290 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 3, MARCH H d (e j! ;e j!

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1 9 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 3, MARCH 998 Transactons Brefs Two-Dmensonal FIR Notch Flter Desgn Usng Sngular Value Decomoston S.-C. Pe, W.-S. Lu, and C.-C. Tseng Abstract Ths aer s concerned wth the two-dmensonal (-D) lnear-hase FIR notch flter-desgn roblem. Frst, the sngular value decomoston (SVD) s used to reduce the -D notch flter-desgn roblem to two ars of one-dmensonal (-D) flter desgn roblems. Then, we derve an analytc least-squares soluton for the desgn of two ars of - D lnear-hase FIR flters. Agan, the coeffcents of the flter desgned are gven by closed-form formulas and the flter gan at the notch frequences s exactly zero. One examle s ncluded to llustrate the roosed desgn methods. To demonstrate the usefulness of the -D notch flters desgned, the -D notch flter s aled to elmnate the snusodal nterference suermosed on an mage. Index Terms Interference removal, sngular value decomoston, -D FIR notch flter. I. INTRODUCTION Notch flters have been an effectve means for elmnatng narrowband or snusodal nterferences n certan sgnal-rocessng alcatons rangng from ower lne-nterference cancellaton for electrocardograms to multle snusodal nterference removal for corruted mages. For the one-dmensonal (-D) case, several methods for the desgn and erformance analyss of IIR and FIR notch flters have been develoed [] [5]. For the two-dmensonal (-D) case, [6] and [7] roosed a method that reduces the desgn of a stable IIR - D notch flter to the desgn of a -D arallel-lne flter and a -D straght-lne flter. However, the IIR notch flter n [6] s nonlnear hase, so t s meanngful to desgn a lnear-hase notch flter for the mage rocessng alcatons. Thus, an aroach for -D lnear hase FIR notch flter desgn wll be addressed n ths aer. In the -D flter-desgn lterature, the sngular value decomoston (SVD) has been wdely used to reduce the -D flter-desgn roblem to a set of -D subflter-desgn roblems. The desgn examles of the low-ass, bandass, and fan dgtal flters have been nvestgated n [8] and [9]. In ths aer, we wll use the SVD method to desgn a -D FIR notch flter. Due to the smlcty of the secfcaton of the notch flter, ths SVD-based desgn has the followng two unque features whch do not exst for general flter-desgn cases n [9]. Frst, the SVD of the samled magntude resonse matrx of the desred -D lnear hase notch flter reveals that the desgn roblem can be solved by only fndng two ars of -D transfer functons. Second, the aroxmaton of each -D transfer functon nvolved has the closedform formula wthout requrng any teratve otmzaton rocedure. Due to these two features, t s nterestng to nvestgate ths desgn roblem n ths aer. Manuscrt receved January, 996; revsed October, 996 and Arl 7, 997. Ths work was suorted by the Natonal Scences Councl, R.O.C., under Contract NSC 87-8-E--49. Ths aer was recommended by Assocate Edtor V. Tavsanoglu. S.-C. Pe and C.-C. Tseng are wth the Deartment of Electrcal Engneerng, Natonal Tawan Unversty, Tae, Tawan, R.O.C. W.-S. Lu s wth the Deartment of Electrcal and Comuter Engneerng, Unversty of Vctora, Vctora, B.C., Canada V8W 3P6. Publsher Item Identfer S 57-7(98)9-. II. THE DESIGN OF -D LINEAR-PHASE FIR NOTCH FILTERS In ths secton, we wll concentrate on the desgn of a -D lnearhase FIR notch flter. The desred magntude resonse s gven by H d (e j! ;e j! ) = ; (! ;! )=(! ;! 3 )and 3 (! 3 ;! 3 ) () ; otherwse. We denote the samled magntude resonse matrx by D R M M. If the notch frequences (! ;! 3 )and 3 (! 3 ;! 3 ) are among the samlng grds, then the entres of matrx D are gven by D(n ;n ) ; (n;n)=(;) and (q; q) = () ; otherwse n M ; n M the ( ; )th and (q ;q )th entres wth q = M + and q = M + corresond to the notch frequences (! ;! 3 ) 3 and (! ; 3! 3 ), resectvely. Evdently, the rank of matrx D s three as t only contans three lnearly ndeendent columns. By usng the SVD, the matrx D can be wrtten as D = w w t u u t v v t (3) the elements of the column vectors nvolved are gven by w k (n k )= u k (n k )= v k (n k )= ; n k = k and q k ; n k = k ; n k = q k. n k M k k =; Ths decomoston suggests that the magntude resonse D can be aroxmated by that of -D FIR flters whose magntude resonses aroxmate w k, u k, and v k (k = ; ). In ths secton, we shall use ths decomoston to desgn -D lnear-hase FIR notch flter. It follows from (3) that the transfer functon of a -D FIR notch flter should be aroxmated by the followng form: [(N )=] [(N )=] H(z ;z )=z z f (z )f (z )+g (z )g (z ) (4) the -D FIR flters are denoted by N f k (z k )= a (k) zk N g k (z k )= b (k) zk ; k =; (5) wth N k odd (k =;). The lnear hase constrants on f k (z k ) and g k (z k )(k= ; ) mly that a (k) N = a(k) b (k) N = b(k) ; N k k =; (6) 57 7/98$. 998 IEEE

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 3, MARCH and the frequency resonse of -D flters f k (z k ) and g k (z k ) are gven by f k (e j! ) (N 3)= = a (k) (N )= + j! [(N )=] e a (k) cos N k! k M f (! k )e j! [(N )=] (7) g k (e j! ) = j (N 3)= j! [(N )=] e b (k) sn N k! k jm g (! k )e j! [(N )=] ; k =;: Substtutng (7) nto (4), the frequency resonse of the roosed -D FIR flter can be wrtten as H(e j! ;e j! )=M(! ;! )e j! [(N )=] e j! [(N )=] (8) the magntude resonse s gven by M (! ;! )=M f (! )M f (! )M g (! )M g (! ): (9) Now, we wll desgn -D flters f k (z k ) and g k (z k ) such that magntude resonse M (! ;! )aroxmates jh d (! ;! )jdefned by () as close as ossble. From (3), we have the followng fact. Fact: Let the two desred magntude functons be then M df (w k )= M dg (! k )= ;! k =! k 3 and! k 3 ;! k =! 3 k ;! k =! k 3 jh d (e j! ;e j! )j = M df (! )M df (! ) M dg (! )M dg (! ): () () () Based on ths fact, the task of desgnng a lnear-hase -D FIR notch flter reduces to the followng two aroxmaton roblems: ) fnd flter coeffcents such that M f (! k ) aroxmates M df (! k ) for k =;, and ) fnd flter coeffcents such that M g (! k ) aroxmates M dg (! k ) for k =;. In order to satsfy jh d (! ;! 3 )j= 3 exactly, the followng constrants need to be mosed: M f (! 3 k)=m df (! 3 k )= M g (! k)=m 3 dg (! k)= 3 ; k =;: (3) We are now n a oston to derve closed-form least squares solutons to these aroxmaton roblems. A. Aroxmaton of M f (! k ) The least-squares aroxmaton of M f (! k ) to M df (! k ) for k = ; can be unfed as one aroxmaton roblem. That s, we shall seek to fnd a arameter vector a =[a a N ] t such that J(a) = w(!)[f(!) f d (!)] d! (4) s mnmzed, and f (!) = f d (!) = N a cos(n )! ;!!3 ;!3 + (5) w(!) = ;!!3 ;!3 + : (6) The wdth n (5) s small and ostve, and s assumed to be gven as a desgn arameter. The weght s farly close to n order to ensure a narrow notch. A formula that determnes the rght value of wll be derved later n ths secton. Usng (4) (6), we can wrte J(a) =() f (!)d! +! +=! = f(!) ( )f (!) d!: (7) Snce s very small, the last ntegral n (7) can be aroxmated by f (! 3 ) ( )f (! 3 ): (8) Hence J(a) can be wrtten n a matrx vector form as J(a) a t Qa a t q + const (9) ( ) Q = Q + ( )Q (! 3 ) q = c(!3 )... Q = Q (! 3 )=c(! 3 )c t (! 3 ) c(! 3 ) = [cos(n! 3 );; cos(! 3 ) ] t () and const s a constant ndeendent of a. Snce the weght s close to one but less than one, matrx Q s ostve defnte, and the global mnmum of J(a) s acheved by a 3 = Q q: () Usng the matrx nverson formula for vectors r and s,.e., we obtan (I + rs t ) = I rst +r t s () a 3 = Q ~c(!3 )~c t (! 3 ) + k^c(! 3 )k c(! 3 ) (3) = ( ) Q =... = ( ) ( ) (4) (5) ~c(! 3 )=Q c(! 3 ) ^c(! 3 )=Q = c(! 3 ): (6)

3 9 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 3, MARCH 998 There are two roblems wth (3) that need to be addressed. Frst, as a common roblem wth a least-squares desgn, the flter wth coeffcents a 3 determned by (3) often exhbts undesrable Gbbs oscllatons n the vcnty of the notch frequency! 3. An effectve way to elmnate these oscllatons s to use a certan wndow functon to modfy the coeffcent vector,.e., to modfy the th coeffcent of the flter to be a() =w h ()a 3 (); N (7) w h () =:54 :46 cos N ; N (8) s from the (N +)-ont Hammng wndow. The factor ntroduced n (7) turns out to be necessary to satsfy a constrant at the notch frequency, as wll be seen next. The second roblem that needs to be dealt wth s that the constrant f (! 3 )= (9) must be satsfed. In what follows we show that ths constrant can be met by roerly choosng a weght. By wrtng (9) as c t (! 3 )a = (3) Fg.. The relaton curves of the arameters and for varous lengths N and! 3 =:5. and usng (3), (7), and (9), we obtan r r + = (3) = k^c(! 3 )k, w c = [w h () cos (N! 3 );;w h (N ) cos (! 3 ) w h (N)] t, r = ~c t (! 3 )w c, and r = [~c t (! 3 )c(! 3 )]c t (! 3 )w c. Usng the defntons of and, straghtforward manulatons lead (3) to A + B + C = (3) A = r + (4 ) 4r Fg.. The resonses f(!) for N =4;! 3 =:5, and several values. B = r ( )+r (3 ) C = ( ): (33) Although (3) s a quadratc form, the root = B + B 4AC A (34) falls nto the nterval (; ) and s the rght choce of that satsfes (9). To verfy ths fact, Fg. shows the relaton curves of the arameters and for varous lengths N when we choose! 3 =:5. From ths result, t s clear that aroaches unty when aroaches zero. In fact, f aroaches zero, we have lm A =! that s, (3) becomes lm B =! lm C =! (35) +=: (36) Thus, the fact that aroaches unty when aroaches zero s obvous. Moreover, Fg. shows the magntude resonses of f (!) for N =4,! 3 =:5and varous. It s clear that the resonses f (!) are almost the same for several values even though aroaches zero whch means aroaches unty. Thus, the choce of value close to (6) would not deterorate the flter characterstcs n the assband because aroaches zero under ths condton. In summary, we have develoed a closed-form least-squares soluton to the aroxmaton roblem (4) subject to constrant (9). Ths (N + )th-order flter has symmetrc coeffcents h = [ a(); ; a(n ) a(n ) a(n ); ; a()] fa(); N g are gven by (3) and (7) wth an determned by (34). B. Aroxmaton of M g (! k ) The least-squares aroxmaton of M g (! k ) to M dg (! k ) for k =;can be unfed as one aroxmaton roblem. That s, we shall seek to fnd a arameter vector b =[b ;;b N] t such that J(b) = w(!)[g(!) g d (!)] d! (37)

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 3, MARCH s mnmzed, g(!) = N b sn(n )!!! g d (!) = ; 3 ;!3 + and w(!) s defned by (6). Wth suffcently small J(b) =() g(!) ( ) g (!)d! +! +=! = ( )g (!) d! g (!) d! + g(! 3 ) (38) ( )g (! 3 ) = b t Pb b t + const (39) (a) ( ) P = I + ( )P (! 3 ) P = d(! 3 )d t (! 3 ) d(!) = [sn(n!);; sn(!)] t = d(!3 ): (4) Wth an close to unty, matrx P s ostve defnte and the global mnmal of J(b) s acheved by b 3 = P (4) whch, by usng matrx nverson formula (), can be exressed as b 3 = d(! 3 ) (4) wth = ( ) + ( )kd(! 3 )k : (43) Lke the aroxmaton of M f (! k ), a Hammng wndow w h () =:54 :46 cos N ; N (44) s used to elmnate the Gbbs oscllatons n the frequency resonse of the flter desgned, and the modfed flter coeffcents become b() =w h ()b 3 (); N (45) b 3 () s the th entry of b 3 n (4). Further, to satsfy the constrant g(! 3 )= (46) we wrte g(! 3 )=d t (! 3 )band use (4), (45), and (46) to obtan an equaton for : d t (! 3 )w d ( ) + ( )kd(! 3 )k = (47) w d =[w h () sn(n! 3 );;w h (N ) sn(! 3 )] t : (48) Solvng (47) for gves kd(! 3 )k = +d t (! 3 )w d 4kd(! 3 )k: (49) (b) Fg. 3. The results of the desgned notch flter. (a) Magntude resonse. (b) -magntude resonse (loss). In summary, we have obtaned a closed-form least-squares soluton to the aroxmaton roblem n (37) subject to constrant (46). Ths (N +)th-order flter has odd symmetrc coeffcents h =[ b(); ; b(n ) b(n ); ; b()] fb(); N g are gven by (4) and (45) wth an determned by (49). III. DESIGN EXAMPLE AND APPLICATION In ths secton, one desgn examle of a lnear-hase FIR notch flter s frst llustrated. Next, we use the -D FIR notch flter to remove snusodal nterferences suermosed on an mage. Examle : FIR Notch Flter Desgn In ths examle, we use the rocedure stated n Secton II to desgn a -D lnear-hase FIR notch flter wth notch frequency (! ;! 3 )=(:5; 3 :5). The flter length s 4 4 and the desgn arameter s chosen to be.. Fg. 3(a) dects the resultant magntude resonse desgned n the least-squares error sense. It s clear that the secfcaton s well satsfed. However, the detals of notches are underneath the unty gan lane. In order to show the

5 94 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 3, MARCH 998 IV. CONCLUSIONS In ths aer, the -D FIR notch flter-desgn roblem has been nvestgated. Frst, the SVD s used to reduce the -D notch flterdesgn roblem to two ars of -D flter-desgn roblems. Then, we rovde the closed-form solutons for the desgn of two ars of -D FIR flters. Fnally, several desgn and alcaton examles are resented to demonstrate the erformance of the roosed desgn methods. However, only fxed notch flters are consdered here. Thus, t s nterestng to develo a -D adatve notch flterng algorthm. Ths toc wll be studed n the future. REFERENCES (a) [] R. Carney, Desgn of a dgtal notch flter wth trackng requrements, IEEE Trans. Sace Electron. Telem., vol. SET-9,. 9 4, Dec [] K. Hrano, S. Nshmura, and S. K. Mtra, Desgn of dgtal notch flters, IEEE Trans. Crcuts Syst., vol. CAS-, , July 974. [3] M. H. Er, Desgnng notch flter wth controlled null wdth, Sgnal Process., vol. 4, no. 3, , Set. 99. [4] G. W. Medln, A novel desgn technque for tunable notch flters, n Proc. Int. Sym. Crcuts Syst., New Orleans, LA, May 99, [5] S. C. Dutta Roy, S. B. Jan, and B. Kumar, Desgn of dgtal FIR notch flters, Proc. Inst. Electr. Eng. Vs. Image Sgnal Process., vol. 4, , Oct [6] S. C. Pe and C. C. Tseng, Two dmensonal IIR dgtal notch flter desgn, IEEE Trans. Crcuts Syst. II, vol. 4,. 7 3, Mar [7], Correctons on two dmensonal IIR dgtal notch flter desgn, IEEE Trans. Crcuts Syst. II, vol. 4, , Set [8] A. Antonou and W. S. Lu, Desgn of two-dmensonal dgtal flters by usng the sngular value decomoston, IEEE Trans. Crcuts Syst., vol. CAS-34,. 9 98, Oct [9] W. S. Lu, H. P. Wang, and A. Antonou, Desgn of two-dmensonal dgtal flters by usng the sngular value decomoston, IEEE Trans. Crcuts Syst., vol. 37, , Jan. 99. (b) Fg. 4. Examle of sngle snusodal nterference removal. (a) corruted mage and (b) mage restored by usng -D FIR notch flter. Analyss of the Behavor of a Dynamc Latch Comarator P. Cusnato, M. Bruccoler, D. D. Cavgla, and M. Valle erformance of the desgned flter better, Fg. 3(b) lots the loss (-gan) nstead of the gan. From the result, we observe that the magntude resonse has small rles n the vcnty of the notches. Examle : Sngle Snusodal Interference Removal In the examle, we wll use the -D FIR notch flter to remove a sngle snusodal nterference suermosed on an mage. The scenaro of the exerment s the same as the one descrbed n [6] excet that the mage sze here s The mage shown n Fg. 4(a) s the Lena mage corruted by a snusodal attern of the form 3 sn(:m + :n): (5) Now we desgn a -D FIR notch flter wth (! 3 ;! 3 )=(:; :) and = : to remove the nterference n satal doman. The fltered mage, shown n Fg. 4(b), s clearly free from nterference. Abstract Ths bref deals wth the behavor of a dynamc latch used as a voltage comarator. A detaled analyss of the fne settlng hase s reorted, uttng n evdence the non-dealtes whch lead to comarson errors. A technque to mnmze such errors s suggested. An exermental ch has been fabrcated and measurements are reorted and dscussed. I. INTRODUCTION The dynamc latch (Fg. ), wdely used as a sense amlfer n dynamc RAM s, s a fast CMOS comarator. However, ts large nut offset voltage, manly caused by the crcut-arameter devatons and by the charge njecton msmatch between swtches S and S, lmts ts resoluton to about 5 b [], []. A lnear amlfer may recede the dynamc latch so that the voltage aled Manuscrt receved March 4, 995; revsed January 7, 997. Ths aer was recommended by Assocate Edtor A. Rodrguez-Vazquez. P. Cusnato and M. Bruccoler are wth SGS-Thomson Mcroelectroncs, I- Cornaredo, Mlan, Italy. D. D. Cavgla and M. Valle are wth DIBE, Unversty of Genoa, I-645 Genoa, Italy. Publsher Item Identfer S 57-7(98) /98$. 998 IEEE