Envelope Probability Density Function Research Articles

Multimodality quantitative ultrasound envelope statistics imaging based support vector machines for characterizing tissue scatterer distribution patterns: Methods and application in detecting microwave-induced thermal lesions

Ultrasound envelope statistics imaging, including ultrasound Nakagami imaging, homodyned-K imaging, and information entropy imaging, is an important group of quantitative ultrasound techniques for characterizing tissue scatterer distribution patterns, such as scatterer concentrations and arrangements. In this study, we proposed a machine learning approach to integrate the strength of multimodality quantitative ultrasound envelope statistics imaging techniques and applied it to detecting microwave ablation induced thermal lesions in porcine liver ex vivo. The quantitative ultrasound parameters included were homodyned-K α which is a scatterer clustering parameter related to the effective scatterer number per resolution cell, Nakagami m which is a shape parameter of the envelope probability density function, and Shannon entropy which is a measure of signal uncertainty or complexity. Specifically, the homodyned-K log10(α), Nakagami-m, and horizontally normalized Shannon entropy parameters were combined as input features to train a support vector machine (SVM) model to classify thermal lesions with higher scatterer concentrations from normal tissues with lower scatterer concentrations. Through heterogeneous phantom simulations based on Field II, the proposed SVM model showed a classification accuracy above 0.90; the area accuracy and Dice score of higher-scatterer-concentration zone identification exceeded 83% and 0.86, respectively, with the Hausdorff distance <26. Microwave ablation experiments of porcine liver ex vivo at 60–80 W, 1–3 min showed that the SVM model achieved a classification accuracy of 0.85; compared with single log10(α),m, or hNSE parametric imaging, the SVM model achieved the highest area accuracy (89.1%) and Dice score (0.77) as well as the smallest Hausdorff distance (46.38) of coagulation zone identification. We concluded that the proposed multimodality quantitative ultrasound envelope statistics imaging based SVM approach can enhance the capability to characterize tissue scatterer distribution patterns and has the potential to detect the thermal lesions induced by microwave ablation.

Nonlinear random vibration of the slender deep-water pier under seismic excitation

The present study investigates the nonlinear random vibration of the deep-water pier exposed to horizontal seismic excitation. First of all, the stochastic dynamic model of the pier is formulated. During the process, the pier is simplified as a cantilever beam fixed on the rock foundation, the seismic excitation is treated as Gaussian white noise, the hydrodynamic pressure is described with the radiation wave theory, and the equation for the nonlinear kinematic of the pier is deduced by the means of Kane’s method. Then, with the application of the stochastic averaging (SA) technique, the Fokker–Plank–Kolmogorov (FPK) equation governing the transient probability density function (PDF) of the amplitude envelope and the backward Kolmogorov (BK) equation for the conditional reliability function (CRF) are derived, respectively. The closed-form stationary PDF can be yielded directly from the reduced FPK equation, while the CRF and the conditional PDF of first-passage time are given after solving the BK equation numerically. Numerical discussions are performed to illustrate the trend of excitation intensity, mass ratio, immersion ratio, and inner and outer hydrodynamic effect on the stationary response and first-passage failure are examined, respectively. It has been shown that increases in the excitation intensity, mass ratio, and immersion ratio can amplify the response and reduce the reliability of the deep-water pier system. The hydrodynamic effect also leads to an amplification of the system response and a reduction in the reliability of the system. Similarly, the presence of inner water can also exacerbate these effects, and this phenomenon becomes more pronounced with increasing immersed ratios. Additionally, the analytical solution is validated by the result obtained by pertained Monte Carlo simulations (MCS). It is noted that this work will be helpful for the optimal seismic design of deep-water piers.

Estimation of Truncation Error in Statistical Description of Communication Signals over mm-Wave Channels

This paper presents an illustration of how knowledge from the field of special functions, orthogonal polynomials and numerical series can be applied to solve a very important problem in the field of modern wireless communications. We present the formulas for the probability density function (PDF) and cumulative distribution function (CDF) of the composite signal envelope over an mm-Wave channel. The formulas for the PDF and CDF are expressed in the convergent infinity series form. The main contribution of the paper is in estimating the upper bounds for absolute truncation error in evaluating PDF and CDF of the signal envelope. We also derive the formulas for the required number of terms in the summation under the condition of achieving a given accuracy for typical values of channel parameters. In deriving these formulas, we use the alternating series estimation theorem, as well as some properties of orthogonal polynomials in order to derive upper bounds for hypergeometric functions. Based on the newly derived formulas, numerical results are presented and commented upon. The analytical results are verified by Monte Carlo simulations. The results are essential in the designing and performance estimating of the fifth-generation (5G) and beyond wireless networks.

A New Look at the $\eta$-$\mu$ Fading Model

In this paper, the $\eta$ - $\mu$ fading model is looked at with a unifying perspective, through which correlation and power imbalance between its quadrature processes are simultaneously taken into account. It is formally shown that these two phenomena, quantified by two respective parameters, do not alter the resulting envelope statistics but have a great impact on the corresponding phase. To this end, an exact, simple, closed-form formulation is found that maps these parameters into a resulting $\eta$ parameter that keeps the envelope probability density function (PDF) unaltered. An exact formulation is also found for the phase PDF, given in easily computable infinite series. Closed-form approximation for such a PDF is also found that yields excellent results as compared with the exact one. As a particular case of the general phase formulation, the phase of the Hoyt process is extended to include power imbalance and correlation, and this is given in simple closed-form. Additionally, and as a by-product, some useful mathematical identities are obtained.

Whistling of deep cavities subject to turbulent grazing flow: intermittently unstable aeroacoustic feedback

Abstract

The α-η-F and α-κ-F Composite Fading Distributions

In this letter, we present the α-η-F and α-κ-F composite fading distributions. The two distributions generalize the two well-known composite fading distributions, namely the η-μ/inverse gamma and the κ-μ/inverse gamma distributions. For both distributions, we derive new exact closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the instantaneous signal-to-noise ratio (SNR). Additionally, the outage probabilities (OPs) of a wireless communication system operating over the new composite fading channels are obtained. Moreover, we derive asymptotic OPs, in the high SNR regime, to gain more insight into the influence of fading parameters. Some numerical results on the envelope PDFs and OPs are provided and compared with Monte-Carlo simulation results to validate the analyses.

The α-η-κ-F Composite Fading Distribution

In this letter, a new general composite fading model, namely $\alpha $ - $\eta $ - $\kappa $ - $\mathcal {F}$ model, is proposed. It considers most of the well-known propagation phenomena in wireless fading channels, such as shadowing, multi-path fading, non-linearity of the propagation medium, power of the dominant components, and power of the scattered waves. Additionally, most of the important statistical fading models are included in the $\alpha $ - $\eta $ - $\kappa $ - $\mathcal {F}$ model as special cases. The envelope probability density function (PDF) and cumulative distribution function (CDF) are derived and then employed to derive the PDF and CDF of the instantaneous signal-to-noise ratio. The performance of a wireless communication system operating under the $\alpha $ - $\eta $ - $\kappa $ - $\mathcal {F}$ composite model is evaluated in terms of outage probability and symbol error rate. Numerical results are supported by Monte-Carlo simulations to validate the analysis.

Random vibration of SDOF vibro-impact oscillators with restitution factor related to velocity under wide-band noise excitations

Hertz contact model has not included the impact of inelastic, while the modified Hertz contact models are too complicated to handle. The classical impact model only uses a material-related constant to describe the relationship between the velocity before and after a collision, without considering the energy dissipation mechanism in the impact process. On the other hand, many recent experiments have shown that the restitution factor was a function of impact velocity. Such impact velocity dependent restitution factor offers a comprehensive and reasonable impact model. In the present paper, the random vibration of single-degree-of-freedom (SDOF) vibro-impact oscillators with the restitution factor related to velocity under wide-band noise excitations is investigated. The given stochastic vibro-impact oscillator is firstly transformed into a nonlinear system without impact by a non-smooth transformation. Then, a Markov approximation is applied to obtain the Fokker–Plank-Kolmogorov (FPK) equation governing the evolution of the probability density function (PDF) of the energy envelope. The stationary PDFs of the energy envelope and amplitude envelope are solvable in a closed-form. Finally, two examples are given to verify the effectiveness of the proposed method. The results showed that the classical impact model can simplify the calculations to some extent. However, the error caused by this method cannot be ignored. The revised model presented in this paper is more in line with the actual working conditions. For complex dynamical system, especially for multi-point impact vibration systems, the work of this paper is helpful to the optimization of the system.

Accuracy evaluation of quantitative diagnosis method of liver fibrosis based on multi-Rayleigh model using optimal combination of input moments

We have been developing a quantitative estimation method of liver fibrosis using the multi-Rayleigh model to express a probability density function of the echo envelope. In the estimation, a combination of moments of the echo amplitude is used as input to estimate model parameters. To realize a robust quantification of liver fibrosis stage, we searched for the optimal combination of moments in the complex space. By using the estimated results in the multi-Rayleigh model parameters’ space, we found the optimal combination of moments in real and imaginary spaces that can improve the estimation accuracy of multi-Rayleigh model parameters. From the results of the accuracy analysis, it was found that it is difficult to estimate the mixture ratio for the early stage of fibrosis even if the optimal combination of moments is used. However, estimation of the variance ratio is accurate enough to quantify fibrosis and can provide useful information. It became clear that the quantification method using the multi-Rayleigh model is effective.

Stochastic Response of SDOF Self-Centering System

As a new type of seismic resisting device, the self-centering system is attractive due to its excellent re-centering capability, but research on such a system under random seismic loadings is quite limited. In this paper, the stochastic response of a single-degree-of-freedom (SDOF) self-centering system driven by a white noise process is investigated. For this purpose, the original self-centering system is first approximated by an auxiliary nonlinear system, in which the equivalent damping and stiffness coefficients related to the amplitude envelope of the response are determined by a harmonic balance procedure. Subsequently, by the method of stochastic averaging, the amplitude envelope of the response of the equivalent nonlinear stochastic system is approximated by a Markovian process. The associated Fokker–Plank–Kolmogorov (FPK) equation is used to derive the stationary probability density function (PDF) of the amplitude envelope in a closed form. The effects of energy dissipation coefficient and yield displacement on the response of system are examined using the stationary PDF solution. Moreover, Monte Carlo simulations (MCS) are used for ascertaining the accuracy of the analytical solutions.

The $\alpha$ -$\eta$ -$\kappa$ -$\mu$ Fading Model: New Fundamental Results

A number of new results aiming at facilitating the use of the $\alpha $ - $\eta $ - $\kappa $ - $\mu $ fading model are presented. These results include: 1) fast convergent, with no recursions, series representations for the envelope probability density function (PDF) and cumulative distribution function (CDF); 2) higher order moments of the envelope; 3) moment generating function (MGF) of the envelope; 4) asymptotic behavior of the envelope PDF and also of the envelope CDF; 5) a procedure for parameter estimation; 6) new closed-form expressions for particular cases of the envelope distribution; 7) new mathematical identity for the generalized Laguerre polynomial and the generalized hypergeometric function; and 8) approximate, but tight, series representation for the phase PDF. As application examples, the following are given: 1) outage probability; 2) outage capacity; 3) amount of fading; and 4) bit error rate, for which the asymptotic behavior is also presented.

Examination of optimal moments as input parameters for evaluation of liver fibrosis based on multi-Rayleigh model

The diagnosis of liver fibrosis using an ultrasound B-mode image has the advantages of real-time observation and noninvasive properties. We proposed a multi-Rayleigh model to express a probability density function (PDF) of echo envelope from a fibrotic liver. By using the multi-Rayleigh model, fibrotic parameters can be estimated. To quantitatively evaluate liver fibrosis using the multi-Rayleigh model, it is important to establish a high estimation accuracy method of multi-Rayleigh model parameters for clinical data. In this paper, by using the simulated ultrasound B-mode image, the relationship between the moments of echo data as input parameters for the evaluation of liver fibrosis based on the multi-Rayleigh model and the estimation accuracy of the multi-Rayleigh model parameters was examined. From the simulation results, we can determine the optimal combination of moments that can improve the estimation accuracy of the multi-Rayleigh model by focusing on the distribution of estimated values in the multi-Rayleigh model parameters’ space.

Initial Results from SQUID Sensor: Analysis and Modeling for the ELF/VLF Atmospheric Noise.

In this paper, the amplitude probability density (APD) of the wideband extremely low frequency (ELF) and very low frequency (VLF) atmospheric noise is studied. The electromagnetic signals from the atmosphere, referred to herein as atmospheric noise, was recorded by a mobile low-temperature superconducting quantum interference device (SQUID) receiver under magnetically unshielded conditions. In order to eliminate the adverse effect brought by the geomagnetic activities and powerline, the measured field data was preprocessed to suppress the baseline wandering and harmonics by symmetric wavelet transform and least square methods firstly. Then statistical analysis was performed for the atmospheric noise on different time and frequency scales. Finally, the wideband ELF/VLF atmospheric noise was analyzed and modeled separately. Experimental results show that, Gaussian model is appropriate to depict preprocessed ELF atmospheric noise by a hole puncher operator. While for VLF atmospheric noise, symmetric α-stable (SαS) distribution is more accurate to fit the heavy-tail of the envelope probability density function (pdf).

Quantitative fibrotic imaging based on multi-Rayleigh model for ultrasound B-mode image of liver fibrosis

A quantitative diagnostic method for liver fibrosis using ultrasound B-mode image is highly required. To quantitatively evaluate liver fibrosis using ultrasound B-mode image, we have been focusing on the echo amplitude distribution properties. A probability density function (PDF) of echo envelope from a normal liver can be approximated by a Rayleigh distribution; however, the PDF of echo envelope from a fibrotic liver deviates from the Rayleigh distribution. To approximate the PDF of liver fibrosis, we have been proposing the multi-Rayleigh model which is a combination of Rayleigh distributions with different variances. By using the multi-Rayleigh model, the ultrasound B-mode image of liver fibrosis could be converted to the quantitative fibrotic probability image. In this study, we applied the fibrotic probability imaging method to clinical data which is classified into F0 (normal liver), F1 (early-stage hepatitis), F2 (moderate hepatitis), and F3 (late hepatitis) in accordance with a new Inuyama classif...

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

On the origin of heavy-tail statistics in equations of the Nonlinear Schrödinger type

We study the formation of extreme events in incoherent systems described by the Nonlinear Schrödinger type of equations. We consider an exact identity that relates the evolution of the normalized fourth-order moment of the probability density function of the wave envelope to the rate of change of the width of the Fourier spectrum of the wave field. We show that, given an initial condition characterized by some distribution of the wave envelope, an increase of the spectral bandwidth in the focusing/defocusing regime leads to an increase/decrease of the probability of formation of rogue waves. Extensive numerical simulations in 1D+1 and 2D+1 are also performed to confirm the results.

Performance of Dual-EGC receivers over TWDP fading channels

The importance of equal gain combining receiver is significant because of its better performance and lower complexity. The expression for the probability density function (PDF) of the output signal to noise ratio of dual branch EGC receiver is obtained from the envelope PDF. The expressions of outage probability and average symbol error rate for both coherent and non-coherent M-ary modulation schemes are derived using PDF based approach. The expressions obtained are numerically evaluated. These analytical results are validated with Monte Carlo simulation. Study have been done to evaluate the impact of fading parameters K and $$\Delta $$Δ on the performance of Dual-EGC receiver.

Unified Error Performance Analysis of Space Shift Keying Modulation for MISO and MIMO Systems Under Generalized Fading

The performance of space shift keying modulation is strongly affected by the signal envelopes as well as by the channel induced phase variations. Motivated by this fact, this letter has a twofold motivation: 1) to propose a Hankel transform approach to transform the average error-rate integral for a Multiple-Input-Single-Output system with Maximum-Likelihood detection and perfect Channel State Information into the Hankel transform domain using Parseval theorem; 2) to derive simple analytical expressions for the Average Bit Error Probability of a generic Multiple-Input-Multiple-Output wireless system employing space shift keying modulation, by using an accurate exponential bound for the Gaussian Q-function along with the theory of Hankel transforms; Both methods, require only the knowledge of the Hankel transform of the probability density function of the signal envelope, which is readily available. The proposed analysis is tested and verified by numerically evaluated results accompanied with Monte-Carlo simulations.

Modelling and simulation of a generalised vehicle‐to‐vehicle fading channel

This study presents statistical properties of the vehicle-to-vehicle Nakagami–Hoyt channel model with diffused line of sight (LOS) under isotropic condition. The envelope and phase probability density functions, spatial-time autocorrelation function, the power spectral density, level crossing rate, the average duration of fade, squared time autocorrelation function and mean-square error of time autocorrelation function of the Nakagami–Hoyt channel have been derived under the assumption that both the transmitter and receiver are mobile having omnidirectional antennas and a diffused LOS component is present. The model has been validated using the inverse fast Fourier transform-based simulator. The simulator and analytical results corroborate.

Stability of Quantitative Evaluation Method of Liver Fibrosis Using Amplitude Distribution Model of Fibrotic Liver

In the clinical diagnosis of liver fibrosis using ultrasound B-mode images, there are some differences between individual doctors' diagnosis results. Hence, the realization of a quantitative diagnosis method using ultrasonic echo signals is strongly required. The probability density function (PDF) of the echo signal envelope is an important factor for ultrasound tissue characterization (TC). To realize the quantitative diagnosis of the stage of liver fibrosis, we proposed an amplitude distribution model using two Rayleigh distributions. We have been studying the possibility of the quantitative estimation of liver fibrosis by using this model. In this paper, we present the evaluation results of liver fibrosis for clinical data using the amplitude distribution model. Then, we present the stability of the quantitative estimation method of liver fibrosis using the amplitude distribution model by a simulation, and compare the results using the simulation and clinical data. Stability evaluation has enabled the estimation of the progress of liver fibrosis considering statistical dispersion.