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Multiuser MIMO Achievable Rates with Downlink Training and Channel State Feedback

Giuseppe Caire, Nihar Jindal, Mari Kobayashi, Niranjay Ravindran

arXiv:0711.2642v2cs.IT

TL;DR

The paper asks how much downlink performance is achievable when receiver and transmitter channel information must be learned and fed back imperfectly. It analyzes ZF beamforming with training and analog or digital feedback under noisy, fading, erroneous, and delayed links, finding substantial throughput and a delayed-feedback multiplexing gain of M(1 − 2F) for F < 1/2.

  • Problem

    The paper addresses achievable rates for MIMO broadcast channels when channel state information is not given a priori and must be acquired through training and feedback.

  • Method

    The paper rigorously analyzes ZF linear beamforming with pilot-based channel estimation, explicit analog or digital feedback, feedback errors, MIMO-MAC uplinks, and feedback delay.

  • Results

    The schemes achieve significant downlink throughput; with delayed feedback and F < 1/2, the achievable multiplexing gain is M(1 − 2F).

  • Takeaways & Limitations

    Properly designed explicit feedback can support substantial throughput, while uncoded digital feedback can have a vanishing high-SNR rate gap under suitable parameters.

Abstract

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We consider a MIMO fading broadcast channel and compute achievable ergodic rates when channel state information is acquired at the receivers via downlink training and it is provided to the transmitter by channel state feedback. Unquantized (analog) and quantized (digital) channel state feedback schemes are analyzed and compared under various assumptions. Digital feedback is shown to be potentially superior when the feedback channel uses per channel state coefficient is larger than 1. Also, we show that by proper design of the digital feedback link, errors in the feedback have a minor effect even if simple uncoded modulation is used on the feedback channel. We discuss first the case of an unfaded AWGN feedback channel with orthogonal access and then the case of fading MIMO multi-access (MIMO-MAC). We show that by exploiting the MIMO-MAC nature of the uplink channel, a much better scaling of the feedback channel resource with the number of base station antennas can be achieved. Finally, for the case of delayed feedback, we show that in the realistic case where the fading process has (normalized) maximum Doppler frequency shift 0 < F < 1/2, a fraction 1 - 2F of the optimal multiplexing gain is achievable. The general conclusion of this work is that very significant downlink throughput is achievable with simple and efficient channel state feedback, provided that the feedback link is properly designed.

I. INTRODUCTION

The paper characterizes achievable ergodic rates for ZF beamforming when channel knowledge is acquired through downlink training and explicit feedback over realistic links. It compares analog and digital feedback, accounts for feedback errors, MIMO-MAC uplinks, and delay, and quantifies the resulting rate and multiplexing-gain effects.

  • A. Contributions of this work: The paper studies achievable rates for fading multiuser MIMO downlinks with imperfect CSIR and CSIT obtained through downlink training and channel state feedback.The analysis focuses on ZF linear beamforming and leaves capacity converses largely open.
  • I. INTRODUCTION: Imperfect receiver channel estimates degrade the CSIT fed back to the base station, unlike models assuming perfect CSIR at the terminals.The paper explicitly accounts for common and dedicated training phases.
  • A. Contributions of this work: The feedback channel is modeled explicitly as a noisy transmission link, enabling comparisons between analog and digital feedback and resource measurements.The paper considers both orthogonal unfaded AWGN feedback and fading MIMO-MAC uplinks.
  • A. Contributions of this work: With delayed feedback and normalized Doppler bandwidth F < 1/2, the achievable multiplexing gain is M(1 − 2F), rather than the full gain M.The loss is tied to the noisy prediction error of the channel process.

III. ACHIEVABLE RATE BOUNDS

The paper derives achievable-rate lower bounds for ZF beamforming with Gaussian inputs under imperfect channel estimation and feedback, expressing performance through a rate gap relative to ideal CSI.

  • Theorem 1 lower-bounds the achievable rate for Gaussian-input ZF beamforming with CSI training and feedback.
  • The conditional interference second moment may be difficult to compute because the interference depends intricately on the estimated useful-signal coefficient.
  • The rate gap compares the achievable rate with imperfect CSI against ideal-CSI ZF performance with equal power allocation.
  • The interference term depends on mismatch between the actual channel and the BS estimate, while dedicated training determines the dedicated-training error term.
  • The resulting corollary provides a lower bound on the achievable rate and also bounds the mutual information using only the useful-signal coefficient estimate.

B. Upper Bounds

The paper develops upper bounds for ZF rates and analyzes analog feedback over an orthogonal AWGN uplink, showing bounded rate loss and preserved multiplexing gain under sufficient feedback resources.

  • Upper Bounds: A genie-aided upper bound gives each terminal perfect knowledge of its interference coefficients after the BS selects the beamforming matrix.
  • Upper Bounds: The bound is easy to evaluate by Monte Carlo simulation and can be approached with sufficiently extensive dedicated training.
  • Analog feedback: Each terminal uses βfbM feedback symbols on an orthogonal AWGN feedback channel, for a total of βfbM^2 feedback-channel uses.
  • Analog feedback: Analog feedback transmits scaled, unquantized versions of each terminal’s common downlink-training observation over the feedback channel.
  • Analog feedback: βfb ≥ 1 is required so every channel coefficient is transmitted at least once, and the analog-feedback rate gap remains uniformly bounded, preserving full multiplexing gain.
  • Analog feedback: Imperfect CSI increases effective noise from the thermal level to thermal noise plus self-noise and interference, making the rate gap the logarithm of their power ratio.
  • Analog feedback: A lower uplink SNR replaces βfb with Γβfb in the analog-feedback bound, changing the rate gap but not the multiplexing gain.
  • Analog feedback: In reciprocal TDD, orthogonal uplink training corresponds to perfect-feedback FDD analysis, but dedicated downlink training remains necessary.

B. Digital feedback

Digital feedback quantizes each terminal’s estimated channel direction and sends its index to the BS; with sufficient feedback resources, it preserves multiplexing gain and can outperform analog feedback asymptotically.

  • Digital feedback: Digital feedback quantizes each estimated channel vector into B bits and transmits the resulting packet to the BS.
  • Digital feedback: The quantization codebook contains unit-norm vectors, and the selected vector minimizes the angle to the estimated channel direction.
  • Digital feedback: Because the quantization vectors are unit-norm, digital feedback conveys channel direction but no channel magnitude information.
  • Digital feedback: Random Vector Quantization uses independently and uniformly generated unit-sphere codewords to characterize average angular distortion.
  • Digital feedback: Theorem 5 upper-bounds the rate gap when RVQ feedback bits are conveyed error-free to the BS.
  • Digital feedback: Under an error-free capacity-achieving feedback-link assumption, B = βfb(M − 1) log2(1 + P/N0) bits per mobile.
  • Digital feedback: βfb ≥ 1 is sufficient for a bounded digital-feedback rate gap and preserved full multiplexing gain.
  • Digital feedback: For βfb > 1, the digital-feedback excess term vanishes at high SNR, unlike analog feedback’s constant 1/βfb term.

C. Effects of feedback errors

Feedback errors create a distortion–reliability tradeoff, but properly choosing the digital feedback rate can make their high-SNR effect vanish or remain minor, even with uncoded QAM.

  • Feedback-channel design: Short-block feedback coding is a joint source-channel problem because channel feedback has a non-standard distortion measure and requires very short blocks.A full treatment is outside the paper’s scope.
  • Uncoded QAM: With uncoded QAM, signaling at α = βfb keeps the symbol error probability from decreasing, whereas α < βfb makes it vanish as SNR increases.The entire feedback-message error probability is built from the QAM symbol errors.
  • Feedback-channel design: Increasing α improves quantization but raises the feedback error probability Pe,fb; decreasing α has the opposite tradeoff.The design balances quantization distortion against channel reliability.
  • Uncoded QAM: Theorem 6 bounds the rate gap when each terminal sends B = α(M −1) log P/N0 RVQ bits over βfb(M −1) uncoded-QAM channel uses.The bound incorporates the feedback error probability Pe,fb.
  • High-SNR behavior: For 1 < α < βfb, feedback errors have vanishing high-SNR effect because both quantization distortion and the weighted error term P/N0 Pe,fb vanish.The feedback error probability decays exponentially when α < βfb, while quantization distortion vanishes when α > 1.
  • Error mitigation: Feedback errors behave like impulsive noise, so detecting and discarding erroneous frames can substantially reduce their effect; naive ZF is also robust because interference depends on the affected terminal.A small number of users with poor feedback quality need not destroy overall performance.

D. Comparison between analog and digital channel feedback

Analog and digital feedback are comparable when one feedback channel use serves each channel coefficient, while digital feedback can outperform analog when additional feedback bandwidth is available.

  • AWGN feedback comparison: For βfb = 1, analog and error-free digital feedback achieve essentially the same rate gap of 1 bit per channel user per terminal.This matches the rate-distortion interpretation for transmitting an i.i.d. Gaussian source over an AWGN channel with equal bandwidth.
  • AWGN feedback comparison: For βfb > 1, the quantized-feedback rate gap vanishes as P/N0 increases, whereas analog feedback is strictly suboptimal relative to a rate-distortion-achieving digital scheme.The advantage arises when feedback bandwidth exceeds source bandwidth.
  • Numerical comparison: With M = 4 and βfb = 1, analog and error-free digital feedback perform virtually identically and remain approximately 3 dB from the perfect-CSIT benchmark.Uncoded-QAM digital feedback has a substantial lower–upper-bound gap in this setting.
  • Numerical comparison: With M = 4 and βfb = 2, uncoded-QAM digital feedback outperforms analog feedback above approximately 5 dB and converges to the ideal rate.The result is shown using genie-aided upper bounds.
  • MIMO-MAC scaling: Orthogonal feedback requires O(M^2) channel uses, while downlink capacity scales at best as O(M), limiting scalability with the number of base-station antennas.The MIMO-MAC uplink offers a way to exploit spatial dimensions for feedback.
  • MIMO-MAC scaling: Grouping users for simultaneous MIMO-MAC feedback can reduce feedback-resource growth to linear in M, making it a fixed fraction of downlink capacity.The construction uses groups of L users transmitting simultaneously.

A. Analog Feedback

Analog feedback over the MIMO-MAC is analyzed with MMSE channel estimation and grouped simultaneous transmission, yielding antenna-dependent gains and a linear feedback-resource strategy.

  • Analog-feedback model: Analog feedback sends each terminal’s scaled noisy downlink channel over the uplink, with the base station estimating channel coefficients from the received vectors.The analysis initially assumes perfect knowledge of the uplink channel matrix A.
  • Rate-gap analysis: Theorem 7 upperbounds the rate gap for groups of L simultaneous analog-feedback terminals using βfbM channel uses per group.The bound is expressed through the average channel-state estimation MMSE and the eigenvalues of A^H A.
  • Rate-gap analysis: For L < M, the rate gap remains bounded and converges at high SNR to a constant.The corresponding expression reveals the effect of simultaneous feedback over the MIMO-MAC.
  • MIMO-MAC advantages: MIMO-MAC feedback provides an SNR gain of M − L and requires L times fewer feedback symbols than orthogonal access.The gain results from receiving L users’ feedback over M antennas while nulling interference.
  • Optimal grouping: For a fixed feedback budget with at least 2M symbols, optimizing L(M − L) gives L* = M/2 and a total of 2βfbM feedback symbols.At large M, the optimized rate gap is dominated by common downlink training, while dedicated training and feedback terms vanish.
  • Optimal grouping: Finite-SNR results also favor L* = M/2 when at least 2M feedback symbols are available, combining TDMA and SDMA.When fewer than 2M symbols are available, L = M is the only option and has an unbounded rate-gap bound despite reasonable finite-SNR performance.
  • Full spatial multiplexing: When L = M, the achievable rate remains uniformly bounded from the genie-aided upper bound for all SNRs, allowing up to M terminals to feed back simultaneously.This conclusion relies on terminals estimating their instantaneous residual interference, whose quality depends critically on dedicated training.
  • Estimator choice: The MMSE-based analysis avoids the infinite error variance produced by a zero-forcing pseudo-inverse at L = M.The paper attributes this improvement to component-wise MMSE processing.

B. Digital Feedback

Digital feedback over a fading MIMO-MAC can achieve bounded rate gaps while using feedback resources that scale linearly with the number of base-station antennas. The comparison also shows that digital feedback can approach optimal sum rates under the stated design.

  • Digital feedback: Digital feedback multiplexes up to L ≤ M users’ codewords simultaneously over the MIMO-MAC feedback channel.The receiver jointly decodes the simultaneously transmitted users’ feedback symbols.
  • Digital feedback: The optimal diversity gain for L single-antenna users equals the diversity of a single user transmitting to an M-antenna receiver.Each user’s error probability therefore decays with SNR as if it transmitted alone using TDMA on the feedback link.
  • Digital feedback: A bounded rate gap requires βfb > M/(M−1) and α < M−1/M βfb in the analyzed MIMO-MAC setting.These conditions are stricter than the corresponding βfb > 1 and α < βfb conditions for the previously analyzed unfaded orthogonal feedback channel.
  • Digital feedback: Feedback-channel uses scale linearly with M when digital feedback is designed over the MIMO-MAC uplink.The paper notes that explicit optimal diversity–multiplexing codes are generally difficult, but simple constructions exist for M single-antenna users.
  • Digital feedback: For M = 4 and 24 total feedback symbols, digital feedback achieves near-optimal sum rate across the plotted SNR range.The comparison uses analog feedback with L = 2, βfb = 3 and digital feedback with L = 4, βfb = 8, α = 4.

A. Estimation Error at UT

The UT’s channel-prediction error depends sharply on the fading process and feedback delay. Doppler processes permit vanishing one-step prediction error at high training SNR, whereas regular processes retain nonzero prediction error and can make delayed ZF interference limited.

  • Estimation Error at UT: The UT forms an MMSE estimate of the channel at frame t from noisy observations available through frame t−d, then feeds that estimate to the BS.The BS uses the received estimate to choose beamforming vectors for frame t.
  • Estimation Error at UT: Doppler processes are band-limited to [−F,F] with F < 1/2, while regular processes have strictly positive one-step prediction error even as observation noise vanishes.For Doppler processes, the one-step prediction error tends to zero as δ → 0; for regular processes, it remains positive.
  • Estimation Error at UT: With no feedback delay, the rate gap remains bounded for both Doppler and regular fading processes.The high-SNR filtering error scales so that (P/N0)ϵ0(N0/(β1P)) tends to 1/β1.
  • Estimation Error at UT: With one-frame delay, regular fading makes the achievable ZF rate saturate at high SNR because prediction error grows linearly with P/N0.The corresponding genie-aided upper bound is also bounded, so naive ZF becomes interference limited even with CSIR.
  • Estimation Error at UT: For Doppler fading with delayed feedback, the rate-gap growth is 2F log P/N0 and multiplexing gain M(1−2F) is achievable.Thus, the Doppler bandwidth determines the fraction of multiplexing gain retained under delay.
  • Estimation Error at UT: The paper contrasts this with perfect CSIR, under which Doppler processes permit perfect prediction and full multiplexing gain M despite feedback imperfections.Finite common-training resources instead yield the reduced gain M(1−2F).

C. Examples

Numerical examples compare filtering and prediction under Jakes and Gauss–Markov fading, illustrating the practical distinction between Doppler and regular processes. The conclusions also identify throughput improvements from scheduling and resource scaling with antenna number.

  • C. Examples: TDMA avoids interference limitation under delayed feedback but its sum rate grows like log(P/N0), rather than M log(P/N0).This provides the practical boundary for abandoning simultaneous ZF transmission in the cited setting.
  • C. Examples: For M = 4, v = 10 km/hr, and β1 = 1, optimal filtering improves achievable rates over block-by-block estimation across a wide SNR range.The advantage disappears near 30 dB for Gauss–Markov fading but persists beyond the plotted range for Jakes’ fading.
  • C. Examples: Under one-step prediction, Jakes’ fading stays close to the perfect-CSI rate, whereas Gauss–Markov rates saturate at sufficiently high SNR.The saturation reflects the unpredictability inherent in the regular Gauss–Markov model.
  • C. Examples: Both analog and digital feedback can achieve potentially high multiplexing gain under common Doppler fading models with noisy and delayed feedback.This is the principal conclusion drawn from the numerical and analytical examples.
  • C. Examples: Training and feedback channel uses scale linearly with the number of base-station antennas, and eventually with downlink throughput, even in FDD systems.The conclusion states that explicit CSIT feedback can therefore be implemented with this resource scaling.
  • C. Examples: Greedy user scheduling with K = 10 and M = 4 achieves a very small gap to optimal dirty-paper coding with perfect CSIT.The paper presents this as a throughput improvement over the equal-power, no-selection system analyzed in the main bounds.

APPENDIX I PROOF OF THEOREM 1

The appendix derives a mutual-information lower bound by controlling conditional entropy and exploiting Gaussianity, independence, MMSE estimation, and concavity. The resulting bound supports the paper’s achievable-rate analysis.

  • APPENDIX I PROOF OF THEOREM 1: The proof lower-bounds mutual information by upper-bounding the conditional entropy of the transmitted symbol given the received signal and beamforming information.The entropy bound uses an arbitrary deterministic estimator, conditioning reduction, and Gaussian maximum-entropy arguments.
  • APPENDIX I PROOF OF THEOREM 1: Independence and zero-mean Gaussian assumptions make the desired and interference-plus-noise terms uncorrelated, enabling the conditional second-moment calculation.This step is used after substituting the received-signal decomposition into the entropy bound.
  • APPENDIX I PROOF OF THEOREM 1: Choosing the estimator as the linear MMSE estimate tightens the entropy bound.The corresponding MMSE expression is then substituted into the rate calculation.
  • APPENDIX I PROOF OF THEOREM 1: Spatial whiteness and beamformer selection independent of the channel justify the random-variable substitutions used to complete the bound.The proof sets A = P/(N0M), λ = σ2 and X = |h_k^H v_k|2 before applying the lemma.
  • APPENDIX I PROOF OF THEOREM 1: The proof applies the lemma E[log(1 + XA)] ≤ E[log(1 + (λ + (1−λ)X)A)] for nonnegative X with E[X] = 1.The lemma follows by defining a concave interpolation function and applying Jensen’s inequality.

APPENDIX III PROOF OF THEOREM 4

The proof bounds rate gaps by decomposing interference into feedback-error-free and feedback-error contributions, then applying channel-estimation representations and asymptotic feedback-bit scaling.

  • Rate-gap evaluation: The channel is represented as the sum of a quantized channel estimate and an independent estimation error to evaluate the rate-gap bound.The derivation uses Gaussianity, independence, and the independence of channel norm and direction.
  • Rate-gap evaluation: The final rate-gap expression follows by substituting the interference bound into the rate-gap formula and setting B = α(M −1) log2(P/N0).The beta-function bound is used in the final step.
  • Feedback-error decomposition: The interference variance is decomposed according to whether channel-state feedback contains errors.The error-free contribution matches the previously derived expression, while the feedback-error contribution is bounded separately.
  • Feedback-error decomposition: The expected interference coefficient is bounded by one for feedback-error events, while the error-free term retains its original characterization.This separates the impact of feedback errors from the baseline quantization and estimation effects.

APPENDIX VI PROOF OF THEOREM 7

The proof characterizes the expected interference coefficient through the uplink-channel MMSE, derives its high-SNR asymptotics, and uses these bounds to show a bounded genie-aided rate gap.

  • MMSE characterization: The expected interference coefficient equals the channel-estimation-error variance averaged over the uplink channel matrix.By symmetry, the average is expressed using mmse(ρ), defined earlier in the paper.
  • High-SNR asymptotics: A closed-form expression for mmse(ρ) enables characterization of ρ mmse(ρ) as ρ tends to infinity.The asymptotic expansion uses exponential-integral identities for n = 1 and n > 1.
  • Genie-aided rate bound: The proof bounds the genie-aided rate by dropping nonnegative terms, conditioning on the uplink channel matrix, and applying Jensen’s inequality.The conditional interference variance is expressed through (M −1)Pσ²_k(A).
  • Genie-aided rate bound: The gap between the ideal ZF rate and the genie-aided rate is upper-bounded using symmetry, the MMSE derivation, and monotonicity of the logarithm.The resulting expression depends on λmin, the minimum eigenvalue of A^HA.
  • High-SNR boundedness: For i.i.d. complex Gaussian A, λmin is chi-squared with 2 degrees of freedom and mean 1, while the remaining terms are bounded independently of SNR.The logarithmic P/N0 terms cancel in the upper bound, establishing boundedness.

APPENDIX VIII GENIE-AIDED UPPER BOUND FOR REGULAR PROCESSES WITH DELAYED FEEDBACK

The appendix proves uniform boundedness of the genie-aided upper bound under positive noiseless prediction error, assuming perfect common training and perfect delayed feedback.

  • Assumptions: The analysis assumes perfect common training and perfect delayed feedback, leaving prediction error as the only CSIT noise source.The proof therefore isolates the effect of temporal channel prediction error.
  • Prediction-error model: The current channel is decomposed into its one-step prediction and a jointly Gaussian prediction error.The channel, prediction, and error are spatially i.i.d. with per-component variances 1, 1 − ϵ1(0), and ϵ1(0), respectively.
  • Boundedness result: When the noiseless prediction error is positive, the genie-aided upper bound is uniformly bounded for every SNR.The proof represents the error as ϵ1(0)∆(t) and takes the high-SNR limit using monotonicity of the bound.
  • Bound evaluation: The limiting bound is evaluated using Jensen’s inequality and the chi-square and beta distributions of the prediction-error quantities.The resulting expression also involves the Euler-Digamma function ψ(M).
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