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Multi-Antenna Communication in Ad Hoc Networks: Achieving MIMO Gains with SIMO Transmission

Nihar Jindal, Jeffrey G. Andrews, Steven Weber

arXiv:0809.5008v2cs.IT

TL;DR

The paper asks whether receive antennas alone can produce linear throughput gains in ad hoc networks, despite prior SIMO results showing logarithmic or sublinear gains. It analyzes MMSE and partial zero forcing in an outage-constrained transmission-capacity framework and finds linear scaling in receive antennas, with extensions covering imperfect CSI, multihop transmission, and regular interferer geometries.

  • Problem

    The paper addresses whether ad hoc network throughput can scale linearly with nR when each transmitting node uses only one antenna.

  • Method

    The paper analyzes MMSE and partial zero forcing using transmission-capacity bounds, then examines imperfect CSI, end-to-end throughput, and interferer geometry.

  • Results

    Network-wide throughput and supported transmission density increase linearly with nR under single-transmit/multiple-receive communication.

  • Takeaways & Limitations

    Receive degrees of freedom are best used to combine interference suppression with array gain and increase simultaneous-transmission density rather than per-hop rate or range.

Abstract

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The benefit of multi-antenna receivers is investigated in wireless ad hoc networks, and the main finding is that network throughput can be made to scale linearly with the number of receive antennas nR even if each transmitting node uses only a single antenna. This is in contrast to a large body of prior work in single-user, multiuser, and ad hoc wireless networks that have shown linear scaling is achievable when multiple receive and transmit antennas (i.e., MIMO transmission) are employed, but that throughput increases logarithmically or sublinearly with nR when only a single transmit antenna (i.e., SIMO transmission) is used. The linear gain is achieved by using the receive degrees of freedom to simultaneously suppress interference and increase the power of the desired signal, and exploiting the subsequent performance benefit to increase the density of simultaneous transmissions instead of the transmission rate. This result is proven in the transmission capacity framework, which presumes single-hop transmissions in the presence of randomly located interferers, but it is also illustrated that the result holds under several relaxations of the model, including imperfect channel knowledge, multihop transmission, and regular networks (i.e., interferers are deterministically located on grids).

I. INTRODUCTION

The paper studies whether receive antennas alone can provide linear network-wide throughput gains in ad hoc networks. It models single-antenna transmitters, multi-antenna receivers, randomly located interferers, and outage-constrained transmitter density, using receive processing to balance desired-signal enhancement and interference cancellation.

  • Motivation: The paper targets linear network-wide throughput scaling with nR receive antennas even when every node uses one transmit antenna and sends one stream.This setting uses SIMO communication and linear receive processing without requiring transmit channel state information.
  • Performance metrics: Performance is measured by the maximum interferer density λϵ that keeps P[SINR < β] no larger than the fixed outage constraint ϵ.Transmission capacity is λϵ(1−ϵ)b, with b = log2(1+β).
  • Analysis: The paper develops analytical bounds for MMSE and partial zero forcing to characterize how supported transmitter density scales with nR.Partial zero forcing allocates some receive degrees of freedom to cancel nearby interferers and the remainder to desired-signal enhancement.
  • Motivation: Prior ad hoc receiver designs achieve only sublinear density growth: nR^2/α with maximal-ratio combining and nR^(1−2/α) with full zero-forcing.The paper contrasts these designs with combining array gain and interference cancellation.
  • System model: The model places single-antenna transmitters and nR-antenna receivers in a homogeneous 2-D Poisson network, with each receiver paired at fixed distance d.The Poisson model represents uncoordinated networks such as those using ALOHA.
  • Receive filters: MMSE reception balances desired-signal power boosting against interference cancellation using knowledge of the desired and interfering channels.The receive filter depends on the interference-plus-noise covariance Σ.

III. MAIN RESULTS: DENSITY SCALING WITH RECEIVE ANTENNAS

The main result is that both the MMSE-supported density and a partial-zero-forcing lower bound increase linearly with the number of receive antennas. The proof establishes matching linear scaling through separate lower- and upper-bound arguments.

  • Main result: λmmseϵ(nR) increases linearly with nR.This is established through an upper-bound argument for the MMSE receiver.
  • Main result: The partial-zero-forcing lower bound λpzf−kϵ(nR) increases linearly with nR.The proof first establishes this lower bound, then derives the MMSE upper bound.

A. Lower Bound: Achievability of Linear Scaling with Partial Zero Forcing

The paper proves that partial zero forcing can achieve transmitter density scaling linearly with nR by allocating a fixed fraction of receive degrees of freedom to interference cancellation and the remainder to array gain.

  • Receiver construction: Partial zero forcing cancels interference from the k nearest transmitters while using the remaining receive degrees of freedom to boost desired-signal power.The signal coefficient has χ2 2(nR−k) behavior, while canceled and uncanceled interferers have distinct characterized statistics.
  • Proof strategy: The proof derives an outage upper bound by bounding E[1/SINR] and applying Markov’s inequality, then inverts the bound to obtain a density lower bound.The derivation uses the characterized aggregate interference and signal statistics.
  • Linear achievability: Choosing k = θnR with 0 < θ < 1 makes the PZF density lower bound scale linearly with nR for sufficiently large nR.The theorem conditions hold for any ϵ > 0 and SNR > β > 0.
  • Scaling mechanism: With k = θnR, signal and interference powers both grow linearly with nR, allowing approximately constant SINR while transmitter density also grows linearly.The signal grows as χ2 2(1−θ)nR, while the interference bound grows linearly when density is linear in nR.
  • Scaling boundaries: Keeping k constant limits density growth to nR2/α, while using all but a constant number of degrees of freedom for cancellation limits it to nR1−2/α.Both extremes fail to provide linear scaling because one leaves interference growth too rapid and the other leaves signal power independent of nR.

B. Upper Bound: The MMSE Receiver

The MMSE receiver admits an upper bound on allowable transmitter density that scales linearly with nR, matching the PZF achievability order and establishing linear scaling for MMSE as well.

  • MMSE scaling: The MMSE density upper bound scales linearly with nR, and combined with the PZF lower bound establishes linear-order scaling for MMSE.The bound is obtained from an MMSE outage lower bound and applies by the stated relation to PZF.
  • Outage bound: The MMSE outage lower bound is extended to random interferer locations and thermal noise by averaging over the Poisson geometry and using outage monotonicity in noise power.The resulting bound uses iid unit-mean exponential fading variables independent of the other random quantities.
  • Bound interpretation: The MMSE bound corresponds to an idealized combination of full zero-forcing in the denominator and maximal-ratio combining in the numerator.This relationship connects the MMSE bound to the PZF characterization.
  • Numerical comparison: Numerical results indicate that MMSE and PZF densities both scale linearly with nR, while MMSE achieves a non-negligible constant-factor advantage.The cited comparison concerns the numerically computed densities and their associated bounds.
  • Receiver comparison: MRC and full zero-forcing do not achieve linear scaling: their upper bounds are O(nR2/α) and O(nR1−2/α), respectively.These bounds complement matching lower bounds reported for the two receivers.

C. Array Gain v. Interference Cancellation

Receive degrees of freedom are divided between array gain and interference cancellation, with the optimal balance changing with the path-loss exponent and reflected by MMSE behavior.

  • MMSE interpretation: The MMSE receiver implicitly balances array gain and interference cancellation, whereas PZF implements the allocation explicitly.For roughly equal interference-covariance eigenvalues MMSE aligns near the desired channel; for disparate eigenvalues it projects away from strong-interference directions.
  • PZF optimization: PZF density bounds depend on θ through (1 − θ)α, enabling optimization of the cancellation fraction.The upper and lower bounds share this dependence after minimizing the upper bound over the auxiliary parameter.
  • Path-loss dependence: As α → 2, the optimal cancellation fraction θ∗ approaches zero because far-away interference is significant and array gain is more beneficial.In this regime, canceling a few nearby interferers provides a smaller benefit than boosting desired-signal power.
  • Path-loss dependence: As the path-loss exponent increases, θ∗ approaches one because nearby interferers dominate and interference cancellation becomes more beneficial.The receive degrees of freedom are therefore shifted toward cancellation as α increases.
  • Receiver interpretation: For nR = 8, the normalized MMSE-filter correlation measures the fraction of receive degrees of freedom used for array gain and is consistent with the optimized PZF allocation.For PZF, the corresponding metric equals nR − k, or 1 − θ∗ when k = θ∗nR.

D. Improved Lower and Upper Bounds

The section tightens outage-probability bounds for PZF and MMSE receivers and uses them to bound achievable transmitter densities. The refined bounds are obtained with Chebyshev’s inequality and solved numerically for density.

  • Improved bounds: Chebyshev-based bounds replace looser Markov-based bounds for more accurate density estimates.The earlier bounds remain sufficient to establish linear scaling, but their looseness is attributed primarily to Markov’s inequality.
  • PZF bound: Theorem 3 upper bounds the PZF receiver’s outage probability.
  • Receiver behavior: Fig. 2 compares average squared correlation between h0 and the MMSE filter against path-loss exponent α, alongside the approximation 2/α.
  • MMSE bound: Theorem 4 lower bounds the MMSE receiver’s outage probability using terms involving nR and special functions.The bound includes InR and the Euler-Mascheroni constant and poly-Gamma function.
  • Density bounds: Equating the outage bounds to ϵ and solving numerically yields lower and upper bounds on PZF and MMSE densities, respectively.

E. Numerical Results

The numerical results compare computed MMSE densities with analytical bounds and benchmark receivers, then examine robustness to imperfect channel-state information. The results include density penalties from covariance estimation and show that scaling the training duration with nR preserves performance within a constant factor.

  • Numerical density results: Figures 3 and 4 plot numerically computed MMSE maximum densities against nR for α = 3 and α = 4 on log-log axes.They also show PZF lower bounds, MMSE upper bounds, and MRC and full zero-forcing densities.
  • Model extensions: The model extensions test imperfect CSI, end-to-end antenna benefits, and interferer geometry to assess whether linear scaling depends on the original assumptions.
  • Covariance estimation: Using K observations to estimate the interference covariance reduces expected SINR by a factor that increases with K and converges to one as K approaches infinity.For K = 2nR − 3, the expected SINR decreases by 3 dB.
  • Covariance estimation: The covariance-estimation results approximately translate into a reduced maximum density because density scales with the SINR threshold as β−2/α.
  • Training overhead: Choosing K = 2nR − 3 keeps the loss factor constant across nR, so scaling K linearly with nR maintains performance within a constant factor of perfect CSI.
  • Imperfect CSI: Estimating h0 from pilots does not significantly reduce density, and choosing K around 10 or 20 gives density reasonably close to the perfect-CSI benchmark in the six-antenna example.The example uses SNR = 10 dB and two interference-free pilots for channel estimation.

B. End-to-End Throughput

Linear transmission-capacity scaling translates into linear end-to-end throughput when receive antennas increase the density of simultaneous transmissions. Expected forward progress confirms this benefit in multihop networks, while regular networks achieve only a constant-factor density improvement over Poisson networks.

  • End-to-end scaling: λϵd^2β^(2/α) ∝ nR, so receive antennas can increase transmission range as √nR or rate only logarithmically.The paper concludes that increasing simultaneous-transmission density is more efficient than increasing per-hop range or rate.
  • Expected forward progress: The expected forward progress metric combines transmitter density, expected relay progress, and per-hop rate.Under ALOHA, λ = νp, and opportunistic routing selects the successful relay offering the greatest geographic progress.
  • Expected forward progress: For nR = 1 to 8, the optimizing transmission probability increases approximately linearly with nR, while expected forward progress is itself linear in nR.The optimizing per-hop distance remains approximately constant, supporting density growth rather than range growth.
  • Interference geometry: Regular networks permit a larger simultaneous-transmission density than Poisson networks, but only by a constant factor independent of nR.The regular-network comparison supports the conjecture that linear scaling extends beyond Poisson interference geometry.

V. CONCLUSION

The paper concludes that receive-only antennas with linear processing can produce very large ad hoc-network throughput gains by combining interference cancellation and array gain. The resulting linear density scaling is robust to imperfect channel knowledge, multihop operation, and interferer geometry, while density is more effective than per-hop rate or range.

  • Conclusion: Receive antennas can increase the density of simultaneous transmissions linearly with nR when transmitters use a single antenna.The mechanism combines interference cancellation with some array gain and requires channel state information at the receiver.
  • Conclusion: The linear-scaling conclusion is robust to the particular interferer geometry.The paper compares regular and Poisson networks and reports a larger regular-network density only by a constant factor independent of nR.
  • Conclusion: Linear density growth translates into linear network-wide throughput, making density a more efficient use of receive antennas than per-hop rate or range.The conclusion contrasts linear density growth with weaker gains from increasing rate or range.

APPENDIX III PROOF OF THEOREM 2

The proof derives an outage bound by retaining selected interference contributions and bounding their probability, then chooses the cancellation parameter to obtain a density bound. Setting the resulting bound equal to the outage constraint yields the theorem.

  • Proof strategy: The outage upper bound is formed by retaining the nearest l uncancelled interferers and applying Markov’s inequality.Additional bounds use the interference distribution and moments to control the remaining terms.
  • Proof strategy: Choosing l = nR + 1 yields the desired outage lower bound, after which setting it equal to ϵ and solving gives the density upper bound.This connects the number of receive antennas to the resulting density constraint.
  • Proof strategy: The proof combines independence, Jensen’s inequality, covariance bounds, and Chebyshev’s inequality to control signal and interference moments.These steps establish the intermediate expectations and variance bounds used in the outage analysis.
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