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The Interplay between Massive MIMO and Underlaid D2D Networking

Xingqin Lin, Robert W. Heath Jr., Jeffrey G. Andrews

arXiv:1409.2792v3cs.ITmath.IT

TL;DR

D2D underlay causes mutual interference, and massive MIMO's multi-user uplink can intensify cellular-to-D2D interference. The paper studies cellular and D2D spectral efficiency in this setting and finds that underlay contamination further degrades cellular performance with imperfect CSI, while average D2D performance is relatively robust to more cellular users.

  • Problem

    In underlaid cellular networks, D2D and cellular transmissions interfere with each other, and the impact of multi-user uplink transmission and D2D activity during channel estimation remains uncertain.

    Source: A. Background · p. 2

  • Method

    The paper evaluates cellular and D2D spectral efficiency, including analytical lower bounds and comparisons with simulation.

    Source: C. D2D Spectral Efficiency · p. 14, §V · p. 20

  • Results

    With σ = 7 dB and πR^2 cλ = 4, underlay contamination reduces asymptotic cellular spectral efficiency from 6 bps/Hz to about 3.8 bps/Hz; increasing K from 10 to 20 reduces average D2D spectral efficiency by less than 0.5 bps/Hz.

    Source: §V · pp. 23–24

  • Takeaways & Limitations

    The study concludes that D2D links are relatively robust on average to cellular-to-D2D interference, while D2D activity during training can hurt cellular CSI accuracy; deactivating D2D links during training is one proposed mitigation.

    Source: §VI · p. 25

Abstract

from arXiv · show

In a device-to-device (D2D) underlaid cellular network, the uplink spectrum is reused by the D2D transmissions, causing mutual interference with the ongoing cellular transmissions. Massive MIMO is appealing in such a context as the base station's (BS's) large antenna array can nearly null the D2D-to-BS interference. The multi-user transmission in massive MIMO, however, may lead to increased cellular-to-D2D interference. This paper studies the interplay between massive MIMO and underlaid D2D networking in a multi-cell setting. We investigate cellular and D2D spectral efficiency under both perfect and imperfect channel state information (CSI) at the receivers that employ partial zero-forcing. Compared to the case without D2D, there is a loss in cellular spectral efficiency due to D2D underlay. With perfect CSI, the loss can be completely overcome if the number of canceled D2D interfering signals is scaled with the number of BS antennas at an arbitrarily slow rate. With imperfect CSI, in addition to pilot contamination, a new asymptotic effect termed underlay contamination arises. In the non-asymptotic regime, simple analytical lower bounds are derived for both the cellular and D2D spectral efficiency.

I. INTRODUCTION

D2D underlay creates mutual interference, while massive MIMO can suppress D2D-to-BS interference but may increase cellular-to-D2D interference. The paper analyzes this tradeoff through a multi-cell model and spectral-efficiency results under perfect and imperfect CSI.

A. Background

  • Underlaying D2D transmissions creates mutual interference, and massive MIMO may leave cellular-to-D2D interference significant as uplink users scale with BS antennas.The extent of the D2D impact and the tradeoff with uplink capacity are open questions; D2D transmissions during channel estimation may also reduce CSI accuracy.Source: A. Background · p. 2

B. Contributions and Outcomes; 1) A tractable hybrid network model:

  • The paper extends a PPP-based D2D model to multi-cell SIMO uplink networks and applies partial zero-forcing (PZF) to study spectral efficiency.The model represents random D2D transmitter locations with a Poisson point process and uses spectral efficiency as its metric.Source: 1) A tractable hybrid network model · p. 3

2) Spectral efficiency with perfect CSI:

  • With perfect CSI and M → ∞, cellular-user SINR grows without bound, while noise, fast fading, and cochannel cellular and D2D interference vanish.With cellular transmit power scaled as Θ(1/M), a constant number of canceled D2D signals leaves a cellular spectral-efficiency loss relative to no D2D underlay.Source: 2) Spectral efficiency with perfect CSI · p. 3

3) Spectral efficiency with imperfect CSI:

  • With imperfect CSI, D2D underlay adds underlay contamination to pilot contamination and can make cellular spectral efficiency vanish when transmit power is scaled down.Deactivating D2D links during training avoids this contamination, but data-phase D2D-to-cellular interference can still cause a loss if cellular power is scaled down.Source: 3) Spectral efficiency with imperfect… · p. 4

A. Network Model

  • The network model has B + 1 cells with K cellular UEs per cell and conditions on cellular UE positions when analyzing a particular link.Cellular UEs are assumed uniformly distributed in each cell, with that assumption used in simulation.Source: A. Network Model · p. 4
  • The analysis assumes single-antenna transmitters, M-antenna BSs, N-antenna D2D receivers, and M ≫ K, leaving constant K/M asymptotics for future work.Source: A. Network Model · p. 5

B. Baseband Channel Models

  • The channel model assumes independent i.i.d. complex Gaussian channel elements and uses different pathloss exponents for UE-BS and UE-UE links.The differing exponents reflect distinct propagation characteristics associated with macro-BS and low UE antenna heights.Source: B. Baseband Channel Models · p. 6

C. Receive Filters

Partial zero-forcing divides receiver degrees of freedom between boosting the desired signal and canceling selected interferers. The paper focuses on PZF for its generality and analytical tractability, while noting practical CSI demands.

C. Receive Filters

  • PZF uses some receiver degrees of freedom to boost desired-signal power and the remainder to cancel interference.Source: C. Receive Filters · p. 7
  • At the BS, cancellation targets the nearest cellular and D2D interferers; at a D2D receiver, it targets nearest cellular and D2D interferers within the stated degree-of-freedom constraints.The BS choices satisfy mc ≤ (B + 1)K − 1 and mc + md ≤ M − 1; the D2D choices satisfy nc ≤ (B + 1)K and nc + nd ≤ N − 1.Source: C. Receive Filters · p. 7
  • PZF ranges from MRC with no cancellation to fully ZF, and offers analytical tractability, but requires CSI for each canceled interferer.The paper notes that MMSE is optimal among linear filters and may be preferable in practice; general PZF at D2D receivers is described as impractical given its CSI and coordination requirements.Source: C. Receive Filters · pp. 7–8

III. SPECTRAL EFFICIENCY WITH PERFECT CHANNEL STATE INFORMATION

  • The next section derives cellular and D2D spectral efficiency assuming perfect CSI, before treating imperfect CSI separately.Source: §III · p. 8

A. Asymptotic Cellular Spectral Efficiency

With perfect CSI, cellular spectral efficiency lost to D2D underlay can be recovered by increasing the number of canceled D2D interferers, even at an arbitrarily slow rate. With a fixed cancellation count, uncanceled D2D interference remains a source of spectral-efficiency loss.

A. Asymptotic Cellular Spectral Efficiency

  • With perfect CSI, cellular SINR grows unboundedly in probability as M increases, while noise, fast fading, cellular interferers, and D2D interference vanish.The result holds even with infinitely many D2D transmitters and infinite mean aggregate D2D interference; the proof says MRC with m_c = m_d = 0 suffices.Source: A. Asymptotic Cellular Spectral… · pp. 9–10
  • With fixed cancellation parameters, uncanceled D2D signals cause a cellular spectral-efficiency loss represented in the lower bound by ρ(m_d, α_c).The residual-interference term increases with D2D transmit power and density and decreases as more D2D interferers are canceled.Source: A. Asymptotic Cellular Spectral… · p. 11
  • The lower bound for cellular spectral efficiency is meaningful only when m_d + 1 > α_c; otherwise its residual-interference term diverges.When that term becomes infinite, the bound reduces to zero.Source: A. Asymptotic Cellular Spectral… · p. 11
  • If m_d grows to infinity more slowly than M, cellular spectral efficiency converges to the no-underlay limit under perfect CSI and cellular power scaling P_c/M.The required growth is m_d ∈ ω(1) and m_d ∈ o(M).Source: A. Asymptotic Cellular Spectral… · pp. 11–12
  • Scaling cellular transmit power as Θ(1/M) does not mean UEs can transmit at vanishing power; several hundred antennas yield about 20–30 dB power savings in the cited numerical results.Further scaling may be fragile when non-ideal effects such as spurious emission become significant.Source: A. Asymptotic Cellular Spectral… · p. 12

B. Non-asymptotic Cellular Spectral Efficiency

A Jensen-based analytical lower bound characterizes non-asymptotic cellular spectral efficiency and exposes tradeoffs in choosing the partial zero-forcing parameters. The bound captures how cancellation reduces interference while also reducing desired-signal power.

B. Non-asymptotic Cellular Spectral Efficiency

  • The Jensen-based lower bound on cellular spectral efficiency closely matches simulation, with a closer match at larger m_d.The passage attributes this to lower D2D-interference variance when more D2D interferers are canceled.Source: B. Non-asymptotic Cellular Spectral… · p. 12
  • The lower bound requires M ≥ m_c + m_d + 1 and m_d > α_c; its received-signal power gain is proportional to M − m_c − m_d − 1.Fading prevents achieving the larger proportional gain M − m_c − m_d.Source: B. Non-asymptotic Cellular Spectral… · pp. 12–13
  • The lower bound makes the PZF tradeoff explicit: increasing m_c or m_d reduces the corresponding uncanceled interference, but also reduces desired-signal power.These tradeoffs disappear in the asymptotic results.Source: B. Non-asymptotic Cellular Spectral… · p. 13
  • Although globally optimizing the PZF parameters requires exhaustive search, practical constraints and prior results reduce the search to a few small values of m_c and m_d.The passage notes that BSs typically cancel intra-cell cellular interference and that canceling a few Poisson-distributed interferers can be close to optimal.Source: B. Non-asymptotic Cellular Spectral… · p. 13

C. D2D Spectral Efficiency

The section derives a non-asymptotic lower bound for D2D spectral efficiency and describes the channel-estimation assumptions used in the analysis. It also notes that pilot contamination prevents perfect channel estimation even with infinitely long training sequences.

C. D2D Spectral Efficiency

  • D2D spectral efficiency is defined using the post-processing SINR, with the expectation taken over fast fading, shadowing, and random UE locations.Source: C. D2D Spectral Efficiency · pp. 13–14
  • With perfect CSI, Proposition 5 gives a lower bound on D2D spectral efficiency when N ≥ nc + nd + 1 and nd > αd.The bound characterizes the impact of system parameters on D2D spectral efficiency.Source: C. D2D Spectral Efficiency · p. 14

A. Estimating UE-BS Channels

  • Each BS estimates channels from its cellular UEs and up to md,b nearest in-cell D2D transmitters using orthogonal training sequences, without estimating other-cell transmitters.Other D2D transmitters may send independent symbols during training.Source: A. Estimating UE-BS Channels · p. 15
  • Even infinitely long training sequences cannot make channel estimation perfect because of pilot contamination.Source: A. Estimating UE-BS Channels · p. 16

B. Asymptotic Cellular Spectral Efficiency

With imperfect CSI, D2D underlay adds residual interference that persists as the BS antenna count grows, limiting cellular SINR and spectral efficiency. The section describes the mechanism and mitigation options, including deactivating D2D links during training when cellular transmit power is scaled down.

B. Asymptotic Cellular Spectral Efficiency

  • D2D underlay leaves residual D2D-to-cellular interference because the same D2D signal correlates across channel estimation and data transmission.Unlike receiver noise, the correlated interference does not disappear as the antenna count grows.Source: B. Asymptotic Cellular Spectral… · p. 17
  • Mitigations trade off D2D range, cellular energy or interference, training resources, and D2D transmission time.The discussed options include lowering D2D power, raising cellular power, lengthening training, or deactivating D2D during training.Source: B. Asymptotic Cellular Spectral… · p. 18
  • With D2D underlay contamination, scaling down cellular transmit power causes cellular spectral efficiency to vanish.The section contrasts this with the imperfect-CSI SINR limit arising from both pilot contamination and D2D underlay contamination.Source: B. Asymptotic Cellular Spectral… · p. 18
  • With D2D links deactivated during training, cellular transmit power can scale as Pc/M while achievable cellular spectral efficiency converges as M → ∞.The section notes that separating active D2D sets between training and data phases also solves underlay contamination, but its non-asymptotic performance is unclear.Source: B. Asymptotic Cellular Spectral… · p. 19
  • Proposition 7 gives a more explicit expression for asymptotic cellular spectral efficiency to support efficient numerical evaluation.Source: B. Asymptotic Cellular Spectral… · p. 19

V. SIMULATION AND NUMERICAL RESULTS

The numerical results evaluate cellular and D2D spectral efficiency under different PZF choices and show how interference, cancellation, and transmit-power scaling affect performance. They also compare analytical lower bounds with simulations and examine D2D underlay contamination.

Fig. 2

  • The cellular spectral-efficiency gain is about 1.6 bps/Hz for (mc, md) = (3, 2) versus (0, 2).The simulated cellular spectral efficiency closely matches lower bound (17), with a closer match at larger md.Source: §V · p. 20, Figure 2 · p. 21

Fig. 3

  • With cellular transmit power scaled as Θ(1/M), constant md leaves a fixed cellular spectral-efficiency loss from D2D underlay that increasing BS antennas cannot overcome.The result is reported for perfect CSI and contrasts with the unscaled-power case.Source: §V · p. 22, Figure 3 · p. 22

Fig. 4

  • Increasing D2D range from 20 m to 35 m lowers D2D spectral efficiency by about 3 bps/Hz.For (nc, nd) = (0, 2), lower bound (22) closely matches simulation when N ≥6 and is somewhat loose when N < 6.Source: §V · p. 22, Figure 4 · p. 23

Fig. 5

  • When K increases from 10 to 20, average D2D spectral efficiency falls by less than 0.5 bps/Hz, even with the MRC receiver.The comparison uses perfect CSI and (nc, nd) = (0, 0).Source: §V · p. 23, Figure 5 · p. 23

Fig. 6

  • For cellular spectral efficiency, canceling about 2 to 4 D2D interferers is sufficient for near-optimal performance, while canceling more cellular interferers yields only marginal additional gain after mc = 3.The useful number of D2D cancellations depends on transmitter density λ and antenna count M; the marginal cellular gain may not justify extra training and coordination overhead.Source: §V · p. 24, Figure 6 · p. 24

Fig. 7

  • With σ = 7 dB and πR2 cλ = 4, underlay contamination reduces asymptotic cellular spectral efficiency from 6 bps/Hz to about 3.8 bps/Hz.When πR2 cλ ≥22, underlay contamination dominates the combined effect of pilot and underlay contamination.Source: §V · p. 24, Figure 7 · p. 25

VI. CONCLUSIONS

The study finds that massive MIMO handles D2D-to-cellular interference efficiently, while D2D links are relatively robust on average to cellular-to-D2D interference. D2D transmissions can nevertheless impair CSI estimates and cellular spectral efficiency.

Conclusions

  • D2D interference makes massive-MIMO CSI estimates less accurate and thereby hurts cellular spectral efficiency.Deactivating D2D links during cellular training is presented as one simple way to alleviate this effect.Source: §VI · p. 25
  • The paper uses spectral efficiency as its sole metric and leaves throughput analysis with training, scheduling, and retransmission overhead for future work.It also identifies comparisons with MMSE and successive-interference-cancellation receivers as future work.Source: §VI · p. 25

APPENDIX

The appendix proof establishes that a BS using an MRC receiver suffices for Proposition 1. It shows the normalized D2D interference vanishes by separating transmitters into near and far groups.

A. Proof of Proposition 1

  • A PZF receiver with mc = md = 0, the MRC receiver, suffices at the BS.This is the stated conclusion of the proof of Proposition 1.Source: A. Proof of Proposition 1 · p. 26
  • The proof shows that D2D interference normalized by M^2 converges to 0 as M →∞.It partitions transmitters into those within distance ro of the BS and those farther away, then makes both terms arbitrarily small by taking M large enough.Source: A. Proof of Proposition 1 · pp. 26–27
  • The interference bound's derivation uses the condition αc > 2.The proof identifies this condition in its final equality.Source: A. Proof of Proposition 1 · p. 27

B. Proof of Proposition 2

The proof of Proposition 2 analyzes interference under cellular transmit-power scaling and derives a cellular spectral-efficiency limit. It also obtains a lower bound using Jensen’s inequality.

B. Proof of Proposition 2

  • For finite fixed md, the D2D-to-cellular interference term converges in distribution to its stated limit.The proof uses a result on the asymptotic behavior of D2D interference.Source: B. Proof of Proposition 2 · p. 29
  • With cellular transmit powers scaled as Pc/M, cellular UE k's spectral efficiency converges as in (11).The proof establishes the convergence after analyzing the normalized noise and interference terms.Source: B. Proof of Proposition 2 · pp. 28–29
  • Jensen’s inequality gives lower bound (12), obtained by substituting the derived expression into the bound.The proof identifies the inequality as the basis of the lower bound and then plugs in the expression to obtain it.Source: B. Proof of Proposition 2 · pp. 29–30

C. Proof of Proposition 4

The proof of Proposition 4 uses a projection orthogonal to canceled-interferer channel vectors and evaluates expectations using distributional properties and Jensen’s inequality.

C. Proof of Proposition 4

D. Proof of Lemma 1

The proof establishes the MMSE estimate and error covariances, then analyzes convergence terms under D2D-deactivated training to conclude that (31) is achievable.

D. Proof of Lemma 1

  • The MMSE estimate of h(s) has zero mean and covariance ξ(s)_k I_M, while its estimation error has covariance (1 − ξ(s)_k)I_M.Source: D. Proof of Lemma 1 · p. 32

E. Proof of Proposition 6

0. The fifth term is zero mean and has variance P

  • The proof uses the worst-case noise argument to conclude that (31) is achievable.Source: §0 · p. 33

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