MVDR

MVDR, short for minimum variance distortionless response, is an adaptive beamforming technique used to extract a signal from an array of sensors while suppressing interference and noise from other directions. The goal is to minimize the beamformer's output power subject to a distortionless constraint in the desired look direction, preserving the signal of interest.

Formally, with a steering vector a(θ0) for the desired direction and the sample covariance matrix R of the received data, MVDR solves the optimization: minimize w^H R w subject to w^H a(θ0) = 1. The solution is w = R^{-1} a(θ0) / (a(θ0)^H R^{-1} a(θ0)). This approach assumes a narrowband signal from a far-field source and requires an estimate of R, typically derived from data snapshots.

MVDR is credited to J. Capon, who introduced the distortionless minimum-variance principle in 1969 for spectral estimation and later for array processing. It is widely applied in radar, sonar, wireless communications, microphone arrays, and seismic sensing, where spatial filtering is used to enhance signals from a target direction while suppressing others.

Advantages include strong interference suppression in directions other than the look direction and robustness to noise when the covariance estimate is reliable. Limitations involve sensitivity to errors in the steering vector, limited or biased covariance estimates, and performance degradation in non-stationary environments. Variants such as robust MVDR and diagonal-loading MVDR address these issues by improving stability and mismatch tolerance.