Showing papers on "Fourier transform published in 2017"
TL;DR: In this article, the authors present an updated summary of the penalized pixel-fitting (pPXF) method, which is used to extract the stellar and gas kinematics, as well as the stellar population of galaxies via full spectrum fitting.
Abstract: I start by providing an updated summary of the penalized pixel-fitting (pPXF) method, which is used to extract the stellar and gas kinematics, as well as the stellar population of galaxies, via full spectrum fitting. I then focus on the problem of extracting the kinematic when the velocity dispersion $\sigma$ is smaller than the velocity sampling $\Delta V$, which is generally, by design, close to the instrumental dispersion $\sigma_{\rm inst}$. The standard approach consists of convolving templates with a discretized kernel, while fitting for its parameters. This is obviously very inaccurate when $\sigma<\Delta V/2$, due to undersampling. Oversampling can prevent this, but it has drawbacks. Here I present a more accurate and efficient alternative. It avoids the evaluation of the under-sampled kernel, and instead directly computes its well-sampled analytic Fourier transform, for use with the convolution theorem. A simple analytic transform exists when the kernel is described by the popular Gauss-Hermite parametrization (which includes the Gaussian as special case) for the line-of-sight velocity distribution. I describe how this idea was implemented in a significant upgrade to the publicly available pPXF software. The key advantage of the new approach is that it provides accurate velocities regardless of $\sigma$. This is important e.g. for spectroscopic surveys targeting galaxies with $\sigma\ll\sigma_{\rm inst}$, for galaxy redshift determinations, or for measuring line-of-sight velocities of individual stars. The proposed method could also be used to fix Gaussian convolution algorithms used in today's popular software packages.
866 citations
Journal Article•
TL;DR: The main idea of SET is to only retain the TF information of STFT results most related to time-varying features of the signal and to remove most smeared TF energy, such that the energy concentration of the novel TF representation can be enhanced greatly.
Abstract: In this paper, we introduce a new time-frequency (TF) analysis (TFA) method to study the trend and instantaneous frequency (IF) of nonlinear and nonstationary data. Our proposed method is termed the synchroextracting transform (SET), which belongs to a postprocessing procedure of the short-time Fourier transform (STFT). Compared with classical TFA methods, the proposed method can generate a more energy concentrated TF representation and allow for signal reconstruction. The proposed SET method is inspired by the recently proposed synchrosqueezing transform (SST) and the theory of the ideal TFA. To analyze a signal, it is important to obtain the time-varying information, such as the IF and instantaneous amplitude. The SST is to squeeze all TF coefficients into the IF trajectory. Differ from the squeezing manner of SST, the main idea of SET is to only retain the TF information of STFT results most related to time-varying features of the signal and to remove most smeared TF energy, such that the energy concentration of the novel TF representation can be enhanced greatly. Numerical and real-world signals are employed to validate the effectiveness of the SET method.
310 citations
TL;DR: A generalization of the short-time Fourier-based synchrosqueezing transform using a new local estimate of instantaneous frequency enables not only to achieve a highly concentrated time-frequency representation for a wide variety of amplitude- and frequency-modulated multicomponent signals but also to reconstruct their modes with a high accuracy.
Abstract: This paper puts forward a generalization of the short-time Fourier-based synchrosqueezing transform using a new local estimate of instantaneous frequency. Such a technique enables not only to achieve a highly concentrated time-frequency representation for a wide variety of amplitude- and frequency-modulated multicomponent signals but also to reconstruct their modes with a high accuracy. Numerical investigation on synthetic and gravitational-wave signals shows the efficiency of this new approach.
282 citations
TL;DR: In this paper, the Fast Spectral Correlation (FSC) estimator is proposed, which is based on the short-time Fourier transform (STFT) for cyclostationary signals.
228 citations
TL;DR: In this article, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type, which is efficient in modeling a class of concentrations in the complex transport process.
Abstract: In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed. The new formulation is efficient in modeling a class of concentrations in the complex transport process.
217 citations
TL;DR: A new strategy to increase the speed of FSI by two orders of magnitude is reported, which binarize the Fourier basis patterns based on upsampling and error diffusion dithering to find broad imaging applications at wavebands that are not accessible using conventional two-dimensional image sensors.
Abstract: Fourier single-pixel imaging (FSI) employs Fourier basis patterns for encoding spatial information and is capable of reconstructing high-quality two-dimensional and three-dimensional images. Fourier-domain sparsity in natural scenes allows FSI to recover sharp images from undersampled data. The original FSI demonstration, however, requires grayscale Fourier basis patterns for illumination. This requirement imposes a limitation on the imaging speed as digital micro-mirror devices (DMDs) generate grayscale patterns at a low refreshing rate. In this paper, we report a new strategy to increase the speed of FSI by two orders of magnitude. In this strategy, we binarize the Fourier basis patterns based on upsampling and error diffusion dithering. We demonstrate a 20,000 Hz projection rate using a DMD and capture 256-by-256-pixel dynamic scenes at a speed of 10 frames per second. The reported technique substantially accelerates image acquisition speed of FSI. It may find broad imaging applications at wavebands that are not accessible using conventional two-dimensional image sensors.
175 citations
TL;DR: A new integral transform operator, which is similar to Fourier transform, is proposed for the first time, and an application to the one-dimensional heat-diffusion problem is discussed.
152 citations
TL;DR: A novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory, is proposed, and its efficacy for the analysis of nonlinear and non-stationary time series is demonstrated.
Abstract: for many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory, and demonstrate its efficacy for the analysis of nonlinear and non-stationary time series. The proposed FDM decomposes any data into a small number of ‘Fourier intrinsic band functions’ (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM. The proposed MFDM generates a finite number of band-limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.
147 citations
TL;DR: New stabilization techniques are introduced and unconditional energy stability for modified energy functionals for 2D Cahn–Hilliard equations is proved under relaxed constraints on the size of the time steps.
Abstract: We consider several seconder order in time stabilized semi-implicit Fourier spectral schemes for 2D Cahn---Hilliard equations. We introduce new stabilization techniques and prove unconditional energy stability for modified energy functionals. We also carry out a comparative study of several classical stabilization schemes and identify the corresponding stability regions. In several cases the energy stability is proved under relaxed constraints on the size of the time steps. We do not impose any Lipschitz assumption on the nonlinearity. The error analysis is obtained under almost optimal regularity assumptions.
126 citations
TL;DR: This chapter surveys methods to guarantee uniqueness in Fourier phase retrieval and presents different algorithmic approaches to retrieve the signal in practice, and outlines some of the main open questions in this field.
Abstract: The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science Fourier phase retrieval poses fundamental theoretical and algorithmic challenges In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude, and therefore the problem is ill-posed Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval We then present different algorithmic approaches to retrieve the signal in practice We conclude by outlining some of the main open questions in this field
114 citations
TL;DR: A Fourier pseudo-spectral method that conserves mass and energy is developed for a two-dimensional nonlinear Schrodinger equation and it is proved that the optimal rate of convergence is in the order of O in the discrete L 2 norm without any restrictions on the grid ratio.
Posted Content•
TL;DR: This study supports the hypothesis that time-frequency representations are valuable in learning useful features for sound classification and observes that the optimal window size during transformation is dependent on the characteristics of the audio signal and architecturally, 2D convolution yielded better results in most cases compared to 1D.
Abstract: Recent successful applications of convolutional neural networks (CNNs) to audio classification and speech recognition have motivated the search for better input representations for more efficient training. Visual displays of an audio signal, through various time-frequency representations such as spectrograms offer a rich representation of the temporal and spectral structure of the original signal. In this letter, we compare various popular signal processing methods to obtain this representation, such as short-time Fourier transform (STFT) with linear and Mel scales, constant-Q transform (CQT) and continuous Wavelet transform (CWT), and assess their impact on the classification performance of two environmental sound datasets using CNNs. This study supports the hypothesis that time-frequency representations are valuable in learning useful features for sound classification. Moreover, the actual transformation used is shown to impact the classification accuracy, with Mel-scaled STFT outperforming the other discussed methods slightly and baseline MFCC features to a large degree. Additionally, we observe that the optimal window size during transformation is dependent on the characteristics of the audio signal and architecturally, 2D convolution yielded better results in most cases compared to 1D.
TL;DR: In this paper, the authors consider the survival probability of a thermofield double state under unitary time evolution which is related to the analytic continuation of the partition function and provide an exponential lower bound to the survival probabilities with a rate governed by the inverse of the energy fluctuations of the initial state.
Abstract: Quantum speed limits set an upper bound to the rate at which a quantum system can evolve, and as such can be used to analyze the scrambling of information. To this end, we consider the survival probability of a thermofield double state under unitary time evolution which is related to the analytic continuation of the partition function. We provide an exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state. Further, we elucidate universal features of the nonexponential behavior at short and long times of evolution that follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and for systems with a discrete energy spectrum. We find the spectral form factor in a number of illustrative models; notably, we obtain the exact answer in the Gaussian unitary ensemble for any $N$ with excellent agreement with recent numerical studies. We also discuss the relationship of our findings to models of black hole information loss, such as the Sachdev-Ye-Kitaev model dual to ${\mathrm{AdS}}_{2}$, as well as higher-dimensional versions of AdS/CFT.
TL;DR: It is shown that TSPR can provably recover most discrete-time sparse signals and the recovery is robust in the presence of measurement noise, and these recovery guarantees are asymptotic in nature.
Abstract: The problem of signal recovery from its Fourier transform magnitude is of paramount importance in various fields of engineering and has been around for more than 100 years. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the signal of interest. In this paper, we focus our attention on discrete-time sparse signals (of length $n$ ). We first show that if the discrete Fourier transform dimension is greater than or equal to $2n$, then almost all signals with aperiodic support can be uniquely identified by their Fourier transform magnitude (up to time shift, conjugate flip, and global phase). Then, we develop an efficient two-stage sparse-phase retrieval algorithm (TSPR), which involves: identifying the support, i.e., the locations of the nonzero components, of the signal using a combinatorial algorithm; and identifying the signal values in the support using a convex algorithm. We show that TSPR can provably recover most $O(n^{1/2-{\epsilon }})$ -sparse signals (up to a time shift, conjugate flip, and global phase). We also show that, for most $O(n^{1/4-{\epsilon }})$-sparse signals, the recovery is robust in the presence of measurement noise. These recovery guarantees are asymptotic in nature. Numerical experiments complement our theoretical analysis and verify the effectiveness of TSPR.
Proceedings Article•
17 Jul 2017
TL;DR: In this paper, the spectral properties of random Fourier features are studied from a spectral matrix approximation point of view, and tight bounds on the number of features required to achieve a spectral approximation are given.
Abstract: Random Fourier features is one of the most popular techniques for scaling up kernel methods, such as kernel ridge regression. However, despite impressive empirical results, the statistical properties of random Fourier features are still not well understood. In this paper we take steps toward filling this gap. Specifically, we approach random Fourier features from a spectral matrix approximation point of view, give tight bounds on the number of Fourier features required to achieve a spectral approximation, and show how spectral matrix approximation bounds imply statistical guarantees for kernel ridge regression.
Qualitatively, our results are twofold: on the one hand, we show that random Fourier feature approximation can provably speed up kernel ridge regression under reasonable assumptions. At the same time, we show that the method is suboptimal, and sampling from a modified distribution in Fourier space, given by the leverage function of the kernel, yields provably better performance. We study this optimal sampling distribution for the Gaussian kernel, achieving a nearly complete characterization for the case of low-dimensional bounded datasets. Based on this characterization, we propose an efficient sampling scheme with guarantees superior to random Fourier features in this regime.
18 Sep 2017
TL;DR: The proposed Fourier Convolution Neural Network (FCNN) is proposed whereby training is conducted entirely within the Fourier domain and shows a significant speed up in training time without loss of effectiveness.
Abstract: The Fourier domain is used in computer vision and machine learning as image analysis tasks in the Fourier domain are analogous to spatial domain methods but are achieved using different operations. Convolutional Neural Networks (CNNs) use machine learning to achieve state-of-the-art results with respect to many computer vision tasks. One of the main limiting aspects of CNNs is the computational cost of updating a large number of convolution parameters. Further, in the spatial domain, larger images take exponentially longer than smaller image to train on CNNs due to the operations involved in convolution methods. Consequently, CNNs are often not a viable solution for large image computer vision tasks. In this paper a Fourier Convolution Neural Network (FCNN) is proposed whereby training is conducted entirely within the Fourier domain. The advantage offered is that there is a significant speed up in training time without loss of effectiveness. Using the proposed approach larger images can therefore be processed within viable computation time. The FCNN is fully described and evaluated. The evaluation was conducted using the benchmark Cifar10 and MNIST datasets, and a bespoke fundus retina image dataset. The results demonstrate that convolution in the Fourier domain gives a significant speed up without adversely affecting accuracy. For simplicity the proposed FCNN concept is presented in the context of a basic CNN architecture, however, the FCNN concept has the potential to improve the speed of any neural network system involving convolution.
TL;DR: A general model to characterize MCCSs, where instantaneous frequencies (IFs) and instantaneous amplitudes (IAs) of the intrinsic chirp components (ICCs) are modeled as Fourier series are developed and the decomposition problem boils down to identifying the developed model.
TL;DR: The use of the short-frequency Fourier transform (SFFT) for fault diagnosis of induction machines working under transient regimes is proposed, which keeps the resolution of traditional techniques, but also achieves a drastic reduction of computing time and memory resources, making this proposal suitable for on-line fault diagnosis.
Abstract: Transient-based methods for fault diagnosis of induction machines (IMs) are attracting a rising interest, due to their reliability and ability to adapt to a wide range of IM’s working conditions. These methods compute the time–frequency (TF) distribution of the stator current, where the patterns of the related fault components can be detected. A significant amount of recent proposals in this field have focused on improving the resolution of the TF distributions, allowing a better discrimination and identification of fault harmonic components. Nevertheless, as the resolution improves, computational requirements (power computing and memory) greatly increase, restricting its implementation in low-cost devices for performing on-line fault diagnosis. To address these drawbacks, in this paper, the use of the short-frequency Fourier transform (SFFT) for fault diagnosis of induction machines working under transient regimes is proposed. The SFFT not only keeps the resolution of traditional techniques, such as the short-time Fourier transform, but also achieves a drastic reduction of computing time and memory resources, making this proposal suitable for on-line fault diagnosis. This method is theoretically introduced and experimentally validated using a laboratory test bench.
TL;DR: This work presents the mathematical implementation of a tomographic algorithm, termed GENeralized Fourier Iterative REconstruction (GENFIRE), for high-resolution 3D reconstruction from a limited number of 2D projections and demonstrates that it can produce superior results relative to several other popular tomographic reconstruction techniques.
Abstract: Tomography has made a radical impact on diverse fields ranging from the study of 3D atomic arrangements in matter to the study of human health in medicine. Despite its very diverse applications, the core of tomography remains the same, that is, a mathematical method must be implemented to reconstruct the 3D structure of an object from a number of 2D projections. Here, we present the mathematical implementation of a tomographic algorithm, termed GENeralized Fourier Iterative REconstruction (GENFIRE), for high-resolution 3D reconstruction from a limited number of 2D projections. GENFIRE first assembles a 3D Fourier grid with oversampling and then iterates between real and reciprocal space to search for a global solution that is concurrently consistent with the measured data and general physical constraints. The algorithm requires minimal human intervention and also incorporates angular refinement to reduce the tilt angle error. We demonstrate that GENFIRE can produce superior results relative to several other popular tomographic reconstruction techniques through numerical simulations and by experimentally reconstructing the 3D structure of a porous material and a frozen-hydrated marine cyanobacterium. Equipped with a graphical user interface, GENFIRE is freely available from our website and is expected to find broad applications across different disciplines.
TL;DR: An explicitly time-dependent approach to equation-of-motion (EOM) coupled-cluster theory with single and double excitations (CCSD) is implemented for simulating near-edge X-ray absorption fine structure in molecular systems, providing converged spectra in much shorter simulation times than are required by the Fourier form.
Abstract: An explicitly time-dependent (TD) approach to equation-of-motion (EOM) coupled-cluster theory with single and double excitations (CCSD) is implemented for simulating near-edge X-ray absorption fine structure in molecular systems. The TD-EOM-CCSD absorption line shape function is given by the Fourier transform of the CCSD dipole autocorrelation function. We represent this transform by its Pade approximant, which provides converged spectra in much shorter simulation times than are required by the Fourier form. The result is a powerful framework for the blackbox simulation of broadband absorption spectra. K-edge X-ray absorption spectra for carbon, nitrogen, and oxygen in several small molecules are obtained from the real part of the absorption line shape function and are compared with experiment. The computed and experimentally obtained spectra are in good agreement; the mean unsigned error in the predicted peak positions is only 1.2 eV. We also explore the spectral signatures of protonation in these molecules.
TL;DR: The accuracy and efficiency of the TT-SOFT method is demonstrated as applied to propagation of 24-dimensional wave packets, describing the S1/S2 interconversion dynamics of pyrazine after UV photoexcitation to the S2 state.
Abstract: We introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) method for simulations of multidimensional nonadiabatic quantum dynamics. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wave function in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S1/S2 interconversion dynamics of pyrazine after UV photoexcitation to the S2 state. Our results show that the TT-SOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-...
TL;DR: In this article, a nonlinear finite element (FE) solver was proposed to solve nonlinear problems in a general history-dependent and time-dependent material model, where the kernel is derived from an auxiliary homogeneous linear problem, which renders the extension of FFT-based schemes to nonlinear problem conceptually difficult.
Abstract: Fourier solvers have become efficient tools to establish structure–property relations in heterogeneous materials. Introduced as an alternative to the finite element (FE) method, they are based on fixed-point solutions of the Lippmann–Schwinger type integral equation. Their computational efficiency results from handling the kernel of this equation by the fast Fourier transform (FFT). However, the kernel is derived from an auxiliary homogeneous linear problem, which renders the extension of FFT-based schemes to nonlinear problems conceptually difficult. This paper aims to establish a link between FE-based and FFT-based methods in order to develop a solver applicable to general history-dependent and time-dependent material models. For this purpose, we follow the standard steps of the FE method, starting from the weak form, proceeding to the Galerkin discretization and the numerical quadrature, up to the solution of nonlinear equilibrium equations by an iterative Newton–Krylov solver. No auxiliary linear problem is thus needed. By analyzing a two-phase laminate with nonlinear elastic, elastoplastic, and viscoplastic phases and by elastoplastic simulations of a dual-phase steel microstructure, we demonstrate that the solver exhibits robust convergence. These results are achieved by re-using the nonlinear FE technology, with the potential of further extensions beyond small-strain inelasticity considered in this paper.
TL;DR: In this article, the authors presented fractional Fourier transform (FRFT) based feature extraction as a new technique for classification of power quality disturbances (PQDs) for mitigation of PQDs.
Abstract: Proper mitigation of power quality disturbances (PQDs) requires a fast, accurate and highly noise immune classification technique. This study, therefore, presents fractional Fourier transform (FRFT) based feature extraction as a new technique for classification of PQDs. FRFT is a generalised version of Fourier transform (FT) with an additional order control and can give time, frequency and intermediate time-frequency representations for a signal. The order control offers multi-domain feature extraction, such that most robust feature matrix is utilised for classification under any condition. An expression is derived for the optimal classification order corresponding to maximum overall accuracy. Based on IEEE-1159 standards, 15 PQDs are simulated and a database of pure and noisy signals is prepared. Features extracted from FRFT processed signals are tested with decision trees (DTs) and bagging predictors (BPs). The proposed technique shows better performance in most of the cases, when compared with Stockwell transform based classification under similar conditions. The classification accuracies of FRFT-DT and FRFT-BP are impressive even with significant reduction in training and features. Further, a validation using real PQDs obtained from an experimental setup is shown. The corresponding results closely resemble the simulation outcomes.
Book•
29 Apr 2017TL;DR: A survey of the existing literature on super-oscillations can be found in this article, where it is shown that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces.
Abstract: In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schr\"odinger equation and other equations.
10 Jul 2017
TL;DR: In this paper, the authors presented different computational and experimental vibrational biospectroscopy methods and techniques such as Fourier Transform-Near Infrared (FT-NIR), Fourier transform-Short-Wavelength Infrared(FT-SIR), MIR, MIR and FIR for human cancer cells in tumor tissues simulation, modeling, research, diagnosis and treatment.
Abstract: In the current image article, we present different computational and experimental vibrational biospectroscopy methods and techniques such as Fourier Transform–Near– Infrared (FT–NIR), Fourier Transform–Short–Wavelength Infrared (FT–SIR), Fourier Transform–Mid–Wavelength Infrared (FT–MIR), Fourier Transform–Long–Wavelength Infrared (FT–LIR), Fourier Transform–Far–Infrared (FT– FIR), Attenuated Total Refl ectance–Fourier Transform Infrared (ATR–FTIR) and Fourier Transform–Raman (FT– Raman) spectroscopies for human cancer cells in tumor tissues simulation, modeling, research, diagnosis and treatment (Figure 1) [1–90].
TL;DR: In this paper, the authors examined the efficiency of various mathematical methods in thermographic data processing, with respect to the thermal excitation method and the type of artificial defect in the CFRP specimens.
TL;DR: In this paper, the power spectral density (PSD) of a light source and the interferogram measured with the Si-FTS can be related through a simple Fourier transform (FT), provided the optical frequency and time delay are corrected to account for dispersion, thermo-optic nonlinearity and thermal expansion.
Abstract: The integration of miniaturized optical spectrometers into mobile platforms will have an unprecedented impact on applications ranging from unmanned aerial vehicles (UAVs) to mobile phones. To address this demand, silicon photonics stands out as a platform capable of delivering compact and cost-effective devices. The Fourier transform spectrometer (FTS) is largely used in free-space spectroscopy, and its implementation in silicon photonics will contribute to bringing broadband operation and fine resolution to the chip scale. The implementation of an integrated silicon photonics FTS (Si-FTS) must nonetheless take into account effects such as waveguide dispersion and non-linearity of refractive index tuning mechanisms. Here we present the modeling and experimental demonstration of a silicon-on-insulator (SOI) Si-FTS with integrated microheaters. We show how the power spectral density (PSD) of a light source and the interferogram measured with the Si-FTS can be related through a simple Fourier transform (FT), provided the optical frequency and time delay are corrected to account for dispersion, thermo-optic non-linearity and thermal expansion. We calibrate the Si-FTS, including the correction parameters, using a tunable laser source and we successfully retrieve the PSD of a broadband source. The aforementioned effects are shown to effectively enhance the Si-FTS resolution when properly accounted for. Finally, we discuss the Si-FTS resilience to chip-scale fabrication variations, a major advantage for large-scale manufacturing. Providing design flexibility and robustness, the Si-FTS is poised to become a fundamental building-block for on-chip spectroscopy
TL;DR: It is found that while modulation on the real part of the eigenvalue induces pulse timing drift and leads to neighboring pulse interactions and nonlinear inter-symbol interference (ISI), it is more bandwidth efficient than modulation in practical settings.
Abstract: In this paper, we experimentally investigate high-order modulation over a single discrete eigenvalue under the nonlinear Fourier transform (NFT) framework and exploit all degrees of freedom for encoding information. For a fixed eigenvalue, we compare different 4 bit/symbol modulation formats on the spectral amplitude and show that a 2-ring 16-APSK constellation achieves optimal performance. We then study joint spectral phase, spectral magnitude and eigenvalue modulation and found that while modulation on the real part of the eigenvalue induces pulse timing drift and leads to neighboring pulse interactions and nonlinear inter-symbol interference (ISI), it is more bandwidth efficient than modulation on the imaginary part of the eigenvalue in practical settings. We propose a spectral amplitude scaling method to mitigate such nonlinear ISI and demonstrate a record 4 GBaud 16-APSK on the spectral amplitude plus 2-bit eigenvalue modulation (total 6 bit/symbol at 24 Gb/s) transmission over 1000 km.
TL;DR: In this paper, a broad family of test function spaces and their dual (distribution) spaces is considered, including Gelfand-Shilov spaces and a family of Test Function Spaces introduced by Pilipovic.
Abstract: We consider a broad family of test function spaces and their dual (distribution) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipovic. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.
TL;DR: This work demonstrates the retrieval of three sparse input signals by collecting data from restricted sets of MZIs and applying common CS reconstruction techniques to this data, and shows that this retrieval maintains the full resolution and bandwidth of the original device, despite a sampling factor as low as one-fourth of a conventional (non-compressive) design.
Abstract: We demonstrate compressive-sensing (CS) spectroscopy in a planar-waveguide Fourier-transform spectrometer (FTS) device. The spectrometer is implemented as an array of Mach–Zehnder interferometers (MZIs) integrated on a photonic chip. The signal from a set of MZIs is composed of an undersampled discrete Fourier interferogram, which we invert using l1-norm minimization to retrieve a sparse input spectrum. To implement this technique, we use a subwavelength-engineered spatial heterodyne FTS on a chip composed of 32 independent MZIs. We demonstrate the retrieval of three sparse input signals by collecting data from restricted sets (8 and 14) of MZIs and applying common CS reconstruction techniques to this data. We show that this retrieval maintains the full resolution and bandwidth of the original device, despite a sampling factor as low as one-fourth of a conventional (non-compressive) design.