Showing papers on "Spectral density estimation published in 2014"

GESPAR: Efficient Phase Retrieval of Sparse Signals

Abstract: We consider the problem of phase retrieval, namely, recovery of a signal from the magnitude of its Fourier transform, or of any other linear transform. Due to the loss of Fourier phase information, this problem is ill-posed. Therefore, prior information on the signal is needed in order to enable its recovery. In this work we consider the case in which the signal is known to be sparse, i.e., it consists of a small number of nonzero elements in an appropriate basis. We propose a fast local search method for recovering a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude which we refer to as GESPAR: GrEedy Sparse PhAse Retrieval. Our algorithm does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images. Simulation results indicate that GESPAR is fast and more accurate than existing techniques in a variety of settings.

A Review of Multitaper Spectral Analysis

Abstract: Nonparametric spectral estimation is a widely used technique in many applications ranging from radar and seismic data analysis to electroencephalography (EEG) and speech processing Among the techniques that are used to estimate the spectral representation of a system based on finite observations, multitaper spectral estimation has many important optimality properties, but is not as widely used as it possibly could be We give a brief overview of the standard nonparametric spectral estimation theory and the multitaper spectral estimation, and give two examples from EEG analyses of anesthesia and sleep

On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data

Abstract: This paper is concerned about sparse, continuous frequency estimation in line spectral estimation, and focused on developing gridless sparse methods which overcome grid mismatches and correspond to limiting scenarios of existing grid-based approaches, e.g., $\ell_1$ optimization and SPICE, with an infinitely dense grid. We generalize AST (atomic-norm soft thresholding) to the case of nonconsecutively sampled data (incomplete data) inspired by recent atomic norm based techniques. We present a gridless version of SPICE (gridless SPICE, or GLS), which is applicable to both complete and incomplete data without the knowledge of noise level. We further prove the equivalence between GLS and atomic norm-based techniques under different assumptions of noise. Moreover, we extend GLS to a systematic framework consisting of model order selection and robust frequency estimation, and present feasible algorithms for AST and GLS. Numerical simulations are provided to validate our theoretical analysis and demonstrate performance of our methods compared to existing ones.

STFT phase reconstruction in voiced speech for an improved single-channel speech enhancement

Abstract: The enhancement of speech which is corrupted by noise is commonly performed in the short-time discrete Fourier transform domain. In case only a single microphone signal is available, typically only the spectral amplitude is modified. However, it has recently been shown that an improved spectral phase can as well be utilized for speech enhancement, e.g., for phase-sensitive amplitude estimation. In this paper, we therefore present a method to reconstruct the spectral phase of voiced speech from only the fundamental frequency and the noisy observation. The importance of the spectral phase is highlighted and we elaborate on the reason why noise reduction can be achieved by modifications of the spectral phase. We show that, when the noisy phase is enhanced using the proposed phase reconstruction, instrumental measures predict an increase of speech quality over a range of signal to noise ratios, even without explicit amplitude enhancement.

Spectral estimation—What is new? What is next?

Abstract: Spectral estimation, and corresponding time-frequency representation for nonstationary signals, is a cornerstone in geophysical signal processing and interpretation. The last 10-15 years have seen the development of many new high-resolution decompositions that are often fundamentally different from Fourier and wavelet transforms. These conventional techniques, like the short-time Fourier transform and the continuous wavelet transform, show some limitations in terms of resolution (localization) due to the trade-off between time and frequency localizations and smearing due to the finite size of the time series of their template. Well-known techniques, like autoregressive methods and basis pursuit, and recently developed techniques, such as empirical mode decomposition and the synchrosqueezing transform, can achieve higher time-frequency localization due to reduced spectral smearing and leakage. We first review the theory of various established and novel techniques, pointing out their assumptions, adaptability, and expected time-frequency localization. We illustrate their performances on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event, and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method. Finally, their outcomes are discussed and possible avenues for improvements are proposed.

Resolving multipath interference in time-of-flight imaging via modulation frequency diversity and sparse regularization

Abstract: Time-of-flight (ToF) cameras calculate depth maps by reconstructing phase shifts of amplitude-modulated signals. For broad illumination of transparent objects, reflections from multiple scene points can illuminate a given pixel, giving rise to an erroneous depth map. We report here a sparsity-regularized solution that separates K interfering components using multiple modulation frequency measurements. The method maps ToF imaging to the general framework of spectral estimation theory and has applications in improving depth profiles and exploiting multiple scattering.

Sparse Discrete Fractional Fourier Transform and Its Applications

Abstract: The discrete fractional Fourier transform is a powerful signal processing tool with broad applications for nonstationary signals. In this paper, we propose a sparse discrete fractional Fourier transform (SDFrFT) algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional Fourier domain. The proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise. In addition, we apply the SDFrFT to the synchronization of high dynamic direct-sequence spread-spectrum signals. Furthermore, a sparse fractional cross ambiguity function (SFrCAF) is developed, and the application of SFrCAF to a passive coherent location system is presented. The experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision.

MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution

Abstract: This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho.

Interpolated-DFT-Based Fast and Accurate Frequency Estimation for the Control of Power

Abstract: The energy produced by renewable energy systems must fulfill quality requirements as defined in the respective standards and directives. Improvement of the quality could be achieved through a more accurate estimation of the frequency of the grid's signal that is used to control an inverter. This paper presents an overview of a method for spectrum interpolation and frequency estimation, and a generalized method for very accurate frequency grid estimation using the fast Fourier transform procedure coupled with maximum decay sidelobe windows. An important feature of this algorithm is the elimination of the impact associated with the conjugate's component on the estimation's outcome (i.e, the possibility of designating the frequency even if the signal's measurement time is on the order of 2.5 periods), and the implementation of the algorithm is straightforward. The results of the simulation show that the algorithm could be successfully used for a fast and accurate estimation of the grid signal frequency. The systematic frequency estimation error is approximately 5·10 -11 Hz for a 5-ms measurement window. The algorithm could be used not only for a single sinusoidal signal, but also for a multifrequency signal. This is assuming that the appropriate spectrum leakage reduction (by a time window) will be performed.

A Novel Method for Parameter Estimation of Space Moving Targets

Abstract: This letter proposes a novel parameter estimation method based on keystone transform (KT) and Radon-Fourier transform (RFT) for space moving targets with high-speed maneuvering performance. In this method, second-order KT is used to correct the range curvature and part of the range walk for all targets simultaneously. Then, fractional Fourier transform is employed to estimate the targets' radial acceleration, followed by the quadric phase term compensation. Finally, RFT and Clean technique are carried out to correct the residual range walk, and the initial range and radial velocity of moving targets are further obtained. The advantage of the proposed method is that it can overcome the limitation of Doppler frequency ambiguity and correct range curvature for all targets in one processing step, which simplifies the operation procedure. Simulation results are presented to demonstrate the validity of the proposed method.

A New Family of High-Resolution Multivariate Spectral Estimators

Abstract: In this paper, we extend the Beta divergence family to multivariate power spectral densities. Similarly to the scalar case, we show that it smoothly connects the multivariate Kullback-Leibler divergence with the multivariate Itakura-Saito distance. We successively study a spectrum approximation problem, based on the Beta divergence family, which is related to a multivariate extension of the THREE spectral estimation technique. It is then possible to characterize a family of solutions to the problem. An upper bound on the complexity of these solutions will also be provided. Finally, we will show that the most suitable solution of this family depends on the specific features required from the estimation problem.

System and method for estimating high time-frequency resolution eeg spectrograms to monitor patient state

Abstract: A system and method for monitoring a patient includes a sensor configured to acquire physiological data from a patient and a processor configured to receive the physiological data from the at least one sensor. The processor is also configured to apply a spectral estimation framework that utilizes structured time-frequency representations defined by imposing, to the physiological data, a prior distributions on a time-frequency plane that enforces spectral estimates that are smooth in time and sparse in a frequency domain. The processor is further configured to perform an iteratively re-weighted least squares algorithm to perform yield a denoised time-varying spectral decomposition of the physiological data and generate a report indicating a physiological state of the patient.

Resolving Multi-path Interference in Time-of-Flight Imaging via Modulation Frequency Diversity and Sparse Regularization

Abstract: Time-of-flight (ToF) cameras calculate depth maps by reconstructing phase shifts of amplitude-modulated signals. For broad illumination or transparent objects, reflections from multiple scene points can illuminate a given pixel, giving rise to an erroneous depth map. We report here a sparsity regularized solution that separates K-interfering components using multiple modulation frequency measurements. The method maps ToF imaging to the general framework of spectral estimation theory and has applications in improving depth profiles and exploiting multiple scattering.

A Sinusoidal Frequency Modulation Fourier Transform for Radar-Based Vehicle Vibration Estimation

Abstract: The intricate vibration of a working vehicle provides an important signature to the vehicle type. Small vibrations introduce phase modulation in radar echoes, which is referred to as micro-Doppler (m-D) phenomenon and can be modeled as sinusoidal frequency-modulated (SFM) signal. Such phase modulation induced by vibrations consists of multiple frequency components; moreover, the modulation is usually rather weak. Present parametric estimators are difficult to estimate so many parameters of every frequency component, while nonparametric approaches suffer from low precision. This paper considers the analysis of SFM signal with weak and multiple frequency components modulation on phase term. We first define the SFM signal space to bridge a gap between the SFM signal analysis and classical signal processing methods. Based on the defined signal space, a novel m-D analysis method, i.e., the sinusoidal frequency modulation Fourier transform (SFMFT), is presented. With the operations acting directly on the phase term of SFM signal, SFMFT gives the frequency spectrum of vibration traces. Unlike the existing methods, which apply a sliding short-time window to perform an instantaneous approximation, the proposed method makes use of the global data, which can provide a longer integral period gain, and consequently improves the estimation performance significantly. Simulation results indicate that the proposed method outperforms the existing methods in the spectrum accuracy, the range of estimable vibration amplitude/frequency, and the computation complexity.

Analysis of Power Spectrum Estimation Using Welch Method for Various Window Techniques

Abstract: In this paper, Power Spectral Estimation (PSE) scheme for variable data length using Rectangular, Blackman, Hanning, Bartlett and Hamming window with Welch Method. Welch method includes the periodogram having advantage of possible implementation using the Fast Fourier Transform (FFT). This method gives good resolution when optimal selection of data length samples is done. The PSE based on both Rectangular as well as Hamming window has been designed and simulated using MATLAB. It can be observed that the Rectangular and Hamming give better results than other windows like Bartlett, Hanning and Blackman window.

Frequency Estimation of Distorted and Noisy Signals in Power Systems by FFT-Based Approach

Abstract: This paper focuses on the accurate frequency estimation of power signals corrupted by a stationary white noise. The noneven item interpolation FFT based on the triangular self-convolution window is described. A simple analytical expression for the variance of noise contribution on the frequency estimation is derived, which shows the variances of frequency estimation are proportional to the energy of the adopted window. Based on the proposed method, the noise level of the measurement channel can be estimated, and optimal parameters (e.g., sampling frequency and window length) of the interpolation FFT algorithm that minimize the variances of frequency estimation can thus be determined. The application in a power quality analyzer verified the usefulness of the proposed method.

Super-Resolution Compressed Sensing: An Iterative Reweighted Algorithm for Joint Parameter Learning and Sparse Signal Recovery

Abstract: In many practical applications such as direction-of- arrival (DOA) estimation and line spectral estimation, the sparsifying dictionary is usually characterized by a set of unknown parameters in a continuous domain. To apply the conventional compressed sensing to such applications, the continuous parameter space has to be discretized to a finite set of grid points. Discretization, however, incurs errors and leads to deteriorated recovery performance. To address this issue, we propose an iterative reweighted method which jointly estimates the unknown parameters and the sparse signals. Specifically, the proposed algorithm is developed by iteratively decreasing a surrogate function majorizing a given objective function, which results in a gradual and interweaved iterative process to refine the unknown parameters and the sparse signal. Numerical results show that the algorithm provides superior performance in resolving closely-spaced frequency components.

A New Frequency Estimation Method for Equally and Unequally Spaced Data

Abstract: Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples The method relies on Kronecker's theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix This matrix is generated by sequences approximating the underlying sum of complex exponentials Unequally spaced sampling is accounted for through a proper choice of interpolation matrices The resulting optimization problem is then cast in a form that is suitable for using the alternating direction method of multipliers (ADMM) The method can easily include either a nuclear norm or a finite rank constraint for limiting the number of complex exponentials The usage of a finite rank constraint makes, in contrast to the nuclear norm constraint, the method heuristic in the sense that the problem is non-convex and convergence to a global minimum can not be guaranteed However, we provide a large set of numerical experiments that indicate that usage of the finite rank constraint nevertheless makes the method converge to minima close to the global minimum for reasonably high signal to noise ratios, hence essentially yielding maximum-likelihood parameter estimates Moreover, the method does not seem to be particularly sensitive to initialization and performs substantially better than standard subspace-based methods

Rational approximations of spectral densities based on the Alpha divergence

Abstract: We approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by selecting the spectral density having minimum “distance” from under the constraint corresponding to imposing the given second-order statistics. We analyze the structure of the optimal solutions as the minimized “distance” varies in the Alpha divergence family. We show that the corresponding approximation problem leads to a family of rational solutions. Secondly, such a family contains the solution which generalizes the Kullback–Leibler solution proposed by Georgiou and Lindquist in 2003. Finally, numerical simulations suggest that this family contains solutions close to the non-rational solution given by the principle of minimum discrimination information.

Fast missing-data IAA with application to notched spectrum SAR

Abstract: Recently, the spectral estimation method known as the iterative adaptive approach (IAA) has been shown to provide higher resolution and lower sidelobes than comparable spectral estimation methods. The computational complexity is higher than methods such as the periodogram (matched filter method). Fast algorithms have been developed that considerably reduce the computational complexity of IAA by using Toeplitz and Vandermonde structures. For the missing-data case, several of these structures are lost, and existing fast algorithms are only efficient when the number of available samples is small. In this work, we consider the case in which the number of missing samples is small. This allows us to use low-rank completion to transform the problem to the structured problem. We compare the computational speed of the algorithm with the state of the art and demonstrate the utility in a frequency-notched synthetic aperture radar imaging problem.

Spectral Estimation of Covolatility from Noisy Observations Using Local Weights

Abstract: We propose localized spectral estimators for the quadratic covariation and the spot covolatility of diffusion processes, which are observed discretely with additive observation noise. The appropriate estimation for time-varying volatilities is based on an asymptotic equivalence of the underlying statistical model to a white-noise model with correlation and volatility processes being constant over small time intervals. The asymptotic equivalence of the continuous-time and discrete-time experiments is proved by a construction with linear interpolation in one direction and local means for the other. The new estimator outperforms earlier non-parametric methods in the literature for the considered model. We investigate its finite sample size characteristics in simulations and draw a comparison between various proposed methods.

Time-series Analysis Methods

Abstract: This chapter presents methodologies that examine data series in terms of their frequency content. Time-series analysis is aimed at separating deterministic periodic oscillations in the data from random and aperiodic fluctuations associated with unresolved background noise or with instrument error. Discrete or continuous random time series have a number of fundamental statistical properties that help characterize the variability of the series and make it easily possible to compare one time series against another. Time series can be viewed as linear combinations of periodic or quasi-periodic components that are superimposed on a long-term trend and random high-frequency noise. The periodic components are assumed to have fixed, or slowly varying amplitudes and phases over the length of the record. The trends might include a slow drift in the sensor characteristics or a long-term component of variability that cannot be resolved by the data series. Most oceanographic time or space series, whether they were collected in analog or digital form, are eventually converted to digital data that may then be expressed as series expansions. These expansions are then used to compute the Fourier transform (or periodogram) of the data series.

Polynomial Implementation of the Taylor–Fourier Transform for Harmonic Analysis

Abstract: Recently, the Taylor-Fourier transform (TFT) was proposed to analyze the spectrum of signals with oscillating harmonics. The coefficients of this linear transformation were obtained through the calculation of the pseudoinverse matrix, which provides the classical solution to the normal equations of the least-squares (LS) approximation. This paper presents a filtering design technique that obtains the coefficients of the filters at each harmonic by imposing the maximally flat conditions to the polynomials defining their frequency responses. This condition can be used to solve the LS problem at each particular harmonic frequency, without the need of obtaining the whole set, as in the classical pseudoinverse solution. In addition, the filter passband central frequency can follow the fluctuations of the fundamental frequency. Besides, the method offers a reduction of the computational burden of the pseudoinverse solution. An implementation of the proposed estimator as an adaptive algorithm using its own instantaneous frequency estimate to relocate its bands is shown, and several tests are used to compare its performance with that of the ordinary TFT.

Power System Frequency Estimation by Reduction of Noise Using Three Digital Filters

Abstract: Accurate estimation of power system frequency is essential for monitoring and operation of the smart grid. Traditionally, this has been done using discrete Fourier transform (DFT) coefficients of the positive fundamental frequency. Such DFT-based frequency estimation has been used successfully in phasor measurement units and frequency disturbance recorders in North America. Frequency errors in DFT-based algorithms for single-phase signals arise mainly due to noise and the leakage effect of the negative fundamental frequency. In this paper, a DFT-based frequency estimation algorithm is proposed to introduce three digital filters for reduction of estimate error due to noise and the leakage effect. This algorithm calculates the frequency estimate from the magnitude ratios of DFT coefficients to avoid the leakage effect. It compensates the estimate error, which is induced from the DFT magnitude ratios of three filtered outputs. The enhancement of signal-to-noise ratios is verified through simulations.

Time-frequency-based instantaneous frequency estimation of sparse signals from incomplete set of samples

Abstract: The estimation of time-varying instantaneous frequency (IF) for monocomponent signals with an incomplete set of samples is considered. A suitable time-frequency distribution (TFD) reduces the non-stationary signal into a local sinusoid over the lag variable prior to the Fourier transform. Accordingly, the observed spectral content becomes sparse and suitable for compressive sensing reconstruction in the case of missing samples. Although the local bilinear or higher order auto-correlation functions will increase the number of the missing samples, the analysis shows that an accurate IF estimation can be achieved even if we deal with only few samples, as long as the auto-correlation function is properly chosen to coincide with the signals phase non-linearity. In addition, by employing the sparse signal reconstruction algorithms, ideal time-frequency representations are obtained. The presented theory is illustrated on several examples dealing with different auto-correlation functions and corresponding TFDs.

Multi-eigenvalue communication via the nonlinear Fourier transform

Abstract: Information transmission using only the discrete component of the nonlinear Fourier transform is studied and multi-eigenvalue signal sets are presented that achieve spectral efficiencies greater than 3 bits/s/Hz.

A sparse Bayesian learning algorithm with dictionary parameter estimation

Abstract: This paper concerns sparse decomposition of a noisy signal into atoms which are specified by unknown continuous-valued parameters. An example could be estimation of the model order, frequencies and amplitudes of a superposition of complex sinusoids. The common approach is to reduce the continuous parameter space to a fixed grid of points, thus restricting the solution space. In this work, we avoid discretization by working directly with the signal model containing parameterized atoms. Inspired by the “fast inference scheme” by Tipping and Faul we develop a novel sparse Bayesian learning (SBL) algorithm, which estimates the atom parameters along with the model order and weighting coefficients. Numerical experiments for spectral estimation with closely-spaced frequency components, show that the proposed SBL algorithm outperforms state-of-the-art subspace and compressed sensing methods.