Behind the algebra of statistical signal processing there lies a fascinating geometry. Our aim in this keynote talk is to develop this geometry, and use it to illuminate a number of problems in beamforming, detection, and estimation. In the realm of beamforming, we revisit the Capon, or MVDL beamformer, and establish its connections to more conventional beamformers like the Bartlett. We then offer several extensions to the Capon beamformer, making it suitable for the imaging of distributed targets from broadband data. For detection, we review the geometry of matched and adaptive subspace detectors. These detectors are matched to multidimensional subspaces, making them suitable for detecting distributed or broadband sources.
They generalize much of what is known about matched filters, which are matched only to one-dimensional subspaces. Our review of estimation exploits the geometry of oblique projections. These processors have the virtue that they image sources with perfect nulling of nearby sources. Of course there is a price to pay in noise gain. This price depends on principal angles between subspaces, explaining why super-resolution is a risky enterprise.