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Error correction coding (ECC) for optical communication and persistent storage systems require high rate codes that enable high data throughput and low residual errors. Recently, different concatenated coding schemes were proposed that are based on binary Bose-Chaudhuri-Hocquenghem (BCH) codes that have low error correcting capabilities. Commonly, hardware implementations for BCH decoding are based on the Berlekamp-Massey algorithm (BMA). However, for single, double, and triple error correcting BCH codes, Peterson's algorithm can be more efficient than the BMA. The known hardware architectures of Peterson's algorithm require Galois field inversion. This inversion dominates the hardware complexity and limits the decoding speed. This work proposes an inversion-less version of Peterson's algorithm. Moreover, a decoding architecture is presented that is faster than decoders that employ inversion or the fully parallel BMA at a comparable circuit size.
Generalized concatenated (GC) codes with soft-input decoding were recently proposed for error correction in flash memories. This work proposes a soft-input decoder for GC codes that is based on a low-complexity bit-flipping procedure. This bit-flipping decoder uses a fixed number of test patterns and an algebraic decoder for soft-input decoding. An acceptance criterion for the final candidate codeword is proposed. Combined with error and erasure decoding of the outer Reed-Solomon codes, this bit-flipping decoder can improve the decoding performance and reduce the decoding complexity compared to the previously proposed sequential decoding. The bit-flipping decoder achieves a decoding performance similar to a maximum likelihood decoder for the inner codes.
The introduction of multiple-level cell (MLC) and triple-level cell (TLC) technologies reduced the reliability of flash memories significantly compared with single-level cell flash. With MLC and TLC flash cells, the error probability varies for the different states. Hence, asymmetric models are required to characterize the flash channel, e.g., the binary asymmetric channel (BAC). This contribution presents a combined channel and source coding approach improving the reliability of MLC and TLC flash memories. With flash memories data compression has to be performed on block level considering short-data blocks. We present a coding scheme suitable for blocks of 1 kB of data. The objective of the data compression algorithm is to reduce the amount of user data such that the redundancy of the error correction coding can be increased in order to improve the reliability of the data storage system. Moreover, data compression can be utilized to exploit the asymmetry of the channel to reduce the error probability. With redundant data, the proposed combined coding scheme results in a significant improvement of the program/erase cycling endurance and the data retention time of flash memories.
Automotive computing applications like AI databases, ADAS, and advanced infotainment systems have a huge need for persistent memory. This trend requires NAND flash memories designed for extreme automotive environments. However, the error probability of NAND flash memories has increased in recent years due to higher memory density and production tolerances. Hence, strong error correction coding is needed to meet automotive storage requirements. Many errors can be corrected by soft decoding algorithms. However, soft decoding is very resource-intensive and should be avoided when possible. NAND flash memories are organized in pages, and the error correction codes are usually encoded page-wise to reduce the latency of random reads. This page-wise encoding does not reach the maximum achievable capacity. Reading soft information increases the channel capacity but at the cost of higher latency and power consumption. In this work, we consider cell-wise encoding, which also increases the capacity compared to page-wise encoding. We analyze the cell-wise processing of data in triple-level cell (TLC) NAND flash and show the performance gain when using Low-Density Parity-Check (LDPC) codes. In addition, we investigate a coding approach with page-wise encoding and cell-wise reading.
Spatial modulation is a low-complexity multipleinput/ multipleoutput transmission technique. The recently proposed spatial permutation modulation (SPM) extends the concept of spatial modulation. It is a coding approach, where the symbols are dispersed in space and time. In the original proposal of SPM, short repetition codes and permutation codes were used to construct a space-time code. In this paper, we propose a similar coding scheme that combines permutation codes with codes over Gaussian integers. Short codes over Gaussian integers have good distance properties. Furthermore, the code alphabet can directly be applied as signal constellation, hence no mapping is required. Simulation results demonstrate that the proposed coding approach outperforms SPM with repetition codes.
Spatial modulation (SM) is a low-complexity multiple-input/multiple-output transmission technique that combines index modulation and quadrature amplitude modulation for wireless communications. In this work, we consider the problem of link adaption for generalized spatial modulation (GSM) systems that use multiple active transmit antennas simultaneously. Link adaption algorithms require a real-time estimation of the link quality of the time-variant communication channels, e.g., by means of estimating the mutual information. However, determining the mutual information of SM is challenging because no closed-form expressions have been found so far. Recently, multilayer feedforward neural networks were applied to compute the achievable rate of an index modulation link. However, only a small SM system with two transmit and two receive antennas was considered. In this work, we consider a similar approach but investigate larger GSM systems with multiple active antennas. We analyze the portions of mutual information related to antenna selection and the IQ modulation processes, which depend on the GSM variant and the signal constellation.
Error correction coding based on soft-input decoding can significantly improve the reliability of flash memories. Such soft-input decoding algorithms require reliability information about the state of the memory cell. This work proposes a channel model for soft-input decoding that considers the asymmetric error characteristic of multi-level cell (MLC) and triple-level cell (TLC) memories. Based on this model, an estimation method for the channel state information is devised which avoids additional pilot data for channel estimation. Furthermore, the proposed method supports page-wise read operations.
Reliability is a crucial aspect of non-volatile NAND flash memories, and it is essential to thoroughly analyze the channel to prevent errors and ensure accurate readout. Es-timating the read reference voltages (RRV s) is a significant challenge due to the multitude of physical effects involved. The question arises which features are useful and necessary for the RRV estimation. Various possible features require specialized hardware or specific readout techniques to be usable. In contrast we consider sparse histograms based on the decision thresholds for hard-input and soft-input decoding. These offer a distinct advantage as they are derived directly from the raw readout data without the need for decoding. This paper focuses on the information-theoretic study of different features, especially on the exploration of the mutual information (MI) between feature vector and RRV. In particular, we investigate the dependency of the MI on the resolution of the histograms. With respect to the RRV estimation, sparse histograms provide sufficient information for near-optimum estimation.
Four-Dimensional Hurwitz Signal Constellations, Set Partitioning, Detection, and Multilevel Coding
(2021)
The Hurwitz lattice provides the densest four-dimensional packing. This fact has motivated research on four-dimensional Hurwitz signal constellations for optical and wireless communications. This work presents a new algebraic construction of finite sets of Hurwitz integers that is inherently accompanied by a respective modulo operation. These signal constellations are investigated for transmission over the additive white Gaussian noise (AWGN) channel. It is shown that these signal constellations have a better constellation figure of merit and hence a better asymptotic performance over an AWGN channel when compared with conventional signal constellations with algebraic structure, e.g., two-dimensional Gaussian-integer constellations or four-dimensional Lipschitz-integer constellations. We introduce two concepts for set partitioning of the Hurwitz integers. The first method is useful to reduce the computational complexity of the symbol detection. This suboptimum detection approach achieves near-maximum-likelihood performance. In the second case, the partitioning exploits the algebraic structure of the Hurwitz signal constellations. We partition the Hurwitz integers into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is larger than in the original set. This enables multilevel code constructions for the new signal constellations.
In this article, we give the construction of new four-dimensional signal constellations in the Euclidean space, which represent a certain combination of binary frequency-shift keying (BFSK) and M-ary amplitude-phase-shift keying (MAPSK). Description of such signals and the formulas for calculating the minimum squared Euclidean distance are presented. We have developed an analytic building method for even and odd values of M. Hence, no computer search and no heuristic methods are required. The new optimized BFSK-MAPSK (M = 5,6,···,16) signal constructions are built for the values of modulation indexes h =0.1,0.15,···,0.5 and their parameters are given. The results of computer simulations are also provided. Based on the obtained results we can conclude, that BFSK-MAPSK systems outperform similar four-dimensional systems both in terms of minimum squared Euclidean distance and simulated symbol error rate.