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     

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     

Multidimensional fourier transforms by systolic architectures

In this paper, time-efficient systolic and semi-systolic architectures for the implementation of multidimensional DFTs are proposed that allow modularity and easy expansibility while keeping throughput independent of the dimension of the DFT. We shall call an array semi-systolic if the input data is to be preloaded into every cell of the array or if the output data can be calculated not only in the boundary cells of the array. DFT algorithms are represented by FORTRAN-like code, in order to explicitly display the suggested rotational transformations in the index space. After the transformation, multidimensional DFT algorithm can be mapped onto a systolic structure. The area-time² complexity of the proposed design is within logk factor of the lower bound for ak-point DFT (AT ²=(k ² log² k)), i.e. equals 0(k ² log³ k)=0(N ^2M log³ N) fork=N ^2M point DFT;A is the area complexity andT is the system throughput.     

Decomposition theorem andk-hyper connection expressions for generalk-order cofactors

Twok-hyperconnection expressions of a generalk-order cofactorY(_i,j) are presented for the indefinite parameter matrixY of a linear system by applying directed hypergraph theory, and based on it a decomposition theorem ofY(_i,j) is derived. By this theorem, the multi-level tearing and analysis can be carried out easily for any linear large system. This is a new multilevel topological analysis method. Using proposed method the scale of systems which can be topologically analysed by a computer will be enlarged.     

FIR system identification based on subspaces of a higher ordercumulant matrix

A new linear algebraic approach for identification of a nonminimum phase FIR system of known order using only higher order (>2) cumulants of the output process is proposed. It is first shown that a matrix formed from a set of cumulants of arbitrary order can be expressed as a product of structured matrices. The subspaces of this matrix are then used to obtain the parameters of the FIR system using a set of linear equations. Theoretical analysis and numerical simulation studies are presented to characterize the performance of the proposed methods     

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.     

Cumulant-based blind identification of linearmulti-input-multi-output systems driven by colored inputs

The blind identification problem of a linear multi-input-multi-output (MIMO) system is widely noticed by many researchers in diverse fields due to its relevance to blind signal separation. However, such a problem is ill-posed and has no unique solution. Therefore, we can only find a solution of the problem within an equivalence class. We address the blind identification problem of the linear MIMO system driven by unobservable colored inputs using higher order statistics (HOS), particularly the fourth-order cumulants, of the outputs, where the unobservable inputs are mutually independent but temporally colored linear processes. We first define the set, which is denoted by S, of stable scalar transfer functions and then define the notion of a generalized permutation matrix (which is abbreviated by a g-matrix) over S. Then, it is shown that the transfer function matrix of an unknown system is identified only up to post-multiplication by a g matrix. This result is applied to identifying FIR systems for blind signal separation     

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.     

Sample cumulants of stationary processes: asymptotic results

In this paper, we present the formulas of the covariances of the second-, third-, and fourth-order sample cumulants of stationary processes. These expressions are then used to obtain the analytic performance of FIR system identification methods as a function of the coefficients and the statistics of the input sequence. The lower bound in the variance is also compared for different sets of sample statistics to provide insight about the information carried by each sample statistic. Finally, the effect that the presence of noise has on the accuracy of the estimates is studied analytically. The results are illustrated graphically with plots of the variance of the estimates as a function of the parameters or the signal-to-noise ratio. Monte Carlo simulations are also included to compare their results with the predicted analytic performance     

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     

THE FOURTH-ORDER CUMULANTS BASED SPECTRALESTIMATION METHOD AND ITS APPLICATIONTO DIRECTION-FINDING

Traditional approaches of spatial spectral estimation are usually based on the second-order statistics. The higher-order cumulants and the poly-spectrum contain more information and are capable of reducing the Gaussian noise. In this paper, we present a new spectrum estimation method for direction-finding, the FOMUSIC algorithm, which is based on the eigen-structure analysis of the fourth-order cumulants. The derivation of the algorithm is given in detail and its performance is illustrated by both the computer simulations and the experiments of a direction-finding system. The obtained results demonstrate that the fourth-order cumulants based method outperforms the traditional methods, especially when the noise is an unknown colored one.     

房间环境下基于四阶累积量的语音时延估计

主要研究了利用四阶累积量在房间环境下的语音信号时延估计。根据模拟实验环境和实际实验环境下的仿真结果,分别比较了在高斯白噪声情况下与有色高斯噪声情况下基于二阶统计量的广义互相关法和基于四阶累积量的时延估计法性能,验证了四阶累积量法的有效性。仿真结果表明四阶累积量法具有很好的鲁棒性。     

Exact reliability formulas for linear and circular m-consecutive-k-out-of-n:F systems

A linear (circular) m-consecutive-k-out-of-n:F system, consists of n components ordered on a line (on a circle). The system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. Four theorems are given. Theorem 1 is an exact formula of the reliability of the linear system given in terms of multinomial coefficients. Theorem 2 is an exact formula of the reliability of the linear system in terms of binomial coefficients. Theorems 3 and 4 are exact formulas of the reliability of the circular system, given in terms of multinomial and binomial coefficients, respectively.     

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。     

FBMC-OQAM信号子载波盲估计

为了解决滤波器组多载波(filter-bank multi-carrier, FBMC)信号的子载波参数盲估计问题，提出一种基于四阶累积量的方法来估计FBMC-OQAM信号子载波参数。首先，截获交错正交幅度调制(offset quadrature amplitude modulation，OQAM)的FBMC信号并检测信号频谱来确定循环频率的范围；然后，分析循环频率与四阶累积量的代数关系；最后，得出四阶累积量的峰值受原型滤波器影响且仅在其循环频率等于子载波频率处存在。通过检测四阶累积量峰值的位置和个数能有效估计子载波频率和数目。仿真表明，当信噪比大于2 dB时，算法对FBMC-OQAM信号的子载波参数盲估计正确概率为98%。      

Bounds for reliability of k-within connected-(r, s)-out-of-(m, n) failure systems

A k-within linear connected-(r, s)-out-of-(m, n) failure system is a two-dimensional grid whose components are ordered like the elements of an (m, n)-matrix. A k-within circular connected-(r, s)-out-of (m, n) failure system consists of the intersection points of m circles centered at the same point with n rays starting from that point and crossing the circles. The components of both systems either operate or fail. By definition, a k-within (linear or circular) connected-(r, s)-out-of-(m, n) failure system fails if at least one (r, s)-submatrix contains k or more failed components. These systems are used as mathematical models for design and operation of many engineering systems. For systems with statistically independent and identically distributed components, a lower and upper bound of system reliability are derived using improved Bonferroni inequalities. These bounds are easy to compute and provide good estimates for system reliability. New bounds for the reliability of other connected-(r, s)-out-of-(m, n) failure systems existing in the current literature are also obtained. Several failure systems with various values of the parameters k, r, s, m, n and p are used as numerical examples for comparison and illustrative purposes.     

Reliability of a connected-(r1, s1)-or-(r2, s2)-or-..-or-(rk, sk)-out-of-(m,n): F lattice system

A (linear or circular) connected-(r₁, s₁)-or-(r₂, s₂)-or-^{. .}-or-(r_k, s_k)-out-of-(m, n): F lattice system is the (linear or circular) (m, n)-lattice system if the system fails whenever all components in a connected-(r₁, s₁)-submatrix or all components in a connected-(r₂, s₂)-submatrix or ^{. .} or all components in a connected-(r_k, s_k)-submatrix fail. This paper presents a recursive algorithm for the reliability of the (linear or circular) connected-(r₁, s₁)-or-(r₂, s₂)or-^{. .}-or-(r_k,s_k)-out- of-(m, n):F lattice system. The recursive algorithm requires time and time in the linear case and the circular case, respectively Furthermore, we can reduce the more computing time in the statistically independent and identically distributed case or considering some special systems. Especially, the closed formula is given for the reliability of the linear connected-(2, 1)-or-(1, 2)-out-of-(m, 2): F lattice system in the statistically independent and identically distributed case.     

基于四阶累积过的阵列扩展

在阵列测向问题中应用高阶累积量除了信息利用更充分和抑制高斯噪声等优点之外还具有阵列扩展的功能.本文推导了基于四阶累积量的阵列扩展方法,研究了虚拟阵的结构.放宽了协方差方法对信源数目的限制.但该方法在窄带多信号方向估计应用中存在局限.本文还推导了四阶相干问题和阵列扩展的限制条件.最后给出了实验举例.     

四阶累积量的递推估计及其应用

本文推导一种复数或实数零均值平稳随机过程的高阶累积量估计的递推算法,随机信号的“峰度”递推估计公式,在递推估计公式中引入了随时间变化的“修正”项,使得该递推算法还可适合非平稳过程高阶累积量的估计计算。并且,本文大量计算了多类型。多航速、多舰船辐射噪声的“峰度”值,分析了舰船辐射噪声特性。     

Estimation of multipath channel response in frequency selectivechannels

A novel discrete-time method is proposed for estimating the impulse response of a frequency-selective digitally modulated communication channel. The received signal is first demodulated and sampled and then the fourth-order cumulants of the resulting discrete-time sequence are estimated. The method estimates the channel impulse response from the complex cepstrum of the aforementioned fourth-order cumulants (i.e. tricepstrum). The method depends only on the second- and fourth-order statistics of the transmitted sequence and is capable of reconstructing nonminimum-phase impulse responses. Monte Carlo simulation results demonstrate the effectiveness of the method, its low sensitivity to observation noise, and its improved performance in terms of probability of error or the reconstructed transmitted sequence