Summing received signal powers with arbitrary probability density functions on a logarithmic scale

Received signal power in mobile wireless communications is typically modeled as a product of three factors: distance-dependent average path loss law, variation in the local mean power (shadow fading), and small-scale fading. Of these three factors, the least investigated is the shadow fading, which is usually explained as a result of multiplication of large number of random attenuating factors in the radio channel. In this paper, the authors propose an additive model as an alternative physical basis for shadow fading within an extended local area where path loss is constant. Starting from a sum-of-sinusoids signal model, they show that under mild statistical assumptions on the powers of the sinusoids, the resulting signal power will have approximately Gaussian distribution in logarithmic scale. A cluster-based model for shadow fading emerges as a special instance of the general result. They present simulation and measurement results that support their theoretical findings. The new physical basis for shadow fading also provides insights into simulation and modeling of radio channels

A function approximation based on an orthonormal wave function expansion in a complex space is derived. Although a probability density function (PDF) cannot always be expanded in an orthogonal series in a real space because a PDF is a positive real function, the function approximation can approximate an arbitrary PDF with high accuracy. It is applied to an actor-critic method of reinforcement learning to derive an optimal policy expressed by an arbitrary PDF in a continuous-action continuous-state environment. A chaos control problem and a PDF approximation problem are solved using the actor-critic method with the function approximation, and it is shown that the function approximation can approximate a PDF well and that the actor-critic method with the function approximation exhibits high performance.

The main argument of this paper is the study of the asymptotic behaviour for the sequence \(\{ \frac{1} {n}\sum _{j=1}^{n}Y _{ j}: n\geqslant 1\}\) when \(\{Y _{j}: j\geqslant 1\}\) are independent or pairwise uncorrelated real random variable having possibly different and finite expectations \(\{E(Y _{j}): j\geqslant 1\}\). The usual method of replacing each Y j with the difference (Y j − E(Y j )) is not interesting for our purposes: in fact the differences (Y j − E(Y j )) work in the opposite direction giving random variables with the same null expectation and convergence of $$\displaystyle{ \frac{1} {n}\sum _{j=1}^{n}(Y _{ j} - E(Y _{j}))}$$ to zero, in the general case, gives no information about the convergence of \(\frac{1} {n}\sum _{j=1}^{n}Y _{ j}\). Thus the procedure here adopted is directly based on the sequence \(\frac{1} {n}\sum _{j=1}^{n}Y _{ j}\) where the expectations \(\{E(Y _{j}): j\geqslant 1\}\) define a general sequence of finite real values satisfying no regularity assumptions, i.e. all E(Y j )’s are arbitrarily different values and the sequence {E(Y j )} is not a convergent one. The first three sections below deal with a general and intuitive presentation of the method, whereas the remaining sections contain the rigorous proofs.

We present a theory that quantifies the interplay between intrapore and interpore flow variabilities and their impact on hydrodynamic dispersion. The theory reveals that porous media with varying levels of structural disorder exhibit notable differences in interpore flow variability, characterised by the flux-weighted probability density function (PDF), $\hat {\psi }_\tau (\tau ) \sim \tau ^{-\theta -2}$ , for advection times $\tau$ through conduits. These differences result in varying relative strengths of interpore and intrapore flow variabilities, leading to distinct scaling behaviours of the hydrodynamic dispersion coefficient $D_L$ , normalised by the molecular diffusion coefficient $D_m$ , with respect to the Péclet number $Pe$ . Specifically, when $\hat {\psi }_\tau (\tau )$ exhibits a broad distribution of $\tau$ with $\theta$ in the range of $(0, 1)$ , the dispersion undergoes a transition from power-law scaling, $D_L/D_m \sim Pe^{2-\theta }$ , to linear scaling, $D_L/D_m \sim Pe$ , and eventually to logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$ , as $Pe$ increases. Conversely, when $\tau$ is narrowly distributed or when $\theta$ exceeds 1, dispersion consistently follows a logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$ . The power-law and linear scaling occur when interpore variability predominates over intrapore variability, while logarithmic scaling arises under the opposite condition. These theoretical predictions are supported by experimental data and network simulations across a broad spectrum of porous media.

By means of the multifractal analysis (MFA), the expressions of the probability density functions (PDFs) are unified in a compact analytical formula which is valid for various quantities in turbulence. It is shown that the formula can explain precisely the experimentally observed PDFs both on log and linear scales. The PDF consists of two parts, i.e., the {\it tail} part and the {\it center} part. The structure of the tail part of the PDFs, determined mostly by the intermittency exponent, represents the intermittent large deviations that is a manifestation of the multifractal distribution of singularities in physical space due to the scale invariance of the Navier-Stokes equation for large Reynolds number. On the other hand, the structure of the center part represents small deviations violating the scale invariance due to thermal fluctuations and/or observation error.

In this work, we have proposed a generative model, called VAE-KRnet, for density estimation or approximation, which combines the canonical variational autoencoder (VAE) with our recently developed flow-based generative model, called KRnet. VAE is used as a dimension reduction technique to capture the latent space, and KRnet is used to model the distribution of the latent variable. Using a linear model between the data and the latent variable, we show that VAE-KRnet can be more effective and robust than the canonical VAE. VAE-KRnet can be used as a density model to approximate either data distribution or an arbitrary probability density function (PDF) known up to a constant. VAE-KRnet is flexible in terms of dimensionality. When the number of dimensions is relatively small, KRnet can effectively approximate the distribution in terms of the original random variable. For high-dimensional cases, we may use VAE-KRnet to incorporate dimension reduction. One important application of VAE-KRnet is the variational Bayes for the approximation of the posterior distribution. The variational Bayes approaches are usually based on the minimization of the Kullback-Leibler (KL) divergence between the model and the posterior. For high-dimensional distributions, it is very challenging to construct an accurate density model due to the curse of dimensionality, where extra assumptions are often introduced for efficiency. For instance, the classical mean-field approach assumes mutual independence between dimensions, which often yields an underestimated variance due to oversimplification. To alleviate this issue, we include into the loss the maximization of the mutual information between the latent random variable and the original random variable, which helps keep more information from the region of low density such that the estimation of variance is improved.

In this paper we provide a review for the time domain analysis of the normalized timing error dynamics of a wireless receiver Early-Late tracking loop (A loop that controls the timing of a pseudo-noise code that is locally generated in the receiver). This review shows that the timing error dynamics modelling is cumbersome and can be simplified by linearizing the tracking loop. The linearization assumption makes the use of Laplace transform to model the timing error dynamics possible. The Laplace transform is used to derive formulas for the normalized timing error first and second moments for the case of linear varying time delay inputs of first order Early-Late tracking loops. The derivation of these moments facilitates modelling the timing error dynamics by using probability density functions that implement these moments to calculate the probability that the tracking loop will fail to handle the timing error dynamics. A simulation is carried out and verified that the numerical evaluation of the proposed formulas matches the simulation output to some tolerable error. The formulas are used to assess the performance of Early-Late tracking loops when subjected to linear variable time delay. The results show that the velocity of the wireless receiver, the received signal power and the chip rate are the main factors that affect the timing error first moment (the mean). The second moment (the variance) of the timing error is affected by the received signal power and the tracking loop gain.

This paper aims at channel modeling and bit error rate (BER) performance improvement with diversity reception for in-body to on-body ultra wideband (UWB) communication for capsule endoscope application. The channel characteristics are firstly extracted from 3.4 to 4.8GHz by using finite difference time domain (FDTD) simulations incorporated with an anatomical human body model, and then a two-path impulse response channel model is proposed. Based on the two-path channel model, a spatial diversity reception technique is applied to improve the communication performance. Since the received signal power at each receiver location follows a lognormal distribution after summing the two path components, we investigate two methods to approximate the lognormal sum distribution in the combined diversity channel. As a result, the method matching a short Gauss-Hermite approximation of the moment generating function (MGF) of the lognormal sum with that of a lognormal distribution exhibits high accuracy and flexibility. With the derived probability density function (PDF) for the combined diversity signals, the average BER performances for impulse-radio (IR) UWB with non-coherent detection are investigated to clarify the diversity effect by both theoretical analysis and computer simulation. The results realize an improvement around 10dB on Eb/No at BER of 10-3 for two-branch diversity reception.

Assessment of industrial components in high temperature plant using the “ALIAS-HIDA” – A case study

Introduction and summary. From the naive viewpoint of a nonprobabilist, the remarkable fact about the central limit theorem is that (under suitable conditions) the average of a large number of random variables converges (in a suitable sense) to a limit which is independenit of the things being averaged, but is completely determined by the averaging process itself. In the present paper, this situation is abstracted from its connection with probability. As we shall see, the restriction to positive functionals which characterizes probability is in no way necessary for validity of the limit theorems. Moreover, we are able to dispense completely with measure theory, in both definitions and proofs. In a purely algebraic and function-space context, we find a whole sequence of averaging processes, each with its own limit function; the first two of them generalize the weak law of large numbers and the central limit theorem. We prove convergence in three different senses, under three different sets of assumptions: first convergence of moments, then uniform convergence of characteristic functions, and finally weak convergence of distribution functions. Our first theorem is purely combinatorial; our next, essentially a calculus theorem; in our third we use function-space techniques borrowed from the modern theory of partial differential equations. By doing so, we obtain rather refined and precise results on the strongest norm in which we can assert convergence. In this respect our results may be of interest even in the classical probabilistic cases. In particular (see Corollary 2 below) we show that if a very mild extra regularity condition is imposed on the summands in the classical central limit theorem, then not only do the partial sums converge in law to the Gaussian, but every derivative of the c.d.f. of the partial sum exists and converges to the corresponding derivative of the Gaussian. Theorem 3 possesses an interpretation in terms of electrical engineering, which is given in the Remark following Corollary 2. It also has a surprising application to numerical analysis, which is described in the Postscript. We begin with an abstract algebra A over the real numbers, equipped with a linear functional Y. (If a E A were a random variable, Y(a`) would be its kth

The presence of missing values is common in real‐world datasets, so modeling and uncertainty quantification (UQ) of incomplete datasets have gained increasing attention in various research areas, including structural health monitoring (SHM). However, modeling and UQ utilizing incomplete datasets are nontrivial tasks. On the other hand, prediction based on a set of incomplete measured input variables is also an important task, but most existing methods, which are discriminative models, do not possess this capability. Aiming to tackle these two challenges, we propose the two‐stage analytical Bayesian copula‐based uncertainty quantification (A‐BASIC‐UQ) using incomplete SHM data. In the modeling stage, the copula‐based multivariate joint probability density function (PDF) is modeled directly according to an incomplete dataset without imputation or disposal of any data points. For the univariate marginal PDF, using the measured (nonmissing) values of the corresponding random variable (RV), Bayesian model class selection is conducted to select the most suitable model class. For the Gaussian copula PDF, using the bivariate complete data points of entry‐by‐entry pairwise data, the optimal parameter vector is obtained from the estimation of the Pearson correlation coefficient. In the prediction stage, the analytical expressions of the predictive PDF, the predicted value and the credible region of the output variables are derived according to a set of incomplete measured input variables. The analytical expression of the predictive PDF is obtained based on the analytical operations on the auxiliary RVs and that of the predicted value and the credible region are obtained based on the analysis of multivariate Gaussian distribution. Therefore, the proposed method does not require numerical integration nor Monte Carlo simulation and does not suffer from computational burden even when there are many variables (say 4 or above). Examples using simulated data and real SHM data are presented to illustrate the capability of the proposed A‐BASIC‐UQ.

In this paper, the k-μ-g random variable is considered. The k-μ-g random process arises from the κ-μ random process when the power of the k-μ random variable follows Gamma distribution. The closed form expressions for probability density function (PDF) and cumulative distribution function (CDF) of k-μ-g random variable are determined. Then, PDFs of the moment of n-th order of k-μ-g random variable, the ratio and product of two k-μ-g random variables are calculated in the closed form. Random variables are of crucial significance, not only for the statistical but also for the deterministic modeling of wireless mobile radio channels. The main contribution of using derived expressions is possibility to make easier performance analysis of wireless communication systems in the presence of k-μ-g short term fading, Gamma shadowing and k-μ-g cochannel interference.

Uncertainty quantification of voice signal production mechanical model and experimental updating

The risk assessment in many engineering applications is hampered by a lack of hard data. Under these conditions the selection of probability density function (PDF) seems arbitrary. Quite often the data are not only sparse but also vague expert opinion or conflicting. Several non-probabilistic methods have been proposed in the literature to perform a risk assessment under these conditions. We propose to use probabilistic techniques using uncertain PDFs. The uncertainty on the PDF is characterized by treating the parameters in the PDF as random variables. We expand the classical Bayesian updating scheme to make use of vague or imprecise interval data. Each expert opinion is considered to be a random sample from a parent distribution of expert opinions. Consequently, a conflict between experts is accounted for through the likelihood function. The uncertain PDFs can be used in both simulation-based and MPP-based reliability methods. Because of the uncertainty on the PDF of the random variables, the risk or reliability index itself will be a random variable. Design decisions are made on the basis of the risk assessment and an incorrect risk assessment increases the total cost of the design. Since a cost can be associated with either an over or underestimation of the risk, an optimal reliability index can be determined, which minimizes this cost. The probabilistic framework we present in this paper establishes a direct link between the amount and quality of the available data and the optimal reliability estimate. This link allows the decision maker to weigh expected value of additional data collection efforts against the expected optimal reliability index improvement.

In this study, product of two independent and non-identically distributed (i.n.i.d.) random variables (RVs) for κ-μ fading distribution and α-μ fading distribution is considered. The novel exact series formulas for the product of two i.n.i.d. fading distributions κ-μ and α-μ are derived instead of Fox H-function to solve the problem that Fox H-function with multiple RVs cannot be implemented in professional mathematical software packages as MATHEMATICA and MAPLE. Exact close-form expressions of probability density function (PDF) and cumulative distribution function (CDF) are deduced to represent provided product expressions and generalized composite multipath shadowing models. At last, these analytical results are validated with Monte Carlo simulations, it shows that for provided κ-μ/α-μ model nonlinear parameter has more important influence than multipath component in PDF and CDF when the ratio between the total power of the dominant components and the total power of the scattered waves is same.