# Designing Structured Tight
Frames Via An Alternating Projection
Method

### Authors:

J. A. Tropp,

I.
Dhillon,

R. W.
Heath, Jr., and

T. Strohmer
### Reference:

*IEEE Trans. on Info. Theory*, vol. 51, no. 1, pp. 188-209, January 2005.

### Abstract:

Abstract÷Tight frames, also known as general Welch-Bound-
Equality sequences, generalize orthonormal systems. Numerous
applications÷including communications, coding and sparse
approximation÷require finite-dimensional tight frames that possess
additional structural properties. This paper proposes an
alternating projection method that is versatile enough to solve
a huge class of inverse eigenvalue problems, which includes the
frame design problem. To apply this method, one only needs to
solve a matrix nearness problem that arises naturally from the
design specifications. Therefore, it is fast and easy to develop
versions of the algorithm that target new design problems.
Alternating projection will often succeed even if algebraic constructions
are unavailable.

To demonstrate that alternating projection is an effective tool
for frame design, the article studies some important structural
properties in detail. First, it addresses the most basic design
problem÷constructing tight frames with prescribed vector
norms. Then, it discusses equiangular tight frames, which are
natural dictionaries for sparse approximation. Last, it examines
tight frames whose individual vectors have low peak-to-averagepower
ratio (PAR), which is a valuable property for CDMA
applications. Numerical experiments show that the proposed
algorithm succeeds in each of these three cases. The appendices
investigate the convergence properties of the algorithm.

This paper as a .IEEE Xplore .