Recursive least-squares lattices and trigonometry in the spherical triangle

  • Authors:

  • F. Desbouvries

  • Affiliations:

  • Institut National des Télécommunications, Evry, France

  • Venue:
  • ICASSP'93 Proceedings of the 1993 IEEE international conference on Acoustics, speech, and signal processing: digital speech processing - Volume III
  • Year:
  • 1993

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Abstract

The 3 fundamental planar biorthogonalization steps which underlie the geometric derivation of the FRLS adaptive lattices are gathered into a unit-length 3D 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 brand new geometric interpretation is that we can take advantage of the well-trodden path of spherical trigonometry to derive unoticed recursions among RLS quantities. This leads, for instance, to an original "dual" version of YPII.