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Spatial Interference Cancellation for Multi-Antenna Mobile Ad Hoc Networks

Kaibin Huang, Jeffrey G. Andrews, Dongning Guo, Robert W. Heath, Jr., Randall A. Berry

arXiv:0807.1773v2cs.ITmath.IT

TL;DR

Interference limits peer-to-peer throughput in MANETs, and the paper studies whether multiple antennas can mitigate it. It analyzes zero-forcing cancellation in a Poisson network with Rayleigh fading, deriving transmission-capacity scaling and CSI training requirements. The results show improved scaling with more canceled interferers, including with imperfect CSI, and simulations report a large gain from canceling one interferer.

  • Problem

    The paper addresses the lack of theoretical characterization of how spatial interference cancellation affects transmission capacity in MANETs.

    Source: A. Prior Work and Motivation · p. 3

  • Method

    The paper analyzes zero-forcing receivers that cancel strongest interferers in a Poisson-transmitter network with Rayleigh fading and derives transmission-capacity bounds and scaling.

    Source: §I · p. 1, B. Contributions and Organization · p. 3, A. Network Model · p. 3

  • Results

    Transmission capacity decays as ϵ^(1/(L+1)) as outage probability vanishes, and for fixed outage probability grows as L^(1−2/α); imperfect CSI has the same vanishing-outage scaling when training length is appropriately chosen.

    Source: B. Contributions and Organization · p. 3, §I · p. 26

  • Takeaways & Limitations

    Simulations show that canceling one interferer can yield more than an order-of-magnitude capacity gain even with imperfect CSI, while two to four cancellations capture most available gain.

    Source: B. Contributions and Organization · p. 3

Abstract

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Interference between nodes is a critical impairment in mobile ad hoc networks (MANETs). This paper studies the role of multiple antennas in mitigating such interference. Specifically, a network is studied in which receivers apply zero-forcing beamforming to cancel the strongest interferers. Assuming a network with Poisson distributed transmitters and independent Rayleigh fading channels, the transmission capacity is derived, which gives the maximum number of successful transmissions per unit area. Mathematical tools from stochastic geometry are applied to obtain the asymptotic transmission capacity scaling and characterize the impact of inaccurate channel state information (CSI). It is shown that, if each node cancels L interferers, the transmission capacity decreases as the outage probability to the power of 1/(L+1) as the outage probability vanishes. For fixed outage probability, as L grows, the transmission capacity increases as L to the power of (1-2/alpha) where alpha is the path-loss exponent. Moreover, CSI inaccuracy is shown to have no effect on the transmission capacity scaling as the outage probability vanishes, provided that the CSI training sequence has an appropriate length, which we derived. Numerical results suggest that canceling merely one interferer by each node increases the transmission capacity by an order of magnitude or more, even when the CSI is imperfect.

I. INTRODUCTION

The paper addresses the lack of theoretical characterization of how spatial interference cancellation affects transmission capacity in MANETs. It analyzes receivers that use multiple antennas to cancel strong interferers and derives capacity scaling and training requirements under a modeled random-access network.

I. INTRODUCTION

  • Receivers use zero-forcing beamforming to cancel the strongest interferers while transmitters choose random beams, requiring only limited local coordination.The analysis assumes Poisson distributed transmitters and independent Rayleigh fading channels.Source: §I · p. 1, B. Contributions and Organization · p. 3

C. Transmission Capacity

  • Transmission capacity is the maximum density of successful transmissions under an outage-probability constraint for a target SINR.The model requires received SIR to exceed a common threshold θ.Source: §I · p. 1, C. Transmission Capacity · p. 4

A. Prior Work and Motivation

  • The paper addresses the lack of theoretical characterization of the relationship between MANET transmission capacity and spatial interference cancellation.Prior work considered other multi-antenna approaches, including interference averaging through diversity techniques.Source: A. Prior Work and Motivation · pp. 2–3

B. Contributions and Organization

  • Canceling L interferers makes transmission capacity decay as outage probability ϵ^(1/(L+1)) as ϵ vanishes, and for fixed ϵ it grows as L^(1−2/α).The latter scaling makes cancellation more effective for larger path-loss exponent α.Source: B. Contributions and Organization · p. 3
  • The required CSI training sequence length is derived to constrain outage increases and data-rate losses from imperfect cancellation, and is proportional to log(1/ϵ) for optimal TC scaling.The training length is subject to M ≥ L and integer-valued constraints.Source: B. Contributions and Organization · p. 3
  • Simulations report that canceling two to four interferers captures most available capacity gain, while canceling one yields more than an order-of-magnitude gain even with imperfect CSI.A moderate CSI training-sequence length is also observed to be sufficient.Source: B. Contributions and Organization · p. 3

A. Network Model

  • Potential transmitters form a two-dimensional Poisson point process and transmit independently with fixed probability, yielding an active process of density λ=Ptλo.Each transmitter has an associated receiver at unit distance.Source: A. Network Model · pp. 3–4

B. Channel Model

  • The channel model uses an N × N matrix of i.i.d. unit circularly symmetric complex Gaussian elements and path loss with exponent α>2.The paper models narrowband frequency-flat block fading.Source: B. Channel Model · p. 4

D. Zero-Forcing Beamforming

Receivers use zero-forcing beamforming to cancel their L strongest interferers, while retaining remaining spatial degrees of freedom for signal combining. With imperfect CSI, receivers estimate interferer channels from training sequences, and estimation errors add interference.

D. Zero-Forcing Beamforming

  • Each receiver nulls its L strongest interferer channels and uses the remaining N−L spatial degrees of freedom for maximum-ratio combining.The receiver beamformer is orthogonal to the canceled channels; the transmitter uses a fixed beamformer to avoid transmitters and receivers chasing each other’s beams.Source: D. Zero-Forcing Beamforming · p. 5

E. The Effective SIR Model: Perfect CSI

F. The Effective SIR Model: Imperfect CSI

  • Receivers identify the strongest interferers using random training signatures, then request training sequences to estimate their interference channels.The paper uses least-squares estimation; MMSE estimation would require accurately measuring aggregate interference covariance from weak transmitters, which is difficult in the presence of strong interferers.Source: F. The Effective SIR Model: Imperfect… · pp. 5–6
  • With imperfect CSI, estimation errors add interference relative to perfect cancellation.The estimated CSI is used to compute the beamformer, so channel-estimation errors leave residual interference after beamforming.Source: F. The Effective SIR Model: Imperfect… · pp. 6–7

III. TRANSMISSION CAPACITY WITH PERFECT CSI

This section analyzes outage probability and transmission capacity under perfect spatial interference cancellation. It characterizes residual interference by separating the strongest uncanceled interferer from the weaker interferers, then derives outage-probability bounds.

III. TRANSMISSION CAPACITY WITH PERFECT CSI

  • The section derives transmission-capacity scaling with the number of canceled interferers and the outage probability under perfect CSI.Its analysis focuses on outage probability and transmission capacity with perfect CSI.Source: §III · p. 7

A. Point Processes and Auxiliary Results

  • Conditioned on the primary interferer’s pre-cancellation power, the secondary interferers form a Poisson point process.The primary interferer is the (L+1)-st strongest by pre-cancellation power; treating it alone gives a lower bound on outage probability.Source: A. Point Processes and Auxiliary Results · pp. 7–8
  • After cancellation, the primary interference is its pre-cancellation power multiplied by a beta(1, N−1) random variable, while weaker interferers contribute secondary shot noise.The secondary-interference distribution has no known closed form except in simple cases, so the analysis obtains its conditional first and second moments.Source: A. Point Processes and Auxiliary Results · pp. 8–9

B. Bounds on Outage Probability

  • The outage-probability bounds use the primary interferer for a lower bound and bound secondary-interference exceedance using conditional moments and Chebyshev’s inequality.The paper derives bounds on outage probability from these steps.Source: B. Bounds on Outage Probability · p. 9

C. Asymptotic Transmission Capacity

The section characterizes how transmission capacity scales with outage probability and the number of canceled interferers. More cancellation slows capacity decay as outage vanishes, while increasing CSI estimation overhead.

C. Asymptotic Transmission Capacity

  • For fixed outage probability, transmission capacity grows in the order of L^(1−2/α), with faster growth for steeper path loss.The theorem assumes perfect CSI and fixed array gain (N − L).Source: C. Asymptotic Transmission Capacity · p. 10
  • For L + 1 > α, the exact capacity-scaling exponent is unknown because the available Chebyshev-based outage bound is not tight enough.A tighter bound may require the secondary-interference distribution, which has no known closed-form expression in this case.Source: C. Asymptotic Transmission Capacity · p. 11
  • For L + 1 ≤ α, transmission capacity scales as outage probability to the power 1/(L + 1), and reducing outage by two orders of magnitude lowers capacity 10-, 3.2-, and 1.8-fold for L = 1, 3, and 7, respectively.The scaling is also observed in simulations for outage probabilities ϵ ≤ 0.1.Source: C. Asymptotic Transmission Capacity · p. 11
  • Using more antennas for cancellation can slow capacity decay as outage vanishes, but increases CSI-estimation overhead for local coordination.The paper contrasts this balance with centralized scheduling, which requires global coordination and may incur much higher overhead.Source: C. Asymptotic Transmission Capacity · p. 11

IV. TRANSMISSION CAPACITY WITH IMPERFECT CSI

The section analyzes transmission capacity when CSI is imperfect, including QoS degradation and training-sequence requirements. With suitable training, the perfect-CSI asymptotic scaling remains attainable, while simulations indicate that a few canceled interferers and short training can capture most of the gain.

IV. TRANSMISSION CAPACITY WITH IMPERFECT CSI

  • One analyzed imperfect-CSI scenario allows higher outage probability and a lower data rate to relax users’ QoS requirements.The modified outage probability is defined using the corresponding SIR threshold.Source: §IV · p. 12
  • Theorem 3 gives a sufficient training-sequence length to satisfy bounds on outage-probability and data-rate degradation, with the resulting normalized capacity loss bounded.The sequence length also accounts for M ≥ L and integer-valued M.Source: §IV · p. 12
  • For stringent QoS-degradation constraints, training length is approximately proportional to the logarithms of the reciprocal outage and rate-degradation tolerances.Normalized capacity loss is more sensitive to outage tolerance when ϵ is large, and to rate-degradation tolerance when log2(1 + θ) is small.Source: §IV · p. 12
  • With suitable training-sequence growth as outage vanishes, imperfect CSI achieves the same asymptotic transmission-capacity scaling as perfect CSI.The required length grows sub-linearly with 1/ϵ, with an arbitrary positive exponent.Source: §IV · pp. 12–13
  • Simulations indicate that canceling only a few strongest interferers and using short training sequences can approach perfect-CSI capacity; overhead is insignificant when mobility is low.The paper measures CSI-estimation overhead as a capacity decrease by a factor of M/tc, where tc is channel coherence time.Source: §IV · p. 13

V. SIMULATION AND DISCUSSION

Simulations examine outage bounds and transmission-capacity scaling under perfect and imperfect CSI, then assess gains from canceling more interferers. Results show substantial capacity gains from cancellation, while CSI accuracy and the outage constraint shape the observed performance.

V. SIMULATION AND DISCUSSION

  • Simulations use a disk network with a Poisson number of 200 transmitter-receiver pairs on average, with θ = 3 or 4.8 dB, N − L = 2, and α = 4 unless specified otherwise.The typical receiver is at the disk center, and disk area is adjusted to node density.Source: §V · p. 13

A. Bounds on Outage Probability

  • Outage probability is approximately proportional to λ^(L+1); the bounds tighten as L or λ decreases and converge to exact values as λ decreases.The passages attribute bound looseness to secondary interference relative to primary interference.Source: A. Bounds on Outage Probability · pp. 13–14
  • At L = 3, imperfect-CSI outage rapidly approaches the perfect-CSI result as training length M increases; at M = 11, the increase is less than two-fold.Figure 3 compares outage against transmitting-node density for M = 3, 5, and 11.Source: A. Bounds on Outage Probability · p. 14, Figure 3 · p. 14

B. Scaling of Transmission Capacity

  • For both L + 1 ≤ α and L + 1 > α, small-outage TC scaling follows a power law with exponent 1/(L + 1); simulated slopes converge to asymptotic upper-bound slopes only for L + 1 > α.For L + 1 ≤ α, the asymptotic upper bound is tight even for outage probabilities from 0.01 to 0.1.Source: B. Scaling of Transmission Capacity · p. 14

C. Transmission Capacity vs. Size of Antenna Array

  • At outage probability ϵ = 10^-2, canceling two interferers per node provides a 25-time transmission-capacity gain over no cancellation.More generally, canceling a few interferers yields an order-of-magnitude-or-more gain.Source: C. Transmission Capacity vs. Size of… · p. 14
  • Capacity gains diminish as more interferers are canceled because secondary interference becomes limiting, and the outage constraint affects capacity more at smaller L..Source: C. Transmission Capacity vs. Size of… · p. 15
  • For imperfect CSI, capacity loss decreases as M increases; at M = 11 and L = 7, the reduction is 25%, while M = 3 still permits gains above an order of magnitude.The authors describe this as confirming the practicality of interference cancellation.Source: C. Transmission Capacity vs. Size of… · p. 15

APPENDIX

The appendix establishes intermediate distributional and interference results used in the outage analysis. It proves conditional Poisson properties, characterizes selected-interferer statistics, and derives secondary-interference moments.

A. Proof for Lemma 1

  • Conditioned on J_P = g, Π(g) is Poisson distributed with the stated mean measure, and counts on disjoint sets are independent.The proof establishes the conditional count distribution and independence for disjoint measurable sets.Source: A. Proof for Lemma 1 · p. 17

B. Proof for Lemma 2

  • The proof uses the chi-square distribution of ρ_T to obtain the cumulative distribution and density of J_P.ρ_T has a chi-square distribution with N complex degrees of freedom.Source: B. Proof for Lemma 2 · pp. 17–18

C. Proof for Lemma 3

  • For uncanceled interferers, the normalized channel projections yield beta(1, N − 1) variables independent of the cancellation criterion and of one another.The proof derives these properties from isotropic channel vectors and their independence.Source: C. Proof for Lemma 3 · p. 18

D. Proof of Lemma 4

  • Campbell’s Theorem and the projection distribution are used to derive the mean and conditional variance of secondary interference.The proof uses independence from J_P and the expected projection value 1/N.Source: D. Proof of Lemma 4 · pp. 18–19

E. Proof for Lemma 5

The proof derives lower and upper bounds on outage probability by conditioning on the primary-interference quantities and bounding residual interference.

E. Proof for Lemma 5

F. Proof of Theorem 1

The proof bounds total interference and derives a scaling for the admissible transmitter density. It then uses that scaling and the transmission-capacity definition to obtain the theorem’s result.

F. Proof of Theorem 1

  • The proof obtains the expected total interference from the primary-interference distribution and secondary-interference moments, then applies Markov’s inequality..Source: F. Proof of Theorem 1 · p. 20
  • A lower bound on transmitter density is derived by considering the weakest of the L + 1 strongest uncanceled interferers and applying Markov’s inequality.The density scaling is then derived using Kershaw’s inequality.Source: F. Proof of Theorem 1 · pp. 20–21
  • The proof states that the desired transmission-capacity result follows from the derived limits and the transmission-capacity definition.The derivation also invokes the strong law of larger numbers.Source: F. Proof of Theorem 1 · p. 21

G. Proof of Lemma 6

The proof analyzes the bounds for large L or vanishing outage probability using asymptotic expressions for the Λ terms. It splits Λ3 into two terms and uses a condition on L and α to establish a finite Gamma-function term.

G. Proof of Lemma 6

  • For λ → 0, the proof obtains Λ1 from (32) and uses the distributions of W and δP to establish inequalities in (36) and (39).Source: G. Proof of Lemma 6 · p. 22
  • For λ → 0, the proof splits Λ3 into Λ3,1 and Λ3,2 to derive its asymptotic expression.The bound for Λ3,1 is of order λ^(L+1), with a remainder of order λ^(L+2).Source: G. Proof of Lemma 6 · p. 22
  • The second inequality in (36) follows; for L − α + 1 ≥ 0, Γ(L − α + 2) is finite, and substituting (85), (87), (90), and (96) into (94) gives the second equality in (39).Source: G. Proof of Lemma 6 · p. 23

H. Proof of Theorem 3

The proof derives a training-sequence length for imperfect CSI by bounding outage and data-rate changes. It uses the resulting bounds to upper-bound the capacity loss.

H. Proof of Theorem 3

  • Conditioned on Φ and {h_T}, residual-interference randomness depends only on the i.i.d. data symbols, so its conditional variance can be expressed accordingly.The derivation then uses the stated distributions of {z_T} and ζ.Source: H. Proof of Theorem 3 · p. 24
  • Solving the sufficient constraints on outage increase and data-rate loss gives the training-sequence length in (47).The proof first bounds the outage probability for imperfect CSI and conditions residual-interference variance on the network and channel realizations.Source: H. Proof of Theorem 3 · pp. 23–25
  • The proof upper-bounds the capacity loss defined in (46) and concludes the desired result in (48).Source: H. Proof of Theorem 3 · p. 25

I. Proof of Theorem 4

The proof chooses training parameters to make the outage increase and data-rate loss vanish as the target outage probability vanishes. It concludes that imperfect CSI has the same transmission-capacity scaling as perfect CSI.

I. Proof of Theorem 4

  • The proof chooses Z and M as functions of ϵ to establish the required vanishing error terms.Source: §I · pp. 25–26
  • The imperfect-CSI transmission-capacity scaling is identical to the perfect-CSI scaling when the chosen training parameters make the outage increase and data-rate loss vanish as ϵ → 0.The proof states that the training-sequence length achieves this scaling.Source: §I · pp. 25–26
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