A geometrical problem arising in a signal restoration algorithm

Abstract

Let ℤ2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ2N→ℝ is given by φf(k)=arg $$\hat f(k)$$ for k ∈ ℤ2N, where $$\hat f$$ denotes the discrete Fourier transforms of f. For a fixed s∈L let Ks denote the cone of all f:ℤ2N→ℝ which satisfy φf ≡ φs and let Ms be its linear span. The angle αs between Ms and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove the following conjectures of Urieli et al. [7] who verified them for the N≤3 case: