Multi-window Gabor systems
Translated title
Multi-vindue Gabor systemer
Author
Al-Jawahri, Zeineb Hadi Ibrahim Sadik
Term
4. term
Education
Publication year
2017
Submitted on
2017-01-06
Pages
66
Abstract
This thesis presents a mathematical study of multi-window Gabor systems within time–frequency analysis, without pursuing applications. It begins with a review of frame theory and operators on the Hilbert space L2(R), then introduces the Zak transform and the piecewise Zak transform together with their key properties, notably the unitarity of the Zak transform. Building on this, the thesis analyzes multi-window Gabor systems in the Zak domain: undersampling, critical sampling, and oversampling are addressed, and frame bounds as well as the dual frame are computed. In the special case where the product of the time and frequency parameters is rational, the associated frame operator is shown to admit a matrix-valued representation. The work further generalizes the system to include a countable family of generators and presents an alternative construction via quilted Gabor frames, establishing frame properties in a specific setting. Overall, the thesis assembles a coherent set of tools and results for characterizing and understanding multi-window Gabor systems.
Dette speciale beskæftiger sig med en matematisk analyse af multi-window Gabor-systemer inden for tids-frekvensanalyse, uden at inddrage konkrete anvendelser. Arbejdet indleder med en oversigt over frame-teori og operatorer i Hilbertrummet L2(R) og introducerer dernæst Zak-transformationen og den stykvise Zak-transformation samt deres centrale egenskaber, herunder at Zak-transformationen er unitær. Med dette fundament analyseres multi-window Gabor-systemer i Zak-domænet: undersampling, kritisk sampling og oversampling behandles, og der gives beregninger af framegrænser og den duale frame. I det særlige tilfælde hvor produktet af tids- og frekvensparametrene er et rationelt tal, vises det, at den tilhørende frame-operator kan repræsenteres ved en matrix-værdiert funktion. Specialet udvider derudover systemet til at omfatte en tællelig mængde generatorer og præsenterer en alternativ konstruktion via quiltede Gabor-frames, hvor frameegenskaber etableres i en specifik situation. Samlet giver specialet et struktureret sæt af værktøjer og resultater til at karakterisere og forstå multi-window Gabor-systemer.
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