On the construction of discrete orthonormal Gabor bases on finite dimensional spaces

Zhou, Weiqi

doi:10.1016/j.acha.2021.06.001

Cited by 4 publications

(1 citation statement)

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“…We conclude by observing that subgroups (or structured subsets) S ⊂ Z N ×Z N play also an important role in the construction of orthonormal basis of ℓ 2 (Z N ) of the type {π(j, k)g : (j, k) ∈ S}, for some g ∈ ℓ 2 (Z N ). This problem has a long tradition, for which we address to the recent contribution [33] and the references therein. Also, the study of uncertainty principles involving the cardinality of the support of discrete time-frequency distributions has interesting applications in signal recovery theory, for which we address to [6,19].…”

Section: Introduction and Discussion Of The Main Resultsmentioning

confidence: 99%

The uncertainty principle for the short-time Fourier transform on finite cyclic groups: cases of equality

Nicola¹

2022

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A well-known version of the uncertainty principle on the cyclic group Z N states that for any couple of functions f, g ∈ ℓ 2 (Z N ) \ {0}, the short-time Fourier transform V g f has support of cardinality at least N . This result can be regarded as a time-frequency version of the celebrated Donoho-Stark uncertainty principle on Z N . Unlike the Donoho-Stark principle, however, a complete identification of the extremals is still missing. In this note we provide an answer to this problem by proving that the support of V g f has cardinality N if and only if it is a coset of a subgroup of order N of Z N ×Z N . Also, we completely identify the corresponding extremal functions f, g. Besides translations and modulations, the symmetries of the problem are encoded by certain metaplectic operators associated with elements of SL(2, Z N/a ), where a is a divisor of N . Partial generalizations are given to finite Abelian groups.

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Section: Introduction and Discussion Of The Main Resultsmentioning

confidence: 99%