On the spectrum of distances of a multilevel code, decoded by a multistage decoder
TL;DR: In this correspondence, the error coefficient, available in the literature for some multilevel codes with an infinite size signal constellation, is shown to be significantly different from that of a practical finite constellation.
Abstract: In this correspondence we are concerned with the error rates contributed by component codes in a multilevel code, decoded by a multistage decoder. These error rates are required for estimating the performance of the structure, but more importantly, they can serve as a design tool for achieving high coding gain. The latter objective can be reached by approaching a balance among these error rates. For evaluating the error rate contributed by a component code, it is essential to calculate the corresponding spectrum of Euclidean distances. A method for computing this spectrum is developed in this correspondence. The method is general and applicable to any signal constellation, and convolutional as well as block component codes. The error coefficient, available in the literature for some multilevel codes with an infinite size signal constellation, is shown to be significantly different from that of a practical finite constellation. Note that the complete spectrum is essential when codewords (of a component code) with Hamming distance beyond the minimum distance affect the error rate (see, e.g., Turbo codes).
Citations
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TL;DR: The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied and it is shown that an increased minimum Hamming distance can be obtained by using a structured interleavers.
Abstract: The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied. The known turbo codes consist of long random interleavers, whose purpose is to reduce the value of the error coefficients. It is shown that an increased minimum Hamming distance can be obtained by using a structured interleaver. For low bit-error rates (BERs), we show that the performance of turbo codes with a structured interleaver is better than that obtained with a random interleaver. Another important advantage of the structured interleaver is the short length required, which yields a short decoding delay and reduced decoding complexity (in terms of memory). We also consider the use of turbo codes as component codes in multilevel codes. Powerful coding structures that consist of two component codes are suggested. Computer simulations are performed in order to evaluate the reduction in coding gain due to suboptimal iterative decoding. From the results of these simulations we deduce that the degradation in the performance (due to suboptimal decoding) is very small.
44 citations
Cites background or methods from "On the spectrum of distances of a m..."
...In [12], we developed a method for evaluating the BER, contributed by each component code....
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...In [12], we were concerned with the error rates contributed by the component codes in a multilevel code, decoded by a multistage decoder....
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...A method for computing this spectrum is developed in [12]....
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...In [12], it is shown that the BER contributed by the th partition level is upper bounded by the following union bound...
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...A design rule for multilevel coding, suggested in [12], is to approach a balance among the error rates contributed by the component codes at the desired BER....
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29 Jun 1997
TL;DR: In this paper, the impact of the interleaver, embedded in the encoder for a parallel concatenated code is studied and a structure interleavers are suggested, called a reverse block interlever.
Abstract: In this paper, the impact of the interleaver, embedded in the encoder for a parallel concatenated code is studied. In the case where the length of the interleaver is restricted (and especially when the desired BER is small), we suggest using a structure interleaver, called a reverse block interleaver. Using this interleaver, we construct powerful multilevel codes with parallel concatenated codes as component codes.
16 citations
Cites methods from "On the spectrum of distances of a m..."
...In [ 3 ] we were concerned with the error rates contributed by the component codes in a multilevel code, decoded by a multistage decoder (see e.g., [2])....
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...A method for computing this spectrum is developed in [ 3 ]....
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TL;DR: The exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes are derived and an estimate of the decoding error probability is estimated and a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels is shown.
Abstract: In this paper, we derive the exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes. Though we do not compute the distance spectrum of polar codes, the results allow us to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels. Also, we establish the minimal distance between two cosets associated with two paths that differ in two positions. This can be regarded as a first step toward analyzing the weight distributions for successive cancellation list decoding.
14 citations
Cites background from "On the spectrum of distances of a m..."
...3020959 a multistage decoder, it is natural to calculate the weight distribution (WD) between cosets (or the distance spectrum of component codes) at all the stages [5]....
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Proceedings Article•
01 Jan 1988
TL;DR: A coset code is defined by a lattice partition Lambda / Lambda and by a binary encoder C that selects a sequence of cosets of the lattice Lambda as discussed by the authors.
Abstract: Practically all known good constructive coding techniques for bandlimited channels, including lattice codes and various trellis-coded modulation schemes, can be characterized as coset codes. A coset code is defined by a lattice partition Lambda / Lambda ' and by a binary encoder C that selects a sequence of cosets of the lattice Lambda '. The fundamental coding gain of a coset code, as well as other important parameters such as the error coefficient, the decoding complexity, and the constellation expansion factor, are purely geometric parameters determined by C Lambda / Lambda '. The known types of coset codes, as well as a number of new classes that systematize and generalize known codes, are classified and compared in terms of these parameters. >
13 citations
TL;DR: This paper analyzes a multilevel coded modulation system using phase-shift keying (PSK) signaling and transmitting over the additive white Gaussian noise (AWGN) channel using the concept of a "genie-aided communication system" to simplify the analysis.
Abstract: For pt.I see ibid., vol.47, no.7, p.2782-92 (2001). In this paper we analyze a multilevel coded modulation system using phase-shift keying (PSK) signaling and transmitting over the additive white Gaussian noise (AWGN) channel. We reduce the analysis of the overall system to the analysis of each decoding level, considering them as a set of related channels. The concept of a "genie-aided communication system" is introduced to simplify the analysis. A comparison with the conventional coded modulation system, studied in part I of this work, and some results concerning the synthesis and analysis of practical multilevel modulation schemes concludes the paper.
13 citations
References
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Proceedings Article•
01 Jan 1993
7,742 citations
23 May 1993
TL;DR: In this article, a new class of convolutional codes called turbo-codes, whose performances in terms of bit error rate (BER) are close to the Shannon limit, is discussed.
Abstract: A new class of convolutional codes called turbo-codes, whose performances in terms of bit error rate (BER) are close to the Shannon limit, is discussed. The turbo-code encoder is built using a parallel concatenation of two recursive systematic convolutional codes, and the associated decoder, using a feedback decoding rule, is implemented as P pipelined identical elementary decoders. >
5,963 citations
IBM1
TL;DR: A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance.
Abstract: A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth. This is achieved by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance. Soft maximum--likelihood (ML) decoding using the Viterbi algorithm is assumed. Following a discussion of channel capacity, simple hand-designed trellis codes are presented for 8 phase-shift keying (PSK) and 16 quadrature amplitude-shift keying (QASK) modulation. These simple codes achieve coding gains in the order of 3-4 dB. It is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals, which we call "mapping by set partitioning." Based on a new distance measure between binary code sequences which efficiently lower-bounds the Euclidean distance between the corresponding channel signal sequences, a search procedure for more powerful codes is developed. Codes with coding gains up to 6 dB are obtained for a variety of multilevel/phase modulation schemes. Simulation results are presented and an example of carrier-phase tracking is discussed.
4,091 citations
"On the spectrum of distances of a m..." refers methods in this paper
...[ 6 ] C. Schlegel, S. Roy, P. Alexander, and Z. Xiang, “Multiuser projection...
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...A typical code of this kind is based on a set and an level partition chain [ 6 ]....
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Book•
01 Jan 1979
TL;DR: The principles of digital communication and coding are presented and a practical application of these principles, called "Principles of Digital Communication and coding", are presented.
Abstract: Principles of digital communication and coding , Principles of digital communication and coding , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی
1,189 citations
"On the spectrum of distances of a m..." refers methods in this paper
...[ 14 ] I. Weiss, “Limiting distributions in some occupancy problems,” Ann....
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...In these cases , where is the length of . For convolutional codes, the set can be calculated by using the transfer function of the code [ 14 ]....
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TL;DR: A new multilevel coding method that uses several error-correcting codes that makes effective use of soft-decisions to improve the performance of decoding and is superior to other multileVEL coding systems.
Abstract: A new multilevel coding method that uses several error-correcting codes is proposed. The transmission symbols are constructed by combining symbols of codewords of these codes. Usually, these codes are binary error-correcting codes and have different error-correcting capabilities. For various channels, efficient systems can be obtained by choosing these codes appropriately. Encoding and decoding procedures for this method are relatively simple compared with those of other multilevel coding methods. In addition, this method makes effective use of soft-decisions to improve the performance of decoding. The decoding error probability is analyzed for multiphase modulation, and numerical comparisons to other multilevel coding systems are made. When equally complex systems are compared, the new system is superior to other multilevel coding systems.
1,070 citations
"On the spectrum of distances of a m..." refers background or methods in this paper
...Multilevel coding, conceived by Imai and Hirakawa in an early paper [ 2 ], is a combination of several error-correction codes employing a partition of some signal constellation into subsets....
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...[ 2 ] R. Lupas and S. Verd´ u, “Linear multiuser detectors for synchronous...
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