Effect of Different Deconvolution Methods on Structure Function Calculation

The reliability and thermal issues on electronic components are more and more frequent, complex and crucial for product lifecycle management. Indeed the component temperatures are close to exceed their operating limits. The integration of precise components models at board level simulation is mandatory to analyze their mutual thermal interactions in real concurrent cooling environment. The multi-port RC network approach to model components offers accurate temperature prediction at an affordable simulation time. Nevertheless, establishing a fine representation of the internal structure as well as setting its material properties is required for this approach. This article proposes a way to calibrate the model on a lifelike numerical dataset obtained by different simulations of a perfect numerical model. The study focuses on the deconvolution methods to identify RC ladders network. An iterative Bayesian deconvolution method is preferred to classic Fourier one for this application, thus a fast extraction of a Foster model from discrete time constant spectrum is achieved. Then a Foster-to-Cauer network transformation with many stages of parallel RC ladders is presented with a special attention to performances and limitations of this network conversion. A physical interpretation is possible with this new network model, and its behavior demonstrates a high agreement with the step response impedance of the studied component. This Cauer network finally allows creating the Cumulative Structure Function (CSF) and the Differential Structure Function (DSF) of the component with a great resolution. The spatial superposition of these functions on a numerical dataset of the same component mounted on different configurations of a test board permits to quantify the thermal resistance of vias structures, imperfect report contact and calibrate thermal material properties. This complete calibration procedure permits fixing model discrepancies in order to create a relevant multipath RC network including real manufacturing impact on electronic package and its assembly on PCB. The full process is validated on a usual literature case, and its relevance highlighted on numerical simulations of an electronic component.

Measurement of the time-constant spectrum: Systematic errors, correction

In this paper, we propose a Bayesian methodology for receiver function analysis, a key tool in determining the deep structure of the Earth's crust. We exploit the assumption of sparsity for receiver functions to develop a Bayesian deconvolution method as an alternative to the widely used iterative deconvolution. We model samples of a sparse signal as i.i.d. Student-t random variables. Gibbs sampling and variational Bayes techniques are investigated for our specific posterior inference problem. We used those techniques within the expectation-maximization (EM) algorithm to estimate our unknown model parameters. The superiority of the Bayesian deconvolution is demonstrated by the experiments on both simulated and real earthquake data.

Avoiding Distortion in Corporate Valuation Litigation: An Application of Discounted Cash Flow

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

We perform a global analysis of the new combined data on inclusive structure functions in electron-proton scattering at small value of Bjorken x (x ≤ 0.01) in the framework of the color glass condensate. The new combined HERA data at small-x are studied by an analytic proton structure function. It is shown that the theoretical results of the analytic proton structure function are in agreement with the new combined experimental data. The longitudinal proton structure function is investigated with the analytic expression. It is demonstrated that our approach shows a very good description of the longitudinal structure function experimental data, which is evidence showing the saturation effect is a domination mechanism during the rapidity evolution of the gluon density at high energy.

The frequency response function of structures can be modeled with pole-zero functions or as modal summations. Either representation can be used to obtain probability distributions for the magnitude of the frequency response functions. These probabiltiy functions range from trigonometric functions for simple structures to log-normal distributions for complex structures. Examples are given for simulated and measured frequency response functions of beams, plates, and vehicle structures. The statistical measures obtained from frequency averages are similar to those obtained from ensemble averages in complex structures. This suggests that a sufficiently complex structure can be considered ergodic in the frequency domain.

Coupled DGLAP equations involving singlet quark and gluon distributions are explored by a Taylor expansion as two first order partial differential equations in two variables. The system of equations are then solved by the Lagrange's method and Method of Characteristics. We obtain the proton structure function by combining the corresponding non-singlet and singlet structure functions by both the methods. Analytical solutions for proton structure function thus obtained are compared with the data available and their compatibility is checked. Data favors the analytical solution by Lagrange's method.

By using the Laplace transformation, the analytical solutions of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations at the leading-order and next-to-leading-order (NLO) approximations for the gluon, singlet, and nonsinglet quark polarization inside the nucleon are obtained. At NLO approximation, a second Laplace transformation is required. Complete NLO calculations need an iteration method, which we try to employ. As an efficiently mathematical tool, we employ the Jacobi polynomials to extract the polarized nucleon structure, using the analytical solutions for the polarized parton densities. Optimum nucleon structure functions are determined by a ${\ensuremath{\chi}}^{2}$ analysis of the available experimental data, and their uncertainties are estimated by the Hessian method. Our results for nucleon and deuteron structure functions are in good agreement with all available experimental data and fitting parametrization models.

Deconvolution methods have been widely used in machinery fault diagnosis. However, their application would be confined due to the heavy noise and complex interference since the fault feature in the measured signal becomes rather weak. Time synchronous averaging (TSA) can enhance the periodic components and suppress the others by the comb filter function. And in the iteration process of the deconvolution methods, the filtered signal after each iteration can be further processed using TSA, and the time delay with maximum Gini index value is refined as the iterative period for the next iteration. Benefitting from these advantages, a period-refined maximum second-order cyclostationarity blind deconvolution (PRCYCBD) using TSA is proposed for the weak fault detection of rolling element bearings (REBs) in this paper. Firstly, without any prior knowledge, the proposed method which can estimate the period more accurately is more suitable for the weak fault detection of REBs, especially incipient fault. Secondly, TSA is firstly applied to estimate the iterative period rather than just depending on the Signal Noise Ratio (SNR) of the filtered signal in the iterative process . Furthermore, the new improvement frame can be expanded to other deconvolution methods using iterative algorithms, especially under heavy noise. Finally, a simulation with a slight bearing fault as well as two real experimental data including the vibration signal with the wind turbine bearing fault and the acoustical signal with the locomotive wheel bearing fault is used to verify the superiority of the proposed PRCYCBD compared with the traditional minimum entropy deconvolution and the traditional autocorrelation-improved cyclostationarity blind deconvolution.

The knowledge of mechanical properties of materials is based on a precise analysis of their relaxation spectra. The development of methods to deconvolve spectra from measured data, and the assessment of their reliability, is therefore of paramount importance. We present a novel Bayesian deconvolution method based on a physically grounded parameterization of the spectra. We use a Metropolis-Hastings Markov-chain Monte Carlo fitting algorithm, with a full posterior analysis to obtain the best-fitting spectrum and its uncertainties. We test its performance on simulated data, finding that it is unbiased, reliable, and gives precise results even under strong noise.

Analytic approach to small x structure functions

Films and thin films are exploited by an ever increasing number of technologies. The properties of films can be different from those of the same material in bulk form, and can depend on the preparation process, and on thickness. Specific techniques are needed for their measurement. Whenever films or thin layers have structural functions, as in micro electro-mechanical systems (MEMS), a precise characterization of their stiffness is crucial for the design of devices. The same can be said for the design of devices which exploit thin layers to support surface acoustic waves (surface acoustic wave filters). More generally, knowledge of the elastic properties is interesting because such properties depend on the structural properties. The most widespread technique for the mechanical characterization of films, instrumented indentation, induces both elastic and inelastic strains. It also characterizes irreversible deformation, but the extraction of the information concerning the elastic behaviour is non trivial. To overcome this difficulty, several measurement methods have been developed, which exploit vibrations as a probe of the material behaviour. These methods intrinsically involve only elastic strains, and are non destructive. This is true at any length scale, and is peculiarly useful at micrometric and sub-micrometric scales, where the exploitation of other types of probes can become critical. Both propagating waves and standing waves can be exploited, with excitation which can be either monochromatic (e.g. resonance techniques) or impulsive, therefore broadband, requiring an analysis of the response either in the time domain (the so called picoseconds ultrasonics) or in the frequency domain (the so called laser ultrasonics). The propagation velocities of vibrational modes are obtained, from which the stiffness is derived if an independent value of the mass density (the inertia) is available. Older resonance techniques have been developed to be operated with thin slabs, also exploiting optical measurements of displacement. In the measurement of films and small structures, the advantages of light, a contact-less and inertia-less probe, are substantial, and are increasingly exploited. The laser ultrasonics technique, commercially available since some years ago, measures waves travelling along the film surface. The so called picosecond ultrasonics technique measures waves travelling across the film thickness; it is a relatively sophisticated optical technique, which exploits femtosecond laser pulses in a pump-and-probe measurement scheme. For best performance it needs the deposition of

A method is proposed for direct observation of a systematic frequency drift of a quantum (laser) standard in the time domain and for determination of the coefficients in the expansion of the spectral density of the flicker noise power in the frequency domain. The results of application of this method are compared with the results deduced from the structure functions.

The thermal modeling of electronic components is more and more crucial to prevent ageing phenomena when the component temperatures exceed their operating limits.At board level, the analysis of their mutual thermal interactions is done using multi-port RC networks as thermal models. Those compact models are usually extracted from a full physical representation of the component and its thermal behavior for a set of boundary conditions. Unfortunately, that fine description of the device requires a set of information about the package that is often not available.To complete the missing thermal properties, transient measurements of the junction-to-case behavior proved to be very useful. Using step function responses, a set of network identification by deconvolution of RC thermal model was conducted using iterative methods such as Bayesian ones.As a main result, a practical procedure is proposed that allows a direct extraction from discrete time constant spectrum of a very low-stage RC thermal model in the form of Foster ladder. The derived RC model demonstrates a high agreement with experimental data. The comparison is done on a set of devices mounted on a test vehicle.Further, the network identification procedure is conducted on the detailed numerical model of each device for calibration prospect. That calibration procedure of unidirectional responses permits to fix model discrepancies with the aim of creating relevant Boundary-Condition-Independent multipath RC networks.