Duality of Channel Encoding and Decoding - Part I: Rate-1 Binary Convolutional Codes

TL;DR

This paper explores the duality between encoding and decoding in rate-1 binary convolutional codes, revealing explicit relationships and novel dual encoder structures for various code types.

Contribution

It introduces a new perspective on the relationship between convolutional code encoding and decoding, deriving dual encoder structures for different code classes.

Findings

Forward and backward BCJR decoders can be represented by dual SISO encoders.

Bidirectional decoding is achieved by combining dual encoder outputs.

Explicit dual encoder structures are derived for recursive and non-recursive codes.

Abstract

In this paper, we revisit the forward, backward and bidirectional Bahl-Cocke-Jelinek-Raviv (BCJR) soft-input soft-output (SISO) maximum a posteriori probability (MAP) decoding process of rate-1 binary convolutional codes. From this we establish some interesting explicit relationships between encoding and decoding of rate-1 convolutional codes. We observe that the forward and backward BCJR SISO MAP decoders can be simply represented by their dual SISO channel encoders using shift registers in the complex number field. Similarly, the bidirectional MAP decoding can be implemented by linearly combining the shift register contents of the dual SISO encoders of the respective forward and backward decoders. The dual encoder structures for various recursive and non-recursive rate-1 convolutional codes are derived.