Phase Retrieval in Hardy Space

Phase retrieval for wide-band signals

This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given f $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x|} dx), we find all functions g $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x| dx}), such that |f (x)| = |g(x)| for all x $\in$ R. To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem with additional constraints involving some transforms of f and g, and determine if these …

Multiscale decompositions of Hardy spaces

An inspiration at the origin of wavelet analysis (when Grossmann, Morlet, Meyer and collaborators were interacting and exploring versions of multiscale representations) was provided by the analysis of holomorphic signals, for which the images of the phase of Cauchy wavelets were remarkable in their ability to reveal intricate singularities or dynamic structures, such as instantaneous frequency jumps in musical recordings. Our goal is to follow their seminal work and introduce recent developments in nonlinear analysis. In particular we sketch methods extending conventional Fourier analysis, exp…

Blaschke Decompositions on Weighted Hardy Spaces

Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of a complex function $F: \partial\mathbb{D}\to \mathbb{C}$, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was studied by Coifman and Stei…

整函数的相位恢复及其对Gabor相位恢复的意义

Phase retrieval of entire functions and its implications for Gabor phase retrieval

本文利用Hadamard因子分解定理刻画了在两条任意直线及无穷多条等距平行线上模相等的整函数对，并证明三条有理无关距离的平行线上的模可唯一确定整函数（至多相差全局相位），同时给出了一阶整函数在特定直线族上无法唯一确定的例子，为Gabor相位恢复提供了直接理论依据。

Conjugate Phase Retrieval in Paley-Wiener Space

We consider the problem of conjugate phase retrieval in Paley-Wiener space $PW_π$. The goal of conjugate phase retrieval is to recover a signal $f$ from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning $f(t)$ and $\overline{f(t)}$ are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling both $f$ and $f^{\prime}$ only on the …

揭示采样Gabor相位恢复中唯一性的极限：$L^2(\mathbb{R})$中的稠密反例集

Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$

本文研究采样Gabor相位恢复的唯一性问题，证明其反例集在$L^2(\mathbb{R})$中稠密但一般不等于全空间，从而揭示了该问题在信号恢复中的根本局限性。