Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Spectral Estimation by Geometric, Topological and Optimization Methods
KTH, Superseded Departments, Mathematics.
2001 (English)Doctoral thesis, comprehensive summary (Other scientific)
Place, publisher, year, edition, pages
Stockholm: KTH , 2001. , xii, 32 p.
Series
Trita-MAT, ISSN 1401-2286 ; 01-OS-03
Keyword [en]
Spectral Estimation, ARMA models, Covariance analysis, Cepstral analysis, Markov parameters, Global analysis, Convex Optimization, Continuation methods, Entropy maximization
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-3118OAI: oai:DiVA.org:kth-3118DiVA: diva2:8880
Public defence
2001-04-06, 00:00 (English)
Note
QC 20100601Available from: 2001-03-29 Created: 2001-03-29 Last updated: 2010-06-01Bibliographically approved
List of papers
1. A homotopy approach to rational covariance extension with degree constraint
Open this publication in new window or tab >>A homotopy approach to rational covariance extension with degree constraint
2001 (English)In: International journal of mathematics and computer science, ISSN 1641-876X, Vol. 11, no 5, 1173-1201 p.Article in journal (Refereed) Published
Abstract [en]

The solutions to the Rational Covariance Extension Problem (RCEP) are parameterized by the spectral zeros. The rational filter with a specied numerator solving the RCEP can be determined from a known convex optimization problem. However, this optimization problem may become ill-conditioned for some parameter values. A modication of the optimization problem to avoid the illconditioning is proposed and the modified problem is solved effciently by a continuation method.

Keyword
stochastic realization theory, rational covariance extension problem, ARMA model design, continuation method, optimization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-13171 (URN)
Note
QC 20100601Available from: 2010-06-01 Created: 2010-06-01 Last updated: 2010-06-01Bibliographically approved
2. Cepstral coefficients, covariance lags, and pole-zero models for finite data strings
Open this publication in new window or tab >>Cepstral coefficients, covariance lags, and pole-zero models for finite data strings
2001 (English)In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 49, no 4, 677-693 p.Article in journal (Refereed) Published
Abstract [en]

One of the most widely used methods of spectral estimation in signal and speech processing is linear predictive coding (LPC). LPC has some attractive features, which account for its popularity, including the properties that the resulting modeling filter i) matches a finite window of n + 1 covariance lags, ii) is rational of degree at most n, and iii) has stable zeros and poles. The only limiting factor of this methodology is that the modeling filter is "all-pole," i.e., an autoregressive (AR) model. In this paper, we present a systematic description of all autoregressive moving-average (ARMA) models of processes that have properties i)-iii) in the context of cepstral analysis and homomorphic filtering. Indeed, we show that each such ARMA model determines and is completely determined by its finite windows of cepstral coefficients and covariance lags. This characterization has an intuitively appealing interpretation of a characterization by using measures of the transient and the steady-state behaviors of the signal, respectively. More precisely, we show that these nth-order windows form local coordinates for all ARMA models of degree n and that the pole-zero model can be determined from the windows as the unique minimum of a convex objective function. We refine this optimization method by first noting that the maximum entropy design of an LPC filter is obtained by maximizing the zeroth cepstral coefficient, subject to the constraint i). More generally, we modify this scheme to a more well-posed optimization problem where the covariance data enters as a constraint and the linear weights of the cepstral coefficients are "positive"-in a sense that a certain pseudo-polynomial is positive-rather succinctly generalizing the maximum entropy method. This new problem is a homomorphic filter generalization of the maximum entropy method, providing a procedure for the design of any stable, minimum-phase modeling filter of degree less or equal to n that interpolates the given covariance window We conclude the paper by presenting an algorithm for realizing these biters in a lattice-ladder form, given the covariance window and the moving average part of the model. While we also show how to determine the moving average part using cepstral smoothing, one can make use of any good a priori estimate for the system zeros to initialize the algorithm. Indeed, we conclude the paper with an example of this method, incorporating an example from the literature on ARMA modeling.

Keyword
autoregressive moving average processes, cepstral analysis, covariance analysis, identification, maximum entropy methods, optimization methods, spectral analysis, speech analysis, HOMOMORPHIC PREDICTION, REALIZATION, ALGORITHMS, SPECTRA
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13172 (URN)10.1109/78.912912 (DOI)000167587600001 ()
Note
QC 20100601Available from: 2010-06-01 Created: 2010-06-01 Last updated: 2017-12-12Bibliographically approved
3. Identifiability and well-posedness of shaping-filter parameterizations: A global analysis approach
Open this publication in new window or tab >>Identifiability and well-posedness of shaping-filter parameterizations: A global analysis approach
2002 (English)In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 41, no 1, 23-59 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we study the well-posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients, covariance lags, or Markov parameters. For example, we determine whether there exists a shaping filter with a prescribed window of Markov parameters and a prescribed window of covariance lags. We show that several such problems are well-posed in the sense of Hadamard; that is, one can prove existence, uniqueness (identifiability), and continuous dependence of the model on the measurements. Our starting point is the global analysis of linear systems, where one studies an entire class of systems or models as a whole, and where one views measurements, such as covariance lags and cepstral coefficients or Markov parameters, from data as functions on the entire class. This enables one to pose such problems in a way that tools from calculus, optimization, geometry, and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm-independent fashion. In this language, we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona de coordinate system on the space of shaping filters, thereby establishing existence, uniqueness, and smooth dependence of the model parameters on the measurements from data.

Keyword
identifiability, parameterization, well-posedness, foliations, Caratheodory extension, spectral estimation, cepstrum, NEVANLINNA-PICK INTERPOLATION, CEPSTRAL COEFFICIENTS, COVARIANCE EXTENSION, REALIZATION PROBLEM, DEGREE CONSTRAINT
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13173 (URN)10.1137/S0363012900383077 (DOI)000176312200002 ()
Note
QC 20100601Available from: 2010-06-01 Created: 2010-06-01 Last updated: 2017-12-12Bibliographically approved
4. A convex optimization approach to arma(n,m) model design from covariance and cepstral data
Open this publication in new window or tab >>A convex optimization approach to arma(n,m) model design from covariance and cepstral data
2004 (English)In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 43, no 3, 1011-1036 p.Article in journal (Refereed) Published
Abstract [en]

Methods for determining ARMA(n, m) filters from covariance and cepstral estimates are proposed. In [C. I. Byrnes, P. Enqvist, and A. Lindquist, SIAM J. Control Optim., 41 ( 2002), pp. 23-59], we have shown that an ARMA( n, n) model determines and is uniquely determined by a window r(0), r(1),..., r(n) of covariance lags and c(1), c(2),..., c(n) of cepstral lags. This unique model can be determined from a convex optimization problem which was shown to be the dual of a maximum entropy problem. In this paper, generalizations of this problem are analyzed. Problems with covariance lags r(0), r(1),..., r(n) and cepstral lags c(1), c(2),..., c(m) of different lengths are considered, and by considering different combinations of covariances, cepstral parameters, poles, and zeros, it is shown that only zeros and covariances give a parameterization that is consistent with generic data. However, the main contribution of this paper is a regularization of the optimization problems that is proposed in order to handle generic data. For the covariance and cepstral problem, if the data does not correspond to a system of desired order, solutions with zeros on the boundary occur and the cepstral coefficients are not interpolated exactly. In order to achieve strictly minimum phase filters for estimated covariance and cepstral data, a barrier-like term is introduced to the optimization problem. This term is chosen so that convexity is maintained and so that the unique solution will still interpolate the covariances but only approximate the cepstral lags. Furthermore, the solution will depend analytically on the covariance and cepstral data, which provides robustness, and the barrier term increases the entropy of the solution.

Keyword
cepstrum, covariance, ARMA, entropy, convex optimization
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-13176 (URN)10.1137/S0363012901399751 (DOI)000225642700013 ()2-s2.0-19944383824 (Scopus ID)
Note
QC 20100601Available from: 2010-06-01 Created: 2010-06-01 Last updated: 2017-12-12Bibliographically approved

Open Access in DiVA

fulltext(554 kB)542 downloads
File information
File name FULLTEXT01.pdfFile size 554 kBChecksum MD5
639d1d01b8ec53faa54dd1d2e3e977286871d5f81d88e27b39f89f202e4e1879071a58d7
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Enqvist, Per
By organisation
Mathematics
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 542 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 753 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf