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.     

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.     

On Gram-Charlier Approximations

A Gram-Charlier series involves the expansion of one probability density function in terms of the derivatives of another density function. This paper generalizes the original series introduced by Gram, an expansion that minimizes a weighted square error. Two new Gram-Charlier expansions for the density of a finite variate are derived and recursion formulas for series coefficients are given.     

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.     

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.     

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.     

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.     

Analysis of homodyne crosstalk in optical networks usingGram-Charlier series

Homodyne crosstalk with the same wavelength as the signal causes severe system performance degradation in optical networks by beating with the desired signal. Gaussian approximation is found to overestimate the system degradation. A correction to Gaussian approximation, Gram-Charlier series is used to analyze homodyne crosstalk. Both bit error rate (BER) and power penalty are calculated for multiple homodyne interferers     

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.     

Approximate inter-cell interference modeling for cellular network

In this paper,we develop and analyze several inter-cell interference modeling methods for cellular network.The models can analyze multiple interfering signals under different fading scenarios.Incoherent addition of summing multiple interfering signals is assumed.We propose an approximate method that uses the generalized Gram-Charlier series to analyze the error of the reference model.The approximate method is not only simple but also with neglectable errors.The methods proposed could be very useful in practical system design and analysis.     

Fiber Optics Receiver Error Rate Prediction Using the Gram-Charlier Series

This paper applies the Gram-Charlier series method to the caculation of error probabilties in digital optical receivers. This method allows the calculation of "exact" error probabilities including the effects of avalanche noise, thermal noise, and arbitrary posidetection processing filter. The predictions of this method are compared with those of a simple Gaussian approximation and with the Chernoff bounds. Finally, the effects of modal noise are included in the theory, and some specific cases are explored numerically.     

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.     

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.     

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.     

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。     

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.     

基于镜面散射的雷达探测浅海水下地形研究

海面镜面散射建模及其应用方法是小入射角时雷达海洋遥感的关键问题.深入研究了基于格拉母一卡利尔(Gram-Charlier)分布的海面镜面散射建模方法,分析了其极化特性,将风速统一以海面上10 m高度为基准.通过与实测数据的比较,证明了其正确性和优越性.提出了采用Wu J方法估计海面均方斜率可改进其在中低风速时的性能,指...     

Modulations numériques à grand nombre d’états dans un canal de transmission hertzienne

The implementation of high capacity (140 Mbit/s) digital radio links leads to the use of high level modulation schemes in order to reduce spectrum occupation. A comparison between these modulations (with 16, 32, 64 and 128 levels) in the presence of impairments of a real channel is interesting to determine the spectral efficiency we can reach. The error probability is given by a method based upon a Gram-Charlier expansion using only the moments of the intersymbol interference random variable.     

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     

A Novel Statistical Technique for the Estimation of DC Stability in Higher-Order Σ-Δ A/D Converters

The existing techniques available for the statistical estimation of the dc input signal stability in general-order Σ-Δ analog-to-digital (A/D) converters are based on the assumption that the constituent quantizer input signal has a Gaussian distribution. However, empirical investigations reveal that this assumption holds adequately true only for the special case of conventional first-order Σ-Δ A/D converters. This paper presents an alternative technique for the accurate estimation of the dc input signal stability for higher-order Σ-Δ A/D converters. This estimation technique is based on the practical assumption that the constituent quantizer operates in its overload-free region, permitting the characterization of the quantizer output signal digit-pattern for the determination of the statistical moments of the corresponding quantizer input signal. The resulting statistical moments are subsequently incorporated in a Gram-Charlier series for an accurate quasi-linear modeling of the quantizer. A typical application example is given to demonstrate the accuracy of the proposed statistical technique for predicting the existence of multiple regions of instability and stability in the Σ-Δ A/D converter operation, and particularly for predicting the point where the A/D converter operation becomes unstable.