Fast orthogonalization networks

A numerically efficient and stable algorithm is developed to adaptively filter multiple input channels into desired multiple output channels. The algorithm is a type of adaptive lattice filter that employs a fast orthogonalization network (FON) algorithm for numerical efficiency. Past researchers have concentrated on developing efficient lattice algorithms to process stationary input channels; i.e., the input covariance matrix is Toeplitz in form. The algorithm developed herein is designed to adaptively filter nonstationary input channels. Various implementations of the FON algorithm are given.     

Adaptive Step-Size Natural Gradient ICA Algorithm with Weighted Orthogonalization

To improve the stability of the traditional natural gradient independent component analysis (ICA) algorithm and the accuracy of its separated results, a adaptive step-size natural gradient ICA algorithm with weighted orthogonalization is proposed. First, to take advantage of the pre-whitening pre-processing and keep the equivariance property of the ICA algorithm, based on the weighted orthogonal constraint on the separating matrix without pre-whitening of observed signals, weighted orthogonalization is introduced after the traditional gradient update. Then, according to the error estimation from the smoothed distance between separated outputs and optimal outputs, we obtain two adaptive step sizes based, respectively, on an unconstrained natural gradient ICA process and a weighted orthogonalization process. Simulation experiment results show that the speed of convergence of the adaptive step-size natural gradient ICA algorithms with weighted orthogonalization are faster than the traditional one; also, the stability of the algorithms and the accuracy of the separated results are improved observably.     

AUTOMATIC ORTHOGONALIZATION DETECTION APPROACH FOR GS ALGORITHM

Gram-Schmidt orthogonalization algorithm is an interesting theme in the field of adaptive beam-forming and filtering as a fast algorithm. However, a key problem associated with this algorithm is that the number of orthogonalization, namely, the dimensions of interference subspace, is required to know prior. In this paper we derive a threshold and adopt it to detect the number of orthogonalization in the procedure of Gram-Schmidt(GS) orthogonalization decomposition, and this detection approach is simpler and faster than the approach based on eigenanalysis. Finally, computer simulation results were presented too.     

用基于空间平滑的正交化算法分离相干信号

正交化超分辨算法在大信噪比时性能优良且运算量小,但由于正交化算法是一种信号子空间方法,因而在干扰相干时失效,而空间平滑方法已广泛应用于自适应阵列相干信号的分离.文中提出了3种基于空间平滑的改进正交化算法并给出了计算机仿真结果.第1种方法具有较小的运算量,但当信噪比降低时,性能会严重下降.第2种和第3种方法在运算量增加不多的情况下性能比第1种方法有所改善,尤其是当信噪比较小时.理论分析和计算机仿真结果表明这3种方法是有效的.     

Adaptive lattice IIR filtering revisited: convergence issues andnew algorithms with improved stability properties

Several algorithms for adaptive IIR filters parameterized in lattice form can be found in the literature. The salient feature of these structures when compared with the direct form is that ensuring stability is extremely easy. On the other hand, while computing the gradient signals that drive the direct form update algorithms is straightforward, it is not so for the lattice algorithms. This has led to simplified lattice algorithms using gradient approximations. Although, in general, these simplified schemes present the same stationary points as the original algorithms, whether this is also true for convergent points has remained an open problem. This also applies to nongradient-based lattice algorithms such as hyperstability based and the Steiglitz-McBride algorithms. Here, we answer this question in the negative, by showing that for several adaptive lattice algorithms, there exist settings in which the stationary point corresponding to identification of the unknown system is not convergent. In addition, new lattice algorithms with properties are derived. They are based on the cascade lattice structure, which allows the derivation of sufficient conditions for local stability     

Adaptive array beamforming for cyclostationary signals

This paper concerns the adaptive array beamforming using signal cyclostationarity. We present recursive algorithms for implementing the self-cohesive restoral (SCORE) approaches which were originally developed by Agee, Schell and Gardner (1990) to perform blind adaptive signal extraction using antenna arrays. Utilizing the theory of matrix factorization in conjunction with a power normalization scheme, we first reformulate the original problems considered by the SCORE approaches as simpler constrained optimization problems. By employing the modular Gram-Schmidt orthogonalization (GSO) structure, adaptive algorithms are then developed for finding the optimal weight vector. It is shown that although the GSO structure used requires a computation load O(N² ) for an adaptive-beamformer with N elements, the computing time required for producing an output data vector is reduced to O(N) due to the pipeline implementation suitable for the GSO process. The convergence property for each of the proposed algorithms is considered. A simple and efficient method is proposed for choosing appropriate initial guesses to initiate the adaptive algorithms. Simulation examples are given to show the effectiveness of the proposed algorithms     

Efficient lattice realizations of adaptive IIR algorithms

Computationally efficient adaptive IIR-filter algorithms are presented based on lattice realizations allowing the adaptive filter stability to be easily monitored. New simplified recursive-in-order equations relating the parameters of the direct-form realization with the ones of two lattice realizations are presented. These equations lead to a simplified technique to compute the regressor vector and to a general method to implement any adaptive IIR algorithm using lattice realization. Results indicate that the proposed lattice-based algorithms converge to a set of parameters that realize the same transfer function as the corresponding direct-form algorithms     

Application of Least Squares Lattice Algorithms to Adaptive Equalization

In many applications of adaptive data equalization, rapid initial convergence of the adaptive equalizer is of paramount importance. Apparently, the fastest known equalizer adaptation algorithm is based on a recursive least squares estimation algorithm. In this paper we show how the least squares lattice algorithms, recently introduced by Morf and Lee, can be adapted to the equalizer adjustment algorithm. The resulting algorithm, although computationally more complex than certain other equalizer algorithms (including the fast Kalman algorithm), has a number of desirable features which should prove useful in many applications.     

Adaptive lattice bilinear filters

Two fast least-squares lattice algorithms for adaptive nonlinear filters equipped with bilinear system models are presented. The lattice filter formulation transforms the nonlinear filtering problem into an equivalent multichannel linear filtering problem, thus using multichannel lattice filtering algorithms to solve the nonlinear filtering problem. The computational complexity of the algorithms is an order of magnitude smaller than that of previously available methods. The first of the two approaches is an equation error algorithm that uses the measured desired response signal directly to compute the adaptive filter outputs. This method is conceptually very simple, but results in biased system models in the presence of measurement noise. The second is an approximate least-squares output error solution; the past samples of the output of the adaptive system itself are used to produce the filter output at the current time. Results indicate that the output error algorithm is less sensitive to output measurement noise than the equation error method     

Adaptive blind equalization using second- and higher orderstatistics

This paper presents two classes of adaptive blind algorithms based on second- and higher order statistics. The first class contains fast recursive algorithms whose cost functions involve second and third- or fourth-order cumulants. These algorithms are stochastic gradient-based but have structures similar to the fast transversal filters (FTF) algorithms. The second class is composed of two stages: the first stage uses a gradient adaptive lattice (GAL) while the second stage employs a higher order-cumulant (HOC) based least mean squares (LMS) filter. The computational loads for these algorithms are all linearly proportional to the number of taps used. Furthermore, the second class, as various numerical examples indicate, yields very fast convergence rates and low steady state mean square errors (MSE) and intersymbol interference (ISI). MSE convergence analyses for the proposed algorithms are also provided and compared with simulation results     

Adaptive Laguerre-lattice filters

Adaptive Laguerre-based filters provide an attractive alternative to adaptive FIR filters in the sense that they require fewer parameters to model a linear time-invariant system with a long impulse response. We present an adaptive Laguerre-lattice structure that combines the desirable features of the Laguerre structure (i.e., guaranteed stability, unique global minimum, and small number of parameters M for a prescribed level of modeling error) with the numerical robustness and low computational complexity of adaptive FIR lattice structures. The proposed configuration is based on an extension to the IIR case of the FIR lattice filter; it is a cascade of identical sections but with a single-pole all-pass filter replacing the delay element used in the conventional (FIR) lattice filter. We utilize this structure to obtain computationally efficient adaptive algorithms (O(M) computations per time instant). Our adaptive Laguerre-lattice filter is an extension of the gradient adaptive lattice (GAL) technique, and it demonstrates the same desirable properties, namely, (1) excellent steady-state behavior, (2) relatively fast initial convergence (comparable with that of an RLS algorithm for Laguerre structure), and good numerical stability. Simulation results indicate that for systems with poles close to the unit circle, where an (adaptive) FIR model of very high order would be required to meet a prescribed modeling error, an adaptive Laguerre-lattice model of relatively low order achieves the prescribed bound after just a few updates of the recursions in the adaptive algorithm     

Lattice filters for adaptive processing

This paper presents a tutorial review of lattice structures and their use for adaptive prediction of time series. Lattice filters associated with stationary covariance sequences and their properties are discussed. The least squares prediction problem is defined for the given data case, and it is shown that many of the currently used lattice methods are actually approximations to the stationary least squares solution. The recently developed class of adaptive least squares lattice algorithms are described in detail, both in their unnormalized and normalized forms. The performance of the adaptive least squares lattice algorithm is compared to that of some gradient adaptive methods. Lattice forms for ARMA processes, for joint process estimation, and for the sliding-window covariance case are presented. The use of lattice structures for efficient factorization of covariance matrices and solution of Toeplitz sets of equations is briefly discussed.     

Algorithmes en treillis adaptatifs

This paper is composed of three parts. First, a unified review of various adaptive lattice algorithms is presented. The second part is devoted to a performance comparison of these algorithms in terms of arithmetic complexity and convergence properties. Finally the effect of quantization errors is examined. The main results allowing to improve the numerical robustness of the adaptive lattice algorithms with regard to the quantization errors are presented. Some simulation results showing the effect of quantization on data and filter coefficients are given for the identification of an AR model. Thelms algorithm is also considered for comparison.     

Fast adaptive algorithms for AR parameters estimation using higherorder statistics

Time-varying statistics in linear filtering and linear estimation problems necessitate the use of adaptive or time-varying filters in the solution. With the rapid availability of vast and inexpensive computation power, models which are non-Gaussian even nonstationary are being investigated at increasing intensity. Statistical tools used in such investigations usually involve higher order statistics (HOS). The classical instrumental variable (IV) principle has been widely used to develop adaptive algorithms for the estimation of ARMA processes. Despite, the great number of IV methods developed in the literature, the cumulant-based procedures for pure autoregressive (AR) processes are almost nonexistent, except lattice versions of IV algorithms. This paper deals with the derivation and the properties of fast transversal algorithms. Hence, by establishing a relationship between classical (IV) methods and cumulant-based AR estimation problems, new fast adaptive algorithms, (fast transversal recursive instrumental variable-FTRIV) and (generalized least mean squares-GLMS), are proposed for the estimation of AR processes. The algorithms are seen to have better performance in terms of convergence speed and misadjustment even in low SNR. The extra computational complexity is negligible. The performance of the algorithms, as well as some illustrative tracking comparisons with the existing adaptive ones in the literature, are verified via simulations. The conditions of convergence are investigated for the GLMS     

A structural view of asymptotic convergence speed of adaptive IIRfiltering algorithms. I. Infinite precision implementation

An ordinary differential equation (ODE) approach is used to study the focal convergence speed of adaptive IIR filtering and system identification algorithms of various structures: the direct form, the transform domain, and the lattice. The eigenvalue spreads of the associated information matrices for the algorithms are calculated and compared. Their limits as the unknown system poles approach the unit circle are obtained. For each of the three basic structures, two basic types of adaptive algorithms have been studied: the simple constant gain (SCG) type and the Newton type. It is found that the Newton-type algorithms for the direct form and the lattice have the best local convergence speed, regardless of the unknown system pole locations. For the transform-domain Newton type algorithms, however, local convergence speed depends on the orthogonality of the transformation. It is found that SCG-type algorithms are suitable for identifying poles that are well inside the unit circle     

Order-recursive RLS Laguerre adaptive filtering

This paper solves the problem of designing recursive-least-squares (RLS) lattice (or order-recursive) algorithms for adaptive filters that do not involve tapped-delay-line structures. In particular, an RLS-Laguerre lattice filter is obtained     

Cascade lattice IIR adaptive filters

The feedback lattice filter forms, including the two-multiplier form and the normalized form, are examined with respect to their relationships to the feedback direct form filter. Specifically, the transformation matrix between the lattice forms and the direct form is derived; parameter and state relationships between the lattice forms and the direct form are therefore obtained. An IIR filter structure-the cascade lattice IIR structure-is constructed. Based on this structure, three IIR adaptive filtering algorithms in the two-multiplier form can then be developed following the gradient approach, the Steiglitz-McBride approach and the hyperstability approach. Convergence of these algorithms is theoretically analyzed using either the ODE approach or the hyperstability theorem. These algorithms are then simplified into forms computationally as efficient as their corresponding direct form algorithms. Relationships of the simplified algorithms to the direct form algorithms are also studied, which disclose a consistency in algorithm structure regardless of the filter form. Three normalized lattice algorithms can also be derived from the two-multiplier lattice algorithms. Experimental results show much improved performance of the normalized lattice algorithms over the two-multiplier lattice algorithms and the direct form algorithms     

Multiprocessor Implementation of Adaptive Digital Filters

Adaptive filters, employing the transversal filter structure and the least mean square (LMS) adaptation algorithm, or its variations, have found wide application in data transmission equalization, echo cancellation, prediction, spectral estimation, on-line system identification, and antenna arrays. Recently, in response to requirements of fast start-up, or fast tracking of temporal variations, fast recursive least squares (FRLS) adaptation algorithms for both transversal and lattice filter structures have been proposed. These algorithms offer faster convergence than is possible with the LMS/ transversal adaptive filters, at the price of a five-to-tenfold increase in the number of multiplications, divisions, and additions. Here we discuss architectures and implementations of the LMS/transversal, fast-converging FRLS filter, and lattice filter algorithms which minimize the required hardware speed. We show how each of these algorithms can be partitioned so as to be realizable with an architecture based on multiple parallel processors.     

Adaptive Volterra filtering with complete lattice orthogonalization

The article presents a new recursive least squares (RLS) adaptive nonlinear filter, based on the Volterra series expansion. The main approach is to transform the nonlinear filtering problem into an equivalent multichannel, but linear, filtering problem. Then, the multichannel input signal is completely orthogonalized using sequential processing multichannel lattice stages. With the complete orthogonalization of the input signal, only scalar operations are required, instability problems due to matrix inversion are avoided and good numerical properties are achieved. The avoidance of matrix inversion and vector operations reduce the complexity considerably, making the filter simple, highly modular and suitable for VLSI implementations. Several experiments demonstrating the fast convergence properties of the filter are also included     

一种改进的自适应格型陷波算法及其收敛性分析

现有自适应格型陷波器中的引入算子取自局部误差信号,在迭代过程收敛后存在陷波频率偏移的问题,影响了滤波的效果.本文对级联格型陷波滤波器的自适应算法进行了讨论,分析了陷波参数估计与引入算子的关系,在推导原算法迭代误差的数学期望方程基础上,提出一种改进的自适应滤波算法.该算法在不增加计算量的前提下,克服了原算法在收敛后存在的陷波频率偏移的不足.仿真结果与理论分析相一致,证实了该算法的收敛性能优于原有的算法.