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Massive MU-MIMO Downlink TDD Systems with Linear Precoding and Downlink Pilots
Hien Quoc Ngo, Erik G. Larsson, Thomas L. Marzetta
TL;DR
Massive MU-MIMO downlink users need effective-channel CSI, but conventional downlink training can scale with the many BS antennas. The paper proposes beamforming training with precoded pilots, derives rate lower bounds for MRT and ZF, and finds the scheme preferable in moderate and low-mobility settings.
Problem
Downlink users need CSI for reliable coherent detection, while full-channel downlink training can have overhead proportional to the number of BS antennas.
Method
The BS precodes short pilot sequences, users estimate effective channel gains with MMSE estimation, and achievable-rate lower bounds are derived for MRT and ZF.
Results
The beamforming training scheme improves spectral efficiency over no beamforming training and is preferable at moderate and large coherence intervals, including moderate and low-mobility environments.
Takeaways & Limitations
Beamforming training provides user CSI with overhead independent of the number of BS antennas and proportional to the number of users.
Abstract
from arXiv · showhide
We consider a massive MU-MIMO downlink time-division duplex system where a base station (BS) equipped with many antennas serves several single-antenna users in the same time-frequency resource. We assume that the BS uses linear precoding for the transmission. To reliably decode the signals transmitted from the BS, each user should have an estimate of its channel. In this work, we consider an efficient channel estimation scheme to acquire CSI at each user, called beamforming training scheme. With the beamforming training scheme, the BS precodes the pilot sequences and forwards to all users. Then, based on the received pilots, each user uses minimum mean-square error channel estimation to estimate the effective channel gains. The channel estimation overhead of this scheme does not depend on the number of BS antennas, and is only proportional to the number of users. We then derive a lower bound on the capacity for maximum-ratio transmission and zero-forcing precoding techniques which enables us to evaluate the spectral efficiency taking into account the spectral efficiency loss associated with the transmission of the downlink pilots. Comparing with previous work where each user uses only the statistical channel properties to decode the transmitted signals, we see that the proposed beamforming training scheme is preferable for moderate and low-mobility environments.
I. INTRODUCTION
Massive MU-MIMO uses many BS antennas to serve multiple users with simple linear processing, but accurate CSI is needed. The proposed beamforming training scheme targets downlink CSI acquisition with overhead that scales with users rather than BS antennas.
- Massive MU-MIMO simultaneously serves several users with many BS antennas and can use simple linear processing such as MRT or ZF.
- Accurate CSI is required at the BS and/or users to exploit massive MU-MIMO's data-rate, reliability, and energy-efficiency benefits.
- FDD downlink training and CSI feedback become challenging in massive MU-MIMO because their resource and bandwidth requirements scale with the number of BS antennas.
- The paper proposes beamforming training, in which the BS precodes a short pilot sequence so users estimate effective channel gains.
- The scheme's channel-estimation overhead is proportional to the number of users, and the paper derives capacity lower bounds for MRT and ZF.
B. Downlink Transmission
Users need effective-channel CSI for coherent detection, but estimating the full M × K channel through downlink pilots is inefficient when M is large. Because each user needs only a scalar effective gain, short downlink training can provide the required CSI.
- The BS precodes transmitted symbols using a linear precoding matrix based on its channel estimate.
- Each user's received signal contains its desired symbol, interference from other users, and additive noise after precoding.
- Coherent detection requires CSI, but estimating the full M × K channel at each user with downlink pilots has overhead proportional to M.
- For detecting its symbol, user k needs only the scalar effective channel gain a_kk rather than the full channel matrix H.
- A small portion of the coherence interval can therefore be used for downlink training to acquire a_kk at each user.
C. Beamforming Training Scheme
The beamforming training scheme precodes pilot sequences with the BS's linear precoder and lets each user estimate its effective channel gains from the received pilots. Orthogonal pilot design supports the training procedure, while independent gain estimation is used for analytical simplicity.
- The downlink training duration is τ_d symbols, and the pilot design uses pairwise orthonormal rows requiring τ_d ≥ K.
- The BS beamforms the pilot matrix by transmitting W S_p, where W is the precoding matrix and S_p is the pilot matrix.
- Each user receives a pilot vector containing observations of its effective gains a_ki and corresponding noise.
- Users estimate the effective-gain vector a_k from the received pilots, using each pilot observation to estimate the corresponding gain independently.
- Under MMSE estimation, the effective-channel estimate and estimation error are uncorrelated.
III. ACHIEVABLE DOWNLINK RATE
The achievable downlink rate is evaluated using each user's estimate of its effective channel gains. The paper derives a lower bound on this rate and simplifies it for MRT and ZF precoding.
- User k detects its transmitted signal using the estimated effective-channel vector â_k.
- The achievable rate is expressed as the mutual information I(s_k; y_k, â_k) between the transmitted signal, received signal, and known channel estimate.
- The paper derives a lower bound on the achievable downlink rate using the received-signal model and channel estimate.
- The capacity lower bound is specialized to maximum-ratio transmission and zero-forcing precoding.
A. Maximum-Ratio Transmission
The section specifies maximum-ratio transmission (MRT) precoding and gives a lower bound on the achievable rate under the BS transmit-power constraint.
- A. Maximum-Ratio Transmission: With MRT, the precoding matrix uses the estimated channel matrix and a normalization constant.The normalization enforces the BS transmit-power constraint.
- A. Maximum-Ratio Transmission: Proposition 1 gives the MRT lower bound on the achievable rate.The bound is obtained from the general rate expression in the paper.
- A. Maximum-Ratio Transmission: Figure 2 concerns spectral efficiency versus SNR for a single-user setup with K = 1, p_u = 0 dB, and T = 200.The figure provides the single-user setting used in the spectral-efficiency evaluation.
- B. Zero-Forcing: The section also presents the corresponding zero-forcing precoder and its power-normalization constant.These expressions provide the comparison baseline for the rate analysis.
IV. NUMERICAL RESULTS
The numerical results evaluate spectral efficiency with beamforming training across SNR, precoding choices, and coherence intervals. Beamforming training improves performance in the reported single-user and multiuser settings, but is unfavorable for short coherence intervals.
- Evaluation setup: The numerical comparisons use τ_u = τ_d = K, p_u = 0 dB, and SNR ≜ p_d.The reported figures compare proposed beamforming training with no beamforming training and include MRT and ZF cases.
- SNR dependence: Beamforming training outperforms the scheme without beamforming training in the single-user setup, with the gap increasing as SNR rises.The reported explanation is that higher downlink power produces more accurate channel estimates.
- Multiuser performance: In the multiuser setup with K = 5, beamforming training improves spectral efficiency for both MRT and ZF precoding.The comparison uses the proposed scheme against the no-training case.
- Multiuser performance: With beamforming training, MRT is more efficient than ZF in the reported multiuser setup.The paper attributes this to greater randomness in the effective channel gain under MRT, making channel estimation more valuable for detection.
- Coherence interval: For short coherence intervals, beamforming-training overhead is large relative to the interval, so beamforming training should not be used for CSI estimation.The figure uses M = 50, K = 5, and p_d = 20 dB.
- Coherence interval: At moderate and large coherence intervals, beamforming training is preferable because its duration is relatively small compared with the interval.The spectral-efficiency expression accounts for uplink and beamforming-training symbols within the coherence interval.
V. CONCLUSION AND FUTURE WORK
The paper concludes that beamforming training is an efficient downlink CSI-acquisition scheme for massive MU-MIMO. It avoids antenna-dependent overhead and adds robustness to beamforming that otherwise depends on prior Bayes assumptions.
- Conclusion: Beamforming training acquires downlink CSI by linearly precoding the pilot sequence before transmission to users.The scheme is presented for CSI acquisition at each user.
- Conclusion: Its channel-estimation overhead is small and does not depend on the number of BS antennas.This is the stated efficiency advantage for massive MU-MIMO systems.
- Conclusion: Downlink pilots add robustness to the beamforming process when validity of prior Bayes assumptions is uncertain.The conclusion connects this robustness to the use of downlink pilots.
APPENDIX
The appendix derives moments needed for the capacity analysis by separately computing expectations and variances of the effective channel terms.
- Moment calculations: The derivation begins by computing E{a_ki} from the channel-estimation model.The referenced variables include columns of the channel estimate and estimation-error matrices.
- Moment calculations: For i ≠ k, the appendix computes Var(a_ki) using the relevant preceding expressions and uncorrelatedness properties.The derivation invokes the fact that the estimated-channel and error-related terms are uncorrelated.
MRT E
This derivation substitutes previously obtained moment expressions into the variance and rate formulas to obtain the stated proposition.
- MRT derivation: The resulting expressions are substituted into an intermediate variance relation.The appendix explicitly identifies substitution of equations (25) and (29).
- MRT derivation: Further substitutions into the rate expression yield the stated result in Proposition 1.The final step uses equations (25), (26), and (30).
B. Proof of Proposition 2
The proof computes intermediate variance and expectation terms, substitutes them into earlier expressions, and concludes Proposition 2.
- The proof computes Var(a_ki) from equations (33) and (34).
- A result from Lemma 2.10 is used in the derivation.
- The proof similarly obtains an expectation expression before applying equation (36).
- The derivation concludes with the result stated in Proposition 2.