Ary Symbol Research Articles

Design of Carrier Index Keying-Aided M-Ary Differential Chaos Cyclic Shift Keying for D2D Communications

Device-to-device (D2D) communications can allow different devices to communicate in short range, which greatly meets the demand of effective transmissions. In this paper, a spectral and energy-efficient carrier index keying-aided <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary differential chaos cyclic shift keying (CIK-MDCCSK) system is proposed for D2D communications, where multiple cyclic-shifted <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary information-bearing signals are superimposed and transmitted by a subcarrier, and extra keying bits are transmitted by indices of the selected subcarriers. Furthermore, a carrier keying detection and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary de-mapping algorithm is proposed to retrieve the information bits transmitted by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary symbols and indices of carrier keying. The bit error rate (BER) expressions of the proposed CIK-MDCCSK system are derived over the additive white Gaussian noise (AWGN) and D2D channels, and the optimal power coefficient is deduced for performance enhancement. Then, the spectral efficiency, energy efficiency, and complexity of CIK-MDCCSK are analyzed and compared to benchmark systems to show its superiority. Next, the proposed CIK-MDCCSK system is expanded into its enhanced version for more effective transmission without degrading the BER performance. It is demonstrated by performance comparisons that the proposed CIK-MDCCSK system can achieve more than twice energy efficiency and up to 3 dB gain in BER performance compared to state-of-the-art benchmarks.

Efficient Encoding of Constrained Block Codes

We present coding methods for generating $\ell$ -symbol constrained codewords taken from a set, $\mathcal{\mathcal { S }}$ , of allowed codewords. In standard practice, the size of the set $\mathcal{S}$ , denoted by $M=|\mathcal{S}|$ , is truncated to an integer power of two, which may lead to a serious waste of capacity. We present an efficient and low-complexity coding method for avoiding the truncation loss, where the encoding is accomplished in two steps: first, a series of binary input (user) data is translated into a series of $M$ -ary symbols in the alphabet $\mathbb{M}=\{0, \ldots, M-1\}$ . Then, in the second step, the $M$ -ary symbols are translated into a series of admissible $\ell$ -symbol words in $\mathcal{S}$ by using a small look-up table. The presented construction of Pearson codes and fixed-weight codes offers a rate close to capacity. For example, a 255B320B balanced code, where 255 source bits are translated into 32 10-bit balanced codewords, has a rate 0.1% below capacity.

Differential Permutation Index DCSK Modulation for Chaotic Communication System

This letter proposes a differential permutation index differential chaos shift keying (DPI-DCSK) modulation for wireless communications. The conventional PI-DCSK has the disadvantage of wasting transmission energy and the loss of system performance. In $M$ -ary DPI-DCSK, one chaotic sequence to be sent represents current modulated ${M}$ -ary symbol as well as the reference of next modulation symbol in a data frame, which can fully utilize the transmission energy with the same hardware implementation as PI-DCSK. The bit error performance of the proposed DPI-DCSK over the multipath Rayleigh fading channel is theoretically analyzed. The simulation results show that the proposed DPI-DCSK achieves a performance gain of more than 3 dB over PI-DCSK, especially in high-delay multipath fading channels. The performance advantage of the proposed scheme is also illustrated in practical ultra-wideband (UWB) channels. Thus, DPI-DCSK is an efficient alternative transmission scheme for chaos-based wireless communication, such as indoor applications of transmitted-reference UWB communications.

Joint-Mapping-Based Variable Active Antenna Spatial Modulation

Spatial modulation (SM) maps information to both spatial dimensions (active antenna indices) and conventional 2D ${M}$ -ary symbols. The existing SM schemes use fixed number of active antennas in each transmit vector as varying them causes detection ambiguity. Motivated by this, we introduce a new joint-mapping based variable active antenna spatial modulation scheme, which varies the number of active antennas in each transmit vector. It alleviates detection ambiguity by jointly mapping information to the complete 3D constellation consisting of active antenna indices (spatial dimension) and the 2D ${M}$ -ary symbols. This is unlike the existing SM schemes which separately map information to the active antenna indices and the 2D symbols. This idea enables us to i) avoid detection ambiguity; and ii) achieve low bit error rate and comparable maximum likelihood detection complexity as that of existing SM schemes.

LDPC Codes Over the $q$ -ary Multi-Bit Channel

In this paper, we introduce a new channel model termed as the $q$ -ary multi-bit channel. This channel models a memory device, where $q$ -ary symbols ( $q=2^{s}$ ) are stored in the form of current/voltage levels. The symbols are read in a measurement process, which provides a symbol bit in each measurement step, starting from the most significant bit. An error event occurs when not all the symbol bits are known. To deal with such error events, we use GF( $q$ ) low-density parity-check (LDPC) codes and analyze their decoding performance. We start with iterative-decoding threshold analysis and derive optimal edge-label distributions for maximizing the decoding threshold. We later move to a finite-length iterative-decoding analysis and propose an edge-labeling algorithm for the improved decoding performance. We then provide a finite-length maximum-likelihood decoding analysis for both the standard non-binary random ensemble and LDPC ensembles. Finally, we demonstrate by simulations that the proposed edge-labeling algorithm improves the finite-length decoding performance by orders of magnitude.

Repeated MCIK-OFDM With Enhanced Transmit Diversity Under CSI Uncertainty

This paper investigates the opportunity for a repetition coded multi-carrier index keying—orthogonal frequency division multiplexing (MCIK-OFDM), termed repeated MCIK-OFDM (ReMO), which can provide superior performance over existing schemes at the same spectral efficiency. Unlike the classical scheme, the proposed scheme activates a subset of sub-carriers and modulates them with the same $M$ -ary data symbol, while additional information is conveyed by the active sub-carrier indices. This approach not only provides the frequency diversity gains in the $M$ -ary symbol detection but also improves the index detection, leading to considerable improvement in the transmit diversity. For performance analysis, we derive tight closed-form expressions for the symbol error probability and the bit error rate, under both perfect and imperfect channel state information (CSI). These expressions provide insight into the achievable performance gains, system designs, and impacts of various CSI conditions. Finally, simulation results are given to illustrate the superior performance achieved by our scheme over existing schemes under different CSI uncertainties.

A Serial Concatenation of Binary-Input Nonbinary-Output Convolutional Code and Recursive Convolutional Lattice Code

The recently proposed recursive convolutional lattice code (RCLC) can form a signal with pseudo-Gaussian constellations, and their parallel concatenation is shown to approach the Shannon limit. A practical limitation is that its input symbol is limited to $L^{2}$ -ary quadrature amplitude modulation (QAM), which has non-power-of-two constellation points when $L$ is chosen from the odd numbers. Therefore, encoding binary information by the RCLC is not straightforward. Furthermore, the information rate is limited to $\log _{2}~L$ bits per complex dimension due to their parallel concatenation. In this paper, we tackle these issues by introducing a serial concatenation of binary-input nonbinary-output convolutional code (CC) and the RCLC, where the outer CC outputs an $L$ -ary symbol that is matched to the input of the inner RCLC. We demonstrate that even with $L=3$ , the proposed approach can achieve 2 bits per complex dimension and still is able to approach the Shannon limit with lower decoding complexity compared with its parallel concatenation counterpart. As is demonstrated through theoretical analysis, the major practical drawback of the constellation generated by the RCLC is its Gaussian-like distribution, which has large peak-to-average power ratio. Therefore, we further introduce an approach to reduce the signal dynamic range for the proposed system. It is shown that a remarkable gain can be achieved in terms of capacity compared with the conventional QAM signals under the constraint of comparable power amplifier efficiency.

Channel decomposition for multilevel codes over multilevel and partial erasure channels

We introduce the Multilevel Erasure Channel (MEC) for binary extension field alphabets. The channel model is motivated by applications such as non-volatile multilevel read storage channels. Like the recently proposed \begin{document}$q$\end{document} -ary partial erasure channel (QPEC), the MEC is designed to capture partial erasures. The partial erasures addressed by the MEC are determined by erasures at the bit level of the \begin{document}$q$\end{document} -ary symbol representation. In this paper we derive the channel capacity of the MECand give a multistage decoding scheme on the MEC using binary codes. We also present a low complexity multistage \begin{document}$p$\end{document} -ary decoding strategy for codes on the QPEC when \begin{document}$q = p^k$\end{document} .We show that for appropriately chosen component codes, capacity on the MEC and QPEC may be achieved.

Optimal Download Cost of Private Information Retrieval for Arbitrary Message Length

A private information retrieval scheme is a mechanism that allows a user to retrieve any one out of $K$ messages from $N$ non-communicating replicated databases, each of which stores all $K$ messages, without revealing anything about the identity of the desired message index to any individual database. If the size of each message is $L$ bits and the total download required by a PIR scheme from all $N$ databases is $D$ bits, then $D$ is called the download cost and the ratio $L/D$ is called an achievable rate. For fixed $K,N\in\mathbb{N}$, the capacity of PIR, denoted by $C$, is the supremum of achievable rates over all PIR schemes and over all message sizes, and was recently shown to be $C=(1+1/N+1/N^2+\cdots+1/N^{K-1})^{-1}$. In this work, for arbitrary $K, N$, we explore the minimum download cost $D_L$ across all PIR schemes (not restricted to linear schemes) for arbitrary message lengths $L$ under arbitrary choices of alphabet (not restricted to finite fields) for the message and download symbols. If the same $M$-ary alphabet is used for the message and download symbols, then we show that the optimal download cost in $M$-ary symbols is $D_L=\lceil\frac{L}{C}\rceil$. If the message symbols are in $M$-ary alphabet and the downloaded symbols are in $M'$-ary alphabet, then we show that the optimal download cost in $M'$-ary symbols, $D_L\in\left\{\left\lceil \frac{L'}{C}\right\rceil,\left\lceil \frac{L'}{C}\right\rceil-1,\left\lceil \frac{L'}{C}\right\rceil-2\right\}$, where $L'= \lceil L \log_{M'} M\rceil$.

High-Speed Parallel Decodable Nonbinary Single-Error Correcting (SEC) Codes

This paper presents a novel construction scheme for nonbinary single-error correcting (SEC) codes that yields high-speed parallel decoding. The proposed scheme utilizes two methods, namely, Improved and Reordered; these methods can be also combined. Both of these methods reduce the number of 1's in the parity-check matrix (H-matrix) by reducing the 1's in every row vector. This results in a reduction in the gate depth in the syndrome generator, thus achieving a shorter delay time for parallel decoding. In the proposed Improved method, for a single $b$ -bit byte (i.e., $\mbox{2}^{b}$ -ary symbol) error correcting code, the submatrix of the H-matrix corresponding to every $b$ -bit byte is multiplied with a regular matrix. The so-called improved submatrix is generated using a heuristic (greedy) algorithm. The proposed Reordered method selects the correct $b$ -bit bytes for deletion when shortening is performed. Simulation results show that the proposed scheme accomplishes a faster parallel decoding time than existing schemes. Furthermore, the proposed scheme is applicable to any class of linear SEC codes, whereas existing schemes are applicable only to specific codes. Extensive simulation results are provided to substantiate the viability of the proposed codes for faster parallel decoding (albeit incurring for most cases in modest increases of area and power dissipation due to additional circuitry).

Coded Distributed Diversity: A Novel Distributed Reception Technique for Wireless Communication Systems

In this paper, we consider a distributed reception scenario where a transmitter broadcasts a signal to multiple geographically separated receive nodes over fading channels, and each node forwards a few bits representing a processed version of the received signal to a fusion center. The fusion center then tries to decode the transmitted signal based on the forwarded information from the receive nodes and possible channel state information. We show that there is a strong connection between the problem of minimizing a symbol error probability at the fusion center in distributed reception and channel coding in coding theory. This connection allows us to design a unified framework for coded distributed diversity reception. We focus linear block codes such as simplex codes or first-order Reed-Muller codes that achieve the Griesmer bound with equality to maximize the diversity gain. Due to its simple structure, no complex offline optimization process is needed to design the coding structure at the receive nodes for the proposed coded diversity technique. The proposed technique can support a wide array of distributed reception scenarios, i.e., arbitrary $M$-ary symbol transmission at the transmitter and received signal processing with multiple bits at the receive nodes. Numerical studies show that the proposed coded diversity technique can achieve practical symbol error rates even with moderate signal-to-noise ratio and numbers of the receive nodes.

A Fresh Look at Coding for $q$-ary Symmetric Channels

This paper studies coding schemes for the q-ary symmetric channel based on binary low-density parity-check (LDPC) codes that work for any alphabet size q=2m, m ∈N, thus complementing some recently proposed packet-based schemes requiring large q . First, theoretical optimality of a simple layered scheme is shown; then, a practical coding scheme based on a simple modification of standard binary LDPC decoding is proposed. The decoder is derived from first principles and using a factor-graph representation of a front end that maps q -ary symbols to groups of m bits connected to a binary code. The front end can be processed with a complexity that is linear in m=log2q. An extrinsic information transfer chart analysis is carried out and used for code optimization. Finally, it is shown how the same decoder structure can also be applied to a larger class of q-ary channels.

An embedded transmission scheme using ppm signaling with symmetric error-correcting codes for optical CDMA

An embedded transmission (ET) scheme is proposed to easily apply error-correcting codes into optical code-division multiple-access (CDMA) systems for immunity from multiple-access interference (MAI). The ET scheme offers high transmission capability over the traditional scheme using pulse position modulation (PPM) signaling, because a 2/sup J/-ary symbol of each user is embedded in the signature sequence with 2/sup J/ weighted positions. Furthermore, the ET scheme with 2/sup J/-ary PPM signaling makes the optical CDMA system J parallel transmission systems, because J bits consisting of 2/sup J/-ary symbol are separately decided. Since such a separate decision is a comparison decision, the effect of MAI added in the optical channel is converted to symmetric errors in the individual parallel transmission systems. Using the symmetric error-correcting (SEC) code immunizes the individual parallel transmission systems against MAI more easily than the embedded-modulation scheme described in because the ET scheme avoids using the asymmetric error correcting code, which is difficult to implement. We analyze the bit error rate under Poisson photon counting channel and show that the ET scheme has an advantage of good energy efficiency over the traditional scheme in applying SEC codes.

A practical reliability metric for block codes used on binary-input channels

A reliability metric (RM) for a block code is defined to be a function that operates on both the decoder input (a block of channel output) and the decoder output (the codeword estimate) and produces a real number as a measure of the reliability of the decoder decision. The best RM has the disadvantage of depending on the codeword probabilities. Thus, the ideal RM is defined as the value that would be computed for equally likely codewords. The implementation of the ideal RM is too costly for most applications. The author proposes an easily implemented RM for binary-input memoryless channels (for state-observable channels with a freely evolving state such as some fading channels) when the codewords consist of n 2/sup m/-ary symbols, each of which is transmitted serially by m uses of the channel. Simulation results for some BCH codes and some Reed-Solomon codes used in a simple ARQ system show that the proposed RM performs nearly as well as the ideal RM and much better than a previously proposed practical RM. >

A Practical Receiver Structure for Multi-h CPM Signals

Multi- h schemes are bandwidth-efficient CPM signaling formats which offer means of coding without explicit redundancy. However, the required (optimal) receiver structures tend to be very complex [1], [4], [8] requiring a large number of filters of correlators. Here a reduced complexity receiver for the joint estimation of M -ary data, carrier phase, and symbol timing is realized through the use of an approximate representation of the likelihood function. Evaluation of receiver performance indicates a fraction of a decibel degradation in coding gain and almost no loss in synchronization performance.

Generalized 'write-once' memories

Storage media such as digital optical discs, PROM's, or punched cards consist of a number of write-once bit positions (WIT's); each WIT initially contains a "0" that may later be irreversibly overwritten with a "r'. Rivest and Shamir have shown that such write-once memories (WOM's) can be reused very efficiently. Generalized WOM's are considered, in which the basic storage element has more than two possible states and the legal state transitions are described by an arbitrary directed acyclic graph. The capabilities of such memories depend on the depth of the graphs rather than on their size, and the decision problem associated with the generalized WOM's in NP-hard even for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</tex> -ary symbols rewritten several times or multiple values rewritten once.

The Performance of FSK Permutation Modulations in Fading Channels and Their Comparison Based on a General Method for the Comparison of M-ary Modulations

A problem that arises in multipath fading channels is that of finding a modulation which enables one to obtain high data rates and at the same time does not lead to an excessively complicated system implementation or the requirement of excessive transmitter power. In many dispersive channels the product of time dispersion and Doppler spread is large, so that coherent modulation systems are not feasible. Examples of such overspread or nearly overspread channels are the West Ford channel, the endoionospheric channel when using aurora injection, and, in some instances, the moon relay channel. For such channels consideration must be given to the use of incoherent modulation systems. Consideration is given to the performance in fading channels of an incoherent modulation system which uses a subclass of permutation modulation called frequency-shift keying (FSK) permutation (PFSK) modulation. The channel is assumed to have the following characteristics. First, the (Gaussian) noise for each propagation mode is white and independent, with equal noise density spectrums; and second, the fadings for the different propagation modes are Rayleigh distributed and independent with equal variance. Extensive curves are given for the performance of various PFSK modulations in terms of the modulation signal-waveform error probabilities (often referred to as the word error probabilities) for the case of frequency-selective fading. Bounds on the modulation performance are given for the case of flat fading. Consideration is given to the problem of comparing the PFSK modulations with each other and with other available modulations to determine which is best for communicating a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary message sequence, with respect to the transmitter power and bandwidth required. Toward this end, a relatively simple method is presented for comparing modulations, which circumvents the problem of requiring detailed knowledge of the mapping from the original information data source (which, for greatest generality, is assumed to consist of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s L_{i}</tex> -state Markov chains, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i = 1,...,s,</tex> which are to be transmitted simultaneously) to the modulation waveforms. The method provides upper and lower bounds for the required receiver signal energy to noise density ratio per <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary symbol for a specified <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary error rate (or else, bounds for the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary error rates for a specified receiver signal energy to noise density ratio per <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary symbol). The upper bound is based on the performance of the modulation for the ensemble average of all possible mappings, or, equivalently, for a random mapping (analogous to random coding). The lower bound corresponds to the performance of the modulation for an optimum mapping, which is in effect a generalization of the Gray-code-type mapping used for phaseshift keying (PSK) modulation. (To obtain the lower bound on the modulation performance, the details of the optimum mapping need not be known; moreover, it remains to prove the existence of this optimum mapping.) The method is general and applicable to a wide class of modulations and channels. Applying the technique to FSK, FSK permutation, and amplitude-keyed (AK) carrier modulations yields results which conveniently show, for different <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L_{i}</tex> -ary source sequences, the cost in increased transmitter power required to achieve higher data rates with various modulations. The comparison data indicates that, in some instances, relative to the FSK modulation, one can obtain with PFSK modulations increased data rates for a given signal bandwidth without requiring an increased received signal energy to noise density ratio per <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary source symbol.As a consequence of these results, PFSK modulations should perhaps be of interest to the designer of fading communications channels not having large time-dispersion times Doppler-spread products, as well as to the designer of highly perturbed channels. Finally, a means is given for transmitting PFSK modulations in time-dispersive fading channels, which does not require the use of a linear power amplifier.