Fractional Power Control for Decentralized Wireless Networks

Transcription

1 Fractiona Power Contro for Decentraized Wireess Networks Nihar Jinda, Steven Weber, Jeffrey G. Andrews Abstract 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. We aso show numericay that this choice of s is robust to most reasonabe variations in the network parameters. Intuitivey, s = 2 disadvantaged users whie making sure they do not food the network with interference. baances between heping 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 Manuscript date: December 2, 2007.

2 2 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 []. 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 [2]. Considerabe work has deepy expored the properties of these agorithms, incuding deveoping a framework that describes a power contro probems of this type [3], as we as studying the feasibiity and impementation of such agorithms [4], [5], incuding with varying channes [6]; see the recent monographs [7][8] 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 [9], [0], [], [2], [3], [4]. 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. 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 con-

3 3 sider iterative agorithms and their convergence. Rather, motivated by the poor performance of channe inversion in decentraized networks [5], 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 [6] 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

4 4 artificia, but is significanty easier to hande anayticay, and has been shown to preserve the integrity of concusions even with random transmit distances, e.g. [7], [5]. 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 =. The exponent 2 s = 2 is shown to be optima for an approximation to the outage probabiity/transmission, but our numerica resuts confirm the vaidity of this approximation and thus show that s = 2 is neary optima for the true outage probabiity for typica choices of the system parameters. 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 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. II. PRELIMINARIES A. System Mode We consider a set of transmitting nodes at an arbitrary snapshot in time with ocations specified by a homogeneous Poisson point process (PPP), Π(λ), 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)

5 5 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. 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 Π(λ) P ih i0 X α i + 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 [8]. 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 β, or equivaenty whenever the instantaneous mutua information fas beow og 2 ( + β). Therefore, the system-wide outage probabiity is q(λ) = P(SINR 0 < β) (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 an increasing function of the intensity λ. Define λ(ɛ) as the maximum intensity of attempted transmissions such that the outage probabiity is no arger than ɛ, i.e., λ(ɛ) is the unique soution of q(λ) = ɛ. The transmission capacity is then defined as c(ɛ) = λ(ɛ)( 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.

6 6 For the sake of carity, we define the constants δ = 2/α < and = pd α. Now consider η a path-oss ony environment (H i0 = for a i) with constant transmission power (P i = p for a i). The main resut of [7] is given in the foowing theorem. Theorem ([7]): 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 tight bounds on the outage probabiity and transmission attempt intensity for λ, ɛ sma: { ( q p (λ) q p (λ) = exp λπd 2 β ) } δ, (3) λ p (ɛ) λ p u (ɛ) = og( ɛ) ( πd 2 β ) δ. (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 tight approximations for sma λ, ɛ respectivey, which is the usua regime of interest. Note aso that og( ɛ) = ɛ + O(ɛ 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 β δ}, (5) λ p (ɛ) λ p u (ɛ) = og( ɛ) πd 2 β. 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

7 7 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. A main resut of [5] 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 β ) [ { ( E exp λπd 2 E[H δ H00 ] β ) } ] δ H00 β ( q cp (λ) = P H 00 β ) { [ (H00 exp λπd 2 E[H δ ]E β ) ]} δ H00 β ( ) [ (H00 λ cp (ɛ) λ cp ɛ (ɛ) = og P ( ) H 00 πd 2 E[H δ ] E β ) ] δ H00 β. (7) β Channe inversion. Consider the same network with P i = p 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 ci (λ) = exp λπd 2 E[H δ ]E[H δ ] β E[H ] (8) λ ci (ɛ) λ ci u (ɛ) = og( ɛ) ( ) δ πd 2 E[H δ ]E[H δ ] β E[H ]. (9) Proof: The SINR at the reference receiver for a generic power vector {P i } is SINR 0 = P 0 H 00 d α i Π(λ) P ih i0 X α i + η, (0) and the corresponding outage probabiity is ( ) P 0 H 00 d α q(λ) = P(SINR 0 < β) = P i Π(λ) P ih i0 X α i + η < β. ()

8 8 Rearranging yieds: q(λ) = P i Π(λ) Note that outage is certain when P 0 H 00 denote the density of P 0 H 00 yieds: q(λ) = P (P 0 H 00 ηβd α )+ ηβd α P i Π(λ) P i H i0 X α i P 0H 00 d α β η. (2) < ηβd α. Conditioning on P 0 H 00 and using f( ) to P i H i0 X α i p 0h 00 βd η P0 H α 00 = p 0 h 00 f(p 0 h 00 )d(p 0 h 00 ). Reca the generic ower bound from [5]: if Π(λ) = {(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 Π(λ) Z i X α i Appying here with Z i = P i H i0 and y = p 0h 00 η: βd ( α { q(λ) P (P 0 H 00 ηβd α ) + ηβd α (3) > y exp { πλe[z δ ]y δ}, (4) exp πλe[(p i H i0 ) δ ] ( p0 h 00 βd α { ( ) } δ = exp πλe[(p i H i0 ) δ p0 h 00 ] ηβd βd η f(p α α 0 h 00 )d(p 0 h 00 ) [ { ( = P (P 0 H 00 ηβd α ) E exp λπd 2 E[(P i H i0 ) δ P0 H 00 ] β ) }) δ η f(p 0 h 00 )d(p 0 h 00 ) η ) } ] δ P0 H d α 00 ηβd α (5). The Jensen approximation for this quantity is: { [ (P0 q(λ) P (P 0 H 00 ηβd α ) exp λπd 2 E[(P i H i0 ) δ H 00 ]E η ) ]} δ P0 H β d α 00 ηβd 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 For channe inversion, P 0 H 00 = (λ) and q cp (λ) in (7). To obtain λ cp (ɛ), we sove q cp (λ) = ɛ for λ. 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, the effective interference- ηe[h ] free after taking into account the power cost of inversion, is arger than the SINR threshod

9 9 β. 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: Coroary 2: When η = 0 the expressions in Theorem 2 simpify to: q cp (λ) q cp (λ) = E [ exp { λπd 2 β δ E [ H δ] }] H00 δ q cp (λ) = exp { λπd 2 β δ E [ H δ] E [ H δ]}, q ci (λ) q ci (λ) = exp { λπd 2 β δ E [ H δ] E [ H δ]}, λ cp (ɛ) λ cp (ɛ) = og( ɛ) πd 2 β δ E [H δ ] E [H δ ], λ ci (ɛ) λ ci u (ɛ) = og( ɛ) πd 2 β δ E [H δ ] E [H δ ]. (7) Note that these expressions match Theorem 3 and Coroary 3 of the SIR-anaysis performed in [5]. 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 H ii. 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.

10 0 B. Transmission capacity under fractiona power contro In this section we generaize the resuts of Theorem 2 by introducing fractiona power contro (FPC) with parameter s [0, ]. Under FPC the transmission power is set to P i = for each i. The received power at receiver i is then P i H ii d α = p E[H s ] H s ii 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 κ(s)) [ { E exp λπd 2 E[H sδ ]E[H δ ] ( H s 00 β ) δ } E[H s ] H00 κ(s)] q fpc (λ) = P (H 00 κ(s)) { exp λπd 2 E[H sδ ]E[H δ ]E ( ) λ fpc (ɛ) λ fpc ɛ (ɛ) = og P (H 00 κ(s)) ( where κ(s) = ( β E[H s ] ) s. [ (H s 00 β πd 2 E[H sδ ]E[H δ ] [ (H s 00 E β ) δ E[H s ] H00 κ(s)]} ) δ E[H s ] H00 κ(s)]) Proof: Under FPC, the transmit power for each user is constructed as P i = p 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) hods because e x is neary inear for sma x. 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.

11 Coroary 3: When η = 0 the expressions in Theorem 3 simpify to: [ { q fpc (λ) q fpc (λ) = E exp λπd 2 β δ E [ H δ] E [ H sδ] }] H ( s)δ 00 (λ) = exp { λπd 2 β δ E [ H δ] E [ H sδ] E [ H ( s)δ]}, λ fpc (ɛ) λ fpc (ɛ) = og( ɛ) πd 2 β 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 q fpc 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 =. Hence, the fractiona power contro transmission 2 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 =. To do this, we use the foowing genera resut, 2 which we prove in the Appendix. For any non-negative random variabe X, the function h(s) = E [ X s] E [ X s ], (2)

12 2 is convex in s for s R with a unique minimum at s =. Appying this resut to random 2 variabe X = H δ 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 overcompensates 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. 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. ( [ ]) 2 ( Appying this to X = H δ we get E H δ 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

13 3 the system parameters, and the choice of a robust FPC exponent s wi be proposed. As defaut parameters, the simuations assume α = 3, β = (0 db), d = m, = pd α η = 00 (20 db), λ = 0.0 users m 2. (22) 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 = [H δ ] E [H sδ ] E [H ( s)δ ] = Γ( + δ) Γ( sδ) Γ( ( s)δ) (24) 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 =, and that the cost of not 2 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).

14 4 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 [7], [9], [5]. Second, Jensen s inequaity is used to bound E[exp(X)] exp(e[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 Jensen approximation 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, or the target SINR β can have a significant effect on the bounds as we wi see, athough in a but a coupe anomaous cases, the two approximations hod reativey we. 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

15 5 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 between and β), a significant portion of outages occur because the signa power is so sma that the interference-free received fas beow the target β; this probabiity is captured by the P (H 00 κ(s)) terms in Theorem 3. On the other hand, if is much arger than the target β, 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 κ(s)) 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 β 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 β is so arge that therma noise can effectivey be negected. Target SINR. A defaut SINR of β = 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 β = 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 β and thus therma noise is essentiay negigibe. On the other hand, if β = 0 the bounds are sti reasonabe, but the outage behavior is very simiar to the earier case where = 0 db and β = 0 db because there is again a 0 db gap between and β. 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 = is sti a robust choice for the FPC exponent. 2

16 6 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. 6 8, we pot the truy optima choice of s for the defaut parameters, whie varying α,, and β, 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 β, 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. The expanation for the atter two findings is due to the dependence of outage behavior on the difference between and β. 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 = ) woud be near-optima for networks with very 2 ow eves of therma noise. On the other hand, outage probabiity behaves quite differenty when is ony sighty arger than β. 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. In short, it is expected that in many decentraized networks of interest, an offine design choice of s = 2 appears quite robust and appropriate. 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

17 7 has potentia for many appications due 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 [6], 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 = is optima in terms of outage 2

18 8 probabiity and transmission capacity, in contrast to constant transmit power (s = 0) or channe inversion (s = ). This choice is shown numericay to be robust for most network regimes of interest. 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 h(s) = E [ X s] E [ X s ], (25) is convex in s for s R with a unique minimum at s =. Define the functions 2 f(s) = E [ X s], g(s) = E [ X s ], (26) so that h(s) = f(s)g(s). Reca that a function h is said to be og-convex if og h is convex. Two reevant properties of og-convex functions are: i) if h is og-convex then h is convex (athough the converse need not hod), and ii) the product of two og-convex functions is og-convex. Thus the theorem wi be proved if we show that f, g are og-convex. The functions f, g are easiy shown to be convex by the fact that f (s), g (s) are non-negative for a s: f (s) = E [X s og X], f (s) = E [ X s (og X) 2] g (s) = E [X s og X], g (s) = E [ X s (og X) 2], (27) Define F (s) = og f(s), G(s) = og g(s), and H(s) = og h(s). Then: F (s) = E[X s og X] E[X s ], F (s) = E [X s ] E [X s (og X) 2 ] (E [X s og X]) 2 (E [X s ]) 2 G (s) = E[Xs og X] E[X s ], G (s) = E[Xs ]E[X s (og X) 2 ] (E [X s og X]) 2 (E [X s ]) 2. (28) We empoy the Cauchy-Schwarz inequaity in the form (E [a(x)b(x)]) 2 E [ a(x) 2] E [ b(x) 2], (29)

19 9 for arbitrary functions a, b of a random variabe X. [ (X ) ] [ E s 2 (X F 2 E s 2 og X ) ] 2 (E [X s og X]) 2 (s) = (E [X s ]) 2 G (s) = E [ X s 2 X s 2 og X ] 2 (E [X s og X]) 2 (E [X s ]) 2 = 0 E [ ( ) ] 2 X s 2 E [ ( X s 2 og X ) ] 2 (E [X s og X]) 2 (E [X s ]) 2 [ ] 2 E X s 2 X s 2 og X (E [X s og X]) 2 (E [X s ]) 2 = 0 (30) This estabishes the og-convexity of f, g, and hence the og-convexity of h, and thus the convexity of h. The derivative of h is h (s) = E [ X s] E [ X s og X ] E [ X s ] E [ X s og X ]. (3) It can easiy be seen that s = 2 is the unique maximizer satisfying h (s) = 0. REFERENCES [] A. Godsmith and P. Varaiya, Capacity of fading channes with channe side information, IEEE Trans. on Info. Theory, pp , Nov [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] M. Schubert and H. Boche, QoS-Based Resource Aocation and Transceiver Optimization. NOW: Foundations and Trends in Communications and Information Theory, [8] M. Chiang, P. Hande, T. Lan, and C. W. Tan, Power Contro in Ceuar Networks. NOW: Foundations and Trends in Networking, To appear [9] T. EBatt and A. Ephremides, Joint scheduing and power contro for wireess ad hoc networks, in Proc., IEEE INFOCOM, June 2002, pp

20 20 [0] R. Cruz and A. V. Santhanam, Optima routing, ink scheduing and power contro in mutihop wireess networks, in Proc., IEEE INFOCOM, Apr. 2003, pp [] M. Haenggi, The impact of power ampifier characteristics on routing in random wireess networks, in Proc., IEEE Gobecom, San Francisco, CA, Dec. 2003, pp [2] 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 [3] V. Kawadia and P. R. Kumar, Power contro and custering in ad hoc networks, in Proc., IEEE INFOCOM, [4] 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 [5] 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 [6] 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. [7] 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 [8] F. Baccei, B. Baszczyszyn, and P. Muhethaer, An Aoha protoco for mutihop mobie wireess networks, IEEE Trans. on Info. Theory, pp , Feb [9] 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

21 2 Mutipicative Effect of Fading α=4 α=3 α= Power Contro Exponent (s) Fig.. respectivey. 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, simuation ower bound Jensen approx 0.08 Outage probabiity (q) Fractiona power contro parameter (s) Fig. 2. parameters. The outage probabiity (simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for the defaut

22 simuation ower bound Jensen approx α = 2.2 Outage probabiity (q) simuation ower bound Jensen approx Fractiona power contro parameter (s) α = 5.0 Outage probabiity (q) Fractiona power contro parameter (s) Fig. 3. and α = 5 (right). The outage probabiity (simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for α = 2.2 (eft)

23 simuation ower bound Jensen approx = 0 db Outage probabiity (q) simuation ower bound Jensen approx Fractiona power contro parameter (s) = 30 db Outage probabiity (q) 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).

24 simuation ower bound Jensen approx β = -0 db Outage probabiity (q) simuation ower bound Jensen approx Fractiona power contro parameter (s) β = 0 db Outage probabiity (q) Fractiona power contro parameter (s) Fig. 5. The outage probabiity (simuated, ower bound, and Jensen s approximation) vs. FPC exponent s for β = 0 db (eft) and β = 0 db (right). s,su (Δ = 0%) s,su (Δ = %) s opt Fractiona power contro parameter (s) Pathoss attenuation constant (α) Fig. 6. The optima choice of FPC exponent s vs. PL exponent α, with ±% and ±0% seections for s.

25 25 Fractiona power contro parameter (s) s,su (Δ = 0%) s,su (Δ = %) s opt Transmitter signa to noise ratio (, db) Fig. 7. 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 (β, db) Fig. 8. The optima choice of FPC exponent s vs. SINR constraint β, with ±% and ±0% seections for s.