Showing papers on "Spectral density estimation published in 2001"

Noise power spectral density estimation based on optimal smoothing and minimum statistics

Abstract: We describe a method to estimate the power spectral density of nonstationary noise when a noisy speech signal is given. The method can be combined with any speech enhancement algorithm which requires a noise power spectral density estimate. In contrast to other methods, our approach does not use a voice activity detector. Instead it tracks spectral minima in each frequency band without any distinction between speech activity and speech pause. By minimizing a conditional mean square estimation error criterion in each time step we derive the optimal smoothing parameter for recursive smoothing of the power spectral density of the noisy speech signal. Based on the optimally smoothed power spectral density estimate and the analysis of the statistics of spectral minima an unbiased noise estimator is developed. The estimator is well suited for real time implementations. Furthermore, to improve the performance in nonstationary noise we introduce a method to speed up the tracking of the spectral minima. Finally, we evaluate the proposed method in the context of speech enhancement and low bit rate speech coding with various noise types.

Rotation, scale, and translation resilient public watermarking for images

Abstract: A method for detecting a watermark signal in digital image data. The detecting method includes the steps of: computing a log-polar Fourier transform of the image data to obtain a log-polar Fourier spectrum; projecting the log-polar Fourier spectrum down to a lower dimensional space to obtain an extracted signal; comparing the extracted signal to a target watermark signal; and declaring the presence or absence of the target watermark signal in the image data based on the comparison. Also provided is a method for inserting a watermark signal in digital image data to obtain a watermarked image. The inserting method includes the steps of: computing a log-polar Fourier transform of the image data to obtain a log-polar Fourier spectrum; projecting the log-polar Fourier spectrum down to a lower dimensional space to obtain an extracted signal; modifying the extracted signal such that it is similar to a target watermark; performing a one-to-many mapping of the modified signal back to log-polar Fourier transform space to obtain a set of watermarked coefficients; and performing an inverse log-polar Fourier transform on the set of watermarked coefficients to obtain a watermarked image.

Application of empirical mode decomposition to heart rate variability analysis.

Abstract: The analysis of heart rate variability, involving changes in the autonomic modulation conditions, demands specific capabilities not provided by either parametric or non-parametric spectral estimation methods. Moreover, these methods produce time-averaged power estimates over the entire length of the record. Recently, empirical mode decomposition and the associated Hilbert spectra have been proposed for non-linear and non-stationary time series. The application of these techniques to real and simulated short-term heart rate variability data under stationary and non-stationary conditions is presented. The results demonstrate the ability of empirical mode decomposition to isolate the two main components of one chirp series and three signals simulated by the integral pulse frequency modulation model, and consistently to isolate at least four main components localised in the autonomic bands of 14 real signals under controlled breathing manoeuvres. In addition, within the short time-frequency range that is recognised for heart rate variability phenomena, the Hilbert amplitude component ratio and the instantaneous frequency representation are assessed for their suitability and accuracy in time-tracking changes in amplitude and frequency in the presence of non-stationary and non-linear conditions. The frequency tracking error is found to be less than 0.22% for two simulated signals and one chirp series.

Maximum a posteriori sequence estimation using Monte Carlo particle filters

Abstract: We develop methods for performing maximum a posteriori (MAP) sequence estimation in non-linear non-Gaussian dynamic models. The methods rely on a particle cloud representation of the filtering distribution which evolves through time using importance sampling and resampling ideas. MAP sequence estimation is then performed using a classical dynamic programming technique applied to the discretised version of the state space. In contrast with standard approaches to the problem which essentially compare only the trajectories generated directly during the filtering stage, our method efficiently computes the optimal trajectory over all combinations of the filtered states. A particular strength of the method is that MAP sequence estimation is performed sequentially in one single forwards pass through the data without the requirement of an additional backward sweep. An application to estimation of a non-linear time series model and to spectral estimation for time-varying autoregressions is described.

Method for whitening colored noise in a communication system

Abstract: A filter settings generation operation includes sampling colored noise present at the input of a receiver to produce a sampled signal. The sampled signal is spectrally characterized across a frequency band of interest to produce a spectral characterization of the sampled signal. This spectral characterization may not include a signal of interest. The spectral characterization is then modified to produce a modified spectral characterization. Filter settings are then generated based upon the modified spectral characterization. Finally, the input present at the receiver is filtered using the filter settings when the signal of interest is present to whiten colored noise that is present with the signal of interest. In modifying the spectral characterization, pluralities of spectral components of the spectral characterization are independently modified to produce the modified spectral characterization. Modifications to the spectral characterization may be performed in the frequency domain and/or the time domain. Particular modifications include amplifying spectral components, weighting spectral components based upon prior spectral components, and averaging spectral components with prior spectral components.

Cepstral coefficients, covariance lags, and pole-zero models for finite data strings

Abstract: One of the most widely used methods of spectral estimation in signal and speech processing is linear predictive coding (LPC), LPC has some attractive features, which account for its popularity, including the properties that the resulting modeling filter (i) matches a finite window of n+1 covariance lags, (ii) is rational of degree at most n, and (iii) has stable zeros and poles. The only limiting factor of this methodology is that the modeling filter is "all-pole," i.e., an autoregressive (AR) model. In this paper, we present a systematic description of all autoregressive moving-average (ARMA) models of processes that have properties (i)-(iii) in the context of cepstral analysis and homomorphic filtering. We show that each such an ARMA model determines and is completely determined by its finite windows of cepstral coefficients and covariance lags. We show that these nth-order windows form local coordinates for all ARMA models of degree n and that the pole-zero model can be determined from the windows as the unique minimum of a convex objective function. We refine this optimization method by first noting that the maximum entropy design of an LPC filter is obtained by maximizing the zeroth cepstral coefficient, subject to the constraint (i). More generally, we modify this scheme to a more well-posed optimization problem where the covariance data enter as a constraint and the linear weights of the cepstral coefficients are "positive"-in a sense that a certain pseudo-polynomial is positive-rather succinctly generalizing the maximum entropy method. This new problem is a homomorphic filter generalization of the maximum entropy method. providing a procedure for the design of any stable, minimum-phase modeling filter of degree less or equal to n that interpolates the given covariance window. We present an algorithm for realizing these filters in a lattice-ladder form, given the covariance window and the moving average part of the model, While we also show how to determine the moving average part using cepstral smoothing, one can make use of any good a priori estimate for the system zeros to initialize the algorithm. We conclude the paper with an example of this method, incorporating an example from the literature on ARMA modeling.

Spectral estimation via selective harmonic amplification

Abstract: The state-covariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter, and also imposes a generalized interpolation constraint on the power spectrum of the input process. This algebraic property and the relationship between state-covariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis. Several algorithms for spectral estimation are developed with resolution higher than state of the art.

Iterative Fourier transform algorithm for phase-only pulse shaping.

Abstract: We demonstrate the adaptation of an iterative Fourier transform algorithm for the calculation of theoretical spectral phase functions required for pulse shaping applications. The algorithm is used to determine the phase functions necessary for the generation of different temporal intensity profiles. The performance of the algorithm is compared to two exemplary standard approaches. i.e. a Genetic Algorithm and a combination of a Simplex Downhill and a Simulated Annealing algorithm. It is shown that the iterative Fourier transform algorithm converges much faster than both alternative methods.

DFT-based frequency estimators with narrow acquisition range

Abstract: The authors propose new frequency estimators, with narrow frequency acquisition range, which are based on the calculation of the discrete Fourier transform (DFT) of the input signal. The algorithms consist of two steps: coarse and fine search of the periodogram peak. This frequency estimator structure employing either parabolic interpolation, dichotomous search, two-rate spectral estimation or their combination for fine search allows a considerable decrease in computational load with respect to known estimators. The proposed estimators are shown to have accuracy performance close to that of the maximum likelihood (ML) estimator.

Robust Wigner distribution with application to the instantaneous frequency estimation

Abstract: The Wigner distribution (WD) produces highly concentrated time-frequency (TF) representation of nonstationary signals. It may be used as an efficient signal analysis tool, including the cases of frequency modulated signals corrupted with the Gaussian noise. In some applications, a significant amount of impulse noise is present. Then, the WD fails to produce satisfactory results. The robust periodogram has been introduced for spectral estimation of this kind of noisy signals. It can produce good concentration for pure harmonic signals. However, it is not so efficient in the cases of signals with rapidly varying frequency. This is the motivation for introducing the robust WD. It is a reliable TF representation tool for wide class of nonstationary signals corrupted with impulse noise. This distribution produces good accuracy of the instantaneous frequency (IF) estimation. Using the Huber (1981) loss function, a generalization of the WD is presented. It includes both the standard and the robust WD as special cases. This distribution can be used for TF analysis of signals corrupted with a mixture of impulse and Gaussian noise. The presented theory is illustrated on examples, including applications on the IF estimation and time-varying filtering of signals corrupted with a mixture of the Gaussian and impulse noise. The case study analysis of the IF estimators' accuracy, based on the standard and the robust WD forms, is performed. In order to improve the IF estimation, a median filter is applied on the obtained IF estimate.

Windows for ADC dynamic testing via frequency-domain analysis

Abstract: In this paper, the frequency-domain dynamic test of analog-to-digital converters (ADCs) is considered under the assumption of noncoherent sinewave sampling. A procedure is described which is based on the windowed discrete Fourier transform (WDFT), optimized for the achievement of high estimation accuracy. With this aim, the class of windows belonging to the set of discrete prolate spheroidal sequences is adopted for the reduction of the effects of spectral leakage. Practical suggestions are given for a straightforward applicability of derived results and for an efficient estimation of ADC spectral parameters. Finally, experimental results are presented in order to validate the proposed testing approach.

Fast frequency-domain pitch estimation

Abstract: A method for estimating a pitch frequency of an audio signal includes computing a first transform of the signal to a frequency domain over a first time interval (42), and computing a second transform of the signal of the frequency domain over a second time interval (44), which contains the first time interval. A line spectrum of the signal is found, based on the first and second transforms, the spectrum including spectral lines having respective line amplitudes and line frequencies. A utility function (130) that is periodic in the frequencies of the lines in the spectrum is then computed. This function is indicative (158), for each candidate pitch frequency in a given pitch frequency range, of a compatibility of the spectrum with the candidate pitch frequency. The pitch frequency of the speech signal is estimated responsive to the utility function (176, 178).

High-resolution SAR imaging with angular diversity

Abstract: We propose to use the APES (amplitude and phase estimation) approach for the spectral estimation of gapped data and synthetic aperture radar (SAR) imaging with angular diversity. A relaxation-based algorithm, referred to as GAPES (Gapped-data APES), is proposed, which includes estimating the spectrum via APES and filling in the gaps via a least squares (LS) fitting. For SAR imaging with angular diversity data fusion, we perform one-dimensional (1-D) windowed fast Fourier transforms (FFTs) in range, use the GAPES algorithm to interpolate the gaps in the aperture for each range, apply 1-D inverse FFTs (IFFTs) and dewindow in range, and finally apply the two-dimensional (2-D) APES algorithm to the interpolated matrix to obtain the 2-D SAR image. Numerical results are presented to demonstrate the effectiveness of the proposed algorithm.

Digital modulation signal receiver with adaptive channel equalization employing discrete fourier transforms

Abstract: Techniques for calculating the system characteristic of the adaptive filtering used for equalization and echo-suppression in a digital communications receiver, such as one used for receiving over-the-air broadcast digital television signal, are described. In these techniques, the system characteristic of the adaptive filtering is calculated from the discrete Fourier transform of successive portions of the input signal supplied to the adaptive filtering and from the discrete Fourier transform of corresponding portions of the transmitted signal, as estimated in the receiver. Receivers for implementing these techniques in various ways are also disclosed.

Spectral analysis of atmospheric radar signal using higher order spectral estimation technique

Abstract: Higher order spectral estimation techniques have been applied to the backscattered signals received from the troposphere and lower stratosphere by the Gadanki mesosphere-stratosphere-troposphere (MST) radar. These techniques allow identification of signals that have non-Gaussian probability distribution. To understand these processes and their effect on estimation of the atmospheric parameters, power spectrum, and bispectrum analyses have been performed on the signals received in both vertical and off-vertical directions. The results show that the backscattered echoes received in the vertical direction are significantly non-Gaussian, while those received in the off-vertical direction are inferred to have predominant Gaussian component. It is demonstrated that the bispectral analysis technique has capability for estimating the vertical wind component with greater accuracy than that derived from the commonly employed fast Fourier transform based power spectral technique.

Regularized estimation of mixed spectra using a circular Gibbs-Markov model

Abstract: Formulated as a linear inverse problem, spectral estimation is particularly underdetermined when only short data sets are available. Regularization by penalization is an appealing nonparametric approach to solve such ill-posed problems. Following Sacchi et al. (see ibid., vol.46, no.1, p.32-38, 1998), we first address line spectra recovering in this framework. Then, we extend the methodology to situations of increasing difficulty: the case of smooth spectra and the case of mixed spectra, i.e., peaks embedded in smooth spectral contributions. The practical stake of the latter case is very high since it encompasses many problems of target detection and localization from remote sensing. The stress is put on adequate choices of penalty functions: following Sacchi et al., separable functions are retained to retrieve peaks, whereas Gibbs-Markov potential functions are introduced to encode spectral smoothness. Finally, mixed spectra are obtained from the conjunction of contributions, each one bringing its own penalty function. Spectral estimates are defined as minimizers of strictly convex criteria. In the cases of smooth and mixed spectra, we obtain nondifferentable criteria. We adopt a graduated nondifferentiability approach to compute an estimate. The performance of the proposed techniques is tested on the well-known Kay and Marple (1982) example.

Schrödinger Operators with Sparse Potentials: Asymptotics of the Fourier Transform¶of the Spectral Measure

Abstract: We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional Schrodinger operators with sparse potentials. We find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model. We also present an improved version of known results on the spectrum of such operators.

Spectral analysis of biased LDA data

Abstract: The spectral density function of turbulent velocity fluctuations can be estimated from randomly sampled LDA data by first computing a discretized autocorrelation function (using the slotting technique) followed by a Fourier transform of this function. The spectral estimates obtained in this way have a large statistical scatter in the high-frequency range. This work focuses on a spectral estimator with a much reduced statistical scatter (approximately 1 decade), enabling the retrieval of the spectral density function up to higher frequencies. This spectral estimator is based on a modification of the standard slotting technique, known as ‘local scaling’ or ‘local normalization’, in conjunction with a window of variable width. A series of benchmark tests for spectral estimators indicated that this estimator yields good overall results. This paper explores the characteristics of the new spectral estimator with regard to the effects of velocity bias and the presence of uncorrelated noise in the velocity data.

The Nonuniform Discrete Fourier Transform

Abstract: In many applications, when the representation of a discrete-time signal or a system in the frequency domain is of interest, the Discrete-Time Fourier Transform (DTFT) and the z-transform are often used. In the case of a discrete-time signal of finite length, the most widely used frequency-domain representation is the Discrete Fourier Transform (DFT), which is simply composed of samples of the DTFT of the sequence at equally spaced frequency points, or equivalently, samples of its z-transform at equally spaced points on the unit circle. A generalization of the DFT, introduced in this chapter, is the Nonuniform Discrete Fourier Transform (NDFT), which can be used to obtain frequency domain information of a finite-length signal at arbitrarily chosen frequency points. We provide an introduction to the NDFT and discuss its applications in the design of 1-D and 2-D FIR digital filters. We begin by introducing the problem of computing frequency samples of the z-transform of a finite-length sequence. We develop the basics of the NDFT, including its definition, properties and computational aspects. The NDFT is also extended to two dimensions. We propose NDFT-based nonuniform frequency sampling techniques for designing 1-D and 2-D FIR digital filters, and present design examples. The resulting filters are compared with those designed by other existing methods.

A time-varying complex AR speech analysis based on GLS and ELS method.

Abstract: We have already developed three kinds of time-varying complex AR (TV-CAR) parameter estimation algorithms for analytic speech signal, which are based on minimizing mean square error (MMSE), Huber's robust Mestimation and Instrumental Variable (IV) method. This paper presents novel robust TV-CAR model parameter estimation algorithms on the basis of a Generalized Least Square (GLS) and Extended Least Square (ELS) method, in which the equation error is modeled by complex AR model with white Gaussian input to whiten the equation error. The experiments with natural speech corrupted by white Gaussian demonstrate that the proposed methods achieve robust spectral estimation against additive white Gaussian.

The Frequency Domain

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Parametric TFR via windowed exponential frequency modulated atoms

Abstract: We propose a new atom, namely, the dilated and translated windowed exponential frequency modulated functions (FM/sup m/let) for compactly characterizing both the signal's time-invariant and time-varying spectral contents. The superiority of the proposed method to some existing time-frequency distributions (TFDs) is demonstrated using a bat sonar signal.

Imaging from real scattered field data using a linear spectral estimation technique

Abstract: A linear spectral estimation technique is used to image targets of simple geometric shape from bistatic scattered field microwave data Reconstructions of both metallic and dielectric objects are presented The algorithm we use to recover images of the scattering targets allows one to incorporate prior knowledge about the target into the inverse scattering algorithm to obtain image estimates with improved resolution We apply our method to real data sets provided by the Institut Fresnel, Marseille, France, which illustrates the strength, as well as limitations, of linearized inverse scattering schemes

A Bayesian revolution in spectral analysis

Abstract: The discrete Fourier transforms (DFT) is ubiquitous in spectral analysis as a result of the introduction of the Fast Fourier transform by Cooley and Tukey in 1965. In 1987, E. T. Jaynes derived the DFT using Bayesian Probability Theory and provided surprising new insights into its role in spectral analysis. From this new perspective the spectral resolution achievable is directly dependent on the signal to noise ratio and can be orders of magnitude better than that of a conventional Fourier power spectrum or periodogram. This was the starting point for an ongoing Bayesian revolution in spectral analysis which is reviewed in this paper, with examples taken from physics and astronomy. The revolution is based on a viewpoint of Bayesian Inference as extended logic.