T the formulation of specifications arising from the application

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1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 40. NO 8. AGST METEOR: A Costrait-Based FIR Filter Desig Program Keeth Steiglitz, Fellow, IEEE, Thomas W. Parks, Fellow, IEEE, ad James F. Kaiser, Fellow, IEEE Abstract-The usual way of desigig a filter is to specify a filter legth ad a omial respose, ad the to fid a filter of that legth which best approximates that respose. I this paper we propose a differet approach: specify the filter oly i terms of upper ad lower limits o the respose, fid the shortest filter legth which allows these costraits to be met, ad the fid a filter of that order which is farthest from the upper ad lower costrait boudaries i a miimax sese. Previous papers have described methods for usig a exchage algorithm for fidig a feasible liear-phase FIR filter of a give legth if oe exists, give upper ad lower bouds o its magitude respose. The resultig filters touch the costrait boudaries at may poits, however, ad are ot good fial desigs because they do ot make best use of the degrees of freedom i the coefficiets. We use the simplex algorithm for liear programmig to fid a best liear-phase FIR filter of miimum legth, as well as to fid the miimum feasible legth itself. The simplex algorithm, while much slower tha exchage algorithms, also allows us to icorporate more geeral kids of costraits, such as cocavity costraits (which ca be used to achieve very flat magitude characteristics). We give examples that illustrate how the proposed ad the usual approaches differ, ad how the ew approach ca be used to desig filters with flat passbads, filters which meet poit costraits, miimum phase filters, ad badpass filters with cotrolled trasitio bad behavior. I. INTRODCTION HE FIR liear-phase filter desig problem begis with T the formulatio of specificatios arisig from the applicatio at had. Typical specificatios iclude the desired stop-bad atteuatio, passbad deviatio, locatio of zeros of trasmissio, etc. Two methods for desigig a filter to meet these specificatios iclude the approximatio approach ad the limit approach. I the approximatio approach, the legth of the filter ad a desired frequecy respose are specified. The filter coefficiets are determied to miimize the maximum weighted error betwee the desired ad actual resposes over the frequecy Mauscript received July 27, 1990; revised April 20, This work was supported i part by NSF Grat MIP by the.s. Army Research Office, Durham, uder Cotract DAAG29-85-K-0191, ad i part by ONR Cotract N A earlier versio of this paper was preseted at the Priceto Coferece o Iformatio Scieces ad Systems, Priceto, NJ, March K. Steiglitz is with the Departmet of Computer Sciece, Priceto iversity, Priceto, NJ T. W. Parks is with the Departmet of Electrical Egieerig. Corell iversity, Ithaca, NY J. F. Kaiser is with Bell Commuicatios Research, Morristow. NJ IEEE Log Number bads of iterest. I the limit approach, a set of upper ad lower limits are specified for the frequecy respose. The ecessary umber ad values of filter coefficiets for which the frequecy respose remais withi the prescribed limits are the determied. The limit approach was used i the earliest work o aalog filter desig more tha 50 years ago. Cauer [l] desiged aalog filters to meet prescribed, limit type tolerace schemes usig elliptic fuctios. It is possible to use the approximatio approach to meet limit costraits ad to use the limit approach described i this paper to solve approximatio problems. I 1970, Herrma published a article describig the equatios which must be solved to obtai a filter with the maximum possible umber of equal ripples [2] (later called extra-ripple [3] or maximal-ripple [4] filters). This maximal ripple desig is either a approximatio approach or a limit approach. Rather, it is a hybrid approach where the filter legth ad ripple size (equivalet to limits o the frequecy respose) are specified ad the badedges are determied by the algorithm. Schussler, i 1970, preseted the work he ad Herrma had bee doig o the desig of maximal-ripple filters at the Arde House Workshop [5]. Hofstetter developed a efficiet algorithm for solvig the equatios proposed by Herrma ad Schussler ad preseted papers with Oppeheim ad Siege1 at the 1971 Priceto Coferece [6] ad the 1971 Allerto House Coferece [7] describig the algorithm ad relatig it to the Remes exchage algorithm. Several papers o the Chebyshev approximatio approach to filter desig appeared at about the same time. Helms, i [8], described techiques, icludig liear programmig, to solve the Chebyshev approximatio problem for filter desig. Parks ad McClella used the Remes exchage algorithm [9], [lo] to solve the Chebyshev approximatio problem. Hersey et al., described, at about the same time, a iteractive method for desigig filters with upper ad lower costraits o the magitude of the frequecy respose [ 111. The limit approach was also used by Mc- Callig ad Leo i 1978 [ 121 ad by Greez i 1983 [13]. Whe a low-pass filter is desiged usig the Chebyshev approximatio approach, the five iterrelated parameters are the filter legth N, the passbad edge Fp, the stopbad edge F,, the passbad error S,, ad the stopbad error 6,. Relatios amog these parameters have bee determied umerically for the Chebyshev approximatio problem X/92$ IEEE

2 ~ 1902 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. NO. 8, AGST 1992 ad desig formulas have bee published [ 141, [ 151. With the help of these desig formulas it is possible to fix ay four of these parameters ad optimize the remaiig parameter. Sice these desig formulas are ot exact, several iteratios of the desig process are usually ecessary. For example, whe the badedges ad deviatios are give, a estimate of the ecessary filter legth ca be calculated usig the desig formulas. sually the filter with this estimated legth will ot be exactly the miimum legth required to meet the specificatios ad the filter will be desiged agai with a slightly differet legth util the miimum-legth filter is obtaied. The use of trasitio bads will give good low-pass desigs but may cause problems for multibad badpass filters [ 161. The frequecy respose is ot cotrolled i the trasitio bad ad may make large, uexpected, excursios which make the desig useless. The desig formulas ca be used to modify the stopbad specificatios to elimiate the uwated excursios i most cases, but the choice of stopbad edges ad appropriate error weightig fuctios is more of a art tha a sciece. The limit approach offers a way to avoid uwated excursios i multibad filter desig. pper ad lower limits are imposed o the respose for all frequecies. The limits imposed o the bads which otherwise would be urestricted trasitio bads elimiate the possibility of large peaks i the magitude of the frequecy respose, but do ot impose ay particular shhpe o the respose i these bads. I this paper we describe a very flexible desig program which combies most of the useful characteristics of the approximatio approach ad the limit approach to FIR filter desig. We use the simplex algorithm for liear programmig to fid the liear-phase filter of miimum legth which meets prescribed limits o the frequecy respose ad the maximize the distace from the costraits. For a fixed legth filter, the badedges ca be adjusted to maximize or miimize the width of a frequecy bad while still meetig prescribed limits o the frequecy respose. The bads ca cosist of just oe frequecy so that the locatio of the zeros ca be fixed i the stopbad. Additioal costraits, such as cocavity of the respose to give flat magitude characteristics, ca be imposed i appropriate frequecy bads. First, we describe the algorithm ad the Pascal program ad the we give examples to show how this ew approach ca be used i a variety of situatios. 11. THE ALGORITHM There are four differet types of liear-phase filters. For both eve ad odd symmetry of the impulse respose, we obtai liear phase with either eve or odd umber of coefficiets. I Rabier ad Gold [17] it is show that the frequecy respose for each of the four types of liearphase filters has the form where the real-valued amplitude fuctio A(Q) is a weighted sum of trigoometric fuctios, where N is the legth of the filter ad L = 0 or L = 1, depedig o the filter symmetry. For coveiece, i the followig discussio we assume that the filter model is the followig sum of cosies, correspodig to a odd-legth, eve symmetric impulse respose, although ay liear combiatio of kow fuctios ca be used: - 1)/2 A@,) = c uj cos (&I,). i=o A@,) is the real-valued frequecy respose of the filter at frequecy Q,, ad the frequecy poits at which specificatios are made, Q,, k = 1, 2, 3, * *, eed ot be equally spaced. A upper limit costrait at Qk has the form We itroduce a parameter y which represets the distace betwee the frequecy respose ad the upper boud, so that some of the costraits look like A(Q,) + 4 I (Q2,). Sice we are maximizig y, we call those costraits which have y i them optimized costraits, ad those that do ot, hugged costraits. Similarly, lower bouds o the frequecy respose result i costraits of the form or depedig o whether the costrait is hugged or optimized. Puttig costraits o the secod derivative of the frequecy respose has bee show to be a effective way to obtai filters that are very flat [15]. The secod derivative is a liear fuctio of the coefficiets, amely, for the case 1 filters cosidered here m-1 = - I c = ) i2ai cos (iq2,) so that cocavity ca be writte as liear iequalities of the form A (Q2,) I 0 for a cocave dowward fuctio, or A (Q2,) L 0 for a cocave upward fuctio. Whe all the costraits are writte dow, we obtai the liear programmig problem subject to (PRIMAL) max y CTu + hy 5 b

3 STEICLITZ et al.: METEOR: A CONSTRAINT-BASED FIR FILTER DESIGN PROGRAM I903 where the matrix C is determied from the sampled trigoometric fuctios, the vector a is made up of the coefficiets a;, the vector b cotais the bouds, ad the vector h has a 1 wherever a costrait is optimized, ad a 0 wherever it is hugged. The variables a ad y are urcostraied i sig. We will call this the PRIMAL problem. The dual of this liear program is i stadard form, the most coveiet for umerical solutio: subject to (DAL) mi b?r Cx = 0, htx = 1, adx L 0. We solve DAL usig the stadard two-phase simplex algorithm [ 181. Phase I searches for a feasible solutio to DAL, startig from a artificial basis, ad phase I1 searches for a optimal solutio. It is a fudametal fact of liear programmig theory that the cost fuctio of the DAL always satisfies b x 1 y, the cost fuctio of the PRIMAL, with equality if ad oly if x ad y are both optimal i their respective programs. Therefore, if the DAL cost b Tx ever falls below zero durig pivotig, the optimal PRIMAL cost must be egative. This meas that the origial filter approximatio problem is ifeasible, ad we stop the simplex algorithm wheever this coditio is obtaied. Applicatio of the simplex algorithm to the DAL problem therefore termiates i oe of the followig coditios: a) Negative cost reached, implyig that the origial desig problem is feasible; b) Optimality is reached i DAL with oegative cost, i which case the origial desig problem has a feasible solutio; c) DAL is ubouded, which implies that PRIMAL (ad the origial desig problem) is ifeasible; d) DAL is ifeasible, which implies that PRIMAL (ad the origial desig problem) is either ifeasible or ubouded. A commet is i order as to why the variable y is itroduced i those situatios whe we are iterested oly i whether there is a feasible solutio to lower ad upper boud costraits. Computatioal experiece has show that with a trivial cost fuctio i the primal, the simplex method applied to the dual sometimes cycles i realistic filter-desig problems, because of degeeracy. A otrivial cost fuctio seems to provide eough directio to the simplex algorithm to avoid such stagatio. Rather tha take special precautios to avoid cyclig, we chose always to maximize the distace y from the respose to the costrait boudaries. (As we saw above, it is ot always ecessary to complete the optimizatio whe the origial problem is ifeasible.) This has the additioal advatage of beig useful for the fial desig whe the legth is kow, ad also does ot iterfere with the resolutio of ties based o size of the pivot elemets, which is importat for umerical stability (see [19]). A special case arises uavoidably, however, whe there are o costraits desigated as optimized. I that case, h = 0 ad DAL is always ifeasible. However, the costrait matrix of DAL i this case is ot of full rak, havig a zero row, ad phase I eds with a artificial basis elemet remaiig i the basis. The redudat row is disregarded i phase 11, ad the optimizatio fids a solutio to the origial problem (if ay exist) with zero cost, correspodig to a respose that is allowed to touch ay of the costrait boudaries. Thus, the algorithm fuctios i a useful way, eve if a zero row is preset i the DAL costrait matrix. The optimal value of the dual variable x has a wellkow ad iterestig iterpretatio. Suppose the costrait values b are chaged a small amout to b + db. This chages the cost fuctio i the dual a small amout, but will ot i geeral chage the optimal solutio x to the dual. The ew value of the optimal cost fuctio becomes y = b Tx + db Tx. Thus, x is the partial derivative of the optimal value of y with respect to the costrait values b. Simplex fids a optimal value for x that has at most m + 1 positive etries, ad, by complemetary slackess. each of these correspods to a extremum of the distace betwee the frequecy respose ad costraits (a ripple ) i the case of a upper or lower boud, or to a poit where the secod derivative is zero i the case of a cocavity costrait. The simplex algorithm is used i the followig three modes, depedig o what desig task is desired: a) Give m, < m2, fid the miimum-legth r betwee them such that the origial desig problem is feasible (that is, such that DAL has a oegative optimal solutio), ad optimize y for that miimum legth; b) Solve the origial optimizatio problem for fixed legth mo; c) Give a particular right (left) badedge ad a set of costraits i which it occurs, fid the largest (smallest) value for that badedge for which the origial desig problem is feasible, ad optimize y for that badedge value. (The optimum value of y will i geeral be positive because the badedge value is rouded to the earest gridpoit.) What is the best search strategy to use i fidig the miimum legth i a)? We might expect, because the cost of testig feasibility icreases with m, that the strategy with least expected cost (assumig uiformly distributed aswers) probes to the left of the midpoit betwee the curret left ad right boudaries. However, computatio of the optimal strategies for probe-cost fuctios that grow as a low-order polyomial i m shows that biary search is surprisigly ear optimal. More work o this problem is i progress [20], but biary search appears adequate for this applicatio. Mode b) allows us to do thigs like fid

4 19W IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 8, AGST 1992 the best stopbad rejectio, while keepig passbad ripple withi limits. Mode c) allows us to do thigs like exted the ed of a stopbad as far as possible, while keepig the other costraits fixed. Biary search is also used i c) THE PROGRAM The algorithm described above was implemeted i Pascal, ad the curret versio is available from the authors. The authors itet is that the program be read ad modified by users, rather tha used as a static package, ad Pascal seems well suited to this purpose: it is widely available, clealy desiged, allows careful structurig, ad hopefully, good readability. As might be expected, the critical parts of the program ivolve the treatmet of tests which theoretically determie whether quatities are positive, egative, or zero. These tests determie whe each of the various termiatio coditios is reached, ad roudoff error requires us to decide o how small a positive umber is cosidered zero, how small a egative umber is cosidered egative, ad so o. Experiece has show that a sigle parameter eps ca be used for these tests at several differet places i the program, ad that eps ca be fixed at lo- for the rage of problems used as examples i this paper. The oly cases observed so far where serious accumulatio of roudoff error occurs is whe a wide bad of frequecies is ucostraied, ad the frequecy respose is allowed to grow very large i those bads, say as large as lo6. The problem is maifested by the cost i phase I reachig relatively large egative umbers before detectig optimality, eve though the cost i phase I is theoretically oegative. Of course, these desigs are impractical, ad the accuracy problem irrelevat, but the program cotiues to fuctio i these cases. Tradig off space for ruig time is a serious issue i the program desig. At oe extreme, we ca precompute ad store the tableau etries, which avoids recomputatio, but uses a great deal of storage. At the other extreme, we ca geerate the tableau etries o the fly, usig the least space, but the most time. As a compromise betwee the two, we ca precompute ad store tables of the trigoometric fuctios used for the tableau etries. We chose the first alterative because it appears that executio time is a more serious limitatio tha storage for the kids of desig problems likely to be solved. If storage is a serious problem, refereces to the tableau etries must be replaced by procedure calls that compute the required values. IV. EXAMPLES We preset a series of eight examples illustratig various features of the algorithm. Two separate programs are Pascal ad C versios of source code are available to aoymous ftp at priceto.edu i the directoryipub as meteor.p, form.p, mete0r.c. ad form. c. used to desig a filter. The program FORM is a iteractive program which requests iformatio from the user ad creates a iput file for METEOR which solves the liear programmig problem. The desired frequecy respose is specified by two kids of specificatios: limit ad cocavity. We call these limit specificatios ad cocavity specificatios. Limit Specz$catios: Each limit specificatio cosists of the followig iformatio: Iformatio pper or lower? Left badedge, right badedge Boud at left edge, boud at right edge Hugged or ot hugged? Arithmetic or geometric iterpolatio? Form + or - IF,. F21, real [B,, BJ, real h or &*a or 9 A upper boud o the frequecy repsose is idicated by a +, ad a lower by a -. The left ad right badedges, F, ad F2, are expressed i uits of cycles/ samplig iterval, so that the Nyquist frequecy correspods to 0.5. The frequecy respose is costraied by the value BI at the left badedge, ad by B, at the right badedge; the values i betwee are iterpolated by the program either arithmetically (liearly) ( a ), or geometrically ( g ), (liearly i decibels). Fially, if a limit specificatio is hugged, it is ot icluded i the optimizatio criterio of the fial liear program, ad is icluded if it is ot hugged. Thus, the fial desig is pushed away as much as possible from those limit specificatios that are ot hugged, but may be arbitrarily close to the hugged limit specificatios. Cocavity Spec@catios: Each cocavity specificatio is determied by the followig iformatio: Iformatio Cocave up or dow Left badedge, right badedge Form + or - [F,, Fz], real The frequecy respose is costraied to be cocave up or dow i the idicated bad. Mode: The desig program has three modes, miimum-legth, optimize, ad push. I the miimum-legth mode, the miimum legth that satisfies the give costraits is foud. The user specifies either eve or odd legth ad either eve or odd symmetry of the impulse respose. I the optimize mode, the respose is pushed away from the ohugged costraits for the fixed legth specified by the user. If the desig is ot realizable at all for this fixed legth, the program reports ifeasibility. I the push mode, a set of badedges are pushed as far as possible while still respectig the costraits for the fixed legth specified by the user. The set is pushed either to the left or the right. I the followig examples we first display the specifi-

5 STEIGLITZ er al.: METEOR: A CONSTRAINT-BASED FIR FILTER DESIGN PROGRAM 1905 catio file i the format produced by the program FORM, the graph the resultig frequecy respose. Example I: Low-Pass, Miimum-Legth Filter: Here we use four limit specificatios, which are displayed as follows by FORM: \ -I 1 # Type Sese edgel edge2 boudl boud2 hugged? iterp 1 limit limit limit limit FINDING MINIMM LENGTH ODD LENGTHS from 7 to 2 1 COSINE MODEL (eve symmetric coefficiets) 201 grid poits The lower ad upper limits NI ad N2 (7 ad 21 i this case) for the filter legth are estimated usig formulas developed i [14] ad [15]. Sice we are specifyig limits for the real-valued amplitude fuctio which may be egative we must specify a egative lower limit i the stop bad. Fig. 1 shows the resultig amplitude respose; the miimum legth satisfyig the specificatios is 17. Note i Fig. 1 that sice the costraits are ot hugged the optimized respose is strictly withi the limits. The resultig equiripple respose is equivalet to that obtaied with the Parks-McClella algorithm. I Fig. 1 we have show the amplitude respose to clearly display the egative as well as the positive limits. For the remaiig examples we will display the magitude of the frequecy respose. Example 2: Flat Passbad, Low-Pass, Miimum-Legth Filter: Suppose we wat a low-pass filter with the same badedges as i example 1, but we wat the passbad to be flat. Oe simple way to do this is to add a cocavity specificatio that forces the frequecy respose to be cocave dow ("-") i the passbad. We ca also relax the upper limit specificatio i the passbad to be hugged, ad chage the upper limit to 1.0, so that the frequecy respose ca decrease mootoically i the passbad from a value of 1. The ew specificatios are show below. # Type Sese edgel edge2 boudl boud2 hugged? iterp 1 limit h 2 limit limit limit cocave FINDING MINIMM LENGTH ODD LENGTHS from 2 1 to 3 1 COSINE MODEL (eve symmetric coefficiets) 201 grid poits The resultig frequecy respose, show i Fig. 2, has a zero frequecy gai of exactly 1 because the upper limit i the passbad is hugged. The stopbad has the same upper ad lower limits as example 1. The price we pay for the flat passbad is a icrease i filter legth from N = 17 for example 1, to N = 29 for this example O! C frequecy Fig. 1. Frequecy respose for example I, a legth-17 low-pass filter. 1.2, 1 frequecy Fig. 2. Legth-29 filter with mootoically decreasig passbad respose. Example 3: Flat Passbad, Miimum-Phase Filter: If a miimum-phase filter is desired with the same magitude performace as the liear-phase filter i Example 2, the factorizatio approach of Herrma ad Schiissler [21] ca be used begiig with the legth-43 filter which resulted from the followig specificatios: # Type Sese edgel edge2 boudl boud2 hugged? iterp 1 limit + O.Oo h a 2 limit a 3 limit a 4 limit h 5 cocave FINDING MINIMM LENGTH ODD LENGTHS from 37 to 55 COSINE MODEL (eve symmetric coefficiets) 201 grid poits The lower limit of 0.81 i the passbad ad the upper limit of 0.01 i the stopbad are used i aticipatio of the square root ivolved i the miimum-phase desig, while the lower limit of 0.0 i the stopbad guaratees a i

6 1906 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 8, AGST 1992 o-egative respose. Half of the 42 roots of the legth-43 filter, 10 roots iside the uit circle ad 1 each of the 11 double roots o the uit circle, are retaied to give the legth-22 miimum-phase filter with respose show i Fig. 3. This miimum-phase filter, legth-22 is slightly shorter tha the liear-phase, legth-29, filter of example 2 which meets the same magitude specificatios. Example 4: Poit Costraits with a Flat Passbad Filter: If there are specific frequecies i the stopbad where zeros are desired to ull out iterferece, the followig specificatios which require zeros at frequecies of 0.3 ad 0.4 would be appropriate. 12 I # Type Sese edgel edge2 boudl boud2 1 limit limit limit limit cocave limit I limit limit limit FINDING MINIMM LENGTH ODD LENGTHS from 2 1 to 3 1 COSINE MODEL (eve symmetric coefficiets) 201 grid poits hugged? iterp h I this case there was o icrease i legth over the legth of 29 i example 2, required to meet these additioal poit costraits; the zeros of the respose were simply shifted as show i Fig. 4. Geerally, however a icrease i legth would be required to meet these additioal costraits. Example 5: Partial-Bad Differetiator, Pushig the Stopbad: Suppose ext we wat a liearly icreasig magitude respose, followed by rejectio at higher frequecies. We kow we wat a liearly icreasig respose up to 0.25 cycles/sample, ad we wat as wide a stopbad as possible with a legth of 16. We do this by specifyig the differetiatig bad by a upper costrait liearly iterpolated from 0.01 to 0.26 that is optimized (pushed away from), ad a lower costrait from 0.0 to 0.25 that is huqged. The left badedges of the upper ad lower stopbad costraits are the pushed left i the mode push. a a 1-, frequecy Fig. 4. Legth-29 filter with poit costraits # Type Sese edgel edge2 boudl boud2 hugged? iterp 1 limit limit h 3 limit a 4 limit a PSHING 2 BANDEDGES LEFT, fixed legth = 16, bads: 3 4 SINE MODEL (odd symmetric coefficiets) 201 grid poits The resultig badedge is , ad Fig. 5 shows the frequecy respose. Note that the badedges i the stopbad have bee pushed to lower frequecies as far as possible util the frequecy 5. Frequecy respose for example 5, a legth-16 low-pass differetiator with a miimum-width trasitio bad. costraits are hugged. The specificatios of ay oe badedge for ay type of filter, low-pass, badpass, etc., ca be pushed i this maer.

7 STElGLlTZ c f al.: METEOR: A CONSTRAINT-BASED FIR FILTER DESIGN PROGRAM 1.2, I907, frequecv Fig. 6. Legth-25 badpass filter with a trasitio-bad excursio frequecy Fig. 7. Legth-27 badpass filter with trasitio limits. Example 6: Badpass Filter: This example shows how to fid the miimum-legth liear-phase filter which meets the frequecy specificatios listed below ad has well-behaved trasitio bads i # Type Sese edge1 edge2 boud1 boud2 hugged? iterp 1 limit limit ti 3 limit ti (I 4 limit ,900,900 ti 5 limit limit FINDING MINIMM LENGTH ODD LENGTHS from 21 to 29 COSINE MODEL (eve symmetric coefficiets) 201 grid poits The iitial desig usig METEOR, maximizig the distace from all the costraits, produces a filter of legth 25, the shortest legth that meets these specificatios, ad a deviatio of The frequecy respose for this desig is show i Fig. 6, ad is essetially the same as that produced by the Parks-McClella program [4]. O the scale of Fig. 6, there appears to be a problem i the trasitio bad, but i the bads where the Chebyshev error was miimized, the respose looks good. To elimiate the trasitio bad excursio, ew limits were itroduced which costraied the respose i the first trasitio bad to lie betwee ad The ew algorithm foud that the filter legth must be icreased to 27 i order to meet these ew, stricter, limits. The respose of this legth-27 filter is show i Fig. 7. As i Fig. 6, the distace from all the origial costraits is maximized, but the respose is allowed to touch the ew costraits. Aother way to elimiate the trasitio bad peak is to fix the legth at 27 ad push the upper edge of the lower stopbad to the right, maximizig the width of the first stopbad, thus reducig the width of the first trasitio bad ad elimiatig the trasitio bad peak. The badedge foud is cycles/sample, ad correspods to I I I frequecy Fig. 8. Legth-27 badpass filter with miimum-width trasitio bad. a deviatio of The resultig frequecy respose is show i Fig. 8. V. TIMING COMPARISON WITH THE PARKS-MCCLELLAN PROGRAM The Parks-McClella program rus much faster tha METEOR, as we would expect give METEOR S greater geerality, ad the fact that METEOR uses the simplex algorithm istead of the Remes exchage algorithm. However, the ruig time of METEOR o preset-day computers is ot prohibitive eve for reasoably large problems. To illustrate this, we give some timig comparisos o a SPARCstatio 1+ (Model 4/65) usig a f 77 compiler at optimizatio level 03, ad a Pascal compiler at optimizatio level 2. The examples ru were simple fixed-legth lowpass filters of legth L, with L = 2, i = 4, * - a, 8; passbad [0, 0.11; ad stopbad [x, 0.51, where x = 0.1 * (1 + 2(4- )). Thus the filters were of legth-16 with stopbad [0.2, 0.51; legth-32 with stopbad [O. 15, 0.51; etc. The left edge of the stopbad was moved left as the filter was made loger to keep the deviatio from specificatios

8 1908 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 8, AGST 1992 TABLE I TIMING COMPARISON SPARCstatio 1 + Model 4165 Parks-McClella Meteor Meteor (covex passbad) Legth Deviatio CP secods Deviatio CP secods Deviatio CP secods I I roughly costat over the examples. The umber of grid poits was kept comparable i the programs by choosig grid desity 10 i the Parks-McClella program, ad usig 10 * (L/2) + 1 grid poits i METEOR. The upper ad lower bouds i the passbad were 1.5 ad 0.5; ad 0.5 ad -0.5 i the stopbad. Table I shows the optimal deviatios from 1 i the passbad (or 0 i the stopbad) foud by each program, as well as the user CP times. The deviatios of the two programs check to six sigificat figures. The Parks-McClella program is clearly much faster, by a factor icreasig with filter legth, to a factor of 16 for the legth-128 problem, ad 47 for the legth-256 problem. However, the CP time of about 3 mi o a modem workstatio for a legth-256 filter would hardly be prohibitive i most situatios. Of course, whe the miimum legth is sought, with o prior estimate, biary search o the legth will result i as may as 8 = log (256) istaces of optimizatios. Also show are the deviatios ad ruig times for the same problems, usig METEOR, but with cocave-dow passbads i [0, 0. I]. The passbad was specified by a sigle-poit upper boud at the left-edge, a sigle-poit lower boud at the right edge, ad a cocavity costrait i the etire bad. The upper ad lower bouds are redudat withi the passbad, ad this strategy reduces the umber of colums geerated by METEOR. The ruig times are roughly the same as those for the traditioal desig, ad the results illustrate the price paid i icreased deviatio by costraiig the passbad to be cocave dow. VI. CONCLSION A ew approach to filter desig, usig the simplex method of liear programmig, was proposed which is very geeral ad ca icorporate a wide variety of costraits o the frequecy respose of the filter. Several examples were preseted to illustrate the wide rage of applicatios of this approach to liear-phase filter desig. We are presetly workig o extesios of this approach to the desig of filters with costraits o group delay ad/or phase as well as magitude. ACKNOWLEDGMENT The authors would like to thak Prof. H. W. Schiissler for his careful review of this mauscript ad for his helpful suggestios. REFERENCES Berli: V. D. I. Verlag G.M.B.H., [I] W. Cauer, Siebschaltuge Herrma, Desig of orecursive digital filters with liear phase, Electro Lett., vol. 6, pp , May 28, 1970; also reprited i Digital Sigal Processig, L. R. Rabier ad C. M. Rader, Eds. New York: IEEE, [3] T. W. Parks, L. R. Rabier, ad J. H. McClella, O the trasitio width of fiite impulse-respose digital filters, IEEE Tras. Audio Electroacoustic., vol. A-21, pp. 1-4, Feb [4] L. R. Rabier, J. H. McClella, ad T. W. Parks, FIR digital filter desig techiques usig weighted Chebyshev approximatio, Proc. IEEE, vol. 63, o. 4, pp , Apr [5] 0. Herrma ad H. W. Schiissler, O the desig of orecursive digital filters, preseted at the IEEE Arde House Workshop Digital Filterig, Ja. 12, [6] E. M. Hofstetter, A. V. Oppeheim, ad I. Siegel, A ew techique for the desig of orecursive digital filters, i Proc. Fifth Au. Priceto Coj Iform. Sci. Syst. (Priceto, NJ), Mar. 1971, pp ; also reprited i Digital Sigal Processig, L. R. Rabier ad C. M. Rader, Eds. New York: IEEE, (71 E. M. Hofstetter, A. V. Oppeheim, ad J. Siegel, O optimum orecursive digital filters, i Proc. Nith Au. Allerto Co$ Circuits Sysr. Theory, Oct. 1971, pp ; also reprited i Dig- ital Sigal Processig, L. R. Rabier ad C. M. Rader, Eds. New York, NY: IEEE, [8] H. D. Helms, Digital filters with equiripple or miimax resposes, IEEE Tras. Audio Electroacoust., vol. A-19, pp , Mar. 1971; also reprited i Digital Sigal Processig, L. R. Rabier ad C. M. Rader, Eds. New York: IEEE, T. W. Parks, Extesios of Chebyshev approximatio for fiite impulse respose filters, preseted at the IEEE Arde House Workshop Digital Filterig, Ja. 10, [IO] T. W. Parks ad J. H. McClella, Chebyshev approximatio for orecursive digital filters with liear phase, IEEE Tras. Circuit Theory, vol. CT-19, pp , Mar [I I] H. S. Hersey, D. W. Tufts, ad J. T. Lewis, Iteractive miimax desig of liear phase orecursive digital filters subject to upper ad lower fuctio costraits, IEEE Tras. Audio Electroacoust., vol. A-20, pp , Jue [I21 M. T. McCallig ad B. J. Leo, Costraied ripple desig of FIR digital filters, IEEE Tras. Circuits Syst., vol. CAS-25, pp , Nov [ 131 F. Greez, Costraied Chebyshev approximatio for FIR filters, i Proc Ir. Cof: Acousr., Speech, Sigal Processig (Bosto, MA), pp Hema, L. R. Rabier, ad D. S. K. Cha, Practical desig rules for optimum fiite impulse respose low-pass digital filters, Bell Syst. Tech J., vol. 52, pp , July-Aug [15] J. F. Kaiser ad K. Steiglitz, Desig of FIR filters with flatess costraits, i Proc IEEE It. Co$ Acoust., Speech, Sigal Processig (Bosto, MA), Apr , 1983, pp L. R. Rabier, J. F. Kaiser, ad R. W. Schafer, Some cosideratios i the desig of multibad fiite-impulse-respose digital filters, IEEE Tras. Audio Electroacoust., vol. ASSP-22, pp , Dec L. R. Rabier ad B. Gold, Theory ad Applicatio of Digital Sigal Processig. Eglewood Cliffs, NJ: Pretice-Hall, 1975, pp C. H. Papadimitriou ad K. Steiglitz, Combiatorial Oprimizario: Algorithms ad Complexity. Eglewood Cliffs, NJ: Pretice-Hall, 1982.

9 STEIGLITZ er al.: METEOR: A CONSTRAINT-BASED FIR FILTER DESIGN PROGRAM 1909 [19] K. Steiglitz. Optimal desig of FIR digital filters with mootoe passbad respose, I Tras. Acousr., Speech, Sigal Processig, vol. ASSP-27. o. 6, pp , Dec [20] W. J. Kight, Search i a ordered array havig variable probe cost, SIAMJ. Comput., vol. 17. o. 6, pp, , Dec [21] 0. Herrma ad H. W. Schussler, Desig of orecursive digital filters with miimum phase, Electro. Lett., vol. 26, o. I I, May 28, 1970; also reprited i Digital Sigal Processig, L. R. Rabier ad C. M. Rader, Eds. New York: IEEE, [22] K. Steiglitz ad T. W. Parks, What is the filter desig problem?, i Proc Priceto Cof. Iform. Sci. Syst. (Priceto. NJ), Mar. 1986, pp Keeth Steiglitz (S 57-M 64-SM 79-F 8 I) was bor i Weehawke, NJ, o Jauary 30, He received the B.E.E. (maga cum laude), M.E.E.. ad Eg.Sc.D. degrees from New York iversity, New York, i 1959, 1960, ad 1963, respectively. Sice September 1963 he has bee at Priceto iversity, Priceto, NJ, where he is ow Professor of Computer Sciece, teachig ad coductig research o parallel architectures. sigal Drocessip. odtimizatio algorithms. ad cellular automata. He is the author of Itroductio to Discrete Systems (New York: Wiley, 1974), ad coauthor, with C. H. Papadimitriou, of Combiatorial Optimizatio: Algorithms ad Complexity (Eglewood Cliffs, NJ: Pretice- Hall, 1982). Dr. Steiglitz served two terms as a member of the IEEE Sigal Processig Society s Admiistrative Committee. as Chairma of its Techical Directio Committee, a member of its VLSI Committee, its Digital Sigal Processig Committee, ad as its Awards Chairma. He is a Associate Editor of the joural Networks, ad is a former Associate Editor of the Joural of the Associatio for Computig Machiery. A member of Eta Kappa Nu, Tau Beta Pi, ad Sigma Xi, he received the Techical Achievemet Award of the Sigal Processig Society i its Society Award i 1986, ad the IEEE Ceteial Medal i Thomas W. Parks (S 66-M 67-SM 79-F 82) received the Ph.D. degree from Corell iversity i He joied Rice iversity, Housto, TX, i 1967, where he was a Professor of Electrical Egieerig util He the became Professor of Electrical Egieerig i the School of Electrical Egieerig at Corell iversity. He also serves as a cosultat to idustry. He is the coauthor of a book o the fast Fourier trasform ad a book o digital filter desig, both published by Joh Wiley ad Sos. He is the coauthor of laboratory mauals for digital sigal processig usig the TMS32010 ad the TMS320C25, published by Pretice-Hall. He is egaged i research o sigal theory ad digital sigal processig. His research iterests are i the areas of time frequecy ad wavelet aalysis, sigal recostructio, array processig for soar ad seismic applicatios, digital filter desig, patter classificatio, ad eural etworks. Prof. Parks received the Society Award ad the Techical Achievemet Award from the IEEE Acoustics, Speech, ad Sigal Processig Society, as well as awards for papers published i the IEEE TRANSACTIONS ACOSTICS, SPEECH, AND SIGNAL PROCESSING. He has bee a member of the Admiistrative Committee of that society ad a Associate Editor for the IEEE TRANSACTIONS ACOSTICS, SPEECH, AND SIGNAL PROCESSING. James F. Kaiser (S 5O-A 52-SM 7O-F 73) was bor i Piqua, OH, i He received the E.E. degree from the iversity of Ciciati, Ciciati, OH, i 1952 ad the S.M. ad Sc.D. degrees i 1954 ad 1959, respectively, from M.I.T., Cambridge, MA, all i electrical egieerig. Curretly he is a Distiguished Member of the Techical Staff i the Speech ad Image Processig Research Divisio of Bell Commuicatios Research, Ic., which he joied i He was formerly a Member of the Techical Staff at Bell Laboratories, Murray Hill, NJ, for 25 years where he worked i the areas of speech processig, system simulatio, digital sigal processig, computer graphics, ad computer-aided desig. He is the author of more tha 50 research papers ad the coauthor ad editor of seve books i the sigal processig ad automatic cotrol areas. Dr. Kaiser is a Registered Egieer i Massachusetts, a member of ASA, AAAS, ACM, ERASIP, ad SIAM. He is a member of Sigma Xi, Tau Beta Pi, ad Eta Kappa Nu. He has served i a umber of positios i both the Sigal Processig ad the Circuits ad Systems Societies.