10.6 Simplex Symbol Assignment in Circular Trellis-Coded Modulation

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Simplex Symbol Assignment in Circular Trellis-Coded Modulation Frank A. Alder, Jeffrey C. Dill

Alan R. Lindsey

Electrical Engineering & Computer Science Ohio University Athens, Ohio 45701, USA

US Air Force Research Laboratory 525 Brooks Road Rome, New York 13441, USA II. BACKGROUND

Abstract - This paper presents a method for the assignment of simplex signals in a circular trellis-coded modulation (CTCM) scheme. Background is given on both CTCM and simplex signaling. The CTCM trellis is then shown to have various properties that allow symbol assignment to be carried out in a systematic manner.

I. INTRODUCTION Proposed by Ungerboeck in 1982 [1], trellis-coded modulation (TCM) has become a very popular research area. TCM provides a means by which band-limited channels can reap the benefits of error control coding by combining coding and modulation into a single step. This coding/modulation step is accomplished by integrating a multi-level or multiphase signaling constellation with a state-oriented encoder, such as a convolutional encoder. Where a binary block code might use a binary phase-shift keyed (BPSK) modulator to transmit each of the coded bits individually, a TCM scheme would choose one signal from the constellation to represent a number of the coded bits at one time. The tradeoff that exists in TCM is that it achieves coding gain at the expense of an increase in the complexity of the decoder. Recently, circular trellis-coded modulation (CTCM) has been proposed [2-9]. Also known as high-dimensional trellis-coded modulation (HDTCM), CTCM takes the basic concepts of TCM (such as signal partitioning) and applies them to achieve coding gain on a power-limited channel, such as a spread-spectrum channel, by using a high-dimensional signaling constellation. A CTCM system can be viewed as a block code with trellis structure in the sense that source data is encoded block-by-block independently. Additionally, CTCM satisfies a so-called “state constraint,” which specifies that the starting state of a particular source data block must equal the ending state. This property alleviates the need to set tail information bits to zero to drive the encoder to the “all-zero” state, a necessary procedure in conventional TCM. In [5], the simplex signaling constellation is investigated for use in CTCM, where a source alphabet size of two is emphasized. This paper takes that premise and develops rules for the assignment of simplex channel symbols to a CTCM trellis employing a source alphabet size of four. Several properties of the trellis are exposed that allow the assignment to be carried out in a systematic manner.

A. Simplex Signaling For arbitrary dimensions N ≥ M –1, M simplex signals exist for unit pulse energy if there exist M signals of S = {s1, s2,…,sM} in N dimensions such that:

si − s j = 2 M

i ≠ j, i ≥ 1, j ≤ M

(1)

where si = [si1,si2,…,siN], and sik ∈ {-1,0,1}, 1≤ i ≤ M, 1≤ k ≤ N. A simplex signal constellation can be loosely defined as a set of signals having equal energy and equidistant from each other in the Euclidean sense [10]. Simplex signals are not orthogonal, but they achieve the same error probability as an equally likely orthogonal signaling set while using less energy. Hence, simplex signaling is employed when transmission energy is constrained. The three-dimensional simplex will be emphasized here, where it is desired to create three-dimensional simplex signals occupying N-dimensional signal spaces (N>3). For example, one three-dimensional simplex signal might be s = [1,0,-1,0,-1,0,0,0].

(2)

In this case, eight dimensions exist and dimensions 1, 3, and 5 are occupied. Additionally, shorthand notation can be used to describe simplex signals. Equation (2) can be denoted as [1,-3,-5], for example. This notation indicates non-zero pulses in dimensions one, three, and five, and their associated polarity. There is only one simplex that utilizes only dimensions 1, 3, and 5 and contains the signal in (2) as a member. Using shorthand notation, the matrix expression for this simplex is

3 5  1  1 − 3 − 5 . S= − 1 3 − 5   5 − 1 − 3

(3)

Note that the negative of (3), indeed, the negative of any simplex, is also a simplex.

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By using all 8 dimensions, many simplexes can be formed that contain (2). An example is the following:

 1 − 3 − 5  −1 4 6 . S=  2 3 − 6   5 − 2 − 4

(4)

Equation (4) is just one of dozens of simplexes that contain (2) as a member.

The transmission symbol table is a look-up table analogous to the state table. Both the state and transmission symbol tables contain entries for all trellis states and all possible source symbols. However, where the state table defines the next state transition given the current state and current input, the transmission symbol table describes what symbol is transmitted when transiting to that next state. Given the state table and the transmission symbol table, the trellis is completely described. A trellis diagram can then be constructed with all trellis states, their associated transitions, and the related symbols [2,3]. III. CONSTRUCTION OF THE TRANSMISSION SYMBOL TABLE

B. Circular Trellis-Coded Modulation The trellis-coded modulation scheme introduced in [2,3], the so-called circular trellis-coded modulation (CTCM), is the backbone of this research. A straightforward way to think of CTCM is as a block code with trellis structure. That is, individual source data blocks are mapped to a particular path through the trellis instead of an output data block. The mapping of the source data blocks to trellis paths is one-toone. The CTCM scheme is characterized by the following four parameters: N (dimensionality of the transmitted signal space), n (size of the source alphabet), D (trellis depth), and B (input source symbol sequence length, or block length). The CTCM system can be described by these four parameters in the form (N,n,D,B). The number of trellis states in CTCM is determined by S = nD. Source alphabet size of n=4 is emphasized in this paper. Trellis depth refers to the number of transitions needed for a given state to reach any other state in the trellis. The block size denotes the size of the input source block as well as the number of transitions in a legal trellis path. The block size must be greater than or equal to the trellis depth plus one (B ≥ D + 1). One drawback of conventional TCM is that the decoder must know the starting state of the encoder before transmission. This is usually accomplished by “padding” the source data with additional zeros to force the encoder to an all-zero state before additional source data is encoded. CTCM alleviates this problem by forcing the starting and ending states of a particular block to be the same. This property of CTCM is known as the state constraint [2,7,8]. The state constraint is satisfied through proper design of the state table. A state table lists, for every state in the trellis, what the next state transition will be for any given input symbol. Since a 4-ary source is emphasized, the state table will have a number of rows equal to the number of states in the trellis, and four columns which correspond to each of the four source symbols. The design of the state table is achieved through the use of a Zech logarithm table, and is discussed in detail in [2], which also presents computer code capable of generating state tables for arbitrary source alphabet size n and trellis depth D. Only one state table exists for a given pair of n and D. An example of a state table is shown in Table I.

With the problem of formulating the state table solved, rules for constructing the transmission symbol table will be presented in this section. However, before assignment rules can be discussed, some general properties of the CTCM trellis must be explored. One of the most important of these is known as the “butterfly structure.” In general, an n-fly is a subset of the trellis in which n initial states transit to the same n next states, which need not be the same as the initial states. In CTCM, the number of n-flys embedded in the trellis is equal to S/n [6]. For example, for a source alphabet size of n = 4, four 4flys are embedded in the 16-state trellis. This can be seen from Table I, where, in one 4-fly, states 1, 3, 9, and 16 all transit to states 1, 2, 4, and 10. Similarly, a 64-state trellis would have sixteen 4-flys. Another important CTCM concept is that of the “minimum error event.” In CTCM, minimum error events occur for the smallest block size that can be employed for a given size trellis, which is B = D + 1 [2,5,6]. Since a source alphabet B size of n=4 is being used, there are 4 legal

TABLE I CTCM STATE TABLE FOR 16-STATE TRELLIS (n=4, D=2). Current Next State State 0 input 1 input 2 input 3 input 1 1 2 4 10 2 3 6 7 12 3 4 10 1 2 4 5 16 8 15 5 6 3 12 7 6 7 12 3 6 7 8 15 5 16 8 9 11 14 13 9 10 4 2 1 10 11 9 13 14 11 12 7 6 3 12 13 14 11 9 13 14 13 9 11 14 15 8 16 5 15 16 5 15 8 16 2 1 10 4

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D

paths and 4 states in the trellis, and thus 4 legal paths associated with each state when using the minimum block size. If one path associated with a state is chosen as a reference, then the minimum error events are the other three legal paths. The goal in assigning transmission symbols is to make the distance between these four paths as large as possible, which will improve the error performance of the system [6]. To increase the distance between the minimum error events, symbols need to be assigned to the butterflies such that all groups of transitions in the minimum error events are simplexes [5]. Forcing the symbols associated with both the incoming branches of a state and the outgoing branches of a state to be simplexes satisfies the distance maximization requirement for the first and last transitions of the minimum error events. Additionally, each of the B-2 intermediate transitions must also be assigned symbols that have simplex distance. This can be summarized in the following rules: Rule 1: The four branches leaving a state must use symbols that form a simplex (called a “source simplex”). Rule 2: The B-2 sets of intermediate branches in the minimum error events associated with a state must use symbols that form a simplex (called an “intermediate simplex”). Rule 3: The four branches entering a state must use symbols that form a simplex (called a “final simplex”). Each of the source simplexes utilize only three dimensions, in the form of (3). Each of the intermediate and final simplexes utilize six dimensions, in the form of (4). The first step in assigning the symbols is the construction of an assignment input matrix (AIM). The AIM contains one symbol associated with each state in the system. If the AIM is constructed properly, it can be used to determine all of the channel symbols used in the system. The second step involves construction of a signal index vector (SIV). The SIV conveys the correspondence between a symbol in the AIM and the state it is associated with. A.

Construction of the AIM

The AIM is an S-by-3 matrix with each row representing, in short-hand notation, a symbol associated with each state in the trellis. Determining which symbol corresponds to which state is unimportant when constructing the AIM and will be addressed later by the SIV. The main focus in constructing the AIM is the set of forthcoming rules. Note that once the AIM is constructed, all of the 4S symbols used in the system can be easily determined. This is due to the fact that each AIM symbol is a member of a source simplex that utilizes only the three dimensions used by the symbol. A computer program can be written that uses an

AIM symbol as an input and then determines the corresponding source simplex. The AIM can be partitioned into four sub-matrices with size (S/4)-by-3. Each of these sub-matrices can be partitioned into four sub-matrices of size (S/16)-by-3, and so on. The partitioning continues until S/4 sub-matrices are obtained, which are 4-by-3 in size. Each of these smallest submatrices defines the sign assignment for one of the S/4 butterflies embedded in the trellis The signs of the symbols in the AIM must be chosen such that each of the smallest sub-matrices is a simplex. Further, signs must also be chosen such that the i-th rows of any four sub-matrices contained within a larger sub-matrix, when taken together, form a simplex. This process continues through the four largest sub-matrices. This process can be summarized as follows: Rule 4: When any four sub-matrices are contained within a larger sub-matrix, the i-th rows of those four sub-matrices, when taken together, should form a simplex. This will ensure that Rule 2 is satisfied. Rule 5: The S/4 sub-matrices, of size 4-by-3, should each form a simplex. This will ensure that Rule 3 is satisfied. Note that, when considering Rule 4, the AIM itself is considered a sub-matrix. That is, the four largest submatrices must adhere to Rule 4. An example of an AIM is shown in Fig. 1. The most straightforward way to create the AIM is to start by assigning the symbols without the signs. This can be done by following Rules 4 and 5, since each of the simplexes defined by those two rules utilize six dimensions. For example, an AIM without signs might look like the absolute value of the AIM shown in Fig. 1. Once this is accomplished, the signs can then be assigned such that Rules 4 and 5 are obeyed. From Rule 5, it can be seen that there are only S/4 simplexes to be applied to the final branches of the minimum error events associated with each state. Since there are S states, it would seem that only one quarter of the trellis has been assigned. This is not the case, and is explained by Rule 6: Rule 6: The simplexes defined by Rules 4 and 5 also define three other simplexes. Rule 6 arises from the following. Each of the simplexes defined by Rules 4 and 5 are constructed of four signals, each of which are members of four different source simplexes. There are a total of sixteen signals defined by these four source simplexes. Additionally, there is only one way to arrange the sixteen signals such that four new simplexes are formed [4]. The simplexes defined by Rules 4 and 5 simply present one of the four new simplexes.

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one of the final simplexes in the AIM. Fig. 2 shows one possible assignment, using one of the final simplexes of Fig. 1. The intermediate groups are best determined by writing out the minimum error paths. Again, from Table I, the group of states 1, 2, 4, and 10 are the origins of the intermediate transitions of the minimum error events associated with state 1. The symbol associated with state 1 has already been assigned, as shown in Fig. 2. Therefore, since states 1, 2, 4, and 10 must form a simplex, states 2, 4, and 10 must be assigned to row 1 of the remaining three final simplexes to satisfy Rule 4. This process continues through examination of the other three intermediate and final groups. The SIV can then be fully constructed. The transpose of an SIV for a 16-state trellis is as follows:

1, 3, 5 -1, 4, 6 2,-3,-6 -2,-4,-5 -1, 4, 7 1, 3, 8 -2,-4,-8 2,-3,-7 -2,-3,-7 2,-4,-8 -1, 3, 8 1, 4, 7 2,-4,-5 -2,-3,-6 1, 4, 6 -1, 3, 5

T

SIV = [1,3,16,9,2,6,5,11,4,7,15,14,10,12,8,13].

Fig. 1. AIM for 16-state trellis with proper partitioning.

However, since there is only one arrangement, the other three simplexes are easily determined, since the twelve signals from which they are formed are already known. Therefore, given the AIM, all symbols used in a CTCM trellis can be determined. B.

Construction of the SIV

As previously stated, one member from each source simplex is included in the AIM. It is now desirable to determine which AIM symbol is associated with which trellis state. This is accomplished through the formulation of an Sby-1 matrix known as a signal index vector (SIV). The SIV entries are one-to-one correspondent with the rows of the AIM. That is, the symbol defined in row i of the AIM is associated with the state contained in SIV(i,1). The SIV is determined through examination of the state table. It is known that the 4-by-3 sub-matrices define the simplexes associated with the final branches of the four legal paths associated with each state (from Rules 3 and 5). It is also known that three other sub-matrices are also defined by these smallest sub-matrices (from Rule 6). Since each of the 4-by-3 sub-matrices consist of one signal from four different states, a group of four states provides the final branches for four different states in the trellis. This is also the case for the intermediate branches. This grouping rule gives rise to Rule 7: Rule 7: Groups of four states appear together four times in providing branches in both intermediate and final transitions in the minimum error event. The final groups can be determined by looking at the butterflies that are embedded in the trellis. From Table I, states 1, 3, 9, and 16 are one final group since they provide the final transition in the minimum error events of states 1, 2, 4, and 10. Therefore, states 1, 3, 9, and 16 can be assigned to

C.

(5)

Assignment Properties

The final rule provides the most important aspect of the assignment process. This rule gives the final piece of information needed to construct the transmission symbol table: Rule 8: If the symbols of an arbitrary state are chosen and assigned, the assignment of all other symbols will be fixed. For any given state, there are symbols that represent source data symbols 0, 1, 2, and 3, which could be expressed in binary as 00, 01, 10, and 11. The only amount of freedom that exists in making the symbol assignment is the act of choosing which symbols of the source simplex will be associated with which source data symbols for one given state. It is easily seen that there are 4! = 24 possible ways to make this assignment. This reason this method fixes all other symbols is as follows. Say a symbol, s, is chosen from the source simplex for an arbitrary state, state X, and assigned to represent any arbitrary source symbol. Assume this source symbol is associated with the transition from state X to some other state in the trellis, say, state Y. The X-to-Y transition occurs B-2

State 1 State 3 State 16 State 9

1, 3, 5 -1, 4, 6 2,-3,-6 -2,-4,-5

Fig. 2. A final simplex and the associated states.

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times in the intermediate branches and again in a final branch. If the AIM has been constructed properly, it can be used to determine all intermediate simplexes and all final simplexes. A computer program can be used to compute all of these simplexes and store them in a look-up table [4]. Assume these look-up tables have been formed and consider the first X-to-Y intermediate branch transition. There is an intermediate simplex that contains s as a member. This simplex can be found in the look-up table and assigned to this intermediate group. Three other symbols have now been assigned (the other three intermediate group members), for a total of four assigned symbols. Each of these four symbols also occur in the next intermediate branch transition. Using the method described above, each of the four symbols “point” to an intermediate simplex in the look-up table. These intermediate simplexes can be assigned to the appropriate groups, and thus sixteen symbols have now been assigned. This process continues through the final branches, at which time 4S/4 = S symbols have been assigned. This method can be continued with each of the three remaining source symbols associated with state X. At this point, all trellis symbols will have been assigned. A 16-state trellis is simple enough that the transmission symbol assignment could be completed by hand in a modest amount of time. However, the reader should keep in mind that the target dimensionality for applications of CTCM, i.e. spread spectrum communication systems with very wide (perhaps ultra wide if computationally feasible) bandwidths, is on the order of 1024 dimensions and greater. Indeed, CTCM becomes a strategic alternative to turbo codes at or above this level – first because the bit error rates are similar in this operating region and also because the block decoding strategy is non-iterative. IV. CONCLUSIONS

assignment is shown to be capable of being carried out in a systematic manner by exploiting various properties of the trellis. Though space does not permit proof, maximum Euclidean distance between error events (erroneous trellis paths) is attained, ensuring optimal error performance. REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

This paper has examined the assignment of simplex signals in a circular trellis-coded modulation scheme. The

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Transactions on Information Theory, vol. 28, pp. 55-67, January 1982. Y.C. Lo, “Circular Trellis-Coded Modulation in Spread Spectrum Communications,” Ph.D. Dissertation, School of Electrical Engineering and Computer Science, Russ College of Engineering and Technology, Ohio University, Athens, OH, August 1997. C. Chen, “The Performance Analysis and Decoding of High Dimensional Trellis-Coded Modulation for Spread Spectrum Communications,” Ph.D. Dissertation, School of Electrical Engineering and Computer Science, Russ College of Engineering and Technology, Ohio University, Athens, OH, August 1997. F.A. Alder, “Symbol Assignment and Performance of Simplex Signaling in High Dimensional Trellis-Coded Modulation,” Master’s Thesis, School of Electrical Engineering and Computer Science, Russ College of Engineering and Technology, Ohio University, Athens, OH, August 1998. J.C. Dill, Y.C. Lo, and A.R. Lindsey, “Circular trellis-coded modulation with high-dimensional simplex signals,” MILCOM98 Conference, Bedford, MA, October 1998. J.C. Dill, Y.C. Lo and A.R. Lindsey, “Butterfly structure in Trellis Coded Modulation,” Globecom 99 – to appear, Rio De Janiero, BRAZIL, November 1999. Y.C. Lo, J.C. Dill, C. Chen, and S.R. Lopez-Permouth, “The encoding of a new trellis-coded modulation with state permutation structure,” 4th International Symposium on Communication Theory and Applications, Ambleside, U.K., July 1997. Y.C. Lo, J.C. Dill, and A.R. Lindsey, “Circular trellis-coded modulation with permuted-state structure,” Defense Applications of Signal Processing, Adelaide, Australia, August 1997. J.C. Dill, A.R. Lindsey, Y.C. Lo, and C. Chen, “A novel errorcorrecting codec for m-ary orthogonal modulations,” Asilomar Conference on Signals, Systems, and Computers, Monterey, CA, November 1996. J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering. John Wiley and Sons, New York, NY, 1965.

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