PaperlayerView exact revision on arXiv

Source-linked AI summary

Massive MIMO Multicasting in Noncooperative Cellular Networks

Zhengzheng Xiang, Meixia Tao, Xiaodong Wang

arXiv:1312.1134v2cs.ITmath.IT

TL;DR

The paper studies multicell massive-MIMO transmission of common information to multiple mobile users. It derives an asymptotically optimal beamformer and proposes composite-channel pilot estimation, which eliminates pilot contamination and performs close to perfect CSI with pilot power control.

  • Problem

    The paper studies transmitting common information to multiple mobile users in a noncooperative multicell massive-MIMO network.

    Source: §I · p. 1, §VII · p. 11

  • Method

    It derives a closed-form asymptotically optimal beamformer for perfect CSI and proposes a shared-pilot scheme to estimate a composite channel for multicasting.

    Source: §I · p. 2

  • Results

    The composite-channel pilot scheme completely eliminates pilot contamination and, with user pilot power control, performs close to the perfect-CSI scenario.

    Source: §VII · p. 12

  • Takeaways & Limitations

    The estimated composite channel has the structure of the asymptotically optimal multicast beamformer and can itself serve as the beamformer.

    Source: §I · p. 2

Abstract

from arXiv · show

We study the massive multiple-input multiple-output (MIMO) multicast transmission in cellular networks where each base station (BS) is equipped with a large-scale antenna array and transmits a common message using a single beamformer to multiple mobile users. We first show that when each BS knows the perfect channel state information (CSI) of its own served users, the asymptotically optimal beamformer at each BS is a linear combination of the channel vectors of its multicast users. Moreover, the optimal combining coefficients are obtained in closed form. Then we consider the imperfect CSI scenario where the CSI is obtained through uplink channel estimation in timedivision duplex systems. We propose a new pilot scheme that estimates the composite channel which is a linear combination of the individual channels of multicast users in each cell. This scheme is able to completely eliminate pilot contamination. The pilot power control for optimizing the multicast beamformer at each BS is also derived. Numerical results show that the asymptotic performance of the proposed scheme is close to the ideal case with perfect CSI. Simulation also verifies the effectiveness of the proposed scheme with finite number of antennas at each BS.

I. INTRODUCTION

The paper frames massive-MIMO multicasting as common-message delivery in noncooperative cells, where uplink channel estimation is vulnerable to pilot contamination. It presents a closed-form perfect-CSI beamformer and a shared-pilot composite-channel approach for imperfect CSI.

I. INTRODUCTION

  • The paper focuses on noncooperative multicell transmission because intercell coordination requires substantial CSI exchange over limited-capacity backhaul links.Source: §I · pp. 1–2
  • Under conventional uplink channel estimation, reused or nonorthogonal pilots cause pilot contamination that interferes with channel estimates and degrades network performance.The paper also characterizes contamination’s impact on each user’s SINR.Source: §I · p. 2
  • With perfect CSI, each BS’s asymptotically optimal beamformer is a linear combination of its served users’ channels, with closed-form coefficients depending on large-scale attenuation.Source: §I · p. 2
  • A shared-pilot scheme estimates a composite channel that can serve as the multicast beamformer, eliminating pilot contamination and achieving a small, power-independent gap to perfect CSI.Pilot power control optimizes the composite channel; the gap does not depend on BS transmitting power.Source: §I · p. 2

II. SYSTEM MODEL

  • The system has N cells with K users per cell, each served by an M-antenna BS; channels are flat and quasi-static, and each cell transmits a common message.Source: §II · p. 2

III. ASYMPTOTIC ANALYSIS WITH PERFECT CSI

With perfect CSI, the paper formulates multicast beamforming as maximizing the minimum SINR and derives an asymptotically optimal beamformer from the served users’ channels. The structure relies on asymptotic channel orthogonality and supports power scaling with the antenna count.

III. ASYMPTOTIC ANALYSIS WITH PERFECT CSI

A. Asymptotically Optimal Beamformer Structure

  • As M grows, channel vectors become asymptotically orthogonal, enabling the paper to derive the beamformer structure.The proof argues that components outside the span of served-user channels add no useful signal and cause extra interference to other cells.Source: A. Asymptotically Optimal Beamformer… · p. 3
  • The asymptotically optimal beamformer is a linear combination of the channels to the BS’s served multicast users.Its combining coefficients are denoted {ξi,k}.Source: A. Asymptotically Optimal Beamformer… · p. 3
  • Scaling each BS’s transmit power as Ei/M allows reduced power consumption as the antenna count increases for a given SINR target.The asymptotic user SINR is derived under this power scaling.Source: A. Asymptotically Optimal Beamformer… · p. 4

B. Optimal Beamforming Parameters

The paper derives optimal combining coefficients and the resulting asymptotic SINR under a max-min criterion. It also shows that users need only learn a scalar effective channel with slowly changing large-scale parameters.

B. Optimal Beamforming Parameters

  • The asymptotically optimal beamforming vector and resulting user SINR are given in closed form, with the optimal solution equalizing all K users’ SINRs.The optimization of the combining coefficients is equivalent to optimizing the SINR-related parameters {λi,k}.Source: B. Optimal Beamforming Parameters · p. 4
  • The optimal combining coefficients depend only on large-scale channel parameters, and users with larger βi,i,k receive lower beamformer weight.Performance decreases as the number of multicast users K increases.Source: B. Optimal Beamforming Parameters · p. 4
  • Each receiver needs only a scalar effective channel containing slowly changing large-scale fading parameters, not the rapidly changing small-scale channel vector.This reduces the channel information the user must learn to decode its message.Source: B. Optimal Beamforming Parameters · p. 5

IV. ANALYSIS WITH IMPERFECT CSI AND PILOT CONTAMINATION

In the TDD setting, the paper analyzes conventional uplink channel estimation with pilot reuse across cells. The resulting estimates include other-cell interference, and the asymptotic SINR reaches a ceiling as BS transmit power grows.

IV. ANALYSIS WITH IMPERFECT CSI AND PILOT CONTAMINATION

  • With conventional estimation, each cell reuses the same K orthogonal pilots, so users in different cells transmit the same pilot sequence.The BS estimates uplink channels and treats them as downlink CSI through reciprocity, assuming the channel varies slowly enough for reciprocity to hold.Source: §IV · p. 5
  • The channel estimate contains interference from users in other cells that reuse the pilot, contaminating the desired CSI.The estimate also includes additive noise.Source: §IV · p. 5
  • As every BS’s transmit power tends to infinity, the asymptotic SINR reaches a ceiling rather than continuing to increase.The paper identifies this ceiling as an effect of pilot contamination.Source: §IV · p. 6
  • Under the imperfect-CSI SINR expression, obtaining asymptotically optimal beamforming parameters is difficult, unlike in the perfect-CSI case.The analysis uses TDD and uplink pilots.Source: §IV · pp. 5–6

V. A CONTAMINATION-FREE PILOT SCHEME

The section proposes a new pilot scheme for multicell multicast with large antenna arrays to eliminate pilot contamination.

V. A CONTAMINATION-FREE PILOT SCHEME

  • The paper proposes a new pilot scheme for multicell multicast with large antenna arrays to eliminate pilot contamination.Assigning a distinct orthogonal pilot to every user in the network is described as requiring a prohibitive pilot sequence length.Source: §V · p. 6

A. The New Pilot Scheme

The scheme assigns one orthogonal pilot sequence to each cell, shared by all multicast users there, so each base station estimates a composite channel for its served users. This estimate has the same linear-combination structure as the asymptotically optimal beamformer and excludes channels from other cells.

A. The New Pilot Scheme

  • All K users in a cell share one pilot, with orthogonal sequences assigned across cells, allowing each base station to estimate its served users’ composite channel.The estimate is a linear combination of the users’ channels in that cell, plus noise.Source: A. The New Pilot Scheme · p. 6
  • The composite-channel estimate matches the linear-combination structure of the asymptotically optimal beamformer, so the base station can use it to design its beamforming vector.The estimate contains no channels from users in other cells, eliminating intercell interference for this estimation scheme.Source: A. The New Pilot Scheme · p. 6

B. Pilot Power Control

Pilot power control optimizes the beamformer coefficients when composite-channel estimation removes pilot contamination but leaves the base station without coefficient flexibility. The derived solution equalizes user SINRs and approaches the perfect-CSI beamformer as peak pilot power increases.

B. Pilot Power Control

  • Composite-channel estimation eliminates pilot contamination but leaves the base station without flexibility to optimize beamformer coefficients directly.Source: B. Pilot Power Control · p. 6
  • Pilot powers are optimized under a max-min SINR objective to tune the beamformer coefficients, and the resulting optimization is a geometric program with a closed-form solution.The optimization can also be transformed into a convex problem and solved optimally.Source: B. Pilot Power Control · pp. 6–7
  • At the optimum, all users achieve the same SINR, and the user with the minimum channel gain transmits its pilot at peak power.The paper derives the optimal pilot powers in closed form.Source: B. Pilot Power Control · p. 7
  • The beamformer under optimal pilot power control is almost the same as in the perfect-CSI case, except for noise terms.Pilot-side power optimization plays a role similar to optimizing the linear-combination coefficients at the base station.Source: B. Pilot Power Control · p. 8
  • The SINR gap from the perfect-CSI case decreases as peak pilot power increases and is independent of base-station transmit power.Source: B. Pilot Power Control · p. 8
  • The composite pilot sequence must be at least N symbols long, and optimal power control requires knowledge of users’ large-scale channel attenuation factors.Without that knowledge, users transmit at peak power and performance decreases; the proposed scheme still outperforms individual-channel estimation in the reported simulations.Source: B. Pilot Power Control · p. 8

C. Discussion On Synchronization

The effect of pilot-arrival asynchrony depends on its size relative to a symbol duration. Negligible delays act as phase shifts, while considerable delays destroy pilot orthogonality and degrade the proposed composite-channel estimate.

C. Discussion On Synchronization

  • When pilot-arrival time differences are negligible relative to one symbol duration, the delay acts as a phase offset that does not affect system performance.Source: C. Discussion On Synchronization · p. 8
  • For considerable arrival-time differences, the received pilot symbols combine two consecutive transmitted symbols, producing asynchrony-polluted pilots.The model uses a pilot stream and a matched filter at the base station to form the received signal.Source: C. Discussion On Synchronization · pp. 8–9
  • When arrival-time differences are not negligible, pilot orthogonality is destroyed, causing both scaling loss and pilot contamination in the composite-channel estimate.Source: C. Discussion On Synchronization · p. 9
  • Under this nonorthogonality, the asymptotic SINR reaches a ceiling, and overcoming the asynchrony effect is left for future work.Source: C. Discussion On Synchronization · p. 9

VI. NUMERICAL AND SIMULATION RESULTS

The numerical and simulation evaluation considers a multicell multicast network under specified propagation and fading assumptions, with asymptotic and finite-size performance assessed.

VI. NUMERICAL AND SIMULATION RESULTS

  • The simulated cells are hexagonal with a 1000-meter center-to-vertex radius, and users are uniformly distributed outside a 100-meter inner circle.Source: §VI · p. 9
  • The channel model uses 20 MHz bandwidth, distance-dependent path loss, 8 dB log-normal shadowing, 20 dB penetration loss, and normalized Rayleigh small-scale fading.Source: §VI · p. 9
  • Unless otherwise specified, evaluation uses seven cells and the central cell, with 1000 large-scale channel realizations and 1000 small-scale fading samples per realization for simulations.Source: §VI · p. 9

A. Asymptotic SINR Performance with Perfect CSI

With perfect CSI, optimal combining coefficients improve the proposed beamformer’s minimum asymptotic SINR over equal combining; the minimum SINR decreases as the user count increases.

A. Asymptotic SINR Performance with Perfect CSI

B. Asymptotic SINR Performance with Imperfect CSI

The proposed composite-channel pilot scheme eliminates pilot contamination, and optimal pilot power control brings asymptotic performance close to perfect CSI. Finite-antenna simulations show small gaps to the asymptotic result and a modest advantage over noncooperative SDR, while cooperative SDR performs better in the stated comparison.

B. Asymptotic SINR Performance with Imperfect CSI

C. Actual SINR Performance at Finite Number of Antennas

  • With perfect CSI and the considered antenna counts, the proposed beamformer slightly outperforms noncooperative SDR and needs no numerical computation.The SDR approach uses iterative interior-point optimization and bisection, while the proposed beamformer has a closed-form solution.Source: C. Actual SINR Performance at Finite… · p. 11
  • In the N = 3 comparison, cooperative SDR outperforms the proposed beamformer, but requires global CSI and has much higher computation and signaling overhead.The proposed beamformer is asymptotically optimal and needs only local CSI.Source: C. Actual SINR Performance at Finite… · p. 11
  • At M = 300, actual minimum SINR is 1.5dB below the asymptotic result; at M = 500, the gap is 1dB.Source: C. Actual SINR Performance at Finite… · p. 11

VII. CONCLUSIONS

  • The conclusion identifies pilot power control as the means for bringing the contamination-free composite-channel scheme close to perfect-CSI performance.Source: §VII · p. 12

APPENDIX A PROOF OF LEMMA 1

The Appendix A proof uses contradiction to establish condition (19), then combines it with (17) to obtain (20); Appendix B similarly argues from objective improvement to the pilot-power condition (40).

APPENDIX A PROOF OF LEMMA 1

  • Assuming condition (19) fails allows a parameter change that increases Q’s objective, contradicting optimality and establishing condition (19).Source: Appendix A · p. 12
  • Combining conditions (17) and (19) yields (20), completing the lemma’s proof.Source: Appendix A · p. 12
  • The proof notes that the extension to multiple equal maximum or minimum SINRs is straightforward and omitted.Source: Appendix A · p. 12

APPENDIX B PROOF OF LEMMA 2

  • For the pilot-power optimization, increasing feasible user pilot powers would increase the objective, so optimality requires condition (40).Source: Appendix B · p. 12
Loading 1312.1134v2…