AN DIGITAL FILTER DESIGN SCHEME BASED ON MATRIX TRANSFORMATION
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1 Jourl of heoretil d Applied Iformtio ehology 5 th November. Vol. 5 No. 5 - JAI & LLS. All rights reserved. ISSN: E-ISSN: AN DIGIAL FILER DESIGN SCHEME BASED ON MARIX RANSFORMAION ZHANG LI ZHANG DAN ZHANG WEIXI College of Computer Egieerig Jigsu ehers College of ehology Chghou 5 Jigsu Chi Uiversity of Exeter he Quee's Drive Exeter Devo UK EX QJ UK Eletroi & Iformtio Egieerig Deprtmet Jigsu ehers College of ehology Chghou 5 Jigsu Chi ABSRAC his work disussed the desig steps of digitl filter d methods of trsformtio from log domi to digitl domi. I order to use omputer to solve the problem of reursive futio whose order is N this work derivtes trsformed lgorithm. he proposed sheme shifts system trsformtio to oeffiiet mtrix d vrible mtrix. his work explis how to fix oeffiiet mtrix d vrible mtrix i detil. Ad t lst it tkes dvtge of fesibility d vlidity of simultio lgorithm to verify it. he merits re s follows. First it is very diret d it pplies the lultio betwee mtrix d mtrix. Ad seod it is very geerl so o mtter it is from simultive low pss to digitl low pss or from simultive high pss to digitl high pss or from simultive bd pss to digitl bd pss this method be used. hird it is the test sie there re fewest umbers of lultios. Keywords: Bilier rsform Digitl Filter Mtrix Algorithms Z rsform INRODUCION Lier trsformtio is widely used i digitl simultio sigl proessig utomti otrol d system idetifitio. he so-lled lier trsformtio [] is to use: Z- s ( or Z S S o relie the trsformtio of trsfer futio from S domi to Z domi or the trsformtio of trsfer futio from Z domi to S domi. However the bilier trsformtio is kid of frtiol trsformtio therefore with the irese of system order the mout of omputtio will be iresig. If tht to lulte by hd will be big problem d it is ioveiet to use this trsformtio. I order to use omputer to solve the problem of order N s trsfer futio trsformtio here preset the lier trsformtio lgorithm the followig re the fetures [ ]: (rsformtio is relied through trsformtio mtrix d oeffiiet mtrix; it is esy to rry out by omputer ( he oeffiiet mtrix used i trsformtio be diretly derived; the elemets of oeffiiet mtrix be doe by dditio. ( he digitl low pss high pss bd pss bd stop be diretly trsferred. HEORIES o give log low pss filter prototype [ 5]: A As As As As As H( S B Bs Bs Bs Bs Bs o order oeffiiet vetor s: A A A A A A A B B B B B B B ( ( o ormlie the log filter the it hged to be: Z S f ( ot (
2 Jourl of heoretil d Applied Iformtio ehology 5 th November. Vol. 5 No. 5 - JAI & LLS. All rights reserved. ISSN: E-ISSN: f is the ut-off frequey s frequey [6]. Put H( Z S f is the smplig i H( s d there ome out Z ( ( / S C ( Z H H s H s If H( s is low pss filter trsfer from the bove the H( is lso digitl low pss filter. he prototype of digitl filter is H( Next to order H( b b b b Ad the oeffiiet vetor is: ( 5 b ( b b b b b b5 b Whe A A ( A A H( b b B B ( B B ( Here to let mtrix to preset the omputig proess of H( s H( ht is Whe H( b b b A A A A A (A A ( A A A B B B (B B ( B B B Here to let mtrix to preset the omputig proess of H( s H( ht is A A A Ad so whe 56 we get the trsfer mtrix to express with formul s: Here P A ( A A A A... A Now let s see the P mtrix whe P Whe P Whe P herefore to y P he elemets of the first olum i P omply with lw tht is Yg Hui trigle rrgemet. Next to itrodue Yg Hui trigle rrgemet: 5 6 From the fetures [7] of Yg Hui trigle rrgemet:. he figures i eh lie re symmetril d tur lrge from d the tur smll d go bk to.. he figure umbers of the umber lie re.. he sum of figures i umber lie is. eh figure is the sum of its left d right figure i the previous lie whih is ( C i C i C i 5. he first figure of umber lie is he seod is ( ( ( he third is he fourth is ( ( ( -
3 Jourl of heoretil d Applied Iformtio ehology 5 th November. Vol. 5 No. 5 - JAI & LLS. All rights reserved. ISSN: E-ISSN: Ad so EXAMPLES (! Exmple to use bilier trsformtio to he umber figure is desig Butterworth digitl low pss filter[9] of ( i!( i! order smplig frequey is khz Wht s more to extrt ll the elemets of first (smplig period is 5us d the db ut off olum i trsfer mtrix P d get the follows: frequey f khz d log Butterworth filter of order is H( s s s s ( ( ( 6 Here tully the is the previous give - (! thus use to isted the the it ( i!( i!! beomes whih is lso ( i!( i!! P ( i!( i! (5 As result to y P it is oly eed to kow its d the the elemets i the first olum be writte dow. FEAURES From the bove disussio we get fetures of trsfer mtrix [8]: First the first lie elemets of P oly be d ll re Seod the first olum elemets if i d re kow the P i be kow. ht is P i! ( i!( i! i (6 hird there is speil reltio i the mtrix ht is d P P P P (7 i j i j i j i j i j From these fetures if the order is kow the y trsfer mtrix P be lulted out. f Solve: to order ot d H( s s s s from formul ( A A A A B B B B Apply formul (6 d (7 it is esy to get: P o use the previous formul ( d put P A i A A A A b B 6 b B b B b B ht is b 6 b b b hus get the system futio of digitl filter: H( 6 Exmple : smplig frequey[] is khz us it is required to desig Butterworth digitl bd pss filter of order d its up d dow db ut off frequey is f 7.5kHZ f.5khz.
4 Jourl of heoretil d Applied Iformtio ehology 5 th November. Vol. 5 No. 5 - JAI & LLS. All rights reserved. ISSN: E-ISSN: Solve: first to get the eh ritil frequey of the eeded digitl filter. Ad the up d dow ut off frequey is 6 f f ω ω Alog low pss ut off frequey is ω [t( ω t( ] [t( t( ] From the ppedix the trsfer formul of trsformtio from log low pss prototype to digitl bd pss is (8 to use this to get H( Z whih is the eeded digitl bd pss filter prototype. E s D[ ] We get D is ω ω D ot( ot( We get E is ω ω os( os( E ω ω os( os( (8 We use this to substitute d get H( whih is the eeded digitl bd pss filter prototype. his reltioship formul is lso the trsformtio formul from log low pss prototype to digitl bd pss. Ad from N get the system futio of Butterworth filter of order is H( s s s s ( ( ( We put it bk i the S H( H( s s ( ( ( 6 H( 6 Now hge to lulte i other wy firstly to trsfer the log low pss to log bd pss d the use the trsfer mtrix P from the ppedix the formul of trsformtio from log low pss prototype to log bd pss is: s hl s (9 s( h l : they re the tul bd pss s up d dow ut-off frequey. From formul ( the ω d ω is kow the orrespodig log low pss orer frequey is ω ω t( t( Now let s ormlie to order tht is thus Ad from the ppedix the formul should be: h l s s ( s Put the bove formul bk to system futio of log low pss filter the: H( s s s s ( ( ( s s s s s s s s s s Here 6 from previous trsfer mtrix whe 6 P P herefore the orrespodig oeffiiet of digitl filter prototype be hieved from formul ( they re: 5
5 Jourl of heoretil d Applied Iformtio ehology 5 th November. Vol. 5 No. 5 - JAI & LLS. All rights reserved. ISSN: E-ISSN: uderstd. his whole proess be doe by 6 6 omputer t oe REFRENCES: [] Deg hogyi he filter s pst preset d future Globl Eletrois Chi Vol. No. 6 pp [] Federio Fot Use of the Nyquist Stbility b /8 6 b 6 6 / Criterio i the Desig of Itertive Audio Digitl Filters IEEE Sigl Proessig Letters b / 8 Vo. 8 No. pp b [] Zhg Weixi Digitl Sigl Proessig Chi b / 8 Mhie Press. b [5] Pei Dg d o Qi Alyti Phse b /8 6 Derivtives All-Pss Filters d Sigls of Miimum Phse IEEE Sigl Proessig Get Letters Vol. 59 No. pp [6] Jiyu Li Mrk J.. Smith wo-bd Hybrid b 6 b b b b b5 b6 FIR-IIR Filters for Imge Compressio IEEE rstios o imge proessig Vol. he write the digitl bd pss filter s system No. pp futio: 6 [7] Serge Proveher Prmeters Estimtio of H( Complex Multitoe Sigl i the DF Domi 6 IEEE rstios o sigl proessio Vol. 59 No. 7 pp. - 5 CONCLUSIONS [8] K Lug Lw N. Do. Mih Multidimesiol Filter Bk Sigl Reostrutio From From the previous lysis mtrix omputig Multihel Aquisitio IEEE rstios is the key of bilier trsformtio its worklod o Imge Proessig Vo. No. is: ( d the worklod of ommo pp.7-6 omputig is : ( therefore whe is [9] Ami Chebir Mtthew Fikus d Dusti G.Mixo Filter Bk Fusio Frmes IEEE lrge umber the mtrix omputig is muh ter rstios o Sigl Proessig Vol. 59 No. th the regulr omputig. Obviously it is esy to pp relie by omputer proedure beuse mtrix omputig is esy to rry out by omputer. his [] E. Rodrigue-Villegs A. J. Csso P. method lso be used to desig digitl filter it is Corbishley A Subhert Nopwer Low-Pss oly eeded to desig the idel log low pss Filter IEEE rstios o iruits d high pss bd pss d bd stop filter d the systems-ii: Express Brie Vo. 58 No. 6 to diretly bilier trsfer to get digitl low pss pp high pss bd pss d bd stop filter by mtrix. If there be little hge the it output grphs d dt d it be very ler d esy to 6
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