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45
A Unifying Construction of Orthonormal Bases for System Identification
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 1994
"... In this paper we develop a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of all known orthonormal bases since the common FIR and recently popular Laguerre and Kautz model structures are restrictive spe ..."
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Cited by 79 (20 self)
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In this paper we develop a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of all known orthonormal bases since the common FIR and recently popular Laguerre and Kautz model structures are restrictive special cases of our construction as is another construction method based on balanced realisations of all pass functions. However, in contrast to these special cases, the basis vectors in our unifying construction can have nearly arbitrary magnitude frequency response according to the prior information the user wishes to inject into the problem. We also provide results characterising the completeness properties of our bases.
The Fundamental Role of General Orthonormal Bases in System Identification
 IEEE Transactions on Automatic Control
, 1997
"... The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance express ..."
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Cited by 21 (11 self)
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The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros. This is achieved by employing certain new results on generalised Fourier series and the asymptotic properties of Toeplitzlike matrices in such a way that the new variance expressions presented here encompass preexisting ones as special cases. Via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification. Namely, that orthonormal bases are much more than an implementational option offering improved numerical properties. In fact, they are an intrinsic part of estimation since, as shown here, or...
Discrete Frequency Warped Wavelets: Theory and Applications
 IEEE Trans. Signal Processing
, 1998
"... In this paper, we extend the definition of dyadic wavelets to include frequency warped wavelets. The new wavelets are generated and the transform computed in discretetime by alternating the Laguerre transform with perfect reconstruction filterbanks. This scheme provides the unique implementation of ..."
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Cited by 17 (8 self)
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In this paper, we extend the definition of dyadic wavelets to include frequency warped wavelets. The new wavelets are generated and the transform computed in discretetime by alternating the Laguerre transform with perfect reconstruction filterbanks. This scheme provides the unique implementation of orthogonal or biorthogonal warped wavelets by means of rational transfer functions. We show that the discretetime warped wavelets lead to welldefined continuoustime wavelet bases, satisfying a warped form of the twoscale equation. The shape of the wavelets is not invariant by translation. Rather, the "wavelet translates" are obtained from one another by allpass filtering. We show that the phase of the delay element is asymptotically a fractal. A feature of the warped wavelet transform is that the cutoff frequencies of the wavelets may be arbitrarily assigned while preserving a dyadic structure. The new transform provides an arbitrary tiling of the timefrequency plane, which can be designed by selecting as little as a single parameter. This feature is particularly desirable in cochlear and perceptual models of speech and music, where accurate bandwidth selection is an issue. As our examples show, by defining pitchsynchronous wavelets based on warped wavelets, the analysis of transients and denoising of inharmonic pseudoperiodic signals is greatly enhanced.
Generalised Fourier and Toeplitz Results for Rational Orthonormal Bases
, 1997
"... This paper provides a generalisation of certain classical Fourier convergence and asymptotic Toeplitz matrix properties to the case where the underlying orthonormal basis is not the conventional trigonometric one, but a rational generalisation which encompasses the trigonometric one as a special cas ..."
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Cited by 11 (8 self)
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This paper provides a generalisation of certain classical Fourier convergence and asymptotic Toeplitz matrix properties to the case where the underlying orthonormal basis is not the conventional trigonometric one, but a rational generalisation which encompasses the trigonometric one as a special case. These generalised Fourier and Toeplitz results have particular application in dynamic system estimation theory.
Frequency Domain Estimation Using Orthonormal Bases
 in Proceedings of the 13th IFAC World Congress
, 1996
"... . This paper examines the use of general orthonormal bases for system identification from frequency domain data. This idea has been studied in great depth for the particular case of the orthonormal trigonometric basis. Here we show that the accuracy of the estimate can be significantly improved by r ..."
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Cited by 8 (2 self)
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. This paper examines the use of general orthonormal bases for system identification from frequency domain data. This idea has been studied in great depth for the particular case of the orthonormal trigonometric basis. Here we show that the accuracy of the estimate can be significantly improved by rejecting the trigonometric basis in favour of a more general orthogonal basis that is able to be adapted according to prior information that is available about the system being identified. The usual trigonometric basis emerges as a special case of the general bases employed here. Keywords. Frequency Response Estimation, System Identification, Parameter Estimation 1. INTRODUCTION The bulk of system identification theory addresses the problem of estimating system models on the basis of observed time domain data(Ljung, 1987; T.Soderstrom and P.Stoica, 1989). However, in many cases the available data involves measurements of the systems frequency response(Pintelon et al., 1994; Ljung, 1993). Ind...
Orthonormal Basis Functions for ContinuousTime Systems and L_p Convergence
, 1999
"... In this paper, model sets for continuoustime linear time invariant systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalise the well known Laguerre and twoparameter Kautz cases. It is shown that the obtained model sets are norm dense in the Hardy space H ..."
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Cited by 8 (4 self)
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In this paper, model sets for continuoustime linear time invariant systems that are spanned by fixed pole orthonormal bases are investigated. These bases generalise the well known Laguerre and twoparameter Kautz cases. It is shown that the obtained model sets are norm dense in the Hardy space H 1 (#) under the same condition as previously derived by the authors for the norm denseness in the (# is the open right half plane) Hardy spaces H p (#), 1 < p < #.
Generalized Fourier and Toeplitz results for rational orthonormal bases
 SIAM Journal on Control and Optimization
, 1999
"... ..."
Selection of Best Orthonormal Rational Basis
, 1997
"... . This contribution deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parameterized by a prespecified set of poles. Given this structure and experimental data a model can be estimated using linear regress ..."
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Cited by 5 (1 self)
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. This contribution deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parameterized by a prespecified set of poles. Given this structure and experimental data a model can be estimated using linear regression techniques. Since the variance of the estimated model increases with the number of estimated parameters, the objective is to find structures that are as compact/parsimonious as possible. A natural approach would be to estimate the poles, but this leads to nonlinear optimization with possible local minima. In this paper, a best basis algorithm and a coefficient decomposition scheme are derived for the generalized orthonormal rational bases. Combined with linear regression and thresholding this leads to compact transfer function representations. Key words. System identification, model structures, orthonormal basis functions, best basis AMS subject classifications. 39A30, 41A20, 42C20, 05...
Physically Inspired Models for the Synthesis of Stiff Strings with Dispersive Waveguides
 EURASIP JOURNAL ON APPLIED SIGNAL PROCESSING 2004:7, 964–977 C ○ 2004 HINDAWI PUBLISHING CORPORATION
, 2004
"... We review the derivation and design of digital waveguides from physical models of stiff systems, useful for the synthesis of sounds from strings, rods, and similar objects. A transform method approach is proposed to solve the classic fourthorder equations of stiff systems in order to reduce it to t ..."
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Cited by 4 (1 self)
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We review the derivation and design of digital waveguides from physical models of stiff systems, useful for the synthesis of sounds from strings, rods, and similar objects. A transform method approach is proposed to solve the classic fourthorder equations of stiff systems in order to reduce it to two secondorder equations. By introducing scattering boundary matrices, the eigenfrequencies are determined and their n² dependency is discussed for the clamped, hinged, and intermediate cases. On the basis of the frequencydomain physical model, the numerical discretization is carried out, showing how the insertion of an allpass delay line generalizes the KarplusStrong algorithm for the synthesis of ideally flexible vibrating strings. Knowing the physical parameters, the synthesis can proceed using the generalized structure. Another point of view is offered by Laguerre expansions and frequency warping, which are introduced in order to show that a stiff system can be treated as a nonstiff one, provided that the solutions are warped. A method to compute the allpass chain coefficients and the optimum warping curves from sound samples is discussed. Once the optimum warping characteristic is found, the length of the dispersive delay line to be employed in the simulation is simply determined from the requirement of matching the desired fundamental frequency. The regularization of the dispersion curves by means of optimum unwarping is experimentally evaluated.
Alternatives for Warped Linear Predictors
 In Proc. 12th ProRISC Workshop
, 2001
"... Linear Prediction (LP) is a wellknown compression tool for coding speech and audio signals. By employing a frequencywarping technique, an extension has been proposed called Warped Linear Prediction (WLP). WLP enables controlling the degree at which details in specific regions in the spectral envel ..."
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Cited by 4 (3 self)
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Linear Prediction (LP) is a wellknown compression tool for coding speech and audio signals. By employing a frequencywarping technique, an extension has been proposed called Warped Linear Prediction (WLP). WLP enables controlling the degree at which details in specific regions in the spectral envelope can be conserved. Therefore, the coder can be tuned to a particular application. However, the design of the synthesis filters in WLP is not as straightforward as in LP. Therefore, another linear prediction scheme, called pure linear prediction, is proposed and has been tested. In particular, it has been found that the proposed prediction scheme with a properly parameterised Laguerre or Kautz system shows a behaviour similar to that of WLP while avoiding its problems.