An analytical method for approximate performance evaluation of binary linear block codes

An analytical method for approximate performance evaluation of binary linear block codes using an additive white Gaussian noise channel model with binary phase-shift keying modulation is presented. We focus on the probability density function of the bit log-likelihood ratio (LLR), which is expressed in terms of the Gram-Charlier series expansion. This expansion requires knowledge of the statistical moments of the bit LLR. We introduce an analytical method for calculating these moments. This is based on some recursive calculations involving certain weight enumerating functions of the code. It is proved that the approximation can be as accurate as desired, if we use enough terms in the Gram-Charlier series expansion. Numerical results are provided for some examples, which demonstrate close agreement with simulation results.     

A Convenient Multivariate Gram-Charlier Type A Series

It is often necessary to approximate the probability density function of a random variable from given statistical moments. The Gram-Charlier Type A series is one well known method for such representations. In this note, the Gram-Charlier Type A series is generalized to the multidimensional case.     

A recursive method for computing the coefficients of the Gram-Charlier series

The Gram-Charlier series is a known tool for approximating a probability density function when the moments or the cumulants of a random variable are known. A recursive procedure is presented which is well suited for the numerical computation of the coefficients of the series.     

Generalized Gram-Charlier series with application to the sum of log-normal variates (Corresp.)

A generalized Gram-Charlier series, applicable to non-Gaussian problems, is developed. Expressions are given for the first six error coefficients. The high inherent accuracy of the series is demonstrated by development of the expansion for the sum of independent, identically distributed log-normal variates.     

Online Algorithm of Blind Source Separation Based on Conjugate Gradient Method

A novel online algorithm is proposed for blind source separation based on the conjugate gradient method. The probability density function is first estimated using a Gram-Charlier expansion, and then the score function is calculated to form the algorithm. The conjugate gradient method is then used in the novel algorithm, and the line search method is applied to find the best learning rate. Simulation and comparison show the algorithm's ability to perform the separation even with an ill-conditioned mixed matrix.     

Error probability in digital fiber optic communication systems

In a digital fiber optic transmission system, during photodetection process, a shot noise is produced that is neither stationary nor independent of the digital message. The evaluation of the average error probability in the presence of such a message-dependent shot noise, of additive Gaussian noise, and of intersymbol interference is considered. Two methods of calculation are outlined: an exhaustive method and a Gram-Charlier series expansion method. The latter is preferred when the number of interferers is moderately large. Some numerical examples for binary independent-symbol transmission are presented.     

Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series

The probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded as a Gram-Charlier (G-C) series, leading to arbitrarily accurate systematic evaluation of bit-error rates (BERs) and optimal decision thresholds of optical communication systems. The method displays negligible computational complexity and is applicable whenever the moment or cumulant generating functions of the decision statistics are analytically available. We applied the technique to a birth-and-death Markovian model of a direct-detection receiver with optical preamplifier in a two-level amplitude-shift keying system. The modal expansion series rapidly converged, whereas the alternative saddlepoint approximation method predicted a BER which deviated by 7% from the G-C result.     

A Gram-Charlier series method of calculating the probability of error in lightwave transmission systems

A Gram-Charlier series method has previously been applied to the calculation of the conditional probability of error, for a fixed data sequence, in the presence of intersymbol interference, detector multiplication noise, shot noise, and thermal noise. The probability of error is obtained by averaging the conditional probability of error over all possible data sequences. In this paper it is shown how the computational efficiency of the Gram-Charlier series method can be improved by calculating the probability of error without the need for an exhaustive averaging procedure.     

Generalization of the Gram-Charlier/Edgeworth Series and Application to Time-Frequency Analysis

We derive a general equation relating probability densities and as special cases we the obtain Gram-Charlier and Edgeworth series. This allows us to generalize these methods and clarify a number of issues pertaining to both probability theory and time-frequency analysis. In particular we show how the Gram-Charlier and Edgeworth series are related to the kernel method of time-frequency analysis. The approach allows us to construct densities that satisfy given constraints such as joint moments or conditional moments. Also, we show that the kernel has to be signal dependent and that to obtain a proper distribution it should be the ratio of two characteristic functions.     

Sensitivity Study of the Cumulant Method for Evaluating Reliability Measures of Two Interconnected Systems

The rationale for using the cumulant method to take advantage of its computational efficiency is well known among power system planners. However, although an analysis of the sensitivity of the univariate Gram-Charlier series has been investigated, an equivalent analysis of the sensitivity of the bivariate Gram-Charlier series has not yet been reported in the literature. This paper investigates the sensitivity of the bivariate Gram-Charlier series in the evaluation of reliability for several types of interconnected systems. The impact of different number of terms in the series on the accuracy of the results as well as on the computational requirements is also investigated. Load correlation between the interconnected systems is considered. As anticipated, the cumulant method is much faster than the commonly used recursive method. However, the reliability indexes, obtained using this method for interconnected systems with low reserve margin and with units of low forced outage rates can not be trusted. The relative error in the calculation of the loss of load probabilities increases with the increase of tie line capacity. However, the error is greatly reduced if the systems have units of higher forced outage rate. The use of additional terms in the bivariate Gram-Charlier series increases somewhat the accuracy of the results but also increases the computational time.     

Evaluation of Gram-Charlier coefficients

A formulation of the Gram-Charlier coefficients for the expansion of a frequency distribution of data in terms of Hermite polynomials is presented. The formulation has been found to be particularly amenable to computer programming.     

基于二维概率密度函数比较的SAR图像变化检测方法

该文将传统区域统计分布特征变化检测方法拓展到2维特征空间,提出一种基于2维概率密度函数比较的SAR图像变化检测方法。该方法首先将观测区域内相邻像素的灰度值组合成2维观测矢量,而后采用2维Gram- Charlier展开式对观测矢量在不同时相图像中的2维概率密度函数分别进行估计,在此基础上,借助K-L散度理论对2维概率密度函数在不同时相图像间的变化大小进行定量分析以实现变化检测。实验结果表明,与传统方法相比,该文方法具有更优的检测性能。     

非线性混叠信号的可分离性及分离方法研究

该文分析了非线性混叠信号的可分离性及分离条件,指出现阶段非线性混叠信号盲分离的局限性。将Edgeworth展开代入信息后向传输算法中,通过一种新的自适应累积量估计方法,克服了原算法指出的Edgeworth展开在盲信号分离中的缺限。计算机仿真结果表明了所提算法在特定非线性混叠模型信号分离的效果,我们还对不同的方法进行了分析对比,指出了累积量对不同算法的影响。     

Considerations on Error Probability in Correlated-Symbol Systems

An approach to evaluate the error probability in conventional PAM digital data transmission systems with correlated symbols in the presence of intersymbol interference and additive noise is formulated in general and it is applied to the Gram-Charlier series expansion method. It is shown that the technique of conditioning few symbols before and/or after the symbol to be detected increases substantially the range of signal-to-noise ratios with an acceptable increase of numerical work. This technique also improves existing bounds on error probability, as, e.g, Glave's bound.     

Analysis of equal-gain diversity with partially coherent fadingsignals

An analytical technique based on Gram-Charlier series expansion is presented for the computation of the error probability of equal-gain combiner (EGC) with partially coherent fading signals. Imperfect carrier recovery is attributed to the random noise present in the carrier recovery loops. The resulting noisy phase references are assumed to satisfy Tikhonov distribution. The fades on the diversity branches are assumed to be slowly varying and statistically independent with Rayleigh-distributed envelopes. The error-rate performance of coherent and differentially coherent phase-shift keying (PSK) systems are compared and the phase precision requirement for a reliable coherent detection is computed. Detection loss caused by carrier phase errors is computed for several signal-to-noise ratio (SNR) reliability and bit error probability levels. It is demonstrated that the effect of carrier phase errors on the mean SNR is negligible compared to their effect on deep fades or small bit error probabilities. It is also shown that the carrier phase precision requirement can be reduced through signal combination     

Statistics of the scattering cross-section of a small number ofrandom scatterers

Complex radar targets are often modeled as a number of individual scattering elements randomly distributed throughout the spatial region containing the target. While it is known that as the number of scatterers grows large the distribution of the scattered signal power or intensity is asymptotically exponential, this is not true for a small number of scatterers. The authors study the statistics of measured power or intensity, and hence scattering cross section, resulting from a small number of constant amplitude scatterers each having a random phase. They derive closed-form expressions for the probability density function (pdf) of the scattered signal intensity for one, two, and three scatterers having arbitrary amplitudes. For n>3 scatterers, they derive expressions for the pdf when the individual scatterers have identical constant amplitudes and independent random phases; these expressions are Gram-Charlier type expansions with weighting functions determined by the asymptotic form of the intensity pdf for a large number of scatterers n. The Kolmogorov-Smirnov goodness-of-fit test is used to show that the series expansions are a good fit to empirical pdfs computed using Monte-Carlo simulation of targets made up of a small number of constant amplitude scatterers with random phase     

On Fourier Series Expansion of the Phase Density Function

This note presents a new Fourier series expansion of the probability density function of noise at the output of a phase-shift keyed receiver. A recurrence relation for the coefficients is derived and a bound on the series truncation error is obtained. The convergence rate of the series is then estimated and discussed.     

On the Fourier coefficients for the phase-shift keyed phase density function (Corresp.)

A closed-form expression is derived for the Fourier coefficients in the series expansion of the phase-shift keyed (PSK) receiver output phase probability density function.     

Estimating the Engineering Properties of Electronic Packaging Materials

Silica-filled epoxy composites represent an important class of electronic packaging materials. In this paper, a series of semi-empirical equations are proposed for estimating the density, temperature-dependant modulus, expansion coefficient and Poisson's ratio of silica-filled epoxy composites as a function of the silica content and glass transition temperature. The density and expansion coefficients are calculated using the rule of mixtures, while the composite moduli in the glassy and rubbery plateaus are derived using the Halpin-Tsai equation, the theory of rubber visco-elasticity, and elementary considerations of the polymer cross-link density. A four-parameter sigmoidal function is shown to account well for the composite stiffness in the transition region between the glassy and rubbery states, while a three-parameter single rise to maximum equation expresses the change in the composite's Poisson ratio with silica content. The models are corroborated against a large data library of actual packaging materials. Their usefulness in calculating e.g., the warpage in a plastic ball-grid array package is demonstrated in a worked example.     

Image clustering using higher-order statistics

Traditional algorithms for clustering image data have used Euclidean or Mahalanobis distance. Here, a more general higher-order statistics-based closeness measure derived from a series expansion for a multivariate probability density function in terms of the Gaussian function and the Hermite polynomials is proposed for clustering. The superiority of this measure is demonstrated with an example application