Group Frames With Few Distinct Inner Products and Low Coherence

Handle URI:
http://hdl.handle.net/10754/598436
Title:
Group Frames With Few Distinct Inner Products and Low Coherence
Authors:
Thill, Matthew; Hassibi, Babak
Abstract:
Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. High-performance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original “group frame” design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian's original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with n elements from [(n(n-1))/2] to n-1. All of our constructions further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When n is prime, for example, we use cyclic groups to construct m-dimensional frame vectors with at most [(n-1)/m] distinct inner products. We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance.
Publisher:
Institute of Electrical and Electronics Engineers (IEEE)
Journal:
IEEE Transactions on Signal Processing
Issue Date:
Oct-2015
DOI:
10.1109/TSP.2015.2450195
Type:
Article
ISSN:
1053-587X; 1941-0476
Sponsors:
This work was supported inpart bythe National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASAÕs Jet Propulsion Laboratory through the President and DirectorÕs Fund, by King Abdulaziz University, and by King Abdullah University of Science and Technology
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorThill, Matthewen
dc.contributor.authorHassibi, Babaken
dc.date.accessioned2016-02-25T13:20:41Zen
dc.date.available2016-02-25T13:20:41Zen
dc.date.issued2015-10en
dc.identifier.issn1053-587Xen
dc.identifier.issn1941-0476en
dc.identifier.doi10.1109/TSP.2015.2450195en
dc.identifier.urihttp://hdl.handle.net/10754/598436en
dc.description.abstractFrame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. High-performance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original “group frame” design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian's original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with n elements from [(n(n-1))/2] to n-1. All of our constructions further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When n is prime, for example, we use cyclic groups to construct m-dimensional frame vectors with at most [(n-1)/m] distinct inner products. We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance.en
dc.description.sponsorshipThis work was supported inpart bythe National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASAÕs Jet Propulsion Laboratory through the President and DirectorÕs Fund, by King Abdulaziz University, and by King Abdullah University of Science and Technologyen
dc.publisherInstitute of Electrical and Electronics Engineers (IEEE)en
dc.titleGroup Frames With Few Distinct Inner Products and Low Coherenceen
dc.typeArticleen
dc.identifier.journalIEEE Transactions on Signal Processingen
dc.contributor.institutionDepartment of Electrical Engineering, Caltech, Pasadena, CA 91125 USAen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.