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RF Direction Finding, Geolocation & Coherent Arrays

An angle is an inferred parameterA sensor never reports direction directly. It measures phase, arrival time, amplitude, polarization or spectral structure; a calibrated physical model converts that observable into a bearing. Model errors can dominate thermal-noise limits.

1. Array phase and geometry

For a plane wave with propagation unit vector $\hat{\mathbf u}$ and element locations $\mathbf r_n$, the relative complex response may be modeled as $a_n=\exp(-jk\hat{\mathbf u}\cdot\mathbf r_n)$ with a stated received-field sign convention. On a uniform linear array with spacing $d$, adjacent-element phase difference is

\[\Delta\phi=-kd\sin\theta\quad (\mathrm{mod}\ 2\pi).\]

Multiple angles can map to the same wrapped phase when element spacing or observation geometry is ambiguous. Finite array dimensions, polarization, embedded-element patterns and coupling must be included in rigorous calibration.

2. Methods and observables

Method Primary observable Principal limitation
Amplitude comparison Pattern-weighted received power Requires known angular pattern; multipath sensitivity
Interferometric AoA Interelement phase Phase wrapping and RF-chain offsets
MUSIC / ESPRIT Covariance/eigenspace structure Snapshot count, coherence, array model errors
TDOA Differential propagation delay Bandwidth and absolute timestamp uncertainty
FDOA Differential Doppler Relative velocity model and clock stability
Polarization/spectral sensing Anisotropic amplitude or spectral response Orientation and calibration dependence

For a multichannel signal covariance $\mathbf R=\mathbf A\mathbf R_s\mathbf A^H+\sigma_n^2\mathbf I$, idealized MUSIC searches directions whose steering vectors are orthogonal to the estimated noise subspace. Coherent multipath and correlated source signals can violate assumptions; spatial smoothing or additional model structure may be required.

3. Geolocation

AoA bearings intersect in geometry; TDOA yields difference-of-range surfaces; FDOA adds velocity-sensitive surfaces. A bearing intersection with a small crossing angle is poorly conditioned even if each individual bearing has low variance. Solve for position using uncertainty-weighted nonlinear least squares and visualize position-error ellipses.

Advanced exercise: doa_array_demo.py compares a two-sensor phase-difference model with Bartlett beamforming for an ideal ULA. Add calibrated interchannel phase errors and correlated multipath to evaluate bias.

4. Laboratory requirements

Use coherent RF receivers, calibrated delay/phase at the antenna reference planes, anechoic or characterized propagation, frequency sweeps, full azimuth sweeps and stated angle convention. Compare measured phase-only and amplitude-only AoA error; test off-plane arrivals and polarization changes.

5. Readings