Unique wavelet sign retrieval from samples without bandlimiting
Abstract
We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multi-wavelet frame coefficients \[ \ Wφi f(αmβ n,αm) : i∈\1,2,3\, m,n∈Z\ \] for every α>1,β>0 with β(α)≤ 4π/(1+4p), p>0, when the three wavelets φi are suitable linear combinations of the Poisson wavelet Pp of order p and its Hilbert transform HPp. For complex-valued signals we find that this is not possible for any choice of the parameters α>1,β>0, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients.