8th European Sgnal Processng Conference EUSIPCO- Aalborg, Denmar, August -7, DESIGN OF STABLE TWO-DIMENSIONAL IIR NOTC FILTER USING ROOT MAP Soo-Chang Pe and Chen-Cheng Tseng Depart. of Electrcal Engneerng, Natonal Tawan Unversty Tape, Tawan pe@cc.ee.ntu.edu.tw Depart. of Computer and Communcaton Engneerng Natonal Kaohsung Frst Unversty of Sc. and Tech. Kaohsung, Tawan tcc@ccms.nfust.edu.tw ABSTRACT In ths pape the two-dmensonal -D IIR notch flter desgn problem s presented. Frst, the lnear fractonal transformaton LFT n complex analyss s used to construct the desred root map of -D polynomal. Then, the numerator and denomnator of transfer functon of -D IIR notch flter can be obtaned from the LFT used n the constructon of root map. Ths desgn not only has closed-form transfer functon but also satsfes the bounded nput/bounded output BIBO stablty condton. Fnally, desgn example and mage applcaton are demonstrated to show the effectveness of the proposed approach.. INTRODUCTION In many sgnal processng applcatons, there s a need for a notch flter whch s charactered by a unt gan at all frequences except at the snusodal frequences where ther gan s ero. These applcatons nclude communcaton, control, mage processng and bomedcal engneerng etc. In one-dmensonal -D case, a typcal example s to cancel 5 or 6 power lne nterference n the recordng of electrocardograms ECGS []. In two-dmensonal -D case, two examples are to elmnate a -D snusodal nterference pattern supermposed on an mage [] and to reduce blocng artfact from DCT coded mage []. Thus, t s nterestng to desgn notch flter to remove the snusodal nterferences corrupted on a desred sgnal. The deal frequency response of -D notch flter s gven by, ± N N D, otherwse where N N s the prescrbed notch frequency. The problem s how to desgn -D flter, to approxmate D as well as possble. So fa the desgn of - D notch flter can be classfed nto two categores. One s FIR flter desgn, the other s IIR flter desgn. In the FIR case, the desgn methods nclude least squares approach [4], transformaton technques [5], sngular value decomposton approach [6] and constraned nonlnear optmaton method [7]. In the IIR case, the desgn approaches contan decomposton method [8] and outer product expanson [9]. So fa the -D adaptve IIR notch flters based on outer product expanson have been developed for removng snusodal nterference wth unnown or tme-varyng frequences [][]. On the other hand, the root map of -D polynomal has been a useful tool for checng the stablty of -D IIR dgtal flters []-[4]. Due to the success of ths tool n stablty checng, we wll use the root map to desgn -D notch flter n ths paper. The proposed desgn not only has closed-form transfer functon but also satsfes the BIBO stablty condton. Compared wth conventonal methods n [8][9], our method does not need to decompose the -D notch flter desgn nto several sub-flter desgns, so the proposed approach s easer to use than conventonal methods. Now, the desgn detals are descrbed n next sectons.. ROOT MAP AND LINEAR FRACTIONAL TRANSFORM In ths secton, the root map of -D polynomal s frst defned. Then, we study the relaton between root map and stablty of -D IIR flter. Fnally, the lnear fractonal transformaton s descrbed.. Defnton of Root Map Consder the -D polynomal wth order N, below: N N P, p,, N If P s regarded as a mappng between the plane and plane, the root map s defned as the mage of the unt crcle of one varable n the plane of other varable under ths mplct mappng P,. Now, let us descrbe the detals of these two maps below: Frst, the -D polynomal P, can be rearranged as the followng form: N N P, p, P[ ] [ ] The above expresson P can be nterpreted as -D polynomal n varable whose coeffcents are polynomal n varable. When s fxed, the P [ ] can be factored to yeld N roots. Thus, the frst root map Γ s defned as the loc of N roots of P [ ] when parameter traverses the unt crcle e. Second, the -D EURASIP, ISSN 76-465 685
polynomal P, can be rearranged as the followng form: N N P, p, 4 P[ ] [ ] where the expresson P can be nterpreted as -D polynomal n varable whose coeffcents are polynomal n varable. When s fxed, the P [ ] can be factored to yeld N roots. Thus, the second root map Γ s defned as the loc of N roots of P [ ] when parameter traverses the unt crcle e.. Stablty and Root Map For the -D frst quadrant IIR flter wth order M, M, N,, ts transfer functon s gven by N, M M N N q p,, Q, P, In [], Shans has shown that, s stable f and only f the followng condtons are true: P, f, 6a P, f, 6b It s obvous that the condton n Eq.6a mples that the root map Γ of the unt crcle of varable les nsde the unt crcle n the plane. And, the condton n Eq.6b means that the root map Γ of the unt crcle of les wthn the unt crcle n the plane. Thus, stablty condton of Shan's theorem wll be satsfed f two root maps le wthn unt crcle. So, we can control the stablty of -D IIR flter by controllng two root maps to le nsde the unt crcle.. Lnear Fractonal Transform In [5], the lnear fractonal transformaton LFT from complex varable to s defned by c + c 7 c + c4 where c, c, c and c are constants. It can be shown that 4 the LFT transforms crcles and lnes nto crcles and lnes. Moreove the Eq.7 can be rewrtten n the form c + c4 c c 8 Ths expresson s the same as P, wth order, f we choose p, c, p, c, 4 p, c, and p, c. Thus, we can use the propertes of LFT n complex analyss to construct the desred root maps of P, wth order,. In next 5 secton, the detals that desgn -D IIR notch flter usng root map and LFT wll be descrbed.. DESIGN OF -D IIR NOTC FILTER In ths secton, we wll summare some mportant root maps. Then, these root maps are used to desgn -D IIR notch flter. The frst mportant root map s descrbed by the followng fact: Fact : Choosng parameters ρ r r a, ρ b 9 r + r + and let -D polynomal wth order, be B, a + b + b + then we have the results: The two root maps Γ and Γ are the same. The two root maps are both crcles wth center at ρ and radus r. The equaltes B, ρ + r and B ρ + hold. Proof: Because B, s a specal LFT n complex varable textboo n [5], ths fact can be proved by usng technques n [5]. The detals are descrbed below. Snce the symmetry property B, B, s vald, two root maps are the same. So, we only need to fnd the root map Γ because Γ s the same as Γ. To fnd Γ, let complex varables and be represented by x + jy u + jv Substtutng Eq. nto B,, t yelds b + u x vy bu a vx + b + u y bv Snce s on the unt crcle, we have the constrant x + y. Based on ths constrant and Eq., we get b a a b u + v b b Substtutng Eq.9 nto Eq., we have u ρ + v r 4 Ths means that the root map Γ s a crcle wth center at ρ and radus r. Fnally, t s easy to show that the equaltes B, ρ + r and B ρ + hold by usng drect substtuton. Based on the results n Fact, we have the further results of root map below: Fact : Let -D polynomal wth order, be F Φ,, ρ, r,, [,, ] 5 where matrx Φ s 686
b cos b a cos + 6 Φ b cos ab cos + b cos b ab cos a wth parameters ρ r r a and ρ b, then we r + r + have the results: The root map Γ n plane conssts of two crcles wth center at ρe ± and rad r. The root map Γ n plane conssts of two crcles wth center at ρe ± and rad r. The equaltes F e, ρ + r e, ρ,, and e, ρ + r e, ρ,, hold. F ρ + r e, e, ρ,, ρ + r e, e, ρ,, hold. F 4 The equaltes and F Proof: The -D polynomal n Eq.5 can be factored as F,, ρ,, B, B, 7 where B, a e e + b e 8a + b e + B, a e e + b e 8b + b e + Thus, F,, ρ,, mples that B, or B,. Usng ths factoraton and the results n Fact, the proof can be shown easly. Now, let us use the Fact to construct the transfer functon of -D IIR notch flter wth order,,,. The desgn problem s how determne the flter coeffcents p, and q, n Eq.5 such that the frequency response e, e N N approxmates the deal response D n Eq. as well as possble f the notch frequency s gven. In ths pape the ey step to desgn IIR notch flter, s to choose numerator polynomal Q, and denomnator polynomal P, n Eq.5 as the form: Q, F,, ρ, r N N 9a P, F,, ρ, r N N 9b where parameters ρ, r, ρ, r need to be prescrbed. In ths pape we choose these four parameters below: ρ α, r α a ρ α, r α ε b where α s a number n the nterval,, and ε s a very small postve number. Based on ths choce, the root maps of numerator polynomal Q, and denomnator polynomal P, are shown n Fg.. Now, several remars are made below: From Eq.a, we have ρ r. Ths equalty maes the two root map crcles of numerator polynomal Q, be tangent to the unt crcle. Usng Fact and ρ r, t can be shown that N N N N Q e, e Q e, e Thus, there s ero gan at the notch frequency for the desgned IIR notch flter. The reason to choose ε as a postve number s to mae the root map crcles of denomnator polynomal P, le wthn unt crcle. Thus, the desgned IIR notch flter s guaranteed stable. In order to obtan unt gan at all frequences except at the notch frequency, the ε must be chosen as small as possble. When ε s very small, the root maps of P, wll be very close to ones of Q,. 4 Because we choose ρ ρ, the root map crcles of P, and Q, have the same center. Thus, we may refer to ths desgn as "concentrc crcle desgn". Fnally, substtutng Eq.9 and Eq. nto Eq.5, the transfer functon of the desgned -D IIR notch flter s gven by Q,, P, F,, α, α N N F,, α, α ε N N Usng Eq.5, the above transfer functon can be computed easly wthout requrng optmaton and complcated calculaton. In next secton, numercal examples wll be used to llustrate the effectveness of the proposed desgn. 4. DESIGN EXAMPLE AND APPLICATION In ths secton, one desgn example of proposed -D IIR notch flter s frst llustrated. Then, we compare the proposed desgn method wth conventonal outer product expanson method n [9]. Fnally, we use the -D IIR notch flter to remove snusodal nterference supermposed on an mage. Example : -D IIR Notch Flter Desgn In ths example, we wll study the performance of the desgned -D IIR notch flter. The notch frequency s chosen as N N.5π. And, the desgn parameters are α. and ε.. Fg.a show the resultant magntude response. It s clear that j e, e has unt gan at all frequences except at notch frequency.5π where gan s ero. So, the specfcaton s ftted well. oweve the detals of the notch are underneath the unt gan plane. In order to show the performance of the desgned flter bette Fg.b plots the loss j e, e. From ths result, we observe that the magntude response has small rpples n the vcnty of the notch. Fortunately, ths rpple can be reduced by reducng the parameter ε. To llustrate ths fact, Fg.cd show the magntude response and loss of notch flter when ε s reduced to. and other 687
desgn parameters are not changed. From ths result, t s clear that error rpple near notch frequency has been reduced. Moreove t s nterestng to study the shape of -db contour of notch n whch the magntude j e, e s equal to. Fg. shows the -db contours of the desgned -D IIR notch flters, for varous parameters α and ε when notch frequency s chosen as N N.5π. It s clear that the -db contour s symmetrc about the notch frequency.5π. And, the se of -db contour s reduced f parameters α and ε are reduced. Thus, we can use parameters α and ε to control the shape of -db contour of notch. Example : Comparson wth Conventonal -D IIR Notch Flter Desgn In ths example, we wll compare the proposed desgn method wth conventonal outer product expanson method n [9]. Gven the notch frequency and parameter N N BW, the transfer functon of conventonal -D IIR notch flter n [9] s gven by, b b a a where flters a a + b, 4 a + a b + a, 5 + b wth the coeffcents BW cos N tan a, a BW, 6 BW + tan + tan N π 4 N π + 4 sn b, 7 sn Fg.4 shows the magntude response j e, e and the loss j e, e of ths conventonal IIR notch flter for N N.5π and BW.5π. Comparng Fg. wth Fg.4, t can be seen that the performance of the proposed desgn method s smlar to one of the conventonal method n [9]. oweve the order of the proposed flter, n Eq. s,,, and the order of conventonal flter, n Eq. s,,,. So, the mplementaton complexty of proposed flter wll be less than the complexty of conventonal flter f the drect-form realaton s chosen. Example : Snusodal Interference Removal on Image In ths example, we wll use the proposed IIR notch flter to remove the snusodal nterference supermposed on an mage wth se 5 5. The mage shown n Fg.5a s the Lae mage corrupted by a snusodal nterference below: 5 sn.π m +.πn 8 Now, let us desgn a -D IIR notch flter wth notch frequency N N.π,.π, α. and ε. to remove ths snusodal nterference n spatal doman. The fltered mage s shown n Fg.5b. It s clear that the nterference has been removed by the proposed -D IIR notch flter. In the above, only the case of sngle snusod s studed. For the case of multple snusodal nterferences, the cascade notch flter can be used to remove the multple nterferences f the notch frequences are chosen as the frequences of snusods. 5. CONCLUSIONS In ths pape the root map has been presented to desgn a stable -D IIR notch flter. Frst, the lnear fractonal transformaton LFT n complex analyss s used to construct the desred root map of -D polynomal. Then, the numerator and denomnator of transfer functon of -D IIR notch flter can be obtaned from the LFT used n the constructon of root map. Fnally, desgn example and mage applcaton are demonstrated to show the effectveness of the proposed approach. oweve only IIR notch flter desgn s consdered here. Thus, t s nterestng to desgn other -D IIR flters by usng root map n the future. REFERENCES [] S.C. Pe and C.C. Tseng, "Elmnaton of AC nterference n electrocardogram usng IIR notch flter wth transent suppresson" IEEE Trans. on Bomedcal Engneerng, pp.8-, Nov. 995. [] R.C. Gonale and R.E. woods, Dgtal Image Processng, nd Edton, Prentce-all,. [] V.K. Srvastava and G.C. Ray, "Desgn of D-multple notch flter and ts applcaton n reducng blocng artfact from DCT coded mage," Proc. of the nd Annual EMBS Internatonal Conference, pp.89-8, July. [4] S.C. Pe and C.C. Tseng, "Two-dmensonal IIR and FIR dgtal notch flter desgn" Proc. of the ISCAS 99, pp.89-89, May 99. [5] V.L. Narayana Murthy and A. Mau "Desgn of some -D flter through the transformaton technque," IEE Proc. Vson, Image and Sgnal Processng, vol.4, pp.84-9, June 996. [6] S.C. Pe, W.S. Lu and C.C. Tseng, "Two-dmensonal FIR notch flter desgn usng sngular value decomposton," IEEE Trans. on Crcuts and Systems-I, pp.9-94, Mar. 998. [7] F. Wysoca-Schlla, "Desgn of equrpple -D lnear-phase FIR notch flters," EUROCON 7 The Internatonal Conference on "Computer as a Tool", pp.-5, Sept. 7. [8] S.C. Pe and C.C. Tseng, "Two dmensonal IIR dgtal notch flter desgn," IEEE Trans. on Crcuts and Systems-II, pp.7-, Mar. 994. [9] S.C. Pe, W.S. Lu and C.C. Tseng, "Analytcal twodmensonal IIR notch flter desgn usng outer product expanson," IEEE Trans. on Crcuts and Systems-II, pp.765-768, Sept. 997. [] T. namoto, N. Ieda, S. Nshmura and A. Do, "Desgn of two-dmensonal adaptve notch flters," Proc. of 5th Internatonal Conference on Sgnal Processng, pp.58-54, Aug.. [] S.C. Pe, C.L. Wu and J.J. Dng, "Smplfed structures for twodmensonal adaptve notch flters" Proc. of the ISCAS, pp.iv-46-iv-49, May. [] J. L. Shans, S. Tretel and J.. Justce, "Stablty and synthess of two-dmensonal recursve flters," IEEE Trans. on Audo Electroacoust., pp.5-8, June 97. 688
[] D.E. Dudgeon and R.M. Mersereau, Multdmensonal Dgtal Sgnal Processng, Prentce-all, 984. [4] J.W. Woods, Multdmensonal Sgnal, Image, and Vdeo Processng and Codng, Academc Press, 6. [5] J.W. Brown and R.V. Churchll, Complex Varables and Applcatons, 7th Edton, McGraw-ll, 4. frequency x π.6 α. and ε. frequency x π.6 α. and ε.....6 frequency x π..6 frequency x π α. and ε.5 α. and ε. a b frequency x π.6. frequency x π.6...6 frequency x π..6 frequency x π Fg. The -db contours of the desgned -D IIR notch flters, for varous parameters α and ε when notch frequency s chosen as.5π,.5 N N π. c d Fg. The root maps of numerator polynomal Q, and denomnator polynomal P, of -D IIR notch flter a Root map Γ of Q,. b Root map Γ of Q,. c Root map Γ of P,. d Root map Γ of P,. a b Fg.4 The magntude response e, e and loss j e, e of the conventonal -D IIR notch flter for N N.5π and BW.5 π. a b 4 4 c d Fg. The magntude response e, e and loss j e, e of the desgned IIR notch flter ab The results wth α. and ε.. cd The results wth α. and ε.. 5 4 5 5 4 5 a b Fg.5 Example of snusodal nterference removal on an mage a The corrupted mage b The mage restored by usng -D IIR notch flter. 689