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Nowadays, most digital modulation schemes are based on conventional signal constellations that have no algebraic group, ring, or field properties, e.g. square quadrature-amplitude modulation constellations. Signal constellations with algebraic structure can enhance the system performance. For instance, multidimensional signal constellations based on dense lattices can achieve performance gains due to the dense packing. The algebraic structure enables low-complexity decoding and detection schemes. In this work, signal constellations with algebraic properties and their application in spatial modulation transmission schemes are investigated. Several design approaches of two- and four-dimensional signal constellations based on Gaussian, Eisenstein, and Hurwitz integers are shown. Detection algorithms with reduced complexity are proposed. It is shown, that the proposed Eisenstein and Hurwitz constellations combined with the proposed suboptimal detection can outperform conventional two-dimensional constellations with ML detection.
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.
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.
Multi-dimensional spatial modulation is a multipleinput/ multiple-output wireless transmission technique, that uses only a few active antennas simultaneously. The computational complexity of the optimal maximum-likelihood (ML) detector at the receiver increases rapidly as more transmit antennas or larger modulation orders are employed. ML detection may be infeasible for higher bit rates. Many suboptimal detection algorithms for spatial modulation use two-stage detection schemes where the set of active antennas is detected in the first stage and the transmitted symbols in the second stage. Typically, these detection schemes use the ML strategy for the symbol detection. In this work, we consider a suboptimal detection algorithm for the second detection stage. This approach combines equalization and list decoding. We propose an algorithm for multi-dimensional signal constellations with a reduced search space in the second detection stage through set partitioning. In particular, we derive a set partitioning from the properties of Hurwitz integers. Simulation results demonstrate that the new algorithm achieves near-ML performance. It significantly reduces the complexity when compared with conventional two-stage detection schemes. Multi-dimensional constellations in combination with suboptimal detection can even outperform conventional signal constellations in combination with ML detection.
This work proposes a construction for low-density parity-check (LDPC) codes over finite Gaussian integer fields. Furthermore, a new channel model for codes over Gaussian integers is introduced and its channel capacity is derived. This channel can be considered as a first order approximation of the additive white Gaussian noise channel with hard decision detection where only errors to nearest neighbors in the signal constellation are considered. For this channel, the proposed LDPC codes can be decoded with a simple non-probabilistic iterative decoding algorithm similar to Gallager's decoding algorithm A.
This work proposes a lossless data compression algorithm for short data blocks. The proposed compression scheme combines a modified move-to-front algorithm with Huffman coding. This algorithm is applicable in storage systems where the data compression is performed on block level with short block sizes, in particular, in non-volatile memories. For block sizes in the range of 1(Formula presented.)kB, it provides a compression gain comparable to the Lempel–Ziv–Welch algorithm. Moreover, encoder and decoder architectures are proposed that have low memory requirements and provide fast data encoding and decoding.
This letter proposes two contributions to improve the performance of transmission with generalized multistream spatial modulation (SM). In particular, a modified suboptimal detection algorithm based on the Gaussian approximation method is proposed. The proposed modifications reduce the complexity of the Gaussian approximation method and improve the performance for high signal-to-noise ratios. Furthermore, this letter introduces signal constellations based on Hurwitz integers, i.e., a 4-D lattice. Simulation results demonstrate that these signal constellations are beneficial for generalized SM with two active antennas.
It is well known that signal constellations which are based on a hexagonal grid, so-called Eisenstein constellations, exhibit a performance gain over conventional QAM ones. This benefit is realized by a packing and shaping gain of the Eisenstein (hexagonal) integers in comparison to the Gaussian (complex) integers. Such constellations are especially relevant in transmission schemes that utilize lattice structures, e.g., in MIMO communications. However, for coded modulation, the straightforward approach is to combine Eisenstein constellations with ternary channel codes. In this paper, a multilevel-coding approach is proposed where encoding and multistage decoding can directly be performed with state-of-the-art binary channel codes. An associated mapping and a binary set partitioning are derived. The performance of the proposed approach is contrasted to classical multilevel coding over QAM constellations. To this end, both the single-user AWGN scenario and the (multiuser) MIMO broadcast scenario using lattice-reduction-aided preequalization are considered. Results obtained from numerical simulations with LDPC codes complement the theoretical aspects.
The computational complexity of the optimal maximum likelihood (ML) detector for spatial modulation increases rapidly as more transmit antennas or larger modulation orders are employed. Hence, ML detection may be infeasible for higher bit rates. This work proposes an improved suboptimal detection algorithm based on the Gaussian approximation method. It is demonstrated that the new method is closely related to the previously published signal vector based detection and the modified maximum ratio combiner, but can improve the detection performance compared to these methods. Furthermore, the performance of different signal constellations with suboptimal detection is investigated. Simulation results indicate that the performance loss compared to ML detection depends heavily on the signal constellation, where the recently proposed Eisenstein integer constellations are beneficial compared to classical QAM or PSK constellations.
This work introduces new signal constellations based on Eisenstein integers, i.e., the hexagonal lattice. These sets of Eisenstein integers have a cardinality which is an integer power of three. They are proposed as signal constellations for representation in the equivalent complex baseband model, especially for applications like physical-layer network coding or MIMO transmission where the constellation is required to be a subset of a lattice. It is shown that these constellations form additive groups where the addition over the complex plane corresponds to the addition with carry over ternary Galois fields. A ternary set partitioning is derived that enables multilevel coding based on ternary error-correcting codes. In the subsets, this partitioning achieves a gain of 4.77 dB, which results from an increased minimum squared Euclidean distance of the signal points. Furthermore, the constellation-constrained capacities over the AWGN channel and the related level capacities in case of ternary multilevel coding are investigated. Simulation results for multilevel coding based on ternary LDPC codes are presented which show that a performance close to the constellation-constrained capacities can be achieved.
