Transforms of Deterministic Continuous-Time Signals

Time-domain autocovariance processing is widely accepted as a computationally efficient method to estimate the first three spectral moments of Doppler weather radar signals (i.e., mean signal power, mean Doppler velocity, and spectrum width). However, when signals with different frequency content (e.g., ground clutter) contaminate the weather signal, spectral processing using the periodogram estimator of the power spectral density (PSD) is the preferred tool of analysis. After spectral processing (i.e., filtering), a PSD-based autocorrelation estimator is typically employed to produce unbiased estimates of the weather-signal spectral moments. However, the PSD does not convey explicit phase information, which has the potential to aid in the spectral analysis of radar signals. In this paper, the autocorrelation spectral density (ASD) is introduced for spectral analysis of weather-radar signals as a generalization of the classical PSD, and an ASD-based autocorrelation estimator is proposed to produce unbiased estimates of the weather-signal spectral moments. A significant advantage of the ASD over the PSD is that it provides explicit phase information that can be exploited to identify and remove certain types of contaminant signals. Thus, the ASD provides an alternative means for spectral analysis, which can lead to improved quality of meteorological data from weather radars.

In seismic engineering, the earthquake-induced ground motion is generally represented in the form of pseudo-acceleration or displacement response spectra. There are, however, situations in which the response spectrum is not considered appropriate, and a fully dynamic analysis is required. In this case, the most effective approach is to define artificial spectrum-compatible stationary accelerograms, which are generated to match the target elastic response spectrum. So a Power Spectral Density (PSD) function is derived from the response spectrum. However, the above approach possesses the drawback that the artificial accelerograms do not manifest the variability in time and in frequency observed from the analysis of real earthquakes. Indeed, the recorded accelerograms can be considered sample of a fully non-stationary process. In this study a procedure based on the analysis of a set of accelerograms recorded in a chosen site to take into account their time and frequency variability is described. In particular the generation of artificial fully non-stationary accelerograms is performed in three steps. In the first step the spectrum-compatible PSD function, in the hypothesis of stationary excitations, is derived. In the second step the spectrum-compatible Evolutionary Power Spectral Density (EPSD) function is obtained by an iterative procedure to improve the match with the target response spectrum starting from the PSD function, once a time-frequency modulating function is chosen. In the third step the artificial accelerograms are generated by the well-known Shinozuka and Jan (J Sound Vib 25:111–128, 1972) formula and deterministic analyses can be performed to evaluate the structural response. Once the EPSD spectrum-compatible function is derived, a method recently proposed by the authors (Muscolino and Alderucci in Probab Eng Mech 40:75–89, 2015), is adopted to evaluate the EPSD response function of linear structural systems subjected to fully non-stationary excitations by very handy explicit closed-form.

In order to reveal the unsteady features of gas–solid flow, the pressure fluctuations were measured at different locations along the length of the pipeline while conveying powders through the pipeline. Power spectral density (PSD) functions were obtained for the analysis of the pressure fluctuation. Two types of powders (fly ash and alumina) were used in this analysis. The PSD analysis was conducted by taking into account different aspects such as flow conditions (dilute or dense), location of transmitter (top and bottom transmitters), location of transmitter along the length of the pipeline (three different locations), material property (fly ash or alumina), etc. Analysis of signals from top and bottom transmitters shows that it is not possible to identify the flow mode at upper and lower portions of pipeline. The magnitude of power is found to be higher for alumina as compared to fly ash. PSD parametric analysis reveals that frequency bandwidth and average power decreases exponentially with increase in solid loading ratio.

PurposeObstructive sleep apnea (OSA) is a prevalent sleep-related breathing disorder. Research conducted on patients with OSA using electroencephalography (EEG) has revealed a noticeable shift in the overnight polysomnography (PSG) power spectrum. To better quantify the effects of OSA on brain function and to identify the most reliable predictors of pathological cortical activation, this study quantified the PSG power and its association with the degree of hypoxia in OSA patients.Patients and MethodsThis retrospective study recruited 93 patients with OSA. OSA patients were divided into three groups based on their apnea-hypopnea index (AHI) scores. The clinical characteristics and sleep macrostructure of these patients were examined, followed by an analysis of PSG signals. Power spectral density (PSD) in five frequency bands was analyzed during nonrapid eye movement (NREM) sleep, rapid eye movement (REM) sleep, and wakefulness. Finally, correlation analysis was conducted to assess the relationships among PSD, PSG parameters, and serum levels of S100β and uric acid.ResultsObstructive sleep apnea occurred during both the NREM and REM sleep phases. Except for a decrease in the duration of N2 sleep and an increase in the microarousal index, there were no significant differences in sleep architecture based on disease severity. Compared to the mild OSA group, the theta and alpha band PSD in the frontal and occipital regions during NREM sleep and wakefulness were significantly decreased in the moderate and severe OSA groups. Correlation analysis revealed that theta PSD in N1 and N3 stages were negatively correlated the AHI, oxygen desaturation index, SaO2<90% and microarousal index.ConclusionThese findings imply that patients with more severe OSA exhibited considerable NREM hypoxia and abnormal brain activity in the frontal and occipital regions. Therefore, sleep EEG oscillation may be a useful neurophysiological indicator for assessing brain function and disease severity in patients with OSA.

In order to simulate the non-Gaussian fluctuating wind pressure with prescribed lower-order moments and its target power spectral density (PSD) function, the following framework is proposed in the present paper: Firstly, the PSD function of fluctuating wind pressure is deduced rigorously according to the relationship between the wind pressure and velocity; Then, an efficient yet practical method by which the fluctuating wind pressure may be reproduced with the designated skewness and kurtosis is presented based on the derived PSD function, Johnson translator system, and digital filter; Finally, in order to corroborate the feasibility and correctness of the present methodology, a numerical example of simulating one-dimensional univariate non-Gaussian fluctuating wind pressure is taken into consideration. Numerical results indicate that the statistical parameters, such as the skewness and kurtosis, of the simulated fluctuating wind pressure are in remarkably agreement with their targets. Likewise, the PSDs of the simulated records and target show considerably good agreement with each other.

JPT Forum articles are limited to 1,500 words including 250 words for each table and figure, or a maximum of two pages in JPT. A Forum article may present preliminary results or conclusions of an investigation that the present preliminary results or conclusions of an investigation that the author wishes to publish before completing a full study; it may impart general technical information that does not warrant publication as a full-length paper. All Forum articles are subject to approval by a editorial committee. Letters to the editor are published under Dialogue, and may cover technical or nontechnical topics. SPE-AIME reserves the right to edit letters for style and content. A wave-follower buoy has many advantages over nearly all other types of wave-measuring devices, particularly on the open sea or far offshore. The buoy requires only the earth's gravitational field as a frame of reference instead of a pier, a piling, or perhaps a large spar buoy. Any of the ordinary-capacity or resistive-wave gauges must be attached to a solid place. Any type of wave-follower (accelerometer) buoy will exhibit its own resonant dynamic behavior in the presence of a wave field. The engineer in need of wave information generally needs to know its power spectral density to be able to apply the data, whether the problem is wave forces on a structure or something else. It is not sufficient to simply double-integrate the vertical acceleration of the wave buoy and claim to have the desired wave-height information. The objective of this paper is to compare an actual power spectral density in a wave field with that of the power spectral density in a wave field with that of the accelerometer-output power spectral density produced in the same wave field. We will show an empirical spectral transfer function that can be applied to the observed accelerometer spectrum to yield the desired power spectral density of the driving waves. For this purpose, an ideal linear model for the buoy was proposed to keep the system design simple. Even with this incorrect model, the comparison of the precise spectrum with the compensated spectrum out of the buoy is quite good. The buoy used for this study was a Model WF-100 buoy manufactured by Coastal Data Service, Inc. Let x(t) be the actual vertical displacement of the sea surface as a function of time. If y(t) is the output from the buoy accelerometer, then x(t) and Y\y(t) are related as in Eq. 1. x(t) * h(t) = y(t),...................................(1) where h(t) is the time domain representative of desired transfer function and * denotes convolution. In the frequency domain, we have Eq. 2. X (f) x H(f) = Y(f)...................................(2) Using the Fourier transform theory, one obtains Eq. 3. Gy(f) H(f) 2 = ------------, Gx(f) where Gx(f) = X(f) 2 ..................................(3) In this equation, means absolute value, means expected value, Gx(f) is the power spectral density of waves, and Gy(f) is the power spectral density of the accelerometer output. In this experiment, the two desired power spectral densities were generated by a minicomputer, power spectral densities were generated by a minicomputer, and the ratio of the two in Eq. 3 leads to the compensation function, H(f). The linear model is shown in Eq. 4. 1 1/2 sin (1.26f) H(f) = 32.2 [w2] -------- x ------------- 1 + 2f2 1.26f I II III ...................................(4) Factor I in Eq. 4 includes the accelerometer scale constant and w 2 corresponds to straight double integration of sinusoidal components. P. 652

Background:Non-nutritive suck (NNS) is used to promote ororhythmic patterning and assess oral feeding readiness in preterm infants in the neonatal intensive care unit (NICU). While time domain measures of NNS are available in real time at cribside, our understanding of suck pattern generation in the frequency domain is limited. The aim of this study is to model the development of NNS in the frequency domain using Fourier and machine learning (ML) techniques in extremely preterm infants (EPIs).Methods:A total of 117 EPIs were randomized to a pulsed or sham orocutaneous intervention during tube feedings 3 times/day for 4 weeks, beginning at 30 weeks post-menstrual age (PMA). Infants were assessed 3 times/week for NNS dynamics until they attained 100% oral feeding or NICU discharge. Digitized NNS signals were processed in the frequency domain using two transforms, including the Welch power spectral density (PSD) method, and the Yule-Walker PSD method. Data analysis proceeded in two stages. Stage 1: ML longitudinal cluster analysis was conducted to identify groups (classes) of infants, each showing a unique pattern of change in Welch and Yule-Walker calculations during the interventions. Stage 2: linear mixed modeling (LMM) was performed for the Welch and Yule-Walker dependent variables to examine the effects of gestationally-aged (GA), PMA, sex (male, female), patient type [respiratory distress syndrome (RDS), bronchopulmonary dysplasia (BPD)], treatment (NTrainer, Sham), intervention phase [1, 2, 3], cluster class, and phase-by-class interaction.Results:ML of Welch PSD method and Yule-Walker PSD method measures revealed three membership classes of NNS growth patterns. The dependent measures peak_Hz, PSD amplitude, and area under the curve (AUC) are highly dependent on PMA, but show little relation to respiratory status (RDS, BPD) or somatosensory intervention. Thus, neural regulation of NNS in the frequency domain is significantly different for each identified cluster (classes A, B, C) during this developmental period.Conclusions:Efforts to increase our knowledge of the evolution of the suck central pattern generator (sCPG) in preterm infants, including NNS rhythmogenesis will help us better understand the observed phenotypes of NNS production in both the frequency and time domains. Knowledge of those features of the NNS which are relatively invariant vs. other features which are modifiable by experience will likewise inform more effective treatment strategies in this fragile population.

Signal processing of flutter flight test data enables verification of aircraft flutter design, and the signal from a flutter flight test excited by atmospheric turbulence is a particularly important form of the flutter test. Owing to the randomness of atmospheric turbulence excitation, multi-channel analysis of turbulence responses at various positions in the same component can improve the analytical accuracy of flutter signal processing. The relationship between the maximum singular value of the multi-channel turbulence response power spectral density matrix and the system self-power spectral density function is elucidated herein using a frequency domain decomposition method. However, there is a contradiction in the power spectral density function between the spectral line density and the spectral line smoothing calculated based on the periodogram of the frequency domain decomposition. By applying an autoregressive spectral model, the power spectral density function of the turbulence response is calculated to achieve spectral line smoothing and sufficient spectral line density. Additionally, the power spectral density function is then used to construct the power spectral density function matrix of the multi-channel turbulence response, and the maximum singular value curve is calculated based on the singular value decomposition of each spectral pin. Finally, the modal parameters of the turbulence response signal are estimated via multi-modal frequency domain fitting. The developed approach is validated based on simulations and flutter flight test turbulence response signals.

Power handling and power spectral density (EIRP-SD) considerations in the design of reflectarray antennas are outlined for wireless high power transfer applications. The impact of fast Fourier Transform (FFT) based pattern analysis methods is considered, including the accuracy of EIRP-SD calculations for regulatory considerations, as well as how to compute those limits. A hybrid pattern analysis appropriate for the practical design of high EIRP-SD reflectarray systems is proposed. To illustrate these concepts, a reflectarray is designed and modeled with this method and the results exhibit agreement with predicted patterns from a full-wave simulation while showing dramatic simulation time improvement. Finally, the power handling, EIRP-SD and gain characteristics are related to each other.

Postural control was studied when the subject was kneeling with erect trunk in a quiet posture and compared to that obtained during quiet standing. The analysis was based on the center of pressure motion in the sagittal plane (CPx), both in the time and in the frequency domains. One could assume that postural control during kneeling would be poorer than in standing because it is a less natural posture. This could cause a higher CPx variability. The power spectral density (PSD) of the CPx obtained from the experimental data in the kneeling position (KN) showed a significant decrease at frequencies below 0.3 Hz compared to upright (UP) (P < 0.01), which indicates less sway in KN. Conversely, there was an increase in fast postural oscillations (above 0.7 Hz) during KN compared to UP (P < 0.05). The root mean square (RMS) of the CPx was higher for UP (P < 0.01) while the mean velocity (MV) was higher during KN (P < 0.05). Lack of vision had a significant effect on the PSD and the parameters estimated from the CPx in both positions. We also sought to verify whether the changes in the PSD of the CPx found between the UP and KN positions were exclusively due to biomechanical factors (e.g., lowered center of gravity), or also reflected changes in the neural processes involved in the control of balance. To reach this goal, besides the experimental approach, a simple feedback model (a PID neural system, with added neural noise and controlling an inverted pendulum) was used to simulate postural sway in both conditions (in KN the pendulum was shortened, the mass and the moment of inertia were decreased). A parameter optimization method was used to fit the CPx power spectrum given by the model to that obtained experimentally. The results indicated that the changed anthropometric parameters in KN would indeed cause a large decrease in the power spectrum at low frequencies. However, the model fitting also showed that there were considerable changes also in the neural subsystem when the subject went from standing to kneeling. There was a lowering of the proportional and derivative gains and an increase in the neural noise power. Additional increases in the neural noise power were found also when the subject closed his eyes.

Digital simulation of stochastic wind velocity field is prerequisite to flutter or buffeting analysis of long-span bridges in time-domain. Wavelet method was applied to the simulation of the stochastic wind field, performing the effective simulation of spatial stochastic wind field. According to the idea of multi-resolution analysis and orthonormal wavelets, relationship between wavelet coefficients and PSD(power spectral density)function is deduced , then the wavelet coefficients on each and every scale are obtained from the given PSD function, the intermittency was introduced into wavelet coefficients while at the same time preserves the target spectral characters. As a result, the inverse wavelet transform is applied to generate stationary wind velocity history through the given wavelet coefficients. A numerical example to illustrate the application of the proposed method for the simulation of a bridge wind velocity field is provided.

BACKGROUND: One of the areas for further improvement of spectral filter analysis of random signals is the transition from conventional narrow-band filters to non-stationary narrow-band dynamic filters operating in a transient mode, allowing to increase the measurement accuracy of the estimated spectral power density with conventional and dynamic filters of the same order. AIM: This work aimed to improve procedural algorithms, generalized expressions for the spectral window function, the relative dispersion of the measured spectral power density estimate, and optimal synthesis of the tuning laws for the narrow-band dynamic filters, including the attenuation ratio, carrier frequency, and transmission ratio in the analysis bandwidth, for spectral analysis of random signals. METHODS: To determine the relative dispersion of the estimated spectral power density, a correlation filter method using narrow-band dynamic second-order filters is used. RESULTS: The paper develops and specifies theoretical results in relation to narrow-band second-order dynamic filters to optimize their tuning (measurement) laws with the focus on a more advanced correlation filter method of spectral analysis of random signals. CONCLUSION: The findings show that narrow-band dynamic filters reordered in the analysis bandwidth allow for higher accuracy of spectral analysis compared to conventional steady-state filters of the same order.

The power spectral density (PSD) of any time-dependent stochastic process Xt is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is a fluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.

The present paper focuses on reliability analysis of structural systems under ground motion acceleration taking into account the inherent random nature of the excitation as well as epistemic uncertainties affecting the definition of its spectrum. Specifically, seismic excitation is modelled as a zero-mean stationary Gaussian random process fully characterized by an imprecise power spectral density (PSD) function i.e. with interval parameters. The ranges of such interval parameters are determined by analysing a large set of accelerograms recorded on rigid soil deposits. To discard outliers, the Chauvenet’s Criterion is applied iteratively. The proposed imprecise PSD function may be viewed as representative of the actual accelerograms recorded on rigid soil deposits. Reliability assessment is carried out by incorporating the imprecise PSD function of ground motion acceleration into the formulation of the classical first-passage problem using Vanmarcke’s failure criterion. Due to imprecision of the excitation, the reliability function and the fractile of order p of the selected extreme value response process turn out to have an interval nature. Thus, the aim of reliability analysis is the evaluation of the bounds of such functions which define the range of structural performance. Numerical results show that neglecting imprecision of the PSD may lead to significant overestimation of the safety level.

The isotropic turbulence assumption, though widely used, may not accurately represent real-world conditions in stochastic wind models for turbulence relative to moving vehicles. This study derives the statistical properties of non-isotropic turbulence relative to moving vehicles, focusing on correlation functions, power spectral densities (PSDs), and coherence functions of wind speed fluctuations. A quantitative parameter characterizing the impact of non-isotropic turbulence is introduced, enabling an in-depth analysis of its effects on turbulence statistics. Non-isotropic turbulence notably increases PSDs in frequency regions critical to vehicle stability under crosswinds, underscoring the need to account for these effects in quantitative assessments. Without considering non-isotropic influences, crosswind stability may be overestimated. Non-isotropic turbulence impacts the PSDs both indirectly, by altering the lateral-to-longitudinal turbulence length scale ratio, and directly, through additional influences on PSDs. Prior studies have largely focused on the indirect effect, but both effects should be considered, as omitting the direct effect results in up to a 28.9% underestimation of PSDs at low frequencies, potentially overestimating running safety. The analysis also reveals that the relative wind angle, defined as the difference between the vehicle's running direction wind angle and its yaw angle, is the primary factor influencing turbulence characteristics, consistent for both isotropic and non-isotropic conditions. Additionally, the spatial correlation of wind speed fluctuations depends largely on the separation distance relative to the longitudinal turbulence length scale. Therefore, in qualitative studies of wind fluctuation effects on crosswind stability, non-isotropic turbulence may be disregarded. This work enhances the accuracy of stochastic wind models across different scenarios, providing a foundation for future research on crosswind safety in railway operations under turbulent wind conditions.