# On linear structure and phase rotation invariant properties of block M-PSK modulation codes

TL;DR: Two important structural properties of block M(=2/sup '/)-ary PSK modulation codes, linear structure and phase symmetry, are investigated.

Abstract: Two important structural properties of block M(=2/sup '/)-ary PSK modulation codes, linear structure and phase symmetry, are investigated. An M-ary modulation code is first represented as a code with symbols from the integer group S/sub M-PSK/=(0,1,2,---,M-1) under modulo-M addition. Then the linear structure of block M-PSK modulation codes over S/sub M-PSK/ with respect to modulo-M vector addition is defined, and conditions are derived under which a block M-PSK modulation code is linear. Once the linear structure is developed, the phase symmetry of block M-PSK modulation codes is studied. In particular, a necessary and sufficient condition for a block M-PSK modulation code that is linear as a binary code to be invariant under 2/sup h/180 degrees /M phase rotation, for 1 >

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TL;DR: A systematic approach to partitioning L*MPSK signal sets that is based on block coding is used and an encoder system approach is developed that incorporates the design of a differential precoder, a systematic convolutional encoder, and a signal set mapper.

Abstract: A 2L-dimensional multiple phase-shift keyed (L*MPSK) signal set is obtained by forming the Cartesian product of L two-dimensional MPSK signal sets. A systematic approach to partitioning L*MPSK signal sets that is based on block coding is used. An encoder system approach is developed. It incorporates the design of a differential precoder, a systematic convolutional encoder, and a signal set mapper. Trellis-coded L*4PSK, L*8PSK, and L*16PSK modulation schemes are found for 1 >

268 citations

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24 Jun 1991TL;DR: An upper bound on the number of states of a minimal trellis diagram for a linear block code is derived and a cyclic code or its extended code is shown to be the worst in terms of Trellis state complexity among the linear codes of the same length and dimension.

Abstract: An upper bound on the number of states of a minimal trellis diagram for a linear block code is derived. Using this derivation a cyclic (or shortened cyclic) code or its extended code is shown to be the worst in terms of trellis state complexity among the linear codes of the same length and dimension. The complexity of the minimal trellis diagrams for linear block codes of length 2/sup m/, including the Reed-Muller codes, is analyzed. The construction of minimal trellis diagrams for some extended and permuted primitive BCH codes is presented. It is shown that these codes have considerably simpler trellis structure than the original codes in cyclic form without bit-position permutation. >

109 citations

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TL;DR: The multilevel technique for combining block coding and modulation is investigated, and a technique is presented for analyzing the error performance of block modulation codes for an additive white Gaussian noise channel based on soft-decision maximum likelihood decoding.

Abstract: The multilevel technique for combining block coding and modulation is investigated. A general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures. A specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed. Given a multilevel block modulation code C with no interdependency among the binary component codes, the proposed method gives a multilevel block modulation code C' that has the same rate as C, a minimum squared Euclidean distance not less than that of C, a trellis diagram with the same number of states as that of C, and a smaller number of nearest neighbor codewords than that of C. Finally, a technique is presented for analyzing the error performance of block modulation codes for an additive white Gaussian noise (AWGN) channel based on soft-decision maximum likelihood decoding. Error probabilities of some specific codes are evaluated by simulation and upper bounds based on their Euclidean weight distributions. >

100 citations

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TL;DR: The main building block for the construction of a geometrically uniform coded modulation scheme is a subgroup of G/sup I/, where G is a group generating a low-dimensional signal constellation and I is an index set, and this paper study the properties of these subgroups when G is cyclic.

Abstract: The main building block for the construction of a geometrically uniform coded modulation scheme is a subgroup of G/sup I/, where G is a group generating a low-dimensional signal constellation and I is an index set. In this paper we study the properties of these subgroups when G is cyclic. We exploit the fact that any cyclic group of q elements is isomorphic to the additive group of Z/sub q/ (the ring of integers modulo q) so that we can make use of concepts related to linearity. Our attention is focused mainly on indecomposable cyclic groups (i.e., of prime power order), since they are the elementary "building blocks" of any abelian group. In analogy with the usual construction of linear codes over fields, we define a generator matrix and a parity check matrix. Trellis construction and bounds on the minimum Euclidean distance are also investigated. Some examples of coded modulation schemes based on this theory are also exhibited, and their performance evaluated. >

44 citations

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TL;DR: The computation and simulation results for these codes show that with multistage decoding, significant coding gains can be achieved with large reduction in decoding complexity.

Abstract: Multistage decoding of multilevel block multilevel phase-shift keying (M-PSK) modulation codes for the additive white Gaussian noise (AWGN) channel is investigated. Several types of multistage decoding, including a suboptimum soft-decision decoding scheme, are devised and analyzed. Upper bounds on the probability of an incorrect decoding of a code are derived for the proposed multistage decoding schemes. Error probabilities of some specific multilevel block 8-PSK modulation codes are evaluated and simulated. The computation and simulation results for these codes show that with multistage decoding, significant coding gains can be achieved with large reduction in decoding complexity. In one example, it is shown that the difference in performance between the proposed suboptimum multistage soft-decision decoding and the single-stage optimum decoding is small, only a fraction of a dB loss in SNR at the block error probability of 10/sup -6/. >

38 citations

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IBM

^{1}TL;DR: A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance.

Abstract: A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth. This is achieved by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance. Soft maximum--likelihood (ML) decoding using the Viterbi algorithm is assumed. Following a discussion of channel capacity, simple hand-designed trellis codes are presented for 8 phase-shift keying (PSK) and 16 quadrature amplitude-shift keying (QASK) modulation. These simple codes achieve coding gains in the order of 3-4 dB. It is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals, which we call "mapping by set partitioning." Based on a new distance measure between binary code sequences which efficiently lower-bounds the Euclidean distance between the corresponding channel signal sequences, a search procedure for more powerful codes is developed. Codes with coding gains up to 6 dB are obtained for a variety of multilevel/phase modulation schemes. Simulation results are presented and an example of carrier-phase tracking is discussed.

4,091 citations

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TL;DR: A new multilevel coding method that uses several error-correcting codes that makes effective use of soft-decisions to improve the performance of decoding and is superior to other multileVEL coding systems.

Abstract: A new multilevel coding method that uses several error-correcting codes is proposed. The transmission symbols are constructed by combining symbols of codewords of these codes. Usually, these codes are binary error-correcting codes and have different error-correcting capabilities. For various channels, efficient systems can be obtained by choosing these codes appropriately. Encoding and decoding procedures for this method are relatively simple compared with those of other multilevel coding methods. In addition, this method makes effective use of soft-decisions to improve the performance of decoding. The decoding error probability is analyzed for multiphase modulation, and numerical comparisons to other multilevel coding systems are made. When equally complex systems are compared, the new system is superior to other multilevel coding systems.

1,070 citations

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IBM

^{1}TL;DR: An introduction into TCM is given, reasons for the development of TCM are reviewed, and examples of simple TCM schemes are discussed.

Abstract: rellis-Coded Modulation (TCM) has evolved over the past decade as a combined coding and modulation technique for digital transmission over band-limited channels. Its main attraction comes from the fact that it allows the achievement of significant coding gains over conventional uncoded multilevel modulation without compromising bandwidth efficiency. T h e first TCM schemes were proposed in 1976 [I]. Following a more detailed publication [2] in 1982, an explosion of research and actual implementations of TCM took place, to the point where today there is a good understanding of the theory and capabilities of TCM methods. In Part 1 of this two-part article, an introduction into TCM is given. T h e reasons for the development of TCM are reviewed, and examples of simple TCM schemes are discussed. Part I1 [I51 provides further insight into code design and performance, and addresses. recent advances in TCM. TCM schemes employ redundant nonbinary modulation in combination with a finite-state encoder which governs the selection of modulation signals to generate coded signal sequences. In the receiver, the noisy signals are decoded by a soft-decision maximum-likelihood sequence decoder. Simple four-state TCM schemes can improve. the robustness of digital transmission against additive noise by 3 dB, compared to conventional , uncoded modulation. With more complex TCM schemes, the coding gain can reach 6 dB or more. These gains are obtained without bandwidth expansion or reduction of the effective information rate as required by traditional error-correction schemes. Shannon's information theory predicted the existence of coded modulation schemes with these characteristics more than three decades ago. T h e development of effective TCM techniques and today's signal-processing technology now allow these ,gains to be obtained in practice. Signal waveforms representing information sequences ~ are most impervious to noise-induced detection errors if they are very different from each other. Mathematically, this translates into therequirement that signal sequences should have large distance in Euclidean signal space. ~ T h e essential new concept of TCM that led to the afore-1 mentioned gains was to use signal-set expansion to I provide redundancy for coding, and to design coding and ' signal-mapping functions jointly so as to maximize ~ directly the \" free distance \" (minimum Euclidean distance) between coded signal sequences. This allowed the construction of modulation codes whose free distance significantly exceeded the minimum distance between uncoded modulation signals, at the same information rate, bandwidth, and signal power. The term \" …

874 citations

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TL;DR: The family of Barnes-Wall lattices and their principal sublattices, which are useful in constructing coset codes, are generated by iteration of a simple construction called the squaring construction, and are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum-likelihood decoding algorithms.

Abstract: For pt.I see ibid., vol.34, no.5, p.1123-51 (1988). The family of Barnes-Wall lattices (including D/sub 4/ and E/sub 8/) of lengths N=2/sup n/ and their principal sublattices, which are useful in constructing coset codes, are generated by iteration of a simple construction called the squaring construction. The closely related Reed-Muller codes are generated by the same construction. The principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions, which also exhibit the interrelationships between codes and lattices of different lengths. An extension called the cubing construction generates good codes and lattices of lengths N=3*2/sup n/, including the Golay code and Leech lattice, with the use of special bases for 8-space. Another related construction generates the Nordstrom-Robinson code and an analogous 16-dimensional nonlattice packing. These constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum-likelihood decoding algorithms. >

685 citations

01 Jan 1988

TL;DR: The family of Barnes-Wall lattices and their principal sublattices, which are useful in con- structing coset codes, are generated by iteration of a simple construction called the "squaring construction," and the closely related Reed-Muller codes are generate by the same construction.

Abstract: The family of Barnes-Wall lattices (including D4 and E,) of lengths N = 2" and their principal sublattices, which are useful in con- structing coset codes, are generated by iteration of a simple construction called the "squaring construction." The closely related Reed-Muller codes are generated by the same construction. The principal properties of these codes and lattices, including distances, dimensions, partitions, generator matrices, and duality properties, are consequences of the general proper- ties of iterated squaring constructions, which also exhibit the interrelation- ships between codes and lattices of different lengths. An extension called the "cubing construction" generates good codes and lattices of lengths N = 3.2", including the Golay code and Leech lattice, with the use of special bases for 8-space. Another related construction generates the Nordstrom-Robinson code and an analogous 16-dimensional nonlattice packing. These constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms. General algebraic methods for determining minimal trellis diagrams of codes, lattices, and partitions are given in an Appendix.

274 citations