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Full-Duplex MIMO Relaying: Achievable Rates under Limited Dynamic Range

Brian P. Day, Adam R. Margetts, Daniel W. Bliss, Philip Schniter

arXiv:1111.2618v4cs.IT

TL;DR

The paper addresses full-duplex MIMO relaying when relay self-interference interacts with limited dynamic range and imperfect channel estimation. It models these impairments, derives achievable-rate bounds, and optimizes a lower-bound-based transmission scheme. The analysis and numerical studies characterize tradeoffs across interference, signal power, dynamic range, antenna counts, training, and half-duplex baselines.

  • Problem

    Relay self-interference can overwhelm limited-dynamic-range input circuitry, while limited dynamic range prevents perfect cancellation even when interference is known.

  • Method

    The paper explicitly models transmitter/receiver dynamic-range limitations and pilot-aided channel estimation, derives rate bounds, and maximizes the lower bound using bisection search and Gradient Projection.

  • Results

    The paper derives upper and lower achievable-rate bounds, an analytic approximation, and numerical evaluations across SNR, INR, dynamic range, antenna counts, and training length.

  • Takeaways & Limitations

    Full-duplex MIMO relay performance depends on jointly managing self-interference, dynamic-range impairments, channel estimation, spatial resources, and independently optimized data periods.

Abstract

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In this paper we consider the problem of full-duplex multiple-input multiple-output (MIMO) relaying between multi-antenna source and destination nodes. The principal difficulty in implementing such a system is that, due to the limited attenuation between the relay's transmit and receive antenna arrays, the relay's outgoing signal may overwhelm its limited-dynamic-range input circuitry, making it difficult---if not impossible---to recover the desired incoming signal. While explicitly modeling transmitter/receiver dynamic-range limitations and channel estimation error, we derive tight upper and lower bounds on the end-to-end achievable rate of decode-and-forward-based full-duplex MIMO relay systems, and propose a transmission scheme based on maximization of the lower bound. The maximization requires us to (numerically) solve a nonconvex optimization problem, for which we detail a novel approach based on bisection search and gradient projection. To gain insights into system design tradeoffs, we also derive an analytic approximation to the achievable rate and numerically demonstrate its accuracy. We then study the behavior of the achievable rate as a function of signal-to-noise ratio, interference-to-noise ratio, transmitter/receiver dynamic range, number of antennas, and training length, using optimized half-duplex signaling as a baseline.

I. INTRODUCTION

The paper studies full-duplex MIMO relaying under severe self-interference, limited transmitter/receiver dynamic range, and imperfect channel information. It develops rate bounds and an optimized transmission design while examining tradeoffs involving interference, antennas, training, and half-duplex operation.

  • I. INTRODUCTION: Full-duplex relaying reuses the same time-frequency signal space but faces high relay self-interference from the relay's transmit array.Half-duplex systems avoid this interference by separating source-to-relay and relay-to-destination transmissions in time or frequency.
  • I. INTRODUCTION: Limited dynamic range can make perfect self-interference cancellation impossible even when the interference signal is perfectly known.Receiver distortion and transmitter noise are modeled with variances proportional to received and intended transmit-signal energy, respectively.
  • I. INTRODUCTION: MIMO spatial nulling can suppress self-interference, but consumes spatial degrees of freedom needed for relay-to-destination communication.The design must jointly determine which receive antennas to protect and how strongly to protect them.
  • I. INTRODUCTION: The paper derives achievable-rate upper and lower bounds under pilot-aided channel estimation and proposes maximizing the lower bound subject to a power constraint.The resulting nonconvex optimization is addressed using bisection search and Gradient Projection.
  • II. SYSTEM MODEL: The system model represents source-to-relay, relay self-interference, relay-to-destination, and source-to-destination propagation using Rayleigh-fading MIMO channels with AWGN.The relay and destination SNRs and INRs quantify desired-signal and interference levels in the received-signal equations.
  • II. SYSTEM MODEL: Training and data epochs are divided into periods whose transmission parameters can be optimized independently, providing flexibility when relay interference is large relative to signal power.The same design is reused across data epochs because the modeled channels are time-invariant.

D. Limited Receiver Dynamic Range

The receiver dynamic-range model adds Gaussian distortion proportional to received energy, then incorporates this distortion into pilot-based channel estimation and the equivalent relay system model. Partial self-interference cancellation leaves an effective channel with aggregate noise that includes distortion and estimation error.

  • D. Limited Receiver Dynamic Range: Receiver distortion is modeled as independent zero-mean Gaussian noise at each antenna, with variance β times the antenna's collected energy.The parameter β is typically much smaller than one, and β^-1 characterizes receiver dynamic range.
  • D. Limited Receiver Dynamic Range: The Gaussian distortion model is intended to capture combined effects of AGC noise, ADC and gain-control nonlinearities, and oscillator phase noise.The paper also relates the model to quantization errors from AGC followed by dithered uniform quantization.
  • Channel Estimation: Pilot training uses separate source and relay periods, with one transmitter sending known pilots while the other remains silent.The receiver applies least-squares channel estimation and communicates the estimate to the transmitter.
  • Equivalent Channel: After partial cancellation, the relay's desired signal is represented through an estimated source-relay channel plus aggregate noise conditioned on the channel estimates.The aggregate noise can be non-Gaussian because it includes channel-estimation-error components.
  • Channel Estimation: Increasing the training length can make channel-estimation error terms arbitrarily small under an appropriate choice of T.The data protocol assumes one training epoch followed by many data epochs, reducing relative training overhead as data epochs increase.
  • Equivalent Channel: The destination absorbs the source-destination interference into aggregate noise because that link is assumed much weaker than the relay-destination link.Figure 3 summarizes the resulting equivalent system model.

C. Bounds on Achievable Rate

The paper analyzes source-relay and relay-destination mutual information using an equivalent model that includes estimation error and residual self-interference. Gaussian worst-noise arguments produce a lower bound on the end-to-end rate, alongside an upper bound and an achievable coding interpretation.

  • Rate Bounds: The end-to-end mutual information is expressed for a fixed time-sharing parameter using the source-relay and relay-destination period-specific mutual informations.The covariance design Q[l] determines the signaling in each data period.
  • Rate Bounds: The equivalent-model analysis includes channel-estimation error, residual relay self-interference, and source-destination interference treated as noise.Aggregate noises may be non-Gaussian because of channel-estimation-error components.
  • Rate Bounds: The lower bound follows because Gaussian noise is worst for mutual information among noise distributions with the same covariance.This converts the complicated aggregate-noise analysis into a covariance-based bound.
  • Rate Bounds: The lower-bound rate is achievable with independent Gaussian codebooks at the transmitters and maximum-likelihood detection at the receivers.The stated rate is measured in bits-per-channel-use.
  • Rate Bounds: A corresponding achievable-rate upper bound is also derived for the same end-to-end analysis.

IV. TRANSMIT COVARIANCE OPTIMIZATION

Transmit covariance optimization maximizes the lower-bound bottleneck rate for each time share and then optimizes the time-sharing parameter. The proposed weighted-sum formulation uses link equalization when possible and bisection otherwise.

  • IV. TRANSMIT COVARIANCE OPTIMIZATION: The design jointly optimizes source and relay transmit covariance matrices under per-link power constraints, then optimizes the time-sharing parameter τ.Covariance optimization is equivalent to jointly choosing transmission beam patterns and power levels.
  • IV. TRANSMIT COVARIANCE OPTIMIZATION: A maximin covariance design seeks to balance the source-relay and relay-destination rates, with at least one link-equalizing design in the maximin set.The two link rates are denoted I_sr,τ(Q) and I_rd,τ(Q).
  • IV. TRANSMIT COVARIANCE OPTIMIZATION: The maximin equalizing design is obtained by relaxing the equalization constraint and solving a ζ-weighted-sum-rate optimization over the broader covariance-feasible set.This relaxation enables the weighted optimization to recover a maximin design when an equalizing weight exists.
  • IV. TRANSMIT COVARIANCE OPTIMIZATION: Figure 4 contrasts weighted-sum-rate behavior when a link-equalizing solution exists with the case where no such solution exists.The figure uses τ-specific rate curves and optimal covariance sets indexed by τ and ζ.
  • IV. TRANSMIT COVARIANCE OPTIMIZATION: Bisection searches ζ∈[0,1] using the non-decreasing source-relay rate, while a grid search optimizes τ∈[0,1].If no equalizing ζ exists, the bisection converges toward ζ=0; this includes cases where one link dominates for all weights.

B. Gradient Projection

Gradient Projection solves each τ-specific weighted-sum-rate covariance problem inside the outer bisection search. The algorithm alternates projected updates for relay and source covariances, while initialization and nonconvexity limit formal global-optimality guarantees.

  • B. Gradient Projection: At each bisection step, Gradient Projection solves the τ-specific ζ-weighted-sum-rate optimization problem.The method uses gradient steps followed by projection onto the covariance constraint set.
  • B. Gradient Projection: The updates alternate between relay covariance matrices and source covariance matrices, with an outer loop continuing until covariance changes fall below a threshold.The relay and source updates each cover both data periods.
  • B. Gradient Projection: Projection exploits Hermitian covariance matrices by eigendecomposing the relay gradient-step matrices before projecting their eigenvalues onto the constraint set.The decomposition uses unitary eigenvectors and real-valued eigenvalues.
  • B. Gradient Projection: Because the optimization is generally non-convex, Gradient Projection is guaranteed only to find a local maximum, although tested initializations suggested global solutions in this problem.The implementation tries orthogonal half-duplex and naive full-duplex initializations and retains the one with the larger min-rate.
  • B. Gradient Projection: The covariance projection includes a water-filling operation through a scalar μ chosen to satisfy the power constraint.The positive-part operation is applied elementwise to the shifted eigenvalues.
  • B. Gradient Projection: The algorithm uses Armijo stepsize selection to adjust the gradient-step parameter.The stepsize is selected through a geometric sequence governed by constants σ and ν.

V. ACHIEVABLE-RATE APPROXIMATION

The paper develops an analytic achievable-rate approximation under a symmetric, idealized setting and uses it to characterize when full- or half-duplex signaling is preferable. The approximation is compared with covariance optimization and captures rate limits across interference regimes.

  • Approximation setup: The approximation assumes vanishing channel-estimation error, equal antenna dimensions across nodes, no direct source-to-destination link, and equal time-sharing.It further considers N transmit and M receive antennas at each node, with τ = 1/2.
  • Approximation setup: The simplifying channel model uses diagonal channel matrices with R = min{M, N} identical nonzero diagonal entries.The entry value is selected to preserve the assumed average channel-energy normalization.
  • Operating regimes: When ηr ≪ ρr, interference-dependent terms are negligible and the optimal covariances are full-duplex covariances QFD.This regime yields the full-duplex operating approximation under the power constraint.
  • Operating regimes: When ηr ≫ ρr, the interference-dependent term dominates unless the relay transmit covariance is zero, producing half-duplex covariances QHD.The approximation therefore suppresses relay transmission in the high-interference regime.
  • Approximate rate: For any (ρr, ηr, ρd), the approximate achievable rate is I* ≈ max{I(QFD), I(QHD)}.The approximation selects the larger of the full-duplex and half-duplex rates.
  • Approximate rate: The rate is invariant to ρd when the source-to-relay link is limiting, whereas at sufficiently strong relay-to-destination conditions it is invariant to ρr and ηr.The approximation identifies the limiting hop through distinct SNR and interference regimes.
  • Validation: The approximation is reasonably close to the covariance-optimized achievable rate obtained using bisection and gradient projection.Numerical comparisons show close agreement, including the predicted transition from full-duplex to half-duplex behavior as ηr increases.

VI. NUMERICAL RESULTS

The numerical study evaluates optimized full-duplex and half-duplex relaying across training, interference, SNR, antenna, and dynamic-range conditions. TCO-2-IC generally adapts between full- and half-duplex operation, and the analytic approximation matches numerical optimization closely.

  • Experimental setup: The experiments evaluate bisection/GP-optimized achievable rates across SNR, INR, dynamic range, antenna counts, training length, cancellation, data periods, and time-sharing.Results use the stated propagation, dynamic-range, channel-estimation, and power-constraint models, with τ optimized on a grid and rates averaged over channel realizations.
  • Training length: Training increases TCO-2-IC’s lower-bound rate rapidly at small T, then the rate saturates as channel-estimation error becomes negligible.The lower and upper bounds converge as training grows, and the nominal T = 50 provides nearly equal values for the two bounds.
  • INR and signaling regimes: At low-to-mid INR, TCO-2-IC achieves full-duplex-like performance; at high INR, it transitions toward OHD performance, unlike TCO-1-IC, which falls below OHD.Two distinct data periods allow TCO-2-IC to facilitate half-duplex signaling when high INR makes that regime preferable.
  • Interference cancellation and time sharing: Partial interference cancellation is important for TCO-2 except at extremely low or high INR, while optimizing OHD’s τ produces a small rate gain over τ = 0.5.The comparison uses TCO-2-IC, TCO-2, TCO-1-IC, and both fixed- and optimized-time-share OHD traces.
  • SNR dependence: At ηr = 20dB, TCO-2-IC remains full-duplex across SNR; at ηr = 60dB, it is half-duplex at low SNR and switches to full-duplex above a threshold.These SNR-dependent regimes are predicted by the analytic approximation.
  • Approximation accuracy: The analytic approximation and GP-optimized rate contours show a relatively good match, with the greatest discrepancy when ηr ≈ ρr and both are large.The approximation assumes either ηr ≪ ρr or ηr ≫ ρr, explaining the discrepancy near equality.
  • Number of antennas: Achievable rate increases with both M and N, while the study also examines antenna allocation when the total number of antennas per modem is fixed.The fixed-total comparison uses N + M = 7 in the intermediate full-/half-duplex regime.

VII. CONCLUSION

The paper analyzes decode-and-forward full-duplex MIMO relaying with limited dynamic range and imperfect CSI. It derives rate bounds, optimizes the lower bound numerically, develops a close analytic approximation, and studies key system parameters.

  • Scope: The analysis models limited transmitter/receiver dynamic range, imperfect CSI, background AWGN, and very high self-interference in decode-and-forward full-duplex MIMO relaying.The paper explicitly models transmitter and receiver dynamic-range limitations and pilot-aided channel-estimation error.
  • Rate analysis: Upper and lower bounds on end-to-end achievable rate tighten as the number of pilots increases.The bounds are derived under the explicit dynamic-range and channel-estimation models.
  • Optimization: The proposed transmission scheme maximizes the achievable-rate lower bound through bisection search and Gradient Projection for a nonconvex optimization problem.Gradient Projection implicitly performs water-filling.
  • Approximation and evaluation: The analytic achievable-rate approximation agrees closely with numerical optimization across the studied system parameters.The numerical study varies SNR, INR, dynamic range, antenna count, and number of pilots.

APPENDIX A CHANNEL ESTIMATION DETAILS

Appendix A characterizes pilot-aided channel estimation under transmitter and receiver dynamic-range limitations. It derives the conditional covariance structure of estimation errors and aggregate noise.

  • Signal model: Under limited transmitter dynamic range, the appendix models the received training signal with transmitter noise, receiver distortion, and AWGN.The distorted received signal is represented using these aggregate impairments.
  • Pilot correlations: The pilot signal’s spatial correlation is N^-1 I, which yields transmitter-distortion spatial correlation κN^-1 I.These correlations are used in deriving the conditional covariance expressions.
  • Conditional covariances: Conditioned on H, the aggregate noise is temporally white with a derived spatial correlation that includes transmitter distortion and receiver distortion.The receiver-distortion covariance is obtained from the conditional covariance of the undistorted received signal.
  • Estimation error: The channel-estimation error is temporally white with an H-conditional spatial correlation.The appendix further establishes statistical equivalence to a Gaussian matrix construction.

APPENDIX B INTERFERENCE CANCELLATION DETAILS

Appendix B derives conditional covariance expressions for aggregate interference and receiver distortion under the channel-estimation model. The approximation assumes small transmitter- and receiver-distortion parameters.

  • Aggregate interference: The appendix characterizes the channel-estimate-conditioned covariance of the relay’s aggregate interference vr.The target covariance is conditioned on the estimated source-relay and relay-relay channels.
  • Covariance identity: A Gaussian matrix identity relates conditional quadratic terms to an estimated covariance matrix multiplied by the trace of the input covariance.The identity and its corollary support the subsequent covariance derivations.
  • Receiver distortion: The conditional receiver-distortion covariance is β diag(Φ̂r), where Φ̂r is the covariance of the undistorted received signal conditioned on the channel estimates.The derivation uses the transmit-signal covariances including transmitter distortion.

APPENDIX C GRADIENT DETAILS

This appendix derives gradient expressions by differentiating a determinant involving matrix-valued terms, then applies the result to obtain the gradients used in the main formulation.

  • Gradient derivation: The appendix derives an expression for the gradient ∇Qr[l]I through an intermediate determinant-derivative calculation.The related derivative is ∂det(Y)/∂X, with Y defined as a sum of diagonal, trace, and residual terms.
  • Gradient derivation: The matrix Y combines C diag(X)D, diag(EXF), G tr(X), and Z, whose elementwise form is also considered.The matrix Z represents terms with zero derivative with respect to Qr[l].
  • Gradient derivation: Using ∂det(Y)/∂Y = det(Y)(Y^-1)^T and basis matrices V r,s, the derivation obtains the required determinant derivatives.V r,s is zero except for a unity element at row r and column s.
  • Gradient derivation: Substituting the defined matrices and covariance expression yields Gr[l], with a similar expression obtained for Gs[l].The derivation uses the Hermitian property of Sd[l], Sr[l], Σ̂d[l], and Σ̂r[l].
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