Finite Dimensional Infinite Constellations

In the setting of a Gaussian channel without power constraints, proposed by Poltyrev, the codewords are points in an n-dimensional Euclidean space (an infinite constellation) and the tradeoff between their density and the error probability is considered. The capacity in this setting is the highest achievable normalized log density (NLD) with vanishing error probability. This capacity as well as error exponent bounds for this setting are known. In this work we consider the optimal performance achievable in the fixed blocklength (dimension) regime. We provide two new achievability bounds, and extend the validity of the sphere bound to finite dimensional infinite constellations. We also provide asymptotic analysis of the bounds: When the NLD is fixed, we provide asymptotic expansions for the bounds that are significantly tighter than the previously known error exponent results. When the error probability is fixed, we show that as n grows, the gap to capacity is inversely proportional (up to the first order) to the square-root of n where the proportion constant is given by the inverse Q-function of the allowed error probability, times the square root of 1/2. In an analogy to similar result in channel coding, the dispersion of infinite constellations is 1/2nat^2 per channel use. All our achievability results use lattices and therefore hold for the maximal error probability as well. Connections to the error exponent of the power constrained Gaussian channel and to the volume-to-noise ratio as a figure of merit are discussed. In addition, we demonstrate the tightness of the results numerically and compare to state-of-the-art coding schemes.Comment: 54 pages, 13 figures. Submitted to IEEE Transactions on Information Theor

Perfect Space–Time Block Codes

In this paper, we introduce the notion of perfect space–time block codes (STBCs). These codes have full-rate, full-diversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas

Cyclic division algebras: a tool for space-time coding

Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space–Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank. Extensive work has been done on Space–Time coding, aiming at finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space–Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes. The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space–Time block codes

Capacity of a Nonlinear Optical Channel with Finite Memory

The channel capacity of a nonlinear, dispersive fiber-optic link is revisited. To this end, the popular Gaussian noise (GN) model is extended with a parameter to account for the finite memory of realistic fiber channels. This finite-memory model is harder to analyze mathematically but, in contrast to previous models, it is valid also for nonstationary or heavy-tailed input signals. For uncoded transmission and standard modulation formats, the new model gives the same results as the regular GN model when the memory of the channel is about 10 symbols or more. These results confirm previous results that the GN model is accurate for uncoded transmission. However, when coding is considered, the results obtained using the finite-memory model are very different from those obtained by previous models, even when the channel memory is large. In particular, the peaky behavior of the channel capacity, which has been reported for numerous nonlinear channel models, appears to be an artifact of applying models derived for independent input in a coded (i.e., dependent) scenario

A New Low-Complexity Decodable Rate-5/4 STBC for Four Transmit Antennas with Nonvanishing Determinants

The use of Space-Time Block Codes (STBCs) increases significantly the optimal detection complexity at the receiver unless the low-complexity decodability property is taken into consideration in the STBC design. In this paper we propose a new low-complexity decodable rate-5/4 full-diversity 4 x 4 STBC. We provide an analytical proof that the proposed code has the Non-Vanishing-Determinant (NVD) property, a property that can be exploited through the use of adaptive modulation which changes the transmission rate according to the wireless channel quality. We compare the proposed code to the best existing low-complexity decodable rate-5/4 full-diversity 4 x 4 STBC in terms of performance over quasi-static Rayleigh fading channels, worst- case complexity, average complexity, and Peak-to-Average Power Ratio (PAPR). Our code is found to provide better performance, lower average decoding complexity, and lower PAPR at the expense of a slight increase in worst-case decoding complexity.Comment: 5 pages, 2 figures and 1 table; IEEE Global Telecommunications Conference (GLOBECOM 2011), 201

Fast-Decodable Asymmetric Space-Time Codes from Division Algebras

Multiple-input double-output (MIDO) codes are important in the near-future wireless communications, where the portable end-user device is physically small and will typically contain at most two receive antennas. Especially tempting is the 4 x 2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of (4 x 4) matrices. Unfortunately, such codes are in general very complex to decode, hence setting forth a call for constructions with reduced complexity. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes. In this paper, it will be shown that a family of division algebra based MIDO codes will always result in at least 37.5% worst-case complexity reduction, while maintaining full diversity and, for the first time, the non-vanishing determinant (NVD) property. The reduction follows from the fact that, similarly to the Alamouti code, the codes will be subsets of matrix rings of the Hamiltonian quaternions, hence allowing simplified decoding. At the moment, such reductions are among the best known for rate-two MIDO codes. Several explicit constructions are presented and shown to have excellent performance through computer simulations.Comment: 26 pages, 1 figure, submitted to IEEE Trans. Inf. Theory, October 201

Conditions for a Monotonic Channel Capacity

Motivated by results in optical communications, where the performance can degrade dramatically if the transmit power is sufficiently increased, the channel capacity is characterized for various kinds of memoryless vector channels. It is proved that for all static point-to-point channels, the channel capacity is a nondecreasing function of power. As a consequence, maximizing the mutual information over all input distributions with a certain power is for such channels equivalent to maximizing it over the larger set of input distributions with upperbounded power. For interference channels such as optical wavelength-division multiplexing systems, the primary channel capacity is always nondecreasing with power if all interferers transmit with identical distributions as the primary user. Also, if all input distributions in an interference channel are optimized jointly, then the achievable sum-rate capacity is again nondecreasing. The results generalizes to the channel capacity as a function of a wide class of costs, not only power.Comment: This is an updated and expanded version of arXiv:1108.039

On the Impact of Optimal Modulation and FEC Overhead on Future Optical Networks

The potential of optimum selection of modulation and forward error correction (FEC) overhead (OH) in future transparent nonlinear optical mesh networks is studied from an information theory perspective. Different network topologies are studied as well as both ideal soft-decision (SD) and hard-decision (HD) FEC based on demap-and-decode (bit-wise) receivers. When compared to the de-facto QPSK with 7% OH, our results show large gains in network throughput. When compared to SD-FEC, HD-FEC is shown to cause network throughput losses of 12%, 15%, and 20% for a country, continental, and global network topology, respectively. Furthermore, it is shown that most of the theoretically possible gains can be achieved by using one modulation format and only two OHs. This is in contrast to the infinite number of OHs required in the ideal case. The obtained optimal OHs are between 5% and 80%, which highlights the potential advantage of using FEC with high OHs.Comment: Some minor typos were correcte