Direction finding and phased arrays

Seen from above. The dots are antenna elements; the coloured lines are wavefronts arriving from each emitter, one wavelength apart. A wavefront reaches each element at a slightly different time, so each element sees the signal with a different phase. That pattern of phases across the array is the whole basis of direction finding. Angles are measured clockwise from broadside, the direction straight out of the array.

The simplest direction finder uses two elements a distance d apart (here elements 1 and 2). A wave from angle θ travels an extra d·sin θ to reach one of them, a phase difference Δφ = 2π d sin θ / λ. Measure Δφ, invert, and you have the angle. But a phase is only known modulo 360°, so Δφ, Δφ ± 360°, … all fit. When d is at most half a wavelength only one of them gives a real angle; beyond that, several do, and the pair cannot tell which is right.

With more than two elements, the receiver compares every pair at once. It averages the product of each element's signal with every other's over all the snapshots: the covariance matrix. Its eigenvalues split into a few large ones, one per source (the signal subspace), and the rest, which sit near the noise level (the noise subspace). Counting the large ones estimates how many emitters there are; MDL and AIC are two standard rules for drawing that line.

Bartlett beamforming steers the array's beam across every angle and measures the power that comes in; peaks are emitters. Its angular resolution is limited by the beam width, roughly λ divided by the array length. MUSIC instead asks, for each angle, how nearly that direction's phase pattern is orthogonal to the noise subspace. At an emitter's true angle it is almost exactly orthogonal, so MUSIC's peaks are far sharper and can separate emitters closer together than the beam width. It needs the number of sources, and it fails when two emitters carry the same signal (coherent multipath) unless the covariance is smoothed over subarrays.

Every channel behind every element has its own cable length, filter and amplifier, so each adds its own phase shift and gain. The estimators assume the ideal phase pattern, so these errors bend the answer. Calibration measures them with a reference source from a known direction: the strongest eigenvector of that measurement is the array's real response, and dividing by the ideal response gives each channel's error. A common error on every channel does not matter, so everything is relative to element 1.

Transmitting or receiving, an array can point a beam by giving each element the phase that lines up the wavefront from the chosen direction. The pattern shows the response in every direction. Tapering the element amplitudes lowers the sidelobes at the cost of a wider main beam. Nulls can be placed on chosen emitters by removing those directions from the weights, or found automatically by MVDR, which minimizes total received power while keeping the steered direction at full gain. Elements more than half a wavelength apart produce grating lobes: full-strength copies of the main beam.

How accurate can any estimator be? The Cramér-Rao bound is the smallest error spread an unbiased estimator can reach for a given array, number of snapshots and SNR. This sweep repeats the measurement many times at each SNR with a single emitter and compares the RMS error of each method to the bound. Above a threshold SNR, Bartlett and MUSIC reach the bound for the whole array; the two-element phase difference can only reach the far higher bound for its pair. Below the threshold, errors jump as noise peaks win.

Bound: stochastic Cramér-Rao bound for one noise-like source, CRB = (1 + 1/(M·SNR)) / (2 T · SNR · h), where M is the number of elements, T the number of snapshots and h = |a′|² − |aᴴa′|²/M measures how fast the array's phase pattern a changes with angle. The sweep uses a noise-like signal and a perfectly calibrated array. On the circular array, the two-element pair cannot tell front from back, so the sweep gives it the correct side.