GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION

By Thomas Strohmer, Robert W. Heath. 2003

1,030 citations recorded by the source corpus.

Abstract For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation |〈fk,fl〉| among all frames {f k } k∈ I ∈ F . We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with unit norm tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on unit norm tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to unit norm tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.

Subfields: Applied Mathematics, Electrical and Electronic Engineering, Signal Processing

View Paper: https://www.semanticscholar.org/paper/ab3591ebf1e52134cb2106f1a6f0bfd3b17e7c77

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In Signal Processing

Citations in the corpus

Sparse representations in unions of bases (2003, methodology): “We refer to [6] and [7] for a proof of Theorem 3.”
Block-Sparse Signals: Uncertainty Relations and Efficient Recovery (2009, background): “We first note that the coherence of a dictionary consisting of elements in a vector space of dimension can be lower-bounded as [46]”
Steiner equiangular tight frames (2010, methodology): “Conversely, if one can design an N ×N self-adjoint, positive semidefinite Gram matrix G of rank M whose diagonal entries are one and whose off-diagonal entries are all N−M M(N−1) in squared-modulus, then one can then factor G as F ∗F , where F is an M × N ETF [27].”
Achieving the Welch bound with difference sets (2005, background): “Other closely related problems include the design of equi-angular line sets, the design of antipodal spherical codes, spherical t -designs, and characterization of strongly regular graphs [26].”
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images (2009, background): “In [150] it has been shown that for full-rank matrices of size n × m the mutual coherence is bounded from below by , equality being obtained for a family of matrices named Grassmanian frames .”
On the construction of equiangular frames from graphs (2009, background): “When such frames do not exist (cf [STDH07]) closely related classes of frames have been suggested as substitutes, most notably Grassmannian frames and mutually unbiased bases (see [ SH03 ], [BK06], [KR04]).”
Gabor Frames in Finite Dimensions (2013, methodology): “For the Alltop window function, we have [1, 89]”
Prevalence of neural collapse during the terminal phase of deep learning training (2020, result): “The form in [6] is similar to the one first developed by R.”
Density, overcompleteness, and localization of frames (2006, background): “Frames were rst introduced by Dun and Schaeer [DS52] in the context of nonharmonic Fourier series, and today frames play important roles in many applications in mathematics, science, and engineering, including time-frequency analysis [Gr o01], internet coding [GKK01], speech and music processing [W”
An overview of limited feedback in wireless communication systems (2008, background): “One approach is construction based on difference sets [244], [271].”
The road to equal-norm Parseval frames (2010, methodology): “This type of equal-norm Parseval frames is particularly important because of their applications to signal processing [33, 31, 6, 34, 27] and to quantum information theory [35, 30, 17, 5].”
Upper bounds for $s$-distance sets and equiangular lines (2016, background): “For instance, there is an extensive literature in frame theory devoted to equiangular tight frames [56, 38, 57].”
Tight informationally complete quantum measurements (2006, methodology): “Following the terminology of frame theory, a finite tight rank-one IC-POVM will be called uniform when τ(x) ≡ d/|X |, or equiangular [70] if we additionally have ( π(x) ∣π(y) ) = |〈x|y〉|2 = c for all x 6= y ∈ X and some constant c.”