Multilevel trellis coded modulations for the Rayleigh fadingchannel

The authors present coded 8-phase-shift-keyed (8-PSK) modulations for the Rayleigh fading channel. The schemes are based on multilevel trellis-coded-modulation constructions and utilize maximum free Hamming distance binary convolutional codes as building blocks. A suboptimal multistage decoder that utilizes interstage interleaving and iterative decoding is proposed and evaluated. Examples are constructed to show that the proposed schemes outperform the best modified codes of the Ungerboeck type due to significantly higher implicit time diversity, yielding seven branches of built-in time diversity, whereas the Ungerboeck code yields four branches of time diversity for a 64-state system. The transmission delay is higher, however. The new schemes can provide three levels of unequal error protection when 8-PSK or 8-differential-phase-shift-keying (8-DPSK) modulations are used. They provide 10-14-dB channel signal-to-noise ratio gain over uncoded 4-DPSK at a bit error rate of 10^-3 for a modest decoding complexity     

8-PSK trellis codes for a Rayleigh channel

A suboptimal trellis coding approach based on the concept of combining a good convolutional code and bit interleavers is presented. The aim is to improve the reliability of digital radio communication over a fading channel. It is shown that over a Rayleigh channel and for a fixed code complexity the proposed system is superior to the baseline system. Its performance is analyzed using the generalized R_o and the upper bound on the bit error rate. The results suggest that on a Rayleigh channel, the standard trellis codes may not be the correct approach for improving the reliability of the communication channel. The discussion is restricted to a rate 2/3 coded system with 8-PSK modulation     

Multilevel construction of block and trellis group codes

The theory of group codes has been shown to be a useful starting point for the construction of good geometrically uniform codes. In this paper we study the problem of building multilevel group codes, i.e., codes obtained combining separate coding at different levels in such a way that the resulting code is a group code. A construction leading to multilevel group codes for semi-direct and direct products is illustrated. The codes that can be obtained in this way are identified. New geometrically uniform Euclidean-space codes obtained from multilevel codes over abelian and nonabelian groups are presented     

Matrix representation of CPFSK trellis modulation codes

A matrix representation is presented for M-CPFSK signals when the modulation index h=p/q (q>or=M). It simplifies trellis search algorithms that compute minimum distances and other trellis parameters. The matrix representation has been applied to find self-transparent CPFSK trellis modulation codes.<>     

Geometrically uniform partitions of L×MPSK constellations andrelated binary trellis codes

The theory of geometrically uniform trellis codes is applied to the case of multidimensional PSK (phase shift keying) constellations. The symmetry group of an L×MPSK (M-ary PSK) constellation is completely characterized. Conditions for rotational invariance of geometrically uniform partitions of a signal constellation are given. Through suitable algorithms, geometrically uniform partitions of L×MPSK (M=4,8,16 and L=1,2,3,4) constellations are found, which present good characteristics in terms of the set of distances at a given partition level, the maximum obtainable rotational invariance, and the isomorphism of the quotient group associated with the partition. These partitions are used as starting points in a search for good geometrically uniform trellis codes based on binary convolutional codes     

Some block- and trellis-coded modulations for the Rayleigh fadingchannel

The error performance of a modulation code over a channel depends on several distance parameters and the path multiplicity of the code. For the AWGN channel, the error performance of a modulation code depends mainly on its minimum squared Euclidean distance and path multiplicity. For the Rayleigh fading channel, however, the error performance of a modulation code depends strongly on its minimum symbol distance, minimum product distance, and path multiplicity. It depends on the minimum squared Euclidean distance in a lesser degree. This paper is concerned with the construction of block and trellis MPSK modulation codes for the Rayleigh fading channel. In each construction, the distance parameters are chosen to achieve good error performance with reduced decoding complexity     

Permutation trellis codes

We introduce the new concept of permutation trellis codes and present a generalized construction procedure, applying our technique of distance-preserving mappings. Minimum-distance decoding follows naturally, using the Viterbi algorithm. We furthermore investigate the performance of these codes when combined with multitone frequency-shift keying modulation and noncoherent detection in a diversity scheme, to make transmissions robust against narrowband, broadband, and background noise disturbances, such as those encountered in power-line communications.     

空时网格码的正交构造法

空时编码是实现宽带无线数据通信的一种极有潜力的技术。然而，随着发射天线数目的增加，空时码的构造变得十分困难。本文提出了空时网格码的一种构造方法－正交构造法：这种方法可以根据已知的适用于M根发射天线的空时码来构造适用于KM（K≥2）根天线的空时码。仿真结果表明，对于相同的码率和和相近的译码复杂度，用正交构造法得到的空时码的差错性能甚至优于通过计算机搜索得到的空时卷积码。     

Upper bounds for small trellis codes

An upper bound on the minimum squared distance of trellis codes by packing Voronoi cells is derived and compared with previously known bounds. The authors focus on codes with small memory for modulation formats such as pulse amplitude modulation (PAM), m-ary quadrature amplitude modulation (QAM), and m-ary phase shift keying (PSK). The bound is tight to search results for coset codes with a small number of states     

Performance bounds for space-time trellis codes

We derive the union bound for space-time trellis codes over quasi-static fading channels. We first observe that the standard approach for evaluating the union bound yields very loose, in fact divergent, bounds over the quasi-static fading channel. We then develop a method for obtaining a tight bound on the error probability. We derive the union bound by performing expurgation of the standard union bound. In addition, we limit the conditional union bound before averaging over the fading process. We demonstrate that this approach provides a tight bound on the error probability of space-time codes. The bounds can be used for the case when the fading coefficients among different transmit/receive antenna pairs are correlated as well. We present several examples of the bounds to illustrate their usefulness.     

Robustly good trellis codes

The relationship between the distance properties of trellis codes and the computational effort and error performance of sequential decoding is studied and optimum distance profile (ODP) and optimum free distance (OFD) trellis codes are constructed for 8-PSK and 16 QAM modulation. A comparison of the performance of both the ODP and the OFD trellis codes reveals that neither class of codes results in the best trade-off between error performance and computational effort when sequential decoding is used. A new algorithm is then proposed to construct robustly good trellis codes for use with sequential decoding. New trellis codes with asymptotic coding gains up to 6.66 dB are obtained using this algorithm, and the new codes achieve nearly the same free distances as the OFD codes and nearly the same distance profiles as the ODP codes     

Super-orthogonal space-time trellis codes

We introduce a new class of space-time codes called super-orthogonal space-time trellis codes. These codes combine set partitioning and a super set of orthogonal space-time block codes in a systematic way to provide full diversity and improved coding gain over earlier space-time trellis code constructions. We also study the optimality of our set partitioning and provide coding gain analysis. Codes operating at different rates, up to the highest theoretically possible rate, for different number of states can be designed by using our optimal set partitioning. Super-orthogonal space-time trellis codes can provide a tradeoff between rate and coding gain. Simulation results show more than 2-dB improvements over the codes presented in the literature while providing a systematic design methodology.     

On multilevel coded MPSK for the Rayleigh fading channel

This paper presents a new upper bound on the pair-wise error probability of MPSK sequences for the Rayleigh fading channel when channel state information (CSI) is unavailable. This bound is derived by adding weight factors in computing symbol metrics. Simulation results show that the weight factors which optimize the upper bound likely optimize the error rate as well. Multilevel coded MPSK schemes for the Rayleigh fading channel are also devised. Results show that the added weight factors improve the error performance of these schemes in the case that CSI is unavailable     

On the BCJR trellis for linear block codes

In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algorithm is Θ(e). This result suggests that the “best” trellis representation for a given linear block code is the one with the fewest edges. We will then show that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf (1978), Massey (1978), and Forney (1988), uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine (1995), we will also discuss “trellis-oriented” or “minimal-span” generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it     

The design of trellis coded MPSK for fading channels: performancecriteria

It has been well established that the appropriate criterion for optimum trellis-coded modulation design on the additive white Gaussian noise channel is maximization of the free Euclidean distance. It is shown that when the trellis-coded modulation is used on a Rician fading channel with interleaving/deinterleaving, the design of the code of optimum performance is guided by other factors, in particular, the length of the shortest error-event path, and the product of branch distances (possibly normalized by the Euclidean distance of the path) along the path. Although maximum free distance (d_free) is still an important consideration, it plays a less significant role the more severe the fading is on the channel. These considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over Rician fading channels. If no interleaving/deinterleaving is used, then once again the design of the trellis code is guided by maximizing d_free      

Universal space-time codes from demultiplexed trellis codes

In broadcast scenarios or in the absence of accurate channel probability distribution information, code design for consistent channel-by-channel performance, rather than average performance over a channel distribution, may be desirable. Root and Varaiya's compound channel theorem for linear Gaussian channels promises the existence of universal codes that operate reliably whenever the channel mutual information (MI) is above the transmitted rate. This paper presents two-dimensional trellis codes that provide such universal performance over the compound linear vector Gaussian channel when demultiplexed over two, three, and four transmit antennas. The presented trellis codes, found by exhaustive search, guarantee consistent performance on every matrix channel that supports the information transmission rate with an MI gap that is similar to the capacity gap of a well-designed additive white Gaussian noise (AWGN)-specific code on the AWGN channel. As a result of their channel-by-channel consistency, the universal trellis codes presented here also deliver comparable, or, in some cases, superior, frame-error rate and bit-error rate performance under quasi-static Rayleigh fading, as compared with trellis codes of similar complexity that are designed specifically for the quasi-static Rayleigh-fading scenario.     

Probability of bit error for MPSK modulation with diversityreception in Rayleigh fading and log-normal shadowing channel

In an additive white Gaussian noise channel subject to Rayleigh fading and log-normal shadowing, consideration is given to diversity reception with K-port macroscopic selection and L-branch microscopic maximal-ratio combining, and analytical expressions are derived for upper and lower bounds on the bit error probabilities for BPSK, QPSK, 8-PSK, and 16-PSK modulations. The derived expressions can be evaluated at any location in the serving cell. Error-probability can be calculated by averaging over all possible locations within the serving cell. By evaluating these bounds, numerical results for the following cases are presented: without diversity reception, microdiversity combining, macrodiversity selection, and simultaneous use of macrodiversity selection and microdiversity combining. For symmetrical arrangement of macrodiversity ports against the lognormal shadowing, error probability at the equidistant point gives upper bounds on the error probabilities for most of the area in the serving cell. Error probability at the equidistant point is a good estimate of the error probability obtained by averaging over all possible locations in the cell     

Chip interleaved turbo codes for DS-CDMA mobile radio in fadingchannel

A new channel interleaving method, called chip interleaving, for direct sequence code division multiple access (DS-CDMA) is proposed. A chip interleaver scrambles the chips and transforms the transmission channel into a highly time-selective or highly memoryless channel. It was found that the turbo decoding performance which degrades in a fading channel is significantly improved with chip interleaving     

Rotationally invariant nonlinear trellis codes for two-dimensionalmodulation

A general parity-check equation is presented that defines rotationally invariant trellis codes of rate k/(k+1) for two-dimensional signal sets. This parity-check equation is used to find rate k/(k+1) codes for 4PSK, 8PSK, 16PSK, and QAM signal sets by systematic code searches. The MPSK codes exhibit smaller free Euclidean distances than nonrotationally invariant linear codes with the same number of states. However, since the nonlinear codes have a smaller number of nearest neighbors, their performance at moderate signal to noise ratios is close to that of the best linear codes. The rotationally invariant QAM codes with 8, 32, 64, and 256 states achieve the same free Euclidean distance as the best linear codes. Transparency of user information under phase rotations is accomplished either by conventional differential encoding and decoding, or by integrating this function directly into the code trellis