On signal reconstruction from FROG measurements

doi:10.1016/j.acha.2018.10.003

Applied and Computational Harmonic Analysis

Volume 48, Issue 3, May 2020, Pages 1030-1044

https://doi.org/10.1016/j.acha.2018.10.003 Get rights and content

Abstract

Phase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. Unfortunately, phase retrieval is ill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the ill-posedness, it is common to use a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. The FROG trace results in a system of phaseless quartic Fourier measurements. In this paper, we prove that it suffices to consider only three translations of the signal to determine almost all bandlimited signals, up to trivial ambiguities. In practice, one usually also has access to the signal's Fourier magnitude. We show that in this case only two translations suffice. Our results significantly improve upon earlier work.

Introduction

Phase retrieval is the problem of estimating a signal from its Fourier magnitude. This problem plays a key role in many scientific and engineering applications, among them X-ray crystallography, speech recognition, blind channel estimation, alignment tasks and astronomy [1], [2], [3], [4], [5], [6]. Optical applications are of particular interest since optical devices, such as a charge-coupled device (CCD) and the human eye, cannot detect phase information of the light wave [7].

Almost all one-dimensional signals cannot be determined uniquely from their Fourier magnitude. This immanent ill-posedness makes this problem substantially more challenging than its multi-dimensional counterpart, which is well-posed for almost all signals [8]. Two exceptions for one-dimensional signals that are determined uniquely from their Fourier magnitude are minimum phase signals and sparse signals with non-periodic support [9], [10] ; see also [11], [12]. One popular way to overcome the non-uniqueness is by collecting additional information on the sought signal beyond its Fourier magnitude. For instance, this can be done by taking multiple measurements, each one with a different known mask [13], [14], [15]. An important special case employs shifted versions of a single mask. The acquired data is simply the short-time Fourier transform (STFT) magnitude of the underlying signal. It has been shown that in many setups, this information is sufficient for efficient and stable recovery [16], [17], [18], [19], [20], [21]. For a recent survey of phase retrieval from a signal processing point-of-view, see [8].

In this work, we consider an ultra-short laser pulse characterization method, called Frequency-Resolved Optical Gating (FROG). FROG is a simple, commonly-used technique for full characterization of ultra-short laser pulses which enjoys good experimental performance [22], [23]. In order to characterize the signal, the FROG method measures the Fourier magnitude of the product of the signal with a translated version of itself, for several different translations. The product of the signal with itself is usually performed using a second harmonic generation (SHG) crystal [24]. The acquired data, referred to as FROG trace, is a quartic function of the underlying signal and can be thought of as phaseless quartic Fourier measurements. We refer to the problem of recovering a signal from its FROG trace as the quartic phase retrieval problem. Illustration of the FROG setup is given in Fig. 1.1.

In this paper we provide sufficient conditions on the number of samples required to determine a bandlimited signal uniquely, up to trivial ambiguities, from its FROG trace. Particularly, we show that it is sufficient to consider only three translations of the signal to determine almost all bandlimited signals. If one also measures the power spectrum of the signal, then two translations suffice.

The outline of this paper is as follows. In Section 2 we formulate the FROG problem, discuss its ambiguities and present our main result. Proof of the main result is given in Section 3 and Section 4 provides additional proofs for intermediate results. Section 5 concludes the paper and presents open questions.

Throughout the paper we use the following notation. We denote the Fourier transform of a signal $z \in C^{N}$ by ${\hat{z}}_{k} = \sum_{n = 0}^{N - 1} z_{n} e^{- 2 π ι k n / N}$ , where $ι : = \sqrt{- 1}$ . We further use $\overline{z}$ , $ℜ {z}$ and $ℑ {z}$ for its conjugate, real part and imaginary part, respectively. We reserve $x \in C^{N}$ to be the underlying signal. In the sequel, all signals are assumed to be periodic with period N and all indices should be considered as modulo N, i.e., $z_{n} = z_{n + N ℓ}$ for any integer $ℓ \in Z$ .

Section snippets

Mathematical formulation and main result

The goal of this paper is to derive the minimal number of measurements required to determine a signal from its FROG trace. To this end, we first formulate the FROG problem and identify its symmetries, usually called trivial ambiguities in the phase retrieval literature. Then, we introduce and discuss the main results of the paper.

Preliminaries

We begin the proof by reformulating the measurement model to a more convenient structure. Applying the inverse Fourier transform we write $x_{n} = \frac{1}{N} \sum_{k = 0}^{N = 1} {\hat{x}}_{k} e^{2 π ι k n / N}$ . Then, according to (2.1), we have $\begin{matrix} {\hat{y}}_{k, m} & = \sum_{n = 0}^{N - 1} x_{n} x_{n + m L} e^{- 2 π ι k n / N} \\ = \frac{1}{N^{2}} \sum_{n = 0}^{N - 1} (\sum_{ℓ_{1} = 0}^{N - 1} {\hat{x}}_{ℓ_{1}} e^{2 π ι ℓ_{1} n / N}) (\sum_{ℓ_{2} = 0}^{N - 1} {\hat{x}}_{ℓ_{2}} e^{2 π ι ℓ_{2} n / N} e^{2 π ι ℓ_{2} m L / N}) e^{- 2 π ι k n / N} \\ = \frac{1}{N^{2}} \sum_{ℓ_{1}, ℓ_{2} = 0}^{N - 1} {\hat{x}}_{ℓ_{1}} {\hat{x}}_{ℓ_{2}} e^{2 π ι ℓ_{2} m L / N} \sum_{n = 0}^{N - 1} e^{- 2 π ι (k - ℓ_{1} - ℓ_{2}) n / N} . \end{matrix}$ Since the later sum is equal to N if $k = ℓ_{1} + ℓ_{2}$ and zero otherwise, we get ${\hat{y}}_{k, m} = \frac{1}{N} \sum_{ℓ = 0}^{N - 1} {\hat{x}}_{ℓ} {\hat{x}}_{k - ℓ} e^{2 π ι ℓ m L / N} = \frac{1}{N} \sum_{ℓ = 0}^{N - 1} {\hat{x}}_{ℓ} {\hat{x}}_{k - ℓ} ω^{ℓ m},$ where $ω : = e^{2 π ι / r}$

On the translation symmetry for bandlimited signals

The following proposition shows that if the signal is bandlimited, then the translation symmetry is continuous.

Proposition 4.1

Suppose that x is a B-bandlimited signal with $B \leq N / 2$ . Assume without loss of generality that ${\hat{x}}_{B} = \dots = {\hat{x}}_{N - 1} = 0$ . Then, for any $μ = e^{ι ψ}$ for some $ψ \in [0, 2 π)$ , any signal with Fourier transform $[{\hat{x}}_{0}, μ {\hat{x}}_{1}, μ^{2} {\hat{x}}_{2}, \dots, μ^{B - 1} {\hat{x}}_{B - 1}, 0, \dots, 0],$ has the same FROG trace (2.2) as x.

Proof

Under the bandlimit assumption, we can substitute $p = ℓ$ and $q = k - ℓ$ and write (3.1) as ${\hat{y}}_{k, m} = \frac{1}{N} \sum_{\begin{matrix} p + q = k \\ 0 \leq p, q \leq N / 2 - 1 \end{matrix}} {\hat{x}}_{p} {\hat{x}}_{q} e^{2 π ι p m L / N} .$ Now, if ${\hat{x}}_{p}$ is

Conclusion and perspective

FROG is an important tool for ultra-short laser pulse characterization. The problem involves a system of phaseless quartic equations that differs significantly from quadratic systems, appearing in standard phase retrieval problems. In this work, we analyzed the uniqueness of the FROG method. We have shown that it is sufficient to take only 3B FROG measurements in order to determine a generic B-bandlimited signal uniquely, up to unavoidable symmetries. If the power spectrum of the sought signal

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Cited by (0)

^☆: D. Edidin acknowledges support from Simons Collaboration Grant 315460. Y.C. Eldar acknowledges support from the European Union's Horizon 2020 research and innovation program under grant agreement No. 646804-ERCCOG-BNYQ, and from the Israel Science Foundation under Grant no. 335/14.

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On signal reconstruction from FROG measurements☆

Abstract

Introduction

Section snippets

Mathematical formulation and main result

Preliminaries

On the translation symmetry for bandlimited signals

Conclusion and perspective

Appl. Comput. Harmon. Anal.

Appl. Comput. Harmon. Anal.

Phase retrieval in crystallography and optics

J. Opt. Soc. Amer. A

Benchmark problems for phase retrieval

Fundamentals of Speech Recognition, vol. 14

Blind channel estimation via combining autocorrelation and blind phase estimation

IEEE Trans. Circuits Syst. I. Regul. Pap.

Phase retrieval and image reconstruction for astronomy

Bispectrum inversion with application to multireference alignment

IEEE Trans. Signal Process.

Phase retrieval with application to optical imaging: a contemporary overview

IEEE Signal Process. Mag.

Fourier phase retrieval: uniqueness and algorithms

Phase retrieval from 1D Fourier measurements: convexity, uniqueness, and algorithms

IEEE Trans. Signal Process.

Sparse phase retrieval: convex algorithms and limitations

Ambiguities in one-dimensional discrete phase retrieval from Fourier magnitudes

J. Fourier Anal. Appl.

Phase retrieval via matrix completion

SIAM Rev.

Phase retrieval with masks using convex optimization

Phase retrieval from local measurements: improved robustness via eigenvector-based angular synchronization

Non-convex phase retrieval from STFT measurements

IEEE Trans. Inform. Theory

STFT phase retrieval: uniqueness guarantees and recovery algorithms

IEEE J. Sel. Top. Signal Process.