Spark deficient Gabor frames

Abstract

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 ...

Optimal arrangements of classical and quantum states with limited purity

“…In what follows, we will calculate the spark of an infinite class of frames and for an infinite subclass of these frames, we determine all of the subsets of vectors which are linearly dependent. Most work about linear dependencies in Gabor frames focuses on full spark conditions in general Gabor frames (not necessarily SICs) . However, (inspired in part by ) deal with finding special subsets of vectors in certain SICs which form equiangular tight frames for their span.…”

Dimension towers of SICs. I. Aligned SICs and embedded tight frames

“…The embedding of lower dimensional ETFs in the SIC means that non-trivial linear dependencies are present among the vectors of the latter. The general question under what conditions sets of vectors in Weyl-Heisenberg orbits can be linearly dependent has been studied [33,34], and it is known that linear dependencies do occur, in such orbits, whenever the order of their symmetry group fails to be coprime with the dimension. Some of the linear dependencies that we report here are not covered by these results.…”

Spark Deficient Gabor Frame Provides A Novel Analysis Operator For Compressed Sensing

“…Now, as proven in [21], almost all window vectors generate full spark Gabor frames, so the SDGFs are generated by exceptional window vectors. Indeed, it was proven in [20] and informally stated in [22], for the Zauner matrix Z ∈ SL(2, Z L ) given by…”