Constellation Design With Hypercube Graphs

Kumar Vijay Mishra, Sunder Ram Krishnan, Brian M. Sadler

Abstract: A high peak-to-average power ratio (PAPR) is a major disadvantage of orthogonal frequency division multiplexing (OFDM) communications systems. In this letter, we present a graph-theoretic heuristic to mitigate high PAPR. In particular, we focus on searching for an optimal Gray-coded mapping to encode user messages such that minimum PAPR is obtained for a given message sequence in an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> -ary quadrature amplitude modulation (QAM). We exploit the bijection between vertex-weighted lattice constellations and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hypercube graphs</i> to formulate the OFDM PAPR optimization as a computationally efficient integer linear program (ILP) through the application of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Birkhoff's theorem</i> to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">doubly stochastic matrices</i> . Our numerical experiments show an average PAPR reduction of 9–10 dB using the hypercube-graph-based constellation map over the worst map while still within 0.5 dB of the brute-force method.