# Generalised Welch Bound Equality sequences are tight frames

## Shayne Waldron

## Abstract:

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

## Availability: