Wavelets and stochastic theory: Past and future

Milovanović, Miloš; Tomić, Bojan M.; Saulig, Nicoletta

doi:10.1016/j.chaos.2023.113724

Cited by 6 publications

(7 citation statements)

References 38 publications

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“…The time operator T χ of the system evolving by U χ has been explicitly constructed [17]. Its projection onto the signal space L 2 µ concerns the hierarchy of the Haar base (9). The time operator of any wavelet ( 14) is obtained through conjugation T = CT χ C † by C :…”

Section: Space Of Ensemblesmentioning

confidence: 99%

“…Projectors constitute the Boolean algebra, which is isomorphic to an algebra of sets due to the Stone representation theorem. It is the measurable space corresponding to devices, which an observable has been defined on [9,10]. A measurement state on the other hand corresponds to a density ρ = FF † which is defined upon the same domain.…”

Section: Measurements In the Hierarchical Basementioning

confidence: 99%

“…m ], respectively, values ψ l,m ψ o j,k are negligible if supports do not intersect. This implies an approximate decorrelation of the ensemble, which should mean that correlation between detail coefficients is predominantly concerned by inheritance along branches of the binary tree [9,10].…”

Section: Starting From the Decompositionmentioning

confidence: 99%

“…Correlation between detail coefficients D = D j,k is transmitted only through the Markovian tree of hidden variables S = S j,k , with one attributed to each node, and out of such an interrelation, the ensemble is considered decorrelated. The conditional distribution D|S is supposed to be normal, which implies that D j,k |S j,k are independent variables [9].…”

Section: Starting From the Decompositionmentioning

confidence: 99%

“…A paradigmatic measurement corresponds to commensuration of magnitudes by the Euclidean algorithm, which is an intensional procedure producing real numbers of the unit interval. Regarding that, one comes to a general definition of the process concerning a time series of binary digits [9].…”

Section: Introductionmentioning

confidence: 99%

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The Measurement Problem in Statistical Signal Processing

Milovanović

2023

Mathematics

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Discussing quantum theory foundations, von Neumann noted that the measurement process should not be regarded in terms of a temporal evolution. A reason for their claim is the insurmountability of the gap between reversible and irreversible processes. The time operator formalism that goes beyond such a gap is an adequate framework to elaborate the measurement problem. It considers signals to be stochastic processes, regardless of whether they correspond to variables or distribution densities. Signal processing that utilizes statistical properties to perform tasks is statistical signal processing. The hierarchy of the measurement process is indicated by crossing between states and devices, which implies an evolution in the temporal domain. The concept has been generalized to an open system by the use of duality in frame theory.

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Section: Space Of Ensemblesmentioning

confidence: 99%

Section: Measurements In the Hierarchical Basementioning

confidence: 99%

Section: Starting From the Decompositionmentioning

confidence: 99%

Section: Starting From the Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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