Conf., Sept. 2006, pp. 5. [8] S. Weber, X. Yang, J. G. Andrews, and G. de Veciana, Transmission capacity of wireess ad hoc networks with outage constraints, IEEE Trans. on Info. Theory, vo. 5, no. 2, pp , Dec [9] F. Baccei, B. Baszczyszyn, and P. Muhethaer, An Aoha protoco for mutihop mobie wireess networks, IEEE Trans. on Info. Theory, pp , Feb [20] S. Weber, J. G. Andrews, X. Yang, and G. de Veciana, Transmission capacity of wireess ad hoc networks with successive interference canceation, IEEE Trans. on Info. Theory, vo. 53, no. 8, pp , Aug

13 3 Mutipicative Effect of Fading α=4 α=3 α= Power Contro Exponent s) Fig.. The oss factor L vs. s for Rayeigh fading. Note that L cp and L ci are the eft edge and right edge of the pot, respectivey simuation ower bound Outage probabiity q) Fractiona power contro parameter s) Fig. 2. The outage probabiity simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for the defaut parameters. Outage probabiity q) simuation ower bound α = 2.2 Outage probabiity q) simuation ower bound α = Fractiona power contro parameter s) Fractiona power contro parameter s) Fig. 3. The outage probabiity simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for α = 2.2 eft) and α = 5 right).

14 simuation ower bound = 0 db simuation ower bound = 30 db Outage probabiity q) Outage probabiity q) Fractiona power contro parameter s) Fractiona power contro parameter s) Fig. 4. The outage probabiity simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for = 0 db eft) and = 30 db right) simuation ower bound = -0 db 0.9 simuation ower bound = 0 db Outage probabiity q) Outage probabiity q) Fractiona power contro parameter s) Fractiona power contro parameter s) Fig. 5. The outage probabiity simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for = 0 db eft) and = 0 db right) simuation ower bound λ = simuation ower bound λ = 0.00 Outage probabiity Outage probabiity Fractiona power contro parameter s) Fractiona power contro parameter s) Fig. 6. The outage probabiity simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for λ = eft) and λ = 0.00 right).

15 5 Fractiona power contro parameter s) s,su Δ = 0%) s,su Δ = %) s opt Pathoss attenuation constant α) Fig. 7. The optima choice of FPC exponent s vs. PL exponent α, with ±% and ±0% seections for s. Fractiona power contro parameter s) s,su Δ = 0%) s,su Δ = %) s opt Transmitter signa to noise ratio, db) Fig. 8. The optima choice of FPC exponent s vs. transmitter = ρ, with ±% and ±0% seections for s. η Fractiona power contro parameter s) s,su Δ = 0%) s,su Δ = %) s opt Required receiver SINR, db) Fig. 9. The optima choice of FPC exponent s vs. SINR constraint, with ±% and ±0% seections for s.

16 6 Fractiona power contro parameter s) s,su Δ = 0%) s,su Δ = %) s opt Spatia density of interferers λ, db) Fig. 0. The optima choice of FPC exponent s vs. density λ, with ±% and ±0% seections for s.