Variance reduction techniques for power spectral density estimation with coprime sensor arrays

This paper proposes two high-resolution Direction-of-Arrival (DOA) estimators using coprime sensor arrays (CSA) processing broadband signals. The product processor estimates the broadband spatial power spectral density (PSD) by averaging narrowband spatial PSD estimates. These narrowband PSD estimates are formed by multiplying one CSA subarray scanned response with the complex conjugate of the other. Contrastingly, the min processor estimates the broadband spatial PSD by taking the minimum over all subarray periodograms at all processed frequencies for each bearing. The inverse Fourier transform of the broadband spatial PSD estimates the spatial correlation function, which populates the diagonals of a Toeplitz augmented covariance matrix (ACM). The MUSIC algorithm estimates the source DOAs from this constructed ACM. Combining the CSA narrowband PSD estimates over additional bandwidth reduces the number of snapshots needed to attenuate cross-terms in the spatial PSD estimates, providing processing gains for DOA estimation. The MUSIC pseudo-spectra suggest that the product algorithm performs better in scenarios with more sources than sensors and the min algorithm performs better in scenarios with varying source power levels. Monte Carlo simulations show that the new DOA estimators achieve improved precision over previous broadband CSA DOA estimators in snapshot-challenged scenarios.

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.

Three power spectral density estimation problems are considered in this presentation. These are the estimation of power spectral density for a single scalar time series (or equivalently a line array), estimation of power spectral density matrices for vector valued time series, and the estimation of wavenumber or frequency-wavenumber spectra for random fields. The paper is primarily a tutorial coverage of autoregressive (also called maximum entropy) and maximum likelihood methods of spectral analysis. In addition to standard material on these subjects a recent generalization of the Burg algorithm from scalar to vector time series is included.

Coprime sensor arrays (CSAs) can estimate the directions of arrival of O(MN) narrowband plane wave sources using only O(M+N) sensors with the CSA product processor. Processing data from a finite aperture array effectively smears the true spatial power spectral density (PSD) with a kernel function determined by both the array geometry and the processing of the signals observed by the array. This paper examines the asymptotic behaviors of the kernel functions resulting from two different processors applied to a CSA sampling geometry in the limit of large aperture. The kernel functions of the product processed CSA and conventionally beamformed coprime sensor arrays (CBF CSA) are compared to the baseline of the kernel of a densely populated uniform line array (ULA) of similar aperture. At the limit of large aperture, the product processed CSA estimate is asymptotically unbiased like the ULA, while the CBF CSA estimate is not. The PSD estimates computed from the CSA processors are compared when spatially correlated Gaussian noise is an input to the array to highlight the bias issues.

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.

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.

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 estimation of noise Power spectral density (PSD) is a very crucial issue for speech enhancement as a result of its significant effiect on the quality and intelligibility of the enhanced speech. Most of the existing estimators for noise PSD try to employ Gaussian speech priors, which, however, have been proven inconsistent with the reality. We derived an effiective solution to this problem of estimating noise PSD in the Minimum mean square error (MMSE) sense when the speech component is modeled by a Laplacian distribution. Meanwhile, the soft decision technique instead of the hard Voice activity detection (VAD) is evolved into our algorithm, which can automatically makes the estimation unbiased without requiring a bias compensation. The performance of the proposed method is tested by several objective and subjective measures under various stationary and nonstationary noise environments. The results confirm that our method achieves good performance for all the noise conditions and Signalnoise-ratio (SNR) settings.