A new family of concurrent algorithms for adaptive Volterra andlinear filters

A novel idea for introducing concurrency in least squares (LS) adaptive algorithms by sacrificing optimality has been proposed. The resultant class of algorithms provides schemes to fill the wide gap in the convergence rates of LS and stochastic gradient (SG) algorithms. It will be particularly useful in the real time implementations of large-order linear and Volterra filters for which both the LS and SG algorithms are unsuited     

Filtered-X second-order Volterra adaptive algorithms

The authors propose filtered-X second-order Volterra adaptive algorithms based on a multichannel structure application to active noise control. The developed algorithms can be used as alternatives in the case where the standard filtered-X LMS algorithm does not perform well     

Convergence behavior of affine projection algorithms

A class of equivalent algorithms that accelerate the convergence of the normalized LMS (NLMS) algorithm, especially for colored inputs, has previously been discovered independently. The affine projection algorithm (APA) is the earliest and most popular algorithm in this class that inherits its name. The usual APA algorithms update weight estimates on the basis of multiple, unit delayed, input signal vectors. We analyze the convergence behavior of the generalized APA class of algorithms (allowing for arbitrary delay between input vectors) using a simple model for the input signal vectors. Conditions for convergence of the APA class are derived. It is shown that the convergence rate is exponential and that it improves as the number of input signal vectors used for adaptation is increased. However, the rate of improvement in performance (time-to-steady-state) diminishes as the number of input signal vectors increases. For a given convergence rate, APA algorithms are shown to exhibit less misadjustment (steady-state error) than NLMS. Simulation results are provided to corroborate the analytical results     

Low computational complexity family of affine projection algorithms over adaptive distributed incremental networks

This paper presents the problem of distributed estimation in an incremental network based on the family of affine projection (AP) adaptive algorithms. The distributed selective partial update normalized least mean squares (dSPU-NLMS), the distributed SPU-AP algorithm (dSPU-APA), the distributed selective regressor APA (dSR-APA), the distributed dynamic selection of APA (dDS-APA), dSPU-SR-APA and dSPU-DS-APA are introduced in a unified way. These algorithms have low computational complexity feature and close convergence speed to ordinary distributed adaptive algorithms. In dSPU-NLMS and dSPU-APA, the weight coefficients are partially updated at each node during the adaptation. In dSR-APA, the optimum number of input regressors is selected during the weight coefficients update. The dynamic selection of input regressors is used in dDS-APA. dSPU-SR-APA and dSPU-DS-APA combine SPU with SR and DS approaches. In these algorithms, the weight coefficients are partially updated and the input regressors are optimally/dynamically selected at every iteration for each node. In addition, a unified approach for mean-square performance analysis of each individual node is presented. This approach can be used to establish a performance analysis of classical distributed adaptive algorithms as well. The theoretical expressions for stability bounds, transient, and steady-state performance analysis of various distributed APAs are introduced. The validity of the theoretical results and the good performance of dAPAs are demonstrated by several computer simulations.     

Color image processing using adaptive multichannel filters

New adaptive filters for color image processing are introduced and analyzed. The proposed adaptive methodology constitutes a unifying and powerful framework for multichannel signal processing. Using the proposed methodology, color image filtering problems are treated from a global viewpoint that readily yields and unifies previous, seemingly unrelated, results. The new filters utilize Bayesian techniques and nonparametric methodologies to adapt to local data in the color image. The principles behind the new filters are explained in detail. Simulation studies indicate that the new filters are computationally attractive and have excellent performance.     

Envelope-constrained filters: adaptive algorithms

We consider an envelope-constrained (EC) optimal filter design problem involving a quadratic cost function and a number of linear inequality constraints. Using the duality theory and the space transformation function, the optimal solution of the dual problem can be computed by finding the limiting point of an ordinary differential equation given in terms of the gradient flow. An iterative algorithm is developed via discretizing the differential equation. From the primal-dual relationship, the corresponding sequence of approximate solutions in the original EC filtering problem is obtained. Based on these results, an adaptive algorithm is constructed for solving the stochastic EC filtering problem in which the input signal is corrupted by an additive random noise. For illustration,a practical example is solved for both noise-free and noisy cases     

A projection approach for robust adaptive beamforming

It is well known that calibration errors can seriously degrade performance in adaptive arrays, particularly when the input signal-to-noise ratio is large. The effect is caused by the perturbation of the presumed steering vector from its optimal value. Although it is not as widely known, similar degradation occurs in sampled matrix inversion processing when the covariance matrix is estimated while the desired signal is present in the snapshot data. Under these conditions, performance loss is due to errors in the estimated covariance matrix and occurs even when the steering vector is known exactly. In the paper, a new method based of modification of the steering vector is proposed to overcome both the problems of perturbation and of sample covariance errors. The method involves projection of the presumed steering vector onto the observed signal-plus-interference subspace. An analysis is also presented illustrating that the sample covariance errors can be viewed as a particular type of perturbation error and a simple approximation is derived for the expected beamformer performance as a function of the number of data snapshots. Both analytical and experimental results are presented that illustrate the advantages of the proposed method     

Mean-square performance of a family of affine projection algorithms

Affine projection algorithms are useful adaptive filters whose main purpose is to speed the convergence of LMS-type filters. Most analytical results on affine projection algorithms assume special regression models or Gaussian regression data. The available analysis also treat different affine projection filters separately. This paper provides a unified treatment of the mean-square error, tracking, and transient performances of a family of affine projection algorithms. The treatment relies on energy conservation arguments and does not restrict the regressors to specific models or to a Gaussian distribution. Simulation results illustrate the analysis and the derived performance expressions.     

Transient and steady-state analysis of filtered-x affine projection algorithms

This paper provides an analysis of transient and steady-state behavior of different filtered-x affine projection algorithms. Algorithms suitable for single-channel and for multichannel active noise controllers are treated within a unified framework. Very mild assumptions are posed on the active noise control system model, which is only required to have a linear dependence of the output from the filter coefficients. Therefore, the analysis applies not only to the linear finite impulse response models but also to nonlinear Volterra filters, i.e., polynomial filters, and other nonlinear filter structures. The convergence analysis presented in this paper relies on energy conservation arguments and does not apply the independence theory, nor does it impose any restriction to the signal distributions. It is shown in the paper that filtered-x affine projection algorithms always provide a biased estimate of the minimum mean square solution. Nevertheless, in many cases, the bias is small and therefore these algorithms can be profitably applied to active noise control.     

Systolic architectures for adaptive multichannel least squares lattice filters

A family of systolic array architectures for adaptive multichannel least squares lattice (MLSL) filters is presented. These architectures are based on a recently developed algorithm that provides an efficient, numerically sound, and well-structured set of recursions for realizing MLSL filters. The algorithm is based on the recursive QR decomposition of the forward and backward error correlation matrices. Form input channels andp filter taps,O(pm ²) computations are required per time step. Numerous space-time tradeoffs are available in mapping the algorithm's recursions to systolic architectures, leading to the architectural family presented here.Los Alamos National Laboratory is operated by the University of California for the United States Department of Energy under contract W-7405-ENG-36.     

Intelligent learning algorithms for adaptive digital filters

It is wellknown that gradient search fails in adaptive IIR filters, since their mean-square error surfaces may be multi-modal. In the letter a new approach based on learning algorithms is shown to be capable of performing global optimisation. The new algorithms are suitable for both adaptive FIR and IIR filters.<>     

Eigenstructure algorithms for multirate adaptive lossless FIR filters

This paper addresses the problem of adaptively optimizing a two-channel lossless finite-impulse-response (FIR) filter bank, which finds application in subband coding and wavelet signal analysis. Instead of using a gradient decent procedure-with its inherent problem of becoming trapped in local minima of a nonquadratic cost function-two eigenstructure algorithms are proposed. Both algorithms feature a priori bounds on the output variance at any convergent point, which, based on simulations, lead to solutions that lie acceptably close to a global minimum point of an output variance objective function. Moreover, a sufficient condition for such stationary points based on fixed-point theory is shown. It is shown that the convergence rate of both algorithms increases as the separation of eigenvalues of the input covariance matrix increases. Simulations for synthetic and real data support the conclusions.     

Analysis of gradient algorithms for TLS-based adaptive IIR filters

Steepest descent gradient algorithms for unbiased equation error adaptive infinite impulse response (IIR) filtering are analyzed collectively for both the total least squares and mixed least squares-total least squares framework. These algorithms have a monic normalization that allows for a direct filtering implementation. We show that the algorithms converge to the desired filter coefficient vector. We achieve the convergence result by analyzing the stability of the equilibrium points and demonstrate that only the desired solution is locally stable. Additionally, we describe a region of initialization under which the algorithm converges to the desired solution. We derive the results using interlacing relationships between the eigenvalues of the data correlation matrices and their respective Schur complements. Finally, we illustrate the performance of these new approaches through simulation.     

An adaptive approach for designing phase compensation filters

The objective of this communication is to introduce an adaptive method for designing filters to compensate for nonlinear phase shift introduced into a data stream by fixed (i.e., nonadaptive) recursive digital filters, e.g., those obtained via the bilinear transform.     

A two-stage affine projection algorithm with mean-square-error-matching step-sizes

This paper proposes a two-stage affine projection algorithm (APA) with different projection orders and step-sizes. The proposed algorithm has a high projection order and a fixed step-size to achieve fast convergence rate at the first stage and a low projection order and a variable step-size to achieve small steady-state estimation errors at the second stage. The stage transition moment from the first to the second stage is determined by examining, from a stochastic point of view, whether the current error reaches the steady-state value. Moreover, in order to prevent the sudden drop of convergence rate on switching from a high projection order to a low projection order, a matching step-size method has been introduced to determine the initial step-size of the second stage by matching the mean-square errors (MSEs) before and after the transition moment. In order to continuously reduce steady-state estimation errors, the proposed algorithm adjusts the step-size of the second stage by employing a simple algorithm. Because of the reduced projection orders and variable step-size in the steady-state, the algorithm achieves improved performance as well as extremely low computational complexity as compared to the existing APAs with selective input vectors and APAs with variable step-size.     

A class of frequency-domain adaptive approaches to blind multichannel identification

We extend our previous studies on adaptive blind channel identification from the time domain into the frequency domain. A class of frequency-domain adaptive approaches, including the multichannel frequency-domain LMS (MCFLMS) and constrained/unconstrained normalized multichannel frequency-domain LMS (NMCFLMS) algorithms, are proposed. By utilizing the fast Fourier transform (FFT) and overlap-save techniques, the convolution and correlation operations that are computationally intensive when performed by the time-domain multichannel LMS (MCLMS) or multichannel Newton (MCN) methods are efficiently implemented in the frequency domain, and the MCFLMS is rigorously derived. In order to achieve independent and uniform convergence for each filter coefficient and, therefore, accelerate the overall convergence, the coefficient updates are properly normalized at each iteration, and the NMCFLMS algorithms are developed. Simulations show that the frequency-domain adaptive approaches perform as well as or better than their time-domain counterparts and the cross-relation (CR) batch method in most practical cases. It is remarkable that for a three-channel acoustic system with long impulse responses (256 taps in each channel) excited by a male speech signal, only the proposed NMCFLMS algorithm succeeds in determining a reasonably accurate channel estimate, which is good enough for applications such as time delay estimation.     

A unified balanced approach to multichannel blind deconvolution

In this paper, we explore the application of a common operator used in systems theory, viz., the delta operator, to formulate a unified theory of multichannel blind deconvolution (MBD) which is valid in both discrete and continuous time domains. Apart from providing a unified treatment of MBD problems, this formulation permits a smooth transition of the demixer from a discrete time domain to a continuous time domain when the sampling rate is high. Furthermore we give a unified treatment of a balanced parameterized state space formulation to solving the MBD problem in both discrete and continuous time domains when the number of states is unknown.     

Adaptive algorithms for a two-channel structure employing allpass filters, with applications to polarization mode dispersion compensation

A two-channel structure consisting of multiple cascades of allpass filters and directional couplers has certain desirable attributes. This paper considers adaptive algorithms for adjusting the parameters of such systems. The exemplary algorithms are motivated by an application to the compensation of polarization mode dispersion, which is a key problem in fiber optic systems. Gradient algorithms are shown to exhibit extremely slow convergence. This problem is solved by deriving a Newton-type algorithm. However, possible convergence to a local minimum emerges as a more fundamental problem. One of the conditions for local minima is specifically identified in terms of problematic parameter states.     

Stochastic analysis of an error power ratio scheme applied to the affine combination of two LMS adaptive filters

The affine combination of two adaptive filters that simultaneously adapt on the same inputs has been actively investigated. In these structures, the filter outputs are linearly combined to yield a performance that is better than that of either filter. Various decision rules can be used to determine the time-varying parameter for combining the filter outputs. A recently proposed scheme based on the ratio of error powers of the two filters has been shown by simulation to achieve nearly optimum performance. The purpose of this paper is to present a first analysis of the statistical behavior of this error power scheme for white Gaussian inputs. Expressions are derived for the mean behavior of the combination parameter and for the adaptive weight mean-square deviation. Monte Carlo simulations show good to excellent agreement with the theoretical predictions.