The Dynamic Doppler Spectrum Induced by Nonlinear Sensor Motion: Relativistic Kinematics and 4D Frenet-Serret Spacetime Geometry

Abstract

Fundamental to the analysis of nonlinear relativistic motion is the precise characterization of the induced dynamic Doppler effects. In this work, we analyze the electromagnetic signals observed by non-inertial receivers using two frameworks to describe the relativistic motion. We first consider observer paths described by higher-order kinematic 4 vectors: relativistic acceleration and jolt. The dynamic Doppler effects of relativistic acceleration and jolt are exponential spectral broadening and exponential amplitude growth or decay. We derive compact expressions for the spectrum transformation resulting from relativistic acceleration and jolt. The jolt induces nonlinear skewed chirps in observed signals. Next we consider observer paths described by the 4D Frenet-Serret frame and the curvature and torsion of the observer path. We obtain descriptions of the amplitude and phase fluctuations of the signal in terms of the geometric parameters of curvature and torsion. Concise, interpretable descriptions of non-inertial dynamic Doppler effects provide a useful diagnostic and predictive tool for engineering applications including radar, sensing, and communications systems.

Figures (4)

• Figure 1: Amplitude spectra of the signal received along the spacetime path $z^\mu(c\tau)$ given by Eqs (\ref{['rela_jolt_path0']})-(\ref{['rela_jolt_path1']}) in blue. The frequency and amplitude are normalized by their respective values at $c\tau=0$. The spectrum approximation is obtained using the stationary phase method (\ref{['SPA_rela_jolt_amp']}). In the top panel, proper jolt $j_0$ and proper acceleration $a_0$ are zero (classical Doppler shift). In the middle panel (chirp spectrum), proper jolt $j_0$ is zero. In the bottom panel (skewed chirp spectrum), proper jolt $j_0$ is nonzero. In the bottom two panels, the stationary phase approximation is displayed in red.
• Figure 2: Amplitude spectra of the signal received along the spacetime path $z^\mu(c\tau)$ given by Eqs (\ref{['rela_jolt_path0']})-(\ref{['rela_jolt_path1']}). The frequency and amplitude are normalized by their respective values at $c\tau=0$. The spectrum approximation is obtained using the stationary phase method (\ref{['SPA_rela_jolt_amp']}). The yellow and blue curves correspond to constant proper acceleration and constant proper jolt, respectively. For constant proper acceleration, the amplitude spectrum is nearly constant. The amplitude spectrum is strongly nonlinear for a receiver with constant proper jolt.
• Figure 3: Comparison of the spectrum of the signal received along the spacetime path $z^\mu(c\tau)$ given by Eq. (\ref{['path_4DFSTS']}) (in blue) to the spectrum approximation (\ref{['SPA_4DFSTS']}) obtained from the stationary phase method (in red). The receiver path $z^\mu(c\tau)$ is described by the 4D Frenet-Serret expansion.
• Figure 4: The Airy approximation (\ref{['Airy_spectrum_approx']}) of the spectrum of the signal received along the 4D Frenet-Serret path $z^\mu(c\tau)$ given by Eq. (\ref{['path_4DFSTS']}). The received signal frequency is a non-monotonic function of time creating an interference pattern in the spectrum.