On Closed Form Calculation of Line Spectral Frequencies (LSF)

Research output: Contribution to book/anthology/report/conference proceedingArticle in proceedingResearchpeer-review

Abstract

The mathematical theory of closed form functions for calculating LSFs on the basis of generating functions is presented. Exploiting recurrence relationships in the series expansion of Chebyshev polynomials of the first kind makes it possible to bootstrap iterative LSF-search from a set of characteristic polynomial zeros. The theoretical analysis is based on decomposition of sequences into symmetric and anti-symmetric polynomials defined as a series expansion of reduced Chebyshev polynomials of the first kind. Two variants of closed form functions are presented — each characterised by using a recurrence relationship in Chebyshev polynomials. The first exploits the well known three terms recurrence relationships of Chebyshev polynomials. The second hitherto unused recurrence properties of Chebyshev coefficients defining a set of coefficients and zeros used for bootstrapping calculation of LSFs. The theory is tested using bootstrapped calculation of zeros and by evaluating the complexity of the closed form function. The results of the lower complexity calculations show that real axis zeros are within a given iteration tolerance when compared to results of a standard root-finder.

Full Paper
Original languageDanish
Title of host publicationProceedings of the International Conference on Spoken Language Processing
Number of pages5
PublisherInternational Speech Communications Association
Publication date2014
Pages2390-2394
Publication statusPublished - 2014
EventINTERSPEECH: 15th Annual Conference of the International Speech Communication Association - Singapore, Singapore
Duration: 14 Sep 201418 Sep 2014

Conference

ConferenceINTERSPEECH
CountrySingapore
CitySingapore
Period14/09/201418/09/2014
SeriesProceedings of the International Conference on Spoken Language Processing
ISSN1990-9772

Cite this

Dalsgaard, P., & Andersen, O. (2014). On Closed Form Calculation of Line Spectral Frequencies (LSF). In Proceedings of the International Conference on Spoken Language Processing (pp. 2390-2394). International Speech Communications Association. Proceedings of the International Conference on Spoken Language Processing
Dalsgaard, Paul ; Andersen, Ove. / On Closed Form Calculation of Line Spectral Frequencies (LSF). Proceedings of the International Conference on Spoken Language Processing. International Speech Communications Association, 2014. pp. 2390-2394 (Proceedings of the International Conference on Spoken Language Processing).
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Dalsgaard, P & Andersen, O 2014, On Closed Form Calculation of Line Spectral Frequencies (LSF). in Proceedings of the International Conference on Spoken Language Processing. International Speech Communications Association, Proceedings of the International Conference on Spoken Language Processing, pp. 2390-2394, Singapore, Singapore, 14/09/2014.

On Closed Form Calculation of Line Spectral Frequencies (LSF). / Dalsgaard, Paul; Andersen, Ove.

Proceedings of the International Conference on Spoken Language Processing. International Speech Communications Association, 2014. p. 2390-2394 (Proceedings of the International Conference on Spoken Language Processing).

Research output: Contribution to book/anthology/report/conference proceedingArticle in proceedingResearchpeer-review

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AB - The mathematical theory of closed form functions for calculating LSFs on the basis of generating functions is presented. Exploiting recurrence relationships in the series expansion of Chebyshev polynomials of the first kind makes it possible to bootstrap iterative LSF-search from a set of characteristic polynomial zeros. The theoretical analysis is based on decomposition of sequences into symmetric and anti-symmetric polynomials defined as a series expansion of reduced Chebyshev polynomials of the first kind. Two variants of closed form functions are presented — each characterised by using a recurrence relationship in Chebyshev polynomials. The first exploits the well known three terms recurrence relationships of Chebyshev polynomials. The second hitherto unused recurrence properties of Chebyshev coefficients defining a set of coefficients and zeros used for bootstrapping calculation of LSFs. The theory is tested using bootstrapped calculation of zeros and by evaluating the complexity of the closed form function. The results of the lower complexity calculations show that real axis zeros are within a given iteration tolerance when compared to results of a standard root-finder.Full Paper

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Dalsgaard P, Andersen O. On Closed Form Calculation of Line Spectral Frequencies (LSF). In Proceedings of the International Conference on Spoken Language Processing. International Speech Communications Association. 2014. p. 2390-2394. (Proceedings of the International Conference on Spoken Language Processing).