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