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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., Comment: 40 pages, 6 figures, 10tables

In this paper, we show that the soft-decision input to the main decoder in an SST Viterbi decoder is regarded as the innovation as well from the viewpoint of mutual information and mean-square error. It is assumed that a code sequence is transmitted symbol by symbol over an AWGN channel using BPSK modulation. Then we can consider the signal model, where the signal is composed of the signal-to-noise ratio (SNR) and the equiprobable binary input. By assuming that the soft-decision input to the main decoder is the innovation, we show that the minimum mean-square error (MMSE) in estimating the binary input is expressed in terms of the distribution of the encoded block for the main decoder. It is shown that the obtained MMSE satisfies indirectly the known relation between the mutual information and the MMSE in Gaussian channels. Thus the derived MMSE is justified, which in turn implies that the soft-decision input to the main decoder can be regarded as the innovation. Moreover, we see that the input-output mutual information is connected with the distribution of the encoded block for the main decoder., Comment: 26 pages, 5 figures, 5 tables

We correct errors in Section III-B of the above-titled paper. The corrections are related to the joint distribution of the inputs to the main decoder which corresponds to a branch in the code trellis for the main decoder. Note that as far as the distribution of the input corresponding to a single code symbol is concerned, the results in Section III-B are correct. We have noticed that alpha in Proposition 12 (beta in Proposition 14) has another meaning, which has made it possible to derive the joint distribution of the inputs corresponding to the whole branch. Corrections are based on this important observation., Comment: 6 pages

Suppose that a convolutionally encoded sequence is transmitted symbol by symbol over an AWGN channel using BPSK modulation. In this case, pairs of the signal (i.e., code symbol) and observation are not jointly Gaussian and therefore, a linear estimation method cannot be applied. Hence, in this paper, non-linear estimation of convolutionally encoded sequences is discussed. First a probability measure (denoted Q), whose Radon-Nikodym derivative with respect to the underlying probability measure P is an exponential martingale, is constructed. It is shown that with respect to Q, the observations are mutually independent Gaussian random vectors with zero mean and identity covariance matrix. We see that the relationship between observation noises (with respect to P) and observations (with respect to Q) has a close relation to the Girsanov theorem in continuous case. Next, using the probability measure Q, we calculate the conditional probability of an event related to any encoded symbol conditioned by the observations. Moreover, we transform it into a recursive form. In the process of derivation, the metric associated with an encoded sequence comes out in a natural way. Finally, it is shown that maximum a posteriori probability (MAP) decoding of convolutional codes is realized using the derived conditional probability., Comment: 16 pages, 1 figure

We introduce the notion of innovations for Viterbi decoding of convolutional codes. First we define a kind of innovation corresponding to the received data, i.e., the input to a Viterbi decoder. Then the structure of a Scarce-State-Transition (SST) Viterbi decoder is derived in a natural manner. It is shown that the newly defined innovation is just the input to the main decoder in an SST Viterbi decoder and generates the same syndrome as the original received data does. A similar result holds for Quick-Look-In (QLI) codes as well. In this case, however, the precise innovation is not defined. We see that this innovation-like quantity is related to the linear smoothed estimate of the information. The essence of innovations approach to a linear filtering problem is first to whiten the observed data, and then to treat the resulting simpler white-noise observations problem. In our case, this corresponds to the reduction of decoding complexity in the main decoder in an SST Viterbi decoder. We show the distributions related to the main decoder (i.e., the input distribution and the state distribution in the code trellis for the main decoder) are much biased under moderately noisy conditions. We see that these biased distributions actually lead to the complexity reduction in the main decoder. Furthermore, it is shown that the proposed innovations approach can be extended to maximum-likelihood (ML) decoding of block codes as well., Comment: Accepted for publication in IEEE Trans. Inf. Theory

Basic properties of a characteristic matrix for a tail-biting convolutional code are investigated. A tail-biting convolutional code can be regarded as a linear block code. Since the corresponding scalar generator matrix Gt has a kind of cyclic structure, an associated characteristic matrix also has a cyclic structure, from which basic properties of a characteristic matrix are obtained. Next, using the derived results, we discuss the possibility of trellis reduction for a given tail-biting convolutional code. There are cases where we can find a scalar generator matrix Gs equivalent to Gt based on a characteristic matrix. In this case, if the polynomial generator matrix corresponding to Gs has been reduced, or can be reduced by using appropriate transformations, then trellis reduction for the original tail-biting convolutional code is realized. In many cases, the polynomial generator matrix corresponding to Gs has a monomial factor in some column and is reduced by dividing the column by the factor. Note that this transformation corresponds to cyclically shifting the associated code subsequence (a tail-biting path is regarded as a code sequence) to the left. Thus if we allow partial cyclic shifts of a tail-biting path, then trellis reduction is accomplished., Comment: 25 pages, 3 figures

In this paper, we present an algebraic construction of tail-biting trellises. The proposed method is based on the state space expressions, i.e., the state space is the image of the set of information sequences under the associated state matrix. Then combining with the homomorphism theorem, an algebraic trellis construction is obtained. We show that a tail-biting trellis constructed using the proposed method is isomorphic to the associated Koetter-Vardy (KV) trellis and tail-biting Bahl-Cocke-Jelinek-Raviv (BCJR) trellis. We also evaluate the complexity of the obtained tail-biting trellises. On the other hand, a matrix consisting of linearly independent rows of the characteristic matrix is regarded as a generalization of minimal-span generator matrices. Then we show that a KV trellis is constructed based on an extended minimal-span generator matrix. It is shown that this construction is a natural extension of the method proposed by McEliece (1996)., Comment: 14 pages, 10 figures

In this paper, we discuss the reduction of error-trellises for tail-biting convolutional codes. In the case where some column of a parity-check matrix has a monomial factor (with indeterminate D), we show that the associated tail-biting error-trellis can be reduced by cyclically shifting the corresponding error-subsequence by l (the power of D) time units. We see that the resulting reduced error-trellis is again tail-biting. Moreover, we show that reduction is also possible using backward-shifted error-subsequences., Comment: 5 pages, submitted to the 2014 IEEE International Symposium on Information Theory

In this paper, we present an error-trellis construction for tailbiting convolutional codes. A tailbiting error-trellis is characterized by the condition that the syndrome former starts and ends in the same state. We clarify the correspondence between code subtrellises in the tailbiting code-trellis and error subtrellises in the tailbiting error-trellis. Also, we present a construction of tailbiting backward error-trellises. Moreover, we obtain the scalar parity-check matrix for a tailbiting convolutional code. The proposed construction is based on the adjoint-obvious realization of a syndrome former and its behavior is fully used in the discussion., Comment: 5 pages, submitted to the 2012 IEEE International Symposium on Information Theory

In this paper, we show that the code-trellis and the error-trellis for a convolutional code can be reduced simultaneously, if reduction is possible. Assume that the error-trellis can be reduced using shifted error-subsequences. In this case, if the identical shifts occur in the subsequences of each code path, then the code-trellis can also be reduced. First, we obtain pairs of transformations which generate the identical shifts both in the subsequences of the code-path and in those of the error-path. Next, by applying these transformations to the generator matrix and the parity-check matrix, we show that reduction of these matrices is accomplished simultaneously, if it is possible. Moreover, it is shown that the two associated trellises are also reduced simultaneously., Comment: 5 pages, submitted to the 2011 IEEE International Symposium on Information Theory

We consider the decoding of convolutional codes using an error trellis constructed based on a submatrix of a given check matrix. In the proposed method, the syndrome-subsequence computed using the remaining submatrix is utilized as auxiliary information for decoding. Then the ML error path is correctly decoded using the degenerate error trellis. We also show that the decoding complexity of the proposed method is basically identical with that of the conventional one based on the original error trellis. Next, we apply the method to check matrices with monomial entries proposed by Tanner et al. By choosing any row of the check matrix as the submatrix for error-trellis construction, a 1-state error trellis is obtained. Noting the fact that a likelihood-concentration on the all-zero state and the states with many 0's occurs in the error trellis, we present a simplified decoding method based on a 1-state error trellis, from which decoding-complexity reduction is realized., Comment: 5 pages, submitted to the 2010 IEEE International Symposium on Information Theory