| C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans. Commun., vol. 44, no. 10, pp. 1261--1271, Oct. 1996. |
....prefix computations, turbo coding. I. INTRODUCTION C ALCULATING the soft inverse of a finite state machine (FSM) is a key operation in many data detection decoding algorithms. Perhaps the most appreciated application is iterative decoding of concatenated codes, such as turbo codes [1] [2]. However the soft in soft out (SISO) module [3] is widely applicable in iterative and noniterative receivers and signal processing devices (e.g. 4] 7] The soft outputs generated by a SISO may also be thresholded to obtain optimal hard decisions (e.g. producing the same decisions as the ....
....and or precollapsing yields larger implementation area and is not in keeping with our desire to realistically assess the near term feasibility of these algorithms. In particular, we consider a standard parallel concatenated convolutional code (PCCC) with two four state constituent codes [1] [2]. Each of the recursive systematic constituent codes generates parity using the generator polynomial with parity bits punctured to achieve an overall systematic code with rate 1 2. In order to determine the appropriate value for to be used in the MHW SISOs, we ran simulations where each SISO used ....
C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1261--1271, Oct. 1996.
....low weight input patterns. Divsalar and Pollara [4] also proposed the S random interleaver, which is a randomly created interleaver with some restrictions, designed to break low weight input patterns. Other semi random interleaver designs include the original one used by Berrou and Glavieux [6]. Other early attempts at interleaver design include those by Jung and Na han [7] In this paper, interleavers are chosen from a large set of random interleavers based on the low weight input distance properties of the resulting Turbo code. It was stated that a regular block interleaver performs ....
....performance improvement of the Helical interleaver over the Square and Rectangular interleavers is minimal. Its FER performance, however, is somewhat better, though still far from the uniform interleaver. C. Randomised Interleavers The interleaver originally used by Berrou et al. described in [6], is essentially a Square interleaver with some pseudo random perturbations. However, its performance is signi cantly better than a regular Square interleaver with the same dimensions. This is seen by comparing the Berrou Glavieux interleaver (code D) and the Square interleaver (code A) in Fig. ....
Claude Berrou and Alain Glavieux, \Near optimum error correcting coding and decoding: Turbo-codes," IEEE Transactions on Communications, vol. 44, no. 10, pp. 1261-1271, Oct. 1996.
....entropy code and channel code. We also show that iterative decoding of the proposed concatenated code outperforms iterative decoding of previously reported entropy and channel codes that operate at the same overall rate. I. INTRODUCTION The recent proposal of concatenated channel codes [1], 2] has stimulated the field of error correction. The promising performance of the optimum decoding of concatenated codes and the practicality of its suboptimum iterative (turbo) decoding have triggered the study of similar structures in other communications systems. In particular, it is ....
....different constraints over the sequence. Obviously, all constraints have to be satisfied for the detection process. Iterative decoding suggests satisfying each constraint separately and repeating the process. The use of an interleaver and deinterleaver (see Figure 1) helps the iterative detection [1], 8] In iterative decoding, each decoder processes the noisy received sequence and produces a type of information, called extrinsic information, to be used by the other decoders [1] 8] Extrinsic information represents the additional information obtained by applying the constraint of a ....
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Transactions on Communications, vol. 44, pp. 1261 --1271, October 1996.
....reliable communication of sources with a considerable amount of residual or natural redundancy is an important issue. Several studies (e.g. 1] 4] 10, 13, 16, 20, 21] etc. have shown that appropriate use of the source redundancy can signi cantly improve the system performance. Turbo codes [6, 7] have been regarded as one of the most exciting breakthroughs in channel coding, and excellent performance has been demonstrated for uniform i.i.d. sources over AWGN channels. In [11] the authors considered using Turbo codes for sources with memory. However, to the best of our knowledge, the issue ....
....feedback loop from the second constituent decoder to the rst. Each constituent decoder employs the BCJR algorithm [5] and the decoding process is realized in an iterative fashion by exchanging the extrinsic information between the two constituent decoders. In the original work by Berrou et al. [7], extraordinary performance has been demonstrated by using Turbo codes for uniform i.i.d. sources over AWGN channels. Designing Turbo codes for non uniform i.i.d. sources has been recently studied in [22, 23] in which the Turbo decoder is modi ed to take advantage of the source redundancy in the ....
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C. Berrou and A. Glavieux, \Near optimum error correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., Vol. 44, pp. 1261-1271, Oct. 1996.
....errors by the new scheme. 5. SIMULATION RESULTS In a simulation example the performance of the PaLoP system is analyzed. A block length of L = 4096 bit is assumed. As FECEncoder a parallel concatenation of two rate R = 1 3, constraintlength 5, recursive systematic convolutional codes [14] without termination are used, with the generator polynomials G(D) 1 D 3 D 4 , 1 D 3 . As interleaver a block byblock pseudo random algorithm is used. The coded data c is spread over P = 20 packets using a block by block pseudo random packeting algorithm without buffer (b = 0) ....
C. Berrou and Alain Glavieux, "Near Optimum Error Correcting Coding and Decoding: Turbo-Codes," IEEE Transactions on Communications, vol. 44, no. 10, pp. 1261--1271, Oct. 1996.
....transfer (EXIT) chart, information processing characteristic (IPC) 1. Introduction Extrinsic information transfer (EXIT) charts [1] and information processing characteristics (IPC) 2] have been proposed as tools for analysis and design of parallel concatenated codes (called turbo codes [3]) Although both methods are based on average symbol by symbol mutual information, they address very di erent aspects. EXIT charts describe the iterative decoding process of turbo codes by means of the transfer characteristic of the constituent decoders. For each constituent decoder, the transfer ....
C. Berrou and A. Glavieux, \Near optimum error correcting coding and decoding: Turbo-codes," 1271, Oct. 1996.
....transition to state 1000. The following state transitions would 58 be 1100 0110 0011 0001 1000, and from then on the state transition would be con ned in a short cycle consisting of the above 5 states while all the other states remain dormant. We herein cite the following de nition from [16]. De nition: Let a recursive convolutional encoder initially be in any state other than the all zero state. After introducing a certain number of consecutive 0s at the input, the encoder will return to its initial state. This number of 0s required to drive the encoder to its initial state is ....
C. Berrou, and A. Glavieux, \Re ections on the prize paper: `Near optimum error-correcting coding and decoding: turbo codes'," IEEE Inform. Theory Soc. Newsletter, vol.48, pp. 23-31, Jun. 1998.
....1993 when Berrou, Glavieux, and Thitimajshima [14] made a breakthrough by constructing a scheme named Turbo codes, whose performance is historically unmatched: at a channel coding rate of 1 2, and a BER level of 10 , the Turbo codes performance was only 0. 5 dB away from the Shannon limit [15]. The extraordinary error correcting capability of Turbo codes ignited great enthusiasm in the communications society. Since their debut in 1993, within less than 10 years, there has been a huge volume of publications regarding various aspects of Turbo codes. Turbo codes have indeed blossomed ....
....Finally, performance comparisons with tandem schemes are also provided. Finally, in Chapter 7, the thesis is summarized and future research directions are discussed. 13 Turbo Codes Basics In this chapter, we brie y provide the basics of Turbo codes as originally introduced by Berrou et al. [14, 15]. In particular, we rst introduce the so called recursive systematic convolutional encoders, which are used as the constituent encoders in Turbo codes. We then describe the structures of a Turbo code encoder and decoder, and describe the BCJR algorithm which is used in the iterative decoding ....
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C. Berrou and A. Glavieux, \Near optimum error correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1261-1271, Oct. 1996.
....need to perform the Add Compare Select operation followed by the traceback algorithm. As far as the turbo decoding is concerned the requirements are comparably tough considering the increased data rate and the computational e#ort for iterative decoding algorithms. We use an established scheme [5] where the SoftIn SoftOutput modules comprise a modified BCJR algorithm [6] to find the maximum aposteriori probability (MAP) of a symbol. A popular modification implies the usage of the negative logarithm of the probabilities. A further simplification takes advantage of the relation ln(e x ....
C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans . Com . , vol. 44, no. 10, pp. 1261--1271, Oct 1996.
....same input (although in a di erent order) the systematic output of the second encoder is redundant and does not need to be transmitted. Due to the presence of the interleaver, optimal decoding of turbo codes is incredibly complex and therefore impractical. The suboptimal alternative presented in [38, 43] is the iterative decoding principle where the overall decoding problem is broken into two smaller problems (decoding each of the constituent codes) each with locally optimal solutions. Each decoder is implemented by a SISO decoding algorithm where soft decisions are fed to one another as the a ....
C. Berrou, and A. Glavieux, \Near optimum error correcting coding and decoding: Turbo codes," IEEE Transactions on Communications,vol. 44, no. 10, pp. 1261-71, Oct. 1996.
....as any other available Source Side Information (SSI) and provides the V [1 : T ] output sequence. SISO CC P (S k ) VLC Decoder # 2 2 x V # V y E C E V # # # C # C x C Fig. 4. Proposed iterative decoder. In order that each decoder takes advantage of the iterative process [21], independent information must be exchanged between the two decoders, that is the so called ex trinsic information. We thus define E [t] and E [t] the extrinsic information about the bit t provided respectively by the CC decoder and by the VLC decoder. For r 1, E C [t] # C ....
C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," in IEEE Trans. on Commun., pp. 1261--1271, vol. 44, Oct. 1996.
....with cycles, and present a number of open problems for future research. Index Terms Iterative decoding, linear codes, minimum distance, Tanner graphs. I. INTRODUCTION Iterative decoding algorithms on factor graphs [15] have become a subject of much active research in recent years [1] 2] [4], 5] 9] 15] 18] 22] 29] and [30] For example, the wellknown turbo codes and turbo decoding methods [5] 4] constitute a special case of this general approach to the decoding problem. Factor graph representations for turbo codes were introduced in [29] and [30] where it is also shown ....
....minimum distance, Tanner graphs. I. INTRODUCTION Iterative decoding algorithms on factor graphs [15] have become a subject of much active research in recent years [1] 2] 4] 5] 9] 15] 18] 22] 29] and [30] For example, the wellknown turbo codes and turbo decoding methods [5] [4] constitute a special case of this general approach to the decoding problem. Factor graph representations for turbo codes were introduced in [29] and [30] where it is also shown that turbo decoding is an instance of a general decoding procedure, known as the sum product algorithm. Another ....
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1261--1271, 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans. Commun., vol. 44, no. 10, pp. 1261--1271, Oct. 1996.
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C. Berrou, A. Glavieux, and P. Thitimajshima, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, " Near Optimum Error Correcting Coding And Decoding: Turbo-codes," IEEE Trans.Comm. Vol. 44, N10, pp.1262-1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near-optimum errorcorrecting coding and decoding: Turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1261-1271, Oct. 1996.
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C. Berrou and A. Glavieux, " Near Optimum Error Correcting Coding And Decoding : Turbo-codes," IEEE Trans.Comm. Vol. 44, N10, pp.1262-1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near-optimum error-correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1261-1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbocodes, " IEEE Transactions on Communications, vol. 44, pp. 1261-1271, October 1996.
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C. Berrou and A. Glavieux, "Near optimum error-correcting coding and decoding: Turbo codes," IEEE Trans. Commun., vol. 44, pp. 1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux. Near optimum error correcting coding and decoding: Turbo-codes. IEEE Trans. on Comm., 44(10):1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbocodes, " IEEE Trans. Communications, vol. 44, pp. 1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near Optimum Error Correcting Coding and Decoding: Turbo-Codes," IEEE Trans. on Comm., 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Transactions on Communications, vol. 44, pp. 1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbocodes ", IEEE Trans. Comm., Vol. COM-44, Oct. 1996, pp. 1261-1271.
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Berrou, C. and Glavieux, A. "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1261-1271, October 1996.
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C. Berrou and A. Glavieux, \Near optimum error correcting coding and decoding: Turbo-codes," IEEE Transactions on Communications, vol. 44, pp. 1261-1271, October 1996.
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C. Berrou and A. Glavieux. Near optimum error correcting coding and decoding: Turbo-codes. IEEE Trans. Communications, 44(10):1261-1271, Oct. 1996.
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C. Berrou and A. Glavieux. Near optimum error correcting coding and decoding: Turbo-codes. IEEE Transactions on Communications, 44(10):1261--1271, Oct. 1996.
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C. Berroux and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Commun. Lett., vol. 1, pp. 77--79, May 1997.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans. on Comm., vol. 44, pp. 1261--1271, Oct 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: turbo-codes," IEEE Trans. Commun., vol. 44, no. 10, pp. 1261-- 1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: turbo-codes," IEEE Transactions on Communications, vol. 44, pp. 1261 --1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Transactions on Communications, vol. 44, pp. 1261-- 1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans. Commun., vol. 44, no. 10, pp. 1261--1271, October 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: turbo codes," IEEE Transactions on communications,vol. vol. 44, pp. pp. 1261--1271, October 1996.
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C. Berrou and A. Glavieux. Near optimum error correcting coding and decoding: Turbo-codes. IEEE Transactions on Communications, 44(10):1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: turbo codes," IEEE Transactions on communications, vol. vol. 44, pp. pp. 1261--1271, October 1996.
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C. Berrou and A. Glavieux, "Near Optimum Error Correcting Coding and Decoding: Turbo-Codes," IEEE Transactions on Communications, vol. 44, no. 10, pp. 1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux. Near optimum error correcting coding and decoding: Turbo-codes. IEEE Transactions on Communications, 44(10):1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: turbo-codes," IEEE Trans. on Communications, vol. 44, no. 10, October 1996.
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Claude Berrou and Alain Glavieux. Near optimum error correcting coding and decoding: Turbo-codes. IEEE Transactions on Communications, 44(10):1261--1271, October 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Transactions on Communications, vol. 44, pp. 1261--1271, October 1996.
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C. Berrou and A. Glavieux, "Near-optimum error correcting coding and decoding: turbo codes," IEEE Trans. Commun., vol. 44, pp. 1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Trans. Commun., Vol. 44, pp. 1261--1271, Oct. 1996.
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C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans. Commun., vol. 44, pp. 1261--1271, Oct. 1996.
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C Berrou and A Glavieux, "Near optimum error correcting coding and decoding: Turbo-codes," IEEE Transactions on Communications, vol. 44, pp. 1261-1271, September 1996. t0
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