×

Symmetrical decomposition and transformation. (English) Zbl 0575.93064

Some procedures to design two-dimensional (2-D) digital filters are presented. The filters under consideration are of finite impulse response (FIR) type. As the starting point two main facts are considered: the first is the so-called McClellan transformation (MT) which allows to obtain a 2-D digital filter starting from 1-dimensional filter; the second is the way in which some symmetry properties are carried on from the transfer function of the 1-D digital filter to the transfer function of the new 2-D digital filter when using the MT. Next, coefficient conditions stemming from various symmetries of real two-variable functions are derived, and it results an important decreasing of the number of the distinct coefficients. An important result is contained in Theorem 8 which gives necessary and sufficient conditions for a real two- variable function to be decomposed as the sum of the symmetrical and antisymmetrical part of some operation T. The main problem which is finally asked is the following: if, instead of optimizing the coefficients of a 2-D filter, obtained by the MT, one first decomposes the transfer function into some parts exhibiting particular symmetries and then optimizes the coefficients of the components, is this way leading to less computer time? Some examples seem to give an affirmative answer but the general answer is still to be found.
The paper is very clearly written, the main ideals are illustrated by interesting examples, and this is a fair mathematical investigation in the area of digital filter design.
Reviewer: D.Stanomir

MSC:

93E11 Filtering in stochastic control theory
93B17 Transformations
93C55 Discrete-time control/observation systems
94C99 Circuits, networks
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dudgeon, D. E.; Mersereau, R. M., Multidimensional digital signal processing (1984), Prentice-Hall, Inc: Prentice-Hall, Inc New Jersey, USA · Zbl 0643.94001
[3] Aly, S. A.H.; Fahmy, M. M., Symmetry in two-dimensional rectangularly sampled digital filters, IEEE Trans. Acoust. Speech Signal Processing, vol. ASSP-29, 796-805 (1981)
[4] George, B.; Venetsanopoulos, A. N., Design of two-dimensional recursive digital filters on the basis of quadrantal symmetry, Circuits, Systems and Signal Processing, vol. 3, no. 1 (1984) · Zbl 0567.93067
[5] McClellan, J. H., The design of two-dimensional digital filters by transformations, Proc. 7th Ann. Princeton Conf. Information Sciences and Systems, 247-251 (1973)
[6] Mersereau, R. M.; Mecklenbrauker, W. F.G.; Quatieri, T. F., McClellan transformations for 2-D digital filtering: I-Design, IEEE Trans. Circuits and Systems, vol. CAS-23, 405-416 (1976) · Zbl 0337.93017
[7] Fettweis, A., Symmetry requirements for multidimensional digital filters, Int. J. Circuit Theory, Appl., vol. 5, 343-353 (1977) · Zbl 0375.93040
[8] Rajan, P. K.; Swamy, M. N.S., Quadrantal symmetry associated with 2-D functions, IEEE Trans. Circuits and Systems, vol. CAS-25, 340-343 (1978) · Zbl 0378.93042
[9] Chakrabharti, S.; Mitra, S. K., Design of Two-Dimensional Filters via Spectral Transformations, Proc. IEEE, vol. 65, 378-388 (1977)
[10] Rajan, P. K.; Swamy, M. N.S., Two-Dimensional FIR Filters with Maximally Flat Magnitude, Proc IEEE, vol. 66, 1086-1088 (1978)
[11] Mersereau, R. M., The design of arbitrary 2-D zero-phase FIR filters using Transformations, IEEE Trans. Circuits and Systems, CAS-27, 142-144 (1980)
[12] Rajan, P. K.; Swamy, M. N.S., Design of circularly symmetric two-dimensional FIR digital filters employing transformations with variable parameters, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, 637-642 (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.