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Unified Upper Bounds on the ML decoding Error Probability of Spinal Codes over Fading Channels

Aimin Li, Xiaomeng Chen, Shaohua Wu, Gary C. F. Lee, Sumei Sun

TL;DR

This work develops two deterministic finite-blocklength BLER upper bounds for Spinal codes over fading channels, unifying analysis across complex and real mappings. The first bound adapts a Gallager-based approach to fading, while a refined bound proves tighter by exploiting Spinal-code structure and a new F(L_a,σ,γ) term. The framework yields explicit closed-form bounds for Nakagami-m and Rician fading, and demonstrates that tail transmission pattern (TTP) is optimal under ML decoding. Simulations confirm the bounds are tight and that the refined bound significantly outperforms the Gallager-based one, providing practical guidance for evaluating and designing Spinal codes in diverse fading environments.

Abstract

Performance evaluation of particular channel coding has been a significant topic in coding theory, often involving the use of bounding techniques. This paper focuses on the new family of capacity-achieving codes, Spinal codes, to provide a comprehensive analysis framework to tightly upper bound the block error rate (BLER) of Spinal codes in the finite block length (FBL) regime. First, we resort to a variant of the Gallager random coding bound to upper bound the BLER of Spinal codes over the fading channel. Then, this paper derives a new bound without resorting to the use of Gallager random coding bound, achieving provable tightness over the wide range of signal-to-noise ratios (SNR). The derived BLER upper bounds in this paper are generalized, facilitating the performance evaluations of Spinal codes over different types of fast fading channels. Over the Rayleigh, Nakagami-m, and Rician fading channels, this paper explicitly derived the BLER upper bounds on Spinal codes as case studies. Based on the bounds, we theoretically reveal that the tail transmission pattern (TTP) for ML-decoded Spinal codes remains optimal in terms of reliability performance. Simulations verify the tightness of the bounds and the insights obtained.

Unified Upper Bounds on the ML decoding Error Probability of Spinal Codes over Fading Channels

TL;DR

This work develops two deterministic finite-blocklength BLER upper bounds for Spinal codes over fading channels, unifying analysis across complex and real mappings. The first bound adapts a Gallager-based approach to fading, while a refined bound proves tighter by exploiting Spinal-code structure and a new F(L_a,σ,γ) term. The framework yields explicit closed-form bounds for Nakagami-m and Rician fading, and demonstrates that tail transmission pattern (TTP) is optimal under ML decoding. Simulations confirm the bounds are tight and that the refined bound significantly outperforms the Gallager-based one, providing practical guidance for evaluating and designing Spinal codes in diverse fading environments.

Abstract

Performance evaluation of particular channel coding has been a significant topic in coding theory, often involving the use of bounding techniques. This paper focuses on the new family of capacity-achieving codes, Spinal codes, to provide a comprehensive analysis framework to tightly upper bound the block error rate (BLER) of Spinal codes in the finite block length (FBL) regime. First, we resort to a variant of the Gallager random coding bound to upper bound the BLER of Spinal codes over the fading channel. Then, this paper derives a new bound without resorting to the use of Gallager random coding bound, achieving provable tightness over the wide range of signal-to-noise ratios (SNR). The derived BLER upper bounds in this paper are generalized, facilitating the performance evaluations of Spinal codes over different types of fast fading channels. Over the Rayleigh, Nakagami-m, and Rician fading channels, this paper explicitly derived the BLER upper bounds on Spinal codes as case studies. Based on the bounds, we theoretically reveal that the tail transmission pattern (TTP) for ML-decoded Spinal codes remains optimal in terms of reliability performance. Simulations verify the tightness of the bounds and the insights obtained.
Paper Structure (45 sections, 25 theorems, 83 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 45 sections, 25 theorems, 83 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Related Works and Motivations
  4. Main Results and Contributions
  5. Notations
  6. Preliminaries
  7. Encoding Process of Spinal Codes
  8. Channel Model
  9. Bound Based on Gallager's Results
  10. Gallager Bound over the Fading Channel
  11. Upper Bound on the BLER of Spinal Codes
  12. Refined Upper Bound
  13. Main Result
  14. The Refined Upper Bound is Tighter
  15. Proof of Theorem \ref{['coretheorem']}
  16. ...and 30 more sections

Key Result

Theorem 1

(Relaxed Gallager Bound for Fading Channels) For channel codes with codelength $L$, code rate $R$, and channel input set $\Psi$ transmitted over the fading channel with AWGN variance $\sigma^2$, the average BLER under ML decoding with perfect channel state information (CSI) is upper bounded by: with $\gamma=1$ for complex fading channels and $\gamma=2$ for real fading channelsReal fading channels

Figures (5)

  • Figure 1: The encoding process of Spinal codes.
  • Figure 2: The rule of Right Rieman sum.
  • Figure 3: A dynamic solution process of Algorithm \ref{['Algorithm 1']}. Parameter is set as $n=8$, $k=2$, $r=3$, and $N=19$.
  • Figure 4: Upper Bounds vs. Monte Carlo Simulations (MCS). BLER of Spinal codes with $n = 8, v = 32, pass = 6, c = 8$ and $k = 2$ over complex fading channels with $\Omega = 1$.
  • Figure 5: Upper bounds under different parameters setup. Here $n = 8, v = 32, pass = 6, c = 8, k = 2$ and $\Omega = 1$. The upper bounds are obtained by Theorem \ref{['coretheorem']}.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Lemma 1
  • Corollary 1
  • Theorem 4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 19 more