Recursive least-squares lattices and trigonometry in the spherical triangle

摘要

The three fundamental planar biorthogonalization steps which underlie the geometric derivation of the fast recursive least squares (FRLS) adaptive lattices are gathered into a unit-length 3-D tetrahedron. The inverse of Yule's PARCOR Identity (YPII) then admits a nice geometric interpretation in terms of projections into this tetrahedron. Since tetrahedrons are closely related to spherical triangles, YPII is recognized as the fundamental 'cosine law' of spherical trigonometry. In that framework, the angle-normalized RLS lattice recursions happen to be one particular solution to one of the six spherical triangle problems. The practical interest of this geometric interpretation is that one can take advantage of spherical trigonometry to derive unnoticed recursions among RLS quantities. This leads, for instance, to an original 'dual' version of YPII.