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On Stochastic Geometry Modeling of Cellular Uplink Transmission with Truncated Channel Inversion Power Control
Hesham ElSawy, Ekram Hossain
TL;DR
The paper addresses tractable uplink analysis under realistic UE power constraints in irregular, single- and multi-tier cellular networks. Using stochastic geometry with truncated channel inversion, it identifies performance regimes and shows that cutoff selection and power constraints govern outage and spectral efficiency.
Problem
Irregular cellular deployments and uplink-specific UE power control and maximum-power limitations require tractable stochastic-geometry models beyond regular-grid analyses.
Method
The paper develops a stochastic-geometry framework for single- and multi-tier uplinks using truncated channel inversion, accounting for per-UE power control, maximum transmit power, and cutoff thresholds.
Results
The framework identifies a transfer point governed by (λ, Pu, ρo); binding power constraints make performance depend on λ and ρo, while loose constraints yield intensity-independent behavior.
Takeaways & Limitations
An optimal cutoff threshold can minimize outage probability, while in dense networks additional tiers or BSs may reduce UE power consumption without changing SINR outage or spectral efficiency.
Abstract
from arXiv · showhide
Using stochastic geometry, we develop a tractable uplink modeling paradigm for outage probability and spectral efficiency in both single and multi-tier cellular wireless networks. The analysis accounts for per user equipment (UE) power control as well as the maximum power limitations for UEs. More specifically, for interference mitigation and robust uplink communication, each UE is required to control its transmit power such that the average received signal power at its serving base station (BS) is equal to a certain threshold $ρ_o$. Due to the limited transmit power, the UEs employ a truncated channel inversion power control policy with a cutoff threshold of $ρ_o$. We show that there exists a transfer point in the uplink system performance that depends on the tuple: BS intensity ($λ$), maximum transmit power of UEs ($P_u$), and $ρ_o$. That is, when $P_u$ is a tight operational constraint with respect to [w.r.t.] $λ$ and $ρ_o$, the uplink outage probability and spectral efficiency highly depend on the values of $λ$ and $ρ_o$. In this case, there exists an optimal cutoff threshold $ρ^*_o$, which depends on the system parameters, that minimizes the outage probability. On the other hand, when $P_u$ is not a binding operational constraint w.r.t. $λ$ and $ρ_o$, the uplink outage probability and spectral efficiency become independent of $λ$ and $ρ_o$. We obtain approximate yet accurate simple expressions for outage probability and spectral efficiency which reduce to closed-forms in some special cases.
I. INTRODUCTION
The paper addresses the difficulty of modeling uplink cellular transmission under random topologies, per-UE power control, correlated interferers, and maximum transmit-power limits. It proposes a tractable stochastic-geometry framework for single- and multi-tier networks and identifies operating regimes governed by BS intensity, maximum UE power, and the cutoff threshold.
- Motivation: Stochastic geometry models randomized cellular topologies with tractable point-process abstractions such as the Poisson point process.The motivation includes heterogeneous deployments and spatially varying capacity demand.
- Research gap: Uplink analysis is challenging because neighboring interfering UEs can be closer to a BS than its tagged UE, making per-UE power control essential.Orthogonal channel assignment also correlates the locations of UEs using the same uplink channel.
- Research gap: Existing stochastic-geometry uplink models did not account for the UEs’ maximum transmit power.Prior models also used differing approximations to simplify uplink analysis.
- Contributions: The proposed framework models uplink transmission in single- and multi-tier cellular networks while accounting for per-UE power control, limited UE power, and a cutoff threshold.The analysis partially ignores mutual correlations among interfering sources while retaining correlations between interferers and tagged network elements, with accuracy validated by simulations.
- Contributions: Approximate yet accurate simple expressions are derived for outage probability and spectral efficiency, including closed forms in special cases.The framework is designed to preserve analytical tractability while characterizing both network settings.
- Design insights: The uplink has a transfer point determined by (λ, P_u, ρ_o): binding power constraints make performance depend on λ and ρ_o, whereas non-binding constraints remove that dependence.The cutoff threshold also creates tradeoffs, with an optimal threshold minimizing outage probability.
B. Radio Channel Model
The model uses a common power-law channel with Rayleigh fading and nearest-BS association in open-access Poisson cellular networks. Truncated channel inversion makes transmit powers random and divides UEs into active and inactive subsets.
- Channel and association assumptions: Signal power decays as r^-η, with a common path-loss exponent across tiers and independent unit-mean exponential channel gains under Rayleigh fading.
- Channel and association assumptions: UEs associate with their nearest BS based on average link quality, under an open-access policy across all tiers.
- Transmit power analysis: Random serving distances and truncated channel inversion produce random UE transmit powers, motivating characterization of their distribution and moments before SINR analysis.
- Transmit power analysis: A relatively high cutoff threshold leaves cell-edge UEs inactive when they cannot compensate for path loss, whereas a low threshold can avoid truncation outage.
- Transmit power analysis: Lowering ρo reduces UE transmit power, while truncation outage increases with ρo and decreases exponentially with BS intensity λ.
B. SINR Analysis
The SINR analysis derives outage and spectral-efficiency expressions for active uplink UEs using a tagged BS and tractable interference approximations. It explicitly captures source-to-test-element correlation while approximating interfering UEs as an independent PPP with independent transmit powers.
- SINR outage: SINR outage is analyzed for a tagged BS, with useful received signal power ρoho and interference represented by the aggregate random variable I.
- SINR outage: The outage calculation uses the expectation over I and the Laplace transform of its probability density function.
- Interference approximation: The exact interfering-UE process is correlated because each BS assigns a channel uniquely, but the analysis approximates it by a PPP and ignores mutual transmit-power correlations for tractability.
- SINR outage: Theorem 1 gives the SINR outage probability for a generic active UE under the PPP and independent-power assumptions.
- Spectral efficiency and tradeoff: Theorem 2 similarly gives average uplink spectral efficiency, and increasing ρo improves spectral efficiency and SINR outage while increasing truncation outage and UE transmit power.
1) Special case 1 (infinite Pu):
With infinite maximum transmit power, the uplink maximum-power constraint is non-binding. In this regime, SINR outage and average spectral efficiency are independent of BS intensity, and in the interference-limited case also independent of ρo.
- Non-binding power constraint: Setting Pu = ∞ models dense deployments where serving distances are small enough that UEs remain below the maximum power almost surely.
- Non-binding power constraint: SINR outage and average spectral efficiency become independent of BS intensity when maximum transmit power is not a binding constraint.
- Performance interpretation: In the PPP uplink, increasing BS intensity neither improves nor degrades SINR outage or average spectral efficiency under the non-binding constraint.
- Performance interpretation: Interference management techniques are identified as the mechanisms through which coverage probability and average spectral efficiency can improve in this regime.
2) Special case 2 (η = 4):
The paper develops special-case and multi-tier results for truncated channel inversion. Closed forms arise for selected parameters, while multi-tier performance generally depends on relative cutoff thresholds and BS intensities, except when power is non-binding.
- Special cases: For integer η, the single-tier SINR-outage integral can be expressed in closed form; η = 4 is given as an example.
- Special cases: In the interference-limited, non-binding-power case, outage and spectral efficiency are independent of both ρo and BS intensity.
- Multi-tier extension: The single-tier framework extends naturally to multi-tier Poisson networks with tier-specific intensities and cutoff thresholds.
- Multi-tier extension: The multi-tier analysis approximates each tier’s interfering UEs as an independent PPP with independent transmit powers, separating co-tier and cross-tier interference.
- Multi-tier performance: When Pu is non-binding, multi-tier performance becomes independent of the different tier intensities and the network reduces to a single-tier model with intensity Λ.
- Multi-tier performance: Multi-tier outage and spectral efficiency generally depend on relative cutoff thresholds and relative BS intensities, with performance improving when the target tier’s cutoff rises and other tiers’ thresholds or intensities fall.
B. Different Path-loss Exponents
For multi-tier networks with different path-loss exponents, the framework models transmit-power distributions and derives general outage and spectral-efficiency formulas under tractability assumptions. These formulas are flexible and recover previously presented special cases, but generally do not simplify to closed forms.
- Different path-loss exponents produce weighted Voronoi association regions rather than the single-tier Voronoi tessellation.
- Lemma 3 gives the transmit-power pdf and moments for active UEs in each tier under truncated channel inversion.
- The analysis approximates interfering UE locations as a PPP and ignores correlations among interfering sources and their transmit powers for tractability.
- Theorem 5 provides a general uplink SINR outage formula for a K-tier Poisson network with tier-specific intensities, cutoff thresholds, and path-loss exponents.
- Theorem 6 provides the corresponding average uplink spectral-efficiency formula under the same independent-PPP and independent-power assumptions.
- The multi-tier formulas reduce to earlier special cases, but complicated transmit-power moments prevent simple outage and spectral-efficiency formulas when path-loss exponents differ.
A. Results
The numerical study validates the proposed uplink model against simulations and examines SINR outage, total outage, effective spectral efficiency, and transmit power. Results reveal distinct binding and non-binding maximum-power regimes governed by cutoff threshold, BS intensity, and noise.
- The study validates the model using Poisson-network simulations with randomly placed UEs and 10,000 repetitions.
- The derived model accurately captures SINR outage, while circular coverage approximations can under- or overestimate outage depending on the cutoff threshold.The circular approximation underestimates outage for ρo ≤−70 dBm and overestimates it at ρo = −90 dBm.
- The cutoff threshold trades off SINR outage against truncation outage, with lower thresholds favoring SINR degradation and higher thresholds increasing truncation outage.
- When maximum transmit power is non-binding and noise is negligible, SINR outage is independent of both cutoff threshold and BS intensity.
- Effective spectral efficiency is independent of BS intensity in non-binding regimes, but depends on cutoff threshold under relatively high noise and on both intensity and threshold when power becomes binding.
- An optimal cutoff threshold maximizes effective spectral efficiency, while average transmit power is non-decreasing in the cutoff threshold.
B. Discussions
The paper contrasts uplink and downlink operation, emphasizing that uplink power control is essential and that maximum UE power creates a transfer point in performance. It also identifies tractability benefits and limitations of truncated channel inversion control.
- Uplink power control is essential because nearby interfering UEs can cause severe inter-cell interference, unlike downlink operation with association-based interference protection.
- A binding maximum transmit power constraint makes single-tier uplink performance depend strongly on BS intensity and the cutoff threshold.
- With non-binding maximum power, the uplink can be represented using an equivalent constant-transmit-power network and becomes similar to downlink performance.
- Fixed received power ρ_o in uplink increases SINR outage probability and reduces spectral efficiency relative to downlink, including 0.77 nat/sec/Hz versus 1.49 nat/sec/Hz.
- In multi-tier networks, uplink performance depends on relative cutoff thresholds and BS intensities, but common thresholds, common path loss, and loose power constraints remove BS-intensity dependence.
- The truncated channel-inversion analysis does not capture fractional channel inversion and yields location-independent performance for active users, unlike fractional control.
- The framework identifies an optimal outage-minimizing cutoff threshold and provides simple approximate outage and spectral-efficiency expressions, with closed forms in special cases.
- When maximum UE power is non-binding, SINR outage and spectral efficiency are independent of BS intensity, although denser deployments may reduce UE transmit-power consumption and improve reuse and capacity.
APPENDIX A PROOF OF LEMMA 1
The appendix derives the transmit-power distribution and the interference Laplace transform for the single-tier uplink model. It uses truncated channel inversion, channel-assignment facts, and a PPP approximation for interfering UEs.
- The nearest-BS uplink distance has Rayleigh density f_r_o(r) = 2πλr e^-πλr^2, and truncated inversion sets transmit power to ρ_o r_o^η subject to P ≤ P_u.
- The transmit-power density is normalized under truncated channel inversion, and its αth moment is used in subsequent interference calculations.
- The interference derivation accounts for correlated interfering-UE locations caused by orthogonal channel assignment and correlation with the tagged BS.
- The model assumes the average useful received signal equals ρ_o, each individual interferer contributes less than ρ_o on average, and each channel has one served UE per BS.
- For tractability, interfering UEs are modeled as a PPP with independent transmit powers; the interference Laplace transform follows using independence, the PPP PGFL, and the fading transform.
- The outage expression is obtained by substituting the aggregate-interference Laplace transform and the transmit-power moment into the SINR formulation.
APPENDIX C PROOF OF THEOREM 2
This appendix extends the interference and spectral-efficiency derivation to multi-tier uplinks. It applies tier-specific cutoff thresholds and intensities while using PPP and independent-power assumptions for tractability.
- Average spectral efficiency is obtained by integrating the probability that ln(1 + SINR) exceeds a threshold.
- The spectral-efficiency theorem substitutes the interference Laplace transform and transmit-power moment into the integral after changing variables to x = e^t − 1.
- For tier-k interferers observed at a tier-j tagged BS, the derivation uses tier-specific useful-signal and interference thresholds, one UE per channel per BS, and PPP-independent powers.
- The aggregate interference from tier-k UEs is represented through a PPP of intensity λ_k, with an indicator capturing correlation with the tagged BS.
- The multi-tier outage theorem follows by substituting the tier-specific interference transform and transmit-power moment into the SINR expression.
APPENDIX E PROOF OF LEMMA 3
The appendix derives transmit-power distributions for users associated with multi-tier networks, including tier-dependent link qualities and cutoff thresholds. It notes a closed-form limitation when path-loss exponents differ.
- The UE-to-tier-k nearest-BS distance has density f_r_k(r) = 2πλ_k r e^-πλ_k r^2.
- Association with the tier offering the best link quality determines the serving-tier distance relationships and the transmit power for a tier-j-associated UE.
- The transmit-power CDF and PDF are derived under the constraint 0 ≤ P_j ≤ P_u.
- The αth moment of transmit power lacks a closed form except when all tiers share a common path-loss exponent η.
- For tier-k interference at a tier-j tagged BS, the derivation uses the tier-specific aggregate-interference representation and association-based distance relation.