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Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength

Wei Yang, Giuseppe Durisi, Tobias Koch, Yury Polyanskiy

arXiv:1311.2012v2cs.IT

TL;DR

The paper examines finite-blocklength rates for quasi-static MIMO fading channels, where it is unclear whether outage capacity is meaningful at practical blocklengths. It derives achievability and converse bounds across CSI configurations and finds zero dispersion, with numerical results showing rapid convergence to outage capacity.

  • Problem

    The paper addresses whether outage capacity is a meaningful finite-blocklength performance metric for delay-constrained quasi-static fading channels.

  • Method

    The paper develops achievability and converse bounds on R*(n, ϵ), including a threshold decoder that requires neither CSIR nor knowledge of the fading distribution.

  • Results

    The channel dispersion is zero for all four CSI-availability cases, and numerical results show fast convergence to outage capacity as blocklength increases.

  • Takeaways & Limitations

    Outage capacity is a valid performance metric for some stringent-latency communication systems operating over quasi-static fading channels.

Abstract

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This paper investigates the maximal achievable rate for a given blocklength and error probability over quasi-static multiple-input multiple-output (MIMO) fading channels, with and without channel state information (CSI) at the transmitter and/or the receiver. The principal finding is that outage capacity, despite being an asymptotic quantity, is a sharp proxy for the finite-blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver, at only one of them, or at neither of them. These results follow from analytically tractable converse and achievability bounds. Numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the blocklength increases. In the example of a particular $1 \times 2$ single-input multiple-output (SIMO) Rician fading channel, the blocklength required to achieve $90\%$ of capacity is about an order of magnitude smaller compared to the blocklength required for an AWGN channel with the same capacity. For this specific scenario, the coding/decoding schemes adopted in the LTE-Advanced standard are benchmarked against the finite-blocklength achievability and converse bounds.

I. INTRODUCTION

The paper asks whether outage capacity remains meaningful at finite blocklength for quasi-static MIMO fading channels, where fading is constant over each codeword. It develops achievability and converse bounds and shows zero dispersion across CSI configurations, supporting rapid convergence to outage capacity.

  • Channel model: Quasi-static fading keeps the random fading coefficients constant during each codeword because codewords are shorter than the channel coherence time.This model applies to slowly varying, delay-constrained communication systems.
  • Motivation: For fading distributions that can be arbitrarily small, Shannon capacity is zero because outage occurs with positive probability at every positive rate.Capacity versus outage, or ϵ-capacity, is therefore a more relevant metric when a positive block error probability is acceptable.
  • Contributions: The paper studies the maximal achievable rate R*(n, ϵ) for fixed blocklength and error probability over quasi-static MIMO channels under a per-codeword power constraint.The analysis covers channels with full CSIT and without CSIT, using converse bounds with perfect CSIR and achievability bounds without CSIR.
  • Main result: The achievability and converse bounds show that the 1/√n rate penalty is absent, so the ϵ-dispersion of quasi-static fading channels is zero under mild fading conditions.The result holds with perfect or no CSIT and independently of whether CSIR is available.
  • Numerical evidence: 120–320 channel uses suffice for 90% of Cϵ in the specified 1×2 SIMO Rician channel, about an order of magnitude fewer than for an AWGN channel with the same capacity.The comparison uses Cϵ = 1 bit/channel use and ϵ = 10^-3 in the perfect-CSIR case.
  • Implications and caveat: Outage-capacity-optimal strategies may perform well at finite blocklength, but this need not hold for very small n because the O(n^-1 log n) term may dominate.The paper notes that its bounds can differ from outage capacity in that regime.

III. ASYMPTOTIC RESULTS AND PREVIEW

The paper develops finite-blocklength bounds for quasi-static MIMO fading channels under different CSI configurations and shows that their asymptotic behavior establishes zero dispersion. The no-CSIT case is harder to characterize sharply, while isotropic-code and SIMO bounds permit tighter analysis and numerical evaluation.

  • Asymptotic results: The ϵ-capacity does not depend on receiver CSI because the constant channel can be estimated with pilots whose asymptotic rate penalty vanishes.This invariance applies to the quasi-static setting as blocklength grows.
  • CSI dependence: When multiple transmit antennas are available, spatial water-filling with CSIT yields larger ϵ-capacity than the no-CSIT case.For a single transmit antenna, the ϵ-capacity is unchanged by CSIT availability.
  • Asymptotic results: For CSIT, the paper provides achievability and converse bounds whose gap is O(log(n)/n) under mild fading-distribution conditions.The bounds establish the zero-dispersion result for the CSIT case.
  • Asymptotic results: For no CSIT, the asymptotic zero-dispersion result holds under slightly more stringent assumptions on the fading distribution.The relevant no-CSIT bounds are less sharp because the optimizing input covariance is generally unknown.
  • No CSIT: For no-CSIT Rayleigh fading, isotropic codewords are studied because isotropic ϵ-rate is often an accurate lower bound on ϵ-capacity.The paper notes that the optimal covariance is generally difficult to determine in this setting.
  • Isotropic codes: The paper characterizes isotropic-code performance more accurately with CSIR than in the general no-CSI case, under weaker fading-distribution conditions.This provides a more tractable finite-blocklength analysis for the isotropic-code setting.

IV. CSI AVAILABLE AT THE TRANSMITTER

For CSIT-enabled quasi-static MIMO channels, the paper develops geometric achievability and meta-converse bounds, including a threshold decoder that does not require CSIR. The bounds support finite-blocklength analysis, but some converse forms are difficult to evaluate or loose.

  • 1) Geometric Intuition:: For large n, the outage event corresponds to an angle θ(x1,Y) larger than θϵ, whereas nonoutage realizations produce a smaller angle.This follows from approximate orthogonality and norm concentration of the noise vector.
  • 1) Geometric Intuition:: A threshold decoder declares a codeword when it is the only one whose angle to the received signal is below θϵ, without requiring CSIR.The threshold θϵ is chosen from the outage-capacity geometry and must be selected appropriately for finite blocklength.
  • 2) The Achievability Bound:: The CSIT achievability theorem gives an (n, M, ϵ)tx code using independent Beta-distributed variables Bj and a threshold γn.The theorem is stated in terms of the eigenvalues of HHH and water-filling power gains.
  • 2) The Achievability Bound:: The MIMO achievability decoder compares the received and codeword subspaces using squared sine of their angle and accepts the first codeword below γn.Its analysis applies the κβ bound to a physically degraded channel whose output is span(Y).
  • B. Converse: With CSIT and CSIR, the converse uses the meta-converse theorem after transforming the MIMO channel into parallel quasi-static channels.The resulting bound optimizes over power-allocation functions based on the channel eigenvalues.
  • B. Converse: The converse bound is difficult to evaluate numerically because it contains an infimum over power-allocation functions, while a computable upper bound is generally loose.The paper notes that tighter numerical evaluation is possible in the SIMO or isotropic-code cases, where CSIT is not beneficial.

C. Asymptotic Analysis

The paper derives finite-blocklength asymptotic expansions showing zero dispersion for quasi-static fading channels under several CSI configurations. It also contrasts this convergence behavior with finite-state mixed channels and AWGN channels, while identifying conditions and higher-order effects that limit the approximation.

  • Zero dispersion: The ϵ-dispersion is zero for both the CSIRT and CSIT cases under conditions on fading moments, eigenvalue densities, and outage-capacity growth.The growth condition also implies continuity of the corresponding ϵ-capacity.
  • Zero dispersion: The ϵ-dispersion is zero for both the CSIR and no-CSI cases under the stated regularity conditions.The proof combines converse and achievability analyses of the finite-blocklength bounds.
  • Convergence rate: Finite-state mixed channels converge at O(1/√n), whereas the quasi-static expansion has an O(log(n)/n) remainder associated with zero dispersion.The comparison concerns the rate of convergence to ϵ-capacity.
  • Convergence rate: As the Rician factor K increases, higher-order terms become increasingly important as the fading distribution approaches a step function.This shows that zero dispersion alone does not guarantee fast convergence in the near-nonfading regime.

B. Converse

The converse analysis develops upper bounds for quasi-static MIMO fading channels with CSIR, including general MIMO, SIMO, and isotropic-codeword cases. The general bound is harder to evaluate, while SIMO specialization removes the benefit of CSIT and yields tighter bounds.

  • General MIMO converse: With CSIR but no CSIT, the maximal achievable rate is upper-bounded through an optimization involving the input covariance matrix and auxiliary random variables.The resulting infimum makes the bound more difficult to evaluate numerically and asymptotically.
  • General MIMO converse: The general CSIR converse remains difficult because even minimizing its asymptotic probability expression over Q is an open problem.The paper therefore treats SIMO and isotropic-codeword cases separately.
  • SIMO converse: In the SIMO case, CSIT is not beneficial, and the CSIR converse applies with or without transmitter channel information.The bound is specialized to one transmit antenna using G = ∥H∥2 and a scalar random variable Ln.
  • Isotropic-codeword converse: For isotropic codewords, CSIT is likewise not beneficial because the codewords are restricted to the isotropic set Fiso.A separate converse bound is established for this coding class.

C. Asymptotic Analysis

The asymptotic analysis establishes zero dispersion under regularity conditions for no-CSI, CSIR, CSIT, and CSIRT settings, with milder conditions in SIMO and isotropic-codeword cases. Numerical comparisons use a SIMO Rician example to examine convergence and finite-blocklength bounds.

  • General MIMO asymptotics: Theorem 9 establishes zero dispersion for the no-CSIT case under smoothness, tail, and outage-function conditions on the fading distribution.These conditions are satisfied by commonly used Rayleigh, Rician, and Nakagami models.
  • General MIMO asymptotics: The no-CSIT outage probability may fail to be differentiable when different capacity-achieving covariance matrices have different numbers of nonzero entries.This issue is identified for an i.i.d. Rayleigh MISO example.
  • Special cases: For SIMO channels, milder distributional conditions suffice for the zero-dispersion result, and CSIT is not beneficial.The result follows by specializing the general asymptotic theorem and a SIMO proposition.
  • Special cases: Isotropic-codeword channels also exhibit zero dispersion when the joint density of the nonzero eigenvalues is continuously differentiable.The corresponding outage probability function is used in the theorem.

VI. NUMERICAL RESULTS

Numerical evaluations show fast convergence of finite-blocklength rates toward outage capacity in several quasi-static fading scenarios, while LTE-Advanced turbo codes retain a roughly constant suboptimality gap.

  • MIMO Rayleigh fading: For the MIMO Rayleigh channel with two transmit and three receive antennas, the bounds and normal approximation are compared at ε = 10^-3 and SNR = 2.12 dB.The selected SNR gives C_iso,ε = 1 bit/(ch. use).
  • SIMO Rician fading: The corresponding AWGN channel requires approximately 1420 channel uses to achieve 90% of capacity, validating fast convergence predicted by zero dispersion.The AWGN reference has C = 1 bit/(ch. use).
  • MIMO Rayleigh fading: The blocklength required to achieve 90% of C_iso,ε is less than 500, demonstrating fast convergence to outage capacity.
  • Comparison with coding schemes in LTE-Advanced: LTE-Advanced turbo codes use QPSK and 10 max-log-MAP decoding iterations, and their performance is benchmarked against the achievability and converse bounds.The analysis assumes perfect CSI at the decoder.
  • Comparison with coding schemes in LTE-Advanced: Between blocklengths 500 and 1000, LTE-Advanced codes achieve about 85% of the maximal coding rate, while code suboptimality remains a roughly constant gap.For ϵ = 10^-1, HARQ compensates for outage-related packet loss, and the turbo-code gap remains constant.

APPENDIX II PROOF OF THEOREM 1 (CSIT ACHIEVABILITY BOUND)

The proof develops an achievability bound for CSIT by transforming the channel geometrically, constructing unitary codewords, and evaluating a hypothesis-testing bound through Grassmannian distributions.

  • Code construction: Unitary matrices from the complex Stiefel manifold generate codewords satisfying the power constraint.The construction uses codewords of the form Xj(H) built from unitary Φj and a channel-dependent precoding matrix P(H).
  • Auxiliary channel: A physically degraded Grassmannian-output channel provides a lower bound on the achievable rate of the original channel.The output subspace belongs almost surely to the Grassmannian manifold, and degradation preserves the lower-bound relationship.
  • Hypothesis testing: The κβ bound is applied with an auxiliary output distribution and channel-output distributions indexed by the transmitted subspace.The proof then lower-bounds the resulting expression for numerical evaluation.
  • Distributional evaluation: Isotropic noise makes the relevant principal-angle distribution independent of the transmitted subspace, allowing the analysis to fix Φ = I_n,t.Under the uniform auxiliary distribution, the angle statistics are represented through a complex multivariate Beta matrix and products of independent Beta variables.
  • Channel transformation: The channel is reduced using the singular value decomposition of H into parallel quasi-static channels with independent Gaussian noise vectors.The resulting subchannels correspond to the largest eigenvalues of H^H H.
  • Converse comparison: The converse analysis reduces detection to normalized column energies, which form a sufficient statistic with conditionally independent Gamma-distributed components.The resulting bound depends on a code only through its induced power-allocation function and is integrable independently of the input values.

A. Proof of (169)

The proof establishes the required finite-blocklength bound by applying a refined Cramer-Esseen expansion and controlling the power-allocation-dependent quantities across fading regions.

  • Expansion tool: A Berry-Esseen expansion with an O(1/√n) remainder is insufficient because the proof requires an O(1/n) remainder.The argument therefore invokes a stronger Cramer-Esseen theorem.
  • Probabilistic conditions: The random variables used in the bound are conditionally i.i.d. with zero mean and unit variance given the channel eigenvalues.Their characteristic functions are then shown to satisfy the theorem’s uniform decay condition.
  • Power allocation: The minimization over power-allocation functions is difficult because the relevant objective is neither convex nor concave.A lower bound from Lemma 16 is sufficient for the proof.
  • Region decomposition: Different numbers of active eigenmodes make the density nonsmooth, so the eigenvalue space is partitioned into regions with fixed active-mode counts.Within each region’s interior, the density is continuously differentiable.
  • Conclusion: The resulting bounds control the transition probability uniformly near the threshold and yield the claimed inequality.The conclusion follows by combining the intermediate bounds with the main finite-blocklength expression.

D. Proof of Lemma 16

The proof of Lemma 16 compares the minimizing power allocation with water filling and bounds the standardized threshold according to its distance from the water-filling mean.

  • Water-filling reference: Water filling uniquely maximizes the mean term μ(v, λ) over the feasible power-allocation set.This identifies the reference allocation v* used throughout the lemma.
  • Optimization conditions: The minimizer v_min satisfies the KKT conditions and uses the full available power.Monotonicity in each allocation coordinate implies that the sum of its components equals ρ.
  • Allocation comparison: The difference between the minimizing allocation and water filling is bounded by a finite constant independent of λ, v_min, v*, and γ.This stability estimate supports the subsequent control of the variance term.
  • Variance control: The proof bounds the variance at the minimizing allocation above or below the water-filling variance according to the sign of γ − μ*(λ).These bounds are combined with the allocation comparison to establish the lemma.
  • Regularity: The same decomposition handles the nonsmooth water-filling boundaries by analyzing regions with fixed active eigenmodes.The resulting density regularity permits the asymptotic expansion used in the achievability analysis.

A. Proof of Lemma 18

Lemma 18 controls probabilities involving normalized noisy quadratic forms by decomposing them into cases and applying a smoothing lemma under bounded density conditions.

  • Noise concentration: The chi-square behavior of the accumulated noise terms supplies the concentration estimates used in the conditional probability bounds.The argument also exploits independence of the Beta variables and boundedness of the normalized components.
  • Case decomposition: The proof decomposes the target probability into terms p0, p1, and p2 according to whether G1 exceeds the threshold g_th.Separate lower bounds are then established for the resulting terms.
  • Conditional regularity: The auxiliary random variables satisfy the smoothness conditions needed for Lemma 17 after conditioning on G1 < g_th.The conditional density and its derivatives are bounded on the relevant bounded sets.
  • Conclusion: Combining the casewise estimates proves the intermediate inequalities and completes Lemma 18.The final steps substitute the bounds for p1 and p2 into the decomposition.

APPENDIX VI PROOF OF PROPOSITION 5 (EXISTENCE OF OPTIMAL

The appendix establishes continuity and derives converse and asymptotic-expansion results using symmetry reductions, auxiliary channels, and probability bounds.

  • Existence of optimal input covariance: Continuity of the relevant function in Q follows from convergence in probability and continuity of the fading-induced distribution.Absolute continuity of H makes the cdf of log det(I_r + HHQH) continuous for every real threshold.
  • Converse bound: The CSIR converse evaluates β_{1−ε}(X) through channel symmetries and shows dependence on each codeword only through U(X).A QR decomposition and the Neyman–Pearson lemma identify the relevant distributions under the true and auxiliary channels.
  • Converse bound: The meta-converse uses an auxiliary channel whose output depends on X through U(X), then lower-bounds its error probability for every code.Lemma 19 provides the lower bound for codes with n ≥ r, and the resulting upper bound is code-independent.
  • Asymptotic expansion: The converse asymptotic expansion is obtained by combining conditional i.i.d. sums, uniform O(1/n) control, and regularity properties of the induced density.Lemma 21 supplies the required local differentiability and convergence properties for the density and its derivative.

A. Proof of Lemma 21

The proof of Lemma 21 establishes smoothness and convergence of the density induced by the fading transformation, using geometric measure arguments and uniform bounds.

  • A. Proof of Lemma 21: Lemma 22 expresses the density of a smooth transformation through level sets on an oriented Riemannian manifold.The proof uses the smooth coarea formula and, for derivatives, Stokes’ theorem under compact-support conditions.
  • 1) Uniform Boundness of {f_l} and {f'_l}: Uniform bounds on the conditional densities and their derivatives are established over expanding subsets of the fading-matrix space.The construction uses compactness, continuity of the fading density, geometric bounds, and the assumption controlling tails.
  • 1) Uniform Boundness of {f_l} and {f'_l}: The gradient of the transformation is bounded away from zero locally by determinant expansions and eigenvalue estimates.Choosing δ sufficiently small makes the matrix T invertible, enabling a lower bound on the Frobenius norm of the gradient.
  • 2) Convergence of f_l(u) and f'_l(u): The conditional densities form a Cauchy sequence in C1([−δ,δ]) and converge uniformly, including convergence of their derivatives.The limiting density f_U(γ,Q) is continuously differentiable on the interval.

APPENDIX IX PROOF OF THE ACHIEVABILITY PART OF THEOREM 9

The appendix proves the achievability part of Theorem 9 and uses symmetry and eigenvalue-based analysis to construct the required asymptotic expansion.

  • APPENDIX IX PROOF OF THE ACHIEVABILITY PART OF THEOREM 9: The achievability proof obtains a parameter γ_n = exp(−C_no ε + O(1/n)) satisfying the target bound.The construction relies on a lemma and symmetry arguments analogous to earlier achievability steps.
  • Input construction: The chosen input structure uses U satisfying U^H U = Q^* and aligns with the dominant eigenvalues of HHQ^*H.The decomposition separates unitary factors, the largest m^* eigenvalues, and zero blocks.
  • Scope of assumptions: The conditions required for Proposition 23 are less stringent than those used for the converse part of Theorem 9.The paper states that the stronger converse conditions imply the conditions of Proposition 23.
  • Asymptotic evaluation: The proof evaluates the converse bound in the large-n limit using ordered nonzero eigenvalues and conditional sum approximations.The analysis averages the resulting bounds over the eigenvalue distribution and invokes continuous differentiability of its joint density.
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