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Oblique projection operators are used to project measurements onto a low-rank subspace along a direction that is oblique to the subspace. They may be used to enhance signals while nulling interferences. In the paper, the authors give several basic results for oblique projections, including formulas for constructing oblique projections with desired range and null space. They analyze the algebra and geometry of oblique projections in order to understand their properties. They then show how oblique projections can be used to separate signals from structured noise (such as impulse noise), damped or undamped interfering sinusoids (such as power line interference), and narrow-band noise. In some of the problems addressed, the oblique projection provides an alternative way to implement an already known solution. Expressing these solutions as oblique projections brings geometrical insight to the study of the solution. The geometry of oblique projections enables one to compute performance in terms of angles between signal and noise subspaces. As a special case of removing impulse noise, the authors can use oblique projections to interpolate missing data samples. In array processing, oblique projections can be used to simultaneously steer beams and nulls. In communications, oblique projections can be used to remove intersymbol interference