Angle-of-arrival mathematics for EW receivers: amplitude comparison, Watson-Watt, pseudo-Doppler, phase interferometry, array AOA, triangulation and error geometry.
Direction finding estimates the bearing or angle of arrival of an intercepted emitter. A single DF sensor normally gives a line of bearing, not a position. Position appears when several bearings, time differences, frequency differences or motion-assisted measurements are combined.
In EW this matters because the receiver is often passive. It does not illuminate the target like radar. It listens to hostile or unknown emissions, extracts signal parameters, estimates the wavefront direction and sends that bearing to an operator, tracker or geolocation processor.
A DF sensor converts antenna response, phase or Doppler modulation into a bearing line. Several bearing lines intersect to form an estimated emitter location and an error ellipse.
An angle-of-arrival measurement is a geometric constraint. From a sensor position, the emitter must lie somewhere along the measured bearing, with an angular uncertainty caused by noise, multipath, calibration error and finite aperture.
Formula reading: the emitter position is p = [x, y]. The DF sensor is at ps = [xs, ys]. The measured bearing θ is the true geometric angle plus angular error nθ. The unit vector u(θ) points from the sensor along the measured line of bearing.
Amplitude DF compares received power in two or more directional antenna patterns. It is simple and fast, but accuracy depends strongly on antenna pattern calibration, polarization, multipath and signal fading.
Formula reading: each channel sees the same incident signal power Pr, multiplied by its directional gain pattern. The dB ratio removes much of the unknown emitter power. The normalized difference ε is a left-right angle error signal: zero means the emitter lies on the electrical centerline, positive or negative tells which side.
Watson-Watt systems use two orthogonal directional channels, often represented as sine and cosine patterns, plus an optional sense channel to remove the 180° ambiguity. The bearing comes from the arctangent of the channel voltages.
Formula reading: Vx and Vy are the two orthogonal DF channel voltages. K is receiver gain and E is field strength. Because both channels share K and E, the angle is found from their ratio. atan2 is used instead of a simple tangent so the correct quadrant is retained.
Pseudo-Doppler DF electronically switches around a circular antenna array. The phase of the resulting frequency modulation indicates the incoming wave direction. It is common in compact VHF/UHF DF receivers because it can emulate a rotating antenna without mechanical rotation.
Formula reading: a is the effective rotation radius, frot is switching/rotation rate, and λ is wavelength. The larger the circular aperture relative to wavelength, the stronger the Doppler modulation. The unknown arrival angle θ appears as the phase of that modulation.
A phase interferometer measures the phase difference between spatially separated antennas. It can be very accurate because phase can be measured finely, but baselines longer than half a wavelength introduce ambiguous angle solutions unless multiple baselines or coarse sensors are used.
Formula reading: d is antenna spacing. A wave arriving from angle θ reaches one antenna before the other, creating path difference d sinθ. Dividing that path by wavelength gives phase. Since measured phase wraps every 2π, long baselines create several possible angles.
Digital EW receivers can estimate direction by comparing the received vector across many antenna elements. The same steering-vector mathematics supports conventional beam scanning, maximum likelihood estimators and subspace methods such as MUSIC.
Formula reading: a(θ) predicts the phase progression across an M-element array for a plane wave from angle θ. x is the measured complex channel vector. The beamformer tests candidate angles and picks the angle whose steering vector best matches the received data.
Two or more DF sensors can geolocate an emitter by intersecting bearing lines. In practice the lines do not meet exactly, so least squares or maximum likelihood finds the position that best fits all bearing measurements.
Formula reading: each sensor contributes an angular residual: measured bearing minus predicted bearing from the trial emitter position. The residual is wrapped so 359° and -1° are treated correctly. Measurements with smaller angular variance receive more weight.
A one-degree DF error can be small nearby and huge far away. The lateral position uncertainty grows approximately linearly with range. Geolocation geometry is therefore as important as receiver quality.
Formula reading: angular error must be in radians. A bearing uncertainty creates a wedge. At range R, the cross-range width is roughly R times the angular standard deviation. The error ellipse summarizes how all sensor geometry and measurement errors map into position uncertainty.
DF/AOA is only one passive geolocation family. Time difference of arrival creates hyperbolas; frequency difference of arrival creates Doppler contours. Modern EW geolocation often fuses AOA, TDOA, FDOA, signal ID and platform motion.
Formula reading: a time-difference measurement says the emitter is closer to one sensor than another by c0Δt. The locus of constant range difference is a hyperbola in 2-D.
Formula reading: FDOA uses relative Doppler shift. Moving receivers see slightly different frequencies depending on their velocity projection toward or away from the emitter.