Doppler Shift

The classic U-shaped Doppler power spectral density that defines the statistical distribution of frequency shifts in a rich multipath environment. The Jakes model assumes uniformly distributed scatterers, producing a spectrum with singularities at the maximum Doppler frequency f_d = v·f_c / c, where v is velocity, f_c is carrier frequency, and c is the speed of light.

• Produces a bathtub-shaped spectrum bounded by ±f_d
• Assumes isotropic scattering with no dominant line-of-sight component
• Forms the baseline for Rayleigh fading channel emulation

Extends the Jakes model by adding a deterministic frequency offset representing a dominant line-of-sight path. The Rician K-factor controls the power ratio between the specular LOS component and the diffuse scattered components.

• LOS component introduces a constant frequency shift proportional to cos(θ), where θ is the angle of arrival
• Higher K-factors concentrate spectral energy around the LOS Doppler frequency
• Essential for emulating drone-to-ground or satellite-to-terminal links where a direct path exists

Realistic mobility simulation requires dynamic Doppler profiles that evolve as velocity and scattering geometry change. Static Doppler spectra fail to capture acceleration, deceleration, and turning maneuvers.

• Implements piecewise-linear or spline-interpolated velocity vectors over time
• Updates the channel impulse response at each coherence time interval
• Critical for training models to recognize transient mobility signatures in emitter identification

In a tapped delay line channel emulator, each resolvable multipath component receives an independent Doppler shift based on its angle of arrival. This creates a frequency-dispersive channel where different delay taps fade at different rates.

• Each tap's Doppler shift is drawn from the Jakes or Rician distribution
• Produces frequency-selective fading when combined with delay spread
• Enables simulation of complex environments like urban canyons with distinct scatterer clusters

A domain randomization strategy where Doppler parameters are deliberately varied during synthetic dataset generation to force fingerprinting models to learn velocity-invariant features.

• Randomizes f_d range, K-factor, and scatterer distribution across training samples
• Prevents the model from overfitting to a single mobility profile
• Improves generalization to unseen operational velocities during deployment

The coherence time T_c defines the interval over which the channel impulse response remains approximately constant. It is inversely proportional to the maximum Doppler spread: T_c ≈ 0.423 / f_d.

• Determines the update rate for time-varying channel convolution
• Shorter coherence times at higher velocities demand finer temporal granularity in simulation
• Guides the frame duration design for pilot symbol insertion in coherent receivers