Nyquist Signaling Modulation (NSM): An FTN-Inspired Paradigm Shift in Modulation Design for 6G and Beyond

Nyquist Signaling Modulations (NSMs) are a new signaling paradigm inspired by faster-than-Nyquist principles but based on a distinct approach that enables controlled inter-symbol interference through carefully designed finite-impulse-response filters. NSMs can operate in any number of dimensions, including mixed-dimensional configurations, offering wide flexibility in filter design, optional energy balancing, and preservation of the 2-ASK minimum squared Euclidean distance (MSED). Both real and rational tapped filters are investigated, and closed-form expressions are derived for the optimal real-tap filters in the one-dimensional case (MS-PRS), providing analytical insight and strong agreement with simulated bit-error behavior across wide SNR ranges. The paradigm is conceptually expanded through an analog Low-Density Generator Matrix (LDGM) formulation, which broadens the NSM family and unifies modulation and coding within a single, structurally coherent framework. When combined with LDPC coding, it enables efficient and naturally synergistic interaction between the analog modulation and the digital LDPC code. Alternatively, when analog LDGM is employed for both source coding and modulation, a simple and harmonious joint source-channel-modulation structure emerges. In both configurations, the constituent blocks exhibit dual graph-based architectures suited to message passing, achieving high flexibility and complexity-efficient operation. Collectively, these results establish promising physical-layer directions for future 6G communication systems.