sufficient to generate S starting from an arbitrary initial point of it. The. MPSK Constellation is GU, its symmetrygroup is isomorphic to the dihedral group DM and ...
Geometrically Uniform Multidimensional PSK Constellations S.Benedetto, R.Garello, M.Mondin, G.Montorsi Dipartimento di Elettronica, Politecnico di Torino: C.so Duca degli Abruzzi 24, 10129 Torino, Italy Abstract The theory of geometrically unaform (GU) codes is applied to the case of multidimensional (MO)PSK constellations. The symmetry group of an L x M P S I i is completely characterazed. Conditions f o r rotational invariance of GU partitions of a sagnal constellation are zllustrated. Through suitable algorithms, “good” GU partitaons of L x M P S K (M=4,8,16 and L=1,2,3,4) constellations are found. They are used as starting points in the search for good GU trellzs codes. 1 G U TCM SCHEMES A signal set S is GU [l] if it has a transitive symmetry group r(S),i.e. if for any two points s and s‘ in S , there exists a symmetry of S that sends s to 8 ’ . A generating group G ( S ) of S is a subgroup of P(S) which is minimally sufficient to generate S starting from an arbitrary initial point of it. The MPSK Constellation is GU, its symmetrygroup is isomorphic to the dihedral group DM and, in the case of M even, the only two possible generating . signal sets have the important groups are isomorphic to ZM and D M / ~GU property that the Voronoi regions are congruent, so that the error probability is independent of which signali was transmitted. In [l]this property was shown to hold for signal sequences too, through a suitable extension of the concept of geometrical uniformity. h normal subgroup G‘ of the generating group G(S) induces a partition S/S’of the signal set S, in which each subset of the partition is GU and has G‘as a common generating group. A one-to-one mapping is induced between the quotient group G/G’ and the subsets of the partition S/S’. If we combine a linear code over the label group A Y GIG’, i.e. a subgroup of A’ (with 1 possibly infinite) with the mapping GIG’ -+ S/S’ we obtain a GU code over S. As an example, a linear rate k / n binary The basis for a GU T C M convolutional code may be used if G/G‘ = (22)“. code with good properties in terms of minimum Euclidean distance is a GU partition with a minimum squared Euclidean distance within signal sets a t a given partition level as large as possible. GU PARTITIONS O F MD PSK CONSTELLATIONS 2 We denote a multidimensional PSI< constellation obtained through the L-fold Cartesian product of a 2D MPSK signal set with itself by LxMPSK. It contains M L waveforms formed by L consecutive MPSK signals. We prove that the symmetry group of Lx4PSK constellations is isomorphic to S?r. ( Z 2 ) 2 Land that of .LxMPSK, A4 even larger than 4, is isomorphic to SL ( D M ) ~Starting . from the symmetry group we develop an algorithm able to construct all the posciible generating groups of the constellation. In this way we find generating groups which are not simple Cartesian products of the generating groups of the constituent MPSK constellation. We call G = Go/G1/, . ,/Gn-l/Gn a binary partition cham of a group G with /GI = 2” if G,, . . . , GI are normal subgroups of G and lGpl = 2 lGp+lI Vp. In order to select “good” (in some sense) GU partition chains of the constellation S, we need to associate t o a given partition chain some important psrameters like: the m.inimuin Euclidean intraset squared distance 6; a t the p-th partition level, the zsomorphism of both the normal subgroup generating the partition and the quot.ient group, and the rotational invariance of the partition chain at its various levels. Given S =MPSK we denote by vk the rotation by degrees with respect to the origin and by vf the simmetry of S L =LxMPSI< obtained through L Cartesian products of Q by itself. Introducing the subgroup of r(SL)called the Rotationally Invariant wbGroup: RIG(SL)= {l, r f , (T:)~, . . . , ( ~ f ) ~ - =< l } T? >E ZM, we say that a partition is congruent with respect t o vf E R I G ( S L )if vf induces a permutation among the partition subsets, and (rotationally) invariant with respect to rf if this permutation reduces to the identity. Necessary and sufficient conditions for the congruence and the invariance of a partition are stated. When R I G ( S L )C G ( S L )the partitions are automatically congruent with respect to all r f E RIG(SL)and invariant with respect to r f iff rf E Gj. An algorithm is illustrated which scans all the possible binary partition chains starting from a given generating group G. 11,constructs the tree of all possible binary partition chains induced by norma.1 subgroups of G, identifies each partition level through the parameters aforementioned (minimum Euclidean distance, isomorphisms and rotationally invariance), and chooses the best partition chains as paths through the subgroup tree according to optimality criteria related to the previous parameters. Every partition chain is identified like in Table 1. 3 SEARCH FOR GOOD GU TCM CODES The partitions tables obtained are used to find “good” GU TCM schemes
based on binary as well as more general group convolutional codes. The obtained codes, as well as their performance, are presented in [2] and [3]. As an example, in Table 2 the results of the search for binary 3x8PSK GU codes transmitting 2.33 bit/T for increasing complexity are presented. Some of them improve over known non-GU codes. As for more general group codes, in Table 3 GU TCM codes for 3x8PSK based on the group 2; and transmitting 2 bits/?‘ are presented. They present good characteristics both in terms of Euclidean distance and rotational invariance. Error event probability curves for these codes are shown in Figure 1. References [I] G.D. Forney, Jr., “Geometrically Uniform Codes”, IEEE Trans. Inform. Theory, vol. IT-37, pp. 1241-1260, September 1991. [2] S. Benedetto, R. Garello, M. Mondin and G. Montorsi, “Geometrically Uniform Partitions of Multidimensional PSI< Const,ellations and Related Binary Codes”, submitted for publacatzon, October 1992. [3] S. Benedetto, R. Garello, M. Mondin and G . Montorsi, “Geometrically Uniform TCM Codes over Groups Based on Multidimensional PSI< Constellations”, submitfed f o r pubhealion, October 1992. Signal set 3: level
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