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Fractional Power Control for Decentralized Wireless Networks

Nihar Jindal, Steven Weber, Jeffrey G. Andrews

arXiv:0707.0476v2cs.ITmath.IT

TL;DR

In decentralized wireless networks, transmitters have distinct receivers and interfere with one another, making the choice of power-control policy important. The paper analyzes fractional power control, which interpolates between constant power and channel inversion, and finds s=1/2 optimal in relatively low-density, low-noise conditions.

  • Problem

    The paper asks which power-control exponent is appropriate for mutually interfering users in decentralized networks, where channel inversion is described as performing poorly.

    Source: A. Background and Motivation for… · p. 2

  • Method

    The paper analyzes fractional power control, setting transmit power proportional to intended-link channel quality raised to -s for s in [0, 1].

    Source: A. Background and Motivation for… · p. 2

  • Results

    s*=1/2 is optimal for outage probability and transmission capacity in relatively low-density, low-noise networks, with gains on the order of 50% or larger versus no power control or channel inversion potentially typical.

    Source: §VI · p. 11

  • Takeaways & Limitations

    The inverse-square-root channel-strength policy balances compensating for desired-link fades against amplifying interference in the stated regime.

    Source: §VI · p. 11

    The numerical results assume Rayleigh fading.

    Source: §IV · p. 8

Abstract

from arXiv · show

We consider a new approach to power control in decentralized wireless networks, termed fractional power control (FPC). Transmission power is chosen as the current channel quality raised to an exponent -s, where s is a constant between 0 and 1. The choices s = 1 and s = 0 correspond to the familiar cases of channel inversion and constant power transmission, respectively. Choosing s in (0,1) allows all intermediate policies between these two extremes to be evaluated, and we see that usually neither extreme is ideal. We derive closed-form approximations for the outage probability relative to a target SINR in a decentralized (ad hoc or unlicensed) network as well as for the resulting 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 typical system parameter values, we prove that using an exponent of 1/2 minimizes the outage probability, meaning that the inverse square root of the channel strength is a sensible transmit power scaling for networks with a relatively low density of interferers. We also show numerically that this choice of s is robust to a wide range of variations in the network parameters. Intuitively, s=1/2 balances between helping disadvantaged users while making sure they do not flood the network with interference.

I. INTRODUCTION

Power control in decentralized networks must balance target-link performance against interference from other transmitters. The paper introduces fractional power control and analyzes it under a spatial network model, identifying s=1/2 as optimal in the stated low-density, low-noise regime.

I. INTRODUCTION

  • Channel inversion adjusts power inversely with instantaneous channel quality to meet a target level, contrasting with waterfilling's rate-maximizing policy.The introduction notes that some inversion is used to provide a baseline user experience over long time scales.Source: §I · p. 1

A. Background and Motivation for Fractional Power Control

  • Decentralized networks have distinct receivers in the same geographic area, so mutual interference changes power-control properties relative to cellular settings with a common receiver.Prior multi-user power-control work is described as primarily focused on cellular uplinks.Source: A. Background and Motivation for… · p. 1
  • Fractional power control sets transmit power proportional to H^-s, with s=0 representing constant power and s=1 channel inversion.The paper treats s as a common parameter chosen offline by all users.Source: A. Background and Motivation for… · p. 2

B. Technical Approach

  • The model uses fixed transmitter–receiver distance, channel knowledge only for the intended link, unscheduled transmissions, and Poisson-distributed transmitter locations.The paper notes that fixed distance simplifies analysis and that random distances reduce transmission capacity by a constant factor without changing FPC's relative effect.Source: B. Technical Approach · p. 2

C. Contributions and Organization

  • s=1/2 is identified as optimal for outage probability and transmission capacity under relatively low transmitter density and low noise.The paper reports larger transmission-capacity gains for smaller path-loss exponents, with more modest gains at higher attenuation.Source: C. Contributions and Organization · p. 2

A. System Model

  • The reference-link SINR is modeled from received signal power against interference and noise, with received power determined by transmit power, path loss, and fading.The system model corresponds to independently transmitting nodes, as in slotted ALOHA.Source: A. System Model · p. 3

B. Transmission Capacity

The section defines outage and transmission capacity, then frames the paper's analysis around how fractional power control affects outage and the resulting transmission density.

B. Transmission Capacity

  • An outage occurs when SINR falls below the prescribed threshold β, equivalently when instantaneous mutual information falls below log2(1 + β).The system-wide outage probability is evaluated over transmitter locations and fading.Source: B. Transmission Capacity · p. 3
  • Transmission capacity is the maximum density of successful transmissions subject to an outage constraint, multiplied by each transmission's spectral efficiency.The maximum attempted-transmission intensity λ(ϵ) is defined by requiring outage probability no greater than ϵ.Source: B. Transmission Capacity · p. 3

III. FRACTIONAL POWER CONTROL

  • The paper aims to determine how fractional power control affects an outage-probability lower bound and the corresponding transmission-capacity upper bound.It derives maximum transmission densities for different power-control policies and then seeks the optimal exponent.Source: §III · p. 4

A. Transmission capacity under constant power and channel inversion

The section compares constant power and channel inversion using outage and transmission-capacity approximations, highlighting how fading affects both strategies.

A. Transmission capacity under constant power and channel inversion

  • The paper gives good approximations for constant-power outage and transmission-attempt intensity, and tight bounds for channel inversion, when λ and ϵ are small.The channel-inversion comparison requires its effective interference-free SNR to exceed the SINR threshold β.Source: A. Transmission capacity under… · pp. 4–5
  • Without thermal noise, constant-power outage approximation equals the channel-inversion outage lower bound, and their transmission-attempt intensity approximations also match.The section relates this equivalence to the shared expected transmission power of the two strategies.Source: A. Transmission capacity under… · pp. 4, 6
  • Fading of interfering signals has a positive effect on the loss factor, while fading of the desired signal has a negative effect.For channel inversion, the interfering transmitter's power also depends on its intended-link fading.Source: A. Transmission capacity under… · p. 6
  • For Rayleigh fading with path-loss exponent α=3, the constant-power and channel-inversion loss factor is 0.41.The loss factor captures transmission-capacity reduction relative to the pure path-loss case.Source: A. Transmission capacity under… · p. 6

B. Transmission capacity under fractional power control

Fractional power control generalizes fixed-power and channel-inversion policies using an exponent s. The section provides approximations for outage and allowable transmission intensity, while noting that approximation accuracy depends on relative network density.

B. Transmission capacity under fractional power control

  • Fractional power control sets each transmitter's power using its intended-link channel quality raised to -s, with expected transmission power normalized to p.The endpoints s=0 and s=1 correspond to constant power and channel inversion, respectively.Source: B. Transmission capacity under… · p. 6
  • Theorem 3 gives approximations for outage probability and maximum transmission-attempt intensity when λ and ϵ are small.The intensity approximation is obtained by solving the approximated outage expression for λ at target outage ϵ.Source: B. Transmission capacity under… · pp. 6–7
  • The outage approximation is reasonable when relative density λπd^2 is small, but can be insufficiently accurate otherwise.The text attributes the approximation to near-linearity of e^-x for small x.Source: B. Transmission capacity under… · p. 7
  • The FPC loss factor measures reduction in the transmission-capacity approximation relative to pure path loss and depends on exponent s.The section also presents simplified expressions for the noise-free case.Source: B. Transmission capacity under… · p. 7

C. Optimal Fractional Power Control Exponent

In the no-noise approximation, the outage-minimizing fractional power control exponent is s = 1/2. This balances reduced signal fading against the increased interference caused by power control.

C. Optimal Fractional Power Control Exponent

  • With no noise, the outage-probability approximation is minimized and transmission-attempt intensity is maximized at s = 1/2.The result follows from how the approximation depends on s through a moment of the fading variable.Source: C. Optimal Fractional Power Control… · p. 7
  • The balance is between reducing signal fading and increasing interference: exponents above 1/2 over-compensate and raise interference, while those below 1/2 under-compensate and leave smaller interference.Theorem’s balance explains why neither extreme of channel inversion nor constant power is generally ideal.Source: C. Optimal Fractional Power Control… · p. 7
  • For s < 0, greedy power allocation gives good channels more resources at poor channels’ expense and performs worse than constant power under the SINR-target setup.Waterfilling is given as an example of greedy optimization.Source: C. Optimal Fractional Power Control… · p. 8
  • Fractional power control cannot fully overcome fading, but it performs better than constant power or channel inversion.The argument invokes Jensen’s inequality for a non-negative fading variable.Source: C. Optimal Fractional Power Control… · p. 8

IV. NUMERICAL RESULTS AND DISCUSSION

The numerical section uses plots and analytical discussion to assess bound tightness and propose a robust FPC exponent. Its default simulations assume specified system parameters and Rayleigh fading.

IV. NUMERICAL RESULTS AND DISCUSSION

  • The section uses numerical plots and analytical discussion to examine bound tightness across system parameters and propose a robust exponent s.The stated default parameters are α = 3, β = 1 (0 dB), d = 10 m, and SNR = pd^-α.Source: §IV · p. 8
  • The default path-loss exponent is α = 3, target SINR is β = 1 (0 dB), link distance is d = 10 m, and SNR is pd^-α.These are the stated default simulation parameters.Source: §IV · p. 8
  • The numerical results assume Rayleigh fading.The fading channel power H is the modeled quantity.Source: §IV · p. 8

A. Effect of Fading

For Rayleigh fading, the paper expresses transmission-capacity loss due to fading using gamma-function terms. The loss is greatest at s = 1/2, while the cost of omitting FPC is highest for small path-loss exponents; positive thermal noise can make power normalization costly.

A. Effect of Fading

  • Under Rayleigh fading, transmission-capacity loss due to fading is expressed using gamma-function terms.The discussion uses the channel-power distribution and its moment generating function to obtain the loss expression.Source: A. Effect of Fading · p. 8
  • For each plotted path-loss exponent α = {2.1, 3, 4}, the loss factor is greatest at s = 1/2; the cost of not using FPC is highest for small α.The passage notes that FPC’s gain can be significant in severe fading channels.Source: A. Effect of Fading · p. 8
  • The loss expression assumes no thermal noise; with positive noise, power-normalization cost can be significant, especially when SNR is not large.For Rayleigh fading, s = 1 requires infinite power, whereas Γ(.5) is approximately 2.5 dB; at reasonably large interference-free SNR, normalization is relatively negligible.Source: A. Effect of Fading · p. 8

B. Tightness of Bounds

The outage bounds and approximation are reasonably accurate across typical parameter values, but their tightness depends on network conditions. High density is a notable exception, where the approximations can be loose.

B. Tightness of Bounds

  • The lower bound counts only dominant interferers, while the Jensen-based expression approximates outage and is not generally a lower bound.Because the methods rely on opposing bounds, neither can be assumed more accurate in advance.Source: B. Tightness of Bounds · p. 9
  • Both approximations are reasonably accurate for typical parameters, with high-density networks an important exception.The Jensen approximation is especially accurate at large s and approaches the lower bound as s → 1.Source: B. Tightness of Bounds · p. 9
  • With weaker attenuation, the lower bound keeps the right shape but loses accuracy, while the Jensen approximation becomes very loose at small s; both are accurate at α = 5.At α near 2, non-dominant interferers are less attenuated, weakening the dominant-interferer approximation.Source: B. Tightness of Bounds · p. 9
  • At 10 dB SNR, the bounds are tight, but outage decreases slowly with s before jumping near s = 1; at 30 dB, behavior resembles the 20 dB case.When SNR is only 10 dB above β, signal fading alone can cause outages, and the power cost of FPC grows near s = 1.Source: B. Tightness of Bounds · p. 9
  • For β = 0.1, the bounds are highly accurate and s = 1/2 is a good choice; for β = 10, the bounds remain reasonable and outage resembles the low-SNR case.The latter case again has a 10 dB gap between SNR and β.Source: B. Tightness of Bounds · p. 9
  • In sparse networks the bounds are extremely accurate and s = 1/2 is near-optimal, but in dense networks the nearest-interferer bound is loose and the Jensen approximation misses how outage varies with s.In dense networks, non-dominant interferers contribute substantially to outages.Source: B. Tightness of Bounds · p. 10

C. Choosing the FPC exponent s

The exponent minimizing true outage depends on network conditions. The half exponent is robust across several regimes, while constant power is preferred when SNR is low, the target SINR is high, or density is high.

C. Choosing the FPC exponent s

  • s = 1/2 is robust across attenuation regimes, at high SNR, at low and moderate SINR targets, and at low densities; constant power is preferred in the other stated regimes.The study determines the true outage-minimizing exponent across parameter variations.Source: C. Choosing the FPC exponent s · p. 10
  • The SNR-to-β gap helps explain why the half exponent is near-optimal when noise is negligible, while very small exponents are optimal when SNR only slightly exceeds β.The text identifies a gap larger than approximately 20 dB as one where thermal noise is essentially negligible.Source: C. Choosing the FPC exponent s · p. 10
  • In relatively sparse networks, reducing signal-fading effects balances the cost of more harmful interference, whereas dense networks favor constant power.With many nearby interferers, the interference cost can overwhelm the fading benefit of increasing s.Source: C. Choosing the FPC exponent s · p. 10

V. POSSIBLE AREAS FOR FUTURE STUDY

The paper identifies cellular FPC, alternative objectives, scheduling, and iterative power control as directions for future study. It also qualifies apparent gains from negative exponents as limited to extremely dense networks with unreasonably high outage.

V. POSSIBLE AREAS FOR FUTURE STUDY

  • For s < 0, apparent gains over constant power occur only at extremely high densities where outage is unreasonably large.The text points to a threshold-based policy as a way to avoid transmissions by users unlikely to communicate successfully.Source: §V · p. 10
  • Cellular FPC remains unanalyzed here because regular-grid base stations do not permit the spatial Poisson model’s tractability.The paper notes numerical cellular results in but no analysis.Source: §V · p. 11
  • Optimizing expected spectral efficiency rather than outage could yield optimal exponents below zero.The stated objective is max E[log2(1 + SINR)].Source: §V · p. 11
  • The effect of scheduling on FPC, including whether s = 1/2 remains optimal, is left open; the authors conjecture the gain will be smaller but non-zero for sensible policies.They attribute this conjecture to softened interference inversion.Source: §V · p. 11
  • FPC-motivated partial compensation could be explored within iterative power control when users’ SINR targets cannot all be satisfied.The example raises power by 5 dB for a 10 dB target gap rather than fully compensating by 10 dB.Source: §V · p. 11

VI. CONCLUSIONS

The paper proposes fractional power control for decentralized networks and identifies s=1/2 as optimal for outage probability and transmission capacity under relatively low transmitter density and low noise. This choice balances compensating for desired-signal fading against increasing interference.

VI. CONCLUSIONS

  • Fractional power control is proposed as a general approach to pairwise power control in decentralized networks.The approach uses an exponent to balance reducing desired-signal fading against increasing interference.Source: §VI · p. 11
  • s=1/2 is optimal for outage probability and transmission capacity in relatively low-density networks with low noise, unlike constant power or channel inversion.The paper reports gains on the order of 50% or larger relative to either no power control or channel inversion as typical.Source: §VI · p. 11
  • In low-density, low-noise networks, the reported optimum for outage probability and transmission capacity reflects a balance between compensating desired-signal fades and amplifying interference, unlike constant-power or channel-inversion control.Fractional power control might provide a gain of 50% or larger relative to no power control or channel inversion in a typical wireless channel.Source: §VI · p. 11

APPENDIX

The appendix proves convexity of a function of the exponent and establishes its unique minimum at s=1/2. The proof uses log-convexity and Hölder’s inequality.

APPENDIX

  • For any non-negative random variable X, the function under consideration is convex in s and has a unique minimum at s=1/2.The derivative condition at the minimum is h′(s)=0.Source: C. Optimal Fractional Power Control… · p. 7, APPENDIX · pp. 11–12
  • The proof establishes log-convexity and uses the fact that a log-convex function is convex.It defines H(s) and invokes Hölder’s inequality to show H(s) is convex, which implies convexity of h(s).Source: APPENDIX · pp. 11–12
  • The convexity result and derivative condition identify s=1/2 as the unique minimizer.The appendix states that this minimizer satisfies h′(s)=0.Source: APPENDIX · p. 12
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