Joint Source-Channel Coding of Gaussian Sources over Block Erasure Channels: Nonasymptotic Bounds and Channel-Uniform Normal Approximations

I. INTRODUCTION

• The Gaussian-source, block-erasure-channel achievability bound exactly evaluates the scheme’s ensemble-average excess-distortion probability and improves corresponding specializations of general one-shot bounds.Unlike one cited specialization, it requires no optimization over an auxiliary scalar parameter.Source: §I · p. 1
• Conditioning the converse on erasure pattern and source energy yields spherical-cap and volume bounds, with the spherical-cap term supplying the matching third-order term.The bound is at least as strong as the specified Lebesgue-measure specialization of the list-code converse.Source: §I · p. 2
• The information-balance condition compares channel information with the information required to describe the source at the target distortion, with dispersion terms representing fluctuations around their first-order quantities.The channel-uniform result avoids a separate large-n approximation of the channel.Source: §I · p. 2
• The sufficient and necessary information-balance conditions share the first-order, dispersion, and 1/2 log k terms, with bounded remainders uniform over channel blocklength and erasure profile.The results hold for fixed distortion ratio, target excess-distortion probability, and block size.Source: §I · p. 2
• The numerical evaluation compares the nonasymptotic JSCC bounds and their common third-order approximation with an SSCC benchmark optimized over the interface size.The SSCC benchmark uses a symmetrized fixed-length digital interface and evaluates the specified construction’s exact ensemble-average excess-distortion probability.Source: §I · p. 3

II. PRELIMINARIES

• The notation uses natural logarithms, the Euclidean norm, chi-square CDFs, the standard normal CDF, and ⊥⊥ for independence.The text also specifies log(0) = −∞ for its information-density definitions.Source: §II · p. 3
• The fundamental limits are governed by coupling between the channel information density and the negative logarithm of the source distortion-ball probability.The paper calls these quantities the channel and source random variables, respectively.Source: §II · p. 4

A. Channel Random Variable

• A block erasure channel has binary m-bit inputs, outputs either the input block or an erasure symbol, and erases with probability p.The channel alphabet is {0, 1}^m, with the erasure symbol added to the output alphabet.Source: A. Channel Random Variable · p. 4
• For nonstationary channels, the joint channel law is the product of the per-block laws with erasure probabilities p1 through pn.The resulting channel-output marginal is likewise a product distribution.Source: A. Channel Random Variable · p. 4
• The count of unerased blocks has a Poisson-binomial distribution with parameters 1−p1 through 1−pn.The paper denotes its probability mass function by {πj}.Source: A. Channel Random Variable · p. 4

B. Source Random Variable

• For Gaussian sources under mean-squared error, the source has variance σ² and the reproduction distortion is k⁻¹ times the squared Euclidean error.For distortion d below σ², the rate-distortion-achieving backward representation uses independent Gaussian reproduction and noise components.Source: B. Source Random Variable · pp. 4–5
• The source random variable is the negative log probability that a random reproduction falls in the distortion ball around the source realization.For spherical reproduction distributions, this probability is the spherical-cap fraction of the reproduction sphere.Source: B. Source Random Variable · pp. 5–6
• The exact finite-dimensional single-codeword coverage probability depends on the source sequence through its normalized energy.With independent random reproduction codewords, the random-coding analysis is governed by the magnitude of the negative log of that coverage probability.Source: B. Source Random Variable · p. 6
• A threshold-uniform normal approximation applies to functions of normalized Gaussian source energy that admit a local quadratic expansion near energy 1.The function need not be differentiable or monotone, and its behavior outside the specified neighborhood is unrestricted; a bounded perturbation is permitted.Source: B. Source Random Variable · pp. 6–7
• The spherical-cap specialization gives a robust, threshold-uniform normal approximation for the negative log cap probability, with Kolmogorov error of order k^-1/2.The same approximation remains valid after adding a perturbation uniformly bounded near normalized energy 1, allowing the threshold to be replaced by the independent channel random variable.Source: B. Source Random Variable · p. 7
• The spherical-cap approximation provides a uniform-over-threshold form needed for the subsequent channel-uniform conditioning argument.The cited source-coding comparisons provide one-sided high-probability refinements or fixed-target rate expansions, rather than this threshold-uniform form.Source: B. Source Random Variable · p. 7
• The source-coding setup defines an excess-distortion event by squared reconstruction error exceeding kd, with target probability at most ε.Achievability and converse results respectively give sufficient and necessary conditions for a code to meet that criterion.Source: B. Source Random Variable · p. 8

Coding scheme

• The achievability code uses independently generated channel and reproduction sequences, with uniform channel inputs and a spherical reproduction distribution.The codebook is shared between encoder and decoder and is independent of the source.Source: §III · pp. 8–9

Non-asymptotic achievability

• Theorem 1’s bound exactly evaluates the scheme’s ensemble-average excess-distortion probability and is stronger than direct specializations of general one-shot bounds.The tightening can matter at short blocklengths, but is asymptotically negligible for typical source realizations.Source: §III · p. 9

General source adaptation

• The scheme extends to arbitrary source and reproduction distributions and measurable distortion functions, with an exact ensemble-average excess-distortion expression.Proposition 2 states the extension for arbitrary Borel probability measures and measurable distortion functions.Source: §III · p. 9

Semi-asymptotic achievability

• For fixed distortion ratio, target excess-distortion probability, and block size, Theorem 2 gives a sufficient information-balance condition with constants uniform over channel blocklengths and erasure profiles.The constants and source-blocklength threshold depend only on the fixed distortion ratio, target probability, and block size.Source: §III · pp. 10–11

Uniform approximation method

• The channel-uniform approximation combines the channel fluctuation with the Gaussian source fluctuation before applying Berry–Esseen, preserving a uniform O(k^-1/2) error.The resulting constants do not depend on channel blocklength or the erasure-probability profile.Source: §III · pp. 10–11

Third-order refinement

• The achievability analysis has a 1/2 log k + O(1) third-order term, and the paper’s achievability and converse conditions match through logarithmic third order.The paper attributes the refinement to specializing the analysis to the Gaussian-source/block-erasure-channel pair; the remaining 1/2 log k term is necessary for this pair.Source: §III · pp. 11–12

IV. CONVERSE RESULTS

• Conditioning on the erasure pattern and normalized source energy turns the Gaussian source into a sphere covered by at most 2^mj reproduction caps.A union bound over reproductions and averaging over source energy and erasures gives a necessary code-existence condition; a separate total-volume bound supplies the second term.Source: §IV · p. 13
• The spherical-cap component alone yields the 1/2 log k third-order term, matching the achievability result.Theorem 3 also retains a volume component that can strengthen the nonasymptotic bound.Source: §IV · pp. 13–14
• The volume component can strengthen the nonasymptotic converse and makes it at least as strong as the Lebesgue-measure list-code specialization.The volume argument replaces the union of distortion balls with a centered ball whose volume is 2^mj times that of one distortion ball, relaxing the angular covering constraint.Source: §IV · pp. 13–14
• The sufficient and necessary conditions share the logarithmic third-order term, with bounded remainders uniform over channel blocklength and erasure profile.Only the source blocklength must exceed a threshold depending on distortion ratio, target excess-distortion probability, and block size; for fixed block size, the necessary condition still forces channel blocklength to grow along feasible sequences with k.Source: §IV · p. 14
• The matching 1/2 log k term arises from spherical-cap probabilities with the same centering, kR(d) + 1/2 log k.The achievability analysis uses a fixed reproduction sphere, while the converse maximizes covered fraction over reproduction norms.Source: §IV · p. 14
• The converse uses an exponential random-threshold representation and obtains a 1/2 log k − O(1) correction without showing that BLEC has a smaller optimal third-order term than BSC or AWGN.The comparison bounds for BMS–BSC and GMS–AWGN do not identify their optimal third-order terms.Source: §IV · p. 14

V. COMPARISON WITH SEPARATE SOURCE–CHANNEL CODING

• The SSCC setup first describes the source and channel codes separately, then analyzes their concatenation through a shared random permutation.For M messages, [M] denotes {1, . . . , M}.Source: §V · p. 14

A. Source Code

• A source code maps R^k to one of M indices and maps that index back to a reproduction in R^k.It meets the (d, ϵ_s, M, k) criterion when the probability that squared reconstruction error exceeds kd is at most ϵ_s.Source: A. Source Code · p. 14
• The codebook entries are independently drawn from the uniform spherical distribution and shared by encoder and decoder.The codebook is independent of the source; an excess-distortion event occurs exactly when no reproduction sequence covers the source realization.Source: A. Source Code · pp. 14–15
• The ensemble-average excess-distortion probability of the spherical random source code is evaluated exactly.A deterministic (M, k) source code exists with excess-distortion probability at most A_s(M; k, d).Source: A. Source Code · p. 15
• Every fixed reproduction vector with squared norm k(σ^2 − d) has the stated covering probability.The construction uses the same reproduction distribution as Theorem 1 and the spherical random-coding construction of [10, Theorem 37], but uses the exact spherical-cap probability rather than a lower bound.Source: A. Source Code · p. 15

B. Channel Code

• The codebook entries are independently drawn uniformly from X^n and shared by the channel encoder and decoder.The codebook is independent of both the message and channel erasures.Source: B. Channel Code · p. 15
• The decoder selects the first codeword matching every unerased block; the transmitted codeword guarantees that this set is nonempty.If no match exists, the decoder outputs 1, though that case cannot occur when a codeword traverses the BLEC.Source: B. Channel Code · p. 15
• The ensemble-average error probability under a uniform message is 1 − H_A(M; n).There exists a deterministic (n, M) code with uniform-message average error probability at most that value.Source: B. Channel Code · p. 15

C. Separate Source–Channel Code

• A shared uniform permutation makes each source index’s channel label uniform, enabling use of the uniform-message channel-code success probability despite nonuniform source indices.The permutation is independent of the source and channel erasures and is shared by encoder and decoder.Source: C. Separate Source–Channel Code · p. 16
• For M = 1, the construction uses zero as the reproduction, and the code criterion averages excess-distortion probability over the permutation.For M ≥ 2, the source and channel codebooks are generated independently.Source: C. Separate Source–Channel Code · p. 16
• Proposition 7 gives the exact ensemble-average excess-distortion probability for the symmetrized SSCC construction and an existence condition for deterministic component codes.The probability averages over the source, erasures, independent component codebooks, and shared permutation.Source: C. Separate Source–Channel Code · p. 16
• The exact calculation does not count every channel decoding error as excess distortion when the final reproduction still meets the distortion constraint.Counting only joint source coverage and correct index decoding gives a simpler but weaker sufficient condition.Source: C. Separate Source–Channel Code · pp. 16–17
• Unlike the additive union-bound form in [5, Theorem 6], the SSCC bound retains the joint event that the source is covered and its selected index is decoded correctly.The shared permutation also distinguishes this construction from a criterion based on worst-case average channel error over message distributions.Source: C. Separate Source–Channel Code · pp. 16–17

VI. NUMERICAL EVALUATION

• The converse-certified source blocklength kC(n) is nondecreasing in n for a fixed erasure-probability sequence, while kC(n)/n need not be.The bounds certify kA(n) ≤ k⋆(n) ≤ kC(n), with kA achievable and kC not ruled out by the converse.Source: §VI · p. 17
• The common third-order approximation omits bounded remainder constants, so it is neither a certified achievability nor converse bound and need not lie between kA(n) and kC(n) at finite blocklength.For the plotted parameters, it lies above the converse curve.Source: §VI · pp. 17–18
• For n ≥ 3, the JSCC achievable rate exceeds the symmetrized SSCC achievable benchmark in Figure 1.With m = 100, σ = 1, d = 0.1, ϵ = 0.01, and stationary p = 0.1, the JSCC achievability and converse curves are relatively close except at the shortest blocklengths; this compares specified random-coding constructions, not optimal SSCC performance.Source: §VI · p. 18, Figure 1 · p. 18

APPENDIX A PROOF OF LEMMA 1

• The Appendix A proof reduces a spherical-cap probability to the distribution of the first coordinate of a uniformly distributed point on the unit sphere.It treats the one-dimensional sphere separately and rewrites the integral using the regularized incomplete beta function.Source: Appendix A · pp. 18–19

APPENDIX B PROOF OF PROPOSITION 1

• The Appendix B proof transfers a normal approximation through auxiliary increasing functions and uses them to bracket the distribution of cWk.The argument combines the transformed bounds with a control on the event G and concludes the stated approximation result.Source: Appendix B · p. 21
• Under the stated local boundedness condition, the Appendix B specialization applies the proposition with constants depending only on δ, η1, k1, and C1.The proof first restricts the source-energy variable to an interval where the cap parameter remains in a compact subinterval of (0, 1).Source: Spherical-cap specialization… · pp. 22–23

APPENDIX C PROOF OF THEOREM 1

• Theorem 1’s ensemble-average excess-distortion analysis yields a sufficient condition for a (d, ϵ, n, k) source–channel code.The conditional probability depends on the erasure pattern only through the number of unerased blocks and on the source realization only through its normalized energy; some codebook realization performs no worse than the ensemble average.Source: Appendix C · p. 25

APPENDIX D PROOF OF THEOREM 2

• Condition (23) is sufficient for an optimal source–channel code, with constants depending only on δ, ϵ, and m.The proof’s approximation constants are uniform in channel blocklength n and erasure probabilities p1, …, pn because it approximates only the source term and retains the channel term’s distribution exactly.Source: Appendix D · p. 27

APPENDIX E PROOF OF THEOREM 3

• Theorem 3’s converse bounds reproduction by combining a uniform sphere-covering bound with a volume bound, then averaging over erasures.Conditioning on the erasure pattern gives at most 2^mj decoder outputs; the resulting bound extends from deterministic to randomized codes and yields necessary condition (32).Source: Appendix E · pp. 29–30

A. Comparison with the list-code converse

• Every parameter choice satisfying (32) also satisfies the specialization of the list-code converse in [5, Theorem 5], so this necessary condition is at least as strong.The comparison maps each source–channel code to a list code with list volume vk,d.Source: A. Comparison with the list-code… · p. 31
• Every parameter choice ruled out by the Kostina–Verdú specialization is also ruled out by condition (32).The proof constructs an admissible test whose Q-integral equals the spherical-cap volume, establishing the implication between the converse conditions.Source: A. Comparison with the list-code… · pp. 32–33

APPENDIX F PROOF OF THEOREM 4

• The proof approximates the source term near unit source energy and applies Berry–Esseen to Gaussian and centered Bernoulli summands to obtain source-induced Gaussian smoothing.The stated constants in the intermediate bound depend only on δ and m, not on n or the erasure-probability profile.Source: Appendix F · pp. 33–35
• The proof represents the spherical-cap bound as a tail event involving the source term, unerased-block count, and an independent unit-mean exponential threshold.It writes Bn,k as E[P[Ak(Υ) > Jn + E | Jn, E]], retaining the channel-term distribution exactly in the ensuing bound.Source: Appendix F · p. 34
• Uniform inversion at the target probability yields constants C and k0 depending only on δ, ϵ, and m.The proof handles separate cases for the mean unerased-block count before choosing these constants.Source: Appendix F · pp. 35–36

• For an erasure pattern with j unerased blocks, message t is decoded correctly with probability (1 − 2^-mj)^(t−1); averaging gives HA(M; n), and some deterministic channel codebook achieves at least that ensemble average.Source: Appendix G · p. 36
• Uniform permutation averaging makes label images uniform, and distinct image pairs uniform, yielding the exact ensemble-average excess-distortion identity and a deterministic codebook pair no worse than that average.Source: Appendix G · pp. 37–38
• The weaker sufficient condition follows from integrating 1 − (1 − ρ)^M ≤ Mρ for 0 ≤ ρ ≤ 1.Source: Appendix G · p. 38
• The minimum in (50) is attained at a finite interface size because sufficiently large sizes give a value above the M = 1 choice.Source: Appendix G · p. 38