This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms ofstopping sets. We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-sizeturbo stopping setwhich is also a subset of the set of erased positions. Furthermore, we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results. Finally, we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption, and an efficient enumeration algorithm of small-size turbo stopping sets for a particular interleaver. The solution is based on the algorithm proposed by Garelloet al.in 2001 to compute an exhaustive list of all low-weight codewords in a turbo code.
This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms ofstopping sets. We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-sizeturbo stopping setwhich is also a subset of the set of erased positions. Furthermore, we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results. Finally, we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption, and an efficient enumeration algorithm of small-size turbo stopping sets for a particular interleaver. The solution is based on the algorithm proposed by Garelloet al.in 2001 to compute an exhaustive list of all low-weight codewords in a turbo code.
This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation (BP) decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of stopping sets. In the first part we describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or until the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum size turbo stopping set which is also a subset of the set of erased positions. In the second part we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results
In this correspondence, the stopping set of turbo codes with iterative decoding in the binary erasure channel is defined. Block and bit erasure probabilities of turbo codes are studied by using the stopping set analysis. It is found that block and bit erasure probabilities of turbo codes with iterative decoding are higher than those with maximum-likelihood decoding, where the differences are negligible in the error floor region. It is shown that in the error floor region, block and bit erasure probabilities of turbo codes with iterative decoding are dominated by small stopping sets and are asymptotically dominated by low weight codewords.
Low-density parity-check (LDPC) codes in conjunction with iterative decoding based on message-passing algorithms have been shown to achieve excellent performance over a variety of communication channels. In this dissertation, we investigate LDPC codes under two scenarios which present different challenges for providing robust communications. First, LDPC codes are analyzed and designed for time-selective, frequency-nonselective complex-fading channels where both the amplitude and the phase of the transmitted signal are altered by the channel. To combat the fading, we use a pilot symbol-assisted scheme where known pilot symbols are added to the LDPC code and used to help estimate the channel at the receiver. Using asymptotic, infinite-length analysis, we investigate several iterative, message-passing, joint-estimation-and-decoding strategies; optimal energy distribution between pilot and data symbols; and LDPC code design. Several interesting results are obtained regarding unification of analysis and code design. Next, we address the issue of achieving very low error rates with practical, finite-length LDPC codes for binary erasure channels. Finite-length LDPC codes suffer from error floors which limit the achievable error rate, and this error-floor performance is determined by stopping sets. Asymptotic analysis of weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with lower error floors but does not reflect the behavior of sublinearly-sized stopping sets, which can dominate the iterative-decoding, error-floor performance. Thus, we provide a perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, advantages of protograph structure, and effects of precoding. We show for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly-growing stopping sets, and this polynomial relationship can be approximately captured by a single parameter. Further, we begin to address the question, Given finite stopping-set sizes and finite codelengths, do the stopping sets follow the behavior predicted by linear or sublinear analysis?
In this correspondence, we first investigate some analytical aspects of the recently proposed improved decoding algorithm for low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We derive a necessary and sufficient condition for the improved decoding algorithm to successfully complete decoding when the decoder is initialized to guess a predetermined number of guesses after the standard message-passing terminates at a stopping set. Furthermore, we present improved bounds on the number of bits to be guessed for successful completion of the decoding process when a stopping set is encountered. Under suitable conditions, we derive a lower bound on the number of iterations to be performed for complete decoding of the stopping set. We then present a superior, novel improved decoding algorithm for LDPC codes over the binary erasure channel (BEC). The proposed algorithm combines the observation that a considerable fraction of unsatisfied check nodes in the neighborhood of a stopping set are of degree two, and the concept of guessing bits to perform simple and intuitive graph-theoretic manipulations on the Tanner graph. The proposed decoding algorithm has a complexity similar to previous improved decoding algorithms. Finally, we present simulation results of short-length codes over BEC that demonstrate the superiority of our algorithm over previous improved decoding algorithms for a wide range of bit error rates
Assuming iterative decoding for binary erasure channels (BECs), a novel tree-based technique for upper bounding the bit error rates (BERs) of arbitrary, finite low-density parity-check (LDPC) codes is provided and the resulting bound can be evaluated for all operating erasure probabilities, including both the waterfall and the error floor regions. This upper bound can also be viewed as a narrowing search of stopping sets, which is an approach different from the stopping set enumeration used for lower bounding the error floor. When combined with optimal leaf-finding modules, this upper bound is guaranteed to be tight in terms of the asymptotic order. The Boolean framework proposed herein further admits a composite search for even tighter results. For comparison, a refinement of the algorithm is capable of exhausting all stopping sets of size les 13 for irregular LDPC codes of length n ap 500, which requires (13500) ap 1.67 times 1025trials if a brute force approach is taken. These experiments indicate that this upper bound can be used both as an analytical tool and as a deterministic worst-performance (error floor) guarantee, the latter of which is crucial to optimizing LDPC codes for extremely low BER applications, e.g., optical/satellite communications
This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.