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.     

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.     

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.     

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     

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.     

Concatenated coding performance for FSK modulation ontime-correlated Rician fading channels

We present an analytical method for evaluating the performance of noninterleaved concatenated codes over channels modeled as a nonfrequency selective correlated Rician fading channel with a known power spectral density. The main idea is to model the communication system from the modulator input to the demodulator output as a finite state channel (FSC) model, and apply powerful enumeration techniques to such a discrete channel in order to gain useful information on the system performance. The concatenated scheme makes use of two codes; Reed-Solomon codes are employed for the outer code, and binary block codes are used as the inner code. Next, the method is extended to study the effect on the performance when an interleaving with finite depth is incorporated into the communication system. A comparison between symbol and bit interleaving is made. Finally, we study the potential gain produced when channel information is passed on to the outer decoder in the form of an erasure symbol. In all cases, analytical expressions for the probability of the number of error symbols produced by the FSC model were obtained in terms of a coefficient in a formal power series. This is an interesting alternative approach with respect to computer simulations     

Binary communications through noisy, non-Gaussian channels

New and unifying analytical tools are developed and used to evaluate the bit error probability, false alarm and detection probabilities that result when binary information is communicated through a random channel further disturbed by additive white Gaussian noise. The class of channels modeled here are those which envelop the received electric field with an arbitrary space-time complex envelope. The complex Gaussian envelope, being a special case, yields the Rayleigh and Rice fading statistics. Considerable insight into the problem of communicating through a complex non-Gaussian fading channel is obtained by decomposing the performance measures into the sum of two terms, viz., one attributable to the usually assumed complex Gaussian envelope plus a residual performance term expressed as a series expansion in terms of multidimensional Hermite polynomials whose coefficients are the channel quasi-moments. Finally, a numerical example is presented in which the theory is applied to a specific non-Gaussian channel     

一种利用软信息网络编码在双向中继网络进行估计转发的方法

 在本文中,我们为双向中继网络(Two-way Relay Networks)设计了一种崭新的估计-转发(EF:Estimate-and-Forward)方法.在中继端信道解码后首先得到每一比特的对数似然比(LLR:Log-Likelihood Ratio),然后利用两个方向的比特LLR进行软信息网络编码,并借此构造出中继端的发送信号.因为此方法保留了两端比特信息的软信息并抑制了噪声,所以在任何条件下它的性能都要优于传统的放大-转发(AF:Amplify-and-Forward)方式和解码-转发(DF:Decode-and-Forward)方式,文中详细推导了三种转发方式的最大可达速率,得到EF方式下最大可达速率的上界和下界,还证明了EF方式较其它两种方式节省功率消耗,并通过仿真得到了速率曲线与误码率曲线,明确的验证了EF的优势.     

基于Kaczmarz迭代的大规模MIMO系统低复杂度软输出信号检测

大规模MIMO系统上行链路中,最小均方误差（MMSE）算法能获得接近最优的线性检测性能,但是涉及复杂度较高的矩阵求逆运算.本文基于Kaczmarz迭代提出一种低复杂度软输出信号检测算法,在算法实现中避免了矩阵求逆运算,将实现复杂度由O（K³）降为O（K²）.同时,引入了最优松弛参数进一步加快算法收敛,最后给出了两种用于信道译码的LLR的近似计算方法.仿真结果表明：所提出的Kaczmarz迭代软输出信号检测算法经过两到三次简单的迭代即可较快地收敛,并达到接近MMSE检测算法的误码率性能的水平,其性能与复杂度均优于基于矩阵近似求逆的一类检测算法.     

Bit Error Rate Evaluation for Spread-Spectrum Multiple-Access Systems

A unified method for approximating and bounding the average bit error probability for spread-spectrum multiple-access communication systems is presented. Various forms of direct-sequence spreadspectrum modulation are considered including binary phase-shift keying, quadriphase-shift keying, and minimum-shift keying. The analysis of the multiple-access interference makes use of a number of moments sufficent to evaluate the error probability with a high degree of accuracy. A computationally efficient algorithm for computing the moments is also given. The subsequent transformation from the moments to the average bit error probability is carried out by means of Gauss-type numerical integration formulas. It is shown that the same approach can be exploited for evaluating two classes of upper and lower bounds on the bit error rate. Finally, some results and comparisons are reported.     

Full-wave space-domain analysis of open microstrip discontinuitiesincluding the singular current-edge behavior

A full-wave space-domain analysis is presented for the high-frequency characterization of microstrip discontinuities. This approach solves the electric field integral equation (EFIE) for the surface current density on the microstrip using the method of moments. The current expansion functions incorporate the singular edge behavior of the surface current, yielding a very accurate current modeling. Special attention is devoted to the analytical treatment of the singular terms in the electric field Green's dyadic. The numerical results focus on the S-parameters of some simple microstrip discontinuities and the comparison with results obtained with other techniques and from measurements     

PAPR reduction for LDPC coded OFDM systems using binary masks and optimal LLR estimation

A probabilistic PAPR reduction method using binary masks is proposed for OFDM systems. The binary masks are used to generate multiple signal candidates containing the same information. The candidate with the lowest PAPR is selected for transmission. In the presence of a non-linear amplifier (soft limiter), as the number of candidates increases, the PAPR is reduced, resulting in the reduction of clipping distortion power. Taking into account both distortion and channel noise, we derive the analytical total noise power to estimate the log-likelihood ratio (LLR), which is then used to enhance the decoding performance. We derive a minimum achievable E_b/N₀ and a decoding threshold for LDPC codes in the presence of the soft limiter. Simulation results show that our LLR estimation improves the error performance, and multiple candidate system lowers the error rate.     

Unified Analysis of UWB Transmitted-Reference Schemes in the Presence of Narrowband Interference

Transmitted-Reference (TR) signaling, in conjunction with an autocorrelation receiver (AcR), offers a low- complexity alternative to Rake reception in ultrawide bandwidth systems. This paper provides a unified performance analysis of various TR schemes by developing an analytical framework based on the sampling expansion approach. Specifically, we derive the uncoded bit error probability (BEP) of different TR signaling schemes, including TR and differential TR (DTR) signaling valid for a broad class of fading channels. We consider both AcRs and modified AcRs with noise averaging. We further develop a quasi-analytical method as well as an approximate analytical method to extend the BEP analysis to include the effect of narrowband interference (NBI). We show that the approximate analytical method is particularly useful in obtaining BEP expressions that provide insight into the effect of NBI. We quantify the effects of NBI and channel power dispersion profile on the optimum integration interval of an AcR. Finally, we compare TR and DTR signaling in terms of their sensitivity to NBI.     

General Log‐Likelihood Ratio Expression and Its Implementation Algorithm for Gray‐Coded QAM Signals

A simple and general bit log‐likelihood ratio (LLR) expression is provided for Gray‐coded rectangular quadrature amplitude modulation (R‐QAM) signals. The characteristics of Gray code mapping such as symmetries and repeated formats of the bit assignment in a symbol among bit groups are applied effectively for the simplification of the LLR expression. In order to reduce the complexity of the max‐log‐MAP algorithm for LLR calculation, we replace the mathematical max or min function of the conventional LLR expression with simple arithmetic functions. In addition, we propose an implementation algorithm of this expression. Because the proposed expression is very simple and constructive with some parameters reflecting the characteristic of the Gray code mapping result, it can easily be implemented, providing an efficient symbol de‐mapping structure for various wireless applications.     

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.     

Fast and rigorous calculation method for MoM matrix elements inplanar microstrip structures

An analytical integration technique is proposed for evaluating the elements of the impedance matrix obtained by using the spatial-domain method of moments (MoM) applied to the mixed-potential integral equation (MPIE). This technique is based on a Taylor series expansion of the integrands involving only polynomial functions, and thus allowing immediate analytical Integration     

Paraxial space-domain formulation for surface fields on a large dielectric coated circular cylinder

A new method to evaluate the surface fields excited within the paraxial (nearly axial) region of an electrically large dielectric coated circular cylinder is presented. This representation is obtained by performing the Watson's transformation in the standard eigenfunction solution and using the fact that the circumferentially propagating series representation of the appropriate Green's function is periodic in one of its two variables. Therefore, it can be approximated by a Fourier series where the two leading terms of the expansion yield engineering accuracy in most cases. This work can be used in conjunction with a method of moments solution for the design/analysis of conformal microstrip antennas and arrays. Numerical results are presented and compared with a standard eigenfunction expansion.     

Effects of Thermal Noise, Filtering and Co-Channel Interference on the Probability of Error in Binary Coherent PSK Systems

An analytical method for evaluating the bit error probability in Binary Coherent Phase Shift Keying (BCPSK) systems is presented. The system impairments considered are additive Gaussian noise, intersymbol interference and co-channel interference. The method is based on the power series expansion of the characteristic function of the interferences. Each characteristic function is expanded in a power series and its coefficients are averaged analytically with respect to the random phase and the bit timing alignment. Numerical examples are given to illustrate the method and to provide design data for system designers.     

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.