Capon–Bartlett cross-spectrum and a perspective on robust adaptive filtering

Adaptive filtering / beamforming (ABF) optimized for maximum signal-to-interference-plus-noise ratio (SINR) results in filter weights that depend on the data covariance and the signal array response vector. The effectiveness of practical application of such solutions, however, is limited by (i) extent of data stationarity and (ii) imperfect knowledge of the true signal array response vector. Robust ABF methods attempt to address these two critical issues via a slight reformulation of the ABF problem statement. Many robust ABF solutions result in some form of diagonal loading of the data covariance, and can be interpreted as a hybrid beamformer that engages the tradespace between data adaptive beamforming and conventional beamforming (CBF). In view of this interpretation, an exact joint probability distribution is derived for 1) the data adaptive Capon minimum variance distortionless response (MVDR) spectral estimator, and 2) the conventional Bartlett spectral estimator (a.k.a. the smoothed periodogram) when based on the same data covariance estimate. The resulting joint distribution motivates (i) proposal of a robust adaptive filtering design that constrains filter weight cross coherence (a metric quantifying the statistical coupling between the Capon and Bartlett spectral statistics), and (ii) introduction of a coherence estimate called the Capon–Bartlett cross spectrum.