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Published October 1, 2015 | metadata_only
Journal Article

Group Frames With Few Distinct Inner Products and Low Coherence


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.

Additional Information

© 2015 IEEE. Manuscript received June 06, 2014; revised April 18, 2015; accepted June 02, 2015. Date of publication July 02, 2015; date of current version September 02, 2015. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Zhengdao Wang. This work was supported in part by the National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASA's Jet Propulsion Laboratory through the President and Director's Fund, by King Abdulaziz University, and by King Abdullah University of Science and Technology.

Additional details

August 20, 2023
August 20, 2023