A Recursive Construction for Low-Complexity Non-coherent Constellations

It is known that at high signal to noise ratio (SNR), or for large coherence interval (T ), a constellations of unitary matrices can achieve the capacity of the non-coherent multiple-antenna system in block Rayleigh flat-fading channel. For a single transmit antenna system, a unitary constellation is simply a collection of T -dimensional unit vectors. Nevertheless, except for a few special cases, the optimal constellations are obtained only through exhaustive or random search, and their decoding complexity is exponential in the rate of the constellation and the length of the coherence interval, T . In this work, we propose a recursive construction method for real-valued single transmit antenna non-coherent constellations, in which a T -dimensional unitary constellation is constructed by using a number of (T − 1)-dimensional unitary or spherical constellations as its equi-latitude subsets. Comparison of the minimum distances achieved by the proposed constructions with the best known packings in G(T, 1) [1] shows that, for practical values of T , the recursive constellations are close to optimal. We also propose a simple low-complexity decoding algorithm for the single-antenna recursive constellations. The complexity of the proposed decoder is linear in the total number of the two-dimensional constituent subsets, which is usually much smaller than the number of the constellation points. Nevertheless, the performance of the suboptimal decoder is similar to the optimal decoder. A comparison of the error rate performance of the recursive constellations with the complex-valued systematic designs of [2] shows that the proposed real-valued constellations have similar performance to the complex-valued systematic designs. The recursive designs also show a significant gain over the low-complexity PSK constellations of [3].