Abstract
Given a signal, whether it is a discrete vector or a continuous function, one desires to write it in terms of simpler components. Typically, these components or “building blocks” form what is called a basis. A basis is an optimal set, containing the minimal number of elements needed to uniquely represent any signal in a given space. A frame can be thought of as a redundant basis, having more elements than needed. In fact, in any finite dimensional vector space every finite spanning set is a frame. The redundancy of a frame leads to a non-unique representation, however, this makes signal representation resilient to noise and robust to transmission losses. Frames are now standard tools in signal processing and are of great interest to mathematicians and engineers alike. This chapter presents a brief introduction to frames in finite dimensional spaces, and in particular discusses a highly desirable class of frames called equiangular tight frames. Possible research ideas suitable for an undergraduate curriculum are also discussed.
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Acknowledgements
The authors would like to thank the anonymous reviewer for in-depth comments and helpful suggestions. The authors were partially supported by the NSF under Award No. CCF-1422252.
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Datta, S., Oldroyd, J. (2017). Finite Frame Theory. In: Wootton, A., Peterson, V., Lee, C. (eds) A Primer for Undergraduate Research. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-66065-3_7
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