A classification of the harmonic frames up to unitary equivalence

Chien, Tuan-Yow; Waldron, Shayne

doi:10.1016/j.acha.2010.09.001

Cited by 19 publications

(26 citation statements)

References 21 publications

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“…Calculations of [CW11] indicate that most cyclic harmonic frames are not exceptional, i.e., unitarily equivalent cyclic harmonic frames usually come from multiplicatively equivalent subsets (this is proved in Proposition 4.5). This is the basic principle underlying our results.…”

Section: The Number Of Cyclic Harmonic Framesmentioning

confidence: 93%

On the number of harmonic frames

Marshall

Waldron

2020

Applied and Computational Harmonic Analysis

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There is a finite number h n,d of tight frames of n distinct vectors for C d which are the orbit of a vector under a unitary action of the cyclic group Z n . These cyclic harmonic frames (or geometrically uniform tight frames) are used in signal analysis and quantum information theory, and provide many tight frames of particular interest.Here we investigate the conjecture that h n,d grows like n d−1 . By using a result of Laurent which describes the set of solutions of algebraic equations in roots of unity, we prove the asymptotic estimateBy using a group theoretic approach, we also give some exact formulas for h n,d , and estimate the number of cyclic harmonic frames up to projective unitary equivalence.

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Section: The Number Of Cyclic Harmonic Framesmentioning

confidence: 93%

On the number of harmonic frames

Marshall

Waldron

2020

Applied and Computational Harmonic Analysis

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“…Frames. We conclude this section by defining and stating basic facts about harmonic frames, which have also been studied in [24,14], for example.…”

Section: G-harmonicmentioning

confidence: 99%

Toward the classification of biangular harmonic frames

Casazza

Farzannia

Haas

et al. 2019

Applied and Computational Harmonic Analysis

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Equiangular tight frames (ETFs) and biangular tight frames (BTFs) -sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectivelyare useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames -vector sets defined in terms of the characters of finite abelian groups -because they are characterized by combinatorial objects called difference sets.This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties -all three of which are generalizations of difference sets that fall under the umbrella term "bidifference set" -then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.

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“…Georg Zimmermann [92] and Götz Pfander [unpublished] independently introduced and provided substantial properties of harmonic frames at Bommerholz in 1999, cf. [50] (2000), [21] (2003), [78] (2003), [84] (2005), [55] (2010), [86] (2010), [23] (2011). In [84] it was shown that harmonic frames and geometrically uniform tight frames are equivalent and can be characterized by their Gramian.…”

Section: Frame Multiplicationmentioning

confidence: 99%

Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

Andrews

Benedetto

Donatelli

2018

J Fourier Anal Appl

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Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time-frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory.The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms (STFTs), and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames.In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory. FRAME MULTIPLICATION THEORY AND A VECTOR-VALUED DFT AND AMBIGUITY FUNCTION 3 d. The short-time Fourier transform (STFT) of v with respect to a window function wfor a definitive mathematical treatment. Thus, we think of the window w as centered at t, and we havee. A(v, w) and V w (v) can clearly be defined for functions v, w on R d and for other function spaces besides L 2 (R d ). The quantity |V w (v)| is the spectrogram of v, that is so important in power spectrum analysis, see, e.g., [89], [17], [68], [70], [24], [65], [83].Our goals are the following.• Ultimately, we shall establish the theory of vector-valued ambiguity functions of vector-valued functions v on R d in terms of their discrete periodic counterparts on Z/N Z. • To this end, in this paper, we define vector-valued DFTs and discrete periodic vectorvalued ambiguity functions on Z/N Z in a natural way. The STFT is the guide and the theory of frames, especially the theory of DFT, harmonic, and group frames, is the framework (sic) to formulate these goals. The underlying technology that allows us to obtain these goals is frame multiplication theory.1.3. Outline. We begin with an extended exposition on the theory of frames (Section 2). The reason is that frames are essential for our results, and our results are sometimes not conceived in terms of frames. As such, it made sense to add sufficient background material.The vect...

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A classification of the harmonic frames up to unitary equivalence

Cited by 19 publications

References 21 publications

On the number of harmonic frames

On the number of harmonic frames

Toward the classification of biangular harmonic frames

Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

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