A Novel Approach for Testing Stability of 1-D Recursive Digital Filters Based on Lagrange Multipliers

Transcription

1 Amercan Journal of Appled Scence 5 (5: , 8 ISSN Scence Publcaton A Novel Approach for Tetng Stablty of -D Recurve Dgtal Flter Baed on Lagrange ultpler KRSanth, NGangatharan and Ponnavakko Faculty of Engneerng, Kgal Inttute of Scence and Technology (KIST, Kgal, BP39, Rwanda and Drectorate of Reearch and Vrtual Educaton, SR Unverty, SR Nagar, Kattangulathur, 633, TamlNadu, Kanchepuram Dt, Inda Abtract: Dgtal flter have found ther way nto many product from every day conumer tem uch a moble phone to advanced martme and mltary communcaton and avonc ytem Degn of dgtal flter face two fundamental problem, ther tablty and ynthe Recurve flter have more tablty problem than nonrecurve flter Stablty of a flter can be determned by the locaton of the zero valued regon of the denomnator polynomal of t tranfer functon Stablty of recurve flter ha been tuded by many reearcher for the pat three decade Several theorem on tablty tetng and tablzng recurve dgtal flter have been already propoed We preent a new approach to tet the tablty problem of the one-dmenonal (-D recurve dgtal flter ung Lagrange ultpler Th method not only tet the tablty of recurve dgtal flter, but alo provde the table veron of the flter tranfer functon f found to be untable Keyword: Dgtal Flter, Lagrange ultpler, Autocorrelaton, Stablty, Leat Square Invere INTRODUCTION Dgtal flter are ued n many dgtal gnal proceng (DSP feld, but the recurve veron of whch dffcult to degn due to tablty problem Several flterng applcaton, uch a n mage proceng or n communcaton, requre a lnear-phae charactertc Th can be ealy acheved wth fntempule repone (FIR flter, alo known a nonrecurve flter However, the reultng FIR flter order ometme exceedngly hgh, producng an unacceptable delay and computatonal cot The alternatve to ue nfnte-mpule repone (IIR flter known alo a recurve flter wth approxmate lnear-phae For a gven flter pecfcaton, the requred order for a recurve flter 5 to tme lower than that of a nonrecurve flter [] In recurve flter, the pole of the tranfer functon (equvalently, the zero of the denomnator polynomal of the flter tranfer functon can be placed anywhere nde the unt crcle n the z-plane A a conequence of th degree of freedom, hgh electvty flter can ealy be degned wth lower-order tranfer functon In nonrecurve flter, on the other hand, wth pole fxed a relatvely hgher order tranfer functon [] A general dere n any flter degn that the number of operaton needed to compute the flter repone a low a poble The order of a flter more or le proportonal to the number of operaton Recurve flter preferable, prmarly becaue of t lower order So t mplementaton nvolve fewer parameter, requre le memory and ha lower computatonal complexty than the nonrecurve counterpart [3] Th gve more flexblty and power to mplement hgh electvty recurve flter ung maller number of coeffcent meetng a partcular magntude pecfcaton, but reult n tablty problem n both degn and mplementaton A recurve dgtal flter defned to be Bounded- Input-Bounded-Output (BIBO table f every bounded nput reult n a bounded output Stablty often a derable contrant to mpoe nce an untable ytem can generate an unbounded output, whch can caue ytem overload and other dffculte [4, 6] The tablty tetng and tablzaton of -D recurve dgtal flter ha been tuded by everal reearcher [, 5] In th paper a new method of tablty tetng and tablzng -D recurve dgtal flter baed on Lagrange at orgn, hgh electvty can be acheved only by ung multpler obtaned Correpondng Author: KRSanth, Department of Computer Engneerng and Informaton Technology, Faculty of Engneerng, Kgal Inttute of Scence and Technology (KIST, PO Box 39, Rwanda 49

2 -D RECURSIVE DIGITAL FILTER STABILITY THEORES AND STABILIZATION ETHODS In th ecton, the varou tablty theorem and tablzaton method on -D recurve dgtal flter are revewed and compared Recurve flter are convenently decrbed n term of -D z-tranform It aumed throughout th ecton that z-tranform defned wth negatve power of z Under th aumpton, tablty mple that all pole of the flter tranfer functon mut be wthn the unt crcle Conder the tranfer functon H (z of a certan - D recurve dgtal flter gven by H ( z = = ( z a k k z k= One method of tetng the tablty of the flter decrbed by the tranfer functon a n ( to explctly determne the zero of A (z by a root fndng method [3, 6] and check f all the zero are nde the unt crcle of the complex z-plane If o, the ytem decrbed by the tranfer functon ( table ele untable Th method conceptually very mple, and computatonally effcent for low value of where, the order of the denomnator polynomal of ( For value of up to = 4, cloed-form expreon can be ued to determne the zero Even f the cloedform expreon are not avalable, the zero determned n one tet can be ued a ntal etmate for the zero n the ubequent tet Th becaue the zero obtaned n the -D tablty tet typcally do not vary much from one tet to the other Th method ha been reported [] to be qute ueful n practce for = 8 The explct evaluaton of the zero ung th method provde u conderable nght nto exactly how table a table flter If none of the zero determned very cloe to the unt crcle, the flter ad to be table [3, 6] The econd method of tetng the tablty of the flter decrbed by the tranfer functon ( to ue arden-jury Tet [6] whch proven to be computatonally effcent Th method ha been reported to be computatonally effcent and relable for up to that cover mot recurve flter condered n practce [] However, the dadvantage beng t doe not tell u how table a flter The method can be ued n determnng the number of zero nde the unt crcle, but cannot be ued explctly to determne the locaton of the zero Am J Appled Sc, 5 (5: , 8 49 The thrd method of tetng the flter for tablty to explot the Argument Approach [] Conder the net change n the argument of A (z n ( a we follow the unt crcle contour gven by z = e from ω = to ω = π n a counterclockwe drecton Denotng the net argument change by θ A ( ω :, π, the argument approach tate that θ ω :,π = π ( ( A ( N z where, N Z the number of zero nde the unt crcle When all root are nde the unt crcle, N Z = and therefore, θ A ( ω :,π = (3 From (3 t clear that, to tet the tablty to check f the net phae change zero The net phae change can be determned by unwrappng the phae If the unwrapped phae contnuou and the unwrapped phae at ω = and ω = π are dentcal, the ytem table, ele t untable Th method ha been reported to be relable and computatonally effcent n comparon to other method for > [] It alo noted that the phae unwrappng qute dffcult when ome root are very cloe to the unt crcle The unwrapped phae ha ome qualtatve feature that can be related to the degree of ytem tablty For a table flter, rapd change n the unwrapped phae typcally occur when ome zero are cloe to the unt crcle In -D ytem there are two tandard approache to degn recurve flter One approach to degn the flter from an analog ytem functon and the other to degn drectly n the dcrete doman The drect degn method further clafed nto patal doman and frequency doman method In the patal doman method, the flter are degned by ung an error crteron n the patal doman, converely n the frequency doman method the flter are degned ung an error crteron n the frequency doman The flter degned by both patal doman and frequency doman technque can tll be untable There are few method by whch the -D recurve dgtal flter can be tablzed The popular method are Leat Square Invere (LSI [5, 7] and Dcrete Hlbert Tranform (DHT method [8] It ha been hown [8] that DHT method of tablzaton wll yeld table polynomal f the orgnal polynomal doe not have [5, 9- zero on the unt crcle Lkewe, t well known ] that the LSI of a -D polynomal that doe not have zero on the unt crcle alway table Th fact can be utlzed for the effectve degn of table -D recurve dgtal flter In the proce of obtanng the table LSI jω

3 polynomal correpondng to an untable polynomal, nce the mnmum error [ 5,, 4] gven by Emn = ab (4 the value of b wll alway turn out to be alway the hghet value In th, a and b are the contant term of the orgnal and LSI polynomal repectvely It apparent that the table LSI polynomal wll have the hghet value for t contant term We ue th concept of maxmzng the contant term of a polynomal to tet the tablty or otherwe of a polynomal n the followng ecton SOE IPORTANT PRELIINARIES AND CONCEPTS Unlke the conventonal method of defnng z- tranform of a dcrete gnal [6], t aumed hereafter that potve power of z are ued n the defnton of z- tranform [, 3] Wth th aumpton, we have the followng Theorem [] Theorem : ( The -D polynomal A (z table f and only f A ( z, for all z ( A table A (z ad to be margnally table f A ( z =, for ome z = We alo have the followng defnton on autocorrelaton coeffcent [] Defnton: If z an th degree polynomal a n (5 the coeffcent γ of z z wrtten a z z γ ( z + z = γ + γ ( z + z + γ ( z Am J Appled Sc, 5 (5: , 8 + z + + are called the autocorrelaton coeffcent of A (z The coeffcent are (+ n number For a real polynomal A (z, γ are real Conder the tranfer functon of a certan recurve dgtal flter uch that H ( z = where, z a r a r+ = γ, =,,, (6 We would lke to pont out that for any gven -D polynomal A (z of order, there are number of -D polynomal n total (wthout countng the negatve of thee polynomal whch wll have the ame autocorrelaton coeffcent γ a that of z a hown n later ecton Out of thee number of -D polynomal, whch are ad to form a famly, only one polynomal wll lead to a table ytem It ha been hown that the autocorrelaton functon or coeffcent are related to magntude quare of ytem functon [5] Therefore t obvou all the number of polynomal wll have the ame magntude repone Out of th number of polynomal n total only one polynomal wll be table for whch the contant term wll have the hghet value a dcued n the foregong ecton The remanng polynomal although provde the ame magntude repone, may lead to untable ytem We now tate th n the form of a theorem Theorem : For any -D polynomal z of degree, there are dtnct -D polynomal wth ame autocorrelaton functon that form a famly Only one -D polynomal among the famly whoe contant term the hghet lead to a table ytem Baed on Theorem 3 we ue the Lagrange multpler to fnd out the table polynomal The proof of the above theorem hown n the followng ecton STABILITY TESTING AND STABILIZATION BASED ON LAGRANGE ULTIPLIERS In th ecton, a new method of tetng the tablty and tablzng -D recurve dgtal flter baed on Lagrange multpler explaned Th method am to maxmze the contant term of the -D polynomal and a decon regardng t tablty can be made dependng k upon whether the contant term of the -D polynomal z = a k z (5 (5 maxmzable or not To tet the tablty of a ytem k = be of degree It ha ( + autocorrelaton decrbed by the tranfer functon ( whoe denomnator polynomal A (z, we ue Lagrange functon γ a gven below: multpler 49

4 Let the table veron of A (z be A (z whoe coeffcent are denoted by r To how that the gven polynomal A (z table, we hould maxmze the contant term or multpler Let f be the functon to be maxmzed a Am J Appled Sc, 5 (5: , 8 of A (z by ung Lagrange f = (7 atfyng the contrant g, gven a g = + ar r γ =, =,,, (8 where, γ = arar+ =,,, that, g =, =,,, (9 The above contrant are framed due to the fact that the contant term a of one of the polynomal from the famly wll be equal to the contant term of the aumed table veron of the polynomal A (z f A (z the table polynomal among the famly Here, γ repreent the autocorrelaton coeffcent a n (6 of the orgnal polynomal A (z and γ = a r ar + and λ >, =,,,, ( 493 repreent the autocorrelaton coeffcent of the table veron A (z Let u now form the Lagrange Functon L( a, λ uch that L(, λ = f + λ g ( = where λ are the Lagrange multpler The neceary and uffcent condton for the functon f to be global maxmum and hence to be the greatet n magntude are gven ung Lagrange multpler method a follow [3] f + = g λ = ( For a gven polynomal A (z, all the coeffcent a r are known and hence γ are fxed If we want to tet whether z wll lead to a table ytem or not, we have to do the followng: Obtan the equaton L(, λ = Ung ( n (3 we get, L(, λ = + λ + + λ + (3 (4 + λ = Smplfyng (4, we get, + λ + + λ a = (5 ( + λ The contrant equaton n (8 can be expanded a follow: a + a + a + + a = γ a a + a a + + a a = γ = γ (6 a Subttute the value of whch wa obtaned from (5 n the expanded contrant equaton (6 above Th wll reult n (+ equaton nvolvng (+ number of λ and number of a unknown + λ + + ( + λ ( + λ + + ( + λ λ + a + a + + a λ ( + λ + + ( + λ λ = γ = γ = γ (7 Solve (7 for and λ In th proce all λ hould turn out to be potve and real atfyng ( and ( The value of and λ thu obtaned from (7 are ued to get the maxmum value of by

5 Am J Appled Sc, 5 (5: , 8 ung (5 If the maxmum value of the ame a a of the gven polynomal A (z, then we can conclude that z wll lead to a table ytem NUERICAL RESULTS In th ecton, we frt how that for any gven -D polynomal A (z, there are number of -D polynomal n total whch wll have the ame autocorrelaton coeffcent γ a that of A (z Conder a -D polynomal of econd degree (=, A ( z = z + 7z + 6, whch of the form A ( z = a z + a z + a There wll be 4 polynomal n total n the famly of A (z The autocorrelaton coeffcent of z are a follow: γ = a + a + a = = 89 γ = a a + a a = ( ( 56 = = aa = 6 = γ There are three other polynomal wthout countng the negatve of the polynomal n the famly for the gven A (z They are a follow: A ( z = 3z + 8z + 4 A ( z = 4z + 8z + 3 A 3 ( z = 6z + 7z + 3 It can be found that all thee three polynomal wll alo have the ame autocorrelaton coeffcent a that of A (z, e 89, 56 and A dcued n prevou ecton, the polynomal whch have the ame autocorrelaton coeffcent wll alo have the ame magntude repone and among th only one polynomal wll be table for whch the contant term the hghet (magntude we Now we have to how that the polynomal wth the hghet value for t contant term table It palpable from our dcuon n ecton IV that the Lagrange multpler method maxmze the contant term of the gven polynomal and hence n the proce of maxmzaton f turn out to be equal to a, then one can conclude that the gven polynomal table or not Conder agan the ame -D polynomal of econd degree A ( z = z + 7z + 6 (8 where a = 6, a = 7, a = Let u now check the tablty of th polynomal Let A ( z = z + z + (9 repreent the table veron of the gven polynomal A (z havng the ame autocorrelaton coeffcent a A (z The autocorrelaton contrant equaton of A (z ung (8 are a follow a a + + = + = 56 = 89 a ( By ung (5 n (, we get, ( + λ + + = 89 4( + λ + λ + = 56 ( + λ + λ = ( ( + λ Smplfyng ( we get, ( + λ = ( 89 = ( + λ ( 56 = It clear that (5 and ( can be only atfed only by a = 7 and a = Subttutng thee value n (5 we get, a 6 So the polynomal wth a 6, = = a = 7, a =, namely A z = z + 7 z + 6 a table polynomal In th cae, the gven polynomal found to be table For th polynomal A (z, we fnd that the zero are z = 3 / & z = ( A can be een, th -D polynomal A (z found to be table a all t zero le outde the unt crcle a ad n theorem 3 Now when we check the tablty of the remanng polynomal A (, A ( and z z A3 ( z that belong to the ame famly a that of A (z, we fnd that at leat one root of thee polynomal le nde the unt crcle and hence untable From th, t can be concluded that, out of thee four polynomal that belong to a famly havng the ame ACF and hence magntude repone, only one polynomal found to be table for whch the contant term the hghet It treed that the above procedure to tet the -D polynomal for tablty applcable only f the polynomal nonlacunary, n the ene there are no mng term between the hghet degree and the contant term When the gven 494

6 Am J Appled Sc, 5 (5: , 8 polynomal lacunary, we have to make t nonlacunary and follow the ame procedure CONCLUSIONS In th paper, we propoed a new method for tetng the tablty and tablzng -D recurve dgtal flter baed on autocorrelaton functon and Lagrange multpler It ha been hown that maxmzng the contant term of a denomnator polynomal of a ytem tranfer functon wll lead to a table ytem whoe magntude repone mlar to the magntude repone of the orgnal ytem The propoed method not only tet the tablty of recurve dgtal flter, but alo provde the table veron of the ytem Lke the LSI and DHT tablzaton method, th method alo doe not guarantee tablty f the orgnal denomnator polynomal of the ytem tranfer functon ha zero on the unt crcle If the orgnal polynomal ha a zero on the unt crcle, then all the polynomal n the famly wll have a zero on the unt crcle and hence none of them are table The propoed method can be extended to tetng the tablty of -D recurve dgtal flter REFERENCES AKlouche-Djedd and SSLawon, Degn of doubly complementary flter wth approxmate lnear phae, IEE Proc-Von, Image, and Sgnal Proceng, vol 47, No, pp 3-8 Jae S Lm, 989 Two-dmenonal gnal and mage proceng, Prentce-Hall 3 GProak John and Ganolak Dmtr, 999 Dgtal Sgnal Proceng-Prncple, Algorthm, and Applcaton, Prentce-Hall 4 Dan EDudgeon and Ruell erereau, 984 ultdmenonal Dgtal Sgnal Proceng, Prentce-Hall 5 EA Robnon, 967 Stattcal Communcaton and Detecton, New York, Haftner, pp Andrea Antonou, 993 Dgtal Flter-Analy, Degn and Applcaton, cgraw-hll, Inc 7 GSchmeer, DRaghuramreddy, RUnbehauen, 986 An alternatve proof for the tablty of leat quare nvere polynomal, IEEE Tran Crcut and Sytem, vol 33, Iue 8, pp Nranjan Damera Venkata, ahalakhm Venkatraman, Hrhkeh and PSReddy, 999 Stablzaton of -D recurve dgtal flter by the DHT method, IEEE Tran Crcut and Sytem- II: Analog and Dgtal Proceng, vol 4, No, pp BDOAnderon and EIJury, 976 Proof of a Specal Cae of Shank Conjecture, IEEE Tran Acoutc, Speech, and Sgnal Proceng, pp PKRajan and HCReddy, 985 A note on LSI Polynomal and Comment on a Proof of a odfed Form of Shank Conjecture, IEEE Tran Crcut and Sytem, volcas-3, No 9, pp PSReddy, DRaghu Ram Reddy, and NSSwamy, 984 Proof of a odfed Form of Shank Conjecture on the Stablty of -D Planar Leat Square Invere Polynomal and It Implcaton, IEEE Tran Crcut and Sytem, volcas-3, No John L Shank, Sven Tretel, and Jame H Jutce, 97 Stablty and Synthe of two-dmenonal recurve flter, IEEE Tranacton on Audo and Electro Acoutc, volau-, No, pp SSRao, 987 Optmzaton theory and applcaton, New Delh, Wley Eatern Lmted 4 NGangatharan and PSReddy, 3 The PLSI ethod of Stablzng Two-Dmenonal Nonymmetrc Half Plane Recurve Dgtal Flter, EURASIP Journal on Appled Sgnal Proceng, 9, pp Khan Sheno, 995 Dgtal Sgnal Proceng n Telecommuncaton, Prentce Hall Internatonal Edton, USA 495