Representing and Counting the Subgroups of the Group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>

Hampejs, Mario; Holighaus, Nicki; Tóth, László; Wiesmeyr, Christoph

doi:10.1155/2014/491428

Cited by 23 publications

(21 citation statements)

References 16 publications

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“…Furthermore, it suggests a normal form that allows us to establish a one to one relation between lattices and generating matrices. Further implications of this bijection can be found in [21].…”

Section: Gabor Frames On Subgroups Of the Tf-planementioning

confidence: 88%

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Efficient Algorithms for Discrete Gabor Transforms on a Nonseparable Lattice

Wiesmeyr¹,

Holighaus²,

Søndergaard³

2013

IEEE Trans. Signal Process.

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The Discrete Gabor Transform (DGT) is the most commonly used transform for signal analysis and synthesis using a linear frequency scale. It turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup. Most of the literature focuses on separable subgroups, in this paper we will survey existing methods for a generalization to arbitrary groups, as well as present an improvement on existing methods. Comparisons are made with respect to the computational complexity, and the running time of optimized implementations in the C programming language. The new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download. By summarizing general background information on the state of the art, this article can also be seen as a research survey, sharing with the readers experience in the numerical work in Gabor analysis.

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Section: Gabor Frames On Subgroups Of the Tf-planementioning

confidence: 88%

“…There are possibly other feasible signal lengths than determined by (21). The full set of feasible lengths is determined by…”

Section: Further Optimizationmentioning

confidence: 99%

Efficient Algorithms for Discrete Gabor Transforms on a Nonseparable Lattice

Wiesmeyr¹,

Holighaus²,

Søndergaard³

2013

IEEE Trans. Signal Process.

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show abstract

“…In the paper [7] the identities (5), (6), (10), (13) and (14) were derived using another approach. The identity (13), as a special case of a formula valid for arbitrary finite abelian groups, was obtained by the author [16,17] using different arguments.…”

Section: Number Of Subgroupsmentioning

confidence: 99%

“…Finally, we remark that the functions (m, n) → s(m, n) and (m, n) → c(m, n) are multiplicative, viewed as arithmetic functions of two variables. See [7,18] for details. …”

Section: Number Of Subgroupsmentioning

confidence: 99%

“…Another representation of the subgroups of Z m × Z n , and the formulas of Theorems 4.1, 4.3 and 4.6, but not Theorem 4.5, were also derived in [7] using different group theoretical arguments. That representation and the formula of Theorem 4.1 was generalized to the case of the subgroups of the group Z m × Z n × Z r (m, n, r ∈ N) [8], using similar arguments, which are different from those of the present paper.…”

Section: Introductionmentioning

confidence: 99%

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