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WeightDistsConvBlock_Wolf

# WeightDistsConvBlock_Wolf - IEEE TRANSACTIONS ON...

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Unformatted text preview: IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 9, SEPTEMBER 1996 Transactions Letters 1049 On the Weight Distribution of Linear Block Codes Formed From Convolutional Codes Jack Keil Wolf and Andrew J. Viterbi Abstract— In this note we explain how to obtain the weight enumerator and the performance of linear block codes formed in several distinct ways from a convolutional code. SSUME a rate 1/2 binary convolutional code of coni straint length (m+ 1). Deﬁne A to be a 2’" by 2’" matrix with i—jth element Aij given by Aij 2 D”, if there is an input (either zero or one) that takes the encoder from state i to state j and produces an output pair of Hamming weight h (h = 0, 1, or 2). Otherwise, assume that A“ = 0. For example, for the simple four—state rate 1/2 binary convolutional encoder shown in Fig. 1, the matrix A is given as 1 D2 0 0 0 0 D D A_D2 1 0 0 0 0 D D. The ordering of the rows and columns is easy to deduce for this example. For later examples, we will always assume that the ﬁrst row and column corresponds to the all-zero state and the last row and column corresponds to the all—one state. Note that for this ordering, the only nonzero terms on the diagonal of the matrix A are the ﬁrst and last, where the ﬁrst (i.e., the upper left most term) is always equal to one. For a noncatastrophic encoder, the last (thelower right-most term) can be equal to D or D2 (but not one). Here we consider ﬁve different methods of constructing binary linear block codes from a rate 1/2 binary convolutional code of constraint length (m + 1). All methods except the ﬁrst two result in block codes with rate equal to 1/2. Of these ﬁve methods, the ﬁrst [the zero tail (ZT) method] and the fourth [the Tailbiting (TB) method] are of most practical interest, the others being included for purposes of generality. As an aside, all of these codes are described by a 2'” state trellis, where m can be much smaller than the number of parity digits or information digits in the code word. In what follows assume that m’gm.7 and k are positive integers such that 0 g m’ S m<k. 1) ZT: The code words of the binary linear code formed by this method are all of the output sequences of the encoder when the encoder is initialized to the all-zero Paper approved by T. M. Aulin, the Editor for Coding and Communications Theory of the IEEE Communications Society. Manuscript received July 20. 1995; revised January 20, 1996. The authors are with the Qualcomm, Inc., San Diego, CA 92121—2779 USA. Publisher Item Identiﬁer S 0090-6778(96)07089-4. Fig. l. Founstate. binary, rate l/2 convolutional encoder. state and (k — m) arbitrary data bits are input into the encoder followed by m zeros. The rate of the resultant block code is (k — m)/2k. 2) Generalized ZT: The code words of the binary linear code formed 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’ zeros. The rate of the resultant block code is (k — m’)/2k. For m’ = m, method 2 reduces to method 1 while if m’ = 07 method 2 reduces to method 3 (to bedescribed next). 3) Direct Truncation: The code words of the binary linear code formed by this method are all of the output sequences of the encoder when the encoder is initialized to the all-zero state and 1c arbitrary data bits are input into the encoder. " 4) TB [1]: The code words of the binary linear code formed by this method are all of the output sequences of the encoder When the encoder is initialized to the state corresponding to the last m binary digits of an arbitrary string of k data digits and then those k data digits are input into the encoder. ‘1 5) Generalized TB [1]: The code words of the binary linear code formed by this: method are all of the output sequences of the encoder wheh the ehcher is initialized to the state corresponding to the last m’ binary digits of an arbitrary string of k data digits (with the last (m—m’) digits of the state set to zero) and then those k data digits are input into the encoder. ‘ The i ~ jth element of [A’“] is the weight enumerator for all output sequences when the encoder starts in state i and ends in state 3' and k binary digits are input into the encoder. The Weight enumerators for the binary linear codes formed by the ﬁve different methods described above are obtained 0090—6778/96\$05.00 © 1996 IEEE 1050 x- —x (d) Fig. 2. Terms of [Ak] which contribute to the weight enumerator (a) ZT, (b) generalized ZT with m’ generalized TB with m’ : 2. Fig. 3. Sixteen~state, binary. rate 1/2 convolutional encoder. by summing speciﬁc terms in To illustrate this we consider a slightly more complicated example of an m z 3, (eight-state) rate 1/2 convolutional code, where the rows and columns of the matrix A correspond to encoder states in the following order: {000,100,010,110,001,101,011,111}. The X’s in Fig. 2(a)—(e) indicate the terms of [Ak] that must be added to give the weight enumerator for the ﬁve different types of block codes. For example, for ZT codes, only the term in the upper left most comer of the matrix [Ah] is used while for TB codes we add the terms on the diagonal of [Ak]. The determination of the weight enumerator for any of the block codes requires raising the matripr (with symbolic coefﬁcients) to the power k. This can be done easily using one of several readily available computer programs. The results obtained using one of them for ZT and TB block codes produced from the 16—state convolutional encoder shown in Fig. 3 are given in Table I. All TB codes in the above table have rate 1/2, while the ZT codes for k : 12,15, and 18 have rates (12 — 4)/24 : 1/3,(15 — 4)/30 : 11/30, and (18 — 4) / 36 = 7/18, respectively. It should be noted that the block length 24 TB code has a smaller minimum Hamming distance (dmin = 5) than the ZT code (dmin : 7) but for the two larger block lengths, the two methods both yield block codes with the same minimum Hamming distance (dmin : 7 By examination of the individual elements of [Ak], one can ﬁnd other interesting block codes derived from the convolu~ tional code. For example, if one examines the entries of [A12], one ﬁnds several entries With minimum Hamming weight seven. One can construct a linear block code of minimum lEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 9, SEPTEMBER 1996 xxxxxxxx INI (6) 2, (0) direct truncation, ((1) TB, and (e) Hamming weight seven by choosing the union of the code words in the ZT code and the code words corresponding to one of the entries with Hamming weight seven. The resultant code is linear since it consists of a group code and one of its cosets. It has rate 9/24 : 3/8 (as compared to the ZT code with rate 8/24 : 1/3) and yet has the same minimum distance as the ZT code. Sometimes, one does not need the weight enumerator itself but rather the weight enumerator evaluated at numerical values of D. For example, consider that one wishes to calculate a union bound to the block error rate for a maximum likelihood decoder for bipolar signaling over an additive white Gaussian noise (AWGN) Channel. If T (D) is the weight enumerator for the block code in question, one knows that the probability of block error is upper—bounded as Perror S Q(sqr(2dminEs/Na)) ‘ eXp(dminES/]Vn) T 1)|D:exp(—-Es/No) where Es/NU is the energy per symbol divided by the noise power density and drum is the minimum distance of the block code. The reason that a one is subtracted from T(D) is to remove the effect of the all—zero code word (D0 = 1). This upper~bound can be obtained by simple matrix manipulations on A with numerical values substituted for its nonzero coef— ﬁcients. The results of using this expression for the ZT and TB codes with k = 12 given in Table I are shown in Fig. 4. Here, the probability of block error for each code is plotted versus the energy per bit divided by the noise power density. Note, that even though the TB code has a smaller minimum distance than the ZT code, it has a better performance because of its higher code rate. For the case of TB codes, the upper—bound can be expressed in terms of the eigenvalues of the matrix A. This is the case since the sum of the diagonal elements of the matrix [Ak] is equal to the sum of the eigenvalues of [A19] and the eigenvalues of [Ak] are equal to the eigenvalues of A raised to the power k. Note that this formulation is convenient if one is interested IEEE TRANSACTIONS ON COMMUNICATIONS, VOL, 44. NO. 9, SEPTEMBER 1996 1051 TABLE I WEIGHT DISTRIBUTION or BLOCK CODES FORMED FROM THE SixteeneSTATE CONVOLUTIONAL CODE Hamming Weight ZT TB ZT TB ZT TB k=12 k=15 k=18 0 1 ~ 1 1 1 1 1 5 - 12 - - - - 6 - 30 - - - - 7 13 84 19 75 25 54 8 12 174 21 195 30 126 9 12 316 24 440 36 258 10 36 522 80 990 128 972 11 37 612 108 1620 208 2376 12 30 608 121 2510 282 4677 13 38 612 210 3720 570 9144 14 34 498 276 4470 939 14706 15 21 316 268 4674 1174 20580 16 13 177 276 4485 1581 27837 17 6 84 252 3720 2006 32940 18 2 38 176 2650 2066 34288 19 l 12 112 1620 2016 33192 20 - - 59 858 1824 28170 21 - - 26 440 1398 21240 22 - - 12 210 956 14508 23 - - 5 75 577 8406 24 - - 2 15 308 4644 25 - - - - 150 2466 26 - - - - - 70 1044 27 - - r - - 32 416 28 - - - - 6 81 29 - - - - - - y 30 - - - - 1 18 Block Error Probability vs. Eb/No: k=12 10'3E § u‘i €104 __ Zero Tail 0 E . “6 . §10'5‘E Tailbiting n . 2 m 10’6 r 10'7 4.5 5 5.5 6 6.5 7 7.5 8 Eb/NO Fig. 4. Block error probability Versus the energy per bit divided by the noise powcr‘dcnsity for the TB and ZT Codes of Table I with k : 12. in the upper-bound for a number of different values of k. A of all rates, and also to nonbinary block codes derivable from more complicated method for obtaining the performance of nonbinary convolutional codes these codes was given in a previous paper [1]. Although for simplicity we have restricted this discussion to binary block codes derivable from a rate 1/2 binary con- volutional code, the techniques discussed above apply equally [l] H. A. Ma and J. K. Wolf, “On tailbiting convolutional codes,” IEEE well to binary block codes derivable from convolutional codes TrdnS- Communa VOL COM-34, I10 2, PP- 104411, Felt 1936- REFERENCES ...
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