Elliptical trajectories and uncertainty principles in a geometric theory of generalized signal analysis

Abstract Bivariate signals, representing vector motions in 2D spaces (e.g., oceanography and radar systems), require specialized uncertainty analysis beyond traditional complex signals. This paper introduces the novel uncertainty principles for bivariate signals via quaternion Fourier transform and Hilbert transforms. The main contributions of this paper include: Sharper uncertainty bounds in time-frequency domains; For the first time, a geometric connection between uncertainty principles and bivariate trajectory properties (ellipticity, orientation, and rotational frequency) is established, revealing that faster elliptical motion correlates with reduced time-frequency resolution—a finding aligning with intuitive physics. Numerical experiments and theoretical analysis are shown to validate the proposed frameworks, bridging the classical physical macroscopic motion and quantum-inspired uncertainty principles in information science for the continuous signal space.