Showing papers on "Spectral density estimation published in 2012"

Super-Resolution Power and Robustness of Compressive Sensing for Spectral Estimation With Application to Spaceborne Tomographic SAR

Abstract: We address the problem of resolving two closely spaced complex-valued points from N irregular Fourier do- main samples. Although this is a generic super-resolution (SR) problem, our target application is SAR tomography (TomoSAR), where typically the number of acquisitions is N = 10 - 100 and SNR = 0-10 dB. As the TomoSAR algorithm, we introduce "Scale-down by LI norm Minimization, Model selection, and Estimation Reconstruction" (SL1MMER), which is a spectral estimation algorithm based on compressive sensing, model order selection, and final maximum likelihood parameter estimation. We investigate the limits of SLIMMER concerning the following questions. How accurately can the positions of two closely spaced scatterers be estimated? What is the closest distance of two scat- terers such that they can be separated with a detection rate of 50% by assuming a uniformly distributed phase difference? How many acquisitions N are required for a robust estimation (i.e., for separating two scatterers spaced by one Rayleigh resolution unit with a probability of 90%)? For all of these questions, we provide numerical results, simulations, and analytical approxima- tions. Although we take TomoSAR as the preferred application, the SLIMMER algorithm and our results on SR are generally applicable to sparse spectral estimation, including SR SAR focus- ing of point-like objects. Our results are approximately applicable to nonlinear least-squares estimation, and hence, although it is derived experimentally, they can be considered as a fundamental bound for SR of spectral estimators. We show that SR factors are in the range of 1.5-25 for the aforementioned parameter ranges of N and SNR.

A windowed graph Fourier transform

Abstract: The prevalence of signals on weighted graphs is increasing; however, because of the irregular structure of weighted graphs, classical signal processing techniques cannot be directly applied to signals on graphs. In this paper, we define generalized translation and modulation operators for signals on graphs, and use these operators to adapt the classical windowed Fourier transform to the graph setting, enabling vertex-frequency analysis. When we apply this transform to a signal with frequency components that vary along a path graph, the resulting spectrogram matches our intuition from classical discrete-time signal processing. Yet, our construction is fully generalized and can be applied to analyze signals on any undirected, connected, weighted graph.

A unified approach to sparse signal processing

Abstract: A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, component analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing, and rate of innovation. The redundancy introduced by channel coding in finite and real Galois fields is then related to over-sampling with similar reconstruction algorithms. The error locator polynomial (ELP) and iterative methods are shown to work quite effectively for both sampling and coding applications. The methods of Prony, Pisarenko, and MUltiple SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with sparse frequency domain representations. Specifically, the relations of the approach of Prony to an annihilating filter in rate of innovation and ELP in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony method under noisy environments. The iterative methods developed for sampling and coding applications are shown to be powerful tools in spectral estimation. Such narrowband spectral estimation is then related to multi-source location and direction of arrival estimation in array processing. Sparsity in unobservable source signals is also shown to facilitate source separation in sparse component analysis; the algorithms developed in this area such as linear programming and matching pursuit are also widely used in compressed sensing. Finally, the multipath channel estimation problem is shown to have a sparse formulation; algorithms similar to sampling and coding are used to estimate typical multicarrier communication channels.

Wavelet-based compressed sensing for SAR tomography of forested areas

Abstract: SAR tomography is a thriving three-dimensional imaging modality that is commonly tackled by spectral estimation techniques. As a matter of fact, the backscattered power along the vertical direction can be readily obtained by computing the Fourier spectrum of a stack of multi-baseline measurements. Alternatively, recent groundbreaking work has addressed the tomographic problem from a parametric viewpoint, thus estimating effective scattering centers by means of covariance matching techniques. In this paper, we introduce a compressed sensing based covariance matching approach that allows us to retrieve the complete vertical structure of forested areas. For this purpose, we employ sparse representations in the wavelet domain and propose suitable pre-filtering techniques. Finally, we validate this approach by using fully polarimetric L-band data acquired by the E-SAR sensor of DLR.

Reconciling Compressive Sampling Systems for Spectrally Sparse Continuous-Time Signals

Abstract: The random demodulator (RD) and the modulated wideband converter (MWC) are two recently proposed compressed sensing (CS) techniques for the acquisition of continuous-time spectrally sparse signals. They extend the standard CS paradigm from sampling discrete, finite dimensional signals to sampling continuous and possibly infinite dimensional ones, and thus establish the ability to capture these signals at sub-Nyquist sampling rates. The RD and the MWC have remarkably similar structures (similar block diagrams), but their reconstruction algorithms and signal models strongly differ. To date, few results exist that compare these systems, and owing to the potential impacts they could have on spectral estimation in applications like electromagnetic scanning and cognitive radio, we more fully investigate their relationship in this paper. We show that the RD and the MWC are both based on the general concept of random filtering, but employ significantly different sampling functions. We also investigate system sensitivities (or robustness) to sparse signal model assumptions. Last, we show that “block convolution” is a fundamental aspect of the MWC, allowing it to successfully sample and reconstruct block-sparse (multiband) signals. Based on this concept, we propose a new acquisition system for continuous-time signals whose amplitudes are block sparse. The paper includes detailed time and frequency domain analyses of the RD and the MWC that differ, sometimes substantially, from published results.

A Maximum Entropy Enhancement for a Family of High-Resolution Spectral Estimators

Abstract: Structured covariances occurring in spectral analysis, filtering and identification need to be estimated from a finite observation record. The corresponding sample covariance usually fails to possess the required structure. This is the case, for instance, in the Byrnes-Georgiou-Lindquist THREE-like tunable, high-resolution spectral estimators. There, the output covariance Σ of a linear filter is needed to initialize the spectral estimation technique. The sample covariance estimate Σ, however, is usually not compatible with the filter. In this paper, we present a new, systematic way to overcome this difficulty. The new estimate Σο is obtained by solving an ancillary problem with an entropic-type criterion. Extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances.

Constrained least-squares spectral analysis: Application to seismic data

Abstract: An inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data. The Fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window. The method resulted in spectra that have reduced window smearing for a given window length relative to the discrete Fourier transform irrespective of window shape, and a time-frequency analysis with a combination of time and frequency resolution that is superior to the short time Fourier transform and the continuous wavelet transform. The reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window. The degree of resolution improvement relative to the short time Fourier transform increases as window length decreases. As compared with the continu...

AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series

Abstract: We propose a method for analyzing possibly nonstationary time series by adaptively dividing the time series into an unknown but finite number of segments and estimating the corresponding local spectra by smoothing splines. The model is formulated in a Bayesian framework, and the estimation relies on reversible jump Markov chain Monte Carlo (RJMCMC) methods. For a given segmentation of the time series, the likelihood function is approximated via a product of local Whittle likelihoods. Thus, no parametric assumption is made about the process underlying the time series. The number and lengths of the segments are assumed unknown and may change from one MCMC iteration to another. The frequentist properties of the method are investigated by simulation, and applications to electroencephalogram and the El Nino Southern Oscillation phenomenon are described in detail.

The Estimation of Frequency

Abstract: Numerical methods have been used for fitting sinusoids to data since the middle of the 18th century. Since the discovery of the Fast Fourier Transform by Cooley and Tukey in 1965, the techniques for estimating frequency have become computationally feasible. This review examines various techniques for estimating the frequency or frequencies of sinusoids in additive noise. The techniques fall into two categories – those based on Fourier, or frequency-domain methods, and those derived from a consideration of a small number of sample autocovariances. The Fourier techniques invariably have asymptotic variances of order T −3 , where T is the sample size, and are particularly useful when T is large and the signal is noisy, whereas the other techniques are usually statistically inefficient, with asymptotic variances of order T −1 , and are often biased, but because of their computational efficiency, can be useful when T is small and the signal is relatively noise free.

Compressive sensing off the grid

Abstract: We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressive sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. We propose an atomic norm minimization approach to exactly recover the unobserved samples, which is then followed by any linear prediction method to identify the frequency components. We reformulate the atomic norm minimization as an exact semidefinite program. By constructing a dual certificate polynomial using random kernels, we show that roughly s log s log n random samples are sufficient to guarantee the exact frequency estimation with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method. Our approach avoids the basis mismatch issue arising from discretization by working directly on the continuous parameter space. Potential impact on both compressive sensing and line spectral estimation, in particular implications in sub-Nyquist sampling and superresolution, are discussed.

A Multi-Frame Approach to the Frequency-Domain Single-Channel Noise Reduction Problem

Abstract: This paper focuses on the class of single-channel noise reduction methods that are performed in the frequency domain via the short-time Fourier transform (STFT). The simplicity and relative effectiveness of this class of approaches make them the dominant choice in practical systems. Over the past years, many popular algorithms have been proposed. These algorithms, no matter how they are developed, have one feature in common: the solution is eventually formulated as a gain function applied to the STFT of the noisy signal only in the current frame, implying that the interframe correlation is ignored. This assumption is not accurate for speech enhancement since speech is a highly self-correlated signal. In this paper, by taking the interframe correlation into account, a new linear model for speech spectral estimation and some optimal filters are proposed. They include the multi-frame Wiener and minimum variance distortionless response (MVDR) filters. With these filters, both the narrowband and fullband signal-to-noise ratios (SNRs) can be improved. Furthermore, with the MVDR filter, speech distortion at the output can be zero. Simulations present promising results in support of the claimed merits obtained by theoretical analysis.

Nonparametric Missing Sample Spectral Analysis and Its Applications to Interrupted SAR

Abstract: We consider nonparametric adaptive spectral analysis of complex-valued data sequences with missing samples occurring in arbitrary patterns. We first present two high-resolution missing-data spectral estimation algorithms: the Iterative Adaptive Approach (IAA) and the Sparse Learning via Iterative Minimization (SLIM) method. Both algorithms can significantly improve the spectral estimation performance, including enhanced resolution and reduced sidelobe levels. Moreover, we consider fast implementations of these algorithms using the Conjugate Gradient (CG) technique and the Gohberg-Semencul-type (GS) formula. Our proposed implementations fully exploit the structure of the steering matrices and maximize the usage of the fast Fourier transform (FFT), resulting in much lower computational complexities as well as much reduced memory requirements. The effectiveness of the adaptive spectral estimation algorithms is demonstrated via several numerical examples including both 1-D spectral estimation and 2-D interrupted synthetic aperture radar (SAR) imaging examples.

SAR-Based Vibration Estimation Using the Discrete Fractional Fourier Transform

Abstract: A vibration estimation method for synthetic aperture radar (SAR) is presented based on a novel application of the discrete fractional Fourier transform (DFRFT). Small vibrations of ground targets introduce phase modulation in the SAR returned signals. With standard preprocessing of the returned signals, followed by the application of the DFRFT, the time-varying accelerations, frequencies, and displacements associated with vibrating objects can be extracted by successively estimating the quasi-instantaneous chirp rate in the phase-modulated signal in each subaperture. The performance of the proposed method is investigated quantitatively, and the measurable vibration frequencies and displacements are determined. Simulation results show that the proposed method can successfully estimate a two-component vibration at practical signal-to-noise levels. Two airborne experiments were also conducted using the Lynx SAR system in conjunction with vibrating ground test targets. The experiments demonstrated the correct estimation of a 1-Hz vibration with an amplitude of 1.5 cm and a 5-Hz vibration with an amplitude of 1.5 mm.

Micromotion Parameter Estimation of Free Rigid Targets Based on Radar Micro-Doppler

Abstract: In this paper, an estimation method of micromotion parameters for free rigid targets using micro-Doppler (mD) features is investigated. These parameters include spin rate, precession rate, nutation angle, and inertia ratio. They represent the microdynamic characteristics and intrinsic properties of targets. The time variation of mD frequency is found complicated yet valuable to estimate the micromotion parameters. From the viewpoint of the spectra of mixed mD time-frequency (TF) data sequences, the theoretical analysis and mathematical derivation are conducted in detail according to the scatterer distribution of rigid bodies. We then present an approach to realize the micromotion parameter estimation from radar mD echoes. It mainly consists of TF transform, TF image processing, mixed mD TF data sequence formation, and spectral estimation. Simulation experiments and result discussion are carried out to demonstrate the effectiveness of the proposed estimation method.

Multiple Grazing Angle Sea Clutter Modeling

Abstract: Radar clutter in a non-standard atmosphere usually is modeled based on a single grazing angle at each range. Instead, the angular distribution of incident power can be used to obtain a more accurate model of the clutter. Angular spectral estimation provides the grazing angle distribution of propagating power. However, a large gradient in the refractivity profile, e.g., an evaporation duct, distorts plane wave propagation which in turn violates assumptions of plane wave spectral estimation. Ray tracing is used in these situations, but has its own limitations (e.g., shadow zones). We suggest using curved wave spectral estimation (CWS) that yields reliable results for any refractivity profile, in contrast to plane wave spectral estimation. CWS is used to derive multiple grazing angle clutter, a model for ocean surface clutter in the microwave region that depends on all incident angles at each range and their corresponding powers.

Spectral estimation for locally stationary time series with missing observations

Abstract: Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled time series, or require modifications to cope with irregular or `gappy' data. Additionally, many techniques also assume that the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally stationary wavelet process in the sense introduced by Nason et al. (J. R. Stat. Soc. B 62(2):271---292, 2000) and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the centre of our construction are second generation wavelets built by means of the lifting scheme (Sweldens, Wavelet Applications in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68---79, 1995), designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate our method with real data and simulated examples.

A Novel Time–Frequency Transform Based Spectral Energy Function for Fault Detection During Power Swing

Abstract: Fault during a power swing is a challenging task for the distance relay functioning This article presents a spectral energy function for fault detection during a power swing using a novel time frequency transform known as the S-transform, a variable windowed short-time Fourier transform, which combines the elements of short-time Fourier transform and wavelet transform Initially, the current signal is preprocessed using S-transform to generate the S-matrix and corresponding S-contours (time–frequency contours) The spectral energy content of the S-counters is used to register symmetrical and unsymmetrical faults during a power swing and, based on a set threshold on the spectral energy, the relay blocks during a power swing and issue of the tripping signal during fault The proposed technique is tested for different fault conditions during a power swing with possible variations in operating parameters, including the ability to identify the faults with a response time of 125 cycles from the fault

Efficient phase retrieval of sparse signals

Abstract: We consider the problem of one dimensional (1D) phase retrieval, namely, recovery of a 1D signal from the magnitude of its Fourier transform. This problem is ill-posed since the Fourier phase information is lost. Therefore, prior information on the signal is needed in order to recover it. In this work we consider the case in which the prior information on the signal is that it is sparse, i.e., it consists of a small number of nonzero elements. We propose a fast local search method for recovering a sparse 1D signal from measurements of its Fourier transform magnitude. 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 the proposed algorithm is fast and more accurate than existing techniques.

A Novel Strategy for the Diagnosis of Arbitrary Geometries Large Arrays

Abstract: A general solution strategy for detecting faulty elements in phased arrays of arbitrary geometries is suggested. The proposed deterministic approach assumes as input data the amplitude and phase of near-field distributions and allows to determine the positions of the faulty elements. In particular, the method is founded on the well known Multiple Signal Classification (MUSIC) method, i.e., a spectral estimation technique. The proposed algorithm is also compared with a recently published method by the same authors, against experimental and numerical data. The results fully confirm the usefulness of the proposed technique, highlighting the advantages and the disadvantages of both methods.

On robust phase retrieval for sparse signals

Abstract: Recovering signals from their Fourier transform magnitudes is a classical problem referred to as phase retrieval and has been around for decades. In general, the Fourier transform magnitudes do not carry enough information to uniquely identify the signal and therefore additional prior information is required. In this paper, we shall assume that the underlying signal is sparse, which is true in many applications such as X-ray crystallography, astronomical imaging, etc. Recently, several techniques involving semidefinite relaxations have been proposed for this problem, however very little analysis has been performed. The phase retrieval problem can be decomposed into two tasks — (i) identifying the support of the sparse signal from the Fourier transform magnitudes, and (ii) recovering the signal using the support information. In earlier work [13], we developed algorithms for (i) which provably recovered the support for sparsities upto O(n1/3−∊). Simulations suggest that support recovery is possible upto sparsity O(n1/2−∊). In this paper, we focus on (ii) and propose an algorithm based on semidefinite relaxation, which provably recovers the signal from its Fourier transform magnitude and support knowledge with high probability if the support size is O(n1/2−∊).

Uncertainty Bounds for Spectral Estimation

Abstract: The purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second-order statistics. The family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the "true" power spectrum. Our aim is to quantify the size of this uncertainty set using suitable notions of distance, and in particular, to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the "true" power spectrum. Since the uncertainty set may contain power spectra with lines and discontinuities, it is natural to quantify distances in the weak topology---the topology defined by continuity of moments. We provide examples of such weakly-continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty. We then consider certain high resolution techniques which utilize filter-banks for pre-processing, and compute worst-case a priori uncertainty bounds solely on the basis of the filter dynamics. This allows the a priori tuning of the filter-banks for improved resolution over selected frequency bands.

Computationally Efficient Time-Recursive IAA-Based Blood Velocity Estimation

Abstract: High-resolution spectral Doppler is an important and powerful noninvasive tool for estimation of velocities in blood vessels using medical ultrasound scanners. Such estimates are typically formed using an averaged periodogram technique, resulting in well-known limitations in the resulting spectral resolution. Recently, we have proposed techniques to instead form high-resolution data-adaptive estimates exploiting measurements along both depth and emission. The resulting estimates gives noticeably superior velocity estimates as compared to the standard technique, but suffers from a high computational complexity, making it interesting to formulate computationally efficient implementations of the estimators. In this work, by exploiting the rich structure of the iterative adaptive approach (IAA) based estimator, we examine how these estimates can be efficiently implemented in a time-recursive manner using both exact and approximate formulations of the method. The resulting algorithms are shown to reduce the necessary computational load with several orders of magnitude without noticeable loss of performance.

Background estimation based on Fourier Transform in the energy-dispersive X-ray fluorescence analysis

Abstract: This paper discusses a new method of background estimation in the energy-dispersive X-ray fluorescence (EDXRF) analysis, which is based on Fourier Transform (in this paper, we call it Fourier Transform background estimation method). Compared with the Sensitive Nonlinear Iterative Peak method, the new method has the feature of FWHM independence. It has been proved that a background can be estimated automatically and accurately by the new method in the synthesized spectrum and the spectra from measurement. Fourier Transform background estimation method can estimate the background accurately in the EDXRF spectrum using an X-ray tube source. Copyright © 2011 John Wiley & Sons, Ltd.

Nonlinear spectral density estimation: thresholding the correlogram

Abstract: Traditional kernel spectral density estimators are linear as a function of the sample autocovariance sequence. The purpose of the present paper is to propose and analyze two new spectral estimation methods that are based on the sample autocovariances in a nonlinear way. The rate of convergence of the new estimators is quantified, and practical issues such as bandwidth and/or threshold choice are addressed. The new estimators are also compared to traditional ones using flat-top lag-windows in a simulation experiment involving sparse time series models.

A new DFT-based frequency estimator for single-tone complex sinusoidal signals

Abstract: Frequency estimation for single-tone complex sinusoidal signals under additive white Gaussian noise is a classical and fundamental problem in many applications, such as communications, radar, sonar and power systems. In this paper, we propose a new algorithm by interpolating discrete Fourier transform (DFT) samples. Different from other existing interpolation methods for frequency estimation, our algorithm is based on a much simpler expression and has mathematically tractable bias expression in closed form, which can potentially assist future bias correction. Simulations confirm that our proposed algorithm outperforms all existing alternatives in the literature with comparable complexity.

Atomic norm denoising with applications to line spectral estimation

Abstract: Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.