A geometrical problem arising in a signal restoration algorithm

Let ℤ_2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ_2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ_2N→ℝ is given by φ_f(k)=arg for k ∈ ℤ_2N, where denotes the discrete Fourier transforms of f. For a fixed s∈L let K_s denote the cone of all f:ℤ_2N→ℝ which satisfy φ_f ≡ φ_s and let M_s be its linear span. The angle α_s between M_s 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:

1.	α (Ms, L)≤π/4 for a generic s∈L.
2.	If s∈L is geometric, i.e., s(j)=qj for 0≤j≤N−1 where ±1≠q∈ℝ, then α(Ms, L)=π/4.

Acknowledgments and Notes. Nir Cohen-Supported by CNPq grant 300019/96-3. Roy Meshulam-Research supported by the Fund for the Promotion of Research at the Technion.