Using Pisarenko Harmonic Decomposition for the design of 2-D IIR Notch filters

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1 WSEAS RANSACIONS on SIGNAL PROCESSING Using Prenko Hrmonic Decomposition for the design of -D IIR Notch filters Astrct In th pper, the Prenko Hrmonic Decomposition used for the design of -D (wo- Dimensionl notch filters. An pproprite trnsformtion, recently proposed y the uthor used. Keywords Prenko Hrmonic Decomposition, Notch Filters, -D Notch Filters, Multidimensionl Systems, Multidimensionl Filters, Filter Design, I. INRODUCION He Prenko Hrmonic Decomposition, which well-known frequency estimtion method, widely used in mny res of Signl Estimtion, Signl Reconstruction nd Adptive Filtering. he Prenko method uses the eigenvector ssocited with the smllest eigenvlue to estimte the frequencies of the input signl []. Adptive Notch filters re used in live sound reproduction, in instrument mplifiers design, in electrocrdiogrm (ECG signl processing etc. So, for every cse in signl processing nd communictions tht n elimintion of n undesirle frequency necessry, n pproprite Notch Filter necessry in order to cut-off th unwnted frequency. For exmple in ECG signl processing there need to eliminte the power line noe s it dded in the ndwidth of the ECG signl. In the one-dimensionl (-D cse, severl methods for the design nd performnce nlys of IIR nd FIR notch filters hve een developed [] [3]. In th pper, we use the results of [3] nd n ttempt to extend them in -D cse vi pproprite trnsformtions presented. On the other hnd, the dpttion here chieved y using -D dpttion lw. h pper orgnied s follows: Section II presents First-Order -D IIR Notch Filters design together with numericl exmple. In Section III, the design of fmily of Second-Order -D IIR Notch Filters presented. Some remrks cn e found in IV nd finlly there Conclusion. he dpttion lw sed on Prenko Method nd given in Section IV. II. HE PROPOSED MEHOD FOR FIRS-ORDER IIR -D NOCH FILERS (WIHOU ADAPAION WIHOU PISARENKO MEHOD First we consider the trnsfer function ( K r j with e π π, the Smpling Period, nd 0 << r <. For 0 << r < th -D trnsfer function stle [3]. Fig.. K scling fctor such tht the mximum gin of the filter to e equl to. With the pole rdius lmost equl to, the pole lmost cncels the effect of the ero except in the cse. So, th filter n ll-pss filter tht rejects the frequency of 0 (e.g. DC frequency. he mgnitude response illustrted in Fig.. in the cse of r 0.9, ( without loss of generlity. ( with the echnicl University of Sofi, Industril Engineering Deprtment, Sofi 000, Sofi, BULGARIA, mstor@tu-sofi.g s well s with the Militry Institutions of University Eduction (ASEI, Hellenic Nvl Acdemy, Electricl Engineering nd Computer Science, erm Htikyrikou, 8539, Pireus, GREECE mstor@hn.gr ISSN: Issue 4, Volume 6, Octoer 00

2 WSEAS RANSACIONS on SIGNAL PROCESSING rnsformtion ( remrkle ecuse the Eqution ( + 0 hs the unique solution, j j since e, e nd cn e esily extended to fmily of trnsformtions s follows. Introducing + + with, rel numers or simply + ( with 0 < < one otins ( ( + ( K r + ( ( (3 Fig.. ArgH ( j he Group Dely τ depicted in Fig.. nd shows lmost liner ehvior in ig prt of the frequency domin. he Notch filter of ( presented in [3]. In th section, we extend it to -D cse s follows: In th pper, we propose -D filters, sed on the results of [3] y pplying pproprite trnsformtions. So, for the first-order notch filter of ( considering the trnsformtion we tke ( + ( + K r + ( ( j with e j π π, e, π π, re the smpling periods to horiontl nd verticl direction wheres: 0 << r < j with e j π π, e, π π ( 0 << r < he r < condition gurntees the -D nd -D filter stility in ll the ove cses. Numericl Exmple : Consider without loss of generlity, equl to. hen, for r 0.9 nd in (3, one finds K.0560, the mgnitude response depicted in Fig.., while the Group Delys ArgH ( j, j ArgH ( j, j τ, τ re depicted in Fig.. nd Fig..c. Fig.. Fig.. ISSN: Issue 4, Volume 6, Octoer 00

3 WSEAS RANSACIONS on SIGNAL PROCESSING r( + ( ( cos + ( cos + r ( + ( 0 0 Let s exmine for wht (, we hve H ( Becuse for the frequency response, 0,, we hve to exmine the frequencies (, for which j j j0 j0 e + ( e e ( e 0 or e e e e j j j0 j0 + ( ( 0 Using now: c Fig..c It pprent tht the fmily of the filters of (3 elimintes the -D frequency (, (0,0. Using th First-Order -D Notch filter, the only frequency tht cn e eliminted (, (0,0. If elimintion of nother -D frequency (, ( 0, 0 (0,0 necessry, second-order -D IIR notch filter must e used. As we prove in Section IV the -D first-order Notch filter lso Stle for 0 << r <. III. HE PROPOSED MEHOD FOR SECOND -ORDER IIR -D NOCH FILERS (WIHOU ADAPAION WIHOU PISARENKO MEHOD e + ce e ce (5. j j j0 j0 ( 0 e + ce e ce (5. j j j0 j0 ( 0 We exmine two cses c, tht mens nd c, tht mens he first cse yields the two equtions: j j j0 j0 e + e e e 0 (6. j j j0 j0 e + e e e 0 (6. In th session, we extend (3 s follows in order to crete filter for rejection (, ( 0, 0 H ( ( ( (, K e + e + e ( e r ( ( j0 j0 j0 j ( ( + ( e ( e r( ( j0 j0 e e j0 j0 + + (4 with 0 < <, 0 << r < nd K scling fctor such tht the mximum gin of the filter to e equl to. For 0 << r < th -D trnsfer function stle [3]. Eq. (4 cn e written lso s (, (, A H (, K B where (, A ( ( cos( ( + ( ( cos 0 + ( cos 0 + ( + ( So, from (7. one otins the notch frequencies 0, 0, nd from (7. the notch frequencies nd B, 0, 0 + ( + ( cos( ( 0 0 From (6. one otins the notch frequencies, 0 0 nd the symmetric solution 0, 0 while from (6. two other couple of notch frequencies,, i.e. 0 0, 0 0 re otined. he second cse yields lso two equtions: j j j0 j0 e + ce e ce 0 (7. j j j0 j0 e + ce e ce 0 (7. with c. ISSN: Issue 4, Volume 6, Octoer 00

4 WSEAS RANSACIONS on SIGNAL PROCESSING Evidently, s -D IIR filter we cn use only the cse since the elimintion of the symmetric frequencies ( 0, 0 not required. re depicted in Fig.3. nd Fig.3.c herefore our -D IIR Notch Filter given y (4 tht cn e lso written s (, (, A H (, K B where (, A ( ( cos( ( + ( ( cos( + ( cos( + ( + ( 0 0 nd (, B + ( + ( cos( 0 0 r( + ( ( cos( + ( cos( + r ( + ( 0 0 Fig.3. with 0 << r < 0 < <, 0.5 nd K scling fctor such tht the mximum gin of the filter to e equl to. Using c now, further simplifiction of the second-order -D IIR Notch Filter trnsfer function (, + c + ccos( ( + c (cos( + c cos( + ( + c K cos( ( (cos( cos( ( c + c 0 0 r + c 0 + c 0 + r + c where c (8 Fig.3. Numericl Exmple : Consider the -D IIR Notch Filter of (8. Suppose tht we π π wnt the cncelltion of 0, 0 (nd of course 4 π π the symmetric 0, 0. One cn choose for 4 exmple c, r 0.9. Consider lso without loss of generlity,. hen ( j( + 3 j H (, K ( r + r + 3 j ( r + r 3 j j with e j π π, e, Hence, the mgnitude response depicted in Fig.3., while the Group Delys ArgH ( j, j ArgH ( j, j τ, τ π π Fig.3.c ISSN: Issue 4, Volume 6, Octoer 00

5 WSEAS RANSACIONS on SIGNAL PROCESSING As we prove in the next Section, our -D second-order Notch filter lso Stle for 0 << r <. REMARKS A first remrk tht -D filters with severl notch frequencies cn e esily implemented y cscde design, while y using the new trnsformtions P nd P where P, P re positive integers, except the notch frequencies ± 0, ± 0 the following notch frequencies re otined k k ± 0, 0 P ± P k,,..., P nd k,,..., P herefore periodic -D notch filters cn esily implemented. IV. PISARENKO HARMONIC DECOMPOSIION FOR HE DESIGN OF -D NOCH FILERS he question how to find the Notch frequencies 0, 0. o th end, we pply the usul -D Prenko Method seprtely to ech direction. More detiles out Prenko Method re given in [] nd []. First we strt from the horiontl direction (i.e. to determine 0 We pply the following lgorithm for the horiontl dimension ( Estimte the covrince mtrix R of sie N x N from the N mesured smples of our -D signl in the horiontl dimension. he exct covrince mtrix R given s follows R(0,0 R(,0 R( N,0 R(,0 R(0,0 R( N,0 R R( N,0 R( N,0 R(0,0 ( Compute the eigenvector corresponding to the smllest eigenvlue of the estimted covrince mtrix R. (3 Compute the roots of the polynomil formed y the elements of the ove eigenvector. h polynomil will hve roots locted t exp( ± j0 nd the following lgorithm for the verticl dimension ( Estimte the covrince mtrix R of sie N x N from the N mesured smples of our -D signl in the verticl dimension.. he exct covrince mtrix R given s follows R(0,0 R(0, R(0, N R(0, R(0,0 R(0, N R R(0, N R(0, N R(0,0 ( Compute the eigenvector corresponding to the smllest eigenvlue of the estimted covrince mtrix R. (3 Compute the roots of the polynomil formed y the elements of the ove eigenvector. h polynomil will hve roots locted t exp( ± j0 Except the method of Prenko, we cn use lterntively in ech dimension (See [] MUSIC Algorithm Minimum-Norm Method ESPRI Algorithm V. CONCLUSION A new efficient nd elegnt technique for dptive -D Notch Filter Design investigted in th pper. Some other studies of the uthor for the stility of m-d systems cn e found in [4] [0]. Work in progress y the uthor towrds of sttement new m-d design techniques etter nd more effective thn the McClelln rnsformtions. REFERENCES [] R. Crney, Design of digitl notch filter with trcking requirements, IEEE rns. Spce Electron. elem., vol. SE-9, pp. 09 4, Dec [] K. Hirno, S. Nhimur, nd S. K. Mitr, Design of digitl notch filters, IEEE rns. Circuits Syst., vol. CAS-, pp , July 974. [3].I.Lkso, J. Rnt, S.J. Ovsk, "Design nd implementtion of efficient IIR notch filters with quntition error feedck", IEEE rnsctions on Instrumenttion nd Mesurement, Vol.43, No.3, pp , Jun 994 [4] N.E.Mstork, "A method for computing the -D stility mrgin sed on new stility test for -D systems", Multidimensionl Systems nd Signl Processing, Vol.0, pp.93-99, 998. [5] N.E.Mstork, "New Necessry Stility Conditions for -D Systems", IEEE rnsctions on Circuits nd Systems, Prt I, Vol.47, No.7, pp.03-05, July 000. [6] N.E.Mstork, "A method for computing the -D stility mrgin", IEEE rnsctions on Circuits nd Systems, Prt II, Vol.45, No.3, pp , Mrch 998. [7] N.E.Mstork, "Recursive Algorithms for wo-dimensionl Filters' Spectrl rnsformtions", IEEE rnsctions on Signl Processing. Vol.44, No.0, pp , Oct.996. [8] V. Mldenov, nd N. Mstork, Design of -Dimensionl Recursive Filters y using Neurl Networks, IEEE rns. on Neurl Networks. [9] Mstork, N.E., Gonos, I.F., nd Swmy, M.N.S., Stility of multidimensionl systems using genetic lgorithms, IEEE rns. on Circuits nd Systems, Prt I: Fundmentl heory nd Applictions, Volume 50, Issue 7, July 003, pp [0] Mstork, N.E. Swmy, M.N.S., "Spectrl trnsformtions for two-dimensionl filters vi FF" IEEE rnsctions on Circuits nd Systems I: Fundmentl heory nd Applictions, Vol.49, Issue 6, pp.87-83, Jun 00 [] Murice Bellnger, Adptive Digitl Filters nd Signl Anlys, Mrcel Dekker, Bsel, Switerlnd, 00. [] Dimitr G. Mnolk, Dimitr Mnolk, Viny K. Ingle, Stephen M. Kogon, «Sttticl nd Adptive Signl Processing», Artech House Pulhers, 005. ISSN: Issue 4, Volume 6, Octoer 00

z TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability

z TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSP-G 6. Trnsform Bsics The definition of the trnsform for digitl signl is: -n X x[ n is complex vrile The trnsform