Characteristic Matrices and Trellis Reduction for Tail-Biting Convolutional Codes
Masato Tajima
TL;DR
This paper explores the properties of characteristic matrices for tail-biting convolutional codes and demonstrates how cyclic transformations and polynomial matrix reductions can lead to trellis complexity reduction.
Contribution
It introduces a cyclic structure-based analysis of characteristic matrices and proposes a method for trellis reduction using polynomial matrix transformations and partial cyclic shifts.
Findings
Characteristic matrices have a cyclic structure related to the code's generator matrix.
Trellis reduction can be achieved through polynomial matrix reduction and cyclic shifts.
Partial cyclic shifts of code sequences facilitate trellis complexity reduction.
Abstract
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…
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Taxonomy
TopicsCoding theory and cryptography · Advanced Wireless Communication Techniques · Error Correcting Code Techniques
Characteristic Matrices and Trellis Reduction for Tail-Biting Convolutional Codes
Masato Tajima M. Tajima was with the graduate School of Science and Engineering, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan (e-mail: [email protected]).Manuscript received April 19, 2005; revised August 26, 2015. This paper was presented in part at the IEICE Technical Committee IT Conference in March 2017.
Abstract
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 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 equivalent to based on a characteristic matrix. In this case, if the polynomial generator matrix corresponding to 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 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.
Index Terms:
Tail-biting convolutional codes, tail-biting trellis, characteristic matrix, cyclic shift, trellis reduction.
I Introduction
From the 1980s to 1990s, trellis representations of linear block codes were studied with a great interest [2, 7, 8, 14, 17, 18, 19]. Subsequently, tail-biting trellises of linear block codes have received much attention. Given a linear block code, there exists a unique minimal conventional trellis. This trellis simultaneously minimizes all measures of trellis complexity. However, tail-biting trellises do not have such a property. That is, minimality of tail-biting trellises depends on the measure being used [13]. In general, the complexity of a tail-biting trellis may be much lower than that of the minimal conventional trellis. There have been many contributions to the subject, including [3, 4, 9, 10, 11, 13, 15, 20, 22, 32]. The works [3, 13] had a strong influence on the subsequent studies. A remarkable progress has been made by Koetter and Vardy in their paper [13]. They showed that for a -dimensional linear block code of length with full support, there exists a list of characteristic generators (i.e., a characteristic matrix [13]) from which all minimal tail-biting minimal trellises can be obtained. A different method of producing tail-biting trellises was proposed by Nori and Shankar [20]. They used the Bahl-Cocke-Jelinek-Raviv (BCJR) construction [2]. These works were further investigated by Gluesing-Luerssen and Weaver [9, 10]. In particular, noting that a characteristic matrix for a given code is not necessarily unique, they have refined and generalized the previous works. More recent works [4, 11] provide further research on the subject.
On the other hand, tail-biting convolutional codes were proposed by Ma and Wolf in 1986 [16] (tail-biting representations of block codes were introduced by Solomon and van Tilborg [24]). Tail-biting (abbreviated TB) is a technique by which a convolutional code can be used to construct a block code without any loss of rate. In connection with the subject, there have been also many works, including [1, 16, 23, 25, 28, 33]. Since a TB convolutional code is identified with a linear block code, the results on TB trellises for linear block codes can be used. In particular, we can think of a characteristic matrix of a given TB convolutional code. In this paper, we first investigate a characteristic matrix for a TB convolutional code. And then, based on the derived results, we discuss the possibility of trellis reduction for a given TB convolutional code. An outline of the rest of the paper is as follows:
In Section II, we review the basic notions needed for this paper.
In Section III, we investigate the basic properties of a characteristic matrix for a TB convolutional code. When a TB convolutional code with generator matrix is regarded as a linear block code , a (scalar) generator matrix (denoted by ) for is constructed using the coefficients which appear in the polynomial expansion of . We see that has a kind of cyclic structure. Then it is shown that the (characteristic) span list associated with a characteristic matrix for consists of some basic spans and their right cyclic shifts, from which basic properties of a characteristic matrix are derived.
In Section IV, we deal with transformations of and discuss the relationship between these transformations and the corresponding scalar generator matrices . We see that dividing a column of by a monomial factor corresponds to cyclically shifting a column subsequence of to the left, whereas multiplying a column of by a monomial corresponds to cyclically shifting a column subsequence of to the right. These properties are essentially used for trellis reduction to be discussed in Section V.
In Section V, we discuss the possibility of trellis reduction for a given TB convolutional code (we identify the code with an block code ). As is stated above, we can think of a characteristic matrix for . Consider the case where some characteristic generators, which consist of some basic generators and their right cyclic shifts, can generate the same code . We see that these characteristic generators form a (scalar) generator matrix associated with a (polynomial) generator matrix of another convolutional code. In this case, if the constraint length of the obtained generator matrix is smaller than that of the original one, then trellis reduction is realized. Even if this kind of reduction is not possible, there are cases where a newly obtained generator matrix contains a monomial factor in some column. Then there is a possibility that the generator matrix is reduced by sweeping the monomial factor out of the column. Note that this operation corresponds to cyclically shifting the corresponding code subsequence to the left. In this way, trellis reduction can be accomplished. We also present a trellis reduction method for high rate codes which uses a reciprocal dual encoder. We remark that the (trellis) section length is an important parameter and the proposed method is restricted to TB convolutional codes with short to moderate section length. We give an upper bound for the section length by evaluating the span lengths of characteristic generators.
Finally, conclusions are provided in Section VI.
II Preliminaries
We begin with the basic notions needed in this paper, where the underlying field is assumed to be . Let be an linear block code, where the set of indices for a codeword in is denoted by . Then a codeword is expressed as . is also regarded as the time axis for TB trellises for . Since TB trellises for are considered in this paper, it is convenient to identify with , the ring of integers modulo . Hence, when dealing with TB trellises, all index arithmetic will be implicitly performed modulo [13].
The notion of span is fundamental in trellis theory. Given a codeword , a span of , denoted by , is a semiopen interval such that the corresponding closed interval contains all the nonzero positions of [13]. Due to the cyclic structure of the time axis , we adopt the following interpretation of intervals [9, 10, 13]. For , we define
[TABLE]
and . We call the intervals and conventional if and circular otherwise.
In connection with the construction of minimal TB trellises for , Koetter and Vardy [13] introduced the notion of characteristic generator for . Denote by a cyclic shift to the left by positions [13]. Similarly, denote by a cyclic shift to the right by positions. Let be a basis in minimal-span form [17] for the code . A characteristic generator for is a pair consisting of a codeword and a span such that are nonzero. The set of all the characteristic generators for is given by
[TABLE]
Here we have an understanding that if , then , where .
Assume that has a full support. Then a characteristic matrix for is the matrix having the elements of as its rows. The above definition implies that when we refer to a characteristic matrix, the associated spans are taken into account. Here note that a basis in minimal-span form is not necessarily unique. Hence, may not be uniquely determined. On the other hand, the set of spans (denoted by ) accompanied by is, up to ordering, uniquely determined by the code [9, 10, 13]. is called the characteristic span list of (an element is called a characteristic span of ) [9, 10]. In order to clarify this fact, Gluesing-Luerssen and Weaver introduced the notion of characteristic pair of [9, Definition III.8], where is a generating set of and represents the associated spans. In this paper, we basically follow the definition of Gluesing-Luerssen and Weaver, but in order to emphasize the fact that a characteristic matrix inherently assumes the associated spans, we leave the term characteristic matrix for the definition. Thus we define as follows (cf. [9, Definition III.8]).
Definition II.1
Let be an linear block code with support . A characteristic matrix for with (characteristic) span list is defined to be a pair , where
[TABLE]
[TABLE]
have the properties:
* generates .*
- 2)
* is a span of .*
- 3)
* are distinct and are distinct.*
- 4)
For all , there exist exactly row indices, , such that for .
Remark: Property 3) is derived from [13, Lemma 5.7] and the related remarks. Also, Property 4) is derived from the proof of [13, Theorem 5.10].
In the following, when there is no danger of confusion, we shall use the terms characteristic matrix and characteristic matrix with span list interchangeably.
III Characteristic Matrices for a Tail-Biting Convolutional Code
Let be a polynomial generator matrix of size . Denote by a corresponding polynomial check matrix. Both and are assumed to be canonical [18]. Consider a standard trellis of sections for a convolutional code defined by . Here is assumed, where and are the memory lengths of and , respectively. The TB condition is a restriction that the encoder starts and ends in the same state. That is, only those paths in the trellis that start and end in the same state are admissible. We call such paths TB paths. Let be the set of all TB paths. In the following, we call a TB convolutional code of section length defined by (cf. Fig.1 in Section V). When there is no danger of confusion, we will omit the phrase of section length . can be regarded as a linear block code of length . To simplify the notations, is identified with and is denoted simply by . Let
[TABLE]
be the polynomial expansion of , where are matrices. Then the scalar generator matrix for is given by
[TABLE]
with size [12]. Hence, we can say that a TB convolutional code is generated by . In the following, we call the tail-biting generator matrix (abbreviated TBGM) associated with a TB convolutional code defined by , or simply the TBGM associated with .
III-A Computation of Characteristic Matrices
Koetter and Vardy [13] have given an algorithm which can compute a characteristic matrix for a linear block code. Consider a TB convolutional code generated by . Note that is equivalent to . That is, has a periodic structure of period . Using this property, a characteristic matrix for can be computed efficiently.
Let . is the code generated by . Let be a basis in minimal-span form for the code . Then a characteristic matrix for is defined as follows [13]:
[TABLE]
Since is equivalent to , we have
[TABLE]
where
[TABLE]
Similarly, we have
[TABLE]
In general, for , we have
[TABLE]
Hence,
[TABLE]
is obtained. Thus we have shown the following.
Proposition III.1
A characteristic matrix for a TB convolutional code generated by is given by
[TABLE]
Corollary III.1
If the relation
[TABLE]
holds, then a characteristic matrix is given by
[TABLE]
Proof:
From the assumption, we have
[TABLE]
Similarly, we have
[TABLE]
Then, from Proposition 3.1, it follows that
[TABLE]
∎
We remark that in many practical applications, a characteristic matrix for a TB convolutional code is obtained based on the above corollary.
Example 1: Consider the TB convolutional code of section length defined by
[TABLE]
The associated TBGM is given by
[TABLE]
In this case, we have
[TABLE]
[TABLE]
[TABLE]
By applying and to these matrices, a characteristic matrix is obtained as follows:
[TABLE]
Note that the spans are connected with , whereas the spans are connected with . Hence,
[TABLE]
We see that a characteristic matrix cannot be obtained simply by applying and to .
III-B Structure of the Characteristic Span List
Let , where
[TABLE]
[TABLE]
be a characteristic matrix for with span list , then is a characteristic matrix for with span list [9, Remark III.9 (b)], where
[TABLE]
[TABLE]
Using repeatedly this relation, we see that is a characteristic matrix for with span list . Consider a TB convolutional code generated by and set to . Since is equivalent to , holds. Thus we have the following.
Lemma III.1
Let be a TB convolutional code generated by . If is a characteristic matrix for with span list , then is also a characteristic matrix for with span list . Let
[TABLE]
Then is given by
[TABLE]
Since the characteristic span list is uniquely determined, and coincide up to ordering.
Proposition III.2
The characteristic span list of a TB convolutional code generated by consists of the set of basic spans
[TABLE]
and .
Proof:
Suppose that the spans in are sorted such that
[TABLE]
Then we have
[TABLE]
Here take notice of the following set of spans in :
[TABLE]
In , it is transformed to
[TABLE]
Since and coincide up to ordering,
[TABLE]
holds. Hence, we have
[TABLE]
Similarly, the set of spans
[TABLE]
is transformed to
[TABLE]
Then for the same reason,
[TABLE]
holds. Hence, we have
[TABLE]
Continuing the same argument, we have
[TABLE]
for . ∎
Example 2: Consider the TB convolutional code of section length defined by the rate encoder
[TABLE]
Using the associated TBGM, i.e.,
[TABLE]
a charactreristic matrix is computed as follows:
[TABLE]
We see that the characteristic span list consists of the set of basic spans
[TABLE]
and its right cyclic shifts by and positions.
III-C Counting Characteristic Matrices
Recall the definition of a characteristic matrix for a given code , i.e.,
[TABLE]
where is a basis in minimal-span form for the code . Note that is not necessarily unique. Hence, is not uniquely determined [9]. With respect to this subject, Weaver [32] discussed the relationship between the characteristic span list of and the number of characteristic matrices for .
Let be the characteristic span list of . Define the set as follows [32]:
[TABLE]
represents the number of spans (in ) included in a specified span . Weaver [32] proved the following.
Lemma III.2** (Weaver [32])**
Let be a characteristic span of . Then there exist characteristic generators for having this span.
This fact is derived from the next observation:
Let and consider two characteristic generators and with spans and , respectively. Then is also a characteristic generator with span .
Consider a TB convolutional code generated by . We have already shown that the characteristic span list of consists of the set of basic spans
[TABLE]
and . Hence, it suffices to consider the spans in for the purpose of counting the number of characteristic matrices. Define as follows:
[TABLE]
Also, let
[TABLE]
[TABLE]
Then we have the following:
- •
There exist characteristic generators having span .
- •
There exist characteristic generators having span .
- •
There exist characteristic generators having span .
As a result, the degree of freedom related to the spans in is given by
[TABLE]
Since this degree of freedom is common to other blocks of spans in , the overall degree of freedom related to becomes
[TABLE]
Thus we have shown the following.
Proposition III.3
Let be a TB convolutional code generated by . Let and be as above. Then there exist characteristic matrices for .
Example 2 (Continued): Take notice of the first three rows of the characteristic matrix . We have
[TABLE]
Hence, and there exist characteristic matrices.
III-D Span Lengths of Characteristic Generators
Let be a span of a codeword . Then the span length of is defined by , i.e., the number of elements in the closed interval . When a span alone is referred to without specifying the accompanied codeword, we use the term the span length of a span . Let be the characteristic span list of a TB convolutional code generated by . Suppose that the spans in are sorted such that
[TABLE]
Then by [13, Theorem 5.10],
[TABLE]
holds. Due to the structure of (see Proposition 3.2), the left-hand side of the above equality becomes
[TABLE]
where
[TABLE]
[TABLE]
In the derivation, we also used the relation
[TABLE]
Replacing and by and , respectively, the above equality reduces to
[TABLE]
Thus we have shown the following.
Proposition III.4
Let be the characteristic span list of a TB convolutional code generated by . Denote by the set of basic spans in . Then the sum of span lengths of spans in is given by
[TABLE]
Example 2 (Continued): We have
[TABLE]
[TABLE]
Also, we have
[TABLE]
IV Transformations of and the Corresponding TBGM’s
In this section, we discuss the relationship between transformations of a generator matrix and the corresponding TBGM’s (). We consider the following transformations of :
- a)
Dividing the th column by .
- b)
Multiplying the th column by .
- c)
Adding the th row multiplied by to the th row.
- d)
Implicit transformations.
In the next section, we will see that these transformations play an essential role in trellis reduction for TB convolutional codes.
IV-A Dividing a Column of by
Suppose that the th column of has a monomial factor . We can assume without loss of generality that and . Hence, has the form
[TABLE]
Let
[TABLE]
be the polynomial expansion of . Comparing the entries, we have
[TABLE]
By these equations, we have
[TABLE]
[TABLE]
Dividing the first column of by , let the resulting matrix be . Then has the polynomial expansion:
[TABLE]
Consider the TBGM associated with , denoted by , where
[TABLE]
Note that both and can be regarded as matrices having blocks of columns. Then in view of the entries of and the relation
[TABLE]
is obtained from by cyclically shifting the first column of each block to the left by positions. Thus we have the following.
Proposition IV.1
Regard as a matrix having blocks of columns. Suppose that the th column of has a monomial factor . Then dividing the th column of by is equivalent to cyclically shifting the th column of each block of to the left by positions.
Let be a TB convolutional code of section length defined by . Note that each codeword in consists of blocks of components. Here let us cyclically shift the th component of each block to the left by positions. Denote by the set of resulting (modified) codewords. We have already shown that is obtained from by cyclically shifting the th column of each block to the left by positions. Hence, is generated by . In words, is represented as a TB convolutional code defined by .
IV-B Multiplying a Column of by
Consider multiplication of the th column of by , where . In the following, we assume without loss of generality that C. Hence, the resulting matrix has the form
[TABLE]
Then we have
[TABLE]
Accordingly, the polynomial expansion of becomes
[TABLE]
Consider the TBGM associated with , denoted by , where
[TABLE]
Note that and consist of blocks of columns as above. In view of the entries of , we see that is obtained from by cyclically shifting the first column of each block to the right by positions. Thus we have the following.
Proposition IV.2
Regard as a matrix having blocks of columns. Suppose that . Then multiplying the th column of by is equivalent to cyclically shifting the th column of each block of to the right by positions.
Remark: In order for to be defined, the condition is required.
Let be a TB convolutional code of section length with generator matrix . Let be as in the previous section. In this case, however, the th component of each block is cyclically shifted to the right by positions. We have shown that is obtained from by cyclically shifting the th column of each block to the right by positions. Hence, is generated by . In words, is represented as a TB convolutional code defined by .
IV-C
Consider addition of the th row multiplied by to the th row , denoted by , where . In the following, we assume without loss of generality that and . Let the first row of be
[TABLE]
Also, let
[TABLE]
be the polynomial expansion, where the size of is . Then the polynomial expansion of becomes
[TABLE]
Note that the first row of is expressed as
[TABLE]
Hence, its right cyclic shift by positions, i.e.,
[TABLE]
coincides with the th row of . That is, corresponds to addition of the th row to the second row within the matrix . Note that this is an elementary row operation. Thus we have the following.
Proposition IV.3
Suppose that . Consider the operation . Let the resulting matrix be and the associated TBGM be . Then is equivalent to .
Taking into consideration Proposition 4.3, let us introduce a useful notion. Let and be TB convolutional codes of section length defined by and , respectively. Denote by and the memory lengths of and , respectively, where . Let and be the TBGM’s associated with and , respectively. We see that if and are equivalent, then . All of this leads to the following definition.
Definition IV.1
When and are equivalent, we say that and are “TB-equivalent”.
Thus we have the following.
Proposition IV.4
If and are TB-equivalent, then a TB convolutional code defined by is represented as a TB convolutional code defined by , and vice versa.
Proof:
A direct consequence of the definition of TB-equivalent. ∎
V Trellis Reduction for TB Convolutional Codes
In this section, we will show that for a TB convolutional code of short to moderate section length, the associated TB trellis can be reduced. We begin with an example.
V-A An Example of Trellis Reduction
Consider the TB convolutional code defined by , where the section length is set to . The corresponding TB trellis is shown in Fig.1, where the paths which start and end in the same state are TB paths (i.e., valid codewords). Then is the set of all TB paths. Since has the polynomial expansion
[TABLE]
the TBGM associated with is given by
[TABLE]
Based on , a characteristic matrix for is computed as follows:
[TABLE]
Choosing rows from , let
[TABLE]
We see that the rows of are linearly independent and thus generate , i.e., is equivalent to .
Here note that consists of the first row and its right cyclic shifts by positions. Accordingly, can be regarded as the TBGM associated with
[TABLE]
Hence, is equally represented as a TB convolutional code defined by . We remark that the constraint length of is and is greater than that of .
On the other hand, observe that the first column of has a factor . Then dividing the first column by , we have
[TABLE]
Note that this transformation corresponds to cyclically shifting the first component of each branch (of a TB path) to the left by two branches (cf. Proposition 4.1). By this transformation, the original TB convolutional code is represented using a trellis associated with as well. The trellis for is shown in Fig.2. For example, take notice of the TB path in Fig.1 which starts and ends in state :
[TABLE]
Cyclically shifting the first component of each branch to the left by two branches, it becomes
[TABLE]
We see that the modified path is represented as a path which starts and ends in state (1) in Fig.2.
This example shows that there are cases where a given TB convolutional code is represented using a reduced trellis with less state complexity, if we allow partial cyclic shifts of a TB path.
V-B Trellis Reduction for TB Convolutional Codes
The argument in the previous section, though it was presented in terms of a specific example, is entirely general. Then the method can be directly extended to a general case. Let be as in Section III. Denote by the constraint length of . Consider a TB convolutional code of section length defined by . The trellis reduction procedure becomes as follows.
Procedure for trellis reduction:
- i)
Compute a characteristic matrix for based on the TBGM , where consists of rows and their right cyclic shifts by integer multiple of .
- ii)
Choosing rows from , form , where has the properties:
The rows of are linearly independent and thus generate .
- 2)
consists of rows and their right cyclic shifts by integer multiple of .
- iii)
(Direct reduction) is regarded as the TBGM associated with another generator matrix . Let be the constraint length of . If , then trellis reduction for is realized.
- iv)
(Indirect reduction) Even if , there are cases where has a monomial factor in some (th) column. Then there is a possibility that is reduced by dividing the th column of by (the resulting matrix is denoted by ). Let be the constraint length of . If , then the original TB trellis can be reduced. That is, by cyclically shifting the th component of each branch of a TB path () to the left by branches, the set of modified paths are equally represented as a TB convolutional code defined by (this is justified by Proposition 4.1). Thus trellis reduction is accomplished.
- v)
is not necessarily unique [9]. Hence, if necessary, try i) iv) using another characteristic matrix for .
Remark: For row rate codes, it is rather easy to find which is equivalent to . Also, row rate codes make it easy to determine whether can be reduced or not.
As is stated above, there are some restrictions on the selection of and . We have the following.
Proposition V.1
The number of characteristic matrices in i) is given by , where is defined in Section III-C. For a fixed , the number of which satisfy the condition 2) in ii) is given by .
Proof:
is a candidate for a TBGM associated with an encoder. Hence, the above is a consequence of the structure of TBGM. ∎
Example 3: Consider the TB convolutional code of section lengh defined by
[TABLE]
Using the associated TBGM, i.e., , a characteristic matrix for is computed as follows:
[TABLE]
Choosing rows from , define as
[TABLE]
We see that is equivalent to . Also, we see that is the TBGM associated with
[TABLE]
Note that the constraint length of is not reduced compared to that of . On the other hand, observe that the second column of has a factor . Then dividing the column by , we have
[TABLE]
This transformation corresponds to cyclically shifting the second component of each branch of a TB path to the left by one branch (cf. Proposition 4.1). As a result, the modified paths are represented using the trellis for . Thus trellis reduction for is accomplished.
Remark: As is stated above, is not necessarily unique. For example, if a characteristic matrix
[TABLE]
is used, then trellis reduction cannot be realized using the above procedure.
Using appropriate characteristic matrices, the above reduction method can also be applied to the following cases:
- (1)
[TABLE]
- (2)
[TABLE]
- (3)
[TABLE]
- (4)
[TABLE]
- (5)
[TABLE]
- (6)
[TABLE]
V-C Trellis Reduction Using a Reciprocal Dual Encoder
For high rate codes, may not have a monomial factor in any columns. Then it is not easily determined whether can be reduced or not. In such cases, it is useful to consider a reciprocal dual encoder associated with . A reciprocal dual encoder [21] is defined as follows: Let be as in Section III. Also, let be a corresponding check matrix with size . A reciprocal dual encoder is obtained by substituting for in and by multiplying the th () row of the resulting matrix by , where is the degree of the th row of .
Definition V.1** (McEliece and Lin [18])**
Let be a scalar generator matrix for a terminated convolutional code defined by [18, 21]. is given by
[TABLE]
The matrix
[TABLE]
which repeatedly appears as a vertical slice in except initial and final transient sections, is called the matrix module. Then the trellis module for the trellis associated with corresponds to . If is in minimal-span form, then is minimal. The state complexity profile of is an -tuple consisting of the dimensions of state spaces of .
The meaning of obtaining a reciprocal dual encoder is based on the following result [26, 30, 31].
Proposition V.2** (Tang and Lin [30])**
Consider a minimal trellis module of and that of an associated reciprocal dual encoder . Then their state complex profiles are identical.
Hence, in order to determine whether is reduced or not, we can compute a reciprocal dual encoder associated with . In connection with an encoder and an associated reciprocal dual encoder , we have the following.
Proposition V.3
Let be the TBGM associated with . Then a check matrix corresponding to is obtained as the TBGM (denoted by ) associated with a reciprocal dual encoder .
Proof:
Let the polynomial expansion of be
[TABLE]
where is the memory length of and are matrices. It is known (e.g., [28]) that a check matrix corresponding to is given by
[TABLE]
with size . On the other hand, let the polynomial expansion of be
[TABLE]
Then the TBGM associated with (denoted by ) is defined by
[TABLE]
with size .
Here take notice of the th () row of
[TABLE]
We see that the row is identical to the th row of
[TABLE]
Similarly, the th () row of
[TABLE]
is identical to the th row of
[TABLE]
Due to the cyclic structures of and , similar correspondences hold successively. Hence, is given as a row permutation of . ∎
A procedure for computing is obtained based on the above proposition.
Procedure for computing :
- i)
Compute a characteristic matrix for the dual code based on , where consists of rows and their right cyclic shifts by integer multiple of .
- ii)
Choosing rows from , form , where has the properties:
The rows of are linearly independent and thus generate .
- 2)
consists of rows and their right cyclic shifts by integer multiple of .
- iii)
Let be the polynomial matrix whose TBGM is .
- iv)
Note that and are equivalent to and , respectively. Hence, is a check matrix corresponding to . Then it follows from Proposition 5.3 that is a reciprocal dual encoder associated with .
- v)
is not necessarily unique. Hence, if necessary, try i) iv) using another characteristic matrix for .
The following is an example where trellis reduction is realized using a reciprocal dual encoder.
Example 4: Consider the rate TB convolutional code of section length with generator matrix
[TABLE]
Based on the associated TBGM, i.e.,
[TABLE]
a characteristic matrix for is computed as follows:
[TABLE]
The span list for is given by
[TABLE]
Choosing rows from , let
[TABLE]
The span list for is given by
[TABLE]
We see that the rows of are linearly independent and thus generate , i.e., is equivalent to . Also, note that is the TBGM associated with
[TABLE]
Hence, the original TB convolutional code is equally represented as a TB convolutional code defined by .
Observe that the constraint length of is and is equal to that of . Also, notice that the second column of has a factor . However, is not reduced by dividing the column by . In general, it is difficult to tell a possibility of reduction of just by looking at its entries. So, we will compute a reciprocal dual encoder associated with .
We begin with a reciprocal dual encoder associated with . is given by
[TABLE]
Based on , a characteristic matrix for is computed as follows:
[TABLE]
The span list for is given by
[TABLE]
Note that if the span list for is , then the span list for is given by [10, 13].
Next, choosing rows from , let
[TABLE]
The span list for is given by
[TABLE]
We see that is equivalent to . Thus is a scalar check matrix corresponding to . Also, note that is the TBGM associated with . We already know that is the TBGM associated with . Hence, by Proposition 5.3, a reciprocal dual encoder associated with is given by
[TABLE]
Observe that has a factor in the first column and a factor in the second column. Then sweeping these factors out of the corresponding columns, the constraint length of is reduced to one. This fact implies that the constraint length of can also be reduced.
In the following, we will show that reduction of is actually realized. For the purpose, a check matrix corresponding to , i.e.,
[TABLE]
is used.
Let and be a generator matrix and a corresponding check matrix for a convolutional code, respectively. In the following, this relation is denoted by . It is shown [27] that and can be reduced simultaneously, if reduction is possible, where the relation is retained in the whole reduction process. We apply the method to our case under consideration.
Step 1: For , add the first row multiplied by to the second row. By Proposition 4.3, this is a TB-equivalent transformation. As a result, we have
[TABLE]
Step 2: Divide the second column of by , while divide the first and third columns of by . Then we have
[TABLE]
Step 3: Multiply the third column of by , while divide the third column of by . Then we have
[TABLE]
Step 4: Note that G^{(4)}(D)=\left(\begin{array}[]{ccc}1+D&1&D\\ D&D&D\end{array}\right) is not basic [5]. Using an invariant-factor decomposition [5] of , an equivalent basic matrix
[TABLE]
is obtained. Note that and are TB-equivalent (cf. Proposition 4.4).
In the above reduction process for , except for TB-equivalent transformations, the second column is divided by , whereas the third column is multiplied by . Accordingly, for each TB path, let us cyclically shift the second component of each branch to the left by one branch and cyclically shift the third component of each branch to the right by one branch. Then the modified TB paths are represented as a TB convolutional code defined by (see Propositions 4.1 and 4.2). The trellis for is shown in Fig.3. For example, take an information sequence
[TABLE]
and the corresponding TB path
[TABLE]
By cyclically shifting the second component of each branch to the left by one branch, and by cyclically shifting the third component of each branch to the right by one branch, we have
[TABLE]
We see that \mbox{\boldmathw}_{m} is a TB path which starts and ends in state in Fig.3.
Remark: We remark that in the above argument, it is assumed that is equivalent to (i.e., the equivalence has been checked beforehand). In general, however, is relatively large for high rate codes. Hence, it is preferable that the equivalence of and is derived without checking it beforehand. Actually, the equivalence of and is derived from the equivalence of and using the result of Gluesing-Luerssen and Weaver [10, Theorem IV.3] (see Appendix A).
V-D Relation Between Trellis Reduction and Section Length
In the proposed trellis reduction method, the section length is an important parameter. Actually, the method is effective for TB convolutional codes of short to moderate section length. This is because the span lengths of characteristic generators increase as grows (see Section III-D). We have already shown that a TB trellis with generator matrix can be reduced for the case of . Consider the same trellis. This time, however, is set to . Then is given by
[TABLE]
Note that to each generator in , its span is assigned in the natural manner. Observe that the span lengths of these spans are the same, i.e., . A characteristic matrix is computed as follows:
[TABLE]
Thus the set of basic spans is given by
[TABLE]
With respect to , there are two cases. When the first row of is used as a basic generator of , is identical to . When the second row of is used as a basic generator of , the span lengths of rows of are and are greater than . These facts mean that in either case, trellis reduction is not realized using the proposed method. On the other hand, this example implies that the upper bound for can be estimated by comparing the span lengths of generators in with those of generators in .
Let be a characteristic matrix for a TB convolutional code of section length . We already know that the associated span list consists of the set of basic spans
[TABLE]
and . Also, the sum of span lengths of spans in is given by
[TABLE]
In the proposed method, consists of generators in . From a span viewpoint, this corresponds to choosing spans from . Accordingly, the sum of span lengths of these spans, denoted by , is approximated by
[TABLE]
On the other hand, consider , where to each generator, its span is assigned in the natural manner. Then the span list consists of the set of basic spans and . We evaluate the sum of span lengths of spans in , denoted by . Let be the degree of the th row of . Here take notice of the first block of rows in , i.e.,
[TABLE]
The span length of the th () row is approximated by . Hence, we have
[TABLE]
where is the constraint length of . Since trellis reduction is realized in the case where consists of generators with short span length, we can take the inequality
[TABLE]
as a criterion for trellis reduction. That is, we can estimate the upper bound for using the inequality
[TABLE]
For several concrete cases, we will show the condition .
- (1)
:
[TABLE]
[TABLE]
- (2)
:
[TABLE]
[TABLE]
- (3)
:
[TABLE]
[TABLE]
We observe that the TB convolutional codes presented in Section V-B all satisfy the condition . Also, the rate TB convolutional code discussed in the previous section satisfies the condition .
VI Conclusion
In this paper, we have derived several basic properties of a characteristic matrix for a TB convolutional code. We have shown that the characteristic span list consists of some basic spans and their right cyclic shifts. Using the derived results, we have shown that a trellis associated with a given TB convolutional code can be reduced in some cases. As candidates for trellis reduction, we have taken the generator matrices from the tables in [12, Chapter 8] in principle. For example, the rate encoders in Section V-B were chosen from [12, TABLE 8.1]. On the other hand, good TB convolutional encoders have been obtained [12, 25]. Here, for a given rate , the optimal encoder of memory length produces the largest minimum distance for each section length . We have applied the proposed reduction method to some of such encoders (see [12, TABLE 8.19]). As a result, for example, we have obtained () from (), where the octal notation for generator matrices is used. Similarly, we have obtained () from (). Note that both () and () are listed in the same table.
Finally, we remark that the proposed trellis reduction method depends on the choice of a characteristic matrix for a given convolutional code. Though the number of characteristic matrices to be examined is rather restricted (cf. Proposition 5.1), the method is not fully constructive. Also, a detailed condition that trellis reduction is realized has to be clarified.
Appendix A Proof of the equivalence of and
We first prove the following.
Proposition A.1
Let be as in Section III. Consider a TB convolutional code generated by and the corresponding dual code . Let be a characteristic matrix for with span list . Also, let be a characteristic matrix for with span list . Let and be submatrices of and , respectively, where consists of rows in , whereas consists of rows in . Denote by and the span lists for and , respectively. Here assume the following:
- i)
* consists of rows and their right cyclic shifts by integer multiple of .*
- ii)
Each span in does not include any spans in except itself.
- iii)
The rows of are linearly independent and thus generate .
- iv)
* consists of rows and their right cyclic shifts by integer multiple of .*
- v)
Each span in does not include any spans in except itself.
- vi)
* and satisfy (that is, consists of the spans in whose reverse is not in ).*
Then the rows of are linearly independent, thus generate , i.e., is equivalent to .
Remark: When consists of generators in with short span length, it is probable that the condition ii) holds. Similarly, when consists of generators in with short span length, it is probable that the condition v) holds.
Proof:
From ii), it follows that is common to all the characteristic matrices for . Similarly, from v), it follows that is common to all the characteristic matrices for . Also, by vi), is a dual selection of [10, Definition IV.2]. As a result [10, Theorem IV.3], we have
- 2)
Let . Then the KV trellises [10] based on and are dual to each other.
By iii), . Hence, by 1), . ∎
Let us go back to Example 4. In this example, the code generated by and the dual code generated by are considered. is a characteristic matrix for , whereas is a characteristic matrix for . Note that neither nor are unique. These are observed from the relation of inclusion in the associated span lists and , where
[TABLE]
Next, take notice of the matrices and , which are submatrices of and , respectively. The corresponding span lists are given by
[TABLE]
Here note the following:
- •
Each span in does not include any spans in except itself.
- •
Each span in does not include any spans in except itself.
Moreover, consists of the spans in whose reverse is not in . Actually, by reversing the spans in , we have
[TABLE]
We see that these spans are not in .
All these facts show that the conditions in Proposition A.1 are satisfied, when and are replaced by and , respectively. Hence, the equivalence of and is derived.
Remark: [10, Theorem IV.3] holds only for a pair , where is a characteristic matrix for and is the corresponding dual one for (see [10]). On the other hand, the pair computed above may not be in the duality relation. However, is common to all the characteristic matrices for , and is common to all the characteristic matrices for as well. Hence, the theorem can be applied to our case.
Acknowledgment
The author would like to thank Prof. Heide Gluesing-Luerssen for valuable comments on the duality of Koetter-Vardy (KV) trellises.
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