Phase-only signal reconstruction (original) (raw)
On signal reconstruction without noisy phase
2004
We construct new classes of Parseval frames for a Hilbert space which allow signal reconstruction from the absolute value of the frame coefficients. As a consequence, signal reconstruction can be done without using noisy phase or its estimation. This verifies a longstanding conjecture of the speech processing community.
Stability of unique Fouriertransform phase reconstruction
Journal of The Optical Society of America, 1983
The problem of Fourier-transform phase reconstruction from the Fourier-transform magnitude of multidimensional discrete signals is considered. It is well known that, if a discrete finite-extent n-dimensional signal (n > 2) has an irreducible z transform, then the signal is uniquely determined from the magnitude of its Fourier transform. It is also known that this irreducibility condition holds for all multidimensional signals except for a set of signals that has measure zero. We show that this uniqueness condition is stable in the sense that it is not sensitive to noise. Specifically, it is proved that the set of signals whose z transform is reducible is contained in the zero set of a certain multidimensional polynomial. Several important conclusions can be drawn from this characterization, and, in particular, the zero-measure property is obtained as a simple byproduct.
Phase retrieval of real-valued signals in a shift-invariant space
Applied and Computational Harmonic Analysis, 2018
Phase retrieval arises in various fields of science and engineering and it is well studied in a finite-dimensional setting. In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a shiftinvariant space from its phaseless samples taken either on the whole line or on a set with finite sampling rate. We find the equivalence between nonseparability of signals in a linear space and its phase retrievability with phaseless samples taken on the whole line. For a spline signal of order N , we show that it can be well approximated, up to a sign, from its noisy phaseless samples taken on a set with sampling rate 2N −1. We propose an algorithm to reconstruct nonseparable signals in a shift-invariant space generated by a compactly supported continuous function φ. The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.
Phase Retrieval for Sparse Signals: Uniqueness Conditions
In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: "Under which conditions can we uniquely recover the signal of interest from its measured magnitudes?"
Signal reconstruction from signed Fourier transform magnitude
IEEE Transactions on Signal Processing, 1983
In this paper, we show that a one-dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed Fourier transform magnitude (magnitude and 1 bit of phase information). In addition, we develop a numerical algorithm to reconstruct a one-dimensional or multidimensional sequence from its Fourier transform magnitude. Reconstruction examples obtained using this algorithm are also provided.
Sparse phase retrieval: Uniqueness guarantees and recovery algorithms
The problem of signal recovery from its Fourier transform magnitude, or equivalently, autocorrelation, is of paramount importance in various fields of engineering and has been around for over 100 years. In order to achieve this, additional structure information about the signal is necessary. In this work, we first provide simple and general conditions, which when satisfied, allow unique recovery almost surely. In particular, we focus our attention on sparse signals and show that most O(n)-sparse signals, i.e., signals with O(n) non-zero components, have distinct Fourier transform magnitudes (up to time-shift, time-reversal and global sign). Our results are a significant improvement over the existing identifiability results, which provide such guarantees for only O(n 1/4 )-sparse signals.
Applied and Computational Harmonic Analysis, 2017
Considering the ambiguousness of the discrete-time phase retrieval problem to recover a signal from its Fourier intensities, one can ask the question: what additional information about the unknown signal do we need to select the correct solution within the large solution set? Based on a characterization of the occurring ambiguities, we investigate different a priori conditions in order to reduce the number of ambiguities or even to receive a unique solution. Particularly, if we have access to additional magnitudes of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. Moreover, we prove that an analogous result can be obtained by exploiting additional phase information.
Reconstruction of signal phases for signals closer than the DFT frequency resolution
Advances in Radio Science, 2021
Radar signal processing is a promising tool for vital sign monitoring. For contactless observation of breathing and heart rate a precise measurement of the distance between radar antenna and the patient's skin is required. This results in the need to detect small movements in the range of 0.5 mm and below. Such small changes in distance are hard to be measured with a limited radar bandwidth when relying on the frequency based range detection alone. In order to enhance the relative distance resolution a precise measurement of the observed signal's phase is required. Due to radar reflections from surfaces in close proximity to the main area of interest the desired signal of the radar reflection can get superposed. For superposing signals with little separation in frequency domain the main lobes of their discrete Fourier transform (DFT) merge into a single lobe, so that their peaks cannot be differentiated. This paper evaluates a method for reconstructing the phase and amplitude of such superimposed signals.
Robust Phase Retrieval Algorithm for Time-Frequency Structured Measurements
SIAM Journal on Imaging Sciences
We address the problem of signal reconstruction from intensity measurements with respect to a measurement frame. This non-convex inverse problem is known as phase retrieval. The case considered in this paper concerns phaseless measurements taken with respect to a Gabor frame. It arises naturally in many practical applications, such as diffraction imaging and speech recognition. We present a reconstruction algorithm that uses a nearly optimal number of phaseless time-frequency structured measurements and discuss its robustness in the case when the measurements are corrupted by noise. We show how geometric properties of the measurement frame are related to the robustness of the phaseless reconstruction.
Phase retrieval with Fourier-weighted projections
Journal of the Optical Society of America A, 2008
In coherent lensless imaging, the presence of image sidelobes, which arise as a natural consequence of the finite nature of the detector array, was early recognized as a convergence issue for phase retrieval algorithms that rely on an object support constraint. To mitigate the problem of truncated far-field measurement, a controlled analytic continuation by means of an iterative transform algorithm with weighted projections is proposed and tested. This approach avoids the use of sidelobe reduction windows and achieves full-resolution reconstructions.