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。     

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     

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     

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     

Complex coefficient fir digital filters

The complex FIR digital filter is a filter that has complex coefficients in itsZ-domain transfer function. The set of coefficients is determined, based on some criterion, to meet predefined requirements. On this basis, an algorithm is proposed for designing FIR digital filters with asymmetric amplitude response in conjunction with linear phase. Minimax approximation has been adopted for determining the set of coefficients, where the associated set of overdetermined linear equations is solved by using an efficient linear programming algorithm. Computer simulations show that, to meet prescribed specifications, the proposed design algorithm yields a complex FIR digital filter with the lowest order.     

直接识别系统参数的闭形式解

本文提出了一种直接识别系统参数的闭形式表达式，避免了参数递归迭代方法的误差传播问题。由于本方法仅与高阶累积有关，因此具有抑制加性高斯噪声能力。模拟结果表明，本文的方法具有比参数递归迭代方法更优越的性能。     

(Almost) periodic FIR system identification using third-ordercyclic-statistics

Two new normal equations relating the coefficients of an (almost) periodic FIR system and the time-varying third-order cumulants of the system output are derived, from which two new linear algebraic algorithms are presented for parameter estimation. Simulation results show that the new algorithms perform better than the recently published linear closed-form algorithms     

A direct approach to H2 optimal deconvolution ofperiodic digital channels

This paper studies the H₂ optimal deconvolution problem for periodic finite impulse response (FIR) and infinite impulse response (IIR) channels. It shows that the H₂ norm of a periodic filter can be directly quantified in terms of periodic system matrices and linear matrix inequalities (LMIs) without resorting to the commonly used lifting technique. The optimal signal reconstruction problem is then formulated as an optimization problem subject to a set of matrix inequality constraints. Under this framework, the optimization of both the FIR and IIR periodic deconvolution filters can be made convex, solved using the interior point method, and computed by using the Matlab LMI Toolbox. The robust deconvolution problem for periodic FIR and IIR channels with polytopic uncertainties are further formulated and solved, also by convex optimization and the LMIs. Compared with the lifting approach to the design of periodic filters, the proposed approach is simpler yet more powerful in dealing with multiobjective deconvolution problems and channel uncertainties, especially for IIR deconvolution filter design. The obtained solutions are applied to the design of an optimal filterbank yielding satisfactory performance     

基于倒谱的非最小相位FIR系统自适应辨识

本文介绍了一种在高阶谱一维切片的基础上利用倒谱进行非最小相位FIR系统自适应辨识的算法。这一算法具有计算量小，不需要预先假定系统阶次的特点，而且能够自适应地分别辨识出非最小相位FIR系统的最小相位部分和最大相位部分的冲激响应，同时，系统阶次可以在自适应过程中确定。另外由于本算法利用观测信号的高阶级计量进行系统辨识，所以很好地抑制了高斯有色噪声对辨识结果的影响。数值仿真结果证明了本算法的有效性。     

Blind identification of MISO-FIR channels

In this paper, we address the problem of determining the order of MISO channels by means of a series of hypothesis tests based on scalar statistics. Using estimated 4th-order output cumulants, we exploit the sensitiveness of a Chi-square test statistic to the non-Gaussianity of a stochastic process. This property enables us to detect the order of a linear finite impulse response (FIR) channel. Our approach leads to a new channel order detection method and we provide a performance analysis along with a criterion to establish a decision threshold, according to a desired level of tolerance to false alarm. Afterwards, we introduce the concept of MISO channel nested detectors based on a deflation-type procedure using the 4th-order output cumulants. The nested detector is combined with an estimation algorithm to select the order and estimate the parameters associated with different transmitters composing the MISO channel. By treating successively shorter and shorter channels, it is also possible to determine the number of sources.     

Identification of nonminimum phase FIR systems via fourth-order cumulants and genetic algorithm

In this paper, the problem of estimating the parameters of an FIR system from only the fourth-order cumulants of the noisy system output is considered. The FIR system is driven by a symmetric, independent, and identically distributed (i.i.d) non-Gaussian sequence. We propose a new formula called Weighted Overdetermined C(q, k) (WOC(q, k)) by extending the conventional C(q, k) formula. The optimal selection of the weights in WOC(q, k) is performed by using the Genetic Algorithm (GA) optimization method which minimizes a nonlinear error function based on the fourth-order cumulants alone. Simulations are provided to reveal the effectiveness and the superiority of this novel technique over the C(q, k) and other existing techniques.     

Derivative-controlled design of linear-phase FIR filters via waveform moments

A new method for designing linear-phase finite impulse response (FIR) filters is proposed by using the blockwise waveform moments. The proposed method yields linear-phase FIR filters whose magnitude response and its derivatives to a certain order take the prescribed values at equally spaced frequency points. The design procedure only needs to solve a system of linear equations, whose size is slightly smaller than the degree of the resulting filter. In addition, the inversion of the linear equations can be essentially precomputed. Therefore, the proposed design method is computationally efficient. In particular, for some important cases, i.e., the maximally flat R-regular L/sup th/-band FIR filters, a closed-form formula can be obtained. It is also shown that the resulting R-regular L/sup th/-band FIR filters have the zero intersymbol interference property.     

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.     

Caractérisation des images stationnaires par des modèles non gaussiens bidimensionnels à moyenne ajustée

In this paper, four batch least squares linear approaches are presented for identification of non minimum phase bidimensional non Gaussian moving average (MA) models, and a relationship between autocorrelation and cumulant sequences is given. One of the proposed methods is cumulant-based only but the others use both autocorrelations and m-th order cumulants (m. > 2). Three of them are derived from the Brillinger-Rosenblatt’s non linear relation by using the Tugnait’s closed-form solution. Also, we generalize to m-th order cumulants the 2-D version of Giannakis-Mende’s approach. By simulations, we test and compare the Tugnait’s closed-form solution and the proposed methods, and we evaluate the performance of our relationship in noisy environment. Finally, we propose to characterize textured images by a 2-D ma model witch will be identified using our approaches in noisy and free noise cases.     

Analysis and optimal realization of pole-zero sensitivity for FIR digital filters

Aiming at the deviation of pole and zero in filters which caused by the finite word length (FWL) effects,the sensitivity of pole and zero for FIR digital filters to coefficient errors was studied based on the state-space model.Unlike the IIR filter,the system matrix in state-space model of the FIR filter was defective.A set of generalized eigenvectors of defective matrix was introduced to analyze the pole sensitivity and derive the measure expression,and optimal realizations with respect to pole-zero sensitivity for FIR filters were proposed by finding optimal transformation matrices according to the similarity transformation theory.Theoretical analysis and simulation experiments show that the poles of a FIR filter are more sensitive to coefficient errors,and the proposed optimal realizations can reduce the sensitivity.     

Multidimensional multichannel FIR deconvolution using Gr?bner bases.

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Gr?bner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and Gr?bner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous Gr?bner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.     

Multidimensional Multichannel FIR Deconvolution Using GrÖbner Bases

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using GrÖbner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and GrÖbner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous GrÖbner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.     

一种低复杂度的稀疏FIR陷波滤波器的设计方法

FIR陷波滤波器具有线性相位、精度高、稳定性好等诸多优势,然而当陷波性能要求较高时,通常需要较高的阶数,导致FIR陷波滤波器硬件实现复杂度大大提高。该文基于稀疏FIR滤波器设计算法和共同子式消除的思想,提出一种低复杂度的FIR陷波滤波器设计方法。该方法首先采用稀疏滤波器设计算法得到满足频域性能设计要求的FIR陷波原始滤波器系数,然后对其进行CSD编码,并分析CSD编码量化系数集中所有的2项子式和孤子的灵敏度,最后根据灵敏度的大小依次选择合理的2项子式或孤子直接合成滤波器系数集。仿真结果表明,新算法设计实现的FIR陷波滤波器比已有的低复杂度设计方法最多可减少51%的加法器,有效地降低了硬件实现复杂度,大大节省了硬件资源。     

Strictly passive FIR filtering for state-space models with external disturbance

This paper proposes a new strictly passive filter with a finite impulse response (FIR) structure for linear state-space signal models with external disturbance. This filter is called a strictly passive FIR filter (SPFF). We apply the strict passivity concept to derive a new linear filter with FIR structure and quasi-deadbeat property. The gain matrix of the proposed SPFF in this paper can be determined by resorting to the solution to a linear matrix inequality (LMI) feasibility problem. An illustrative example is presented to show the validity of the SPFF.