IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 8, AUGUST 1999
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A New Technique for Computing the Weight Spectrum of Turbo-Codes Oscar Y. Takeshita, Member, IEEE, Marc P. C. Fossorier, Member, IEEE, and Daniel J. Costello, Jr., Fellow, IEEE
Abstract— We obtain an accurate and simple analytical approximation to the bit error probability of maximum-likelihood decoding of parallel concatenated convolutional codes (Turbocodes) by extending a result from the paper by Fossorier et al., which investigates the weight enumerating function of terminated convolutional codes, to compute their conditional weight enumerating function. Index Terms—Bit-error rate, cutoff rate, terminated convolutional codes, Turbo-codes, weight spectrum.
I. INTRODUCTION
T
HE introduction of the concept of a uniform interleaver in [1] was an important step in the derivation of an analytical upper bound to the bit-error rate (BER) performance of parallel concatenated convolutional codes (PCCC’s) or Turbo-codes [2]. However, approximations and difficult computational procedures were still needed to derive the bound because, in principle, the complete weight-enumerating function (WEF) of a long terminated convolutional code is needed. In [3], a more accurate and systematic approximation to the bound is presented. However, due to a numerical precision problem inherent to the methodology, only block lengths up to about 1000 can be considered. It is also mentioned in [3] that the upper bounds are useful only at signal-to-noise ratios (SNR’s) larger than the channel cutoff rate since, at smaller SNR’s, the usual union bound diverges rapidly, especially for large block lengths. Thus, the traditional union bound is primarily useful for approximating the error-floor region of the BER performance. (We prefer to call these upper bounding techniques approximations because they are computed assuming a uniform interleaver and for large block lengths the complete WEF of the terminated component code is difficult to obtain.) Recently, several new bounds that estimate the BER performance more accurately at SNR’s smaller than the cutoff rate have also been proposed [4]–[6]. In this letter we are interested in approximations to the traditional union bound that significantly reduce the computational
Manuscript received January 4, 1999. The associate editor coordinating the review of this letter and approving it for publication was Prof. N. C. Beaulieu. This work was supported by the National Science Foundation under Grant NCR95-22939 and Grant CCR-97-32959 and by NASA under Grant NAG5557 and Grant NAG5-931. O. Y. Takeshita and D. J. Costello, Jr. are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46656 USA (e-mail:
[email protected]). M. P. C. Fossorier is with the Department of Electrical Engineering, University of Hawaii, Honolulu, HI 96822 USA. Publisher Item Identifier S 1089-7798(99)07363-9.
difficulties. Since the error-floor is mostly affected by a few codewords of small weight, a simple approach that depends on an inspection of the low weight codewords only has been suggested in [7]. In [8], a technique to compute the low weight codeword portion of the WEF of terminated convolutional codes was presented. In this letter, we propose a simple way to obtain an accurate estimate of the BER performance of PCCC’s by extending the result of [8] to compute the low weight codeword portion of the conditional WEF of terminated convolutional codes. This approximation can be written in closed form as a function of the block length. No numerical precision problems have been encountered using this approach. II. CONDITIONAL WEF OF A TERMINATED CONVOLUTIONAL CODE Let be a convolutional code of rate with memory and free distance , generated by a systematic feedback derived from encoder. A terminated convolutional code with information blocks forms an block code. (Note that the redundant bits in the block code may contain some nonzero input bits for terminating the trellis.) A single error event is defined to be a path that diverges from the correct path once and remerges once. In [8], it is shown of all codewords of weight that the WEF in is a linear function of for larger than or equal to (Note that is the maximum length of all a constant which is on the possible single error events of weight in The WEF can then be written as order of
where is a constant greater than or equal to 0. Assuming that the all zero sequence is transmitted, it can be shown that i.e., codewords the set of codewords enumerated by are all single error events [8]. A simple with as a function of is justification of the linearity of is that for a given single error event, its multiplicity in given by one plus its total number of possible shifts, starting with the error event aligned at the beginning of the block and ending with it aligned at the end of the block. (This is true for any single error event regardless of its weight.) is required The conditional WEF instead of the WEF of for the computation of union bounds for PCCC’s. Regardless of its position in the code sequence, a single error event of weight that does not involve the termination bits is associated and a constant redundancy with a constant input weight Therefore, by a similar argument weight such that
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IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 8, AUGUST 1999
TABLE I CODEWORD MULTIPLICITIES FOR TWO TERMINATED CONVOLUTIONAL CODES
as in the derivation of the WEF in [8], we get the following expression relating the WEF to the conditional WEF
Fig. 1. Simulation curves and analytical approximations for a PCCC.
the corresponding conditional WEF of the can be computed as PCCC with component code
(2) where if we assume a uniform interleaver. An approximate bit error probability for the PCCC is then given by
and
This expression is valid for but because of the as a termination bits, to guarantee the linearity of must be larger than or equal to (Note function of ’s is a constant but that can be smaller that each of the as than 0.) We then define the conditional WEF of
(3)
where For large we can drop the constant term the following simplified form:
and obtain
(1)
can be obtained by computing the The coefficients for two difference between the conditional WEF’s of and consecutive block lengths, say i.e., we need two equations to solve for the coefficients This can be accomplished in a straightforward manner, is small for convolutional codes with small since we can use a simple search memory. To determine algorithm. III. WEIGHT SPECTRUM
OF
PCCC’S
Using the technique developed in [1], from the conditional of a rate terminated convolutional code WEF
is the rate of the PCCC.
IV. DIVERGENCE OF THE UNION BOUND AT THE CUTOFF RATE If we substitute (2) in (3), we obtain a very good approximation to the error-floor region of the BER performance curve that coincides with the example bounds found in [1], [3]. However it does not explain the divergence of the union bound beyond the cutoff rate discussed in [3], because the results in A Section II account only for the terms with simple way to take into account the divergence of the bound in our technique is to caused by the terms with add the following term to the conditional WEF of the PCCC (4) which overbounds the contribution to the conditional WEF of the codewords of total weight equal to half of the code length. This is reasonable because a large fraction of the codewords have weight about half the code length. In this overbounding,
TAKESHITA et al.: COMPUTING THE WEIGHT SPECTRUM FOF TURBO-CODES
we assume that we have a total of such codewords and both information bits and parity bits have half of their maximum possible weight.
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the divergence of the bound beyond the cutoff rate:
V. EXAMPLE We use as an example the rate 1/2 convolutional code with and with memory generator matrix Since we can use the technique described in Section II to compute its conditional WEF up to weight We can easily verify that and therefore we compute the conditional WEF for two terminated codes with information block lengths and as shown in Table I. Each row corresponds to an information weight and each column corresponds to For each pair the first entry a redundancy weight corresponds to the multiplicity of codewords for the terminated and the second entry to the multiplicity of code with codewords for the terminated code with Now we must compute the ’s. As an example, to we use the element (20, 23) of Table I and compute solve the following system of linear equations:
i ii
Subtracting (i) from (ii), we obtain Then, using the approximation given in (1), we obtain the following set of conditional WEF’s for
(5)
does not appear be(The term corresponding to can be nonzero, but is always .) cause We can now obtain the approximate conditional WEF of the PCCC by using (2) and (5) and adding (4), which accounts for
By inspecting we can see that the minimum caused by a weight 3 input distance of the PCCC is sequence. Now it is easy to obtain a closed form expression for the approximation to the BER of the PCCC as a function of by applying (3). Approximations to the BER of this PCCC are and along with shown in Fig. 1 for simulation results. We note that the approximation follows the simulation almost exactly at SNR’s above the cutoff rate limit and that the correction term causes the approximation to diverge sharply at lower SNR’s, which is consistent with the behavior of the union bound. REFERENCES [1] S. Benedetto and G. Montorsi, “Unveiling turbo codes: some results on parallel concatenated coding schemes,” IEEE Trans. Inform. Theory, vol. 42, pp. 409-428, Mar. 1996. [2] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes,” in Proc. ICC, May 1993, pp. 1064–1070. [3] D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, “Transfer function bounds on the performance of turbo codes,” JPL TDA Progress Rep., vol. 42-122, pp. 44–55, Aug. 1995. [4] T. M. Duman and M. Salehi, “New performance bounds for turbo codes,” IEEE Trans. Commun., vol. 46, pp. 717–723, June 1998. [5] A. Viterbi and A. Viterbi, “Improved union bound on linear codes for the input-binary AWGN channel, with applications to turbo codes,” in Proc. IEEE Int. Symp. on Information Theory, Boston, MA, Aug. 16–21, 1998, p. 29. [6] I. Sason and S. Shamai (Shitz), “Improved upper bounds on the performance of parallel serial concatenated turbo codes via their ensemble distance spectrum,” in Proc. IEEE Int. Symp. on Information Theory, Boston, MA, Aug. 16–21, 1998, p. 30. [7] P. Robertson, “Illuminating the structure of code and decoder of parallel concatenated recursive systematic (turbo) codes,” in Proc. IEEE Globecom Conf., Dec. 1994, pp. 1298–1303. [8] M. P. C. Fossorier, S. Lin, and D. J. Costello, Jr., “On the weight distribution of terminated convolutional codes,” IEEE Trans. Inform. Theory, vol. 45, pp. 1646–1648, July 1999.
