Generalised Welch Bound Equality sequences are tight frames

Shayne Waldron


This paper shows what are called Welch bound equality (WBE) sequences by the signal processing community are precisely the isometric/equal norm/normalized/uniform tight frames which are currently being investigated for a number of applications, and in the real case are the spherical $2$--designs of combinatorics. Recent applications include wavelet expansions, Grassmannian frames, frames robust to erasures, and quantum measurements.

This is done by giving an elementary proof of a generalisation of Welch's inequality to vectors which need not have equal energy, and then showing that equality occurs in this exactly when the vectors form a tight frame.

Keywords: WBE sequences, Welch bound equality sequences, isometric tight frames, normalized tight frames, spherical 2--designs, Gram matrix, wavelets

Math Review Classification: Primary 05B30, 42C40, 94A12 ; Secondary 42C15, 65D30

Length: 7 pages

Last Updated: 18 March 2003