Modulation Alphabet Research Articles

Geometric approach to the MacWilliams Extension Theorem for codes over module alphabets

The MacWilliams Extension Theorem states that each linear Hamming isometry of a linear code extends to a monomial map. In this paper an analogue of the extension theorem for linear codes over a module alphabet is observed. A geometric approach to the extendability of isometries is described. For a matrix module alphabet we found the minimum length of a code for which an unextendable Hamming isometry exists. We also proved an extension theorem for MDS codes over a module alphabet.

When the extension property does not hold

A complete extension theorem for linear codes over a module alphabet, equipped with the symmetrized weight composition, is proven. It is also shown that the extension property with respect to a general weight function does not hold for module alphabets, that have a noncyclic socle.

Broadcast Coded Modulation: Multilevel and Bit-Interleaved Construction

The capacity of the AWGN broadcast channel is achieved by superposition coding, but the superposition of individual coded modulations expands the modulation alphabet and distorts its configuration. Hierarchical modulations, which appear in the literature mostly in the context of unequal error protection, can approach only a few isolated points on the boundary of the broadcast capacity region. This paper studies multilevel coding (MLC) for constellation-constrained broadcast-coded modulation. The conditions under which multilevel codes can achieve the constellation-constrained capacity of the AWGN broadcast channel are derived. We propose a pragmatic multilevel design technique with near-constellation-constrained-capacity performance where the coupling of the superposition inner and outer codes is localized to each bit-level. It is shown that this can be further relaxed to a code coupling on only one bit-level, with a little or no penalty under natural labeling. The rate allocation problem between the bit levels of the two users is studied and a pragmatic method is proposed, again with near-capacity performance. In further pursuit of lower complexity, a hybrid MLC-bit-interleaved-coded modulation is proposed, whose performance is shown to be very close to the boundary of the constellation-constrained capacity region. Simulation results show that good point-to-point LDPC codes produce excellent performance in the proposed coded modulation framework.

A Class of Non-Linearly Solvable Networks

For each integer $m \geq 2$, a network is constructed which is solvable over an alphabet of size $m$ but is not solvable over any smaller alphabets. If $m$ is composite, then the network has no vector linear solution over any $R$-module alphabet and is not asymptotically linear solvable over any finite-field alphabet. The network's capacity is shown to equal one, and when $m$ is composite, its linear capacity is shown to be bounded away from one for all finite-field alphabets.

A novel artificial bee colony detection algorithm for massive MIMO system

As a hot-spot of 5G, the research on detection algorithms for massive multiple input multiple output (MIMO) system is significant but difficult. The traditional MIMO detection algorithms or their improvements are not appropriate for large scaled antennas. In this paper, we propose artificial bee colony (ABC) detection algorithm for massive MIMO system. As one advanced technology of swarm intelligence, ABC algorithm is most efficient for large scaled constrained numerical combinatorial optimization problem. Therefore, we employ it to search the optimum solution vector in the modulation alphabet with linear detection result as initial. Simulation and data analysis prove the correctness and efficiency. Versus the scale of massive MIMO systems from 64 × 64 to 1024 × 1024 with uncoded four-quadrature-amplitude-modulation signals, the proposed ABC detection algorithm obtains bit error rate of 10 − 5 at low average received signal-to-noise-ratio of 12 dB with rapid convergence rate, which approximates the optimum bit error rate performance of the maximum likelihood and achieves the theoretical optimum spectral efficiency with low required average received signal-to-noise-ratio of 10 dB in similar increasing regularity, over finite time of low polynomial computational complexity of O(NT2) per symbol, where NT denotes the transmitting antennas' number. The proposed ABC detection algorithm is efficient for massive MIMO system. Copyright © 2016 John Wiley & Sons, Ltd.

On modules with cyclic socle

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.

Generalized Space-and-Frequency Index Modulation

Unlike in conventional modulation where information bits are conveyed only through symbols from modulation alphabets defined in the complex plane (e.g., quadrature amplitude modulation (QAM), phase shift keying (PSK)), in index modulation (IM), additional information bits are conveyed through indices of certain transmit entities that get involved in the transmission. Transmit antennas in multi-antenna systems and subcarriers in multi-carrier systems are examples of such transmit entities that can be used to convey additional information bits through indexing. In this paper, we introduce {\em generalized space and frequency index modulation}, where the indices of active transmit antennas and subcarriers convey information bits. We first introduce index modulation in the spatial domain, referred to as generalized spatial index modulation (GSIM). For GSIM, where bits are indexed only in the spatial domain, we derive the expression for achievable rate as well as easy-to-compute upper and lower bounds on this rate. We show that the achievable rate in GSIM can be more than that in spatial multiplexing, and analytically establish the condition under which this can happen. It is noted that GSIM achieves this higher rate using fewer transmit radio frequency (RF) chains compared to spatial multiplexing. We also propose a Gibbs sampling based detection algorithm for GSIM and show that GSIM can achieve better bit error rate (BER) performance than spatial multiplexing. For generalized space-frequency index modulation (GSFIM), where bits are encoded through indexing in both active antennas as well as subcarriers, we derive the achievable rate expression. Numerical results show that GSFIM can achieve higher rates compared to conventional MIMO-OFDM. Also, BER results show the potential for GSFIM performing better than MIMO-OFDM.

Single Antenna Interference Cancellation for GSM/VAMOS/EDGE Using <inline-formula> <tex-math notation="TeX">$L_{p}$</tex-math></inline-formula>-Norm Detection and Decoding

Different schemes for single antenna interference cancellation (SAIC) in the Global System for Mobile Communications (GSM) have been proposed so far. Most of these schemes work best in synchronous interference environments and are limited to Gaussian minimum-shift keying (GMSK) modulation. In this paper, we propose several modifications of the conventional GSM receiver, which target the harmful effects of asynchronous co-channel interference (ACCI). Based on the observation that the impairment caused by ACCI cannot be modeled as additive Gaussian noise, which was assumed for the derivation of the metric used in state-of-the-art receivers, we propose modeling the impairment as Generalized Gaussian noise (GGN). This leads to modifications of the conventional branch metrics of the reduced-state equalizer and the decoder employed at the receiver. Both proposed modifications are applicable to any linear modulation alphabet and do not require any modification of the GSM air interface. Hence, the proposed novel branch metrics are also applicable in Enhanced Data Rates for GSM Evolution (EDGE) systems which employ 8-ary phase-shift keying (8-PSK) modulation and where the SAIC algorithms tailored for GMSK cannot be used. Furthermore, we show that the proposed branch metric modifications also improve the performance of the new Voice Services over Adaptive Multiuser channels on One Slot (VAMOS) extension of GSM in ACCI environments. Based on simulation results, the performance of the proposed schemes is compared with that of an unmodified receiver employing the conventional Gaussian metric. We show that the proposed receivers outperform the unmodified receiver in ACCI environments and discuss the complexity entailed by the proposed branch metric modifications.

Performance limits of conventional and widely linear DFT-precoded-OFDM receivers in wideband frequency-selective channels

This paper describes the limiting behavior of linear and decision feedback equalizers (DFEs) in single/multiple antenna systems employing real/complex-valued modulation alphabets. The wideband frequency-selective channel is modeled using a Rayleigh fading channel model with infinite number of time domain channel taps. Using this model, we show that the considered equalizers offer a fixed post detection signal-to-noise ratio (post-SNR) at the equalizer output that is close to the matched filter bound (MFB). General expressions for the post-SNR are obtained for zero-forcing (ZF)-based conventional receivers as well as for the case of receivers employing widely linear (WL) processing. Simulation is used to study the bit error rate (BER) performance of both minimum-mean-square-error (MMSE) and ZF-based receivers. Results show that the considered receivers advantageously exploit the rich frequency-selective channel to mitigate both fading and inter-symbol interference (ISI) while offering a performance comparable to the MFB.

Adaptive OFDM Modulation for Underwater Acoustic Communications: Design Considerations and Experimental Results

In this paper, we explore design aspects of adaptive modulation based on orthogonal frequency-division multiplexing (OFDM) for underwater acoustic (UWA) communications, and study its performance using real-time at-sea experiments. Our design criterion is to maximize the system throughput under a target average bit error rate (BER). We consider two different schemes based on the level of adaptivity: in the first scheme, only the modulation levels are adjusted while the power is allocated uniformly across the subcarriers, whereas in the second scheme, both the modulation levels and the power are adjusted adaptively. For both schemes we linearly predict the channel one travel time ahead so as to improve the performance in the presence of a long propagation delay. The system design assumes a feedback link from the receiver that is exploited in two forms: one that conveys the modulation alphabet and quantized power levels to be used for each subcarrier, and the other that conveys a quantized estimate of the sparse channel impulse response. The second approach is shown to be advantageous, as it requires significantly fewer feedback bits for the same system throughput. The effectiveness of the proposed adaptive schemes is demonstrated using computer simulations, real channel measurements recorded in shallow water off the western coast of Kauai, HI, USA, in June 2008, and real-time at-sea experiments conducted at the same location in July 2011. We note that this is the first paper that presents adaptive modulation results for UWA links with real-time at-sea experiments.

Adaptive Pilot-Symbol Patterns for MIMO OFDM Systems

Recent standards for cellular transmission systems offer a lot of flexibility, such as the choice of transmission modes, modulation alphabets, coding rates, and precoding matrices. Despite this trend, pilot-symbol patterns in today's standards remain fixed, although such an approach is suboptimal. In this paper, we show how to design optimal pilot-symbol patterns by maximizing an upper bound of a constrained capacity that takes channel estimation errors and Inter Carrier Interference into account. Furthermore, we propose adaptive pilot-symbol patterns that follow changing channel statistics. As a proof of concept, we present throughput simulation results of two competitive systems, a transmission system compliant with the Long Term Evolution (LTE)-standard and an improved system utilizing the proposed adaptive pilot patterns. The transmission system utilizing adaptive pilot patterns outperforms an LTE-standard compliant system in all considered scenarios. The throughput gain for a single input single output system ranges between 3% and 80%. For a 4 × 4 transmission system, the performance gain is significantly higher and can reach up to 850% compared to a conventional LTE system.

Performance Analysis of Combined $$m$$ MCSK– $$m$$ MFSK Modulation in AWGN and Rayleigh Fading Channel

As there are many use cases considered for robotics communications, the data rate variation may be very large. Some sensor applications may require very low data rate, telemetry data may require low-to-medium data rates and e.g. video application will require high data rates. On the other hand, robots may have to operate in very difficult radio propagation environments such as nuclear power plants or industrial facilities. To combat difficult propagation characteristics, an often used and well known mechanism is to use spread spectrum signal structures. Thus in this paper a novel modulation method is considered which offers the inherent signal structure and processing opportunities of spread spectrum signal and at the same time offers an easy mechanism to adapt the data rate from low to high depending on the requirement at hand thus addressing two important communication requirements for robotics. The introduction of $$m$$ MFSK generated the idea of applying the given approach to other modulation methods. To further enlarge the modulation alphabet sizes (with the price of larger spectrum usage) it was realized that combining the $$m$$ MFSK and $$m$$ MCSK would be an interesting choice. The $$m$$ MCSK--- $$m$$ MFSK modulation is hence considered. A method to analyze this two-component modulation is developed and the performance analyses give results for $$m$$ MCSK--- $$m$$ MFSK modulation in AWGN and flat Rayleigh fading channels for both coherent and non-coherent receivers. The performance was also assessed with orthogonal and non-orthogonal code constructs for coherent receiver.

IDMA vs. CDMA: Analysis and Comparison of Two Multiple Access Schemes

This article presents comprehensive comparisons of interleave division multiple access (IDMA) and direct sequence code division multiple access (DS-CDMA) in terms of performance and complexity assuming iterative multiuser detection. IDMA can be seen as a special case of DS-CDMA with spreading gain of one using very low rate code and user-specific interleavers for user separation. We focus on three suboptimum linear detectors: minimum mean square error (MMSE), rake (or matched filter), and soft-rake detectors from practical concerns. We analytically prove that the three detectors are equivalent for asynchronous users of IDMA on frequency flat channels for complex modulation alphabets. Such equivalence has been shown only for binary phase shift keying (BPSK) in the literature. The equivalence guarantees the MMSE solution for IDMA without computationally expensive matrix inversions or matrix-vector multiplications. This is generally not the case for DS-CDMA since DS-CDMA is sensitive to user asynchronism. We also discuss complexity aspects when the MMSE detector is used where we focus on essential differences in complexity between IDMA and DS-CDMA, instead of discussing particular complexity reduction techniques. Computer simulations are performed in various scenarios and the performance is analyzed by bit error rate simulations as well as by extrinsic information transfer (EXIT) charts. The analysis reveals the advantages of IDMA over DS-CDMA in terms of performance and complexity under practical considerations, particularly in highly user loaded scenarios.

Bounds on the Information Rate for Sparse Channels with Long Memory and i.u.d. Inputs

In this paper we propose new bounds on the achievable information rate for discrete-time Gaussian channels with intersymbol interference (ISI) and independent and uniformly distributed (i.u.d.) channel input symbols drawn from finite-order modulation alphabets. Specifically, we are interested in developing new bounds on the achievable rates for sparse channels with long memory. We obtain a lower bound which can be achieved by practical receivers, based on MMSE channel shortening and suboptimal symbol detection for a reduced-state channel. An upper bound is given in the form of a semi-analytical solution derived using basic information theoretic inequalities, by a grouping of the channel taps into several clusters resulting in a newly defined single-input multiple-output (SIMO) channel. We show that the so obtained time-dispersive SIMO channel can be represented by an equivalent single-input single-output (SISO) channel with a significantly shorter channel memory. The reduced computational complexity allows the use of the BCJR algorithm for the newly defined channel. The proposed bounds are illustrated through several sparse channel examples and i.u.d. input symbols, showing that the upper bound significantly outperforms existing bounds. Performance of our lower bound strongly depends on the channel structure, showing best results for minimum-phase and maximum-phase systems.

Numerically Optimized Uniformly Most Powerful Alphabets for Hierarchical-Decode-and-Forward Two-Way Relaying

We address the issue of the parametric performance of the Hierarchical-Decode-and-Forward (HDF) strategy in a wireless 2-way relay channel. Promising HDF, representing the concept of wireless network coding, performs well with a pre-coding strategy that requires Channel State Information (CSI) on the transceiver side. Assuming a practical case when CSI is available only on the receiver side and the channel conditions do not allow adaptive strategies, the parametrization causes significant HDF performance degradation for some modulation alphabets. Alphabets that are robust to the parametrization (denoted Uniformly Most Powerful (UMP)) have already been proposed restricting on the class of non-linear multi-dimensional frequency modulations. In this work, we focus on the general design of unrestricted UMP alphabets. We formulate an optimization problem which is solved by standard non-linear convex constrained optimization algorithms, particularly by Nelder-Mead global optimization search, which is further refined by the local interior-pointsmethod.

Efficient Channel-Adaptive MIMO Detection Using Just-Acceptable Error Rate

This paper proposes a new concept of multiple-input multiple-output (MIMO) detection, aiming at minimizing the average computational cost. The detection methods are adapted according to the estimated channel state information to deliver just-acceptable error rate (JAER) performance. Error rate models for two popular MIMO detection algorithms are derived given the channel matrix. From these models, a channel-adaptive-MIMO (CA-MIMO) receiver with detector-switching strategies is proposed. Simulation results demonstrate that the proposed CA-MIMO detector meets the JAER criterion efficiently. Compared with the sophisticated Sphere Search (SS) MIMO detector, the average saving at moderate signal-to-noise ratio (SNR) is around 58% to 72%, depending on different modulation alphabets. At high SNR, the CA-MIMO detector almost always switch to Zero-Forcing (ZF) detection, where the complexity is several orders lower than the SS detector.

Design Criteria for Hierarchical Exclusive Code with Parameter-Invariant Decision Regions for Wireless 2-Way Relay Channel

The unavoidable parametrization of the wireless link represents a major problem of the network-coded modulation synthesis in a 2-way relay channel. Composite (hierarchical) codeword received at the relay is generally parametrized by the channel gain, forcing any processing on the relay to be dependent on channel parameters. In this paper, we introduce the codebook design criteria, which ensure that all permissible hierarchical codewords have decision regions invariant to the channel parameters (as seen by the relay). We utilize the criterion for parameter-invariant constellation space boundary to obtain the codebooks with channel parameter-invariant decision regions at the relay. Since the requirements on such codebooks are relatively strict, the construction of higher-order codebooks will require a slightly simplified design criteria. We will show that the construction algorithm based on these relaxed criteria provides a feasible way to the design of codebooks with arbitrary cardinality. The promising performance benefits of the example codebooks (compared to a classical linear modulation alphabets) will be exemplified on the minimum distance analysis.

Anti-isomorphisms, character modules and self-dual codes over non-commutative rings

This paper is dedicated to Vera Pless. It is an elaboration on ideas of Nebe, Rains and Sloane: by assuming the existence of an anti-isomorphism on a finite ring and by assuming a module alphabet has a well-behaved duality, one is able to study self-dual codes defined over alphabets that are modules over a non-commutative ring. Various examples are discussed.

Fiber Capacity Limits With Optimized Ring Constellations

In this letter, we calculate theoretical limits to fiber-optic transport capacity under the constraint of requiring the use of concentric multiring modulation alphabets. Subject to these conditions, we vary ring amplitude ratios and occupation probabilities to determine the capacity-optimized ring constellations. The technique used for fiber transmission calculations accurately captures all fiber nonlinear effects, with the accuracy limited by the noise and signal statistical properties only.

A single-symbol-decodable space-time block code with full rate and low peak-to-average power ratio

Three desirable properties of a four-antenna spacetime block code are full rate, full diversity, and single-symbol decodability. Previously reported space-time codes that achieve all three properties do so at the expense of the peak-to-average power ratio (PAPR). A fourth desirable property of a space-time block code is that its PAPR be the same as that of the underlying quadrature-amplitude modulation alphabet. In this letter we introduce space-time codes for three and four transmit antennas that achieve all four properties; these codes use a diversity technique based on constellation stretching. Numerical results for quasistatic Rayleigh-fading channels show that, despite their low PAPR, the proposed codes are comparable in SNR performance to the best-performing single-symbol decodable space-time codes for three and four transmit antennas.