Unique wavelet sign retrieval from samples without bandlimiting

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 $$\{\mid {\mathcal{W}}_{{\phi }_i}f({\alpha }^m\beta n,{\alpha }^m)\mid :i\in \{1,2,3\},m,n\in \mathbb{Z}\}$$ for every $\alpha >1,\beta >0$ with $\beta \ln (\alpha )\le 4\pi \mathrm{/}(1+4p)$, $p>0$, when the three wavelets ${\phi }_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathcal{H}{P}_p$. For complex-valued signals we find that this is not possible for any choice of the parameters $\alpha >1,\beta >0$, 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.