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Training and Feedback Optimization for Multiuser MIMO Downlink

Mari Kobayashi, Nihar Jindal, Giuseppe Caire

arXiv:0912.1987v1cs.IT

TL;DR

The paper addresses how to allocate training and feedback resources in multiuser MIMO downlink systems when CSIT is imperfect and overhead reduces throughput. It uses a tight closed-form rate lower bound across three time-frequency block models to optimize overall spectral efficiency. The resulting analysis provides system-optimization guidelines, including a substantial advantage for digital feedback over analog feedback.

  • Problem

    Improving CSIT benefits downlink rates but consumes training and feedback resources, creating a nontrivial overhead tradeoff.

  • Method

    The paper uses a tight closed-form lower bound on achievable rates with CSIT errors to optimize training and feedback across three time-frequency block models.

  • Results

    The analysis characterizes the relevant spectral-efficiency tradeoffs and provides guidelines for overall system optimization.

  • Takeaways & Limitations

    Digital quantized feedback offers a substantial advantage over analog unquantized feedback.

Abstract

from arXiv · show

We consider a MIMO fading broadcast channel where the fading channel coefficients are constant over time-frequency blocks that span a coherent time $\times$ a coherence bandwidth. In closed-loop systems, channel state information at transmitter (CSIT) is acquired by the downlink training sent by the base station and an explicit feedback from each user terminal. In open-loop systems, CSIT is obtained by exploiting uplink training and channel reciprocity. We use a tight closed-form lower bound on the ergodic achievable rate in the presence of CSIT errors in order to optimize the overall system throughput, by taking explicitly into account the overhead due to channel estimation and channel state feedback. Based on three time-frequency block models inspired by actual systems, we provide some useful guidelines for the overall system optimization. In particular, digital (quantized) feedback is found to offer a substantial advantage over analog (unquantized) feedback.

I. INTRODUCTION

The paper studies how to optimize training and feedback resources for multiuser MIMO downlink systems under the tradeoff between CSIT quality and overhead. It analyzes several time-frequency block models and characterizes operating choices for different system requirements.

  • Accurate CSIT is essential for ZF beamforming, but improving it requires substantial downlink-training and, in FDD, feedback resources.
  • The resulting tradeoff is between the benefits of better CSIT and the overhead of channel estimation and feedback.
  • The work determines the optimal fraction of resources devoted to training and feedback across several time-frequency block models inspired by practical systems.
  • For models where feedback consumes uplink resources, the paper characterizes the uplink/downlink spectral-efficiency region and selects operating points according to traffic demands.
  • The analysis also studies temporally correlated fading and feedback delay using a one-step prediction model.
  • The paper addresses imperfect feedback with ZF beamforming and provides broader system-optimization guidelines than its earlier model-1 analysis.

II. CHANNEL STATE ESTIMATION AND FEEDBACK

This section describes how the base station obtains CSIT through training, feedback, or reciprocity, and how estimation and feedback errors reduce achievable ZF rates. It then formulates joint resource optimization for training and feedback.

  • Closed-loop CSIT is obtained through downlink training followed by channel feedback from each user terminal.
  • CSIT errors arise from both channel-estimation error during common training and distortion introduced during feedback.
  • Residual interference from nonzero leakage coefficients lowers the achievable rate with ZF beamforming.
  • The achievable rate is tightly lower-bounded, with its rate gap depending on training length, feedback length, and feedback strategy.
  • Joint optimization maximizes net downlink spectral efficiency by allocating a total training-and-feedback budget within each coherence block.
  • Model 1 covers reciprocal TDD, nonreciprocal TDD, and FDD cases in which downlink training and feedback occur within the same fading block.

A. TDD with channel reciprocity

With channel reciprocity, uplink pilot training provides open-loop CSIT, and the training-length optimization yields scaling laws for the optimal training duration and its approximation accuracy.

  • Uplink pilot symbols provide open-loop CSIT when channel reciprocity holds, so no CSIT feedback is used.
  • The optimized training length is characterized through a stationary-point equation and an upper bound derived from concavity.
  • For large block lengths, the upper bound becomes a very good approximation to the optimal training length.
  • At fixed SNR, the optimal training length increases with block length, while at fixed block length it decreases as SNR increases.

tr on the net achievable rate. By the definition of T ⋆

The feedback strategy determines the rate gap and feedback-symbol allocation, while TDD with reciprocity provides an upper benchmark for systems using CSIT feedback.

  • Open-loop TDD with channel reciprocity upper-bounds the net rate achievable with the considered CSIT-feedback strategies.
  • B. Analog Feedback: Analog feedback sends channel coefficients through unquantized modulation, with each coefficient transmitted over multiple feedback channel uses and estimated by MMSE.
  • B. Analog Feedback: The analog training-feedback allocation is obtained from a convex minimization subject to a fixed total duration.
  • B. Analog Feedback: The optimal total training duration is obtained by reducing the outer optimization to a single variable and applying the resulting scaling relation.
  • B. Analog Feedback: The optimal downlink training is independent of feedback-channel efficiency, whereas analog feedback increases the rate gap relative to open-loop TDD.

C. Error-Free Digital Feedback

Error-free digital feedback quantizes each user’s channel estimate and allocates feedback symbols under a capacity-based model; its optimized performance nearly reaches the TDD benchmark, while practical QAM feedback remains efficient when errors are controlled.

  • Digital feedback quantizes each user’s estimated channel into a B-bit message and maps it to feedback symbols using directional random vector quantization.
  • Under error-free feedback, the rate gap is characterized for a capacity-limited feedback channel and minimized jointly with training overhead.
  • The feedback-symbol duration grows logarithmically with the Lagrange parameter, much more slowly than the linear growth of training duration.
  • Because feedback duration grows logarithmically and decreases with SNR, its effect on the optimization is negligible, making training optimization similar to TDD.
  • Error-free digital feedback performs almost as well as the TDD open-loop upper bound.
  • With uncoded QAM feedback, optimizing constellation size can make feedback errors sufficiently small when feedback duration is large.
  • At Nt = 4, 10 dB, B = 25 bits, and 4-QAM, the feedback message error probability is Pe,fb = 0.0194.
  • Digital feedback outperforms analog feedback for any block length because it has significantly smaller distortion when Tfb exceeds approximately N_t^2.

IV. SEPARATE UPLINK AND DOWNLINK BANDWIDTHS

With separate uplink and downlink bandwidths, the optimization becomes a tradeoff between downlink spectral efficiency and uplink CSIT-feedback overhead, including training and feedback-delay models.

  • Separate uplink and downlink bandwidths make CSIT-feedback channel uses an uplink overhead rather than a downlink overhead.
  • The section fixes the bandwidths and focuses on the tradeoff between downlink spectral efficiency and uplink CSIT-feedback overhead.
  • For each feedback duration, the optimal number of downlink training symbols is selected while accounting for training overhead in net downlink spectral efficiency.
  • Solving the training optimization gives a tight lower bound on optimal ZF downlink spectral efficiency as a function of feedback uses per block.
  • The analysis treats an AWGN feedback channel and a temporally correlated channel with feedback delay and channel prediction.

A. AWGN feedback link

The section models feedback over an AWGN link and compares analog, error-free digital, and QAM-based digital feedback while accounting for feedback overhead. Digital feedback is analyzed as capturing essential behavior with simpler expressions, and numerical results also include 4QAM feedback.

  • Feedback models: The analysis compares analog feedback, error-free digital feedback, and QAM-based digital feedback over the uplink feedback link.The analytical treatment focuses on error-free digital feedback at AWGN capacity, while numerical results also include 4QAM-based feedback.
  • Feedback efficiency: Digital-feedback rate penalties converge quickly to the optimized-feedback rate, whereas analog-feedback convergence is slower.The lower-bound expressions separate the spectral-efficiency penalties due to training and feedback.
  • System accounting: The net downlink spectral efficiency is characterized as a function of uplink symbols used for CSIT feedback.The analysis then relates those symbols to the uplink bandwidth consumed per coherence block.
  • System accounting: The resulting tradeoff analysis studies how feedback resources affect downlink performance alongside uplink resource consumption.The system-level objective is to understand the fundamental tradeoff between downlink and uplink rate.

Tc Hz of uplink bandwidth, the remaining bandwidth of Wup −Tfb

The system trades uplink resources used for channel feedback against uplink data transmission and downlink rate. Weighted-rate optimization identifies operating points, with digital feedback producing a sharper tradeoff and near-perfect-feedback downlink efficiency except when uplink rate is strongly prioritized.

  • Uplink/downlink tradeoff: As feedback length Tfb increases, downlink rate Rdown increases while uplink rate Rup decreases.This creates a Pareto-optimal uplink/downlink boundary.
  • Uplink/downlink tradeoff: Weighted-sum-rate maximization selects the operating point on the uplink/downlink Pareto boundary.The feedback length is optimized as a function of the rate weight λ.
  • Feedback comparison: Digital feedback produces a sharper tradeoff curve and dominates the analog-feedback curve.The imperfect-feedback rate-loss term is described as marginal for both schemes when T is large and 0 < λ < 1.
  • Numerical example: In the LTE-inspired example, the model uses a 200 kHz bandwidth and 1 ms duration, corresponding to T = 200.The paper presents the corresponding uplink/downlink sum-rate boundary and feedback lengths numerically.
  • Operating point: At the sharp knee of the tradeoff curves, downlink rate is close to its maximum while uplink rate remains reasonably close to its maximum.The paper identifies this region as a typical operating point for a well-designed system.
  • Operating point: Unless uplink rate is very strongly preferred, efficient operation keeps downlink spectral efficiency close to the perfect-feedback case.The training length is 24 symbols for every scheme except when λ is approximately zero.

where ˜hk(t) = E[hk(t)|{sk(t−τ)}] denotes the estimated channel, independent of the estimation

The temporally correlated-channel model accounts for filtering or one-step prediction with delayed feedback and optimizes training and feedback jointly. Higher mobility requires more training and reduces downlink rate, while 4QAM feedback can outperform analog feedback at a fixed feedback allocation.

  • Temporal prediction: The filtering rate-gap upper bound reduces to the AWGN-feedback-link bound for sufficiently large ρ.The section focuses subsequently on the more interesting one-step prediction case.
  • Joint optimization: The analysis maximizes net downlink achievable spectral efficiency for the one-step prediction case by optimizing training and feedback lengths.The objective is concave in training length, enabling optimization through the corresponding first-order condition.
  • Training under mobility: For one-step prediction, the optimal training length increases with Doppler frequency and scales with block length T.In the quasi-static limit, the training length coincides with the block-by-block estimation expression.
  • Numerical mobility results: For mobile speeds v = 6, 50, 80 km/h, the optimized training lengths are 25, 36, 43 symbols, respectively.The feedback length is relatively insensitive to mobile speed, although it tends to decrease as speed increases.
  • Numerical mobility results: Higher mobile speed significantly decreases downlink rate because the larger training length causes significant rate loss.The simulations compare speeds corresponding to Doppler shifts F = 0.011, 0.093, 0.148.
  • Feedback comparison: With Tfb = 30 over T = 200 symbols, allocating 15% of uplink resources to feedback makes uncoded 4QAM outperform analog feedback.The comparison includes analog, error-free digital, and 4QAM-based digital feedback.

V. ALLOWING FOR MANY USERS

With more users than base-station antennas, feedback allocation must jointly select the total feedback length and the users who report channel state. More feedback can enable more user selection and improve downlink efficiency, but it consumes additional uplink bandwidth.

  • Rate tradeoff: Allowing more users to feed back creates a non-negligible downlink spectral-efficiency increase but incurs a larger uplink bandwidth cost.The system should therefore optimize both total feedback symbols and the number of feedback users.
  • Scope and approximation: The lower bound used does not hold when user selection is performed, although numerical verification finds it to be a reasonable approximation with user selection and imperfect CSIT.The many-user discussion focuses on separate uplink/downlink bands, while other models can be adapted similarly.
  • User selection: For Tfb ≤ 24 through Tfb ≥ 42, the spectral-efficiency-maximizing feedback-user counts are 4, 5, 6, 7, and 8, respectively.The intervals are Tfb ≤24, 25 ≤Tfb ≤29, 30 ≤Tfb ≤36, 37 ≤Tfb ≤41, and Tfb ≥42.
  • User selection: The sum spectral efficiency is maximized with approximately 6 users feeding back in the examined range.With K = 4 fixed, increasing Tfb beyond 35 or 40 yields virtually no benefit because the feedback channel is essentially perfect.
  • User selection: For Tfb ≤ 200, no more than 31 users are needed, while adding users can provide a non-negligible rate gain.The marginal benefit decreases as Tfb increases, but benefits persist through the 31st user.
  • Rate tradeoff: When uplink and downlink rates are equally weighted, the operating point is approximately Rup = 828 Kbps and Rdown = 1966 Kbps with K = 11 and Tfb = 63 symbols.At this point, roughly 30% of uplink bandwidth is used for channel feedback.

Model 1: single band

The single-band model examines how training, feedback, block length, and uplink–downlink allocation shape spectral efficiency across several system tradeoffs.

  • The study compares feedback and training behavior across different time-frequency block models.
  • Feedback and training lengths are evaluated as functions of block length for multiple feedback strategies.
  • Sum spectral efficiency is examined as a function of block length.
  • The analysis compares downlink and uplink resource tradeoffs under fixed antenna, SNR, and block-length settings.
  • Feedback length is studied against λ, including delayed-feedback effects and mobile-speed dependence.
  • Downlink sum spectral efficiency is also evaluated against feedback symbols for varying user counts.
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