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Equivalence of Serial and Parallel A-Posteriori Probabilities in the Decoding of DAB Systems

Andrea Di Giusto, Wim van Houtum, Alberto Ravagnani, Yan Wu

TL;DR

The paper tackles whether a-posteriori probabilities (APPs) for information and encoded bits remain unchanged when decoding a CIF composed of multiple convolutional codewords as a single serial codeword in digital audio broadcasting (DAB) systems. It proves a general theorem: if each parallel encoder terminates in the zero state (tail bits are known) and codeword lengths are known, then for all positions $t$ the serial APP $P(b_t=0|Y)$ equals the parallel APP $P(b_t=0|Y_{i(t)})$, with analogous results for encoder outputs; the proof uses a Markov-state model of the encoder and BCJR/LLR reasoning. The authors extend the result to practical decoding tools (BCJR and Viterbi) and show that the serial approach yields identical information about the bits to be decoded, even under puncturing and interleaving. Simulations in both a generic parallel/punctured setup and a DAB-like DE-QPSK/OFDM framework corroborate the theoretical findings, demonstrating BER curves that are unchanged when decoding serially versus in parallel. The work supports treating the CIF as one long codeword in iterative decoding, with potential performance gains for DAB receivers and guidance for future exploration of CIF-based decoding strategies.

Abstract

Motivated by applications to digital audio broadcasting (DAB) systems, we study the a-posteriori probabilities (APPs) of the coded and information bits of the serial concatenation of multiple convolutional codewords. The main result of this correspondence is a proof that the APPs of the input bits do not change when considering the concatenation of multiple codewords as a received sequence. This is a purely theoretical result, which remains valid for every convolutional code, as long as the encoder goes back to the zero state at the end of each codeword. An equivalent heuristic for serial concatenation in Viterbi decoding is described. The applicability of our result to DAB systems, where interleaving and modulation are accounted for, is investigated through Matlab simulations. We show that the Bit Error Rate (BER) of the simulated DAB system does not change when decoding multiple transmitted codewords as one serially concatenated sequence, even when considering all the features of a DAB system.

Equivalence of Serial and Parallel A-Posteriori Probabilities in the Decoding of DAB Systems

TL;DR

The paper tackles whether a-posteriori probabilities (APPs) for information and encoded bits remain unchanged when decoding a CIF composed of multiple convolutional codewords as a single serial codeword in digital audio broadcasting (DAB) systems. It proves a general theorem: if each parallel encoder terminates in the zero state (tail bits are known) and codeword lengths are known, then for all positions the serial APP equals the parallel APP , with analogous results for encoder outputs; the proof uses a Markov-state model of the encoder and BCJR/LLR reasoning. The authors extend the result to practical decoding tools (BCJR and Viterbi) and show that the serial approach yields identical information about the bits to be decoded, even under puncturing and interleaving. Simulations in both a generic parallel/punctured setup and a DAB-like DE-QPSK/OFDM framework corroborate the theoretical findings, demonstrating BER curves that are unchanged when decoding serially versus in parallel. The work supports treating the CIF as one long codeword in iterative decoding, with potential performance gains for DAB receivers and guidance for future exploration of CIF-based decoding strategies.

Abstract

Motivated by applications to digital audio broadcasting (DAB) systems, we study the a-posteriori probabilities (APPs) of the coded and information bits of the serial concatenation of multiple convolutional codewords. The main result of this correspondence is a proof that the APPs of the input bits do not change when considering the concatenation of multiple codewords as a received sequence. This is a purely theoretical result, which remains valid for every convolutional code, as long as the encoder goes back to the zero state at the end of each codeword. An equivalent heuristic for serial concatenation in Viterbi decoding is described. The applicability of our result to DAB systems, where interleaving and modulation are accounted for, is investigated through Matlab simulations. We show that the Bit Error Rate (BER) of the simulated DAB system does not change when decoding multiple transmitted codewords as one serially concatenated sequence, even when considering all the features of a DAB system.

Paper Structure

This paper contains 12 sections, 2 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and problem statement
  3. Notation
  4. Problem statement
  5. Convolutional decoding
  6. Information bit distribution
  7. Encoder output distribution
  8. Viterbi decoding
  9. Simulation results
  10. First simulation
  11. Second simulation
  12. Conclusions and future work

Key Result

Theorem 3.1

Under the assumption that the positions of the tail bits are known to the receiver, the parallel and serial APPs of the information bits coincide. Formally, for all $t=1,\ldots,L_N$ we have

Figures (3)

  • Figure 1: Illustration for the notation.
  • Figure 2: First simulation.
  • Figure 3: Second simulation.

Theorems & Definitions (6)

  • Example 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Corollary 3.1
  • proof