Orthonormal Basis Functions Research Articles

Identification of Wiener Models in the Presence of Structural Disturbances

In empirical system identification, non-stationary structural disturbances, such as trends and outliers, can have a negative effect on the estimation of the system parameters. As it not possible to determine a priori which parts of the measured data stem from structural disturbances and which are due to the system dynamics, the identification of structural disturbances and system model should be done simultaneously. In this study, a method for output error identification of nonlinear Wiener models in the case when the measurement is affected by trends and outliers is presented. The Wiener model can be described by a dynamic linear block followed by an static nonlinear block. In the proposed method the dynamic block is expanded using orthonormal basis functions, while the static nonlinear block is modeled by a kernel model. The kernel parameters and structural disturbances are estimated simultaneously by using sparse optimization, which is solved using l1-regularization and iterative reweighting. The feasibility of the proposed method is demonstrated on a simulated example.

A Scalable Algorithm for Physically Motivated and Sparse Approximation of Room Impulse Responses With Orthonormal Basis Functions

Parametric modeling of room acoustics aims at representing room transfer functions by means of digital filters and finds application in many acoustic signal enhancement algorithms. In previous work by other authors, the use of orthonormal basis functions (OBFs) for modeling room acoustics has been proposed. Some advantages of OBF models over all-zero and pole-zero models have been illustrated, mainly focusing on the fact that OBF models typically require less model parameters to provide the same model accuracy. In this paper, it is shown that the orthogonality of the OBF model brings several additional advantages, which can be exploited if a suitable algorithm for identifying the OBF model parameters is applied. Specifically, the orthogonality of OBF models does not only lead to improved model efficiency (as pointed out in previous work), but also leads to improved model scalability and model stability. Its appealing scalability property derives from a previously unexplored interpretation of the OBF model as an approximation to a solution of the inhomogeneous acoustic wave equation. Following this interpretation, a novel identification algorithm is proposed that takes advantage of the OBF model orthogonality to deliver efficient, scalable, and stable OBF model estimates, which is not necessarily the case for nonlinear estimation techniques that are normally applied.

Identification of System Dynamics with Time Delay: a Two-Stage Frequency Domain Approach

Frequency domain identification of system parameters and time delay is essential for model-based motion control of dynamic systems. Conventional approaches incorporate the delay into the model through an explicit model parametrization, which results in a high amount of nonlocal minima. In this paper, a new strategy is proposed based on a two-stage frequency domain approach to linearly incorporate the delay in the parameter vector of the system. This enables the proposed method to accurately identify both the time delay and system parameters. In a first stage, the system is parametrized and the delay is approximated using rational (all-pass) orthonormal basis functions. The resulting constraints on the individual delay approximation parameters are relaxed through a general all-pass constraint which is enforced iteratively by reformulating the identification algorithm. In a second stage, the continuous time delay term is identified in a convex problem with fixed system parameters using the previous approximates as starting values. Finally, simulations and experimental results validate the effectiveness of this new two-stage method.

Flexible ILC: Towards a Convex Approach for Non-Causal Rational Basis Functions

Iterative learning control (ILC) is subject to a trade-off between effective compensation of repeating disturbances, and amplification of non-repeating disturbances. Although important progress has been made in enhancing the flexibility of ILC to non-repeating tasks by means of basis functions, at present high performance comes at the cost of non-convex optimization. The aim of this paper is to develop a convex approach to ILC with rational basis functions. A key aspect of the proposed approach is the use of orthonormal basis functions in L2, such that non-causal control actions can be utilized. The benefits of using non-causal rational basis functions in ILC are demonstrated by means of a relevant example.

Model Order Reduction of Electrical Machines With Multiple Inputs

In this paper, proper orthogonal decomposition (POD) method is employed to build a reduced-order model from a high-order nonlinear permanent magnet synchronous machine model with multiple inputs. Three parameters are selected as the multiple inputs of the machine. These parameters are terminal current, angle of the terminal current, and rotation angle. To produce the lower-rank system, snapshots or instantaneous system states are projected onto a set of orthonormal basis functions with small dimension. The reduced model is then validated by comparing the vector potential, flux density distribution, and torque results of the original model, which indicates the capability of using the POD method in the multivariable input problems. The developed methodology can be used for fast simulations of the machine.

Detection of a Moving Magnetic Dipole Target Using Multiple Scalar Magnetometers

A detection procedure for a moving magnetic dipole target by using at least three scalar magnetometers is proposed in this letter. The magnetometers are deployed on the ground or seafloor, while the target is traveling along a straight line at a constant speed above the ground. First, the range from each magnetometer to the closest point of approach (CPA) on the flight line as well as the instant that the target passes by the CPA is estimated by adopting the orthonormal basis function expansion method. Then, the moving parameters, including the moving direction, speed, and elevation, are found through basic geometry relations. Finally, the magnetic moment is determined by solving a linear system by using the least-squares method. Both simulated and field experimental data are employed to demonstrate the proposed scheme.

A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes

In this paper, we proposed a new efficient high-order finite volume method for 3D elastic wave simulation on unstructured tetrahedral meshes. With the relative coarse tetrahedral meshes, we make subdivision in each tetrahedron to generate a stencil for the high-order polynomial reconstruction. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. We perform the reconstruction on this stencil by using cell-averaged quantities based on the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, our new method has a very local property like DG and can be written as an inner-split computational scheme which is beneficial to reducing computational amount. Moreover, the stencil in our method is easy to generate for all tetrahedrons especially in the three-dimensional case. The resulting reconstruction matrix is invertible and remains unchanged for all tetrahedrons and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. We show convergence results obtained with the proposed method up to fifth order accuracy in space. The high-order accuracy in time is obtained by the Runge–Kutta method. Comparisons between numerical and analytic solutions show the proposed method can provide accurate wavefield information. Numerical simulation for a realistic model with complex topography demonstrates the effectiveness and potential applications of our method. Though the method is proposed based on the 3D elastic wave equation, it can be extended to other linear hyperbolic system.

A High-Order Time Domain Discontinuous Galerkin Method with Orthogonal Tetrahedral Basis for Electromagnetic Simulations in 3-D Heterogeneous Conductive Media

AbstractWe present a high-order discontinuous Galerkin (DG) method for the time domain Maxwell's equations in three-dimensional heterogeneous media. New hierarchical orthonormal basis functions on unstructured tetrahedral meshes are used for spatial discretization while Runge-Kutta methods for time discretization. A uniaxial perfectly matched layer (UPML) is employed to terminate the computational domain. Exponential convergence with respect to the order of the basis functions is observed and large parallel speedup is obtained for a plane-wave scattering model. The rapid decay of the out-going wave in the UPML is shown in a dipole radiation simulation. Moreover, the low frequency electromagnetic fields excited by a horizontal electric dipole (HED) and a vertical magnetic dipole (VMD) over a layered conductive half-space and a high frequency ground penetrating radar (GPR) detection for an underground structure are investigated, showing the high accuracy and broadband simulation capability of the proposed method.

Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques

In this paper we extend the hierarchical model reduction framework based on reduced basis techniques for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of the adaptive empirical projection method, which is an adaptive integration algorithm based on the (generalized) empirical interpolation method. Different from other partitioning concepts for the empirical interpolation method we perform an adaptive decomposition of the spatial domain. We project both the variational formulation and the range of the nonlinear operator onto reduced spaces. Those reduced spaces combine the full dimensional (finite element) space in an identified dominant spatial direction and a reduction space or collateral basis space spanned by modal orthonormal basis functions in the transverse direction. Both the reduction and the collateral basis space are constructed in a highly nonlinear fashion by introducing a parametrized problem in the transverse direction and associated parametrized operator evaluations, and by applying reduced basis methods to select the bases from the corresponding snapshots. Rigorous a priori and a posteriori error estimators, which do not require additional regularity of the nonlinear operator are proven for the adaptive empirical projection method and then used to derive a rigorous a posteriori error estimator for the resulting hierarchical model reduction approach. Numerical experiments for an elliptic nonlinear diffusion equation demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full-dimensional problem. Runtime experiments verify a close to linear scaling of the reduction method in the number of degrees of freedom used for the computations in the dominant direction.

Identification of a magnetic levitator using NARX-OBF models and genetic algorithm

The Volterra models using orthonormal basis functions (OBFs) are very common in the system identification literature. These models are called Volterra-OBF and they only use polynomial operations with the filtered input signals to capture the behaviour of dynamic systems. The extension of this idea to the filtered output, combined with the filtered input signals, leads to the nonlinear auto regressive with exogenous input - orthonormal basis function (NARX-OBF) models. Within this context, the goal of this paper is to identify a nonlinear system with a NARX-OBF model and compare its results to the one obtained using a Volterra-OBF model. In order to determine the model parameters, some heuristic optimisation methods are presented. The identification of a magnetic levitator is presented in order to exemplify the use of these models. Regarding the comparison between NARX-OBF and Volterra-OBF, in nonlinear system identification, one can conclude that NARX-OBF models have reached smaller mean square error (MSE) in tested cases.

Identification of a magnetic levitator using NARX-OBF models and genetic algorithm

The Volterra models using orthonormal basis functions (OBFs) are very common in the system identification literature. These models are called Volterra-OBF and they only use polynomial operations with the filtered input signals to capture the behaviour of dynamic systems. The extension of this idea to the filtered output, combined with the filtered input signals, leads to the nonlinear auto regressive with exogenous input - orthonormal basis function (NARX-OBF) models. Within this context, the goal of this paper is to identify a nonlinear system with a NARX-OBF model and compare its results to the one obtained using a Volterra-OBF model. In order to determine the model parameters, some heuristic optimisation methods are presented. The identification of a magnetic levitator is presented in order to exemplify the use of these models. Regarding the comparison between NARX-OBF and Volterra-OBF, in nonlinear system identification, one can conclude that NARX-OBF models have reached smaller mean square error (MSE) in tested cases.

Steerable Principal Components for Space-Frequency Localized Images.

As modern scientific image datasets typically consist of a large number of images of high resolution, devising methods for their accurate and efficient processing is a central research task. In this paper, we consider the problem of obtaining the steerable principal components of a dataset, a procedure termed "steerable PCA" (steerable principal component analysis). The output of the procedure is the set of orthonormal basis functions which best approximate the images in the dataset and all of their planar rotations. To derive such basis functions, we first expand the images in an appropriate basis, for which the steerable PCA reduces to the eigen-decomposition of a block-diagonal matrix. If we assume that the images are well localized in space and frequency, then such an appropriate basis is the prolate spheroidal wave functions (PSWFs). We derive a fast method for computing the PSWFs expansion coefficients from the images' equally spaced samples, via a specialized quadrature integration scheme, and show that the number of required quadrature nodes is similar to the number of pixels in each image. We then establish that our PSWF-based steerable PCA is both faster and more accurate then existing methods, and more importantly, provides us with rigorous error bounds on the entire procedure.

Nonlinear acoustic echo cancellation using an adaptive Hammerstein block structure based on a generalized basis function

We investigated adaptive algorithms for a Hammerstein block structure in which a static nonlinear block and dynamic linear block are cascaded. The approach considered here is to use generalized orthonormal basis functions in a Hammerstein block structure by using fixed pole filter banks. We applied the normalized least mean square approach to the developed adaptive algorithm in order to acquire Hammerstein block structure parameters. Performance comparison of the proposed approach was investigated considering convergence speed and parametric complexity for acoustic echo cancellation application. The results indicated that in the developed algorithm along with appropriate selection of fixed poles, the algorithm convergences faster and less parametric complexity is provided when compared to direct adaptive Hammerstein algorithms with IIR and FIR linear blocks.

Illumination Invariant Face Recognition Model using Tetrolet Transform and Truncated Gaussian Mixture Model

Face Recognition is the most focused topic in the field of computer vision and pattern recognition. It has become a major thrust area for research in the last two decades due to the security aspects and demand for video. Face is focus of attention in the social intercourse and it plays a vital role in identification and recognition of individual emotions. Several facial recognition algorithms were developed and discussed in the literature but very little work is focused on facial recognition based on illumination. The appearance of the face image is usually affected by illumination conditions that hinder the facial recognition process. Hence in this paper, we propose and develop a new facial recognition algorithm based on Adaptive Haar Wavelet Transform called Tetrolet Transfrom. In Tetrolet transform, the determined orthonormal basis functions are adapted to geometrical features of the image follow a Truncated Gaussian Mixture Model (TGMM). The truncation on the feature vector has a significant influence in improving the recognition rate of the system.

Unsupervised Image Segmentation with Pairwise Markov Chains Based on Nonparametric Estimation of Copula Using Orthogonal Polynomials

Copula were introduced in Markov models in the early 2000s to better model the relationship between the observation data involved in these models. However, their estimation is difficult. This paper presents a new approach in the estimation of copula in Markov models. The proposed approach is based on a nonparametric method of estimating the density of the copula. The decomposition of an orthonormal basis of the unit interval support polynomials is used to estimate this density. The family of polynomial used is built from the family of Legendre polynomials. Our approach has the major advantage of reducing the problem of unsupervised image segmentation by Pairwise Markov chains to the problem of the estimated marginal distributions unlike the conventional approach which requires both an estimation of the density the marginal distributions and the estimation of the density distribution of the copula. Moreover, the problems of boundary effect encountered in the density estimation of copulas are solved by the use of orthonormal basis functions in the unit interval. The new model is validated from experiments performed on synthetic images and real images including optical and radar satellite images not necessarily affected by the Gaussian noise. The results are encouraging and show the proposed model as an interesting alternative to Pairwise Markov chains commonly used in literature.

Modeling vortex-shedding effects for the stochastic response of tall buildings in non-synoptic winds

This study derives a model for the vortex-induced vibration and the stochastic response of a tall building in strong non-synoptic wind regimes. The vortex-induced stochastic dynamics is obtained by combining turbulent-induced buffeting force, aeroelastic force and vortex-induced force. The governing equations of motion in non-synoptic winds account for the coupled motion with nonlinear aerodynamic damping and non-stationary wind loading. An engineering model, replicating the features of thunderstorm downbursts, is employed to simulate strong non-synoptic winds and non-stationary wind loading. This study also aims to examine the effectiveness of the wavelet-Galerkin (WG) approximation method to numerically solve the vortex-induced stochastic dynamics of a tall building with complex wind loading and coupled equations of motions. In the WG approximation method, the compactly supported Daubechies wavelets are used as orthonormal basis functions for the Galerkin projection, which transforms the time-dependent coupled, nonlinear, non-stationary stochastic dynamic equations into random algebraic equations in the wavelet space. An equivalent single-degree-of-freedom building model and a multi-degree-of-freedom model of the benchmark Commonwealth Advisory Aeronautical Research Council (CAARC) tall building are employed for the formulation and numerical analyses. Preliminary parametric investigations on the vortex-shedding effects and the stochastic dynamics of the two building models in non-synoptic downburst winds are discussed. The proposed WG approximation method proves to be very powerful and promising to approximately solve various cases of stochastic dynamics and the associated equations of motion accounting for vortex shedding effects, complex wind loads, coupling, nonlinearity and non-stationarity.

Asymmetric Volterra Models Based on Ladder-Structured Generalized Orthonormal Basis Functions

In this paper, an improved method to construct and estimate Volterra models using Generalized Orthonormal Basis Functions (GOBF) is presented. The proposed method extends results obtained in previous works, where an exact technique for optimizing the GOBF parameters (poles) for symmetric Volterra models of any order was presented. The proposed extensions take place in two different ways: (i) the new formulation is derived in such a way that each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases (rather than a single, common basis), each of which is parameterized by an individual set of poles responsible for representing the dominant dynamic of the kernel along a particular dimension; and (ii) the new formulation is based on a ladder-structured GOBF architecture that is characterized by having only real-valued parameters to be estimated, regardless of whether the GOBF poles encoded by these parameters are real- or complex-valued. The exact gradients of an error functional with respect to the parameters to be optimized are computed analytically and provide exact search directions for an optimization process that uses only input-output data measured from the dynamic system to be modeled. Computational experiments are presented to illustrate the benefits of the proposed approach when modeling nonlinear systems.

On the generalized Wannier functions

We consider single particle Schrödinger operators with a gap in the energy spectrum. We construct a complete, orthonormal basis function set for the invariant space corresponding to the spectrum below the spectral gap, which are exponentially localized around a set of closed surfaces of monotonically increasing sizes. Estimates on the exponential decay rate and a discussion of the geometry of these surfaces are included.

A New Selection Method of Anthropometric Parameters in Individualizing HRIR

A trend issue in modeling head-related impulse responses (HRIRs) is how to individualize HRIRs models that are convenient for a particular listener. The objective of this research is to show a robust selection method of eight anthropometric parameters out of all 27 parameters defined in CIPIC HRTF Database. The proposed selection method is systematically and scientifically acceptable, compared to ‘trial and error’ method in selecting the parameters. The selected anthropometric parameters of a given listener were applied in establishing multiple linear regression models in order to individualize his / her HRIRs. We modeled the entire minimum phase HRIRs in horizontal plane of 35 subjects using principal components analysis (PCA). The individual minimum phase HRIRs can be estimated adequately by a linear combination of ten orthonormal basis functions.

RF Power Amplifier Behavioral Modeling Based on Takenaka–Malmquist–Volterra Series

In this paper, a Takenaka---Malmquist---Volterra (TMV) model structure is employed to improve the approximations in the low-pass equivalent behavioral modeling of radio frequency (RF) power amplifiers (PAs). The Takenaka---Malmquist basis generalizes the orthonormal basis functions previously used in this context. In addition, it allows each nonlinearity order in the expanded Volterra model to be parameterized by multiple complex poles (dynamics). The state-space realizations for the TMV models are introduced. The pole sets for the TMV model and also for the previous Laguerre---Volterra (LV) and Kautz---Volterra (KV) models are obtained using a constrained nonlinear optimization approach. Based on experimental data measured on a GaN HEMT class AB RF PA excited by a WCDMA signal, it is observed that the TMV model reduces the normalized mean-square error and the adjacent channel error power ratio for the upper adjacent channel (upper ACEPR) by 1.6 dB when it is compared to the previous LV and KV models under the same computational complexity.