## Document Type

Article

## Publication Date

Fall 11-2015

## Department

Mathematics & Statistics

## Abstract

We give a number of explicit matrix-algorithms for analysis/synthesis

in multi-phase filtering; i.e., the operation on discrete-time signals which

allow a separation into frequency-band components, one for each of the

ranges of bands, say N , starting with low-pass, and then corresponding

filtering in the other band-ranges. If there are N bands, the individual

filters will be combined into a single matrix action; so a representation of

the combined operation on all N bands by an N x N matrix, where the

corresponding matrix-entries are periodic functions; or their extensions to

functions of a complex variable. Hence our setting entails a fixed N x N

matrix over a prescribed algebra of functions of a complex variable. In the

case of polynomial filters, the factorizations will always be finite. A novelty

here is that we allow for a wide family of non-polynomial filter-banks.

Working modulo N in the time domain, our approach also allows for

a natural matrix-representation of both down-sampling and up-sampling.

The implementation encompasses the combined operation on input, filtering,

down-sampling, transmission, up-sampling, an action by dual filters,

and synthesis, merges into a single matrix operation. Hence our matrixfactorizations

break down the global filtering-process into elementary steps.

To accomplish this, we offer a number of adapted matrix factorizationalgorithms,

such that each factor in our product representation implements

in a succession of steps the filtering across pairs of frequency-bands; and so

it is of practical significance in implementing signal processing, including

filtering of digitized images. Our matrix-factorizations are especially useful

in the case of the processing a fixed, but large, number of bands.

## Recommended Citation

Song, Myung-Sin and Jorgensen, Palle E. T., "Filters and Matrix Factorization" (2015). *SIUE Faculty Research, Scholarship, and Creative Activity*. 16.

https://spark.siue.edu/siue_fac/16

#### Included in

Harmonic Analysis and Representation Commons, Other Applied Mathematics Commons, Signal Processing Commons

## Comments

This is an accepted manuscript of an article published in

Sampling Theory in Signal and Image ProcessingVolume 14, Number 3, http://stsip.org/.