New minimum error rate algorithms for DFE

Abstract

We propose new adaptive minimum symbol error rate algorithms (MSER) for decision feedback equalization over M‐ary PAM channels. In addition, we take into consideration biased as well as unbiased estimates leading to two major versions respectively called biased MSER (BMSER) and unbiased MSER (UMSER). The exact forms of these algorithms are computationally complex and require channel parameter information and thus must be processed off‐line. We thus modify the exact forms into stochastic and simplified versions to reduce computation load. The stochastic version requires no channel information and hence can be processed on‐line, but at the cost of convergence rate. Merits and characteristics of various versions are discussed and compared.     

The Construction of Good Linear Unbiased Estimates from the Best Linear Estimates for a Smaller Sample Size

A general procedure is found for constructing good unbiased linear estimators of the location and scale parameters of a distribution for use with an uncensored sample of size n.

It is presupposed that the coefficients of the best linear estimates are available for an uncensored sample of size m < n for the distribution under investigation. The coefficients of the proposed estimators are obtained as linear combinations of these with the aid of tabled values of the hypergeometric probability function.     

WOLD decomposition of the backscatter echo in ultrasound images of soft tissue organs

Deals with a method of detecting and estimating the scatterer spacing between the regularly spaced resolvable coherent scatterers in tissue. Scatterer spacing has been successfully used in classifying tissue structure, in differentiating between normal and cirrhotic liver, and in detecting diffuse liver disease. This paper presents a WOLD decomposition of the radio frequency (RF) field into its diffused and coherent components from which maximum likelihood estimates (MLE) or minimum mean square error (MMSE) estimates of the scattering spacing are easily computed. The MLE are efficient and for relatively long record are unbiased. They result in accurate estimates in low signal-to-noise (SNR) ratios. Unfortunately, they require nonlinear minimization and knowledge of the probability density associated with the RF backscatter echo. The MMSE estimates, on the other hand, are computationally simple, yield unique closed form solutions, do not require a-priori knowledge of the probability distribution function of the backscatter echo, and result in accurate estimates in low SNR ratios. This paper also presents an unbiased decision rule to detect whether or not an RF echo exhibits any specular scattering relative to the wavelength of the interrogating ultrasonic pulse. The approach has been tried on simulations as well as on in-vivo scans of liver data, and appears to perform well.     

Unbiased estimates of the Weibull parameters by the linear regression method

For sample sizes from 5 to 100, the bias of the scale parameter was investigated for probability estimators, P = (i − a)/(n + b), which yield unbiased estimates of the shape parameter. A class of unbiased estimators for both the shape and scale parameters was developed for each sample size. In addition, the percentage points of the distribution of unbiased estimate of the shape parameter were determined for all sample sizes. The distribution of the scale parameter was found to be normal by using the Anderson-Darling goodness-of-fit test. How the results can be used to establish confidence intervals on both the shape and scale parameters are demonstrated in the paper.     

Proof testing of ceramics

Theoretical estimates are made of strength distributions after proof testing. Assuming that the crack velocity can be expressed as a power function of the stress intensity factor,v=AK _I ⁿ , an analysis of the amount of strength loss during a load cycle is presented for single-region crack propagation. For multi-region crack propagation, a numerical analysis is used to describe strength loss. In both analyses, the effects of environment and loading rate are studied. For single region crack propagation, the strength after proof testing can be represented by two Weibull curves: one with a slope ofm at high cumulative failure probability levels; the other with a slope ofn–2 at low failure probability levels. Truncation of the strength distribution always occurs as the result of proof testing; the truncation strength depends on the rate of unloading. Multi-region crack propagation results in a more complicated strength distribution after proof testing. Bimodal strength distributions occur as a consequence of region II type crack growth (i.e.n=0). Theoretical results confirm experimental findings that proof tests must be conducted at rapid unloading rates and with good environmental control to be effective.     

Efficient Energy Optimized Faithful Adder with Parallel Carry Generation

Approximate computing has received significant attention in the design of portable CMOS hardware for error-tolerant applications. This work proposes an approximate adder that to optimize area delay and achieve energy efficiency using Parallel Carry (PC) generation logic. For ‘n’ bits in input, the proposed algorithm use approximate addition for least n/2 significant bits and exact addition for most n/2 significant bits. A simple OR logic with no carry propagation is used to implement the approximate part. In the exact part, addition is performed using 4-bit adder blocks that implement PC at block level to reduce node capacitance in the critical path. Evaluations reveal that the maximum error of the proposed adder confines not more than 2^n/2. As an enhancement of the proposed algorithm, we use the Error Recovery (ER) module to reduce the average error. Synthesis results of Proposed-PC (P-PC) and Proposed-PCER (P-PCER) adders with n-16 in 180nm Application Specific Integrated Circuit (ASIC) PDK technology revealed 44.2% & 41.7% PDP reductions and 43.4% & 40.7% ADP reductions, respectively compared to the latest best approximate design compared. The functional and driving effectiveness of proposed adders are examined through digital image processing applications.     

Quantitative Error Assessment in Pattern Matching: Effects of Intensity Pattern Noise, Interpolation, Strain and Image Contrast on Motion Measurements

Abstract: Basic concepts in probability are employed to develop analytic formulae for both the expectation (bias) and variance for image motions obtained during subset‐based pattern matching. Specifically, the expectation and variance in image motions in the presence of uncorrelated Gaussian intensity noise for each pixel location are obtained by optimising a least squares intensity matching metric. Results for both 1D and 2D image analyses clearly quantify both the bias and the covariance matrix for image motion estimates as a function of: (a) interpolation method, (b) sub‐pixel motion, (c) intensity noise, (d) contrast, (e) level of uniaxial normal strain and (f) subset size. For 1D translations, excellent agreement is demonstrated between simulations, theoretical predictions and experimental measurements. The level of agreement confirms that the analytical formulae can be used to provide a priori estimates for the ‘quality’ of local, subset‐based measurements achievable with a given pattern. For 1D strain with linear interpolation, theoretical predictions are provided for the expectation and co‐variance matrix for the local displacement and strain parameters. For 2D translations with bi‐linear interpolation, theoretical predictions are provided for both the expectation and the co‐variance matrix for both displacement components. Theoretical results in both cases show that the expectations for the local parameters are biased and a function of: (a) the interpolation difference between the translated and reference images, (b) magnitude of white noise, (c) decimal part of the motion and (d) intensity pattern gradients. For 1D strain, the biases and the covariance matrix for both parameters are directly affected by the strain parameter p₁ as the deformed image is stretched by (1 + p₁). For 2D rigid body motion case, the covariance matrix for measured motions is shown to have coupling between the motions, demonstrating that the directions of maximum and minimum variability do not generally coincide with the x and y directions.     

Estimation of stochastic environment force for master–slave robotic system

The aim of this work is to obtain the maximum likelihood estimate (MLE) of controller-gain parameters \(\widehat{K} \) of the slave robot to determine the stochastic environment force. This is accomplished by measuring the joint positions of master and slave for a known master torque using stochastic difference equation. Here, the environmental force is modelled as a zero-mean white Gaussian random process. Therefore, the joint probability distribution function (pdf) of the slave angle over a given time duration can be computed as a function of the parameters ‘K’. This pdf is maximized with respect to ‘K’ to obtain the MLE of controller-gain parameters. Subsequently, convergence analysis of error in the estimates is performed. Also, an expression of the Cramer–Rao lower bound (CRLB) is derived to measure accuracy of the estimation. Comparison of CRLB with variance of MLE supports that our estimates are asymptotically efficient. The estimation performance is validated analytically and through simulations carried out on a two-link master–slave robotic system.     

Inference with progressively censored <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system lifetime data

A system with n independent components which works if and only if a least k of its n components work is called a k-out-of-n system. For exponentially distributed component lifetimes, we obtain point and interval estimators for the scale parameter of the component lifetime distribution of a k-out-of-n system when the system failure time is observed only. In particular, we prove that the maximum likelihood estimator (MLE) of the scale parameter based on progressively Type-II censored system lifetimes is unique. Further, we propose a fixed-point iteration procedure to compute the MLE for k-out-of-n systems data. In addition, we illustrate that the Newton–Raphson method does not converge for any initial value. Finally, exact confidence intervals for the scale parameter are constructed based on progressively Type-II censored system lifetimes.     

Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables

Very recently, control charts for monitoring the ratio of 2 normal variables have been investigated in statistical process control. In the two‐sided case, however, these control charts tend to be average run length (ARL) biased, in the sense that some out‐of‐control ARL values are larger than the in‐control ARL. This paper proposes an ARL‐unbiased EWMA control chart for monitoring of this kind of ratio with each subgroup consisting of n?1 sample units. Also, to study the long‐term properties of ARL‐unbiased EWMA‐RZ control chart, we investigate the steady‐state ARL. Several tables and figures are given to show the statistical properties of the proposed control charts. The comparison results show that the proposed ARL‐unbiased chart outperforms other two‐sided control charts in terms of the zero‐state and steady‐state ARL. An example illustrates the use of this chart on a real quality control problem from the food industry.     

Weight functions and stress intensity factors for the single crack round-ended straight lug

Investigations were performed for the round-ended straight attachment lug with a single crack emanating from the hole with the weight function method. The weight functions, covering the geometries from W/D=1.5 to W/D=4.0, were generated from the results obtained with a boundary element method using the approximate weight function technique. The results have been given both in the form of analytical weight functions and tabulated dimensionless stress intensity factors for simple normalized powers of the crack line loading. This is a simple straight forward procedure to calculate stress intensity factors once the crack line loading is approximated by a polynomial. The present method is also valid for deriving stress intensity factors and weight functions for general crack configurations.     

An optimization algorithm based on stochastic sensitivity analysis for noisy objective landscapes

A function minimization algorithm that updates solutions based on approximated derivative information is proposed. The algorithm generates sample points with Gaussian white noise, and approximates derivatives based on stochastic sensitivity analysis. Unlike standard trust region methods which calculate gradients with n or more sample points, where n is the number of variables, the proposed algorithm allows the number of sample points M to be less than n. Furthermore, it ignores small amounts of noise within a trust region. This paper addresses the following two questions: how does the derivative approximation become worse when the number of sample points is small? Can the algorithm find good solutions with inexact derivative information when the objective landscape is noisy? Through intensive numerical experiments using quadratic functions, the algorithm is shown to be able to approximate derivatives when M is about n/10 or more. The experiments using a formulation of the traveling salesman problem show that the algorithm can find reasonably good solutions for noisy objective landscapes with inexact derivatives information.     

The stress intensity factor for grooved DCB specimens loaded by splitting forces

An approximate analysis is used to obtain the stress intensity factors K for DCB specimens with a groove, loaded by splitting forces P. A review is given for the various approaches that have been taken to obtain the stress intensity factor for an ungrooved DCB specimen. The accuracy of the results is evaluated using the exact solution of K, which was finally obtained by Foote and Buchwald [1]. A so-called improved beam analysis is used to obtain K for a rectangular grooved DCB specimen. When the specimen is loaded by bending moments M, the solution of the stress intensity factor is given by Wu et al. [4].     

Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

A refined statistical approach to thermal fatigue life prediction

Finding new applications for ceramic materials requires a better knowledge of thermal fatigue behaviour. However, result-scattering inherent to thermal fatigue and duration of a thermal fatigue cycle lead to a lack of experimental results. For these reasons, we have developed a new approach that permits the determination of a relevant stress intensity factor exponent n with a minimum testing sample number. From knowledge of the distribution function of artificial cracks, the analytical formula of the failure probability F(N) can be completely determined. Thus, it is possible to calculate n from a correlation of F(N) with experimental results obtained for only one temperature difference. Correlations between theoretical curves F(N) and experimental results, conducted for two temperature differences, lead to the same value of n. This and the good agreement between the experimental points and the theoretical curves validate this new approach.     

Accuracy Analysis of the Estimated Process Yield Based on Spk

Process yield has been the most basic and common criterion used in the manufacturing industry as a base for measuring process performance. Boyles considered a measurement formula called S_pk, which establishes the relationship between the manufacturing specification and the actual process performance, providing an exact (rather than approximate) measure of process yield. Unfortunately, the sampling distribution and the associated statistical properties of S_pk are analytically intractable. In this paper, we consider the natural estimator of the measure S_pk. We investigate the accuracy of the natural estimator of S_pk computationally, using a simulation technique to find the relative bias and the relative mean square error for some commonly used quality requirements. Extensive simulation results are provided and analyzed, which are useful to the engineers for factory applications in measuring process performance. Copyright © 2004 John Wiley & Sons, Ltd.     

Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach

We consider goodness-of-fit tests for symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponent α estimated from the data. We treat α as an unknown parameter, but for theoretical simplicity we also consider the case that α is fixed. For estimation of parameters and the standardization of data we use the maximum likelihood estimator (MLE). We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distribution of the test statistic is obtained using complex integration. We find that if the sample size is large the calculated asymptotic critical values of test statistics coincide with the simulated finite sample critical values. Finite sample power of the proposed test is examined. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.     

Condensate Fraction in Liquid 4He

The condensate fraction n ₀ in the ground state of liquid ⁴He is computed by means of an unbiased Quantum Monte Carlo technique. Unlike previous calculations of the same type, the method adopted here is in principle exact, errors being only statistical. At the equilibrium density, our estimate for n ₀ is 0.069 ± 0.005; this is consistent with the most recent experimental measurements, but significantly lower than most existing theoretical estimates. Results are provided in the density range from the spinodal (where n ₀ is maximum, ≈30%) up to the melting density. The condensate fraction remains finite in the high pressure metastable liquid, decaying exponentially with density.     

Effect of fiber damage on the overall electrical conductivity of bare carbon fiber strand

The simple method developed by Kachanov (1985) for multiple interacting cracks in homogenous medium is extended to predict complex stress intensity factor for multiple split type interface cracks. Calculations are implemented for two equal cracks and infinite row of periodic cracks at the interface between two dissimilar isotropic materials. Results for infinite row of cracks are compared against the exact analytical solution provided by Sih (1973). The approximate method leads to the results very close the exact solution for crack density up to 0.90 (relative error is less than 3.8% for real part of stress intensity factor) and material dissimilarity does not have a major influence on the error. For crack densities higher than 0.90, the influence of material dissimilarity is more evident and the error increases as material dissimilarity increases. The promising match between the approximate and exact method proves the capability of the approximate method for solving other interacting interface crack problems, such as multiple penny-shaped interface cracks, in which the solution is not obtained in the literature yet.     

Biaxial-load effects on isopachics for a crack at the interface of dissimilar media

Summary The plane problem of two dissimilar materials, bonded together and containing a crack along their common interface, which were subjected to a biaxial load at infinity, is examined by giving a closed-form expression for the first stress invariant of the normal stresses, which is equally valid everywhere, near to, and far from, the crack-tip region. This exact expression for the first-stress invariant is compared by constructing the respective isopachic-fringe patterns, to the approximate expression with non-singular terms, due to the biaxiality factor, for the same quantity. Significant differences between respective isopachic-patterns were found and their dependence on the elastic properties of both materials and the applied loads was demonstrated. The relative errors between the computedK _I - andK _II -components by using the approximate expression for the first stress-invariant and the accurate one, derived from closed-form solution along either isopachic-fringes or along circles and radii from the crack-tip have been given, indicating in some cases large discrepancies between exact and approximate solutions.With 7 Figures