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Design of Variable Two-Dimensional FIR Digital Filters by McClellan Transformation Jong-Jy Shyu, Member, IEEE, Soo-Chang Pei, Fellow, IEEE, and Yun-Da Huang
Abstract—In this paper, the technique of McClellan transformation is applied to design variable 2-D FIR digital filters. Compared with the conventional transformation, the 2-D transformation subfilter and the 1-D prototype filter are designed such that their frequency characteristics are adjustable. Moreover, they are tunable by the same variable parameter, so the variable characteristics of 1-D prototype filters are coincident with those of 2-D subfilters. Several examples, including variable fan filters, variable circularly symmetric filters, and variable elliptically symmetric filters with arbitrary orientation, are presented to demonstrate the effectiveness and the flexibility of the presented method. Index Terms—Circularly symmetric filter, elliptically symmetric filter, fan filter, least squares approach, McClellan transformation, two-dimensional (2-D) filter, variable digital filter.
I. INTRODUCTION
A
MONG THE existing methods, it is no doubt that the McClellan transformation [1]–[11] is one of the most powerful and popular techniques for designing 2-D FIR digital filters. The transformation consists of mapping 1-D prototype filters into 2-D filters by a change of variables. First, a low-order 2-D transformation subfilter, which is the kernel of the transformation, is designed so that the contour of the cutoff edge can meet the requirement for the designed 2-D filter. Then, a high-order 1-D prototype filter can be designed easily by the existing methods, for example, the Remez exchange algorithm [12]. Furthermore, these 2-D filters can be implemented with highly structured architecture [3], [4]. The McClellan transformation can be used to design 2-D fan-type, circularly symmetric, elliptically symmetric, and diamond-shape filters, as well as complex coefficient filters [9]. Recently, variable 2-D digital filters receive considerable attention for their wide usages in communication systems and image processing where the frequency characteristics need to be adjustable [13]–[19]. Among them, only [18] and [19] deal with the application of McClellan transformation for designing variable 2-D filters.
Manuscript received December 31, 2007; revised April 07, 2008. First published October 31, 2008; current version published March 11, 2009. This work was supported by the National Science Council, Taiwan, under Grant NSC 97-2221-E-390-028. This paper was recommended by Associate Editor H. Johansson. J.-J. Shyu is with the Department of Electrical Engineering, National University of Kaohsiung, Kaohsiung 811, Taiwan (e-mail:
[email protected]). S.-C. Pei is with the Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan (e-mail:
[email protected]). Y.-D. Huang is with the Graduate Institute of Communication Engineering, National Taiwa University, Taipei 106, Taiwan (e-mail:
[email protected]. tw). Digital Object Identifier 10.1109/TCSI.2008.2002119
In this paper, the application of McClellan transformation for designing variable 2-D filters is investigated again. Comparing with the existing methods [18], [19], the 2-D transformation subfilter is concretely designed in this paper, and its frequency characteristic is adjustable by a variable parameter such that the cutoff contour mapping in the subfilter can continuously match with the desired variable 2-D filter. Moreover, a cutoff-frequency orbit function is proposed for the design of variable 2-D transformation subfilter and variable 1-D prototype filter so that they are tunable by the same variable parameter, and the variable characteristics of 1-D prototype filters are coincident with those of 2-D transformation subfilters. This paper is organized as follows. Section II deals with the variable structure of 2-D filters designed by McClellan transformation, which are modified from the Farrow structure [20]. In Section III, the design of variable 2-D fan filters is proposed. First, a cutoff-frequency orbit function is found by solving an overdetermined system, and then, both variable 2-D subfilters and variable 1-D prototype filters are designed by least squares approach [21]–[24]. Finally, the coefficients of variable 2-D filters can be obtained by using the recurrence relations for Chebyshev polynomial. Following the steps in Section III, the designs of variable circularly symmetric filters and variable elliptically symmetric filters with arbitrary orientation are presented in Sections IV and V, respectively. Finally, the conclusions are given in Section VI.
II. STRUCTURE OF THE PROPOSED VARIABLE 2-D FIR FILTERS For a variable zero-phase FIR digital filter, its frequency response is represented by (1) in which the coefficients of
are expressed as the polynomials
(2) Replacing in (1) by the th-order Chebyshev polynomial, (1) becomes
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(3)
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Following the McClellan transformation [1]–[11] and substituting the variable 2-D transformation subfilter
(4) into (3), we can obtain the frequency response of the desired variable 2-D filter (5a)
N
Fig. 1. (a) Structure of the proposed variable 2-D FIR filters ( = 5). (b) Variable-coefficient generator for both variable 1-D prototype filters and variable 2-D subfilters ( ( ) may be as ( ) ( ) or ( )).
c i
a i ; t i; j
s i; j
where III. DESIGN OF VARIABLE 2-D FIR FAN FILTERS (5b)
For designing 2-D fan filters, the original substitution
(5c) As to the choice for the order of the transformation subfilter and are generally adopted in (4), for the design of quadrantally symmetric 2-D filters such as fan and circular filters, while the design of centrosymmetric filters such as elliptical filters with arbitrary orientation takes . Although higher order transformation subfilters can be used for accuracy, it is not recommended because the complexity will increase drastically. In this paper, the variable 2-D subfilter is also designed such that its variable characteristics can be controlled by the same parameter as the variable 1-D prototype filter, i.e., the parameter as in (1) and is used in this paper. For the choice (2). For simplicity, of and in (5), it depends on the accuracy of cutoff contour is used, and the obtained mapping. In this paper, results are satisfactory. Furthermore, due to the recurrence relations for Chebyshev polynomial as follows: (6a) (6b) (6c) the structure of the designed variable 2-D FIR filters is shown in Fig. 1(a), and the variable-coefficient generator for both variable 1-D prototype filters and variable 2-D subfilters is shown in Fig. 1(b). Notice that the proposed structure is different from the general Farrow structure due to the consideration for the special recurrence structure [4]. To avoid enormous computation for the generation of the coefficients of 2-D transformation subfilters and 1-D prototype filters, the variable-coefficient generators shown in Fig. 1(b) can be implemented such that they need to operate only on the demand of variation, and the values of coefficients can be stored in memory.
(7) is used in McClellan transformation. To avoid scaling problem, it is desirable to give the following constraints: 1) the 1-D frequency origin, , is mapped into the point of the 2-D of the 1-D frequency is frequency plane, and 2) the point point of the 2-D frequency plane, which mapped into the result in (8a) (8b) Hence, the transformation becomes
(9) For choosing the proper cutoff frequency of 1-D prototype low-pass filter and the transformation coefficients, the following objective error function is defined as in [8]
(10) in which (11a) (11b) (12a)
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Fig. 2. Specification of a 2-D fan filter within the first quadrant.
Fig. 3. Error curve of (10) when ! varies in [0; ] for
= 35
.
(12b)
(12c) Notice that the superscript denotes the transpose operator; , where is the inclination angle of the 2-D fan filter shown in Fig. 2, and (13) For a given inclination angle of the 2-D fan filter, the cutoff of the 1-D prototype low-pass filter is chosen as frequency , Fig. 3 shows the error curve of follows. For example, if (10) when varies in , in which the errors are computed into (10) for each , and by substituting the corresponding to the smallest error is exactly the desired . When different values of inclination angle are given for designing 2-D fan filters, the corresponding cutoff frequency of the 1-D prototype low-pass filter is also changed according to . To illustrate the relationship between and , Fig. 4(a) shows
Fig. 4. Design of a variable 2-D fan filter. (a) Cutoff-frequency orbit of 1-D prototype low-pass filters. ( and solid line) Individual design. (2 and dotted line) Variable design. (b) Isopotential cutoff edge contours for different integer inclination degrees from 30 to 45 . (c) Magnitude responses of variable 1-D prototype low-pass filter. (d) Magnitude responses of variable 2-D fan filter for p = tan(30 ); tan(35 ); tan(40 ), and tan(45 ).
the cutoff-frequency orbit for individual design with integer inmarked by “ .” clination angles in the range of
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A. Determination of the Cutoff-Frequency Orbit Function To further design variable 2-D transformation subfilters and , which can repvariable 1-D prototype filters, a function resent the cutoff-frequency orbit shown in Fig. 4(a), should be determined. In this paper, the method for solving least squares problems which occur in overdetermined systems [25] is applied as follows. Suppose that the variable range of inclination angle for designing a variable 2-D fan filter is , where and are integer degrees for simplicity, and the variable parameter is defined by
To find the subfilter coefficients in the least squares sense, the objective error function is represented by
(21) where
(14)
(22a)
then the corresponding overdetermined system can be represented by
(22b)
.. .
.. .
.. .
.
.. .
.. .
.. .
..
(22c)
(15) (23a)
which can be expressed in matrix form as (16)
(23b)
It is noted that
in (15) represents the row number of , and , denotes the desired cutoff frequencies marked . Therefore, the least by “ ” in Fig. 4(a). Generally, squares solution of (16) can be obtained by (17) and the cutoff-frequency orbit function by
can be represented
(23c) can be obTherefore, the transformation coefficient vector and setting the tained by differentiating (21) with respect to result to zero, which results in (24)
(18) For example, when cutoff-frequency orbit function marked by “ ” in Fig. 4(a).
and , the obtained is plotted in a dotted line
B. Design of Variable 2-D Transformation Subfilters
, and For example, when , the isopotential cutoff edge contours for different integer inclination degrees from 30 to 45 are shown in Fig. 4(b). To give a measure of how close this approximation is, the root-meansquared deviation error of the variable cutoff isopotentials with the desired ones is computed as follows:
For designing variable 2-D fan filters, the variable transformation
(19) is applied, in which (25) (20a) (20b)
where (15) and
and
, are the cutoff frequencies shown in are set to be 16 and 50, respectively, in this
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TABLE I KEY PARAMETERS AND COEFFICIENTS OF DESIGN EXAMPLES IN SECTIONS III, IV, AND V
example. Moreover, the obtained root-mean-squared deviation . error is
(30b)
C. Design of Variable 1-D Prototype Low-Pass Filters By (1) and (2), the frequency response of a variable zerophase 1-D prototype low-pass filter is characterized by (26) which is used to approximate the desired variable frequency response (27)
(30c) In addition, differentiating (29) with respect to result to zero, the solution is given by
and setting the (31)
where
represents the width of the transition band. Defining For example, when , and , the magnitude responses of the variable 1-D prototype low-pass filter are shown in Fig. 4(c). (28a)
(28b) and following Section III.B, the objective error function is defined by
D. Derivation of Variable 2-D Fan Filters Once the variable 2-D transformation subfilter and the variable 1-D prototype filter have been designed, we can obtain the coefficients of variable 2-D fan filters by (5) after some mathematic manipulations. Fig. 4(d) shows the obtained magnitude responses of the variable 2-D fan filter for , and . To explicitly illustrate the result of the designed filter, the parameters in and the coefficients in are tabulated in Table I, companying with those of other examples. IV. DESIGN OF VARIABLE 2-D CIRCULARLY SYMMETRIC FILTERS
(29) where (30a)
In this section, the transformation of (7) is used for designing 2-D circularly symmetric filters. In addition, there are three conis mapped into ; 2) straints to be considered: 1) is mapped into ; and 3) due to the symmetric contour, which result in (32a) (32b)
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Therefore, the transformation becomes
(33) in which only one transformation coefficient needs to be determined. Following the steps in Section III, we can obtain the cutoff frequencies shown in Fig. 5(a) (marked by “ ”) for individual design when the passband radius of the circularly to in low-pass filter varies from . To find the cutoff-frequency orbit step , the variable parameter is defined by function (34) which is just the variable passband radius, and the corresponding overdetermined system is given by
.. .
.. .
.. .
..
.. .
.
.. .
.. .
(35)
Fig. 5(a) also shows the curve of the cutoff-frequency orbit function in a dotted line marked by “ ” when by using (18). In addition, the variable 2-D transformation subfilter can be designed, which is similar to that in Section III.B, and the objective error function is given by
(36) in which
(37a)
(37b) Fig. 5(b) shows the isopotential cutoff edge contours when the passband radius varies from to .
Fig. 5. Design of a variable 2-D circularly low-pass filter. (a) Cutoff-frequency orbit of 1-D prototype low-pass filters. ( and solid line) Individual design. (2 and dotted line) Variable design. (b) Isopotential cutoff edge contours for different passband radius from 0:35 to 0:6 . (c) Magnitude responses of variable 1-D prototype low-pass filter. (d) Magnitude responses of variable 2-D circularly low-pass filter for p = 0:35; 0:45; 0:55 , and 0:6 .
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Fig. 6. Specification of a 2-D elliptically symmetric low-pass filter.
For the design of the variable 1-D prototype low-pass filter, it is similar to that in Section III.C except that the cutoff orbit in (29) and (30) is replaced by that shown in function , Fig. 5(a). For example, when and , the magnitude responses of the variable 1-D prototype low-pass filter are shown in Fig. 5(c). In addition, the final magnitude responses of the variable 2-D circularly low, and are shown pass filter for in Fig. 5(d), and the root-mean-squared deviation error of the variable cutoff isopotentials is computed by
(38) , are the cutoff frequencies shown in where (35) and and are set to be 41 and 100, respectively, in this example, which result in . V. DESIGN OF VARIABLE 2-D ELLIPTICALLY SYMMETRIC FILTERS Here, the modified McClellan transformation
(39) is used to design variable 2-D elliptically symmetric filters. For axis as an ellipse rotated by an angle with respective to shown in Fig. 6, it can be described by the curve
Fig. 7. Design of a variable 2-D elliptically symmetric low-pass filter. (a) Cutoff-frequency orbit of 1-D prototype low-pass filters. ( and solid line) Individual design. (2 and dotted line) Variable design. (b) Isopotential cutoff edge contours for different inclination angle from 035 to 35 . (c) Magnitude responses of variable 1-D prototype low-pass filter. (d) Magnitude responses of variable 2-D elliptically symmetric low-pass filter for = 035 ; 0 ; 20 , and 35 .
in which (41a)
(40)
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(41b)
where
(41c) (47a) where and are semiminor axis and semimajor axis, respectively. For designing a 2-D elliptically low-pass filter with arbitrary orientation, there are two constraints to be considered: is mapped into (0,0), and 2) is mapped into 1) , which result in (47b) (42a) (42b) Hence, the transformation (39) becomes
(43) To design a variable 2-D elliptically symmetric filter, we first obtain the cutoff frequencies shown in Fig. 7(a) (marked by “ ”) for individual design when the rotated angle varies from to , and . In this section, the variable parameter is defined by
denotes a line integral along the curve of (40) for and a given variable parameter . Fig. 7(b) shows the isopotential cutoff edge contours when the variable parameter varies from to and . As to the design of the variable 1-D prototype low-pass filter, the magnitude responses are shown in and . The final magnitude reFig. 7(c) with sponses of the variable 2-D elliptically symmetric low-pass filter , and are shown in Fig. 7(d). In this for example, the root-mean-squared deviation error of the variable , cutoff isopotentials is given in (48), in which are shown in (45), and are set to be 71 and 180, respectively, are computed by substituting into and in (48) for (41). It is noted that . After numerical computing, . From Table I, it is noted that part of the parameters and transformation coefficients are identical to zero due to the symmetric to 35 . inclination angle from
(44) Like (15) and (35), the corresponding overdetermined system can be formulated as
.. .
.. .
.. .
..
.. .
.
.. .
.. .
(45) (48)
is shown in The obtained cutoff-frequency orbit function . Fig. 7(a) when To design the variable 2-D transformation subfilter, the objective error function is defined by
(46)
VI. CONCLUSION In this paper, the technique of conventional McClellan transformation has successfully been extended to design variable 2-D FIR digital filters. Once the cutoff-frequency orbit function is determined, both variable 2-D transformation subfilter and variable 1-D prototype filter can be designed and are adjustable by the same variable parameter. From the presented numerical examples, the effectiveness and flexibility of the proposed method have been fully depicted by the presented illustrations.
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ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive comments on this paper. REFERENCES [1] J. H. McClellan, “The design of 2-D digital filters by transformations,” in Proc. 7th Annu. Princeton Conf. Inf. Sci., 1973, pp. 247–251. [2] R. M. Mersereau, W. F. G. Mecklenbräuker, and T. F. Quatieri , Jr., “McClellan transformations for 2-D digital filtering—Part I: Design,” IEEE Trans. Circuits Syst., vol. CAS-23, no. 7, pp. 405–414, Jul. 1976. [3] R. M. Mersereau, W. F. G. Mecklenbräuker, and T. F. Quatieri, Jr., “McClellan transformations for 2-D digital filtering—Part II: Implementation,” IEEE Trans. Circuits Syst., vol. CAS-23, no. 7, pp. 414–422, Jul. 1976. [4] J. H. McClellan and D. S. K. Chan, “A 2-D FIR filter structure derived from the Chebyshev recursion,” IEEE Trans. Circuits Syst., vol. CAS-24, no. 7, pp. 372–378, Jul. 1977. [5] M. S. Reddy and S. N. Hazra, “Design of elliptically symmetric 2-D FIR filters with arbitrary orientation,” Electron. Lett., vol. 23, no. 18, pp. 964–966, Aug. 1987. [6] S. C. Pei and J. J. Shyu, “Design of 2-D FIR digital filters by McClellan transformation and least squares eigencontour mapping,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 40, no. 9, pp. 546–555, Sep. 1993. [7] H. C. Lu and K. H. Yeh, “Optimal design of 2-D FIR digital filters by scaling-free McClellan transformation using least-squares estimation,” Signal Process., vol. 58, no. 3, pp. 303–308, May 1997. [8] H. C. Lu and K. H. Yeh, “2-D FIR filters design using least squares error with scaling-free McClellan transformation,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 47, no. 10, pp. 1104–1107, Oct. 2000. [9] S. C. Pei and J. J. Shyu, “Design of 2-D FIR digital filters by McClellan transformation and least-squares contour mapping,” Signal Process., vol. 44, no. 1, pp. 19–26, Jun. 1995. [10] D. T. Nguyen and M. N. S. Swamy, “Approximation design of 2-D digital filters with elliptical magnitude response of arbitrary orientation,” IEEE Trans. Circuits Syst., vol. CAS-33, no. 6, pp. 597–603, Jun. 1986. [11] C.-C. Tseng, “Design of 2-D FIR digital filters by McClellan transform and quadratic programming,” Proc. Inst. Elect. Eng.1/2"Vis. Image Signal Process., vol. 148, no. 5, pp. 325–331, Oct. 2001. [12] J. H. McClellan, T. W. Parks, and L. R. Rabiner, “A computer program for designing optimum FIR linear phase digital filters,” IEEE Trans. Audio Electroacoust., vol. AU-21, no. 6, pp. 506–526, Dec. 1973. [13] T.-B. Deng, “Design of linear phase variable 2-D digital filters using real-complex decomposition,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 45, no. 3, pp. 330–339, Mar. 1998. [14] T.-B. Deng, “Variable 2-D FIR digital filter design and parallel implementation,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 5, pp. 631–635, May 1999. [15] T.-B. Deng and W.-S. Lu, “Weighted least-squares method for designing variable fractional delay 2-D FIR digital filters,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 47, no. 2, pp. 114–124, Feb. 2000. [16] T.-B. Deng, “Design of linear-phase variable 2-D digital filters using matrix-array decomposition,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 50, no. 6, pp. 267–277, Jun. 2003. [17] T.-B. Deng and E. Okamoto, “SVD-based design of fractional-delay 2-D digital filters exploiting specification symmetries,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 50, no. 8, pp. 470–480, Aug. 2003. [18] C. K. S. Pun, S. C. Chan, and K. L. Ho, “Efficient 1-D and circular symmetric 2-D FIR filters with variable cutoff frequencies using the Farrow structure and multiplier-block,” in Proc. ISCAS, 2001, pp. 561–564. [19] K. S. Yeung and S. C. Chan, “Design and implementation of multiplierless tunable 2-D FIR filters using McClellan transformation,” in Proc. ISCAS, 2002, pp. 761–764. [20] C. W. Farrow, “A continuously variable digital delay element,” in Proc. ISCAS, 1988, pp. 2641–2645. [21] H. H. Dam, A. Cantoni, K. L. Teo, and S. Nordholm, “Variable digital filter with least-square criterion and peak gain constraints,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 1, pp. 24–28, Jan. 2007.
[22] H. H. Dam, A. Cantoni, K. L. Teo, and S. Nordholm, “FIR variable digital filter with signed power-of-two coefficients,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 6, pp. 1348–1357, Jun. 2007. [23] T.-B. Deng and Y. Lian, “Weighted-least-squares design of variable fractional-delay FIR filters using coefficient symmetry,” IEEE Trans. Signal Process., vol. 54, no. 8, pp. 3023–3038, Aug. 2006. [24] W.-S. Lu and T.-B. Deng, “An improved weighted least-squares design for variable fractional delay FIR filters,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 8, pp. 1035–1040, Aug. 1999. [25] S. J. Leon, Linear Algebra With Applications, 7E. Englewood Cliffs, NJ: Prentice-Hall, 2006. Jong-Jy Shyu (S’88–M’93) was born in Taiwan, on March 7, 1960. He received the B.S. degree in electrical engineering from Tatung University, Taipei, Taiwan, in 1983 and the M.S. and Ph.D. degrees in electrical engineering from National Taiwan University, Taipei, in 1988 and 1992, respectively. In 1992, he was an Associate Professor with the Department of Computer Science and Engineering, Tatung University, where he was a Professor in 1996. From 1997 to 2000, he was with the Department of Computer and Communication Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan. He is currently with the Department of Electrical Engineering, National University of Kaohsiung, Kaohsiung. His research interests include the design and implementation of digital filters and digital signal processing.
Soo-Chang Pei (SM’89–F’00) was born in Soo-Auo, Taiwan, in 1949. He received B.S.E.E. degree from National Taiwan University, Taipei, Taiwan, in 1970 and the M.S.E.E. and Ph.D. degrees from the University of California, Santa Barbara, Santa Barbara, in 1972 and 1975, respectively. From 1970 to 1971, he was an Engineering Officer with the Chinese Navy Shipyard. From 1971 to 1975, he was a Research Assistant with the University of California, Santa Barbara. From 1981 to 1983, he was a Professor with the Electrical Engineering Department, Tatung Institute of Technology, Taipei. From 1995 to 1998, he was the Chairman with the Electrical Engineering Department, National Taiwan University, where he is currently the Dean of the Electrical Engineering and Computer Science College and the Professor of Electrical Engineering Department. His research interests include digital signal processing, image processing, optical information processing, and laser holography. Dr. Pei was the President of the Chinese Image Processing and Pattern Recognition Society in Taiwan from 1996 to 1998, and a member of the Eta Kappa Nu and Optical Society of America. He became an IEEE Fellow in 2000 for contributions to the development of digital eigenfilter design, color image coding, and signal compression and to electrical engineering education in Taiwan. He was the recipient of the National Sun Yet-Sen Academic Achievement Award in Engineering in 1984, the Distinguished Research Award from the National Science Council from 1990 to 1998, the Outstanding Electrical Engineering Professor Award from the Chinese Institute of Electrical Engineering in 1998, the Academic Achievement Award in Engineering from the Ministry of Education in 1998, the Pan Wen-Yuan Distinguished Research Award in 2002, and the National Chair Professor Award from Ministry of Education in 2002.
Yun-Da Huang received the B.S. and M.S. degrees in electrical engineering from National University of Kaohsiung, Kaohsiung, Taiwan, in 2006 and 2008, respectively. He is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan. His research interests include filter design and digital signal processing.
