Derivation of Mutual Information and Linear Minimum Mean-Square Error for Viterbi Decoding of Convolutional Codes Using the Innovations Method
<|reference_start|>Derivation of Mutual Information and Linear Minimum Mean-Square Error for Viterbi Decoding of Convolutional Codes Using the Innovations Method: We see that convolutional coding/Viterbi decoding has the structure of the Kalman filter (or the linear minimum variance filter). First, we calculate the covariance matrix of the innovation (i.e., the soft-decision input to the main decoder in a Scarce-State-Transition (SST) Viterbi decoder). Then a covariance matrix corresponding to that of the one-step prediction error in the Kalman filter is obtained. Furthermore, from that matrix, a covariance matrix corresponding to that of the filtering error in the Kalman filter is derived using the formula in the Kalman filter. As a result, the average mutual information per branch for Viterbi decoding of convolutional codes is given using these covariance matrices. Also, the trace of the latter matrix represents the linear minimum mean-square error (LMMSE). We show that an approximate value of the average mutual information is sandwiched between half the SNR times the average filtering and one-step prediction LMMSEs. In the case of QLI codes, from the covariance matrix of the soft-decision input to the main decoder, we can get a matrix. We show that the trace of this matrix has some connection with the linear smoothing error.<|reference_end|>
@article{tajima2024derivation,
    title={Derivation of Mutual Information and Linear Minimum Mean-Square Error
  for Viterbi Decoding of Convolutional Codes Using the Innovations Method},
    author={Masato Tajima},
    journal={arXiv preprint arXiv:2403.05740},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.05740},
    primaryClass={cs.IT math.IT}
}