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 $$\{\lvert \mathcal{W}_{\phi_i} f(\alpha^{m}\beta n,\alpha^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\}$$ for every $\alpha>1,\beta>0$ with $\beta\ln(\alpha)\leq 4\pi/(1+4p)$, $p>0$, when the three wavelets $\phi_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathscr{H}P_p$. 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.