A New Design Approach for Recursive Diamond-Shaped Filters

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1 A ew Desgn Approach for Recursve Damond-Shaped Flters RADU MATEI Faculty of Electroncs, Telecommuncatons and Informaton Technology Techncal Unversty Gh.Asach Bd. Carol I no., Ias 756 ROMAIA rmate@etc.tuas.ro Abstract: - Ths paper proposes a new desgn method for two-dmensonal damond-shaped IIR flters. The desgn starts from a desred D prototype flter, to whch a frequency transformaton s appled, whch leads to the D flter wth desred shape. The frequency transformaton s derved by specfyng the flter shape n polar coordnates n the frequency plane. The desgned flters also have the useful zero-phase property. Key-Words: - Two-dmensonal flter desgn, approxmaton theory, complex frequency transformatons Introducton Two-dmensonal flters play a maor role n the vast feld of mage processng applcatons. Varous desgn methods, ether entrely analytc or based on numercal optmzaton have been developed [], []. A commonly-used desgn technque for D flters s to start from a specfed D prototype flter and transform ts transfer functon usng varous frequency mappngs n order to obtan a D flter wth a desred frequency response. Some mportant papers regardng D flter desgn through spectral transformatons are [3] [5]. In [5] the problem of stablty s studed n detal for a large class of D flters, and the authors also approach the desgn of crcular, ellptcal-shaped and fan flters. Damond-shaped flters are currently used as antalasng flters for the converson between sgnals sampled on the rectangular samplng grd and the quncunx samplng grd. Varous aspects and desgn methods for damond flters were approached n [6] []. In ths paper we propose a new analytcal desgn method for damond-shaped flters. The technque s more general and t can be used for desgnng a class of flters whch can be specfed by a perodc functon n polar coordnates. The contour plots of ther frequency response, resulted as sectons wth planes parallel wth the frequency plane, can be defned as closed curves whch can be descrbed n terms of a varable radus whch s a perodc functon of the current angle formed wth one of the axes. Ths perodc radus can be developed as a ratonal perodc functon. The desgn s based on convenent D prototype flters wth a desred shape, whch wll correspond to the vertcal secton of the D flter; ths wll be fnally determned from ts prototype through a D to D frequency transformaton. The D flter functon results drectly n factorzed form, whch s an advantage n mplementaton. A class of more general polygonalshaped flters was approached n []. Two-Dmensonal Flters Defned n Polar Coordnates Ths secton s dvded nto three parts. The frst subsecton wll ntroduce the D to D frequency transformaton on whch s based the desgn of the class of D flters descrbed n polar coordnates, startng from D prototypes. Then some convenent D prototype flters are proposed. Subsecton.3 approaches the effectve desgn method of damond shaped flters.. Frequency Transformaton for the Desgn of D Flters from D Prototypes We wll approach here a partcular class of D flters, namely flters whose frequency response s symmetrc about the orgn and has at the same tme an angular perodcty. For such flters, f we consder any level contour resulted from the ntersecton of the frequency response wth a horzontal plane, the contour has to be a closed curve whch can be descrbed n polar coordnates by: ρ ρ( ϕ ) where ϕ s the angle formed by the radus OP wth the ω - axs, as shown n Fg.(a) for a four-lobe flter. Therefore ρϕ ( ) s a perodc functon of the angle ϕ n the range ϕ [, π ]. The proposed desgn method for ths class of D flters s based on a frequency transformaton of the form: F :, ω F( z ) () ISS: ISB:

2 Through ths transformaton we wll be able to obtan low-pass type flters, n the sense that ther frequency characterstc contans the orgn of the frequency plane, and they are symmetrc about the orgn, as n fact are most D flters currently used n mage processng. The frequency transformaton () s a mappng from the real frequency axs ω to the complex plane ( z ) and t wll be defned ntally through the ntermedate real frequency mappng of the general form: F : () ω F( ω, ω) ω + ω ρ( ω, ω) Here the functon ρ( ω, ω ) plays the role of a radal compressng functon and s ntally determned n the angular varable ϕ as ρϕ ( ). In the frequency plane ( ω, ω ) we can express: cos ϕ ω ω + ω (3) where ϕ s the angle formed by the current radus wth the axs Oω. If the radal functon ρϕ ( ) can be expressed n the varable cosϕ, usng (3) we obtan by substtuton a functon ρ( ω, ω ). Generally the functon ρϕ ( ) wll be determned as a polynomal or a rato of polynomals n varable cosϕ. For nstance, the four-lobe flter whose contour plot s shown n Fg.(a) corresponds to a functon: ρϕ ( ) a+ bcosϕ a+ b 8bcos ϕ+ 8bcos ϕ () whch s plotted n Fg.(b) n the range ϕ [, π ]. Throughout the paper we wll use the noton of template, common n the feld of cellular neural networks (Cs) [], to denomnate the coeffcent matrces correspondng to the numerator and denomnator of a D flter transfer functon H ( z ). We use here only odd-szed templates (3 3, 5 5 etc.) whch correspond to even order flters and allow for usng both negatve and postve powers of z n the transfer functon. It can be Fg.. (a) Contour plot of a four-lobe flter; (b) Perodc functon ρ( ϕ ) shown that n ths general stuaton the cosne of the current angle ϕ wth ntal phase ϕ can be expressed n the form: cos ϕ ω + sn ϕ ω +.5sn ϕ ωω cos ( ϕ + ϕ) ω + ω (5) correspondng to the smple expresson (3). Replacng ω and ω by the complex varables s ω and s ω, cos ( ϕ + ϕ ) can be wrtten n D Laplace doman, as detaled n secton.3.. Zero-Phase D Prototype Flters The proposed desgn method s based on zero-phase flter prototypes whose transfer functon s realvalued and can be expressed as a rato of polynomals n even powers of the frequency ω. In general ths flter wll be descrbed by: M k p( ) k k H ω b ω a ω (6) where M and s the flter order. Ths functon may be obtaned through a ratonal approxmaton of the magntude of an usual prototype, for nstance Chebyshev or ellptc. As the frst D prototype we wll consder a type- Chebyshev dgtal flter wth the followng parameters: order, stopband attenuaton Rs db and passband edge frequency ω p.5, where. s half the samplng frequency. The transfer functon n z for ths prototype flter has the expresson:.77 z.55 z+.77 Hp ( z) (7) z.857 z Usng a Chebyshev-Padé approxmaton (one of the most effcent), we can determne the followng realvalued zero-phase frequency response whch approxmates qute accurately the magntude of the orgnal dgtal flter functon: ω Hp( e ) Ha ( ω) (8) ω +.97 ω ω ω Ths method can be appled for any prototype flter lke (6). More generally, the D flter desgned n polar coordnates s rotated n the frequency plane wth a specfed ntal angle ϕ about one of the frequency axes, e.g. Oω. A maor advantage of ths technque compared to other flter desgn methods s that t naturally extends to ths general case, otherwse dffcult to approach analytcally. ISS: ISB:

3 .3 Damond-Shaped Flter Descrpton n Polar Coordnates n the Frequency Plane In ths secton, we fnd analytcally the mappng whch transforms a crcle of gven radus, n the frequency plane, nto a square, havng ts vertces on the same crcle. We refer to the geometrcal constructon n Fg.. In the frequency plane ( ω,ω ) spanned by the axes Oω, Oω, we consder the crcle of radus R. The default value wll be R π. Let us take an arbtrary pont P stuated on the frst sde of the square ( AA ), and let ϕ be the angle between the segment OP and the axs Oω ; ϕ s the angle between OA and axs Oω (ntal phase), where A s the frst vertex of the square. In the trangle POA we have the angles: OAP π ; POA ϕ ϕ ; OPA 3π ϕ + ϕ Applyng the sne theorem n the trangle POA, we fnd the measure of segment OP as a functon of the radus R and angle ϕ : R sn( OAP ) R OP (9) sn( OPA ) cos( ϕ ϕ π ) Thus we found the measure of OP as a functon of the current angle. However, (9) s vald only for ϕ n the range: ϕ [ ϕ + nπ, ϕ + ( n+ ) π ]. For a standard damond shaped flter we have ϕ, R and n the frst quadrant of the frequency plane we obtan: ρϕ ( ) cos( ϕ π ) () To express the value OP n for an arbtrary angle ϕ, when pont P n s located on any sde of the square, ncludng the vertces, we fnd a perodc functon ρϕ ( ) of the current angle ϕ. Ths functon has the perod Φπ and s plotted n Fg.3(a). The most convenent way to obtan a closed-form perodc approxmaton of ths functon s to use for nstance a ratonal approxmaton. As mentoned earler, the Chebyshev-Padé approxmaton usually gves best results. We look for such a ratonal approxmaton for the functon: ρ ( ϕ) cosϕ () over the phase range ϕ [ π, π ] n powers of the varable cos ϕ, whch s a perodc functon wth perod π. In ths way, the ratonal functon wll actually approxmate the functon ρ ( ϕ ) over Fg.. Square nscrbed n the crcle of radus R n the frequency plane, wth an ntal phase ϕ the entre range [,π ]. Snce the functon ρϕ ( ) s not dfferentable n the ponts ϕ π, π,, π (correspondng to square vertces) as can be notced n Fg.3(a), we wll consder the functon ρ ( ϕ ) on the range ϕ [ π, π ], whch s dfferentable everywhere wthn ths nterval. We frst obtan: ( ϕ ) ρ( ϕ) () cosϕ (.3 ϕ ) At ths step we make the change of varable: x cos( ϕ) ϕ.5 arccos x (3) and we get the ntermedate functon: ρ x+.7 x ( x) x x () Returnng to the ntal varable ϕ.5 arccos x, by substtutng back x cos( ϕ ), we obtan a ratonal approxmaton n powers of cos( ϕ ). In ths expresson we must replace ϕ by ϕ π, n order to get the fnal approxmaton for the radal compressng functon ρϕ ( ) : cos( ϕ)+.897 cos(8 ϕ) ρϕ ( ).5867 cos( ϕ)+.967 cos(8 ϕ) (5) Ths functon s plotted n Fg.3(b) and s a very accurate approxmaton of the orgnal functon n Fg.3(a). Usng trgonometrc denttes, ths can be expressed as a ratonal expresson n (cos ϕ ) n, wth n : ( x+.37)( x+.56)( x.56)( x.37) ρ( x).756 ( x +.3)( x +.36)( x.36)( x.3) (6) where by x we denoted here (cos ϕ ). At ths pont we fnally reach an expresson of the radal functon ISS: ISB:

4 (a) (a) (b) (b) Fg.3. (a) Perodc functon ρϕ ( ) ; (b) ts perodc approxmaton ρ ( ϕ ) ρϕ ( ) of the frequency varables ω and ω,.e. ρ( ω, ω ). Substtutng x (cos ϕ ) from relaton (3) nto (6), we obtan the fnal expresson wrtten n a compact form: ρ( ω, ω ) k ( ω + bω ) ( ω + a ω ) (7) where the numercal values of the coeffcents k, b, a are: k.3875, b.5767, b.537, b 65.53, b , a.89787, a.3, a , a The next step s to use ρ( ω, ω ) from (7) n the frequency transformaton general expresson (). Replacng ω and ω by the complex varables s ω and s ω, the frequency transformaton can be wrtten n the D Laplace doman: ( s + s) ω F( s, s) ρ ( s, s ) ( ) ( + ) k s + s s bs ( s + as) (8) where k Fnally we have to determne a transfer functon of the D flter H ( z ) n the complex plane ( z ). Ths can be acheved f we fnd a dscrete counterpart of the functon ρ( ω, ω ), denoted R( z ). A possble method s to express the functon ρ( ω, ω ) n the complex plane ( s, s ) and then fnd the approprate mappng to the plane (c) (d) Fg.. Frequency responses and contour plots of two damond flters for the radus values: (a), (b) R. ; (c), (d) R.6 ( z ). Ths can be acheved ether usng the Euler approxmatons, or the blnear transform. The blnear transform s a frst-order approxmaton of the natural logarthm functon, whch s an exact mappng of the z plane to the s plane. For our purposes the sample nterval takes the value T so the blnear transform for the two varables s and s n the complex plane ( s, s ) has the form: ( z ) ( z ) s s (9) ( z + ) ( z + ) Substtutng s, s n each factor of the general form ( s + as ) from (8), we fnd after some algebra the frequency transformaton n z wrtten n matrx form: T Bz ( ) Z [ B] Z ω F( z) () T Az ( ) Z [ A] Z where Z and Z are vectors of the form: Z z z z z () Z z z z z where denotes matrx/vector product. Each factor of general form ( s + as ) from (8) wll have an expresson n z correspondng to a symmetrc template, wrtten as a sum of two matrces: A + a A+ a A () R where A R s the matrx A rotated by 9. ISS: ISB:

5 Fnally we obtan the matrces B and A from relaton () as: () B B ( B B B3 B) (3) () A A * ( A A A3 A) where B 8 ; A () Here by * we denoted matrx convoluton and A () A*A. The frequency responses and contour plots for two damond-shaped flters desgned wth the proposed method are shown n Fg. and have the ndcated radus values. They show a very precse damond shape and a relatvely steep transton band. The stop-band regons show no vsble rpple or other dstortons. Concluson We proposed an analytcal desgn method for recursve damond-shaped flters, based on the specfcaton of ther shape n polar coordnates. The method however s more general and apples to any D flter whch can be descrbed n ths way. The desgn technque s based on adequate zero-phase flter prototypes and uses a specfc frequency transformaton. The desgned flters are effcent and have relatvely hgh selectvty. Further research may focus on combnng ths analytcal method wth numercal optmzaton technques n order to obtan more effcent flters of ths type. Proceedngs of the st Internatonal Conference on Mcroelectroncs, -7 Sep. 997, s, Yugoslava, Vol., pp.79-7 [7] Y. C. Lm, S. H. Low, The synthess of sharp damond-shaped flters usng the frequency response maskng approach, IEEE Int. Conf. on Acoustcs, Speech, and Sgnal Processng, ICASSP-97, - Apr. 997, Munch, Germany, Vol.3, pp.8-8 [8] S. H. Low, Y. C. Lm, Synthess of sharp -D flters usng the frequency response maskng technque, Proc. of IEEE Int. Symp. on Crcuts and Systems, ISCAS '97, Vol., pp. 5-8, 9- June 997, Hong Kong [9] S. H. Low, Y. C.Lm, A new approach to desgn sharp damond-shaped flters, Sgnal Processng, Vol. 67 (), May 998, pp [] K. Inu, T. Yoshda, A. shhara,. Fu, Desgn of two-dmensonal maxmally flat damond-shaped half-band fnte mpulse response flters wth rectangular supports of mpulse response, Electroncs&Communcatons n Japan, Vol. 78 Issue, pp [] L.O.Chua, L.Yang, Cellular neural networks: theory, IEEE Transactons on Crcuts and Systems, Vol.35, pp.57-7, 988 [] R. Mate, Desgn method for polygonal spatal flters, The IEEE Regon 8 EUROCO 7 Conference Computer as a Tool, Sept. 9-, 7, Warsaw, Poland, Proceedngs Vols.-6, pp. 3-3 References: [] D. Dudgeon, R. Mersereau, Multdmensonal Dgtal Sgnal Processng, Prentce Hall, 98 [] W.S. Lu, A. Antonou, Two-Dmensonal Dgtal Flters, CRC Press, 99 [3]. A. Pendergrass, S. K. Mtra, E. Jury, Spectral transformatons for two dmensonal dgtal flters, IEEE Transactons on Crcuts& Systems, vol. CAS-3, pp. 6-35, January 976 [] K.Hrano, J.K.Aggarwal, Desgn of twodmensonal recursve dgtal flters, IEEE Trans. Crc. Syst., CAS-5, pp.66-76, Dec. 978 [5] L. Harn, B. A. Sheno, Desgn of stable twodmensonal IIR flters usng dgtal spectral transformatons, IEEE Trans. on Crcuts and Systems, vol. CAS-33, pp. 83 9, May 986 [6] D. V. Tosc et al., Symbolc approach to D borthogonal damond-shaped flter desgn, ISS: ISB: