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Equi-angular tight frame via conference matrix

Kenneth Shum (view profile)

Construction of equi-angular tight frame by Paley's conference matrix and svd

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Description

Input parameters
p : odd prime number
deg: positive integer (default value = 1)

Output is a (p^deg+1) by (p^deg+1)/2 matrix E

When deg > 1, the communication toolbox is required

A collection of d-dimensional vectors of norm 1 is equi-angular
if the absolute values of the inner products between any two
distinct vectors is equal to a constant c.
The equi-angular vectors are called tight if the constant c
attains Welch's lower bound.

The columns of the output matrix E is an equi-angular tight frame
The norm of each column of E is 1
The inner product between each pair of columns is 1/sqrt(p^deg)
Columns of E represent equi-angular lines
in the (p^deg+1)/2 dimensional Euclidean space

Example:

>> tight_frame_paley(5)

ans =

0.0000 0.8944 0.2764 -0.7236 -0.7236 0.2764
-0.0000 -0.0000 -0.8507 -0.5257 0.5257 0.8507
-1.0000 -0.4472 -0.4472 -0.4472 -0.4472 -0.4472

We can check that diagonal entries of E'*E are all 1,
and the off-diagonal entries of abs(E'*E) are all 1/sqrt(q)

Required Products Communications System Toolbox
MATLAB release MATLAB 6.0 (R12)