What Are the Conditions for Equi-angular Tight Frames?

[hellokitten]

The conditions on equi-angular tight frames can be found https://www.math.ucdavis.edu/~strohmer/papers/2007/equi.pdf on the first page. I created optimal grassmannian frames which are equiangular tight frames. But I need to clarify on condition (4) from the link. For example for a ETF(3,7) the frames were created by using the (1/sqrt(3))*{exp(2*pi*i*k*d_j/7)} for j =1 to 3 and k=1,2,3,...,7 d_j ={0,1,5} , there will be two frames equal to another by symmetry. Is this counted as the condition k=/=l ?

 

[jvicens]

Hello,

Thank you for sharing your research on equi-angular tight frames and optimal Grassmannian frames. It is important to clarify the conditions mentioned in the paper to ensure accurate results and interpretations.

Regarding condition (4), it states that for any two distinct indices k and l, the inner product of the corresponding frame vectors should not be equal to the cosine of the minimum angle between the frame vectors. In your example of ETF(3,7), the frame vectors are created using the formula (1/sqrt(3))*{exp(2*pi*i*k*d_j/7)} for j=1 to 3 and k=1,2,3,...,7.

In this case, since the frame vectors are created using different values of k, they will be distinct and will satisfy the condition k=/=l. Therefore, your frames satisfy condition (4) and are valid equi-angular tight frames.

It is important to note that symmetry in your frames does not affect the condition k=/=l. As long as the frame vectors are distinct and satisfy the condition, they are valid equi-angular tight frames.

I hope this clarifies the condition mentioned in the paper. Keep up the good work with your research on equi-angular tight frames and optimal Grassmannian frames!