A BILINEAR TRANSFORMATION METHOD OF ELLIPTIC IIR LOWPASS FILTER (LPF) DESIGN

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1 IJRRAS 8 () August pdf A BILINEAR TRANSFORMATION METHOD OF ELLIPTIC IIR LOWPASS FILTER (LPF) DESIGN Umar,. & Kwaha, B. J. Departmet of SLT, Federal Polytechic, Nasarawa. Nijeria Departmet of Physics, Uiversity of Jos, Nigeria kwaha68@yahoo.com, uyakubu64@yahoo.com ABSTRACT A Elliptic lowpass digital filter was desiged usig the MATLAB tool box. The programme code was based o the biliear trasformatio method through computatio ad simulatio based o the ecessary equatios uderlyig filter desig. At iput samplig frequecy, f s = 400Hz, peak ripple value at passbad, A pass, = Hz ad peak ripple value at badstop, A stop, = 0.45H Z, the Elliptic LPF has its lower 3dB poit at 0Hz ad upper 3dB poit at 85Hz,givig a BW of 85Hz. It was observed that the ripples of the Elliptic LPF is idepedetly adjustable which makes the filter maximally isesitive to compoet variatios due to a faster trasitio gai betwee the passbad ad the stopbad. This makes it suitable for use to process basebad sigals i radio trasmittig statios that employ pulse coded modulatio (PCM). Its steep roll off is highly efficiet i blockig off harmoic distortios arisig from processig image frequecy i superheterodye systems which if eglected causes iterferece with other commuicatio systems at close frequecies. Keywords: Lowpass IIR Digital Filter Desig Usig MATLAB.. INTRODUCTION Mitra (00) defied a filter as a frequecy selective device that fuctios by acceptig certai frequecy iput sigals ad atteuatig others referred to as uwated compoets. Digital filters are the basic buildig block of ay Digital Sigal Processor (DSP) as reported by Kaiser (974). Rabier ad Gold, (975) classified filters ito two groups-aalog filters ad Digital filters, which are categorized ito lowpass, highpass, badpass ad badstop. A lowpass filter passes low frequecies, from dc (zero frequecy) to ω o (lower cut off frequecy) ad atteuates all other frequecies outside this rage. A Elliptic filter is a electroic filter with equalized ripple behavior (equirippled) i both the passbad ad stopbad but with a uique sharp skirt i its trasitio regio. The gai of lowpass elliptic filter, give as a trasfer fuctio of its agular frequecy may be represeted as (Smith, 006; Smith, 004; Smith, 005). G R, o R is the th order of elliptic ratioal fuctio ad ω o is the cutoff frequecy, ε is the ripple factor, ξ is the selectivity factor while ω is the cetre frequecy. Filters are desiged i accordace to what kid of applicatio is evisaged. A first order lowpass IIR digital filters has a trasfer fuctio give by (Mitra, 998): z) z H ( z where < for stability. The trasfer fuctio i equatio () has a zero z = -, where =, which is i the stop bad of the filter. It has a real pole at z =. As icreases from 0 to, the magitude of the zero vector decreases from to. The maximum value of the magitude fuctio is uity at = 0, ad the miimum value is zero at, which may be represeted as (Mitra, 00; Mitra, 998) j j H ( ), H ( ) 0 (3) H ( j ) is a mootoically decreasig fuctio of from = 0 to =. The squared magitude fuctio ca be derived as (Mitra, 00; Mitra, 998): j q cos H ( ) (4) cos The derivative (Mitra, 00; Mitra, 998): () () 34

2 IJRRAS 8 () August 0 j H ( ) si (5) d cos This is o-positive i the rage 0 verifyig agai the mootoically decreasig behaviour of the magitude j fuctio. To determie the 3dB cut off frequecy c, we set H ( e ) i equatio (3) so that cosc or cosc cosc (6) This whe solved yields (Mitra, 00; Mitra, 998): cos c (7) The above quadratic equatio ca be solved for yieldig two solutios. The solutio resultig i the stable trasfer fuctio H (z) is give by (Mitra, 00; Mitra, 998) si H (z) = c (8) cosc This research employs the MATLAB toolbox to desig, maipulate ad aalyze digital filter efficacy. The eed to have a versatile ad flexible tool i the desig ad implemetatio of digital filters ad the rapid evolutio i the field of computig ad commuicatios due to the availability of various hardwares at relatively cheap costs with high performaces traslates to advatages that may be derived especially whe usig DSP.. DIGITAL FILTER RESPONSE I most frequecy domai applicatios frequecy respose is most importat because it icludes the passbad, stopbad ad trasitio bad. The impulse respose is used i implemetig a digital filter. To create a ifiite impulse respose (IIR), the impulse respose y() is covolved with the iput sigal x() (Kale, 004; Mitra, 00). bix aj i j i0 j where x is the iput sigal, y is the output sigal ad the costats b ij = 0,,, m +, a ij = 0,,,. These are called the coefficiets. Impulse respose represetatio of a recursive filter as a LTI system is give as (Soderstorm, 983). X (0) X X X X () This gives rise to the geeral trasfer fuctio H(z) which cotais polyomials i both the umerator ad deomiator. The roots of the deomiator determie the pole locatios of the filter, ad the roots of the umerator determie the zero locatios. The trasfer fuctio of IIR filter is give by as (Smith, 006; Smith, 004; Smith, 003). H z X z z P biz j0 Q ajz i j P is the feed forward filter order, b is the feed forward filter coefficiet, Q is the feedback filter order, a the feedback filter coefficiet ad z - represets uit delay. Smith (004) expressed the frequecy respose as a ratio of DTFTs. M jt DTFTT B jwt M 0 H (3) A jt DTFTT A a 0 8 (9) () 35

3 IJRRAS 8 () August 0 3. DESIGN APPROACH, SPECIFICATIONS AND METHOD I the desig of a IIR digital filter, a aalog IIR filter is coverted to its digital form by applyig discretizatio techique such us biliear trasformatio method. This is doe by covolvig the iput sigal with the impulse respose of the appropriate filter, (Wu, 004). It is the implemeted by trasformig the trasfer fuctio of the aalog prototype filter ito the system fuctio of a digital filter with similar characteristic. The biliear trasformatio from the s-plae to the z-plae is derived by applyig the trapezoidal umerical itegratio approach to the differetial equatio of H a (s) that leads to the differece equatio represetatio of G(z). The parameter T represets the step size i the umerical itegratio. z G (z) Ha (s) (4) T z The iverse biliear trasformatio is applied to the digital filter specificatios to obtai the desired trasfer fuctio G(z) from the aalog trasfer fuctio H a (z). Usig the trasformatio ad otig that s = j 0, we may write j 0 z (5) j0 This has a uity magitude ad this implies that a poit o the liearity axis i the s-plae is mapped oto a poit o the uit circle i the z-plae. I the geeral case for s = z Therefore, z 0 j0 0 j0 = 0 j0 0 j0 0 0 (7) 0 0 A poit o the j axis i the s-plae ( 0 = 0) is mapped oto a poit o the uit circle i the z-plae as z =. I the mappig of the s-plae ito the z-plae via the biliear trasformatio as illustrated, there is o aliasig due to the oe to oe mappig. A computer program is developed for the desig of the IIR lowpass Elliptic filter based o MATLAB program. The digital filter coefficiets, which represet the physical values of resistors, capacitors ad iductors, are determied by computatio. These filter coefficiets are used with the sampled data values to perform the filter calculatios. (6) Fig. A typical Elliptic filter respose (Smith, 006; Smith, 004) I the first step of the desig, the filter order N ad the frequecy scalig factor ω are determied from the give specificatios. The parameters obtaied i step, are used to determie the coefficiets of the trasfer fuctio usig the followig fuctio statemets. [N, ω ] = Ellip ord (ω p, ω s, R p R S ) [N, ω ] = Cheb ord (ω p, ω s, R p R S ) 36

4 IJRRAS 8 () August 0 [N, ω ] = Cheb ord (ω p, ω s, R p R S ) (8) R p ad R S are passbad ripple ad miimum stopbad atteuatio respectively. Both are specified i db. ω is the frequecy scalig factor. ω p, ad ω s are the ormalized passbad ad stopbad frequecies respectively. These frequecy poits are betwee 0 ad ad samplig is equal to. If the samplig frequecy F S, the passbad edge frequecy F pass are specified i Hz, the p Fp ad s Fp (9) Ft Ft Specificatios: Passbad edge frequecy, F pass = 0Hz Stopbad edge frequecy, F stop, = 337Hz Peak ripple value at badpass, A pass, = Hz Peak ripple value at badstop, A stop, = 0.45Hz Samplig rate, F S = 400Hz These specificatios were used with the M-files i the sigal processig tool of the MATLAB software. The LPF has its ow M-files ad specificatios which iclude; [z, p, k] = iirellip(, R P, R S, ω P ) [z, p, k] = iirellip(, R P, R S, ω P, ftype ) [b, a] = iirellip(, R P, R S, ω P,) [b, a] = iirellip(, R P, R S, ω P, ftype ) (0) Table. Parameters of the Desiged Lowpass Elliptic Digital Filter Filter specificatios Badpass frequecy Hz F stop Normalized agular frequecy rad/s o /pί Upper 3dB frequecy Lower 3dB frequecy F s = 400Hz A pass = Hz A stop = 0.45Hz F pass = 0Hz BW = 85Hz o Gai (G) 37

5 IJRRAS 8 () August 0 Fig. The desiged Elliptic filter respose 4. DISCUSSION AND CONCLUSION This work was primarily focused o the desig ad implemetatio of IIR Elliptic digital lowpass filter. Program to desig the filter was developed ad implemeted usig the MATLAB toolbox. The program was writte based o the ecessary equatios uderlyig the operatio of the filter i questio. The biliear trasformatio method was used to develop the program which was tried out severally to ascertai its flexibility ad accuracy before presetatio. For trasfer fuctio, [umd, ded = biliear um, de, f s, f p ] i the call fuctios, coverts s-domai trasfer fuctio give by um (i.e umerator) ad de (i.e deomiator) to a discrete equivalet i the z-domai. Row vectors um ad de specify the coefficiets of the umerator ad deomiator, respectively i descedig powers of s so that at f s of 400H Z, A pass, of H Z ad A stop,of 0.45H Z, the Elliptic LPF desiged has its upper 3dB poit at 85H Z ad its lower 3dB poit at 0H Z,givig a BW of 85H Z. It has a equirippled passbad ad stopbad ad a sharp roll-off rate which is i lie with predicted theory. This is quite perfect for basebad sigals that are pulse coded i format. The Elliptic lowpass filter is also suitable for use where degraded phase resposes ca be tolerated. The desiged IIR elliptic LPF has a equirippled passbad ad stopbad ad a sharp roll-off rate. This is i lie with predicted theory. This type of filter is attractive i commuicatios such as adaptive equalizatio, echo cacellatio, oise reductio, speech aalysis ad sythesis i PCM applicatios. 5. REFERENCES []. Kaiser J.F. (974) No-Recursive Digital Filter Desig usig the Io-Sih Widow Fuctio, Proc. 997 IEEE It. Symp. Circuit Theory, Pp. 0-3, 994. []. Kale I. (004) DES CMSA Web Based Filter Desiger, htm Secod Editio. [3]. Mitra S.K (998) Digital Sigal Processig: A Computer Base Approach, New ork, N: McGraw Hill. [4]. Mitra S.K (00), Digital Sigal Processig: A Computer-Based Approach, McGraw Hill Irwi, Secod Editio. [5]. Rabier, L.R. ad Gold, B. (975) Digital Filter Desig Techiques i Frequecy Domai IEEE proceedigs Vol. 55 Pp [6]. Smith, J.O. (006) Itroductio to Digital Filters. [7]. Smith, J.O. (005) Physical Audio Sigal Processig Digital Wave Guide Modelig of Musical Istrumets ad Audio Effects. [8]. Smith, J.O. (003) DFF. Mathematics of the Discrete Fourier Trasform. [9]. Smith, S.W. (004) A Scietist ad Egieers Guide to Digital Sigal Processig. [0]. Wu, O. (004) Biomedical Sigal ad Image Processig Sprig 00. HST 58J/6.555J/6.456J. 38