Steiner equiangular tight frames

Fickus, Matthew; Mixon, Dustin G.; Tremain, Janet C.

doi:10.1016/j.laa.2011.06.027

Cited by 187 publications

(244 citation statements)

References 33 publications

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“…Also, a more general question can be posed, namely the design of sparse frames with additional design specifications. One example for such specifications is low coherence, which was used as a design criteria for Steiner equiangular tight frames in the work of Fickus et al (2012).…”

Section: Discussion and Future Directionsmentioning

confidence: 99%

Sparse matrices in frame theory

2013

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Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices. The objective for this approach is to ensure not only low-complexity computations, but also high compressibility. We will discuss both existence results and explicit constructions.

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Section: Discussion and Future Directionsmentioning

confidence: 99%

Sparse matrices in frame theory

2013

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“…This is another example that showcases the difference between a nice algebraic property of a Gabor frame (full spark) and a nice geometric one (equiangularity); it is not necessary that both of them can appear, even when the second one appears at all. For unit norm tight frames in general, this is further explained in [18]; see also [12,17] where an infinite family of spark deficient equiangular tight frames is constructed, of arbitrarily high dimension.…”

Section: Spark Deficient Gabor Frames Over Cyclic Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Yet another terminology that appears for this phenomenon is maximal equiangular tight frame (or maximal ETF for short) [11], which is a special case of the packing problem in the setting of projective spaces. The interest of the algebraic construction of families of ETFs has also increased due to applications to signal processing [12,16,17].A conjecture by Heil, Ramanathan, and Topiwala from 1996 [14] states that any finite set of a Gabor frame of a nonzero f ∈ L 2 (R) is linearly independent, and it is still open. Similar questions can be raised when the function f is defined on a finite abelian group G. In this case, the Gabor frame consists of |G| 2 elements in a |G|-dimensional space, so it is not possible that 2010 Mathematics Subject Classification.…”

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confidence: 99%

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Spark deficient Gabor frames

Malikiosis

2018

Pacific J. Math.

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Abstract. The theory of Gabor frames of functions defined on finite abelian groups was initially developed in order to better understand the properties of Gabor frames of functions defined over the reals. However, during the last twenty years the topic has acquired an interest of its own. One of the fundamental questions asked in this finite setting is the existence of full spark Gabor frames. The author proved the existence [21], as well as constructed such frames, when the underlying group is finite cyclic. In this paper, we resolve the non-cyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian non-cyclic groups. We also prove that all eigenvectors of certain unitary matrices in the Clifford group in odd dimensions generate spark deficient Gabor frames. Finally, similarities between the uncertainty principles concerning the finite dimensional Fourier transform and the short-time Fourier transform are discussed. IntroductionThe Gabor frame of a function f ∈ L 2 (R) is the set of all time-frequency translates of f , that is, the set of all functions of the form e 2πixy f (x−t), for y, t ∈ R, and it is a fundamental concept in time-frequency analysis and frame theory [25]. The function f usually represents a signal, t the time delay, and the pointwise multiplication by e 2πixy is the frequency "shift". Through sampling and periodization [8] one passes to the finite version of a Gabor frame, namely the shift-frequency translates of a complex function defined on a finite cyclic group. Even though finite dimensional Gabor frames were studied in order to analyze the properties of continuous signals, they later developed an interest of their own.Up to multiplication by roots of unity, a finite dimensional Gabor frame is the same as a Weyl-Heisenberg orbit, and this terminology is much more prevalent in mathematical physics and quantum information theory. A conjecture by Zauner [33] states that for every dimension N there are vectors (called "fiducials") whose WH orbit is equiangular. This means that the expression | u, v | is constant for every pair of distinct vectors u, v within this orbit. This is also known as the SIC-POVM problem which has attracted a lot of attention lately due to the vast connections to scientific areas such as quantum cryptography [26], quantum tomography [27], and algebraic number theory, especially Hilbert's 12th problem for real quadratic fields [2,3,4,5]. Such a WH orbit would then produce the maximal possible number of vectors in C N that are pairwise equiangular, namely N 2 [29]. Yet another terminology that appears for this phenomenon is maximal equiangular tight frame (or maximal ETF for short) [11], which is a special case of the packing problem in the setting of projective spaces. The interest of the algebraic construction of families of ETFs has also increased due to applications to signal processing [12,16,17].A conjecture by Heil, Ramanathan, and Topiwala from 1996 [14] states that any finite set of a Gabor frame of a ...

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“…Among all frames, equiangular unit norm tight frames (ETFs) have a special structure that makes them particularly important in many fields like signal processing [2,3], quantum information theory [4,5], and communications [6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Approximate equiangular tight frames for compressed sensing and CDMA applications

Tsiligianni

Kondi

Katsaggelos

2017

EURASIP J. Adv. Signal Process.

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Performance guarantees for recovery algorithms employed in sparse representations, and compressed sensing highlights the importance of incoherence. Optimal bounds of incoherence are attained by equiangular unit norm tight frames (ETFs). Although ETFs are important in many applications, they do not exist for all dimensions, while their construction has been proven extremely difficult. In this paper, we construct frames that are close to ETFs. According to results from frame and graph theory, the existence of an ETF depends on the existence of its signature matrix, that is, a symmetric matrix with certain structure and spectrum consisting of two distinct eigenvalues. We view the construction of a signature matrix as an inverse eigenvalue problem and propose a method that produces frames of any dimensions that are close to ETFs. Due to the achieved equiangularity property, the so obtained frames can be employed as spreading sequences in synchronous code-division multiple access (s-CDMA) systems, besides compressed sensing.

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Steiner equiangular tight frames

Cited by 187 publications

References 33 publications

Sparse matrices in frame theory

Sparse matrices in frame theory

Spark deficient Gabor frames

Approximate equiangular tight frames for compressed sensing and CDMA applications

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