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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Trellis coded modulation for 4800-9600 bits/s transmission over a fading mobile satellite channel

The combination of trellis coding and multiple phase-shift-keyed (MPSK) signaling with the addition of asymmetry to the signal set is discussed with regard to its suitability as a modulation/coding scheme for the fading mobile satellite channel. For MPSK, introducing nonuniformity (asymmetry) into the spacing between signal points in the constellation buys a further improvement in performance over that achievable with trellis coded symmetric MPSK, all this without increasing average or peak power, or changing the bandwidth constraints imposed on the system. Whereas previous contributions have considered the performance of trellis coded modulation transmitted over an additive white Gaussian noise (AWGN) channel, the emphasis in the paper is on the performance of trellis coded MPSK in the fading environment. The results will be obtained by using a combination of analysis and simulation. It will be assumed that the effect of the fading on the phase of the received signal is fully compensated for either by tracking it with some form of phase-locked loop or with pilot tone calibration techniques. Thus, results will reflect only the degradation due to the effect of the fading on the amplitude of the received signal. Also, we shall consider only the case where interleaving/deinterleaving is employed to further combat the fading. This allows for considerable simplification of the analysis and is of great practical interest. Finally, the impact of the availability of channel state information on average bit error probability performance is assessed.

Divsalar, Dariush↗
NASA NTRS1988

The design of trellis coded MPSK for fading channels: Performance criteria

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. Athough maximum free distance (dfree) 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 dfree.

Divsalar, Dariush↗
NASA NTRS1989

High rate concatenated coding systems using multidimensional bandwidth-efficient trellis inner codes

A concatenated coding system using two-dimensional trellis-coded MPSK inner codes and Reed-Solomon outer codes for application in high-speed satellite communication systems was proposed previously by the authors (1989). The authors extend their results to systems using symbol-oriented, multidimensional, trellis-coded MPSK inner codes. The concatenated coding systems are divided into two classes according to their achievable effective information rates. The first class uses multidimensional trellis-coded 8-PSK inner codes and achieves effective information rates around 1 b/dimension (spectral efficiency 2 b/s/Hz). The second class employs multidimensional trellis-coded 16-PSK inner codes and provides effective information rates around 1.5 b/dimension (spectral efficiency 3 b/s/Hz). Both classes provide significant coding gains over an uncoded reference system with the same effective information rate as the coded system. The results show that the symbol-oriented nature of multidimensional inner codes can provide an improvement of up to 1 dB in the overall performance of a concatenated coding system when these codes replace bit-oriented two-dimensional codes.

Deng, Robert H.↗
NASA NTRS1990

On complexity of trellis structure of linear block codes

The trellis structure of linear block codes (LBCs) is discussed. The state and branch complexities of a trellis diagram (TD) for a LBC is investigated. The TD with the minimum number of states is said to be minimal. The branch complexity of a minimal TD for a LBC is expressed in terms of the dimensions of specific subcodes of the given code. Then upper and lower bounds are derived on the number of states of a minimal TD for a LBC, and it is shown that a cyclic (or shortened cyclic) code is the worst in terms of the state complexity among the LBCs of the same length and dimension. Furthermore, it is shown that the structural complexity of a minimal TD for a LBC depends on the order of its bit positions. This fact suggests that an appropriate permutation of the bit positions of a code may result in an equivalent code with a much simpler minimal TD. Boolean polynomial representation of codewords of a LBC is also considered. This representation helps in study of the trellis structure of the code. Boolean polynomial representation of a code is applied to construct its minimal TD. Particularly, the construction of minimal trellises for Reed-Muller codes and the extended and permuted binary primitive BCH codes which contain Reed-Muller as subcodes is emphasized. Finally, the structural complexity of minimal trellises for the extended and permuted, and double-error-correcting BCH codes is analyzed and presented. It is shown that these codes have relatively simple trellis structure and hence can be decoded with the Viterbi decoding algorithm.

Lin, Shu↗
NASA NTRS1993

On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes

It was shown earlier that for a punctured Reed-Muller (RM) code or a primitive BCH code, which contains a punctured RM code of the same minimum distance as a large subcode, the state complexity of the minimal trellis diagram is much greater than that for an equivalent code obtained by a proper permutation on the bit positions. To find a permutation on the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is an interesting and challenging problem. This permutation problem is related to the generalized Hamming weight hierarchy of a code, and is shown that for RM codes, the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to Wei. The state complexity of trellis diagram for the extended and permuted (64, 24) BCH code is discussed.

Kasami, Tadao↗
NASA NTRS1995

Trellis complexity bounds for decoding linear block codes

We consider the problem of finding a trellis for a linear block code that minimizes one or more measures of trellis complexity. The domain of optimization may be different permutations of the same code or different codes with the same parameters. Constraints on trellises, including relationships between the minimal trellis of a code and that of the dual code, are used to derive bounds on complexity. We define a partial ordering on trellises: If a trellis is optimum with respect to this partial ordering, if has the desirable property that it simultaneously minimizes all of the complexity measures examined. We examine properties of such optimal trellises and give examples of optimal permutations of codes, most notably the (48,24,12) quadratic residue code.

Kiely, A. B.↗
NASA NTRS1997

Turbo Trellis Coded Modulation With Iterative Decoding for Mobile Satellite Communications

In this paper, analytical bounds on the performance of parallel concatenation of two codes, known as turbo codes, and serial concatenation of two codes over fading channels are obtained. Based on this analysis, design criteria for the selection of component trellis codes for MPSK modulation, and a suitable bit-by-bit iterative decoding structure are proposed. Examples are given for throughput of 2 bits/sec/Hz with 8PSK modulation. The parallel concatenation example uses two rate 4/5 8-state convolutional codes with two interleavers. The convolutional codes' outputs are then mapped to two 8PSK modulations. The serial concatenated code example uses an 8-state outer code with rate 4/5 and a 4-state inner trellis code with 5 inputs and 2 x 8PSK outputs per trellis branch. Based on the above mentioned design criteria for fading channels, a method to obtain he structure of the trellis code with maximum diversity is proposed. Simulation results are given for AWGN and an independent Rayleigh fading channel with perfect Channel State Information (CSI).

Divsalar, D.↗
NASA NTRS1998

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes: A Recursive Maximum Likelihood Decoding - Part 3

The Viterbi algorithm is indeed a very simple and efficient method of implementing the maximum likelihood decoding. However, if we take advantage of the structural properties in a trellis section, other efficient trellis-based decoding algorithms can be devised. Recently, an efficient trellis-based recursive maximum likelihood decoding (RMLD) algorithm for linear block codes has been proposed. This algorithm is more efficient than the conventional Viterbi algorithm in both computation and hardware requirements. Most importantly, the implementation of this algorithm does not require the construction of the entire code trellis, only some special one-section trellises of relatively small state and branch complexities are needed for constructing path (or branch) metric tables recursively. At the end, there is only one table which contains only the most likely code-word and its metric for a given received sequence r = (r(sub 1), r(sub 2),...,r(sub n)). This algorithm basically uses the divide and conquer strategy. Furthermore, it allows parallel/pipeline processing of received sequences to speed up decoding.

Lin, Shu↗
NASA NTRS1985

Combined trellis coding with asymmetric MPSK modulation: An MSAT-X report

Traditionally symmetric, multiple phase-shift-keyed (MPSK) signal constellations, i.e., those with uniformly spaced signal points around the circle, have been used for both uncoded and coded systems. Although symmetric MPSK signal constellations are optimum for systems with no coding, the same is not necessarily true for coded systems. This appears to show that by designing the signal constellations to be asymmetric, one can, in many instances, obtain a significant performance improvement over the traditional symmetric MPSK constellations combined with trellis coding. The joint design of n/(n + 1) trellis codes and asymmetric 2 sup n + 1 - point MPSK is considered, which has a unity bandwidth expansion relative to uncoded 2 sup n-point symmetric MPSK. The asymptotic performance gains due to coding and asymmetry are evaluated in terms of the minimum free Euclidean distance free of the trellis. A comparison of the maximum value of this performance measure with the minimum distance d sub min of the uncoded system is an indication of the maximum reduction in required E sub b/N sub O that can be achieved for arbitrarily small system bit-error rates. It is to be emphasized that the introduction of asymmetry into the signal set does not effect the bandwidth of power requirements of the system; hence, the above-mentioned improvements in performance come at little or no cost. MPSK signal sets in coded systems appear in the work of Divsalar.

Simon, M. K.↗
NASA NTRS1987

The performance of trellis coded multilevel DPSK on a fading mobile satellite channel

The performance of trellis coded multilevel differential phase-shift-keying (MDPSK) over Rician and Rayleigh fading channels is discussed. For operation at L-Band, this signalling technique leads to a more robust system than the coherent system with dual pilot tone calibration previously proposed for UHF. The results are obtained using a combination of analysis and simulation. The analysis shows that the design criterion for trellis codes to be operated on fading channels with interleaving/deinterleaving is no longer free Euclidean distance. The correct design criterion for optimizing bit error probability of trellis coded MDPSK over fading channels will be presented along with examples illustrating its application.

Simon, Marvin K.↗
NASA NTRS1987

Trellis coding with asymmetric modulations

Through the asymmetric design of signal constellations, it often becomes possible to obtain a performance gain over conventional symmetric constellations combined with trellis coding. Attention is given to the joint design of n/(n+1) trellis codes and asymmetric 2 exp (n+1)-point signal constellations having no bandwidth expansion relative to an uncoded 2 exp n-point symmetric signal set. The asymptotic performance gains due to coding and asymmetry are explained in terms of the minimum free Euclidean distance of the trellis; examples are given which show the performance gain due to the asymmetry of the signal set. Since asymmetry does not affect bandwidth or power requirements, these improvements come at little or no cost.

Divsalar, Dariush↗
NASA NTRS1988

The performance of trellis coded multilevel DPSK on a fading mobile satellite channel

The performance of trellis-coded multilevel differential phase-shift-keying (MDPSK) modulation over Rician and Rayleigh fading channels is discussed. For operation at L-band, this signalling technique leads to a more robust system than the coherent system with dual pilot tone calibration previously proposed for UHF. The results are obtained using a combination of analysis and simulation. The analysis shows that the design criterion for trellis codes to be operated on fading channels with interleaving/deinterleaving is no longer free Euclidean distance. The correct design criterion for optimizing bit error probability of trellis coded MDPSD over fading channels is presented along with examples illustrating its application.

Simon, Marvin K.↗
NASA NTRS1988

A lower bound on the minimum Euclidean distance of trellis-coded modulation schemes

A lower bound on the minimum free Euclidean distance of trellis-coded modulation (TCM) is derived that guarantees the existence of good TCM codes of any complexity. The bound is used to compare trellis codes combined with phase-shift keying, pulse amplitude modulation, and quadrature amplitude-shift keying modulation. This random coding bound is the first lower bound on the free distance of trellis codes, is tighter than any upper bound for large constraint lengths, and predicts the asymptotic performance of TCM when the complexity of the code becomes large. The bound can be used with any code rate and any modulation scheme and shows that the free distance increases linearly with the constraint length for large values of the constraint length.

Rouanne, Marc↗
NASA NTRS1990

Multi-level trellis coded modulation and multi-stage decoding

Several constructions for multi-level trellis codes are presented and many codes with better performance than previously known codes are found. These codes provide a flexible trade-off between coding gain, decoding complexity, and decoding delay. New multi-level trellis coded modulation schemes using generalized set partitioning methods are developed for Quadrature Amplitude Modulation (QAM) and Phase Shift Keying (PSK) signal sets. New rotationally invariant multi-level trellis codes which can be combined with differential encoding to resolve phase ambiguity are presented.

Costello, Daniel J., Jr.↗
NASA NTRS1990

Trellis coded modulation for transmission over fading mobile satellite channel

The combination of trellis coding and multiple phase-shift keyed (MPSK) signaling with asymmetry (nonuniform spacing) to the signal set is disclosed with regard to its suitability for a fading mobile satellite communication channel. For MPSK signaling, introducing nonuniformity in the phase spacing between signal points provides an improvement in performance over that achievable with trellis codes symmetric MPSK signaling, all this without increasing the average or peak power, or changing the bandwidth constraints imposed on the system. Block interleaving may be used to reduce error and pilot tone(s) may be used for improving the error correction performance of the trellis decoder in the presence of channel fading.

Simon, Marvin K.↗
NASA NTRS1993

Trellis coding with multidimensional QAM signal sets

Trellis coding using multidimensional QAM signal sets is investigated. Finite-size 2D signal sets are presented that have minimum average energy, are 90-deg rotationally symmetric, and have from 16 to 1024 points. The best trellis codes using the finite 16-QAM signal set with two, four, six, and eight dimensions are found by computer search (the multidimensional signal set is constructed from the 2D signal set). The best moderate complexity trellis codes for infinite lattices with two, four, six, and eight dimensions are also found. The minimum free squared Euclidean distance and number of nearest neighbors for these codes were used as the selection criteria. Many of the multidimensional codes are fully rotationally invariant and give asymptotic coding gains up to 6.0 dB. From the infinite lattice codes, the best codes for transmitting J, J + 1/4, J + 1/3, J + 1/2, J + 2/3, and J + 3/4 bit/sym (J an integer) are presented.

Pietrobon, Steven S.↗
NASA NTRS1995

On the Trellis structure of a (64,40,8) subcode of the (64,42,8) third-order Reed-Muller code

A (64,40,8) subcode of the (64,42,8) third-order Reed-Muller code is proposed to NASA for high-speed satellite communications. This code can be either used alone or used as an inner-code in a concatenated coding system with the NASA standard (255,223,33) Reed-Solomon code as the outer code to achieve high performance with reduced decoding complexity. This Reed-Muller subcode has a relatively simple and parallel trellis structure and consequently can be decoded with a group of identical and relatively simple Viterbi decoders in parallel to achieve high-speed decoding. In this report, the complexities of various sectionalized trellis diagrams are analyzed. Based on this analysis, the trellis diagram with the smallest overall complexity will be used for the implementation of a high-speed decoder.

Moorthy, Hari T.↗
NASA NTRS1998

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes

Decoding algorithms based on the trellis representation of a code (block or convolutional) drastically reduce decoding complexity. The best known and most commonly used trellis-based decoding algorithm is the Viterbi algorithm. It is a maximum likelihood decoding algorithm. Convolutional codes with the Viterbi decoding have been widely used for error control in digital communications over the last two decades. This chapter is concerned with the application of the Viterbi decoding algorithm to linear block codes. First, the Viterbi algorithm is presented. Then, optimum sectionalization of a trellis to minimize the computational complexity of a Viterbi decoder is discussed and an algorithm is presented. Some design issues for IC (integrated circuit) implementation of a Viterbi decoder are considered and discussed. Finally, a new decoding algorithm based on the principle of compare-select-add is presented. This new algorithm can be applied to both block and convolutional codes and is more efficient than the conventional Viterbi algorithm based on the add-compare-select principle. This algorithm is particularly efficient for rate 1/n antipodal convolutional codes and their high-rate punctured codes. It reduces computational complexity by one-third compared with the Viterbi algorithm.

Lin, Shu↗