Signal Processing - September 2017 - 180

TIPS & TRICKS
Peter S. Apostolov, Borislav P. Yurukov,
and Alexey K. Stefanov

An Easy and Efficient Method for Synthesizing Two-Dimensional
Finite Impulse Response Filters with Improved Selectivity

I

t is hard to imagine what the world
would look like without the modern
technologies using digital signal processing. The developments in this technical
field provide an opportunity for building technical devices that implement
mathematical methods unattainable
by analog technology. Many modern
technical devices work with two-dimensional (2-D) signals in a process called
digital image processing, and 2-D digital
finite impulse response (FIR) filters are
basic technical tools in image processing.
FIR filters are extensively used in digital
television, radio astronomy, radio location, biomedicine, and so on.
In this article, we explain an interesting method for 2-D FIR filter design
with low computational complexity. As
a result, the filter design is very simple
and exhibits a fast performance.

Relevance
In recent decades, the theory of 2-D filters has undergone considerable development. The major challenge for this
research was obtaining a linear-phase
2-D digital filter with low computational costs. In addition, some applications
require filters with improved selectivity. Several methods have been used by
researchers in the design of 2-D FIR
digital filters, such as frequency transformation, frequency sampling, windowing, McClellan transformation, and
optimization methods [1], [2].
Digital Object Identifier 10.1109/MSP.2017.2717498
Date of publication: 6 September 2017

180

Generally, the synthesis is carried
out with 2-D approximation. This 2-D
approximation is a complicated and
time-consuming process. In this relation, metaheuristic optimization algorithms have been used to optimize the
design. The literature review in [3] notes
that 63% of the publications analyzed
are devoted to evolutionary algorithms,
31% to swarm intelligence algorithms,
and 6% to other algorithms. The basic
goal of the methods is to obtain an
algorithm with fast, guaranteed convergence of the computing operations,
which directly relates to the computation time. All methods have advantages
and disadvantages, and the authors offer
innovative solutions. However, when
the filter length is large (e.g., of 1,024 ×
1,024 dimensions), the algorithms are
not always convergent, and computational processes take tens to hundreds
of seconds.
This article offers an efficient method for the design of 2-D FIR filters, with
improved selectivity, that does not use
iterative algorithms and, for that reason,
has a high level of performance.

Prerequisites
The only prerequisite for understanding
this article is knowledge of basic algebra and trigonometry. Familiarity with
approximation theory and sigmoidal
functions is helpful.

Problem statement
The filter's design is a mathematical problem for an approximation
IEEE SIGNAL PROCESSING MAGAZINE

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September 2017

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of ideal transfer function of the lowpass filter:
D^ f h = '

1, f # 0.5
; f ! 60, 1@,
0, f 2 0.5

(1)

where f is normalized frequency. This is
a function having a rectangular shape
with two ranges: the passband, where the
function is equal to one, and the stopband,
where the function is equal to zero. In
the middle of its domain ^ f = 0.5h, the
function has an indefiniteness of the
first kind. In practice, such a function
cannot be realized using technical devices. No filter simultaneously has an
attenuation of one and zero because it
would pose a contradiction to basic physical principles. Undesired oscillations
occur near the one-zero transition (an
effect called the Gibbs phenomenon).
Therefore, the ideal function is replaced by others having similar shapes. It
has been proven that the polynomial
minimax (equiripple) approximation (also
known as Parks-McClellan) [4] is the
best solution for this problem. The goal
is to define a function with low computational costs that approximates the ideal
function with a minimal error.

Solution
The concept is to use a function with an
S-shaped graph. Such types are the sigmoidal functions: sigmoid, arcos tangent,
hyperbolic tangent, integral Gaussian
error function, and others. After some
modifications, their graphics become the
shape of magnitude response of a lowpass filter. Of the previously mentioned
1053-5888/17©2017IEEE
Table of Contents for the Digital Edition of Signal Processing - September 2017
