Kautz Filters Research Articles

Room Response Equalization of Non-Minimum Phase Systems Using Kautz Filter and Sparse Autoencoder: A Hybrid Approach

Room Response Equalization (RRE) systems play a vital role in enhancing the hearing experience in different real-time application areas such as cinema theatres, home theatres, hearing aid implementation, car hi-fi systems, etc. Thenceforth, RRE using digital signal processing techniques has been a matter of great interest to researchers for a long time. RRE is intended to provide a listener with a profound audio experience alike the original audio signal. Conventional filtering approaches have been used by several researchers to perform room equalization. However, this article presents a novel technique by cascading a Kautz filter and a sparse autoencoder in performing room equalization. Kautz filters belong to a category of fixed pole IIR filters that use least-square (LS) minimization to generate orthonormal tap-output impulse responses. This LS approximated signal is further trained by a sparse autoencoder, which provides filter coefficients, iteratively optimized, to minimize a cost function. The cost function is considered here in terms of the reconstruction error function. The performance of this hybrid network of equalization is evaluated in terms of mean square error and spectral deviation measures. Computational results show that this hybrid approach yields better results both qualitatively and quantitatively in comparison with the other filtering techniques.

An optimizationless stochastic volterra series approach for nonlinear model identification

Volterra series is a widely used tool for identifying physical systems with polynomial nonlinearities. In this approach, the Volterra kernels expanded using Kautz functions can be identified using several techniques to optimize the filters’ poles. This methodology is very efficient when the system observations are not subject to high noise-induced variabilities (uncertainties). However, this optimization procedure may not be effective when the uncertainty level is increased since the optimal value might be susceptible to small perturbations. Seeking to overcome this weakness, the present work proposes a new stochastic method of identification based on the Volterra series, which does not solve an optimization problem. In this new approach, the Volterra kernels are described as stochastic processes. The parameters of Kautz filters are considered independent random variables so that their probability distribution captures the variabilities. The effectiveness of the new technique is tested experimentally in a nonlinear mechanical system. The results show that the identified stochastic Volterra kernels can reproduce the nonlinear dynamics characteristics and the data variability.

Damage detection through nonparametric models using Kautz filters

The goal of this paper is to present an approach to detect structural changes by using nonparametric models. The impulse response functions (IRFs) of mechanical systems in healthy conditions are identified through the sum of convolution expanded on an orthonormal basis. The Kautz filters with multiple poles optimized are implemented as an efficient orthonormal basis to identify nonparametric reference models. Identifying the IRFs of a smart beam with PZTs actuators/sensors coupled is used to illustrate the necessary procedures. The damages are simulated as a loss of bolts, nuts, and washers attached near a PZT sensor. A statistical procedure of a hypothesis test is also used to confirm the smart beam’s structural state. The experimental results achieved show that it is possible to detect the structural changes with statistical confidence.

Inherently Stable Weighted Least-Squares Estimation of Common Acoustical Poles With the Application in Feedback Path Modeling Utilizing a Kautz Filter

In adaptive feedback cancellation, the feedback path must be modeled precisely using as few adaptive parameters as possible to reduce computational complexity and enable rapid convergence. To reduce the number of adaptive parameters, the feedback path is usually modeled with a transfer function composed of an invariant component and a variable component using all-pole, all-zero, and pole-zero filters. These filters may be inefficient, requiring a large number of parameters for their specification, particularly in reverberant environments. In this letter, we present a weighted least-squares algorithm to precisely estimate the common poles of feedback paths, and we then model the invariant basis functions employing a Kautz filter. The Kautz filter is defined by a set of the fixed poles and a corresponding set of tap output weights. The fixed poles are associated with prominent peaks that are common to the measured feedback path frequency responses. The algorithm guarantees unconditionally the stability of the estimated poles of the inferred model. The experimental results using measured acoustic feedback paths from a two-microphone, behind-the-ear hearing aid show that the proposed method provides an accurate model of the feedback paths for a variety of acoustic environments that were not employed in estimating the common poles.

Damage detection in an uncertain nonlinear beam

The present paper proposes the use of discrete time Volterra series expanded with Kautz filters for predicting the experimental response of a nonlinear beam, taking into account uncertainties that are inherent to the system. Nonlinear behavior is simulated considering large amplitude of vibration in a clamped-free beam, with a magnet near the free end to cause cubic sti_ness e_ect. The Volterra kernels and Kautz functions are estimated in a stochastic way, where the limits of confidence are established to the system response and used as a reference model to detect damage. The detection approach is based on a nonlinear stochastic index and hypothesis test, considering di_erent levels of simulated damage, associated with loss of mass. The results have shown that considering the uncertainties when identifying the model is very important in the process of damage detection and the nonlinear metric proved to be more appropriated to describe the system response with better results mainly in the nonlinear regime of motion.

Modelling and control of nonlinear resonating processes: part I—system identification using orthogonal basis function

Resonating systems show oscillatory characteristics. System identification of resonating systems and design of their model based control strategy always draw special attention. This work presents a Wiener type system identification technique for nonlinear resonating systems. Orthogonal basis function (henceforth termed as OBF) is employed to capture the linear dynamic part of the Wiener structure while the static nonlinear mapping is described by two means, viz., wavelet decomposition and least squares support vector machine. Use of OBF leads to a parsimonious nature in the resulting nonlinear model. Two types of OBFs have been used in this work viz. Laguerre filter and Kautz filter. The Kautz filter has capability of modelling systems with complex conjugate poles. A case study has been performed with continuous stirred tank reactor (henceforth referred as CSTR) which is a reasonably nonlinear resonating systems. Degree of nonlinearity as well as resonance increases with series-connected CSTR. Simulations are carried out using \(\hbox {MATLAB}^\circledR \) software in order to evaluate the performances of various Wiener structures, and identify the OBF–Wiener model best suited for designing a model based controller.

Model predictive control of resonant systems using Kautz model

The scope of this paper broadly spans in two areas: system identification of resonant system and design of an efficient control scheme suitable for resonant systems. Use of filters based on orthogonal basis functions (OBF) have been advocated for modelling of resonant process. Kautz filter has been identified as best suited OBF for this purpose. A state space based system identification technique using Kautz filters, viz. Kautz model, has been demonstrated. Model based controllers are believed to be more efficient than classical controllers because explicit use of process model is essential with these modelling techniques. Extensive literature search concludes that very few reports are available which explore use of the model based control studies on resonant system. Two such model based controllers are considered in this work, viz. model predictive controller and internal model controller. A model predictive control algorithm has been developed using the Kautz model. The efficacy of the model and the controller has been verified by two case studies, viz. linear second order underdamped process and a mildly nonlinear magnetic ball suspension system. Comparative assessment of performances of these controllers in those case studies have been carried out.

Kautz filters based model predictive control for resonating systems

Model predictive control (MPC) of a special class of systems called resonating system is presented in this paper using the recursive Kautz model. Kautz functions are known to have the ability to describe resonating systems. They are formulated in a recursive fashion in order to compute the future incremental control effort of MPC. Use of Kautz functions in the optimization procedure leads to parsimonious representation. The efficacy of the proposed Kautz MPC is tested on two case studies, viz. linear highly oscillatory mechanical system and mildly nonlinear magnetic ball suspension system. Performance of Kautz MPC is compared against already published Laguerre MPC. Stability of the Kautz MPC is established through Lyapunov criteria.

Twin Support Vector Machine Method for Identification of Wiener Models

Twin support vector regression is applied to identify nonlinear Wiener system, consisting of a linear dynamic block in series with static nonlinearity. The linear block is expanded in terms of basis functions, such as Laguerre or Kautz filters, and the static nonlinear block is determined using twin support vector machine regression. Simulation of a control valve model and pH neutralization process have been presented to show the features of the proposed algorithm over support vector machine based algorithm.

Identification of MR Damper Based on Normalized Bouc-Wen Model Using Neural Network

The Magneto-rheological (MR) dampers are favorite mechanical system in dynamic structures. This paper presents an application of Wiener-type nonlinear models for describing the hysteresis behaviors of MR dampers at different operating conditions. In this structure, a linear part consisting discrete-time Kautz filters is cascading by a nonlinear mapping function (feedforward neural network (FFNN)). The pole parameters of Kautz filters were chosen with respect to the poles of best fitted linear model on real system. By defining the parameters of Kautz filter, the nonlinear behaviors of system were identified using neural network model, as the output of filters were considered as the output on NN. In order to assess the performances of the developed models a comparison between the responses of the models and another recent modeling approach was preformed.

Improved Pole Positioning for Parallel Filters Based on Spectral Smoothing and Multi-Band Warping

The use of second-order parallel filters with pre-defined pole locations has been recently proposed for equalization and transfer function modeling. This letter presents an improved method for obtaining the pole positions of the parallel filter. The steps of the new method are the following: first, the target frequency response is smoothed to the required resolution. Then, multiple warped IIR filters with different warping parameters are designed to the smoothed target divided into frequency bands. Finally, the united pole set of the warped IIR designs is used for parallel filter design. The method leads to increased accuracy compared to earlier pole positioning techniques, and can also be used for Kautz filter designs. Examples of loudspeaker-room response modeling and equalization are presented.

Non-parametric identification of mechanical systems by Kautz filter with multiple poles

The main goal of this short communication is to use orthogonal Kautz filters with multiple poles for non-parametric identification of the impulse response functions (IRFs) of mechanical structures. The IRF is estimated by using the covariance method based on the sum of convolution of the input signal processed by Kautz filters. The inclusion of different poles in the Kautz filter allows the use of a wide range of frequency and can improve the non-parametric identification process in the time domain by reducing the number of necessary terms. Some examples are provided to illustrate the simplicity and efficiency of the proposed approach.

Non-linear model updating of a three-dimensional portal frame based on Wiener series

The common metrics used in linear finite element (FE) model updating using vibration test data are generally functions of relationships based on unidimensional convolution, for example, distances involving natural frequencies, frequency response or impulse response functions, modal shapes, etc. When a structure has local elements or geometry, like joints, bolts, gaps, backlash, etc., these approaches can fail once it could to induce non-linear behavior. Thus, the methods for FE model updating, when considering the existence of localized non-linear parameters, have been receiving much attention in the last years. In this sense, the present paper proposes the use of a strategy through objective functions based on multiples convolutions described by the first order and second order discrete-time Volterra kernels. These kernels are effective metrics for a model updating into large FE model with local non-linearity. In order to improve the non-linear coefficient identification, an orthogonal basis involving Kautz filter is used to expand the kernels, called by Wiener kernel. To exemplify in full details the steps of the updating procedure, an FE model of a three-dimensional portal frame with commons non-linearities is simulated with different excitation forces and used to identify the non-linear parameters. These results allow us to characterize the practical applicability and the drawbacks of the proposed method with suggestions and remarks for further use in industrial structures.

Least-squares Support Vector Regression for Identification of Periodic Systems

Least-squares support vector regression is applied to identify linear periodically time-varying (LPTV) systems. In this contribution the LPTV systems are modelled using a Fourier series representation of the transfer function and orthonormal filter expansions, such as Laguerre or Kautz filters. The model parameters are determined using least-squares support vector regression. In order to decrease complexity of the identified model, smoothness priors which penalize fluctuations of the frequency response can be introduced in the support vector procedure. The performance of the identified models are assessed by means of several simulation trials.

Nonlinear identification in structural dynamics based on Wiener series and Kautz filters

The present paper proposes a method to identify localized nonlinear parameters in structural dynamics using vibration data. The approach is based on the identification of the first and second-order Volterra kernels in an orthogonal basis, namely Wiener kernels, while taking the experimental data as exact values. The focus is to identify an already localized nonlinearity with known structure. An optimization procedure is implemented using a metric involving the difference between the experimental kernel and the simulated orthogonal kernel from model updating, which is function of the unknown nonlinear parameters. In order to reduce the problems of convergence, the Kautz filter is used to decrease the number of samples to estimate. The proposed methodology is illustrated on a numerical example using a cantilever beam and its advantages and drawbacks are discussed in detail.

Smoothness priors support vector method for robust system identification

Support vector regression (SVR) is applied to identify linear dynamical systems. The system model is described in terms of basis functions, such as Laguerre or Kautz filters, and the coefficients of the expansion are determined using support vector machine regression. In SVR, the variance of the parameter estimates is bounded by the inclusion of a quadratic regularisation term. Here, model complexity is efficiently reduced by taking the regularisation term as a frequency-domain smoothness prior, defined as the square of the ℒ2-norm of the mth order derivative of the frequency response function.

Support vector method for identification of Wiener models

Support vector regression is applied to identify nonlinear systems represented by Wiener models, consisting of a linear dynamic system in series with a static nonlinear block. The linear block is expanded in terms of basis functions, such as Laguerre or Kautz filters, and the static nonlinear block is determined using support vector machine regression.

Identification of Mechanical Systems through Kautz Filter

This paper presents a novelty approach to estimate the impulse response function (IRF) in lightly damped structures by using classical covariance method. The contribution of the present paper is to employ a Kautz filter to reduce the number of terms in the identification process of a finite impulse response (FIR) model. The Kautz filter is an orthogonal function that includes complex pole pair. So, the method incorporates a priori knowledge into the process that is essential to reach better results and to minimize the number of coefficients in the FIR model. For this reason, the present approach can be classified as a kind of “grey-box” methodology. The numerical and experimental applications of the method confirmed that by considering a priori parametric information the problem can be determined more easily. Moreover, the procedure proposed increased the convergence speed, and, as a consequence the consuming computational cost is lower.

Perceptually Motivated Audio Equalization Using Fixed-Pole Parallel Second-Order Filters

In audio, equalizer design should take into account the frequency resolution of the auditory system. In this paper, this is accomplished by the fixed-pole design of parallel second-order filters. The design process has two steps: first, the poles of the filter are set according to the desired frequency resolution. Then, the feedforward coefficients of the second-order filters are determined by a linear least squares solution. The proposed parallel filter achieves effectively the same equalization results as the Kautz filter, but it requires 33% fewer multiplications and additions.

Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design

DSP-based correction of loudspeaker and room responses is becoming an important part of improving sound reproduction. Such response equalization (EQ) is based on using a digital filter in cascade with the reproduction channel to counteract the response errors introduced by loudspeakers and room acoustics. Several FIR and IIR filter design techniques have been proposed for equalization purposes. In this paper we investigate Kautz filters, an interesting class of IIR filters, from the point of view of direct least squares EQ design. Kautz filters can be seen as generalizations of FIR filters and their frequency-warped counterparts. They provide a flexible means to obtain desired frequency resolution behavior, which allows low filter orders even for complex corrections. Kautz filters have also the desirable property to avoid inverting dips in transfer function to sharp and long-ringing resonances in the equalizer. Furthermore, the direct least squares design is applicable to nonminimum-phase EQ design and allows using a desired target response. The proposed method is demonstrated by case examples with measured and synthetic loudspeaker and room responses.