Finite Frames and Filter Banks
Filter banks are fundamental tools of signal and image processing. A filter is a linear operator which computes the inner products of an input signal with all translates of a fixed function. In a filter bank, several filters are applied to the input, and
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Finite Frames and Filter Banks Matthew Fickus, Melody L. Massar, and Dustin G. Mixon
Abstract Filter banks are fundamental tools of signal and image processing. A filter is a linear operator which computes the inner products of an input signal with all translates of a fixed function. In a filter bank, several filters are applied to the input, and each of the resulting signals is then downsampled. Such operators are closely related to frames, which consist of equally spaced translates of a fixed set of functions. In this chapter, we highlight the rich connections between frame theory and filter banks. We begin with the algebraic properties of related operations, such as translation, convolution, downsampling, the discrete Fourier transform, and the discrete Z-transform. We then discuss how basic frame concepts, such as frame analysis and synthesis operators, carry over to the filter bank setting. The basic theory culminates with the representation of a filter bank’s synthesis operator in terms of its polyphase matrix. This polyphase representation greatly simplifies the process of constructing a filter bank frame with a given set of properties. Indeed, we use this representation to better understand the special case in which the filters are modulations of each other, namely Gabor frames. Keywords Filter · Convolution · Translation · Polyphase · Gabor
10.1 Introduction Frame theory is intrinsically linked to the study of filter banks, with the two fields sharing a great deal of common history. Indeed, much of the modern terminology of frames, such as analysis and synthesis operators, was borrowed from the filter bank literature. And though frames were originally developed for the study of nonharmonic Fourier series, much of their recent popularity stems from their use in M. Fickus () · M.L. Massar Department of Mathematics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA e-mail: [email protected] D.G. Mixon Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA P.G. Casazza, G. Kutyniok (eds.), Finite Frames, 337 Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8373-3_10, © Springer Science+Business Media New York 2013
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Gabor (time-frequency) and wavelet (time-scale) analysis; both Gabor and wavelet transforms are examples of filter banks. In this chapter, we highlight the connections between frames and filter banks. Specifically, we discuss how analysis and synthesis filter banks correspond to the analysis and synthesis operators of a certain class of frames. We then discuss the polyphase representation of a filter bank—a key tool in filter bank design—which reduces the problem of constructing a high-dimensional filter bank frame to that of constructing a low-dimensional frame for a space of polynomials. For the signal processing researcher, these results show how to build filter banks that possess the hallmarks of any good frame: robustness against noise and flexibility with respect to redundancy. Meanwh
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