A Control Theoretical Approach to the Polynomial Spectral-Factorization Problem

Moir, T. J.

doi:10.1007/s00034-011-9270-4

Cited by 9 publications

(11 citation statements)

References 24 publications

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“…Since the pioneering works of Kolmogorov and Wiener in the forties, a variety of methods have been proposed for the analysis and solution of this problem under different assumptions and in different settings, see e.g., [3], [27], [39], [45], [8], [49], [33], [34], [35], to cite but a few. We also refer to the relatively recent survey [42] that contains many other references and different points of view on this problem.…”

Section: Introductionmentioning

confidence: 99%

On the Factorization of Rational Discrete-Time Spectral Densities

Baggio

Ferrante

2016

IEEE Trans. Automat. Contr.

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In this paper, we consider an arbitrary matrix-valued, rational spectral density P(z) . We show with a constructive proof that P(z) admits a factorization of the form P(z)=W'(z^{-1}) W(z) , where W(z) is stochastically minimal. Moreover, W(z) and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic-type conditions. By suitably selecting the analyticity regions, this extremely general result particularizes into a corollary that may be viewed as the discrete-time counterpart of the matrix factorization method devised by Youla in his celebrated work

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Section: Introductionmentioning

confidence: 99%

On the Factorization of Rational Discrete-Time Spectral Densities

Baggio

Ferrante

2016

IEEE Trans. Automat. Contr.

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“…The difference does not show in the form of (17) and further simplification using innovation representations must be found. …”

Section: (Ii) Smoothingmentioning

confidence: 92%

“…Note the normalisation of the spectral factor a.0/ D 1 and that the zeroth element of this polynomial has been absorbed into the variance 2 . Now a Laurent series can be easily factorized using many methods, but here we use a feedback method used in earlier work [16,17], which is shown next. The technique uses negative feedback to force the output of a negative-feedback system to be the same as its input.…”

Section: U N C O R R E C T E D P R O O Fmentioning

confidence: 99%

Filtering, smoothing and prediction using a control‐loop spectral factorization method for coloured noise

Moir

2012

Adaptive Control & Signal

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SUMMARYA method for the linear least‐squares estimation of random signals contaminated with random noise that uses a new method of spectral factorization is shown. It is shown that the optimal filter can be written entirely in terms of the two spectral factors of signal plus noise and noise‐alone, and can be applied to the general case of coloured and white additive noise. The method of spectral factorization used is novel and uses control‐system methodology. Copyright © 2012 John Wiley & Sons, Ltd.

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“…Conversely, if the forgetting factor is too low, then good tracking and a less smooth correlogram vector will result. This was the approach used previously [28]. Hence, from measurements of the system output and computation of the correlogram function, we arrive at an estimate of (13), which is well known to be factorable into the form of (14).…”

Section: A New Frequency-domain Methodsmentioning

confidence: 99%

“…

A method is provided for scalar systems, which uses FFTs and provides spectral factorization directly from the periodogram. The particular approach that concerns this paper is the feedback method proposed by Moir [6] and used for system identification [7]. Although spectral factorization is a mature technique, the current methods available are generally too slow to cope with acoustic problems of the scale discussed here.

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confidence: 99%

Spectral factorization using FFTs for large‐scale problems

Moir

2014

Adaptive Control & Signal

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A method is provided for scalar systems, which uses FFTs and provides spectral factorization directly from the periodogram. The method is block recursive, providing better estimates as time progresses with more data available. Although spectral factorization is a mature technique, the current methods available are generally too slow to cope with acoustic problems of the scale discussed here. SPECTRAL FACTORIZATION USING FFTs 955 problem, some simpler than others. For example, the earliest discrete-time attempt was due to Kolmogorov [3] and later a computational method by Wilson [4]. Since then, there has been a good number of approaches mostly summarized here [5]. The particular approach that concerns this paper is the feedback method proposed by Moir [6] and used for system identification [7]. In fact, the spectral factorization problem appears in all classes of frequency-domain problems where a quadratic cost function is involved. So besides the optimal filtering problem [2], it also has application in optimal control [8-10]. There is another method available for spectral factorization using FFTs, however. For example the cepstrum or kepstrum approach [11][12][13] can be used, but these approaches, when used on real data, need a great deal of smoothing to achieve accurate results. Although such a method could in principle be used for our application, the two methods are totally unrelated. We also note here that the estimation of the polynomial a.´ 1 / is not unique for the more general case when the spectrum has non-minimum-phase zeros. For second-order statistics, it is well established that it is only possible to estimate the minimum-phase equivalent (i.e. the zeros are reflected back within the unit circle of the´-plane). Although this can be a disadvantage, we can also use this as an advantage to obtain stable polynomials as shown in a later section applied to acoustics. Whilst it is possible to apply a similar approach to higher-order statistics, it is known that such an approach can lead to an error surface that is not convex and has local minima.rf .v 0 ; v 1 ; : : :be zero. 956 T. J. MOIR Now, as " k D L V k , we find by differentiating again P j D0w i w i Cj ; i D 0; 1 : : : n. Note that this operation is equivalent in the´-domain to the operation a.´ 1 /a.´/ D ja.´/j 2 . What we are effectively doing is re-forming the Laurent series based on the estimated spectral factor and creating SPECTRAL FACTORIZATION USING FFTs 959 frequency-domain method, which requires a primary and reference input [14]. Although other blind methods exist, for example, [15][16][17][18][19][20], these methods require special input signals associated with communication systems, for example, code division multiple access, or alternatively require signals that have higher-order statistics [21]. The possible exception to this is the Kepstrum method [11,12,22], which has found application in adaptive filtering [23]. It should be made clear however that no method that has Gaussian noise as the driving signal can reconstruct accurate...

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A Control Theoretical Approach to the Polynomial Spectral-Factorization Problem

Cited by 9 publications

References 24 publications

On the Factorization of Rational Discrete-Time Spectral Densities

On the Factorization of Rational Discrete-Time Spectral Densities

Filtering, smoothing and prediction using a control‐loop spectral factorization method for coloured noise

Spectral factorization using FFTs for large‐scale problems

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