Derivation of Mutual Information and Linear Minimum Mean-Square Error for Viterbi Decoding of Convolutional Codes Using the Innovations Method
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917609474097152 |
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| author | Tajima, Masato |
| author_facet | Tajima, Masato |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05740 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derivation of Mutual Information and Linear Minimum Mean-Square Error for Viterbi Decoding of Convolutional Codes Using the Innovations Method Tajima, Masato Information Theory 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. |
| title | Derivation of Mutual Information and Linear Minimum Mean-Square Error for Viterbi Decoding of Convolutional Codes Using the Innovations Method |
| topic | Information Theory |
| url | https://arxiv.org/abs/2403.05740 |