In this paper we study the iterative decoding behavior of time-invariant and time-varying LDPC convolutional codes derived by unwrapping QC LDPC block codes. In particular, for a time-varying LDPC convolutional code, we show that the minimum pseudo-weight of the convolutional code is at least as large as the minimum pseudo-weight of the underlying QC code. We also prove that the unwrapped convolutional codes have fewer short cycles than the QC codes. These results taken together lead to improved BER performance in the low-to-moderate SNR region, where the decoding behavior is influenced by the complete pseudo-codeword spectra and by the Tanner graph cycle histogram, with the time-varying convolutional codes outperforming both the underlying QC block codes and their time-invariant convolutional counterparts.
In this paper we study the iterative decoding behavior of time-invariant and time-varying LDPC convolutional codes derived by unwrapping QC LDPC block codes. In particular, for a time-varying LDPC convolutional code, we show that the minimum pseudo-weight of the convolutional code is at least as large as the minimum pseudo-weight of the underlying QC code. We also prove that the unwrapped convolutional codes have fewer short cycles than the QC codes. These results taken together lead to improved BER performance in the low-to-moderate SNR region, where the decoding behavior is influenced by the complete pseudo-codeword spectra and by the Tanner graph cycle histogram, with the time-varying convolutional codes outperforming both the underlying QC block codes and their time-invariant convolutional counterparts.
LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message passing decoding. In this dissertation, we present several methods of deriving families of time-varying and time-invariant LDPC convolutional codes from LDPC block codes. We demonstrate that the derived LDPC convolutional codes significantly outperform the underlying LDPC block codes, and we investigate the reasons for these 'convolutional gains'. It is well known that cycles in the Tanner graph representation of a sparse code affect the iterative decoding algorithm, with short cycles generally pushing its performance further away from optimum. Hence it is common practice to design codes that do not contain short cycles, so as to obtain independent messages in at least the initial iterations of the decoding process. We show that the derived LDPC convolutional codes have better graph-cycle properties than their block code counterparts. In particular, we show that the unwrapping process that is used to derive LDPC convolutional codes from LDPC block codes can 'break' some cycles of the underlying LDPC block code, but cannot create any shorter cycles. We prove that any cycle in an LDPC convolutional code always maps to a cycle of the same or smaller length in the underlying LDPC block code. Minimum distance is an important code design parameter when the channel quality is high, since codewords that are minimum distance apart are the most likely to cause errors with ML or near-ML decoding. In this case, the minimum distance determines the so-called error floor behavior in the performance curve corresponding to high channel signal-to-noise ratios. Thus studying the minimum distance properties of a code family gives insight into its error floor behavior. We use asymptotic methods to calculate a lower bound on the free distance of several ensembles of asymptotically good LDPC convolutional codes derived from protograph-based LDPC block codes. Further, we show that the free distance to constraint length ratio of the LDPC convolutional codes exceeds the minimum distance to block length ratio of the corresponding LDPC block codes. Message-passing iterative decoders for LDPC block codes are known to be subject to decoding failures due to so-called pseudo-codewords. These failures can cause the large signal-to-noise ratio performance of message-passing iterative decoding to be worse than that predicted by the maximum-likelihood decoding union bound. We address the pseudo-codeword problem from the convolutional code perspective. In particular, we show that the minimum pseudo-weight of an LDPC convolutional code is at least as large as the minimum pseudo-weight of an underlying LDPC block code. This result, which parallels a well-known relationship between the minimum Hamming weight of convolutional codes and the minimum Hamming weight of their quasi-cyclic counterparts, is due to the fact that every pseudo-codeword in the LDPC convolutional code induces a pseudo-codeword in the LDPC block code with pseudo-weight no larger than that of the convolutional pseudo-codeword. More generally, we demonstrate a difference in the weight spectra of LDPC block and convolutional codes that leads to improved performance at low-to-moderate signal-to-noise ratios for the convolutional codes, a conclusion supported by simulation results.
Iterative message-passing decoders for low-density parity-check (LDPC) block codes are known to be subject to decoding failures due to so-called pseudo-codewords. These failures can cause the large signal-to-noise ratio performance of message-passing decoding to be worse than that predicted by the maximum-likelihood decoding union bound. In this paper we study the pseudo-codeword problem for the class of LDPC convolutional codes decoded continuously using an iterative, sliding window, message-passing decoder. In particular, for an LDPC convolutional code derived by unwrapping a quasi-cyclic LDPC block code, we show that the free pseudo-weight of the convolutional code is at least as large as the minimum pseudo-weight of the underlying quasi-cyclic code. This result parallels the well-known relationship between the free Hamming distance of convolutional codes and the minimum Hamming distance of their quasi-cyclic counterparts. Finally, simulation results are included that show improved performance for unwrapped LDPC convolutional codes compared to their underlying quasi-cyclic codes
Low-density parity-check (LDPC) block codes invented by Gallager in 1962 achieve exceptional error performance on a wide variety of communication channels. LDPC convolutional codes are the convolutional counterparts of LDPC block codes. This dissertation describes techniques for the design of LDPC convolutional codes, and analyzes their distance properties and iterative decoding convergence thresholds. The construction of time invariant LDPC convolutional codes by unwrapping the Tanner graph of algebraically constructed quasi-cyclic LDPC codes is described. The convolutional codes outperform the quasi-cyclic codes from which they are derived. The design of parity-check matrices for time invariant LDPC convolutional codes by the polynomial extension of a base matrix is proposed. An upper bound on free distance, proving that time invariant LDPC codes are not asymptotically good, is obtained. The Tanner graph is used to describe a pipelined message passing based iterative decoder for LDPC convolutional codes that outputs decoding results continuously. The performance of LDPC block and convolutional codes are compared for fixed decoding parameters like computational complexity, processor complexity, and decoding delay. In each case, the LDPC convolutional code performance is at least as good as that of LDPC block codes. An analog circuit to implement pipelined decoding of LDPC convolutional codes is proposed. The distance properties of a permutation matrix based (time varying) ensemble of (J,K) regular LDPC convolutional codes is investigated. It is proved that these codes (for J > 2) have free distance increasing linearly with constraint length, i.e., they are asymptotically good. Further, the asymptotic free distance to constraint length ratio for the convolutional codes is several times larger than the minimum distance to block length ratio for corresponding LDPC block codes. Iterative decoding of terminated LDPC convolutional codes, based on the ensemble mentioned above, is analyzed, assuming transmission over a binary erasure channel. The structured irregularity of the codes leads to significantly better convergence thresholds compared to corresponding LDPC block codes. At the calculated thresholds, both the bit and block error probability can be made arbitrarily small. The results obtained suggest that the thresholds approach capacity with increasing J.
Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of time-invariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework. Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the --- mostly moderate --- decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.
Message-passing iterative decoders for low-density parity-check (LDPC) block codes are known to be subject to de- coding failures due to so-called pseudocodewords. These failures can cause the large signal-to-noise ratio (SNR) performance of message-passing iterative decoding to be worse than that predicted by the maximum-likelihood (ML) decoding union bound. In this paper, we address the pseudocodeword problem from the convolutional code perspective. In particular, we compare the performance of LDPC convolutional codes with that of their wrapped quasi-cyclic block versions and we show that the minimum pseudoweight of an LDPC convolutional code is at least as large as the minimum pseudoweight of an underlying quasi-cyclic code. This result, which parallels a well-known rela- tionship between the minimum Hamming weight of convolutional codes and the minimum Hamming weight of their quasi-cyclic counterparts, is due to the fact that every pseudocodeword in the convolutional code induces a pseudocodeword in the block code with pseudoweight no larger than that of the convolutional code's pseudocodeword. This difference in the weight spectra leads to improved performance at low-to-moderate SNRs for the convolutional code, a conclusion supported by simulation results. Index Terms—Convolutional codes, linear programming de- coding, low-density parity-check (LDPC) codes, message-passing iterative decoding, pseudocodewords, pseudoweights, quasi-cyclic codes.
Time-invariant low-density parity-check convolutional codes (TI LDPC-CCs) can be represented by a polynomial-domain parity-check matrix derived from the corresponding quasi-cyclic (QC) LDPC block codes (LDPC-BCs), while time-varying (TV) LDPC-CCs can be obtained by unwrapping the parity-check matrices of LDPC-BCs. The cycle enumerators for TI and TV LDPC-CCs are compared. Based on the analysis of the graphical structures of short cycles in HT(D), we introduce a method of designing the polynomial syndrome former matrix HCRT(D) for LDPC-CCs. It eliminates short cycles and shows improved decoding performance on an additive white Gaussian noise (AWGN) channel with lower bit error ratio (BER) curves.