Phase Retrieval of Bandlimited Signals Using a Morlet Wavelet and an Admissible Companion

In this paper, we demonstrate that complex bandlimited signals can be uniquely recovered (up to a global phase factor) from two families of wavelet transform magnitudes, sampled at twice the Nyquist rate in two frequency bins. These families come from translations of the fixed dilated Morlet wavelet and a fixed companion wavelet. The companion wavelet admits an augmented Morlet--Gaussian representation. We rewrite the two families as two modulated Gaussian convolution measurements of an auxiliary bandlimited signal. Using a Gaussian convolution uniqueness result, we prove that samples on \((4B)⁻¹Z\) determine every signal in \(PW_B^2\) (up to a global phase factor). Compared with earlier wavelet phase retrieval results, our approach requires significantly fewer sampling points.