directionofarrival

Direction of arrival (DOA) refers to the direction from which an incoming signal impinges on an array of sensors, such as microphones or antennas. It is typically described by azimuth and elevation angles relative to a reference coordinate system. DOA estimation relies on differences in signal phase or arrival time across sensors, exploiting spatial information to locate the source.

In sparse terms, a sensor array with M elements receives signals from L sources. The received data can be modeled as x(t) = A s(t) + n(t), where A is the M×L steering matrix whose columns are steering vectors that depend on the source directions, s(t) is the source signal vector, and n(t) is noise. The sample or statistical covariance matrix R = E[x x^H] is often used to extract DOA information. DOA estimation aims to determine the source directions θ1, θ2, ..., θL from the observed data.

Common methods include beamforming and high-resolution techniques. Classical beamformers (Bartlett, Capon/MNF) scan across directions to form synthetic beams. High-resolution subspace methods such as MUSIC and ESPRIT exploit the eigenstructure of the covariance matrix to resolve closely spaced sources. Other approaches include maximum likelihood estimation and sparsity-based methods that leverage the assumption of a few active sources relative to the array size.

Array geometry and hardware considerations influence DOA performance. Arrays can be linear, planar, circular, or conformal, and real systems must account for mutual coupling, calibration, synchronization, and environmental factors. DOA has applications in radar, sonar, wireless communications (MIMO and localization), acoustics (speaker localization, beamforming), and hearing aids. Challenges include multipath, reverberation, noise, and model mismatch. See also angle of arrival.