Windowed Periodogram Research Articles

Wavelet-based information theory in quantitative assessment of AFM images’ quality

The quantitative assessment of the image quality produced by atomic force microscopy (AFM) is an ongoing and challenging task. In our study, we demonstrate Shannon’s application of information theory for measuring image quality. Specifically, we propose quantifying the loss of image information due to the various distortion processes by exploring the relationship between image information based on the information channel capacity (ICC), spectral image representation, and visual quality. Since the ideal image is unavailable, the power and noise spectrum, the critical input information for the image quality evaluation, must be robustly estimated in the proposed method. The classical, most popular Welch method for spectral estimation uses an average of several windowed periodograms and can produce biased spectrum estimates. Therefore, in our work, we discuss an alternative technique based on the wavelet transform that can be applied to solve this challenging problem, specifically in the case of noisy, uncertain AFM measurements. Finally, we validate the performance of the enhanced ICC-wavelet-based algorithm with noisy measurement AFM data.

A fast robust numerical continuation solver to a two‐dimensional spectral estimation problem

This paper presents a fast algorithm to solve a spectral estimation problem for two-dimensional random fields. The latter is formulated as a convex optimization problem with the Itakura-Saito pseudodistance as the objective function subject to the constraints of moment equations. The structure of the Hessian of the dual objective function is exploited in order to make possible a fast Newton solver. Then the Newton solver is incorporated to a predictor-corrector numerical continuation method which is able to produce a parametrized family of solutions to the moment equations. Two sets of numerical simulations are performed to test the algorithm and spectral estimator. The simulations on the frequency estimation problem show that our spectral estimator outperforms the classical windowed periodograms in the case of two hidden frequencies and has a higher resolution. The other set of simulations on system identification indicates that the numerical continuation method is more robust than Newton's method alone in ill-conditioned instances.

Uncertainties of a window mixing estimator for the noise power spectrum of digital signals

The statistical characteristics of several estimators of the noise power spectrum are analysed in this work. Averaged periodogram, Kim’s large subimage and small subimage methods [1] together with windowed periodogram methods using rectangular and Hamming windows and a new window mixing method are studied to obtain their biasing and standard deviation.Sample means and sample standard deviations of the NPS calculations following the different methods are obtained using synthetic images that simulate noise in digital radiography images. In addition, biasing and variance characteristics of the windowed periodograms and the window mixing methods are derived theoretically.Biasing, characteristic of estimators based in periodograms, is eliminated by modifying the periodogram in such a way that it is obtained as the discrete Fourier transform of the unbiased sampled covariance of the signal. Simulations show that Kim’s methods considerably improve the precision of the averaged periodogram, obtaining an important reduction in the sampled standard deviation. Also, the window mixing method, using a convex combination of windowed periodograms with rectangular and Hamming windows, improves the Kim’s methods in terms of standard deviation and has similar biasing.Finally, it is shown that NPS estimators based in the windowed periodogram and in the window mixing methods are unbiased and mean-square consistent, provided that the support of the autocorrelation function of the system PSF is finite.

Measurement of Power Density at Zero Frequency With a Trend Compensation

The noise power spectrum (NPS) at zero frequency can represent a global noise property of a system and is the sum of the covariances with lags. As the lag increases, an estimate of the sum shows a converging curve. However, due to the unpredictable signal trend, the estimate continues to increase. In this paper, a trend model is first introduced to describe the trends. A high-precision estimate algorithm for zero-frequency NPS is then proposed based on a Blackman-Tukey method with a rectangular window, in which windowed periodograms are sample averaged as the Bartlett-Welch method. In order to analyze the covariance and trend components, sequential observations on estimates are conducted with respect to the periodogram and window sizes. Through the analysis, an appropriate range of the window sizes is suggested and a trend-compensation scheme is developed. Experimental results for x-ray images acquired from a radiography detector showed that an estimate precision was improved to 2.63% from 49.1% of a conventional approach with 99.5% covariance components.

On the alleged coherence between the global temperature and the sun׳s movement

It has recently been claimed that there is significant coherence between the spectral peaks of the global temperature series over the last 160 years and those of the speed of the solar center of mass at periods of 10–10.5, 20–21, 30 and 60–62 years. Here it is shown that these claims are based on a comparison between spectral peaks in spectral estimates that assume that the global temperature data contains time-invariant spectral lines. However, time–frequency analysis using both windowed periodograms and the maximum entropy method shows that this is not the case. An estimate of the magnitude squared coherence shows instead that under certain conditions only coherence at a period of 15–17 years can be found in the data. As this result builds on a low number of independent averages and also is unwarranted from any physical model it is doubtful whether it is significant.

Mean square error optimal weighting for multitaper cepstrum estimation

The aim of this paper is to find a multitaper-based spectrum estimator that is mean square error optimal for cepstrum coefficient estimation. The multitaper spectrum estimator consists of windowed periodograms which are weighted together, where the weights are optimized using the Taylor expansion of the log-spectrum variance and a novel approximation for the log-spectrum bias. A thorough discussion and evaluation are also made for different bias approximations for the log-spectrum of multitaper estimators. The optimized weights are applied together with the sinusoidal tapers as the multitaper estimator. Comparisons of the cepstrum mean square error are made of some known multitaper methods as well as with the parametric autoregressive estimator for simulated speech signals.

A note on autoregressive spectral estimates for frequency-wavenumber analysis of strong-motion array data

Frequency-wavenumber (f-k) spectra of seismic strong-motion array data are useful in estimating back-azimuth and apparent propagation velocity of seismic waves arriving at the array. Such estimates are required to model wave passage effects while studying spatial variability of strong ground motion. Although periodogram-based spectral estimates are commonly used, practical applications based on them encounter limitations, such as, lack of objective criteria for selecting a proper smoothing window and its associated bandwidth, and relatively large variance of the estimated spectral quantities. We present an alternative spectral estimate based on parametric time series modelling approach. The well-known autoregressive (AR) time series model is used in a system-based approach to estimate the spectral matrix of auto- and cross-spectral densities. Such spectral estimates are found to be smoother than the windowed periodogram estimates, and can directly be used in f-k spectral analysis. We present an example application of the proposed technique using strong-motion data recorded by the SMART-1 array in Taiwan during the January 29 1981 \(M_{L}\) 6.3 earthquake. Our results, in terms of back azimuth and apparent propagation velocity, are found to be in excellent agreement with those reported in the literature.

A New Regularized Adaptive Windowed Lomb Periodogram for Time–Frequency Analysis of Nonstationary Signals With Impulsive Components

This paper proposes a new class of windowed Lomb periodogram (WLP) for time-frequency analysis of nonstationary signals, which may contain impulsive components and may be nonuniformly sampled. The proposed methods significantly extend the conventional Lomb periodogram in two aspects: 1) The nonstationarity problem is addressed by employing the weighted least squares (WLS) to estimate locally the time-varying periodogram and an intersection of confidence interval technique to adaptively select the window sizes of WLS in the time-frequency domain. This yields an adaptive WLP (AWLP) having a better tradeoff between time resolution and frequency resolution. 2) A more general regularized maximum-likelihood-type (M-) estimator is used instead of the LS estimator in estimating the AWLP. This yields a novel M-estimation-based regularized AWLP method which is capable of reducing estimation variance, accentuating predominant time-frequency components, restraining adverse influence of impulsive components, and separating impulsive components. Simulation results were conducted to illustrate the advantages of the proposed method over the conventional Lomb periodogram in adaptive time-frequency resolution, sparse representation for sinusoids, robustness to impulsive components, and applicability to nonuniformly sampled data. Moreover, as the computation of the proposed method at each time sample and frequency is independent of others, parallel computing can be conveniently employed without much difficulty to significantly reduce the computational time of our proposed method for real-time applications. The proposed method is expected to find a wide range of applications in instrumentation and measurement and related areas. Its potential applications to power quality analysis and speech signal analysis are also discussed and demonstrated.

A New Method to Analyze Distortion Product Otoacoustic Emissions (DPOAEs) in the High-Frequency Range Up to 18 kHz Using Windowed Periodograms

Distortion product otoacoustic emissions (DPOAEs) are widely used as an objective examination procedure to determine cochlear function. In a clinical routine setting, the amplitude of the DPOAE signal at 2f1 .. f2 is applied as an indicator for a potential hearing loss up to 8 kHz. Due to their poor signal to noise ratio, meatal nodes from standing waves and calibration issues, high-frequency DPOAEs > 8 kHz have hardly been addressed in experimental and clinical audiology so far. Here, we present a new method of measuring DPOAE signal levels based on optimal maximum likelihood estimation with windowed power spectral density estimation of stochastic signals and filtering theory. Analysis of simulated data showed that the proposed method effectively reduces the disturbing noise floor compared to conventional averaging techniques. Robust DPOAE signals were measured in 20 ears from 10 normally hearing young adults (21 to 27 years) from 0:5 to 18 kHz. Repeated DPOAE recordings in one individual yielded a good to very good testretest reliability of the proposed method. These observations are discussed in the context of DPOAE signal processing and possible clinical applications of high-frequency DPOAE measurements.

Multitaper Estimation of Frequency-Warped Cepstra With Application to Speaker Verification

Usually the mel-frequency cepstral coefficients are estimated either from a periodogram or from a windowed periodogram. We state a general estimator which also includes multitaper estimators. We propose approximations of the variance and bias of the estimate of each coefficient. By using Monte Carlo computations, we demonstrate that the approximations are accurate. Using the proposed formulas, the peak matched multitaper estimator is shown to have low mean square error (squared bias + variance) on speech-like processes. It is also shown to perform slightly better in the NIST 2006 speaker verification task as compared to the Hamming window conventionally used in this context.

Performance Evaluation of a Spectrum-Sensing Technique for Cognitive Radio Applications in B-VHF Communication Systems

This paper deals with a cognitive radio approach based on a noncooperative spectrum sensing technique for applications in a very-high-frequency (VHF) aeronautical communications system to detect idle VHF channels and allocate them to an additional overlay system or secondary users. In particular, the focus here is on a spectrum sensing technique belonging to the class of energy detectors based on the Welch windowed periodogram. The performance of the proposed detector is presented in terms of the missed detection probability of the primary user under different application conditions. The obtained results and resulting implementation complexity highlight better behavior of the proposed detector with respect to alternative schemes previously proposed in the literature on this subject.

Performance Evaluation of a Precorrelation Interference Detection Algorithm for the GNSS Based on Nonparametrical Spectral Estimation

This paper deals with a signal processing precorrelation method to detect the presence of interference at a global navigation satellite system (GNSS) receiver site. In particular, a nonparametric spectral estimation approach based on the Welch windowed periodogram will be considered here. The performance of the proposed detector, in terms of detection probability, for a given false-alarm probability value, will be derived by means of an analytical approach and resorting to computer simulations. A performance comparison with previously proposed precorrelation methods will also be presented in order to highlight the better behavior of the proposed approach that makes it suitable for safety of life (SoL) applications.

Bayesian interpretation of periodograms

The usual nonparametric approach to spectral analysis is revisited within the regularization framework. Both usual and windowed periodograms are obtained as the squared modulus of the minimizer of regularized least squares criteria. Then, particular attention is paid to their interpretation within the Bayesian statistical framework. Finally, the question of unsupervised hyperparameter and window selection is addressed. It is shown that maximum likelihood solution is both formally achievable and practically useful.

Facts and fiction in spectral analysis

This analysis is limited to the spectral analysis of stationary stochastic processes with unknown spectral density. The main spectral estimation methods are: parametric with time series models, or nonparametric with a windowed periodogram. A single time series model will be chosen with a statistical criterion from three previously estimated and selected models: the best autoregressive (AR) model, the best moving average (MA) model, and the best combined ARMA model. The accuracy of the spectrum, computed from this single selected time series model, is compared with the accuracy of some windowed periodogram estimates. The time series model generally gives a spectrum that is better than the best possible windowed periodogram. It is a fact that a single good time series model can be selected automatically for statistical data with unknown spectral density. It is fiction that objective choices between windowed periodograms can be made.

VARIANCE ESTIMATION FOR SAMPLE AUTOCOVARIANCES: DIRECT AND RESAMPLING APPROACHES

SummaryThe usual covariance estimates for data n‐1 from a stationary zero‐mean stochastic process {Xt} are the sample covariances Both direct and resampling approaches are used to estimate the variance of the sample covariances. This paper compares the performance of these variance estimates. Using a direct approach, we show that a consistent windowed periodogram estimate for the spectrum is more effective than using the periodogram itself. A frequency domain bootstrap for time series is proposed and analyzed, and we introduce a frequency domain version of the jackknife that is shown to be asymptotically unbiased and consistent for Gaussian processes. Monte Carlo techniques show that the time domain jackknife and subseries method cannot be recommended. For a Gaussian underlying series a direct approach using a smoothed periodogram is best; for a non‐Gaussian series the frequency domain bootstrap appears preferable. For small samples, the bootstraps are dangerous: both the direct approach and frequency domain jackknife are better.

Parametric time domain analysis of the multiple input/scalar output problem: The source identification problem

A multiple input/scalar output stationary time series identification problem is considered from a parametric model time domain point of view. Particular emphasis is on the source identification problem. Closed form formula estimates of the individual source power contributions are expressed in terms of sample correlations that are obtained from the observed input and output time series and from parametric models fitted to that data. The estimates of the noise power contributions are asymptotically jointly normally distributed. The mean values and covariance matrix of those estimates yield confidence interval estimates of the individual and joint power contributions. The motivation for developing a rational polynomial transfer function or ARMA model of the multi-input scalar output plus additive noise situation is given. A two correlated input/single output version of this model is considered for a Monte Carlo simulation study. Parametric ARMA and approximate AR models are fitted to the simulated data. The asymptotic normality, and the distribution of the mean and covariances of the source power contribution computed from the ARMA and AR models are appraised. Several facets of the relative performance of windowed periodogram and AR model spectral analysis are examined for the multiple input/scalar output identification problem. The points that are emphasized are that conventional windowed periodogram spectral analysis is subjective, not particularly satisfactory for the sharp spectral peak situation that is commonly encountered in vibration data analysis and very likely not as good as “objective” Akaike criterion order AR modelled spectral analysis.

Automatic classification of multivariate EEGs using an amount of information measure and the eigenvalues of parametric time series model features

Two new classes of features are introduced for the automatic classification of multichannel stationary time series EEG data. The features are the Shannon-Gelfand-Yaglom measure of the amount of information between two sets of stationary Gaussian time series and the eigenvalues computed from a parametric model of the time series. The performance of these features for automatic sleep stage scoring from two EEG data channels, evaluated using the multinomial logistic function, is presented as an example. This parametric modeled EEG time series-two features for classification approach is a radical departure from the more conventional windowed periodogram spectral analysis-discriminant analysis packaged computer program approach.

Parametric time series models for multivariate EEG analysis

The autoregressive (AR) and mixed autoregressive-moving average (AR-MA) parametric models of stationary time series are of current interest for the purposes of spectral analysis and for the extraction of features for automatic EEG classification. Procedures for computing AR models and a new two-stage least-squares procedure for computing AR-MA models of multivariate time series are shown. The results of spectral estimation of EEGs using the multivariate AR, AR-MA, and conventional windowed periodogram analysis are compared.

Spectral analysis of EEG's by autoregressive decomposition of time series

A technique for scalar and multidimensional spectral analysis based on the autoregressive representation of the observed data records is presented and illustrated. In an autoregressive representation the observed data set is regressed on its own past history. This results in a formula that expresses the observed data as the output of a linear filter excited by an uncorrelated sequence (“discrete white noise”). Energy spectral densities, transfer functions, and coherences are computed from the autoregressive formula. The results of this technique are compared with the older windowed periodogram methods of spectral analysis. Two potential advantages over the latter methods are observed. For spectral estimates of comparable statistical performance, the autoregressive method analysis appear smoother and easier to interpret than the older windowed periodogram analysis. Also, in contrast with the expertise required to apply the windowed periodogram analysis, use of the autoregressive representation spectral analysis method appears to require little or no subjective judgment.