Time-Frequency Seminar

Motivated by the use of frames for robust transmission over the Internet, we present a first systematic construction of real tight frames with maximum robustness to erasures. We use concepts from the theory of polynomial algebras and orthogonal polynomials.  Then we use our approach to construct real frames, for all choices of m and n, that combine the properties of equal norm, tightness, and maximal robustness to erasures. To the best of our knowledge, to date only one class of complex frames with these properties have been found.

In n dimensions, a frame with m elements is a spanning set for a given space. Clearly, such a set is redundant---a crucial property of frames.  We deal with finite-dimensional frames only, and look at the properties of interest. These include tightness, as tight frames are natural generalizations of orthonormal bases. Tight frame preserve the norm and are self-dual, thus ensuring efficient reconstruction. We also search for those frames where elements have equal norm, as well as those maximally robust to erasures, that is, those which remain frames up to (m-n) erasures.

We first identify matrix transformation rules which do not destroy properties of interest to us.  We approach the problem in steps: we first construct maximally robust frames by using polynomial transforms. This results in a large class of maximally robust frames. We then add tightness as additional property with the help of orthogonal polynomials.  Finally, we impose the last requirement of equal norm on the frame and construct, to our best knowledge, the first real, tight, equal norm frames that are maximally robust to erasures.

This is joint work with Markus Pueschel from CMU.

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