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IEEE COMMUNICATIONS LETTERS, VOL. 1, NO. 3, MAY 1997

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Performance of Continuous and Blockwise Decoded Turbo Codes Sergio Benedetto, Fellow, IEEE, and Guido Montorsi, Member, IEEE

Abstract— In this letter we define and evaluate the average maximum-likelihood performance of the three ways of codecoding turbo codes. In all cases the information sequence is bits ( being the length of the interleaver split into blocks of used by the turbo code), that are encoded by the first constituent encoder and, after interleaving, by the second encoder. In the first operation mode, both constituent encoders work in a continuous fashion, whereas in the second, at the end of each block, a suitably chosen sequence of bits is appended to the information block in order to terminate the trellises of both constituent codes. In the third mode, finally, the operation is similar to the second, but, instead of trellis termination, both constituent encoders are simply reset.

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Fig. 1. Block diagram of a parallel concatenated convolutional encoder.

A. Continuous Co-Decoding

I. INTRODUCTION

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URBO codes [1], [2], denoted in the following parallel concatenated convolutional codes (PCCC’s), are now widely recognized as an important, highly performing, new class of concatenated codes. Although in the original proposal [1] the suboptimum iterative decoding algorithm worked in a continuous fashion, many subsequent decoding algorithms required the segmentation of the information sequence into adjacent blocks of bits ( being the length of the interleaver used by the PCCC), and the separate encoding and decoding of each block. This operation, which may also be dictated by system constraints, can be performed in three ways, which will be clarified in the next section. In this letter, we extend the analytical tools developed in [2] to evaluate the average performance of maximumlikelihood (ML) decoded PCCC’s in the three co-decoding configurations. The results will be expressed in terms of upper bounds to the average bit-error probability.

II. THREE WAYS FOR OPERATING THE CO-DECODING OPERATIONS OF PCCC’S A parallel concatenated convolutional encoder is shown in Fig. 1. It refers to the case of a rate 1/3 PCCC obtained by concatenating a rate 1/2 and a rate 1 convolutional codes (CC’s). As previously mentioned, the PCCC can work in three different modes, which will be analyzed in the following. Manuscript received January 1, 1997. This work has been supported by Qualcomm Inc. under a research grant. The associate editor coordinating the review of this paper and approving it for publication was Prof. Y. Bar-Ness. The authors are with Dipartimento di Elettronica, Politecnico di Torino. Publisher Item Identifier S 1089-7798(97)04605-X.

The first alternative consists of keeping both constituent encoders in the states they were in at the end of the previous block and starting the encoding process of the new block from those states. Thus, the trellis of each encoder will evolve in a continuous fashion: we call this first operation mode continuous co-decoding. The evolution of the PCC encoder can be described through the hypertrellis with states, as depicted in Fig. 2(a), where and are the number of states of the two CC’s. The state of the hypertrellis corresponds to the pair of states of the CC’s, and each branch represents all paths that start from the pair of states and end in the pair in exactly steps. Thus, when embedded into a PCCC using an interleaver of length , the CC’s contributions to the final codeword derive from -truncated versions of their input information sequences, or, equivalently, from paths in their trellises of length . In [2] we have shown how to evaluate the labels of each branch of the hypertrellis, under the hypothesis of a uniform interleaver, i.e., an interleaver that, for a given input block of bits with weight , outputs all distinct permutations with equal probability. Let us define with the label of the branch , where and are indeterminate whose exponents refer to the weight of the information and code sequences, respectively. is the input–output weight-enumerating function (IOWEF)1 of the equivalent parallel concatenated block code obtained by enumerating all -truncated sequences of the PCCC joining the hyperstates 1 We use here IOWEF’s instead of the input-redundancy weight enumerating functions (IRWEF) defined in [2] because they permit a more general analysis, encompassing nonsystematic constituent encoders.

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IEEE COMMUNICATIONS LETTERS, VOL. 1, NO. 3, MAY 1997

(a) Fig. 2.

(b)

(c)

The hypertrellises of a PCCC for (a) continuous, (b) truncated, and (c) terminated co-decoding.

and . These labels play the same role of those labeling the trellis branches of a standard convolutional code and can be used to obtain the standard transfer function bound to the bit-error probability [3]. Thus, to evaluate the performance of the PCCC, we start from the IOWEF’s of the CC’s, evaluate the labels of the branches of the hypertrellis, compute the transfer function of the hypertrellis, and, finally, upper bound the average ML bit-error probability as (1) B. Trellis Truncated Co-Decoding The second alternative resets the states of both encoders at the end of each block, so that, when the encoded sequences of both decoders end at states different from the zero state, we have a discontinuity at the end of each block: we call this mode truncated co-decoding. We can use the same procedure described before for continuous co-decoding, and obtain a hypertrellis whose branches represent all paths that start from the state and reach all other states in exactly steps [see Fig. 2(b)]. Since the decoder is reset after each block of information bits, the hypertrellis of Fig. 2 enumerates all possible PCCC code words, so that the transfer function is given by

Fig. 3. Bit-error probability for the rate 1/3 PCCC with 16-state CC’s.

From the point of view of the analysis, we can start from the hypertrellis obtained in the previous case of truncated co-decoding, and attach to it a new part consisting of the hyperpaths starting from any hyperstate and leading to in steps [see Fig. 2(c)]. defining the label of the new branches as (there is just one branch for each state , and we only need the indeterminate since the input bits do not carry information, as they are used merely to terminate the trellises and to increase the Hamming weights of code sequences), we obtain the transfer function as

(2) (3) Substituting bit-error probability.

obtained from (2) into (1) yields the

C. Trellis Terminated Co-Decoding The third alternative consists in terminating the CC’s trellises at the end of each block. Various ways for terminating the CC’s trellises have been suggested in the literature, including ad hoc designed interleavers [4]–[6]. The simplest, yet general way [7], requires the transmission of extra data symbols, corresponding to the encoded versions of a number of information bits, for each CC, equal to the constraint length of the CC’s. Thus, the overall PCCC rate is reduced; for example, in the case of a rate 1/3 PCCC, the new rate is .

obtained from (3) into (1) yields the Substituting bit error probability for the blockwise transmission with trellis termination. III. COMPARISONS AMONG THE THREE SOLUTIONS Consider a rate 1/3 PCCC obtained using as CC’s rate 1/2 systematic recursive convolutional encoders with a number states equal to 16 and 32. To show the different behavior of the PCCC with respect to continuous and block (truncated or with trellis termination) encoding and the corresponding ML decoding, we have evaluated the upper bounds to the bit error probabilities, choosing three interleaver lengths, namely,

BENEDETTO AND MONTORSI: CONTINUOUS AND BLOCKWISE DECODED TURBO CODES

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cantly worse than those of the continuous one, whereas trellis termination is only slightly worse. The differences increase with the signal-to-noise ratio, owing to the fact that the three PCCC’s have different free distances. IV. CONCLUSION In conclusion, a block encoder without trellis termination should be avoided, and, if system-compatible, continuous encoding (and decoding) should be used. Although based on ML upper bounds, these final considerations are confirmed when the suboptimum iterative decoding algorithm is used. REFERENCES Fig. 4.

Bit-error probability for the rate 1/3 PCCC with 32-state CC’s.

. The results are reported in Figs. 3 and 4. Each figure contains nine curves, distinct as follows: for each of the three ways to encode (continuous, truncated, and terminated), the performance of the codes using the three interleaver lengths are reported. A careful examination of the figures suggest the following considerations. • Continuous encoding always yields the best performance. The difference in behavior becomes more significant for increasing number of states of the CC’s. • The performance of the truncated encoder are signifi-

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