Infinite convolution products and refinable distributions on Lie groups

Lawton, Wayne

doi:10.1090/s0002-9947-00-02409-0

Cited by 17 publications

(9 citation statements)

References 54 publications

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“…we see that for any z ∈ T, ((S * 0 f 1 )(z), (S * 1 f 1 )(z)) in C is in the orthogonal complement of (A(z), B(z)); indeed with (18) we get…”

Section: Factorization Casesmentioning

confidence: 83%

Filters and Matrix Factorization

Jørgensen

Song

2014

Preprint

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We give a number of explicit matrix-algorithms for analysis/synthesis in multi-phase filtering; i.e., the operation on discrete-time signals which allow a separation into frequency-band components, one for each of the ranges of bands, say N , starting with low-pass, and then corresponding filtering in the other band-ranges. If there are N bands, the individual filters will be combined into a single matrix action; so a representation of the combined operation on all N bands by an N × N matrix, where the corresponding matrix-entries are periodic functions; or their extensions to functions of a complex variable. Hence our setting entails a fixed N × N matrix over a prescribed algebra of functions of a complex variable. In the case of polynomial filters, the factorizations will always be finite. A novelty here is that we allow for a wide family of non-polynomial filter-banks.Working modulo N in the time domain, our approach also allows for a natural matrix-representation of both down-sampling and up-sampling. The implementation encompasses the combined operation on input, filtering, down-sampling, transmission, up-sampling, an action by dual filters, and synthesis, merges into a single matrix operation. Hence our matrixfactorizations break down the global filtering-process into elementary steps. To accomplish this, we offer a number of adapted matrix factorizationalgorithms, such that each factor in our product representation implements in a succession of steps the filtering across pairs of frequency-bands; and so it is of practical significance in implementing signal processing, including filtering of digitized images. Our matrix-factorizations are especially useful in the case of the processing a fixed, but large, number of bands.

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“…we see that for any z ∈ T, ((S * 0 f 1 )(z), (S * 1 f 1 )(z)) in C is in the orthogonal complement of (A(z), B(z)); indeed with (18) we get…”

Section: Factorization Casesmentioning

confidence: 83%

Filters and Matrix Factorization

Jørgensen

Song

2014

Preprint

View full text Add to dashboard Cite

show abstract

“…If A ∈ SU (L ∞ (T)) (i.e., unitary) then L in (18) is 0 and so A = A (1) so the factorization steps.…”

Section: Corollary 32mentioning

confidence: 99%

Filters and Matrix Factorization

Jørgensen

Song

2015

STSIP

View full text Add to dashboard Cite

We give a number of explicit matrix-algorithms for analysis/synthesis in multi-phase filtering; i.e., the operation on discrete-time signals which allow a separation into frequency-band components, one for each of the ranges of bands, say N , starting with low-pass, and then corresponding filtering in the other band-ranges. If there are N bands, the individual filters will be combined into a single matrix action; so a representation of the combined operation on all N bands by an N × N matrix, where the corresponding matrix-entries are periodic functions; or their extensions to functions of a complex variable. Hence our setting entails a fixed N × N matrix over a prescribed algebra of functions of a complex variable. In the case of polynomial filters, the factorizations will always be finite. A novelty here is that we allow for a wide family of non-polynomial filter-banks. Working modulo N in the time domain, our approach also allows for a natural matrix-representation of both down-sampling and up-sampling. The implementation encompasses the combined operation on input, filtering, down-sampling, transmission, up-sampling, an action by dual filters, and synthesis, merges into a single matrix operation. Hence our matrixfactorizations break down the global filtering-process into elementary steps. To accomplish this, we offer a number of adapted matrix factorizationalgorithms, such that each factor in our product representation implements in a succession of steps the filtering across pairs of frequency-bands; and so it is of practical significance in implementing signal processing, including filtering of digitized images. Our matrix-factorizations are especially useful in the case of the processing a fixed, but large, number of bands.

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“…a K and {V j } j∈Z satisfies (2) and (4) of the definition of multiresolution analysis on the Laguerre hypergroup. The characteristic function χ Q of the set Q is a scaling function of multiresolution analysis.…”

Section: Multiresolution Analysis On the Laguerre Hypergroupmentioning

confidence: 99%