Finite Frames pp 337-379

# Finite Frames and Filter Banks

Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## 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

## Notes

### Acknowledgements

The authors thank Amina Chebira and Terika Harris for useful discussions. This work was supported by NSF DMS 1042701, NSF CCF 1017278, AFOSR F1ATA01103J001, AFOSR F1ATA00183G003, and the A.B. Krongard Fellowship. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

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