Adaptive Infinite Impulse Response Research Articles

A Self-Adjusting Simple Hyperstable Adaptive Recursive Algorithm

Adaptive infinite impulse response (IIR), or recursive, filters are less attractive mainly because of the stability and the difficulties associated with their adaptive algorithms. Hyperstability theory when simplified and adapted for digital signal processing offers a new class of IIR filters, simple hyperstable adaptive recursive filters (SHARFs), which is directly related to strictly positive real (SPR) transfer functions. One of the most important drawbacks of the SHARF algorithm is the presence of the unknown denominator in the transfer function that must be SPR in order to ensure convergence. In this paper, SHARF is investigated with SPR transfer functions designed without any priori knowledge of the filter parameters by the pole–zero placement on the unit circle method and made self-adjusting. To demonstrate self-adjustment of the algorithm, SHARF algorithm using constraint least-mean square (LMS) method is applied to a pure four-pole autoregressive process.

Performance Analyses of Adaptive IIR Notch Filters Using a PSD-Based Approach

In this letter we present steady-state analyses of a gradient algorithm (GA) for second-order adaptive infinite impulse response (IIR) notch filters. A method for deriving more accurate estimation mean square error (MSE) expressions than the recently proposed method is presented. The method is based on the estimation error power spectral density (PSD). Moreover, an expression for the estimation bias for the adaptive IIR notch filter with constrained poles and zeros is shown to be obtained from the estimation MSE expression. Simulations are presented to confirm the validity of the analyses.

Fast RLS-type algorithm for unbiased equation-error adaptive IIR filtering based on approximate inverse-power iteration

In this paper, a fast approximate inverse-power (AIP) iteration is applied to compute recursively the total least squares (TLS) solution for unbiased equation-error adaptive infinite impulse response (IIR) filtering, which is established by approximating the well-known inverse-power iteration with Galerkin method. The AIP algorithm is based on an interesting modification of the inverse-power iteration in which the first entry of the parameter vector is constrained to the negative one. The distinctive feature of the proposed algorithm lies in its high computational efficiency, which is characterized by efficient computation of the fast gain vector (FGV), adaptation of the interesting modification of the inverse-power iteration, and rank-one update of the augmented correlation matrix. The shift structure of the input data vector is exploited to develop a fast algorithm for computing the gain vector. This FGV algorithm can be implemented at a numerical cost lower than the well-known fast Kalman algorithm. Since the first entry of the parameter vector has been fixed as the negative one and the weight vector is updated along the input data vector, a very efficient AIP algorithm is obtained by using the FGV algorithm. The proposed AIP algorithm is of computational complexity O(L) per iteration. Moreover, with no need to use the well-known matrix-inversion lemma, the AIP algorithm has another attractive feature of numerical stability. The proposed algorithm is shown to have global convergence. Simulation examples are included to demonstrate the effectiveness of the proposed AIP algorithm.

A new framework for convergence analysis and algorithm development of adaptive IIR filters

A parameterization of an adaptive infinite impulse response (IIR) filter's poles is developed, based on balanced realization theory. From this, we develop a local approximation of the actual adapted pole parameters, in which convergence speed is related to a certain eigenvalue spread. This, in turn, is shown to relate to the Hankel singular values of the system to be identified, as well as certain coefficient sensitivity functions of the adapted filter. The local approximation is not restricted to stationary points. At these points, however, it is equivalent to a Hessian approximation, with the benefit of decomposing the Hessian matrix into terms related to the aforementioned singular values and sensitivity functions. The description of the adaptation process by means of the developed approximation leads to a greater understanding of the effects on convergence speed of factors, such as the Hankel singular values of the system, its order, the distribution of its poles, and the choice of adapted parameters. In particular, the use of direct form and lattice parameters are compared in detail. Based on these properties, a new adaptive IIR algorithm with faster convergence and relatively low computational complexity has recently been proposed, which is briefly mentioned. Results also indicate a potential for variable gain algorithms.

Statistical performance of the memoryless nonlinear gradient algorithm for the constrained adaptive IIR notch filter

Gradient-type algorithms for the adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of both performance and computational requirements for various real-life applications. This paper presents, in detail, a statistical analysis of the memoryless nonlinear gradient (MNG) algorithm applied to the well-known second-order adaptive IIR notch filter with constrained poles and zeros. This analysis is based on a proper use of Taylor series expansion and nonlinearization of output signals of the notch and gradient filters. Two difference equations are derived first for the convergence in the mean and mean square senses, respectively. Two closed-form expressions, one for the steady-state estimation bias and the other for the mean-square error, are then derived based on the difference equations, with the former valid for both fast and slow adaptations and the latter valid for slow adaptation only. A closed-form coarse stability bound for the step size parameter of the algorithm is also derived. Extensive simulations are performed to reveal the validity and limitations of the analytical findings. Comparisons between the MNG and the conventional plain gradient algorithm are also made.

A new method for adaptive IIR filter design based on tabu search algorithm

Adaptive digital filters have proven their worth in a wide range of applications such as channel equalisation, noise reduction, echo cancelling, and system identification. These filters can be broadly classified into two groups: finite impulse–response (FIR) and infinite impulse–response (IIR) filters. IIR filters have become the target of increasing interest because these filters can reduce the filter order significantly as compared to FIR filters. Tabu search is a heuristic optimisation algorithm which has been originally developed for combinatorial optimisation problems. It simulates the general rules of intelligent problem solving and has the ability of discovering the global minima in a multi-modal search space. In this work, a novel method based on tabu search is described for the design of adaptive IIR filters.

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.

Combined Regressor Methods and Adaptive IIR Filtering

An open issue in adaptive infinite-impulse response (IIR) filtering is that of convergence to a global minimum in the presence of observation noise when the system is insufficiently modeled . It is well known , that algorithms based on equation error (EE) formulation contain a single minimum that may be biased whereas, algorithms based on output error (OE) ensure the existence of an unbiased global minimum in presence of local minima. Recently, there have been several attempts to combine these formulations in order to ensure the existence and uniqueness of an unbiased minimum. Works presented here, EEOE and modified EEOE (MEEOE), are such attempts in the context of system identification. We will show, analytically and through simulations, the convergence properties of the MEEOE approach, in the context of system identification.

Global Stability of Adaptive IIR Filters Based on the Output Error Method and Parameter Truncation

This paper presents a rigorous proof of global stability of an adaptive infinite impulse response filter based on the output error method. The underlying idea of the developed analysis is the construction of suitable Lyapunov functions for different phases of adaptation. The considered estimator is an algorithm with a nonvanishing gain sequence. In order to prevent drift along an unstable manifold, a truncation of parameter estimates is incorporated. Such an algorithm is suitable for nonstationary signals.

Pole-cancelation error adaptive IIR filtering

We present an adaptive infinite-impulse response (IIR) filtering structure that is based on a novel performance measure. The proposed adaptive IIR filter consists of two finite-impulse response (FIR) filters that are directly driven by a noise-free input signal. One adaptive filter inversely estimates the poles, and the other filter directly estimates the zeros of an unknown system. Since the filter structure consists of two FIR filters, the performance measure is unimodal, and thus no local minimum exists. The simulation results demonstrate that the proposed method can indeed produce unique unbiased parameter estimates in the presence of both white and colored noise.

Performance analysis of the sign algorithm for a constrained adaptive IIR notch filter

Adaptive infinite impulse response (IIR) notch filters are very attractive in terms of their reasonable performances and low computational requirements. Generally, it is very difficult to assess their performances analytically due to their IIR nature. This paper analyzes in detail the steady-state performance of the sign algorithm (SA) for a well-known adaptive IIR notch filter with constrained poles and zeros. Slow adaptation and Gaussianity of the notch filter output are assumed for the sake of analysis. Two difference equations are first established for the convergences in the mean and mean square in the vicinity of the steady state of the algorithm. Steady-state estimation error or bias and mean square error (MSE) of the SA are then derived in closed forms. A coarse stability bound is also derived for the algorithm. Theory-based comparison between the algorithm and the plain gradient (PG) algorithm is done in some detail. Extensive simulations are conducted to demonstrate the validity of the analytical results for both slow and relatively fast adaptations.

Subband hyperstable adaptive iir filters

Two subband implementations of hyperstability based adaptive infinite-impulse response (IIR) filtering algorithms are presented. These schemes are based on the polyphase decomposition of the adaptive filter and do not require cross filters to avoid aliasing. With this configuration, the larger the number of subbands, the more likely the SPR condition that this class of algorithms requires for convergence is to be satisfied. In addition, it is possible to assign different weights and/or compensating filters across the subbands, thus providing the designer with additional flexibility.

QR-based TLS and mixed LS-TLS algorithms with applications to adaptive IIR filtering

The least squares (LS), total least squares (TLS), and mixed LS-TLS approaches are compared as to their properties and performance on several classical filtering problems. Mixed LS-TLS is introduced as a QR-decomposition-based algorithm for unbiased, equation error adaptive infinite impulse response (IIR) filtering. The algorithm is based on casting adaptive IIR filtering into a mixed LS-TLS framework. This formulation is shown to be equivalent to the minimization of the mean-square equation error subject to a unit norm constraint on the denominator parameter vector. An efficient implementation of the mixed LS-TLS solution is achieved through the use of back substitution and inverse iteration. Unbiasedness of the system parameter estimates is established for the mixed LS-TLS solution in the case of uncorrelated output noise, and the algorithm is shown to converge to this solution. LS, TLS, and mixed LS-TLS performance is then compared for the problems of echo cancellation, noise reduction, and frequency equalization.

Lattice form adaptive infinite impulse response filtering algorithm for active noise control.

In some situations of active noise control, infinite impulse response (IIR) filters are more suitable than finite impulse response (FIR) filters owing to the poles in the transfer function. A number of algorithms have been derived for applying IIR filters in active noise control; however, most of them use the direct form IIR filter structure, which faces the difficulties of checking stability and relatively slow convergence speed for noise composed of narrow-band components with large power disparity. To overcome these difficulties along with using the direct form IIR filters, a new adaptive algorithm is proposed in this paper, which uses and updates the lattice form adaptive IIR filter in an active noise control system. Full mathematical derivations of the proposed algorithm are presented, and the comparison between the proposed algorithm and the commonly used filtered-u LMS and filtered-v LMS algorithms shows the superiority of the proposed algorithm.

Steady‐state analysis of complex adaptive IIR notch filters for the detection of single sinusoid

AbstractIn this paper, we present a new structure for a complex adaptive infinite impulse response (IIR) notch filter which is useful for the detection of complex sinusoids in communication and radar systems. The proposed structure is based on the gradient algorithm. The adaptive detection of a complex sinusoid with additive complex white noise has been investigated. A quantitative analysis for the steady‐state coefficient fluctuation has been developed. Analytical expressions for the steady‐state variance of complex variable coefficient fluctuation have been obtained. It has been shown that the variance of the complex coefficient fluctuation does not depend on the angular frequency of the input sinusoid. Finally, the results of computer simulations which confirm the theoretical predictions are presented. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(1): 32–40, 2002

Steady-state analysis of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros

Gradient-type algorithms for adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of performance and computation cost. However, it is generally quite difficult to assess their performances analytically. There are several trials to analyze adaptive algorithms, such as the sign and the plain gradient algorithms for some types of adaptive IIR notch filters, but the analysis techniques used cannot be directly applied to different types of adaptive IIR notch filters. This brief presents closed form expressions for steady-state estimation bias and mean square error (MSE) of a well known plain gradient (LMS-like) second-order adaptive IIR notch filter with constrained poles and zeros. First, theoretical expressions for output signals of the notch filter and its corresponding gradient filter at their steady states are developed based on the Taylor series expansions of transfer functions of these two filters in the vicinity of the sinusoidal signal frequency difference equations for convergences in the mean and mean square are then established by using these output signals, from which the steady-state bias and MSE of the algorithm are derived. Stability bound of the algorithm is also investigated based on the difference equations. Extensive simulations are provided to support the analytical findings.

Hyperspherical parametrization for unit-norm based adaptive IIR filtering

The bias problem associated with equation error based adaptive infinite impulse response (IIR) filtering can be surmounted by imposing a unit-norm constraint on the autoregressive (AR) coefficients. We propose a hyperspherical parameterization to convert the unit-norm-constrained optimization into an unconstrained optimization. We show that the hyperspherical parameterization does not introduce any additional minima to the equation error surface.

Unbiased equation-error adaptive IIR filtering based on monic normalization

We present a novel may to remove the bias in equation-error based adaptive infinite impulse response (IIR) filtering by conceiving a scheme called monic normalization. It is found that normalizing all the coefficients of the denominator filter by the first coefficient after each adaptation removes the bias and leads to unbiased estimates. The analysis of stationary points is presented to show that the proposed method can indeed produce unbiased parameter estimates in the presence of noise. The computer simulation results also demonstrate that the proposed method performs better than or comparable to existing algorithms, while requiring much lower computational complexity.

A new memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter and its performance analysis

Adaptive infinite impulse-response (IIR) notch filters have been widely studied for many years. However, further efforts are still to be made to pursue new algorithms which work better than the conventional plain gradient algorithm but have little increase in complexity. In this paper, we employ the gradient linearization, Taylor series expansion, and calculus of variations to derive a novel memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter, which improves the estimation performance considerably. Approximate closed-form expressions for the stability bounds on the step size parameter and the steady-state coefficient variance are also derived. Extensive simulations indicate the significant improvement that may be achieved using the new algorithm, and verify the validity of the analytical results.

A global least mean square algorithm for adaptive IIR filtering

In this brief, we develop a least mean square (LMS) algorithm that converges in a statistical sense to the global minimum of the mean square error (MSE) objective function. This is accomplished by estimating the gradient as a smoothed version of the MSE, The smoothed MSE objective begins as a convex functional in the mean. The amount of dispersion or smoothing is reduced, such that over time it becomes the true MSE as the algorithm converges to the global minimum. We show that this smoothing behavior is approximated by appending a variable noise source to the infinite impulse response (IIR)-LMS algorithm. We show, experimentally, that the proposed method does converge to the global minimum in the cases tested. A performance improvement over the IIR-LMS algorithm and the Steiglitz-McBride algorithm has been achieved.