A New Criterion for the Design of Variable Fractional ... - IEEE Xplore

368 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 2, FEBRUARY 2010A New Criterion for the Design of VariableFractional-Delay FIR Digital FiltersJong-Jy Shyu, Member, IEEE, Soo-Chang Pei, Fellow, IEEE, Cheng-Han Chan, Member, IEEE,Yun-Da Huang, and Shih-Hsin LinAbstract—Conventionally, variable fractional-delay finite-impulse-responsedigital filters are generally designed by minimizingthe root-mean-square error of variable frequency response. Inthis paper, a new criterion concerning the minimization of theroot-mean-square error of variable group-delay response is proposed.However, minimization is a highly nonlinear problem, soan iterative method is proposed in this paper to overcome it. Tofurther reduce the maximum absolute group-delay error in theleast squares design, an iterative weighting-updated technique isalso proposed, which constitutes the outer loop of the overall iterativeprocess while the iteration stated earlier makes up the innerloop. Several design examples will be presented to demonstratethe effectiveness of the proposed method.Index Terms—Finite-impulse-response (FIR) filter, iterativeweighted least squares (LS) method, totally equiripple design,variable fractional-delay (VFD) filter.I. INTRODUCTIONFOR the past decade, the design of variable fractional-delay(VFD) digital filters has received considerable attentiondue to its wide applications in signal processing and communicationsystems [1]–[6]. VFD filters belong to a branch of variabledigital filters that are applied in applications in which thefrequency characteristics need to be adjustable online withoutredesigning a new filter. Since the Farrow structure was proposedin 1988 [7], several techniques have been submitted forthe related design [8]–[31].Conventionally, most works concerning the design of VFDfilters are focused on the minimization of the root-mean-squareerror of variable frequency response, and the objective errorfunction can be generally represented byManuscript received December 11, 2008; revised March 19, 2009. First publishedJune 02, 2009; current version published February 10, 2010. This workwas supported by the National Science Council, Taiwan, under Grants NSC98-2221-E-390-019. This paper was recommended by Associate Editor E. A.Barros da Silva.J.-J. Shyu is with the Department of Electrical Engineering, National Universityof Kaohsiung, Kaohsiung 811, Taiwan (e-mail: jshyu@nuk.edu.tw).S.-C. Pei, Y.-D. Huang, and S.-H. Lin are with the Graduate Institute ofCommunication Engineering, National Taiwan University, Taipei 106, Taiwan(e-mail: pei@cc.ee.ntu.edu.tw; d97942016@ntu.edu.tw; r96942113@ntu.edu.tw).C.-H. Chan is with the Department of Computer and Communication Engineering,National Kaohsiung First University of Science and Technology, Kaohsiung824, Taiwan. Also, he is now partly with the Department of Aviation andCommunication Electronics, Air Force Institute of Technology, Kaohsiung 820,Taiwan (e-mail: u9115902@ccms.nkfust.edu.tw).Digital Object Identifier 10.1109/TCSI.2009.2023835(1)where represents the designed region; andare the desired and actual frequency responses,respectively; and is a weighting function. However, inimage processing, it has been shown that the lost information ofmagnitude response can be recovered from phase response, andgroup delay is another way to observe the phase distortion [32].Also, the group-delay function is useful in speech processing[33]. Moreover, for some applications such as discrete-timesignal interpolation and time-delay estimation [1]–[3], timingplays an important role, so it is desirable to have a smallergroup-delay error for the design of VFD filters. In this paper,a new delay-oriented criterion is proposed to minimize theroot-mean-square error of variable group-delay response forthe design of VFD finite-impulse-response (FIR) digital filters,and the objective error function becomeswhere and are the desired and actual groupdelayresponses of the designed system, respectively. However,when the error in (2) is minimized, it is possible that the actualvariable frequency response will deviate from the desired onetoo much, so the error function in (2) is replaced byin this paper, where is a relative weighting constant that mustbe large enough to preserve the approximation of variable frequencyresponse while the root-mean-square error of variablegroup-delay response can be minimized as much as possible.However, the minimization of (3) is a highly nonlinear problem,so an iterative method is proposed to overcome it. Furthermore,to reduce the maximum absolute group-delay error in the leastsquares (LS) design of (3), i.e., , a weighting-updatedtechnique is also proposed in this paper.This paper is organized as follows. In Section II, the design ofVFD FIR digital filters by a conventional LS method is reviewedfor the purpose of comparison and the use of other sections. InSection III, the new delay-oriented criterion is proposed, andthe nonlinear problem to minimize the error in (3) is convertedinto an iterative quadratic problem by replacing some elementsin (3) with the results in the previous iteration. Several designexamples will be presented to demonstrate the effectiveness ofthe proposed method. Then, an iterative weighted LS methodis proposed in Section IV to further reduce the maximum absolutegroup-delay error obtained in Section III. In this paper,a new target error function whose peak error is what we want(2)(3)1549-8328/$26.00 © 2010 IEEE

SHYU et al.: NEW CRITERION FOR THE DESIGN OF VARIABLE FRACTIONAL-DELAY FIR DIGITAL FILTERS 369to minimize is proposed for the design of VFD FIR digital filters.Although we cannot make sure if the maximum absolutegroup-delay error is minimized, the presented example revealsthat it will be reduced drastically. Finally, the conclusions willbe given in Section V.II. REVIEW OF CONVENTIONAL LS DESIGN OF VFD FIRDIGITAL FILTERSFor the purpose of comparison and the use of the followingsections, the conventional LS design of VFD FIR digital filtersis reviewed in this section. The desired response of a VFD FIRfilter is given bywhere is a prescribed mean group delay and is the parameterused to adjust the group delay of a filter online. The used transferfunction is characterized by(4)(5)it is reasonable to choose the coefficients of to be symmetricfor even and antisymmetric for odd , and obviously,. In this paper, only even is used, and the case forodd can be extended in a similar manner. Notice that thefirst subfilter is designed to approximate ,so. Hence, the frequency response of (7)can be written as(10)where (11a)–(11c) are shown at the bottom of the page. Defining(12a)where coefficientsbyare expressed as the polynomials of(6)(12b)hence(7)(12c)where subfiltersare represented by(8)(12d)Obviously, (7) can be implemented by the Farrow structure [7],[15].Equation (4) can be further represented by(9)for sufficiently large . Comparing (7) and (9), it can be foundthat the frequency response of is used inherently to approximatefor . Therefore,Equation (10) can be represented by(13)where the superscript denotes the transpose operator.The conventional objective error function for designing aVFD FIR filter is given by:even: odd;;(11a)(11b)(11c)

370 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 2, FEBRUARY 2010whereis a weighting function(14)For LS design and by applying the technique in [22],the elements in , , , and can be represented in closedform as (17a) and (17b), shown at the bottom of the page, wheredenotes the remainder when integer is divided byinteger , and denotes the largest integer that is less than orequal to real number . In (17), must be chosen large enoughas in [22], and is used in this paper.Once , , , and are obtained, the optimal solutionsin the LS sense can be achieved by differentiating (14) withrespect to and , respectively, and then setting the results tozero as follows:(15a)(18a)(18b)in which(15b)which yield(16a)(19a)(19b)(16b)(16c)(16d)(16e)III. PROPOSED NEW CRITERION FOR DESIGNING VFD FIRDIGITAL FILTERS IN THE LS SENSEIn Section II, the VFD FIR filter is designed such that theroot-mean-square error of variable frequency response can beminimized. In this section, delay-oriented minimization is proposedso that the root-mean-square group-delay error can beminimized as much as possible while the desired variable frequencyresponse can be preserved to a certain extent.The desired group-delay response of a VFD FIR filter can bederived from (4)(16f)(20)(17a)(17b)

SHYU et al.: NEW CRITERION FOR THE DESIGN OF VARIABLE FRACTIONAL-DELAY FIR DIGITAL FILTERS 371and the actual group-delay response of the designed system isgiven by (21), shown at the bottom of the page, where denotesthe argument of a complex number(22a)subscriptare defined byare the results of the previous iteration, which(26a)(26b)(26c)(22b)The objective error function of the proposed method is given byThus, the original nonlinear problem can be converted into aniterative quadratic problem whose error function can be formulatedinto(27)where(23)where is a relative weighting constant, has been definedin (14), and is shown in (24), shown at the bottomof the page. Obviously, minimization of (23) is a highly nonlinearproblem, and an iterative method is proposed in this paperto replace it.The objective error function in the th iteration for the proposediterative method is represented by(28a)(28b)(28c)(28d)(28e)(25)where the coefficient vectors denoted by subscript are to bedetermined in the th iteration and the functions denoted by(28f)(21)(24)

SHYU et al.: NEW CRITERION FOR THE DESIGN OF VARIABLE FRACTIONAL-DELAY FIR DIGITAL FILTERS 373Fig. 3. (a) Curves of (solid line) " and (dash line) " and (b) curves of (solidline) " and (dash line) " when varies from 1 to 40 for the proposedmethod.Fig. 2. Design of an N =50, M =7, and ! =0:9 VFD FIR filter. (a)Group-delay response of the proposed method. (b) Absolute error of variablefrequency response [(top) conventional method; (bottom) proposed method)].(c) Absolute delay error [(top) conventional method; (bottom) proposedmethod)].To compute the errors in (34b)–(34d), frequency and parameterare uniformly sampled at step sizes and 1/60, respectively,in this section.1) Example 1: An , , and VFDFIR filter is designed in this example. When the conventionalmethod in Section II is used, Fig. 2(b) and (c) shows the absoluteerror of variable frequency response and the absolute groupdelayerror, respectively. The first problem that we face for theproposed method is the choice of relative weighting constantin (23). Fig. 3(a) shows the curves of and , and those ofand are shown in Fig. 3(b), when varies from 1 to40. Obviously, there exists a tradeoff relationship betweenand , as well as and . Thus, it is reasonable to choosethe largest integer such that the maximum absolute error ofvariable frequency response is less than that of the conventionalmethod. In this example, is the choice, and thedesign took three iterations. The obtained variable group-delayresponse is shown in Fig. 2(a), while Fig. 2(b) and (c) showsthe absolute error of variable frequency response and the absolutegroup-delay error, respectively, accompanying with thoseof the conventional method for comparison. The errors in (34)are listed as follows:(34c)(34d)

374 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 2, FEBRUARY 2010TABLE INORMALIZED ROOT-MEAN-SQUARE ERROR OF VARIABLE FREQUENCY RESPONSE " , THE MAXIMUM ABSOLUTE ERROR OF VARIABLE FREQUENCYRESPONSE" , THE NORMALIZED ROOT-MEAN-SQUARE ERROR OF VARIABLE GROUP-DELAY RESPONSE " , AND THE MAXIMUM ABSOLUTE GROUP-DELAYERROR " FOR THE DESIGN OF M =7AND ! =0:9 VFD FIR FILTERS WITH DIFFERENT FILTER ORDER N ’s BY THE CONVENTIONAL AND PROPOSEDMETHODSFrom Fig. 2(c) and the aforementioned data, the group-delayperformance of the proposed method is much better than thatof the conventional method under the cost of larger .Toinvestigate the convergence of the iterative method, the trace ofis shown in Fig. 4, in which the sum reduces toin the third iteration.2) Example 2: To illustrate the advantages of the proposedmethod, Table I presents more results for different filter orders’s when and . Also, for each design, thelargest integer is selected such that the maximum absoluteerror of variable frequency response is less than that of the conventionalmethod. The improved ratio shown in Table I is defined,e.g., for by (35), shown at the bottom of the page, andthat for is also defined in the same manner. It is found thatthe improved ratios for both and become more significantfor larger .Fig. 4. Trace of the sum + .IV. TOTALLY EQUIRIPPLE DESIGN OF VFD FIR DIGITALFILTERS IN GROUP-DELAY RESPONSETo further reduce the maximum absolute group-delay error, the modification of the weighting-updated technique in [34]is applied in this section. Peering the absolute group-delay errorof the proposed method and the absolute error of variable frequencyresponse of the conventional method, which are shownin Fig. 2(c) (bottom) and Fig. 2(b) (top), respectively, it can befound that the former is not so regular as the latter. Therefore,the method in [31] cannot be applied here to obtain a minimaxdesign in group-delay response.In this section, the target error function whose peak error wewant to minimize is given by(36)which is the integration of the absolute group-delay error alongthe -axis, so we name the algorithm by “Totally equiripple design.”Minimization of the peak error in (36) is achieved by an(35)

SHYU et al.: NEW CRITERION FOR THE DESIGN OF VARIABLE FRACTIONAL-DELAY FIR DIGITAL FILTERS 375Fig. 6. (a) Absolute error of variable frequency response and (b) absolutegroup-delay error in Example 3.Fig. 5. Flowchart of the totally equiripple design of VFD FIR digital filters ingroup-delay response.iterative weighting-updated method, which constitutes the outerloop of the overall process while the iteration in Section IIImakes up the inner loop. The overall iterative process is shownin Fig. 5 and described in detail as follows.Step 1) Given , , , and , find the initial coefficientvectors and by (19).Step 2) Set the inner iterative counter .Step 3) Increase the inner iterative counter by one andcalculate , , ,, , , , and .Step 4) Find coefficient vectors and by (31).Step 5) Check whether both relative norms and aresmall enough by(37a)(37b)If the condition is satisfied, go to the next step; otherwise,go to Step 3).Step 6) Find the absolute error ripples of in (36), anddenote the th ripple with ripple intervalby , , where is the number of ripples in. Then, search the maximum value and theminimum value of , .Step 7) Check whether error function is nearlyequiripple byStep(38)where is the relative peak error ratio andis a preassigned very small positive constant. If thecondition is satisfied, stop the process; otherwise, goto the next step.8) Compute the unnormalized weighting functionand find its maximum valueThen, update the weighting function by(39)(40)(41)Step 9) Calculate , , , and in (16), and replaceand by and , respectively. Then, go to Step2).1) Example 3: The design in Example 1 is continued so thatthe maximum absolute group-delay error in Section III can bereduced as much as possible. In this example, frequency andparameter are uniformly sampled at step sizes and1/120, respectively. If is used, 12 outer iterationsare needed, and the respective inner iterations are four, three,

376 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 2, FEBRUARY 2010It is noted that the mean group delay in (4) is set assuch that all subfilters , , in (8) can be implementedby symmetric/antisymmetric coefficient FIR filters.However, if a small is desirable, it is difficult to get a high-performanceVFD FIR filter, and maybe, the IIR-type VFD filtercan be considered, which will be the target of related researchin the future.ACKNOWLEDGMENTThe authors would like to thank the anonymous reviewers fortheir constructive comments on this paper.Fig. 7. (a) Total error functions E(!) of the (dash line) first outer iteration andthe (solid line) fourth outer iteration. (b) Trace of relative peak error ratio .three, two, two, three, one, two, two, one, one, and one, respectively,and that would mean 20-five inner iterations are required.The obtained absolute error of variable frequency response andthe absolute group-delay error are shown in Fig. 6(a) and (b),respectively, and the errors in (34) are listed as follows:Once more the maximum absolute group-delay error ismuch smaller than those in Example 1, the sacrifice is thatand will become larger. The error functions of thefirst outer iteration and the twelfth outer iteration are shown inFig. 7(a), and the trace of relative peak error ratio is shownin Fig. 7(b), which show that the convergence of the proposediterative method is satisfactory.V. CONCLUSIONIn this paper, a new criterion concerning the minimizationof the root-mean-square error of variable group-delay responsehas been proposed for the design of VFD FIR digital filters.To overcome the nonlinear optimization for minimization, theproposed iterative method has been successfully used, and theexperimental results show that the performance in group-delayresponse and the convergence of the iterative method are satisfactory.Moreover, an iterative weighting-updated techniquehas also been proposed such that the maximum absolute groupdelayerror can be further reduced drastically.REFERENCES[1] F. M. Gardner, “Interpolation in digital modems—Part I: Fundamental,”IEEE Trans. Commun., vol. 41, no. 3, pp. 501–507, Mar.1993.[2] L. Erup, F. M. Gardner, and R. A. Harris, “Interpolation in digitalmodems—Part II: Implementation and performance,” IEEE Trans.Commun., vol. 41, no. 6, pp. 998–1008, Jun. 1993.[3] K. Rajamani, Y.-S. Lai, and C. W. 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