Group representation theory and radar ambiguity functions

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2022-2

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Kama, Eren Berk

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In this thesis, representations of Heisenberg group are applied to ambiguity functions and their properties are investigated in with an information theoretic perspective with applications in telecommunications and signal processing. Algebraic properties of ambiguity functions were investigated through application of representation theory. Novel approaches on phase space tiling are given and some existing methods for traditional ambiguity functions were extended to MIMO ambiguity functions. Irreducible representations are used to obtain an orthonormal basis of L2(R2). An identity on different functions having same ambiguity function is used in MIMO case. Uncertainty relations on ambiguity functions and certain time frequency distributions are studied. Relations between norms of MIMO ambiguity functions and norms of signals creating them are given. A local uncertainty relation on MIMO ambiguity functions and a bound on delay Doppler support is given. Lieb uncertainty is used in MIMO ambiguity functions to obtain a sharp uncertainty relation. These uncertainty relations are connected with applications in time frequency analysis, compressed sensing and integrated sensing and communication applications. Moreover, uncertainty relations on Wigner distributions and marginalizable time frequency distributions are given. Uncertainty relation of de Bruijn was used on time frequency distributions with signal processing examples. Effect of symplectic transformations on Wigner Distributions was investigated. Actions of generators of SL(2,R) are tied with common signal processing operations and their effect on uncertainty was investigated. This effect and its applications in time frequency analysis and localization tasks are discussed. Furthermore, applications of time frequency analysis in quantum information theory and quantum harmonic analysis are discussed.

Subject Keywords

harmonic analysis, ambiguity functions, time frequency analysis, representation theory, uncertainty relations

URI

https://hdl.handle.net/11511/96314

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Graduate School of Natural and Applied Sciences, Thesis

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Citation Formats

E. B. Kama, “Group representation theory and radar ambiguity functions,” M.S. - Master of Science, Middle East Technical University, 2022.