Geometrical uniformity of a class of space-time trellis codes

Geometrical uniformity of space-time codes sheds more light on their structure, and can assist in designing, or systematically searching for, better space-time codes. In this paper, a family of space-time codes is treated as signal space codes, then proved to be generalized coset codes, and thereby geometrically uniform. It has been observed that good codes exhibit various geometrical properties [ l , 21. In particular, i t is shown in [2] that most of the known good trellis codes share a property called geometrical uniformity. A code is geometrically uniform if for any two codewords c1 and c2 there exists an isometry that leaves the code invariant while mapping c1 to C Z . Geometrical uniformity of a code is a sufficient condition for certain useful properties; for example, distance profiles are transparent to the choice of a reference codeword if a code is geometrically uniform. Recently, space-time codes have been proposed to mitigate the effects of fading inherent to wireless communications (see, for example, [3,4, 51). Geometrical uniformity of space-time codes, is of interest because not only does it provide knowledge about the algebraic structure of space-time codes, but facilitates the performance analysis, as well as the design-perhaps via systematic search-of space-time codes. While the former reason is obvious, the latter needs some justification; this is because the measures of performance for space-time codes seem unrelated to the Euclidean distance between codewords (see, for example, [3]), while geometrical uniformity is usually defined in the context of Euclidean distance (an isometry is defined in a metric space, which is usually the Euclidean space). It was shown in [S, Theorem 51 that the Euclidean distance remains relevant in flat fading-in spite of the multiplicative nature of flat fading distortions, which apparently questions the relevance of distancepreserving properties. Therefore, features derived from Euclidean distance characterizations, including geometrical uniformity, remain potentially meaningful for flat fading channels, and space-time codes in particular. For example, geometrical uniformity can reduce the complexity of systematic searches for good space-time codes. Performance analysis for space-time codes is usually based on pair-wise error probabilities over all possible codeword pairs, and hence a systematic search has to check all possible codeword pairs. In certain cases, it suffices to consider only the case where one of the codewords in the codeword pair is fixed, provided that the space-time code is geometrically uniform. The space-time codes proposed by Tarokh et al. in [3] are proved, therein, to be geometrically uniform by definition; however, a proof by definition would be difficult in the case of more general codes. Since many space-time codes can be viewed as signal space codes [2], the approach in [2] can be used to prove or disprove geometrical uniformity of such codes. This paper treats the space-time trellis coded modulation schemes proposed in [4] and [5] as signal space codes, and presents a constructive proof, via the approach in [2], of ‘This work was supported in part by the National Science Foundation under Grants CCR 99-79381 and ITR 00-85929, and in part by Nokia Corporation. the geometrically uniform nature of these codes. A signal space code [2] is, in essence, a collection of codewords, each of which is a (finite or infinite) sequence of signal points; see [2] for a formal definition and more detailed explanation about signal space codes. Upon any trellis transition for the space-time codes in [4, 51, each of the two transmit antennas sends a complex symbol from the 4PSK signal constellation Q during each of two consecutive complex symbol epochs. Thereby, the space-time codes in [4, 51 can be viewed as signal space codes with a signal point being simply a 2 x 2 matrix, whose entries are from Q. The signal set S of the signal space code consists, in this case, of 32 signal points, as shown in Table I in [4]. By inspection, one can prove Proposition 1 The signal set S is geometrically uniform. That is, for any two points s, s’ E S, there exists an isometry usst which leaves S invariant. The partitioning of the signal set, selection of cosets, and selection of a signal point in a coset are discussed in [4, 51. Following these steps: 0 identify the generating group U ( S ) for S, i.e. a subgroup of the symmetry group of S identify a normal subgroup U’ of U ( S ) , and let the orbit of SO under U’ be S’; U’ is a generating group for S’ and will thereby be denoted U ( S’); 0 identify the geometrically uniform partition S/S’, induced by the factor group U(S) /V(S’ ) ; and 0 prove that the label alphabet A is a group, which is isomorphic to U(S)/U(S’) , one can further show that Proposition 2 The label alphabets of the space-time codes in 14, 51, viewed as signal space codes, are isomorphic to U(S) /U(S’ ) , and the label maps are isometric labelings. Using Proposition 2 and [2, Theorem 51, one obtains directly Theorem 1 The space-time codes in 14, 51 are generalized coset codes in the sense of 121, and thereby geometrically unifarm. REFERENCES [ I ] A. R. Calderbank and N. J. A. Sloane, “New Trellis Codes Based on Lattices and Cosets,” IEEE Trans. Inform. Theory, vol. 33, No.5, pp. 177195, January 1987. [2] G. D. Forney, “Geometrically Uniform Codes,” IEEE Trans. Inform. Theory, vol. 37, NOS, pp. 1241-1260, Sept. 1991. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criteria and code construction:: IEEE Trans. Inform. Theory, vol. 44, No. 2, pp. 744-165, March 1998. [4] D. M. Ionescu, K. K. Mukkavilli, Z. Yan, and J. Lilleberg, “Improved 8and 16-State Space-Time codes for 4PSK with Two Transmit Antennas,” IEEE Commun. Letters, vol. 5, pp. 301-303, July 2001. [5] D. M. Ionescu, “On Space-Time Code Design,” IEEE Truns. Wireless Commun. , vol. 2, pp. 20-28, Jan. 2003. 207 0-7803-7728-1/03/$17.00 02 03 IEEE.