Autoregressive and Maximum Likelihood Spectral Analysis Methods

A coprime sensor array (CSA) is a sparse array geometry that interleaves two spatially undersampled uniform linear arrays (ULAs) with coprime undersampling factors. The CSA spans the same aperture and has equivalent resolution as a fully populated ULA, but uses fewer sensors. Previous works have introduced three CSA processors as spatial power spectral density (PSD) estimators: the Product, Min, and Blend processors. However, two of these processors display increased PSD esti-mate variance over the ULA PSD estimate. This dissertation focuses on reducing the variance of the CSA PSD estimators. PSD estimation traditionally relies on averag-ing uncorrelated coherent measurements (snapshots) to reduce variance. However,non-stationary underwater sonar environments often preclude increasing the num-ber of snapshots required to achieve a desirable PSD variance. In the traditional signal processing literature, there are two alternative methods to reduce variance without additional snapshot cost. The multitaper (MT) method and Welch's over-lapping segment averaging (WOSA) method improve PSD variance by O(K) at the expense of an O(K) resolution decrease by averaging K uncorrelated PSD estimates. Multitaper obtains these uncorrelated estimates by windowing the array with K or-thogonal tapers that span the entire array aperture. In contrast, WOSA obtains the uncorrelated estimates by subdividing the array into K possibly overlapping segments.This dissertation extends these two existing ULA variance reduction techniques to the CSA processors, making four main contributions. The first proposes the multitapered Product processor (MT-Product) that estimates the spatial PSD with reduced variance with respect to the traditional CSA Product processor but still uses fewer sensors than a fully populated ULA. The second proposes the multita-pered Min processor (MT-Min) that reduces the PSD estimate variance further than either MT-Product or Product. The MT-Min estimator has variance comparable to a multitapered ULA. The third proposes the multitapered Blend processor (MT-Blend) that blends MT-Product's attenuation of multiple source cross-terms with MT-Min's low variance while still achieving an O(K) variance reduction over the traditional Blend processor. The final contribution proposes the Welch overlapping segment averaging Product processor (WOSA-Product) that estimates the spatial PSD with reduced variance with respect to the traditional CSA Product processor but still uses fewer sensors than a fully populated ULA. WOSA-Product is gener-ally able to form a larger number of uncorrelated PSD estimates than MT-Product. Closed-form statistics for spatially white Gaussian processes are derived for MT-Product, MT-Min, and WOSA-Product. Simulations verify the variance reduction predicted by the analytical derivations each processor, and the effects of WOSA-Product segment length on resolution, variance reduction, and peak sidelobe levels are discussed.

The aim of this study was to assess whether power spectrum estimates of electroencephalogram (EEG) can be used as a biomarker for autism spectrum disorder (ASD). EEG collected from ASD and control participants performing a short-memory task was preprocessed to remove noise and artefacts, power spectral density (PSD) estimates were obtained by the modified covariance method and used as the study features that were subjected next to the Kruskal-Wallis analysis of differences. After verifying that the features (PSD estimates) were statistically different between the autistic and control subjects, these PSD estimates were classified using the 'k nearest neighbour' (KNN) classification algorithm with the average accuracy of 89.29%. This result indicates that EEG of autistic and control individuals may contain statistically different features; therefore, EEG power spectrum may be used as a biomarker for autism.

Spectrum analysis is one of the momentous fields in signal processing. It has a large variety of applications in radar, sonar, speech and image processing. Parametric methods have been proposed and employed for spectrum analysis including power spectral density (PSD) estimation. These methods estimate the parameters of a statistical model and compute the PSD, afterwards. In some circumstances one is obliged to deal with observations of numerous geographically dispersed sensors, to either increase the precision or based on application demands. Having a set of sensors linked together to take the advantages of cooperation and network topology, one obtains a more comprehensive estimation. In this chapter, the authors propose and study four different algorithms capable of facing spatio-temporal variations for parametric modeling and PSD estimation using wireless sensor networks (WSNs). For this purpose, the authors first validate the proposed algorithms using theoretical and mathematical formulations. Thereafter, performing simulation tasks demonstrates and supports the theoretical achievements. The next section of the chapter illustrates the concepts to a greater degree, the authors analyze and compare the performance of these algorithms with each other, as well as with the simple PSD estimation using individual sensors, wherein there is no cooperation among the nodes.

ABSTRACTA radio frequency interference (RFI) excision algorithm based on spectral kurtosis, a spectral variant of time‐domain kurtosis, is proposed and implemented in software. The algorithm works by providing a robust estimator for Gaussian noise that, when violated, indicates the presence of non‐Gaussian RFI. A theoretical formalism is used that unifies the well‐known time‐domain kurtosis estimator with past work related to spectral kurtosis, and leads naturally to a single expression encompassing both. The algorithm accumulates the first two powers of M power spectral density (PSD) estimates, obtained via Fourier transform, to form a spectral kurtosis (SK) estimator whose expected statistical variance is used to define an RFI detection threshold. The performance of the algorithm is theoretically evaluated for different time‐domain RFI characteristics and signal‐to‐noise ratios η. The theoretical performance of the algorithm for intermittent RFI (RFI present in R out of M PSD estimates) is evaluated and shown to depend greatly on the duty cycle, d = R/M. The algorithm is most effective for d = 1/(4 + η), but cannot distinguish RFI from Gaussian noise at any η when d = 0.5. The expected efficiency and robustness of the algorithm are tested using data from the newly designed FASR Subsystem Testbed radio interferometer operating at the Owens Valley Solar Array. The ability of the algorithm to discriminate RFI against the temporally and spectrally complex radio emission produced during solar radio bursts is demonstrated.

Many well-known and frequently employed Bayesian clean speech estimators have been derived under the assumption that the true power spectral densities (PSDs) of speech and noise are exactly known. In practice, however, only power spectral density (PSD) estimates are available. Simply neglecting PSD estimation errors and handling the estimates as true values leads to speech estimation errors causing musical noise and undesired suppression of speech. In this paper, the uncertainty of the available speech PSD estimates is addressed. The main contributions are the following. First, we summarize and examine ways to model and incorporate the uncertainty of PSD estimates for a more robust speech enhancement performance. Second, a novel nonlinear clean speech estimator is derived that takes into account prior knowledge about the absolute value of typical speech PSDs. Third, we show that the derived statistical framework provides uncertainty-aware counterparts to a number of well-known conventional clean speech estimators such as the Wiener filter and Ephraim and Malah's amplitude estimators. Fourth, we show how modern PSD estimators can be incorporated into the theoretical framework and propose to employ frequency dependent priors. Finally, the effects and benefits of considering the uncertainty of speech PSD estimates are analyzed, discussed, and evaluated via instrumental measures and a listening experiment.

In the study of human cognitive activity using electroencephalogram (EEG), the brain dynamics parameters and characteristics play a crucial role. They allow to investigate the changes in functionality depending on the environment and task performance process, and also to access the intensity of the brain activity in various locations of the cortex and its dependencies. Usually, the dynamics of activation of different brain areas during the cognitive tasks are being studied by spectral analysis based on power spectral density (PSD) estimation, and coherence analysis, which are de facto standard tools in quantitative characterization of brain activity. PSD and coherence reflect the strength of oscillations and similarity of the emergence of these oscillations in the brain, respectively, while the concept of stability of brain activity over time is not well defined and less formalized. We propose to employ the detrended fluctuation analysis (DFA) as a measure of the EEG persistence over time, and use the DFA scaling exponent as its quantitative characteristics. We applied DFA to the study of the changes in activation in brain dynamics during mental calculations and united it with PSD and coherence estimation. In the experiment, EEGs during resting state and mental serial subtraction from 36 subjects were recorded and analyzed in four frequency ranges: θ1 (4.1–5.8 Hz), θ2 (5.9–7.4 Hz), β1 (13–19.9 Hz), and β2 (20–25 Hz). PSD maps to access the intensity of cortex activation and coherence to quantify the connections between different brain areas were calculated, the distribution of DFA scaling exponent over the head surface was exploited to measure the time characteristics of the dynamics of brain activity. Obtained arrangements of DFA scaling exponent suggest that normal functioning of the brain is characterized by long-term temporal correlations in the cortex. Topographical distribution of the DFA scaling exponent was comparable for θ and β frequency bands, demonstrating the largest values of DFA scaling exponent during cognitive activation. The study shows that the long-term temporal correlations evaluated by DFA can be of great interest for diagnosis of the variety of brain dysfunctions of different etiology in the future.

This paper analyzes algorithms which are suitable for spectral estimates of noisy signals in the frequency domain based on the use of the fast Fourier transform (FFT). Several causes of inaccuracy are analyzed and characterized so that the expressions of different components of error on the power spectral density (psd) estimate are given, in terms both of spectral properties of noise and typical parameters of the used filter. These simple expressions point out how an appropriate choice of some window parameters may increase considerably the accuracy of the estimate. The effects of choice on the accuracy are examined. In any case, the performance of the psd estimator can be improved by adopting linear or nonlinear averaging techniques; in the paper the statistical properties of geometric mean of periodograms are particularly examined and compared with those of the more traditional Welch's method. It is proved that, under appropriate conditions, the geometric mean produces a reduction both of bias and variance of psd. Numerical simulations confirm these theoretical results.

Bias compensation methods for minimum statistics noise power spectral density estimation

The multi-channel Wiener filter (MWF) is a well-known multi-microphone speech enhancement technique, aiming at improving the quality of the recorded speech signals in noisy and reverberant environments. Assuming that reverberation and ambient noise can be modeled as a diffuse sound field and the spatial coherence of the residual noise is known, the MWF requires estimates of the relative early transfer function (RETF) vector of the target speaker as well as the power spectral densities (PSDs) of the target, diffuse and residual noise component. RETF vector and PSD estimation is often decoupled, where one quantity is estimated independently of the other quantity. In this paper, we propose to jointly estimate the RETF vector and all PSDs by minimizing the Frobenius norm of a model-based error matrix using an alternating least squares method. Experimental results using different dynamic acoustic scenarios with a moving speaker show that the proposed method leads to a larger MWF performance than a state-of-the-art method based on covariance whitening.

In the paper, two algorithms that allow identification of a parametric models of random time-series from binary-valued observations of their realizations, as well as from quantized measurements of their values, are proposed. The proposed algorithms are based on the idea of time-series decomposition either on a direct power spectral density or autocorrelation function approximation. They use the concepts of randomized search algorithms to recover the corresponding parametric models from calculated estimates of power spectral density or autocorrelation function. The considerations presented in the paper are illustrated with simulated identification examples in which linear and nonlinear block-oriented dynamic models of timeseries are identified from the binary-valued observations and quantized measurements.

BackgroundThe Gini coefficient is a statistical tool generally used by economists to quantify income inequality. However, it can be applied to any kind of data with unequal distribution, including heart rate variability (HRV).ObjectivesTo assess the application of the Gini coefficient to measure inequality in power spectral density of RR intervals, and to use this application as a psychophysiological indicator of mental stress.MethodsThirteen healthy subjects (19 ± 1.5 years) participated in this study, and their RR intervals were obtained by electrocardiogram during rest (five minutes) and during mental stress (arithmetic challenge; five minutes). These RR intervals were used to obtain the estimates of power spectral densities (PSD). The limits for the PSD bands were defined from 0.15 to 0.40 Hz for high frequency band (HF), from 0.04 to 0.15 Hz for low frequency band (LF), from 0.04 to 0.085 Hz for first low frequency sub-band (LF1) and from 0.085 to 0.15 Hz for second low frequency sub-band (LF2). The spectral Gini coefficient (SpG) was proposed to measure the inequality in the power distribution of the RR intervals in each of above-mentioned HRV bands. SpG from each band was compared with its respective traditional index of HRV during the conditions of rest and mental stress. All the differences were considered statistically significant for p < 0.05.ResultsThere was a significant decrease in HF power (p = 0.046), as well as significant increases in heart rate (p = 0.004), LF power (p = 0.033), LF2 power (p = 0.019) and LF/HF (p = 0.002) during mental stress. There was also a significant increase in SpG(LF) (p = 0.009) and SpG(LF2) (p = 0.033) during mental stress. Coefficient of variation showed SpG has more homogeneity compared to the traditional index of HRV during mental stress.ConclusionsThis pilot study suggested that spectral inequality of Heart Rate Variability analyzed using the Gini coefficient seems to be an independent and homogeneous psychophysiological indicator of mental stress. Also, HR, LF/HF, SpG(LF) of HRV are possibly important, reliable and valid indicators of mental stress.

Time delay estimation has important applications in the fields of radar, sonar, seismology, multimedia communication etc. For a distant, stationary radiating source, the optimum time delay vector estimation at a receiving sensor array is realized using Generalized Cross Correlation (GCC) function and its error matrix, expressed in terms of apriori known source signal and sensor background noise spectra. This gives a linear minimum variance unbiased estimate known as the Gauss Markov Estimate. This paper examines the use of Power Spectral Density (PSD) estimation based Gauss Markov Estimate for cases of unknown source spectra and gives a two step time delay estimation algorithm for close range sources. The technique has been analyzed for a three element linear array with wideband source and shows improvement in terms of mean square error (MSE) for inter-sensor time delays, source range and bearing. The corresponding PSD estimation error has been expressed in terms of the spectral distortion measure.

Traditional fast Fourier transform (FFT) has gained enormous recognition for broken rotor bar (BRB) fault detection in induction motors using the sideband features as fault indices, however, the false alarm from inaccurate diagnosis remains a major setback associated with the technique. This paper presents two reliable spectral analysis approaches for BRB fault detection and analysis for induction motors: Thompson Multitaper (MTM) power spectral density (PSD) estimate, and Welch PSD estimate. The two methods are implemented using the simulated stator current signal of an induction motor obtained by the finite element method. The Finite Element analysis software, ANSYS, is used to design and simulate different motor conditions: a healthy motor, a motor with one, two and three BRBs. It is verified that the proposed methods provide robust and reliable BRB fault detection for induction motors.

This paper addresses the problem of extracting a desired speech source from a multispeaker environment in the presence of background noise. A new adaptive beamforming structure is proposed for this speech enhancement problem. This structure incorporates power spectral density (PSD) estimation of the speech sources together with a noise statistics update. An inactive-source detector based on minimum statistics is developed to detect the speech presence and to track the noise statistics. Performance of the proposed beamformer is investigated and compared to the minimum variance distortionless response (MVDR) beamformer with or without a postfilter in a real hands-free communication environment. Evaluations show that the proposed beamformer offers good interference and noise suppression levels while maintaining low distortion of the desired source.

The constant false alarm rate (CFAR) detector based on the FFT filter bank is computationally efficient and widely used for the detection and frequency estimation of narrowband signals embedded in noise. The standard approach for improving its detection performance is to average the power spectral density (PSD) estimates obtained from multiple input data blocks prior to performing a detection decision. Since the variance of the averaged PSD estimates has an inverse linear dependence on the number of input data blocks, the conjecture that a 3 dB performance gain will result from doubling the number of input data blocks processed is plausible. However, this paper shows that the actual performance gain obtained with two input data blocks is less than 3 dB over the single block case and the performance gain for each successive doubling of the number of input data blocks decreases monotonically and appears to converge to a constant around 1.5 dB.