Quasi-Gray labelling for Grassmannian constellations

Abstract

This paper presents a technique for assigning binary labels to the points in an arbitrary Grassmannian constellation in a manner that approximates the Gray labelling. The idea behind this technique is to match the Grassmannian constellation of interest to the points in an auxiliary constellation that can be readily Gray labelled. In order to demonstrate the efficacy of the proposed technique, the labelled constellations are utilized in a BICM-encoded non-coherent MIMO communication system with iterative detection and decoding. Numerical simulations indicate that this labelling technique results in a non-coherent communication system that provides better bit error rate performance than systems that utilize the same constellation but employ labels that are generated either randomly or via a quasi-set-partitioning technique.

Quasi-Gray Labelling for Grassmannian Constellations

“…At a BER of 10 −3 , quasi-setpartitioning provides an SNR gain of 0.15 dB over the random labelling. The basic match-and-label algorithm presented in [28] yields a further gain of 0.35 dB, which improves slightly when the optimal Hungarian matching presented in Sec. III-A is used.…”

“…The match-and-label idea was first presented in [28] as a method for providing quasi-Gray labels to a Grassmannian constellation of interest, . The principle that underlies this algorithm is to match to an auxiliary constellation, , that has the same cardinality and can be readily Gray labelled, but may have a distance spectrum that is less favourable than that of .…”

“…The overall cost of matching the two constellations is then the total sum of the distances between the corresponding points. A suboptimal method for minimizing this cost was proposed in [28]. However, by representing the pairwise distance between the points of each constellation in the form of a matrix, the matching problem can be shown to be equivalent to the combinatorial assignment problem [16] wherein tasks are to be assigned to workers such that the overall cost is minimized.…”

“…II-A provide an approximate performance gain of 2 dB over the Gray-labelled constellation from [8]. Even though the constellation in [8] is perfectly Gray labelled, its distance properties result in a relatively poor BER performance [28]. Below, we will show that the application of quasi-Gray labelled constellations to turbo-coded systems can have an even greater impact on error performance, since, in addition to decreasing the average Hamming distance, the maximum Hamming distance of the nearest neighbours has been substantially reduced, cf.…”