IM CCTD  uropean Conference on Circuit Theory and Design August DJE D spoo inland DigitalMultipleNotchFiltersPerformance Miroslav Vl cek Pavel Zahradn k Abstract  The performance of digital II and FI
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IM CCTD uropean Conference on Circuit Theory and Design August DJE D spoo inland DigitalMultipleNotchFiltersPerformance Miroslav Vl cek Pavel Zahradn k Abstract The performance of digital II and FI

Pro vided the frequency responses of both II0 and FI0 notch 64257lters are comparable we propose that almost equiripple FI0 notch 64257lter exhibits constant group delay and superior output responses over its II0 counterpart The group delay and the

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IM CCTD uropean Conference on Circuit Theory and Design August DJE D spoo inland DigitalMultipleNotchFiltersPerformance Miroslav Vl cek Pavel Zahradn k Abstract The performance of digital II and FI




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Presentation on theme: "IM CCTD uropean Conference on Circuit Theory and Design August DJE D spoo inland DigitalMultipleNotchFiltersPerformance Miroslav Vl cek Pavel Zahradn k Abstract The performance of digital II and FI"— Presentation transcript:


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I/3M :CCTD<91 / :uropean Conference on Circuit Theory and Design, August DJ/E1, D991, :spoo, ?inland DigitalMultipleNotchFiltersPerformance Miroslav Vl cek Pavel Zahradn ık Abstract The performance of digital II0 and FI0 multiple stopband notch filters is presented. Pro- vided the frequency responses of both II0 and FI0 notch filters are comparable we propose that almost equiripple FI0 notch filter exhibits constant group delay and superior output responses over its II0 counterpart. The group delay and the step response of a selective filter are

essential for filtering spurious harmonics of the power supply from weak EEG and E-G signals. The responses for a couple of input pulse train signals together with the step responses of both II0 and FI0 notch filters are included. 1 Introduction 7e would like to recall the fact that in numerous applicationsofdigitalsignalprocessingitisofmain interest to process pulse-like signals by a selective algorithmwhichdoesnotchangesubstantiallytime sequenceattheoutput. Inapartoffrequencyspec- i)cations which guarantee selectivity of a )lter we havetocontroleithergroupdelayorstepresponse

ofthe)lter. Thetimedomainpropertiesofa)lter areoftenomittedemphasizingfrequencyresponses only. In the paper [-] a design technique for multiple IIJnotch)lterwasinvestigated. Amultiplenotch IIJ )lter was then used to eliminate power line (A%)interferenceintherecordingofelectrocardio- grams(E%O).Bowever,removingofharmonicsby themultipleIIJnotch)lterproducesasubstantial distortionoftheoutputresponseduetothegroup delayvariation. 7econ)rmthatduetotherelationbetweenim- pulseresponseandgroupdelayofa)lter,theselec- tivityofthefrequencyresponseprovidesdistortions ofLatregionsintheoutputsignal.

Inordertore- ducetheseripplesintheoutputsignalwepropose FIJmultiplenotch)lterswhichinheritsaconstant groupdelay. In [/] we made a study of FIJ notch )lters and their abridging which led to the e:cient length of a correspondingimpulse response. In [0]we intro- 4zech Technical University in 8rague, Faculty of Trans- portation Sciences, Conviktsk a 20, 4F-110 00 8raha 1, 4zech Republic. E-mail:vlcek:fd.cvut.cz, Tel: +420-2- 24390720, Fax:+420-2-24390702 4zech Technical University in 8rague, Faculty of Electri- cal Engineering, Technick a 2, 4F-166 27 8raha 6, 4zech Re- public.

E-mail:zahradni:fel.cvut.cz, Tel: +420-2-24352039, Fax:+420-2-24310734 ducedthedesignofalmostequirippledouble-notch FIJ )lters. The procedure is based on the obser- vationthattheoddpartofaNolotarevpolynomial hastwoextralobesforwhich p,q p,q )) (-) andwhichareofthesamemagnitude. Substituting theoddpartofaNolotarevpolynomialina%heby- shev polynomial we generate the transfer function ofadouble-notchFIJ)lterusing )G p,q )) (/) The transfer function of a double-notch FIJ )lter oflength/ D-G )D-isthen )G max (0) The ripples in the passband are not exactly equal but they fall within the limit of

ripples of an optimal single notch )lter. Frequency response of suchFIJdouble-notch)lterisshowninFig. -. 0.1 0.2 0.3 0.4 0.5 0.6 0. 0.8 0.! −60 −50 −40 −30 −20 −10 /9 d8 Figure -: Frequency response of the FIJ double- notch )lter is generated by ) where argument is equal to the odd part of Nolotarev polyno- mial 8;/)ofdegree5,with G0 099- G0 003/ and G0 /5-/ . Frequency response/0log j withnotchfrequenciesspec- i)ed by G0 00/9 and G0 8890 is plotted versus the normalised frequency. The rip- plesinthepassbandarelessthan-d*.
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I/59 50 100 150

−0.2 0.2 0.4 0.6 0.8 1.2 Figure /: Step response of the FIJ double-notch )lterfromFig. -. ThisclassofFIJ)lterscanplayanimportantrole in)lteringofthesinusoidalinterferenceharmonics maintainingtheshapeofthepulse-likesignals. 2 3roup-elayandImpulseResponse ThetransferfunctionofanIIJnotch)lterisgiven as notch )G [-D )] (3) 7enotethatthegroupdelay )ofanIIJnotch )lterisdirectlyrelatedtothegroupdelayofacon- stitutingallpasssection notch )G allpass (6) Itsimpli)estheevaluationofthegroupdelayfunc- tion of a digital IIJ notch )lter for which zeros of thetransferfunctionappearontheunitcircle. *y

simple algebra we can conclude that an all- pass section which is the fundamental building block of IIJ notch )lter has the impulse response sequence related to its group delay. For the )rst orderallpass )G az =1 (8) weobtaintheimpulseresponse )G )D( +1 ) (9) orthestepresponse )G =0 )G- +1 ) (;) Thesequence whichdrivestheimpulseresponse function appears in expression for corresponding groupdelayasfollows aw G-D/ =1 ) (5) where )are%hebyshevpolynomialsofthe)rst kind )G )) For the second order allpass section the sequence which controls both the impulse response and

groupdelayinasimilarmannerisbasedonradius of complex conjugate poles jb .7ecan concludethattheimpulseresponse(stepresponse, resp.) and group delay are related to each other and they are both equivalently responsible for the distortionoftheoutputsignal. 3 1xamples Several experiments were performed. Bere we present a couple of responses (see Fig. 3, 6, 9 and;)totheinputrectangularpulsetrainsignals with the different duty cycle. *oth FIJ and IIJ double-notch )lters exhibit comparable frequency responses (see Fig. - and 0) with identical notch frequencies G0 00/9 and G0 8890 The experiments

con)rm that FIJ )lter preserves theshapeofaprocessedpulsetrainbetterthenits IIJcounterpart. 4Conclusion The performance of digital IIJ and FIJ multiple stopbandnotch)lterswaspresented. Arovidedthe frequency responses of both IIJ and FIJ notch )lters are comparable, the almost equiripple FIJ notch )lter exhibits constant group delay and su- perior output responses over its IIJ counterpart. Twoexamplescon)rmthatFIJ)lterpreservesthe shapeofaprocessedpulsetrainbetterthenitsIIJ counterpart.
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I/51 0.1 0.2 0.3 0.4 0.5 0.6 0. 0.8 0.! −60 −50 −40 −30 −20

−10 /9 d8 Figure 0: Frequency response /0log j of a competitiveIIJdouble-notch)lterdesignedbythe methodin[-]. 50 100 150 200 250 300 350 −2 10 12 Figure 3: Double-notch FIJ )lter response to the inputpulsetrainsignalwithdutycycle-/: -0;. 50 100 150 200 250 300 350 −2 10 12 Figure 6: Double-notch FIJ )lter response to the inputpulsetrainsignalwithdutycycle-;: -0/. 50 100 150 0.2 0.4 0.6 0.8 1.2 1.4 Figure 8: Step response of the IIJ double-notch )lterfromFig. 0. 50 100 150 200 250 300 350 −2 10 12 Figure 9: Double-notch IIJ )lter response to the

inputpulsetrainsignalwithdutycycle-/: -0;. 50 100 150 200 250 300 350 −2 10 12 Figure ;: Double-notch IIJ )lter response to the inputpulsetrainsignalwithdutycycle-;: -0/.
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I/5D References [-] S. %. Aei, %. %. Tseng, IIJ ?ultiple Notch Filter Design *ased on Allpass Filter, IEEE Transactions on Circuits and Systems , vol.33, No./,February-559,pp.-00--08. [/] ?.Rl cek,Q.Tire s, FastDesignAlgorithmsfor FIJNotchFilters, Proc. of IEEE International Symposium on Circuits and Systems ISCAS394 Qondon-553,Rol./,pp./59-000. [0] ?.Rl cek,J.Pnbehauen, NolotarevAolynomi-

alsandOptimalFIJFilters, IEEE Trans. on Sig. Proc. , vol.39,pp.9-9-900,?arch-555. [3] ?. Rl cek, A. Nahradn ık, J. Pnbehauen, Ana- lytical Design of FIJ Filters, IEEE Trans. on Sig. Proc. , September/000,pp./906-/905.