I am doing research in compressive sampling for Cognitive Radio applications. While working on a project I came across with the following question: Is there any research about the phase of inner product between $N$ vectors in $\mathbb{C}^M$ which form an Equiangular Tight Frame (or the sign of inner product between $N$ vectors in $\mathbb{R}^M$)? We know that if there is an ETF for a given $N$ and $M$, then the magnitude of inner product will be $\sqrt{\frac{N-M}{M(N-1)}}$, but do we know anything about the phase (or the sign if the vectors are real-valued)? Does it depend on the frame design algorithm? Can we at least determine a distribution for the phase (or sign)?

Kind regards, Ali