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Communication in a Poisson Field of Interferers -- Part I: Interference Distribution and Error Probability

Pedro C. Pinto, Moe Z. Win

arXiv:1001.4519v1cs.ITcs.NImath.IT

TL;DR

The paper addresses how to model network interference beyond a simple AWGN approximation that omits important physical characteristics. It uses a spatial Poisson model for asynchronous interferers and analyzes aggregate interference and error performance across slow- and fast-varying positions. The aggregate interference has stable-distribution characterizations, and the paper derives outage or average error measures for the respective scenarios.

  • Problem

    A simple AWGN approximation of interference does not capture node distribution, transmission characteristics, or propagation effects such as shadowing and multipath fading.

    Source: §I · p. 1, A. Spatial Distribution of the Nodes · p. 2

  • Method

    The paper models asynchronous interferers with a spatial Poisson process and analyzes interference and link error performance for slow- and fast-varying interferer positions.

    Source: §I · p. 1, D. Mobility and Session Lifetime of… · p. 3

  • Results

    The aggregate interference is related to a skewed stable distribution conditioned on interferer positions and a symmetric stable distribution unconditionally; error performance is expressed as outage or average error probability for slow- or fast-varying interferers, respectively.

    Source: §V · p. 9

  • Takeaways & Limitations

    For fixed error performance, increasing interferer energy requires lowering their density, and performance degrades faster as density rises than as transmitted power rises.

    Source: C. Discussion · p. 7

Abstract

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We present a mathematical model for communication subject to both network interference and noise. We introduce a framework where the interferers are scattered according to a spatial Poisson process, and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We consider both cases of slow and fast-varying interferer positions. The paper is comprised of two separate parts. In Part I, we determine the distribution of the aggregate network interference at the output of a linear receiver. We characterize the error performance of the link, in terms of average and outage probabilities. The proposed model is valid for any linear modulation scheme (e.g., M-ary phase shift keying or M-ary quadrature amplitude modulation), and captures all the essential physical parameters that affect network interference. Our work generalizes the conventional analysis of communication in the presence of additive white Gaussian noise and fast fading, allowing the traditional results to be extended to include the effect of network interference. In Part II of the paper, we derive the capacity of the link when subject to network interference and noise, and characterize the spectrum of the aggregate interference.

I. INTRODUCTION

The paper addresses limitations of simpler interference models with a Poisson-field framework that makes spatial, transmission, and propagation characteristics explicit. It studies the resulting interference and link performance for asynchronous nodes under several network and transmission assumptions.

I. INTRODUCTION

  • The two-part paper models asynchronous Poisson-distributed interferers with path loss, shadowing, and multipath fading; Part I analyzes aggregate interference and link error performance.Earlier studies omitted some propagation effects, assumed synchronization, or restricted interferer locations or counts.Source: §I · p. 1

A. Spatial Distribution of the Nodes

  • A homogeneous Poisson point process places nodes over an infinite two-dimensional plane, with density λ counting terminals transmitting in the relevant band and interval.The probe receiver is at the origin, its transmitter is a fixed distance r0 away, and other nodes are interferers at random distances.Source: A. Spatial Distribution of the Nodes · p. 2

B. Transmission Characteristics of the Nodes

  • Interferers use a common linear modulation and power, while the probe transmitter may use arbitrary modulation and power; each interferer transmits with an independent random delay Di ∼ U(0, T).A conventional linear detector is used at the probe receiver.Source: B. Transmission Characteristics of the… · p. 2, Figure 2 · p. 2

C. Propagation Characteristics of the Medium

The model combines distance-dependent path loss, log-normal shadowing, and fast fading, while distinguishing slowly changing interferer positions from frequently changing active sets. The receiver represents the resulting signal as desired transmission, aggregate interference, and noise, then analyzes the interference distribution.

C. Propagation Characteristics of the Medium

  • Received amplitude decays as k/r^b; log-normal shadowing fluctuates around that median path gain, and multipath adds random phase and normalized, arbitrarily distributed amplitude fading.Fast fading is averaged out, and channel effects are assumed independent across nodes and approximately constant for at least one symbol interval.Source: C. Propagation Characteristics of the… · p. 3

D. Mobility and Session Lifetime of the Interferers

  • Slow-varying positions stay approximately constant over a symbol or packet and support conditioned outage analysis, whereas frequent changes in the active interferer set motivate average error analysis.The slow case has long-lived transmitters; the fast case has short sessions with bursts of symbols followed by inactivity.Source: D. Mobility and Session Lifetime of… · p. 3

A. Complex Baseband Representation of the Interference

  • Because interferers transmit asynchronously and independently, their signal components are identically distributed, and the aggregate interference distribution is characterized for position-conditioned and unconditional cases.The received signal also includes the probe transmission and independent AWGN; the receiver demodulates by projecting the aggregate signal onto an orthonormal set.Source: A. Complex Baseband Representation of… · pp. 3–4

B. P-conditioned Interference Distribution

With interferer positions fixed, the aggregate interference is modeled as circularly symmetric complex Gaussian, while the position-dependent factor A has a skewed stable distribution.

B. P-conditioned Interference Distribution

  • Prior work justifies the CS Gaussian approximation by comparing Kullback-Leibler divergence and exact versus approximated error probabilities.The section begins its P-conditioned distribution derivation from results in.Source: B. P-conditioned Interference… · p. 4
  • Conditioned on interferer positions, the sum of independent circularly symmetric complex Gaussian terms is itself CS Gaussian.The cited discussion motivates this approximation using a coded-interferer model near capacity, where each signal is Gaussian.Source: B. P-conditioned Interference… · p. 4
  • For equiprobable, origin-symmetric constellations such as M-PSK and M-QAM, V_X = E/3.The passage defines origin symmetry as having the opposite constellation point for every point.Source: B. P-conditioned Interference… · p. 4
  • A depends on interferer positions and has a skewed stable distribution, plotted for different amplitude loss exponents b and densities λ at σdB = 10.The figure caption notes that stable laws include distributions with heavier algebraic tails than Gaussian distributions.Source: B. P-conditioned Interference… · p. 4, Figure 3 · p. 5

C. Unconditional Interference Distribution

When interferer positions are averaged over, the aggregate interference has a symmetric complex stable distribution. For Rayleigh fading, the paper gives a special-case moment expression and tabulates values across loss exponents and modulations.

C. Unconditional Interference Distribution

  • The unconditional aggregate interference Y has a symmetric complex stable distribution, arising from sums over Poisson-process distances and i.i.d. CS terms.The distribution is averaged over positions P; the text specifies the stable law for b > 1.Source: C. Unconditional Interference… · p. 5
  • For Rayleigh fading, E{|X_i,n|^(2/b)} reduces to k^(2/b)Γ(·)χ(b), with χ(b) depending on b and the interferers’ constellation.Table I gives numerical values of this moment for various b and modulations; for M-PSK it is proportional to E^(1/b).Source: C. Unconditional Interference… · p. 5, Table I · p. 5

D. Discussion

The discussion distinguishes the exactly stable unconditional interference from the approximately CS Gaussian interference when positions are fixed.

D. Discussion

  • The paper considers both slow- and fast-varying interferer positions as part of its framework.The discussion’s conditioning distinction concerns whether positions are random or fixed.Source: D. Discussion · p. 5
  • With random positions and shadowing, the unconditional interference Y is exactly stable-distributed, as in (16).The expressions (16) and (17) hold for a broad class of fading distributions, not only Rayleigh fading.Source: D. Discussion · p. 5
  • With positions fixed, A is fixed and Y is approximately CS Gaussian with total variance 2AV_X.The discussion distinguishes random position outcomes from the P-conditioned case.Source: D. Discussion · p. 5

IV. ERROR PROBABILITY

The section derives link error probabilities under slow-varying interferer positions, including threshold-based outage, and describes how interference enters the error calculation. It also gives the corresponding M-QAM geometry parameters.

IV. ERROR PROBABILITY

  • The error-probability analysis covers both slow- and fast-varying interferer positions, with the slow-varying case conditioned on user positions and probe-link shadowing.The conditional symbol error probability is denoted P_e(G_0, P).Source: §IV · p. 5, A. Slow-varying Interferer Positions P · p. 5

A. Slow-varying Interferer Positions P

  • Conditioned on interferer locations, the analysis reduces to a Gaussian problem; coherent linear-modulation error expressions use 2AV_X + N_0 rather than N_0 as the total noise variance.This substitution is valid for any linear modulation, extending traditional AWGN and fast-fading results to include network interference –.Source: A. Slow-varying Interferer Positions P · pp. 5–6

P MQAM

  • In the general error expressions, 2AV_X accounts for interference: A reflects interferer locations and propagation, while V_X reflects interferer transmission characteristics.Interferer modulation affects V_X, whereas probe modulation determines the form of the error-probability expression.Source: P MQAM · p. 6
  • For M-QAM, the error expression uses constellation decision-region geometry, including boundary angles and the set of signal points sharing a boundary.The treatment assumes a square M-QAM constellation with M = 2^n points and n even.Source: P MQAM · p. 6, Figure 4 · p. 6
  • For slow-varying positions, outage probability is more meaningful than error probability averaged over positions because the conditional error probability varies with positions and probe shadowing.Outage is defined by the probability that P_e(G_0, P) exceeds a target p*.Source: P MQAM · p. 6

B. Fast-varying Interferer Positions P

For fast-varying interferer positions, the analysis averages error probability over possible positions while retaining the probe-link shadowing. Stable-variable decomposition yields a Gaussian combined-noise problem without the cited Gaussian approximation.

B. Fast-varying Interferer Positions P

  • Stable-variable decomposition reduces the fast-varying-position analysis to a Gaussian combined-noise problem without using the Gaussian approximation of (10).The derivation uses the decomposition property of symmetric stable random variables.Source: B. Fast-varying Interferer Positions P · pp. 6–7
  • The average error probability uses coherent AWGN and fast-fading expressions with 2BV_G + N_0 replacing N_0, then averages over B while retaining probe shadowing G_0.The position-averaged quantity is P_e(G_0), since the probe transmitter is immobile and G_0 is slow-varying.Source: B. Fast-varying Interferer Positions P · pp. 6–7

P MQAM

C. Discussion

The discussion examines how error performance depends on interferer density and average symbol energy. It identifies a density–energy tradeoff and relates interference to the product λ^bE.

C. Discussion

  • The discussion analyzes error performance as a function of interferer density λ and average symbol energy E.Source: C. Discussion · p. 7
  • At fixed error performance, increasing interferer energy requires decreasing density λ to maintain the same outage probability.Figure 5 presents INR–λ curves of constant P_e for the stated slow-varying-position scenario.Source: Figure 5 · p. 7, C. Discussion · p. 7
  • The interference term is proportional to λ^bE, with b > 1, so error performance degrades faster as interferer density increases than as their transmitted power increases.This scaling yields the density–energy tradeoff for fixed error performance.Source: C. Discussion · p. 7

D. Numerical Results

The numerical results show that interference density and power worsen outage performance in heterogeneous networks, while common power improves average error probability only marginally at high power in homogeneous networks. The paper also relates the path-loss exponent’s effect to its simultaneous influence on the desired signal and aggregate interference.

D. Numerical Results

  • The error-probability plots use analytical expressions with Monte Carlo simulation over stable random variables A and B, avoiding simulation of interferer positions and waveforms.The plotted probabilities are semianalytical.Source: D. Numerical Results · p. 7
  • In heterogeneous networks, outage error worsens as λ or INR rises at fixed SNR; in homogeneous networks, common power improves average error, with marginal gains as P grows.The heterogeneous case allows probe and interferer powers to differ; the homogeneous case uses equal powers.Source: D. Numerical Results · p. 8, Figure 6 · p. 8, Figure 7 · p. 8
  • The path-loss exponent b can improve or degrade performance depending on link length and other parameters because it affects both the desired signal and aggregate interference.The desired signal changes through 1/r0^b, while interference changes through distribution parameters.Source: D. Numerical Results · p. 9

V. SUMMARY

  • The model relates aggregate interference to a skewed stable distribution conditioned on interferer positions and a symmetric stable distribution unconditionally.It characterizes slow-varying interferers with outage probability and fast-varying interferers with average error probability; the expressions apply to linear modulation schemes.Source: §V · p. 9

APPENDIX A DERIVATION OF THE COMPLEX BASEBAND INTERFERENCE

Appendix A forms complex baseband interference by projecting the received interference onto carrier basis functions. Appendix B derives the distribution of A by mapping planar Poisson node locations to one-dimensional Poisson arrival times and applying a stable-sum result.

APPENDIX A DERIVATION OF THE COMPLEX BASEBAND INTERFERENCE

  • The derivation projects Y(t) onto two carrier basis functions and combines the resulting components into the complex random variable Xi.The first projection uses ψ1(t); the second uses ψ2(t).Source: Appendix A · p. 9

APPENDIX B DERIVATION OF THE DISTRIBUTION OF A

  • For A’s distribution, the derivation applies a stable-sum theorem to nonnegative i.i.d. terms with a finite α-moment for 0 < α < 1.The planar Poisson process is mapped to one-dimensional arrival times with rate λπ.Source: Appendix B · p. 9
  • Using the log-normal moment property, the derivation obtains the stated distribution of A for b > 1.The planar node distances map to Poisson arrival times, enabling the theorem’s application.Source: Appendix B · pp. 9–10

APPENDIX C DERIVATION OF THE DISTRIBUTION OF Y

Appendix C derives the distribution of Y by applying a complex stable-sum theorem to the Poisson-mapped interferer contributions. Uniform phase makes each contribution circularly symmetric, and the derivation concludes for b > 1.

APPENDIX C DERIVATION OF THE DISTRIBUTION OF Y

  • The derivation considers independent, identically distributed circularly symmetric complex random variables Zi with finite α-moment, E{|Zi|α} < ∞, for 0 < α < 2, independent of sequence {τi}.Source: Appendix C · p. 10
  • The Poisson mapping theorem lets the derivation apply the complex stable-sum result to the aggregate interference representation.Source: Appendix C · p. 10
  • Uniform phase makes Xi and its shadowed form circularly symmetric; using the log-normal moment property yields the distribution of Y for b > 1.Source: Appendix C · p. 10
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