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 noncoherent near-optimal linear complexity multistage decoder for NBC-MPSK is proposed and a tree-search ML decoding algorithm is investigated, which is shown to have low complexity and excellent error performance.
Abstract: A novel noncoherent block coding scheme, called noncoherent block-coded MPSK (NBC-MPSK), was proposed recently. In this paper, we present further research results on NBC-MPSK. We first focus on the rotational invariance (RI) of NBC-MPSK. Based on the RI property of NBC-MPSK with multistage decoding, a noncoherent near-optimal linear complexity multistage decoder for NBC-MPSK is proposed. Then we investigate a tree-search ML decoding algorithm for NBCMPSK. The derived algorithm is shown to have low complexity and excellent error performance. In this paper, we also utilize the idea of the NBC-MPSK to design noncoherent space-time block codes, called noncoherent space-time block-coded MPSK (NSTBC-MPSK). For two transmit antennas, we propose a signal set with set partitioning and derive the minimum noncohent distance of NSTBC-MPSK with this signal set. For the decoding of NSTBC-MPSK, we modify the ML decoding algorithm of NBC-MPSK and propose an iterative hard-decision decoding algorithm. Compared with training codes and unitary space-time modulation, NBC-MPSK and NSTBC-MPSK have larger minimum noncoherent distance and thus better error performance for the noncoherent ML decoder.
11 citations
Cites background or methods from "On linear structure and phase rotat..."
...The necessary and sufficient condition for RI block-coded MPSK was derived in [21]....
[...]
...For i ∈ {a, b, c}, because 1 is a codeword of C′ i (the necessary condition for RI [21]) corresponding to the same information bits as the all-zeros codeword 0, it can be shown that dmax(C′ i) = N − di and thus dncH,i = di....
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...If codes C′ a, C ′ b and C ′ c for NBC-MPSK (as defined in Section II) satisfy the necessary and sufficient condition of RI in [21], with a proper differential encoder, the BCM modified from NBC-MPSK can also be an RI code....
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...Block-coded MPSK C is said to be rotationally invariant (RI) under a 2π/M phase shift if for any codeword v in C, we have v + 1 ∈ C where + is component-wise moduloM addition and 1 is the all-ones codeword [21]....
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11 Dec 2006
TL;DR: A tree-search ML decoding algorithm based on the rotational invariance (RI) of NBC-MPSK with multistage decoding, shown to have low complexity and excellent error performance and a new signal set with partitioning for the generalized NBC- MPSK.
Abstract: A novel noncoherent block coding scheme, called noncoherent block-coded MPSK (NBC-MPSK), was proposed in [1] recently. This NBC-MPSK scheme was generalized to multiple-antenna systems in [2] as well. In this paper, we present further research results on NBC-MPSK and its generalization. We first focus on the rotational invariance (RI) of NBC-MPSK. Based on the RI property of NBC-MPSK with multistage decoding, a noncoherent near-optimal linear-complexity multistage decoder for NBC-MPSK is proposed. Then we propose a tree-search ML decoding algorithm for NBC-MPSK. The proposed algorithm is shown to have low complexity and excellent error performance. This ML decoding algorithm can also be applied to the generalized NBC-MPSK for multiple antennas. In addition, we propose a new signal set with partitioning for the generalized NBC-MPSK. Error performance and complexity of decoding algorithms is verified by simulations.
11 citations
Cites background from "On linear structure and phase rotat..."
...But if codes C ′ a, C ′ b and C ′ c (as defined in Section II) satisfy the necessary and sufficient condition of RI in [16], with a proper differential encoder, the BCM modified from NBC-MPSK can also be a RI code....
[...]
...Note that the necessary and sufficient condition for RI block-coded MPSK was derived in [16]....
[...]
...For i ∈ {a, b, c}, because 1 is a codeword of C ′ i (the necessary condition for RI [16]) and correspond to the same information bits as the all-zero codeword 0, it can be shown that C ′ i has dncH,i = di....
[...]
...Block-coded MPSK C is said to be rotationally invariant (RI) under 2π/M phase shift if for any codeword v in C, we have v + 1 ∈ C where + is component-wise moduloM addition and 1 is the all-one codeword [16]....
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TL;DR: The performance of some known BCM codes designed for Gaussian channels is evaluated on slow Rayleigh fading channels and a comparison with corresponding schemes based on trellis coded modulation (TCM) is presented.
10 citations
29 Jun 1997
TL;DR: It is shown that all four-dimensional signal sets matched to Q(2/sup m/) have the same Euclidean distance profile and hence the Euclidea space codes corresponding to each signal set for a given group code over Q( 2/Sup m/) are automorphic Euclideans-distance equivalent.
Abstract: A length n group code over a group G is a subgroup of G/sup n/ under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q(2/sup m/), m/spl ges/3, are constructed using a binary code and a code over Z(2/sup m-1/), the ring of integers modulo 2/sup m-1/ as component codes and a mapping f from Z/sub 2//spl times/Z(2/sup m-1/)to Q(2/sup m/). A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q(2/sup m/). Given the generator matrices of the component codes, the computational effort involved in checking the necessary and sufficient conditions is discussed. Starting from a four-dimensional signal set matched to Q(2/sup m/), it is shown that the Euclidean space codes obtained from the group codes over Q(2/sup m/) have Euclidean distance profiles which are independent of the coset representative selection involved in f. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q(2/sup m/) is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q(2/sup m/) have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q(2/sup m/) are automorphic Euclidean-distance equivalent.
8 citations
TL;DR: Conditions for the construction of 90 degrees phase-invariant QAM are derived and the conditions for phase invariance are specialized for the case of Reed-Muller codes as outer codes of the generalized concatenation.
Abstract: Based on V.A. Zinoviev's (1981) generalized concatenated codes, conditions for the construction of 90 degrees phase-invariant QAM are derived. Furthermore, a proposal for the necessary differential encoding/decoding is made. The conditions for phase invariance are specialized for the case of Reed-Muller codes as outer codes of the generalized concatenation. >
7 citations
References
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IBM1
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
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
IBM1
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
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