Doppler Effects of Nonlinear Sensor Motion in 3-D Space: Curvature, Torsion, Jolts, and Directional Wave Propagation

The estimation of signal distortions caused by sensor movement is a fundamental problem in modern signal processing systems. Standard Doppler analysis estimates signal distortions by assuming that all objects in the system travel at a constant velocity. The increasing speed and complexity of modern systems, however, requires an understanding of how the general 3-D motion of the sensor distorts the signal. In this work, we establish and interpret the spectral perturbations caused by changes in speed, i.e., acceleration and jolt, and by changes in the direction of velocity, i.e., the 3-D geometry of the receiver path, which together form the building blocks of arbitrary nonlinear motion. For constant jolt, the transmitted signal is distorted by a nonlinear chirp, which results in nonuniform spectral broadening and can create Airy-type oscillations in the amplitude spectrum. We identify sensor path approximations that incorporate the nonlinear phases induced by the 3-D geometry: the helical and Frenet–Serret approximations, which result in sinusoidal and cubic phase signals, respectively. We characterize the dependence of the spectrum on curvature, torsion, and the relative direction of wave propagation to the Frenet–Serret frame.