# Phase of the inner product between the elements of an ETF

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

• Hmm you can multiply any frame element with any complex number of modulus one and still get another ETF. But well this changes all the inner products so probably you won't get an arbitrary distribution... – Dirk Jan 25 '15 at 19:01
• Yes, so after multiplying with a complex number of modulus one the resulting distribution is a shifted version of the original distribution. And this is why I guess (and hope!) that it is $\mathrm{unif}(-\pi,\pi)$, as in this case it remains the same for any arbitrary phase shift. But I have no proof. – alira Jan 30 '15 at 19:12