BLOCK CODES FORMED FROM CATASTROPHIC CONVOLUTIONAL ENCODERS : APPLICATIONS TO BCM

This article describes the construction of block codes derived from trellis termination of a binary catastrophic convolutional encoder. Two construction methods are considered: the Zero Tail (ZT) method and a modified Tail Biting (TB) method. 4-PSK block coded modulation (BCM) schemes are designed based on the constructed codes. These schemes outperform similar schemes derived from the best non-catastrupltic encoder.


INTRODUCTION
Methods for convet:sion of convolutional codes into block codes have been considered in the literature [1,2,3).In all of these methods, non-catastrupltic convolutional encoders were considered Not all of these methods can be applied in a simple way to the case when the convolutional encoder is catastropltic.It is known that block codes derived from catastrupltic convolutional encoders may have row distance 1 , d!', greater than that derived from the best non-catastropltic encoders (with the same rate and constraint length,m + 1).A couple of these catastropltic encoders of rate 1/2 are listed in Tab. 1 [4).Note that tltis table lists the catastrophic encoders for values of m for wltich there is no non-catastrophic encoder whose corresponding distance dr reaches the Heller bound (Form= 9, an exhaustive search•indicated that there are non-catastrophic encoders whose distance d!' reach the Heller bound, eg the encoder with generators in octal form 4534 and 6364 has dtree = d!' = 13) .
The fact that block codes derived from catastrophic encoders may have optimal row distances would justify a more detailed investigation on these encoders.There are also other 1 Weight of the minimum weight path that diverges from the zero state and later reconverges to this state [ 4] 22 m Generators (in octal) d! '  4 72 and 56 8 5 no one 9 7 no one 11 11 6731 and 5237 16 Table 1: Some catastrophic encoders of rate 1/2 whose row distance reaches the Heller bound.
reasons that justifies such investigation: 1) The set of catastropltic encoders should be ouly a small subset of the set of all convolutional encoders [5], p. 308.2) It was recently shown [6), that the iterative construction of Reed-Muller (RM) codes [7) can be some times defined through trellis termination of a catastrophic convolutional encoder using the ZT method.
It is worth to notice that any convolutional encoder with termination detennines a simple encoder for the derived block code and also suggests a maximum likelihood decoding algorithm, which can be implemented by applying the Viterbi algorithm to the tenninated trellis [6).

CATASTROPHIC ENCODERS
A convolutional encoder is said to be catastropltic when the input sequence of infinite weight, generates an output sequence of finite weight (Hamming or Euclidian) [8).This means that a finite number of errors in a discrete output channel may cause an infinite number of errors in the decoded information bits.
Consider the convolutional encoder (n,k,m), where the parameters represent, respectively, the number of outputs, of inputs and of memocy units of the encoder.An algebraic way to check if a (n, 1,m) encoder is catastropltic, is by otaining the Greatest Common Divisor (GCD) of its generator polynomials [5).If the GCD is not a power of D, D 1 , where 1 is a natural number, then the encoder is catastrophic.1).The G(;J) of g< 1 l and g( 2 ) is 1 +D, which is not a power of D. Therefore, the encoder is catastrophic.
Another way to check whether an encoder is catastrophic or not, is to analyze its state diagram.If is there any zero weight loop, with the exception of the zero state self-loop, then the encoder is catastrophic.Example 1 (continued): Consider the state diagram of the encoder of Example 1 represented in Fig. 2, where the convention represents input bit/output bits.It is observed that there is a self-loop over state 1 and therefore the encoder is catastrophic.

TERMINATION OF TRELLIS DIA-GRAMS
In the following, two methods for deriving block codes from a convolutional encoder are described [3]: Zero Tail (ZI): The codewords of the binary linear code fom1ed by this method are all of the output sequences of the encoder when the encoder is initialized to the all-zero state and (K-m) arbitrary data bits are input into the encoder followed by m k zeros, where K is greater than m.The rate of the resultant block code is k <;[.K:"' ), where m is now defined as the maximum length of all k shift registers.
The codewords of the derived block code can be represented by paths in the trellis diagram (the state diagram representation in thne) of the encoder.The trellis termination of encoder of Example 1 for the zr method is shown in Fig. 3. Tail Biting (TB ): The codewords of the binary linear code fom1ed by this method are all output sequences of the encoder when the encoder is initialized to the corresponding last m k bits of an arbitrary K bits sequence and then those K bits are ioput into the encoder.The rate of the resultant block code is Consider the encoder trellis of Example 1 with the cut as shown in Fig. 5.The resultant block code is a (N,K,dH) = (16,7,4), where N, K and dH are the block size, the number of information bits and the minimum Hamming distance, respectively.On a similar way, varying only the instant of the cut, Tab. 2 is obtained.Note that the code rate gets closer to ! (which is the rate of the convolutional code) as the code size increaSes.00 00 00 00 00 00 00 ~00 -~~-'.
A very useful tool in the analysis of trellis termination of convolutional encoders is the A matrix described as follows.Its a;; elements are given by Dh if is there an input that takes the encoder from state i to state j and that produces an output of weight h, otherwise they are considered to be zero [3].
Example 1 (continued): Fig. 6 shows the A matrix of the encoder.It is easy to check its elements through the state diagram of Fig. 2.
The a{] element of the A K matrix represents the weight of codewords that start in state i and end in state j when K bits are input into the encoder.For the case of a two memory units convolutional encoder, the weight distributions of binary linear codes formed from the zr and TB methods, are given by the sum of checked terms of AK in Fig. 7 and Fig. 8, respectively [3].
The reader may have already noticed in Fig. 4 that there is a problem with the use of TB method for a catastrophic encoder: distinct information bits sequences may result in the same codeword.The information sequence 1,1,1 generates the all zero codeword if the encoder was in state "1".And the information sequence 0,0,0 also generates the all zero codeword if the encoder was in state "0".Jt dl] AlD resulted from the use of zr method.This exclusion corresponds in matrix A K to the exclusion of element a:i'i, of the main diagonal from the snm that results in the weight distribution.Fortnnately, for encoders with m k > 1, it is possible to exclude some terms of the main diagonal of A K and the code resulted would not be the same as that obtained by using zr method.This exclusion of terms of the main diagonal is called modified TB.The Theorem in the Appendix shows that with an appropriate exclusion, the derived block code is linear.
Figure 7: Terms that contribute to the weight distribution for the zr method, with m=2.

SEARCH FOR GOOD CONVO-LUTIONAL ENCODERS
This and other Reed-Muller codes are the best block codes forfixedK,dH andN s 32 [7].
Figure 9: Encoder of the obtained code.
Figure 10: Trellis of the obtained code.Based on the idea of getting catastrophic encoders, an exhaustive search has taken place form = 4 and rate 1/2.The algorithm proposed by Larsen [4], which calculates the d" of an (n,1,m) encoder, was used.In fact this algorithm is a corrected version of the algorithm proposed by Bah! et al [10].
One of the two encoders that came out was the one with 5. generator polynomials g( 1 ) = 1 + D + D2 + D4, g(2) = APPLICATIONS TO BCM Fig. 12 shows the BCM scheme used.The mapping of the c 0 , c 1 bits is shown in Tab. 3.This mapping is chosen in such a way that the quadratic Euclidean distance, dE 2 (S, S), between two 4-PSK signals SandS, is proportional to the Hamming distance, dH(c, c), between the corresponding pairs of coded bits c=(c 0 ,c 1 ), C=(c 0 ,c 1 ).It is easy to show that dE 2 (S, SJ=2EsdH (c,c), where E. is the average energy of the 4-PSK constellation's signals.This implies that obtaining a code with the maximum distance d H is the same as obtaining a corresponding code in the Euclidean space with maximnm dmin (Euclidean minimnm distance).
1 + D 2 + D 3 + D 4 and d" = 8, which is the same code listed in [11].The best non-catastrophic encoder, with the same rate and constraint length, has d" = 7.Excluding the codewords corresponding to the elements a{f, 9 s i s z<, the remaining codewords form a linear block code (the states of matrix A are in the following order: {0000,0001,• .. , 1110, 1111}).Fig. 9 shows the encoder and Fig. 10 the trellis representation of the encoder.The termination of this trellis in instant 8 is shown in Fig. 11.Note that this terminated code is a (16,4,8) block code.With the use of a coset it is possible to obtain the (16,5,8) code, which is a Reed-Muller code.With this bit mapping and the ZT terminated trellis as shown in Fig. 11, a BCM scheme with ct;,,n=l6E 5 is obtained.This scheme will be called Scheme I.The Euclidean weight distribution of the BCM scheme can be obtained by modifying the matrix method suggested in [3].
It would be interesting now to compare the ZT termination obtained bere for a catastrophic encoder with the termination of the best non-catastrophic encoder with n = 2 and m = 4, for which dr = 7 [5].For K = 8, the BCM scheme derived from trellis termination of this encoder (here called Scheme 11) has cJ;,,n=14E 5 • Therefore, an asymptotic gain of !Olog10 i! = 0.58 dB of Scheme I over Scheme II is expected.
Fig. 13 shows the performance of the two schemes as a function of the signal-to-noise ratio t; (E& is the energy per bit and No/2 is the bilateral power density of the AWGN noise).In Fig. 13, the block error probability P(e) is given by the approximation of Scheme I over Scheme II can be expected.Using the same procedures described before, four schemes forK = 12 were analyzed: • Scheme illuses ZT method with the non-catastrophic encoder of dr = 7.
• Scheme IV -uses ZT method with the catastrophic encoder of dr = 8.
• Scheme V -uses TB method with the non-catastrophic encoder of d" = 7.

Eb/No [dB]
Figure 13: Error performance of Schemes I and II on an AWGN channel.
o Scheme VI -uses the modified TB method with the catastrophic encoder dr = 8. sults in a Rayleigh fading channel for the schemes given here 7. APPENDIX are described in [12].

FINAL REMARKS AND CON-CLUSIONS
In [13], a trellis coded modulation (TCM) scheme with rate equal to 1 bit per 4-PSK symbol is given, whose constituent non-catastrophic binary convolutional code has dfree= 8 and rate 2 J 4. Other TCM schemes with the same rate and d free given in the recent literature are also referenced in [13) (two of them utilize time-invariant convolutional encoders with rates 2/4 and 4/8 and one of them a time-varying encoder with rate 1/2).4-PSK BCM schemes derived from these encoders should also have rate equal to 1 bit per symbol.ill described in Section 4 for the (16, 4, 8) code, we believe that it is possible to add an appropiate coset to the code of Scheme VI in order to obtain a scheme with rate equal to 1 bit per 4-PSK symbol.The decoding process for this new scheme could be implemented with two identical trellises by applying the same algorithm to bnth trellises in parallel.This property can be advantageous if a fast decoding process is needed.The article considered two construction methods for obtaining lioear block codes derived from a catastrophic convolutional encoder: the ZT method and the TB method.In contrast to the ZT method, the TB method cannot be applied in a simple way to the case when the convolutional encoder is catastrophic.A modified TB construction method was then suggested.The performance of several4-PSK BCM schemes constructed from block codes formed from convolutional encoders were analysed.The schemes formed from catastrophic encoders have a better performance when compared with schemes formed from non-catastrophic encoders.For this reason, catastrophic encoders deserve consideration in the design of these schemes.A better study on the modified TB method is under development.
Theorem: Consider the class of binary catastrophic convolutional encoders whose augmented state diagram has only one zero Hamming weight loop: the self-loop around the all-one state.The code words corresponding to the sum of half of the main diagonal elements a{f of A K, form a linear block code with rate k!: J< 1 .

Proof:
In the following, we consider a model for the encoder with k feedback-free shift registers of equal length m.We assume that some of the tap gains can be equal to zero.Let u denote the input sequence of length K k = ( L + m) k. u is of the form: where so, s1, . . ., Smk-l represents the initial state of the encoder.We form the extended input sequence u 0 , which takes the encoder from all-zero state so = s, = ... = Smk-1 = 0, at time -m to state so, s1, ... , Smk-1 at timeL+m, passing through thesamestateso, s1, ... , Smk-1 at time zero.
Let G be the generator matrix of the convolutional code.The output sequences v generated by the TB method are obtained from the sequences v 0 , by deleting the first ( mk )n components.Therefore, the sequences generated by the TB method are the union of 2mk cosets.The code generated by the ZT method is the zero coset (obtained by setting the encoder state to the all-zero state).
Let u' denote the complementary sequence of u, i.e., u; = u, + 1, where + is a modnio 2 sum.Since u and u' determine two complementary sequences of encoder states and the encoder has a self-loop around the all-one state (as stated in theorem), then, v(= uG) +v'(= u'G) = 0, which implies that v = v'.This means that the cosets generated by representatives of two initial complementary states are the same.However, there is oniy one zero weight loop.Therefore, if one of the two identical cosets corresponding to each pair of complementary states, is excluded, then, the remaining set form a block code with only one information bit less than a block code formed from the conventional TB method.By choosing the code words corresponding to the sum of half of the main diagonal elements a{f of A K, whose initial states are labeled by so = 0, s1, ... , Smk-l• then, the derived block code is linear. Q.E.D-

Figure 6 :
Figure 6: A matrix of the encoder of Example 1.

Figure 8 :
Figure 8: Terms that contribute to the weight distribution for the TB method, with m=2.

Fig. 14
Fig.14shows the performance of Schemes ill, IV, V and VI as a function of l.It can be seen in Fig.14that Scheme IV behaves almost like Scheme V.It can also be seen in Fig.14that the modified TB method (Scheme VI) shows a better performance than the traditional TB method (Scheme V) for 15:-> 5.0 dB.It is worth to notice that Scheme V has a rat.;' of 1 bit per 4 -PSK symbol while Scheme VI has a rate of 11/12 bit per 4-PSK symbol.The Theorem in the Appendix shows that applying the modified TB method to a class of binary catastrophic encoders of rate ~ results in a block code of rate k{;i( 1 .

Table 3 :
Mapping of bits in signal space.