  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.
