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Optimal Linear MAP Decoding of Convolutional Codes

Yonghui Li, Chentao Yue, Branka Vucetic

TL;DR

This work introduces linear MAP decoding (LMAP) for rate-1/2 convolutional codes by reformulating BCJR MAP decoding as a pair of dual SISO encoders realized with shift registers. Forward and backward MAP processes are implemented via four-branch, shift-register decoders and offline-designed memory-labeling via a Recursive Label Synthesizer, enabling exact bidirectional MAP outputs with reduced latency. Empirical results show LMAP achieving BCJR-equivalent performance while dramatically lowering complexity and memory-access overhead, yielding substantial speedups and facilitating decoding of very large-state codes (VLSC) near capacity. The approach yields hardware-friendly, low-delay MAP decoding suitable for practical high-performance communication systems and short-blocklength applications.

Abstract

In this paper, we propose a linear representation of BCJR maximum a posteriori probability (MAP) decoding of a rate 1/2 convolutional code (CC), referred to as the linear MAP decoding (LMAP). We discover that the MAP forward and backward decoding can be implemented by the corresponding dual soft input and soft output (SISO) encoders using shift registers. The bidrectional MAP decoding output can be obtained by combining the contents of respective forward and backward dual encoders. Represented using simple shift-registers, LMAP decoder maps naturally to hardware registers and thus can be easily implemented. Simulation results demonstrate that the LMAP decoding achieves the same performance as the BCJR MAP decoding, but has a significantly reduced decoding delay. For the block length 64, the CC of the memory length 14 with LMAP decoding surpasses the random coding union (RCU) bound by approximately 0.5 dB at a BLER of $10^{-3}$, and closely approaches both the normal approximation (NA) and meta-converse (MC) bounds.

Optimal Linear MAP Decoding of Convolutional Codes

TL;DR

This work introduces linear MAP decoding (LMAP) for rate-1/2 convolutional codes by reformulating BCJR MAP decoding as a pair of dual SISO encoders realized with shift registers. Forward and backward MAP processes are implemented via four-branch, shift-register decoders and offline-designed memory-labeling via a Recursive Label Synthesizer, enabling exact bidirectional MAP outputs with reduced latency. Empirical results show LMAP achieving BCJR-equivalent performance while dramatically lowering complexity and memory-access overhead, yielding substantial speedups and facilitating decoding of very large-state codes (VLSC) near capacity. The approach yields hardware-friendly, low-delay MAP decoding suitable for practical high-performance communication systems and short-blocklength applications.

Abstract

In this paper, we propose a linear representation of BCJR maximum a posteriori probability (MAP) decoding of a rate 1/2 convolutional code (CC), referred to as the linear MAP decoding (LMAP). We discover that the MAP forward and backward decoding can be implemented by the corresponding dual soft input and soft output (SISO) encoders using shift registers. The bidrectional MAP decoding output can be obtained by combining the contents of respective forward and backward dual encoders. Represented using simple shift-registers, LMAP decoder maps naturally to hardware registers and thus can be easily implemented. Simulation results demonstrate that the LMAP decoding achieves the same performance as the BCJR MAP decoding, but has a significantly reduced decoding delay. For the block length 64, the CC of the memory length 14 with LMAP decoding surpasses the random coding union (RCU) bound by approximately 0.5 dB at a BLER of , and closely approaches both the normal approximation (NA) and meta-converse (MC) bounds.

Paper Structure

This paper contains 14 sections, 2 theorems, 30 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Linear Representation of MAP Decoding for Rate-1 Convolutional Codes
  3. LMAP Decoding: A 4-State RSC Example
  4. LMAP Decoding of A General Rate 1/2 RSC
  5. LMAP Decoder Structure
  6. LMAP Decoding Process
  7. Register Updates
  8. Output Generation
  9. Complexity Analysis
  10. Simulation Results
  11. Recursive Systematic Codes
  12. Non-Systematic Codes
  13. Decoding Efficiency
  14. Conclusion

Key Result

Theorem 1

For a rate-1 CC, generated by a $g_f(x)=\frac{a_f(x)}{q_f(x)}$, let $z_f(x)$ be the degree-$l$ MCP of $a_f(x)$. The log-domain SISO forward decoding of the code can be simply implemented by its dual encoder with the generator polynomial of where the decoder inputs and outputs are $\hat{x}_{c_k}$ and $\hat{x}_{b_k}$, which are the SSE of received coded symbol ${x_k}$ and information symbol ${b_k}$

Figures (7)

  • Figure 1: The trellis and encoder of $(1, 7_{\mathrm{oct}}/5_{\mathrm{oct}})$
  • Figure 2: The LMAP decoder (forward decoding) of $(1, 7_{\mathrm{oct}}/5_{\mathrm{oct}})$
  • Figure 3: The LMAP decoder of a general rate-$\frac{1}{2}$ RSC
  • Figure 4: Recursive label synthesizer of the LMAP decoding for $\mathbf{g}(x)$
  • Figure 5: BER performance comparison for decoding RSC.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1: Linear Representation of MAP Forward Decoding ETT1
  • Definition 2: Reversed Polynomial
  • Theorem 2: Linear Representation of MAP Backward Decoding ETT1
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Remark 4