Discrete Time Filter Design for Signal
Processing Based on Local Signal Expansions
Borislav Savkovic
Supervisor : Aleksandar Ignjatovic
1. Introduction
Signal processing methods based on harmonic analysis are
inherently global and provide no local signal information.
Wavelets on the other hand provide variable degrees of localization
via multiresolution analyses.
The recently introduced chromatic signal expansion
is a maximally localized signal representation w.r.t to an
orthogonal set of basis functions. The coefficients (chromatic
derivatives) are linear combinations of standard signal derivatives
at a single point in time and thus encode local signal information.
The signal representation is numerically robust (orthonormal
basis functions). A chromatic signal approximation (i.e. truncated
expansion) is shown below :
Due to uniform convergence and excellent local approximation
properties, the action of linear operators, which need not
be shift invariant can be represented with high fidelity using
the chromatic signal expansion. This property allows maximal
localization of operator action on signals. This is illustrated
below, where the operator acts on the chromatic signal derivatives
at a single point in time:
In this project we have developed a series of discrete time
filters, suitable for signal processing based on the chromatic
signal expansion. The developed filters will be used in modulation,
adaptive signal processing and image compression, based on
the chromatic signal representation, being developed at the
School of Computer Science and Engineering at the University
of NSW.
2. Transform Domain Filtering
In order to perform filtering, based on the chromatic signal
representation, the signal is first analyzed via an analysis
filter bank. We then perform the filtering by a linear operator
L w.r.t. the chromatic signal basis, as shown below:
The filtered signal can then be reconstructed back into the
time domain, via a synthesis filter bank. The analysis and
synthesis filter banks are obtained via suitable least squares
optimizations. Such transform domain filtering is inherently
local, since the filtering is done on maximally local signal
coefficients (chromatic derivatives).
3. Transform Domain Interpolation Filters
In planned applications, such as multiresolution analysis,
based on the chromatic signal representation, local signal
information from multiple sampling moments needs to be synthesized
into a smooth global approximation. This is done via a transform
domain interpolation filter, which provides a spline-like
global approximation in the transform domain. The filtered
transform domain signal can then be resynthesized into a smooth
time domain waveform via a global synthesis filter bank, as
show below:
We have implemented a series of transform domain interpolation
filters, based on different optimization criteria. In addition,
we have also implemented a series of two dimensional interpolation
filters, which will be used in applications in image processing
and robotic vision.
4. Future Work
The localization of operator action on the chromatic signal
approximation remains to be explored. We intend to investigate,
as to how suitable linear operators can be decomposed into
local and global components and how they can be implemented.
In adaptive signal processing, this could have important ramifications,
since such operator decomposition could very well accelerate
the rate of convergence by decoupling the adaptation to local
(i.e .transient) and global (i.e. steady state) operator components.
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