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The Circulant Rational Covariance Extension Problem for a Skew Periodic Stochastic Process
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2014 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Det cirkulara rationella kovariansutvidgningsproblemet for skev-periodiskaprocesser (Swedish)
##### Abstract [en]

The Rational Covariance Extension Problem is a problemin applied mathematics where one tries to find a rational spectral density thatmatches a finite covariance sequence. Applications of this can be used in areaslike speech- and image-processing. This problem has been studied intensivelyover the last decades and recently a related problem, the Circulant RationalCovariance Extension Problem, was solved. This version of the problem dealswith periodic stochastic sequences, and was shown to be a natural way toapproximate the solution to the original problem. Here we look at the specialcase when the process in question is skew-periodic, and show that also in thiscase a unique solution to the problem exists. Moreover we develop numerical solversfor both the periodic and the skew-periodic problem, and use these algorithms toapproximate the spectrum from a speech signal.

##### Abstract [sv]

Det Rationella Kovariansutvidgningsproblemet är ett problem inom tillämpad matematik där man försöker hitta en rationell spektraltäthet som matchar en given sekvens av kovarianser. Tillämpningar av problemet finns inom områden som tal- och bildbehandling. Problemet har studerats intensivt under de senaste decennierna, och nyligen har ett relaterat problem lösts - nämligen det Cirkulära Rationella Kovariansutvidgningsproblemet. I detta problem arbetar man med periodiska stokastiska processer, och lösningen visade sig vara ett naturligt sätt att approximera lösningen till det första problemet. I denna uppsats tittar vi på specialfallet när processen är skev-periodisk, och visar att det även i detta fall finns en unik lösning. Dessutom utvecklas numeriska lösare för både det periodiska och skev-periodiska problemet, och dessa algoritmer används tillslut för att approximera spektrumet för en talsignal.

2014.
##### Series
TRITA-MAT-E, 2014:16
##### Keyword [en]
Rational covariance extension problem, Circulant rational covariance
##### Keyword [sv]
Rationella kovariansutvidgningsproblemet, Cirkulära rationella kovariansutvidgningsproblemet, Periodiska processer, Skev-periodiska processer, Talsignalbehandling
Mathematics
##### Identifiers
OAI: oai:DiVA.org:kth-142457DiVA: diva2:704293
##### External cooperation
SJTU, Shanghai Jiao Tong University
##### Subject / course
Optimization and Systems Theory
##### Educational program
Master of Science - Mathematics
##### Examiners
Available from: 2014-03-11 Created: 2014-03-05 Last updated: 2014-03-11Bibliographically approved

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##### By organisation
Optimization and Systems Theory
Mathematics

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Cite
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