1 research outputs found
Geometrically Uniform Frames
We introduce a new class of frames with strong symmetry properties called
geometrically uniform frames (GU), that are defined over an abelian group of
unitary matrices and are generated by a single generating vector. The notion of
GU frames is then extended to compound GU (CGU) frames which are generated by
an abelian group of unitary matrices using multiple generating vectors.
The dual frame vectors and canonical tight frame vectors associated with GU
frames are shown to be GU and therefore generated by a single generating
vector, which can be computed very efficiently using a Fourier transform
defined over the generating group of the frame. Similarly, the dual frame
vectors and canonical tight frame vectors associated with CGU frames are shown
to be CGU.
The impact of removing single or multiple elements from a GU frame is
considered. A systematic method for constructing optimal GU frames from a given
set of frame vectors that are not GU is also developed. Finally, the Euclidean
distance properties of GU frames are discussed and conditions are derived on
the abelian group of unitary matrices to yield GU frames with strictly positive
distance spectrum irrespective of the generating vector.Comment: Submitted to IEEE Transactions on Information Theory. LaTex, 43 page