The cumulant theory of cyclostationary time-series. I. Foundation

The problem of characterizing the sine-wave components in the output of a polynomial nonlinear system with a cyclostationary random time-series input is investigated. The concept of a pure nth-order sine wave is introduced, and it is shown that pure nth-order sine-wave strengths in the output time-series are given by scaled Fourier coefficients of the polyperiodic temporal cumulant of the input time-series. The higher order moments and cumulants of narrowband spectral components of time-series are defined and then idealized to the case of infinitesimal bandwidth. Such spectral moments and cumulants are shown to be characterized by the Fourier transforms of the temporal moments and cumulants of the time-series. It is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts. To put the contributions of the paper in perspective, a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided     

基于高阶累积量和改进BP神经网络的模拟电路故障诊断

提出了基于信号处理领域的高阶累积量诊断模拟电路故障的方法。从待测电路的输出终端提取原始信号,求出其峰度和斜度作为故障特征向量,然后输入给改进的BP神经网络进行故障诊断。对于故障特征向量的构造,相对于一般的以二阶统计量为基础的主元分析方法(PCA),高阶累积量的引入更多的考虑了被PCA所忽略的信息。诊断实例表明该方法生成的故障特征向量,具有高的识别率,诊断精度大大提高。     

基于累计量辨识非最小相位FIR系统的新方法

该文考虑用带有噪声输出数据的累计量实现对非最小相位PIR系统的参数辨识问题。提出一个新的基于高阶累计量的方法。其特点如下,(1)灵活性:采用了两个任意阶次相邻的输出累计量;(2)线性:方法的表达式相对于未知量为线性。这不同于其它一些已存在的算法。因而,避免了额外的滞后处理,可提高参数估计的准确性。本文在ARMA高斯噪声及三种实际噪声情况下,做了大量的实验。结果表明,本文提出的算法不仅能有效地完成参数估计,而且,在低信噪比下,其估计结果比其它已有的算法更准确。     

New approaches without postprocessing to FIR system identificationusing selected order cumulants

In this paper, we address the problem of identifying the parameters of the nonminimum-phase FIR system from the cumulants of noisy output samples. The system is driven by an unobservable, zero-mean, independent and identically distributed (i.i.d) non-Gaussian signal. The measurement noise may be white Gaussian, colored MA, ARMA Gaussian processes, or even real. For this problem, two novel methods are proposed. The methods are designed by using higher order cumulants with the following advantages. (i) Flexibility: method 1 employs two arbitrary adjacent order cumulants of output, whereas method 2 uses three cumulants of output: two cumulants with arbitrary orders and the other one with an order equal to the summation of the two orders minus one. Because of this flexibility, we can select cumulants with appropriate orders to accommodate different applications. (ii) Linearity: both the formulations in method 1 and method 2 are linear with respect to the unknowns, unlike the existing cumulant-based algorithms. The post-processing is thus avoided. Extensive experiments with ARMA Gaussian and three real noises show that the new algorithms, especially algorithm 1, perform the FIR system identification with higher efficiency and better accuracy as compared with the related algorithms in the literature     

基于高阶累积量的突发信号检测技术

突发信号检测是非合作通信中的一项重要工作,是后续处理的前提。针对由高阶累积量作为判决统计量的信号检测方法,分析了高阶累积量法在实际信号检测中存在的问题,提出了相应的改进算法,该算法只用2个符号进行累积量估计,通过在高阶累积量的基础上增加滑动窗,对累积量值进行平滑处理,并通过窗内累积量值的变化自适应调整窗长,减小或消除了因数据过短引起高阶累积量估计值的抖动。仿真结果表明该算法可以提高信号检测性能。     

The cumulant theory of cyclostationary time-series. II. Developmentand applications

For pt.I see ibid., vol.42, no.12, p.3387-3408 (1994). The development of the theory of nonlinear processing of cyclostationary time-series that is initiated in Part I is continued. A new type of cumulant for complex-valued variables is introduced and used to generalize the temporal and spectral moments and cumulants for cyclostationary time-series from real-valued to complex-valued time-series. The relations between the temporal and spectral moments and cumulants at the inputs and outputs of several signal processing operations are determined. Formulas for the temporal and spectral cumulants of complex-valued pulse-amplitude-modulated time-series are derived. Estimators for the temporal moments and cumulants and for the cyclic polyspectra are presented and their properties are discussed. The performance of these estimators is illustrated by several computer simulation examples for pulse-amplitude-modulated time-series. The theory is applied to the problems of weak-signal detection and interference-tolerant time-delay estimation     

基于四阶累积量的DOA估计方法

从一个统一的角度来研究基于高阶累积量的高分辨阵列信号处理方法,用四阶累积量构造了一个较通用的累积量矩阵,该矩阵符合MUSIC算法的结构,从而可进行DOA(Direction ofArrival)估计。对于这种方法在阵列信号处理中的应用,通过计算机仿真与基于二阶矩的MU-SIC算法进行了较全面的比较。仿真结果表明,该方法在高斯噪声中具有良好的统计性能,是实现高分辨方位估计的有效方法。     

Cumulant based identification approaches for nonminimum phase FIRsystems

Recursive and least squares methods for identification of non-minimum-phase linear time-invariant (NMP-LTI) FIR systems are developed. The methods utilize the second- and third-order cumulants of the output of the FIR system whose input is an independent, identically distributed (i.i.d.) non-Gaussian process. Since knowledge of the system order is of utmost importance to many system identification algorithms, new procedures for determining the order of an FIR system using only the output cumulants are also presented. To illustrate the effectiveness of the methods, various simulation examples are presented     

基于高阶累积量的空时分组码盲识别算法研究

针对MISO通信系统的空时分组码盲识别问题,提出了一种基于高阶累积量的空时分组码盲识别算法。首先,给出了MISO接收信号模型,利用高阶累积量的性质分析得到接收信号的四阶累积量的表达式;然后,利用编码矩阵的特性,证明接收信号在不同时延向量下的四阶累积量呈现非零值,其非零值取决于STBC的类型;最后,采用四阶累积量的实验值与理论值的最小欧式距离盲识别空时分组码的类型。仿真结果表明,即使在低信噪比条件下,所提方法能够较好地识别空时分组码。     

Applications of cumulants to array processing .I. Apertureextension and array calibration

An interpretation for the use of cumulants in narrowband array processing problems is proposed. It is shown how fourth-order cumulants of multichannel observations increase the directional information compared with second-order statistics. Based on the interpretation, it is shown how cumulants can be used to increase the effective aperture of an arbitrary antenna array. The amount of partial information necessary to jointly calibrate an arbitrary array and estimate the directions of far-field sources is also investigated. It is proven that the presence of a doublet and use of fourth-order cumulants is sufficient to accomplish this task. The proposed approach is computationally efficient and more general than covariance-based algorithms that have addressed the calibration problem under constraints. A class of beamforming techniques is proposed to recover the source waveforms. Proposed estimation procedures are based on cumulants, which bring insensitivity to the spatial correlation structure of additive Gaussian measurement noise. Simulations are provided to illustrate the use of the proposed algorithms     

Order-recursive blind identification of linear models using mixedcumulants

The problem of determining the AR order and parameters of a nonminimum phase ARMA model from observations of the system output is considered. The model is driven by a sequence of random variables which is assumed unobservable. A novel identification algorithm based on the second- and third-order cumulants of the output sequences is introduced. It performs order-recursively by minimising a well defined cost function. Strong convergence and consistency of the algorithm are proved and the weight of the cost function is balanced between the second-order and the third-order cumulants of output sequences. The influence of the weight on the estimation accuracy is also evaluated. Theoretical analyses and numerical simulations show that the proposed algorithm is satisfactory for both order and parameter identification of an AR model which is subordinate to a nonminimum phase ARMA model     

Extending HOC-based methods for identifying the diagonal parameters of quadratic systems

In this paper, methods developed for the linear case of identifying the diagonal parameters of quadratic systems are extended to nonlinear case. Firstly, nonlinear relationships between model kernels and output cumulants are presented. Secondly, the relationship linking output cumulants and the coefficients of systems presented in the linear case, is extended to the general case of nonlinear quadratic systems identification. According to this concept, two nonlinear approaches are developed, the first use the fourth-order cumulants, and the second combined the third- and fourth-order cumulants. The numerical simulation results, for various signal to noise ratio (SNR) and 200 Monte Carlo runs, show that the proposed approaches achieve better accuracy, as compared with the related algorithm in the literature. Furthermore, the second algorithm is more precise in high noise environment (smallest \(\mathrm{SNR}=0\) dB), but the first algorithm more efficient in the weak noise environment case (highest SNR \(\ge \) 8 dB) comparing to using others methods.     

On the distribution of positive-definite Gaussian quadratic forms

Quadratic signal processing is used in detection and estimation of random signals. To describe the performance of quadratic signal processing, the probability distribution of the output of the processor is needed. Only positive-definite Gaussian quadratic forms are considered. The quadratic form is diagonalized in terms of independent Gaussian variables and its mean, moment-generating function, and cumulants are computed; conditions are given for the quadratic form to bechi^{2}distributed and distributed like a sum of independent random variables having a Gamma distribution. A new method is proposed to approximate its probability distribution using an expansion in Laguerre polynomials for the central case and in generalizedchi^{2}distributions in the noncentral case. The series coefficients and bounds on truncation error are evaluated. Some applications in average power and power spectrum estimation and in detection illustrate our method.     

On MMSE real-time antenna array processing using fourth-orderstatistics in the US cellular TDMA system

The antenna array processing problem in the reverse link of the current US digital cellular communication system is studied and higher-than-second-order-statistics (HOS) baseband processing is proposed as a possible candidate solution. The remarkable difference of our approach as compared to other existing similar techniques is the idea of the minimization of the mean squared error using fourth-order cumulants alone and nonblind criteria. A recursive Jacobi total least squares algorithm is used in the adaptive implementation to mitigate the effects of high error variance in the estimates of the cumulants based on sample statistics. The method is shown to be very effective in a fast fading environment with multiple cochannel interferers     

A parametric modeling of ionic channel current fluctuations usingthird-order statistics and its application to estimation of the kineticparameters of single ionic channels

The case where third-order cumulants of stationary ionic-channel current fluctuations (SICFs) are nonzero, and where SICFs are corrupted by an unobservable additive colored Gaussian noise that is independent of SICFs is considered. First, a virtual synthesizer that yields an output whose third-order cumulants are equivalent to those of SICFs on a specific slice is constructed. The synthesizer output is expressed by the sum of N_s-1 first-order differential equation systems, where N_s denotes the number of states of single ionic channels. Next, discretizing the synthesizer output, a discrete autoregressive [AR(N_s-1)] process driven by the sum of N_s-1 moving average (MA(N_s -2)) processes is derived. Then the AR coefficients are explicitly related to the kinetic parameters of single ionic channels, implying that the kinetic parameters can be estimated by identifying the autoregressive moving-average coefficients using the third-order cumulants. In order to assess the validity of the proposed modeling and the accuracy of parameter estimates, Monte Carlo simulation is carried out in which the closed-open and closed-open-blocked schemes are treated as specific examples     

FIR system identification using third- and fourth-order cumulants

A new set of equations relating the coefficients of a finite-impulse-response (FIR) system and the third- and forth-order cumulants of the system output are derived. Based on these equations, two new methods to estimate FIR parameters are presented. Simulation results show that these methods perform better than other recently published linear methods in the additive coloured Gaussian noise case. This improvement is due to the fact that they do not make use of any correlation information and that they employ several slices of third- and forth-order cumulants     

Low-rank estimation of higher order statistics

Low-rank estimators for higher order statistics are considered in this paper. The bias-variance tradeoff is analyzed for low-rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants. In general, the low-rank estimators have a larger bias and smaller variance than the corresponding full-rank estimator, and the mean-squared error can be significantly smaller. This makes the low-rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics. The low-rank estimators also offer considerable reductions in the computational complexity of such algorithms. The design of subspaces to optimize the tradeoffs between bias, variance, and computation is discussed, and a noisy input, noisy output system identification problem is used to illustrate the results     

Blind identification of quadratic nonlinear models using neuralnetworks with higher order cumulants

A novel approach to blindly estimate kernels of any discrete- and finite-extent quadratic models in higher order cumulants domain based on artificial neural networks is proposed in this paper. The input signal is assumed an unobservable independently identically, distributed random sequence which is viable for engineering practice. Because of the properties of the third-order cumulant functions, identifiability of the nonlinear model holds, even when the model output measurement is corrupted by a Gaussian random disturbance. The proposed approach enables a nonlinear relationship between model kernels and model output cumulants to be established by means of neural networks. The approximation ability of the neural network with the weights-decoupled extended Kalman filter training algorithm is then used to estimate the model parameters. Theoretical statements and simulation examples together with practical application to the train vibration signals modeling corroborate that the developed methodology is capable of providing a very promising way to identify truncated Volterra models blindly     

Cumulant-based method for bearing estimation in the presence ofnonGaussian noise

Bearing estimation algorithms based on the cumulants of array data have been developed to suppress additive spatially correlated Gaussian noises. In practice, however, the noises encountered in signal processing environments are often non-Gaussian, and the applications of those cumulant-based algorithms designed for Gaussian noise to non-Gaussian environments may severely degrade the estimation performance. The authors propose a new cumulant-based method to solve this problem. This approach is based on the fourth-order cumulants of the array data transformed by DFT, and relies on the statistical central limit theorem to show that the fourth-order cumulants of the additive non-Gaussian noises approach zero in each DFT cell. Simulation results are presented to demonstrate that the proposed method can effectively estimate the bearings in both Gaussian and non-Gaussian noise environments     

MA parameter estimation and cumulant enhancement

This paper addresses the problem of estimating the parameters of a moving average (MA) model from either only third- or fourth-order cumulants of the noisy observations of the system output. The system is driven by an independent and identically distributed non-Gaussian sequence that is not observed. The unknown model parameters are obtained using a batch least squares method. Recursive methods are also developed and used to claim the uniqueness of the batch least squares solutions. A novel technique for the enhancement of third-order cumulants of MA processes is introduced. This new technique is based on the concept of composite property mappings and helps reduce the variance of the estimates of third- (or fourth)-order cumulants of MA processes. Simulation results are presented that demonstrate the performance of the new methods and compare them with a range of existing techniques