Good lattice constellations for both Rayleigh fading and Gaussianchannels

Recent work on lattices matched to the Rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency. We present a new family of lattice constellations, based on complex algebraic number fields, which have good performance on Rayleigh fading channels. Some of these lattices also present a reasonable packing density and thus may be used at the same time over a Gaussian channel. Conversely, we show that particular versions of the best lattice packings (D₄, E₆, E₈, K₁₂ , Λ₁₆, Λ₂₄), constructed from totally complex algebraic cyclotomic fields, present better performance over the Rayleigh fading channel. The practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links. Some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results     

Computing upper bounds to error probability of coded modulationschemes

To evaluate an upper bound on error probabilities of signal constellations used for transmission over the additive white Gaussian noise (AWGN) channel, enumeration of all the constellation intradistances is required. These may be infinite in number, for example, when convolutional codes are used and the constellations are lattices. Truncation of the series does not necessarily provide a bound anymore, and must be done with care. Yet the union bound is very simple, as it does not require any further knowledge about the signal constellation than the distance enumerator. In this paper, we describe some methods that can be used to evaluate error probabilities of infinite signal constellations, and that require only a finite number of terms. These methods are applicable, for example, to convolutional codes decoded with a finite-depth Viterbi algorithm and to signal constellations carved from lattices. Coded modulations based on lattices and convolutional or block codes can also be dealt with. As an example of application, we analyze a variable-rate 3-stage coded modulation encoder/decoder, which has been built and is based on a combination of convolutional codes with a single-parity-check block code     

一种基于格理论构造高维星座图的方法

星座图的增益指数是格理论中的一个术语,它可以分解成格的编码增益和星座图边界的成形增益.本文将最大化星座图增益指数的过程构造为一系列优化问题,并将星座图的几何特性作为优化问题的约束条件.由于可通过求解优化问题来得到所需的星座图,因此本文的方法可以作为一种构造高维星座图的通用方法.相比现有算法均只适用于星座点个数较少的情况,本文方法可以简便地构造星座点数目较大的高维星座图.仿真结果显示出：在星座点数目较少时,由本文方法所构造的星座图的误符号率性能与最优值十分接近;而当星座点数目较多时,本文构造的星座图较传统基于整数格的星座图具有更低的误符号率.     

Multilevel coded modulation for unequal error protection andmultistage decoding. II. Asymmetric constellations

In this paper, multilevel coded asymmetric modulation with multistage decoding and unequal error protection (UEP) is discussed. These results further emphasize the fact that unconventional signal set partitionings are more promising than traditional (Ungerboeck-type) partitionings, to achieve UEP capabilities with multilevel coding and multistage decoding. Three types of unconventional partitionings are analyzed for asymmetric 8-PSK and 16-QAM constellations over the additive white Gaussian noise channel to introduce design guidelines. Generalizations to other PSK and QAM type constellations follow the same lines. Upper bounds on the bit-error probability based on union bound arguments are first derived. In some cases, these bounds become loose due to the large overlappings of decision regions associated with asymmetric constellations and unconventional partitionings. To overcome this problem, simpler and tighter approximated bounds are derived. Based on these bounds, it is shown that additional refinements can be achieved in the construction of multilevel UEP codes, by introducing asymmetries in PSK and QAM signal constellations     

Scale-Recursive Lattice-Based Multiple-Access Symbol Constellations

This paper describes scale-recursive symbol constellations for the synchronous multiple-access (MA) channel, which can be seen as a generalization of Voronoi coding. The constellations are related by a fixed scaling and rotation, and are designed so that in the noise-free case, the combined signal always falls onto an appropriate lattice. This allows high throughput and a linear complexity decoding algorithm. Novel constellations of up to 24 dimensions are given that show improved performance over conventional rectangular lattices.     

Nonequiprobable signaling on the Gaussian channel

Signaling schemes for the Gaussian channel based on finite-dimensional lattices are considered. The signal constellation consists of all lattice points within a region R, and the shape of this region determines the average signal power. Spherical signal constellations minimize average signal power, and in the limit as N →∞, the shape gain of the N-sphere over the N-cube approaches πe/6≈1.53 dB. A nonequiprobable signaling scheme is described that approaches this full asymptotic shape gain in any fixed dimension. A signal constellation, Ω is partitioned into T subconstellations Ω₀ , . . ., Ω_τ-1 of equal size by scaling a basic region R. Signal points in the same subconstellation are used equiprobably, and a shaping code selects the subconstellation Ω_i with frequency f_i. Shaping codes make it possible to achieve any desired fractional bit rate. The schemes presented are compared with equiprobable signaling schemes based on Voronoi regions of multidimensional lattices. For comparable shape gain and constellation expansion ratio, the peak to average power ratio of the schemes presented is superior. Furthermore, a simple table lookup is all that is required to address points in the constellations. It is also shown that it is possible to integrate coding and nonequiprobable signaling within a common multilevel framework     

Signal Constellation Design for Symbol?Level Rate Adaptation in AWGN Channels

This paper considers rate adaptation for the additive white Gaussian noise (AWGN)-channel with unknown channel capacity. We show how multi-layer signal constellations can be optimized layer-wise for particular signal-to-noise ratio (SNR)- points, to provide simple and efficient ARQ-based symbol-level rate adaptation with fine granularity. A turbo-coded system is used as an example showing that the proposed design provides actual communication rates that are close to the channel capacity.     

Worst-case power-constrained noise for binary-input channels

Additive noise channels with binary-valued inputs and real-valued outputs are considered. The maximum error probability and the minimum channel capacity achieved by any power-constrained noise distribution are obtained. A general framework which applies to a variety of performance measures shows that the least-favorable noise distribution is, in general, a mixture of two lattice probability mass functions. The framework holds for m-ary input constellations on finite-dimensional lattices.<>     

一种基于高斯近似的极化码打孔算法

现有的极化码打孔算法均未考虑信道构造过程对算法性能的影响,针对这一问题,该文提出一种基于高斯近似的极化码打孔算法(GAPPC)。首先将高斯近似作为极化码构造算法,分析高斯近似与打孔算法的关系,以降低信道构造输出值为目标,引入高斯修正因子,推导出改进的高斯近似函数。然后将改进的高斯近似函数引入信道构造,对极化子信道进行排序获得信道可靠性排序集合。最后依据信道容量关系确定映射规则,选出打孔比特集合和冻结比特集合,完成打孔极化码的构建。实验结果显示,在不同的码长和码率下,误帧率和误码率均获得显著降低。     

Exact Method for the Error Probability Calculation of Three-Dimensional Signal Constellations

The contribution of this letter is the computation of exact symbol error probability (SEP) of three-dimensional (3-D) signal constellations over an additive white Gaussian noise (AWGN) channel. The originality of the proposed method is that it can be applied to any arbitrary 3-D constellation whose decision regions may not meet at right angles. We express the SEP in a triple integral form which is further simplified. The simplified form requires a single integral evaluation of standard "erf" function. Using the derived exact SEP formula, we plot SEP for a number of selected 3-D constellations. The SEP obtained using the proposed formula is validated by simulation results. It is also compared with the union bound approximation, an upper bound for SEP.     

Bhattacharyya bound, cutoff rate, and constellation design for thecompanding channel

In modern voiceband data communication, the received signal is subject to nonlinear quantization noise due to companding. Under certain conditions, this quantization noise may become dominant and cause serious degradation in performance. In this paper, we first calculate the Bhattacharyya bound on the error probability between two signal points, which we then use to obtain some insight into the effects of the companding channel. Next, we compute the cutoff rate, and then use these results to design new “optimal” signal constellations for the companding channel. The bound, cutoff rate, and new constellations are computed assuming a signal-dependent noncircular Gaussian channel transition density. In addition, we obtain “optimal” a priori probabilities for this channel     

Phase noise sensitivity analysis of lattice constellation

Based on the assumption of large number of constellation points and high signal-to-noise ratio (SNR), phase noise sensitivity of lattice constellation is analyzed. The upper bound of symbol error rate (SER) in additive white Gaussian noise (AWGN) channel is derived from pairwise error probability. For small phase noise, phase noise channel is transformed to AWGN channel. With the aid of Wiener model, the obtained upper bound can be extended to phase noise channel. The proposed upper bound can be used as performance criterion to analyze the sensitivity of phase noise in multi-dimensional lattice constellation. Simulation results show that with the same normalized spectral efficiency, higher dimensional lattice constellations are more sensitive than lower ones in phase noise channel. It is also shown that with the same dimension of constellation, larger normalized spectral efficiency means more performance loss in phase noise channel.     

Approximate Error Rate Performance of 8-Point AM-PM Signal Constellations on Voiceband Telephone Channels

The purpose of this paper is a detailed comparison of the error rate performance of six 8-point signal constellations under the conditions of Gaussian noise alone, or combined with either phase jitter, second order harmonic, or third order harmonic distortion for application to the voiceband telephone channel. An approximate evaluation technique is developed for inclusion of the effects of second and third order harmonic distortion which was found to be in agreement with laboratory performance measurements of several of the signal constellations reported here. A signal constellation, which consists of points in a circle surrounding a single point in the center (denoted the 1-7 constellation), was found to be superior to the other 5 constellations considered under the conditions of Gaussian noise combined with second or third order harmonic distortion. Under the condition of phase jitter not exceeding 1.5 degrees RMS, it is also superior to the other constellations. Thus, it appears that the 1-7 constellation in conjunction with a good phase locked loop would offer superior performance.     

QAM constellation classification based on statistical sampling for linear distortive channels

The classification method for phase-amplitude-modulated signals transmitted through linear, frequency-selective fading channels is presented. Utilizing a Monte Carlo (MC) sampling technique, QAM constellation classification is based on Gibbs sampling. Constellation classification is achieved along with joint estimation of unknown system parameters, including channel taps and data symbols. When all channel parameters are known, direct steps are proposed for classification and data estimation without altering the general approach. Moreover, a multistage classification algorithm is also presented to improve computational efficiency. The proposed QAM classification methods are shown to perform well on various constellations with different cardinalities, as well as constellations with symbols.     

The performance of incremental redundancy schemes based on convolutional codes in the block-fading Gaussian collision channel

The throughput performance of incremental redundancy (INR) schemes, based on short constraint length convolutional codes, is evaluated for the block-fading Gaussian collision channel. Results based on simulations and union bound computations are compared to estimates of the achievable throughput performance with random binary and Gaussian coding in the limit of large block lengths, obtained through information outage considerations. For low channel loads, it is observed that INR schemes with binary convolutional codes and limited block length may provide throughput close to the achievable performance for binary random coding. However, for these low loads, compared to binary random coding, Gaussian random coding may provide significantly better throughput performance, which prompts the use of larger modulation constellations. For high channel loads, a relatively large gap in throughput performance between binary convolutional codes and binary random codes indicates a potential for extensive performance improvement by alternative coding strategies. Only small improvements of the throughput have been observed by increasing the complexity through increased state convolutional coding.     

On generalized two-dimensional cross constellations and the opportunistic secondary channel

An optimum bits-to-symbol mapping for square constellations is introduced which allows a simple detection in narrow-sense quadrature amplitude modulation (qam) systems. Then, a family of two-dimensional generalized cross constellations is presented as well as upper and lower bounds on the symbol error probability over an ideal band-limited channel which generalize those previously known for conventional qam. The application of this scheme to the opportunistic secondary channel is analysed and it is shown how fractional rates (in bits per 2-dimensional signal) can be supported on 2D generalized qam systems. These signaling schemes are compared with multidimensional generalized constellations recently proposed by Forney and Wei.     

非理想信噪比估计对BICM-ID系统性能的影响

该文给出一种直观的信噪比估计非理想时的先验信息建模,使用外信息转移图较为全面地分析了非理想信噪比估计对带迭代译码的比特交织编码调制系统性能的影响,分析结果表明非理想信噪比估计对典型的星座类型和标号在高斯和瑞利信道下的影响相似;通过对不同帧长的系统性能比较可知,长帧下信噪比估计偏小在-3dB以下时性能会大大下降,而估计偏大则没有影响;而对于中短帧而言,在相同的偏差下,偏小的影响要大于偏大的影响;此外,还给出由拟合得到的8PSK高斯信道和16QAM高斯与瑞利信道下的信噪比估计算法,最后通过系统仿真验证了该算法的有效性。     

Superposition coding for side-information channels

We present simple, practical codes designed for the binary and Gaussian dirty-paper channels. We show that the dirty-paper decoding problem can be transformed into an equivalent multiple-access decoding problem, for which we apply superposition coding. Our concept is a generalization of the nested lattices approach of Zamir, Shamai, and Erez. In a theoretical setting, our constructions are capable of achieving capacity using random component codes and maximum-likelihood decoding. We also present practical implementations of the constructions, and simulation results for both dirty-paper channels. Our results for the Gaussian dirty-paper channel are on par with the best known results for nested lattices. We discuss the binary dirty- tape channel, for which we present a simple, effective coding technique. Finally, we propose a framework for extending our approach to general Gel'fand-Pinsker channels.     

Universal bound on the performance of lattice codes

We present a lower bound on the probability of symbol error for maximum-likelihood decoding of lattices and lattice codes on a Gaussian channel. The bound is tight for error probabilities and signal-to-noise ratios of practical interest, as opposed to most existing bounds that become tight asymptotically for high signal-to-noise ratios. The bound is also universal; it provides a limit on the highest possible coding gain that may be achieved, at specific symbol error probabilities, using any lattice or lattice code in n dimensions. In particular, it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains. The asymptotic (as n→∞) behavior of the new bound is shown to coincide with the Shannon (1948) limit for Gaussian channels