Parameter estimation of models with many damped complex exponentials

Parameter estimation techniques for data modelled as a sum of damped complex exponentials are proving to be a successful alternative to Fourier transform methods for spectral estimation

Application of the Non-Hermitian Singular Spectrum Analysis to the exponential retrieval problem

We present a new approach to solve the exponential retrieval problem. We derive a stable technique, based on the singular value decomposition (SVD) of lag-covariance and crosscovariance matrices consisting of covariance coefficients computed for index translated copies of an initial time series. For these matrices a generalized eigenvalue problem is solved. The initial signal is mapped into the basis of the generalized eigenvectors and phase portraits are consequently analyzed. Pattern recognition techniques could be applied to distinguish phase portraits related to the exponentials and noise. Each frequency is evaluated by unwrapping phases of the corresponding portrait, detecting potential wrapping events and estimation of the phase slope. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise

Application of the Non-Hermitian Singular Spectrum Analysis to the Exponential Retrieval Problem

Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold.Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold

Inspiral, merger and ringdown of unequal mass black hole binaries: a multipolar analysis

We study the inspiral, merger and ringdown of unequal mass black hole binaries by analyzing a catalogue of numerical simulations for seven different values of the mass ratio (from q=M2/M1=1 to q=4). We compare numerical and Post-Newtonian results by projecting the waveforms onto spin-weighted spherical harmonics, characterized by angular indices (l,m). We find that the Post-Newtonian equations predict remarkably well the relation between the wave amplitude and the orbital frequency for each (l,m), and that the convergence of the Post-Newtonian series to the numerical results is non-monotonic. To leading order the total energy emitted in the merger phase scales like eta^2 and the spin of the final black hole scales like eta, where eta=q/(1+q)^2 is the symmetric mass ratio. We study the multipolar distribution of the radiation, finding that odd-l multipoles are suppressed in the equal mass limit. Higher multipoles carry a larger fraction of the total energy as q increases. We introduce and compare three different definitions for the ringdown starting time. Applying linear estimation methods (the so-called Prony methods) to the ringdown phase, we find resolution-dependent time variations in the fitted parameters of the final black hole. By cross-correlating information from different multipoles we show that ringdown fits can be used to obtain precise estimates of the mass and spin of the final black hole, which are in remarkable agreement with energy and angular momentum balance calculations.Comment: 51 pages, 28 figures, 16 tables. Many improvements throughout the text in response to the referee report. The calculation of multipolar components in Appendix A now uses slightly different conventions. Matches version in press in PR

Parameter Estimation for Superimposed Weighted Exponentials

The approach of modeling measured signals as superimposed exponentials in white Gaussian noise is popular and effective. However, estimating the parameters of the assumed model is challenging, especially when the data record length is short, the signal strength is low, or the parameters are closely spaced. In this dissertation, we first review the most effective parameter estimation scheme for the superimposed exponential model: maximum likelihood. We then provide a historical review of the linear prediction approach to parameter estimation for the same model. After identifying the improvements made to linear prediction and demonstrating their weaknesses, we introduce a completely tractable and statistically sound modification to linear prediction that we call iterative generalized least squares. It is shown, that our algorithm works to minimize the exact maximum likelihood cost function for the superimposed exponential problem and is therefore, equivalent to the previously developed maximum likelihood approach. However, our algorithm is indeed linear prediction, and thus revives a methodology previously categorized as inferior to maximum likelihood. With our modification, the insight provided by linear prediction can be carried to actual applications. We demonstrate this by developing an effective algorithm for deep level transient spectroscopy analysis. The signal of deep level transient spectroscopy is not a straight forward superposition of exponentials. However, with our methodology, an estimator, based on the exact maximum likelihood cost function for the actual signal, is quickly derived. At the end of the dissertation, we verify that our estimator extends the current capabilities of deep level transient spectroscopy analysis

Signal analysis and feature generation for pattern identification of partial discharges in high-voltage equipment

This paper proposes a method for the identification of different partial discharges (PDs) sources through the analysis of a collection of PD signals acquired with a PD measurement system. This method, robust and sensitive enough to cope with noisy data and external interferences, combines the characterization of each signal from the collection, with a clustering procedure, the CLARA algorithm. Several features are proposed for the characterization of the signals, being the wavelet variances, the frequency estimated with the Prony method, and the energy, the most relevant for the performance of the clustering procedure. The result of the unsupervised classification is a set of clusters each containing those signals which are more similar to each other than to those in other clusters. The analysis of the classification results permits both the identification of different PD sources and the discrimination between original PD signals, reflections, noise and external interferences. The methods and graphical tools detailed in this paper have been coded and published as a contributed package of the R environment under a GNU/GPL license

A Comparison of Impedance Eduction Test Rigs with Different Flow Profiles

The experimental characterization of acoustic liners applied for turbofan engines has been in the spotlight of the community for the last few decades. In general, such characterization is done by measurements of the liner acoustic impedance using different techniques in conditions as close as possible to those encountered in turbofan engines. Although a great amount of work has been published related to these techniques, few comparisons between different experimental setups using identical samples are available. The goal of the present study is to provide a comparison between educed acoustic impedances for two nominally identical liner samples in the UFSC Impedance Test Rig and the NASA Langley Research Center Grazing Flow Impedance Tube (GFIT). Due to the geometrical differences between the test rigs, it is possible to consider the effect of different grazing flow profiles on the educed impedance. Impedance measurements between the two facilities show similar results in absence of grazing flow, and different results when the grazing flow is present. Results are presented with both test rigs targeted to two different conditions: (i) same centerline Mach number and; (ii) same average Mach number. Both comparisons suggest a higher acoustic resistance obtained with the UFSC Impedance Test Rig. A comparison using semiempirical predictive models was also conducted. The results suggest that the main source for the observed difference is the grazing flow profile, represented by its boundary layer displacement thicknes