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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

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  • $\begingroup$ 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... $\endgroup$
    – Dirk
    Commented Jan 25, 2015 at 19:01
  • $\begingroup$ 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. $\endgroup$
    – alira
    Commented Jan 30, 2015 at 19:12

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We can say something in the real case. First, note that negating a frame element preserves ETF-ness. The result of interest (see Corollary 5.6 in this paper) posits that for every real ETF, one may negate some of the vectors to form an ETF such that one of the vectors has positive inner product with all the others, and the sign pattern of the Gram matrix of the remaining vectors forms the Seidel adjacency matrix of a strongly regular graph. As such, you can read off the parameters of the graph to establish the distribution of signs.

Very little in known about ETFs in the complex case, so I doubt you can say much about phases in general (though you can certainly analyze the constructions known to date, e.g., harmonic ETFs and Steiner ETFs).

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