Hierarchical generalized Cantor set modulation

Abstract

In this paper, we show that arbitrary hierarchical pulse amplitude modulation (PAM) schemes can be fully described by generalized Cantor sets. Generalized Cantor sets are modified versions of the Cantor ternary set, a famous mathematical construct known for its set-theoretical properties. The fractal nature of generalized Cantor sets allow for a natural reinterpretation as a modulation scheme. The resulting Cantor set description of one-dimensional hierarchical modulation schemes covers the constellation points as well as the boundary points of the decision regions. Furthermore, we derive simple formulas for the average signal power as well as for iterative demodulation. All results can be extended to two dimensions and hierarchical quadrature amplitude modulation (QAM) schemes. As such, this paper offers a novel perspective on the classification and parametrization of practical hierarchical modulation schemes.