Design of Digital IIR Filters with the Advantages of Model Order Reduction Technique

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1 Ittiol Joul of Elctoic Egiig Rch ISSN Volu Nu Rch Ii Pulictio htt:// Dig of Digitl IIR Filt with th Atg of ol O Ructio Tchiqu K. Rh, A. Nilu 3.uuy Lctu i Dtt of Elcticl Elctoic Egiig, VCET, Eo, Tilu, Ii-638 E-il: ych@yhoo.co Pofo H of Elcticl Elctoic Egiig, BIT, Sthygl, Tilu, Ii D i Elcticl Elctoic Egiig, BIT, Sthygl, Tilu, Ii-638 Atct I thi, w ol o uctio hoo i itouc t th ig tg of li h igitl IIR filt. Th colxity of yt c uc y otig th ol o uctio tho i thi ig. I thi ix tho of ol o uctio i oo fo li IIR filt. Th oo tho loy th tg of fcto iiio tchiqu to i th uc o oito olyoil th uc o uto i oti o th ultt oito olyoil. Th o uctio tchiqu i u to uc th ly uit t th ig tg of IIR filt. Th liity of th oo tho i illutt with ig xl i fqucy oi tility i lo xi with hl of yquit lot. Kywo: O uctio, Fcto iiio tho, Nyquit lot, IIR filt, o ix J. Itouctio A iott t i th lot of igitl filt i th titio of lizl tf fuctio z oxitig th gi fqucy o cifictio. If IIR filt i i, it i lo cy to u tht z i tl. Th oc of iig th tf fuctio z i cll igitl filt ig. Aft z h oti, th xt t i to liz it i th fo of uitl filt tuctu. Th two o iu tht to w fo o c

2 9 K. Rh, A. Nilu.uuy lo th igitl tf fuctio z. Th fit foot iu i th lot of ol filt fqucy o cifictio fo th quit of th oll yt i which th igitl filt i to loy. Th co iu i to ti whth FIR o IIR filt i to ig. Th o uctio tchiqu i th itouc to oti quilt uc IIR li h filt. Th thticl ol oti fo th thoticl citi tho lt with th gi yt olly ult high o ol. Such yt ifficult to lyz lo th ig of cotoll co too ifficult. Oiouly, th i fo ll ol. Sll ol ol tht ci th hio of yt ccutly, without th itg of ucy til. Thi l th olig of coul colx ho i ol ti. Fit of ll, th ol houl ll th th oigil ol, ig tht thy couttiolly l ig. Thi hio ut i t lt o wy col to th hio of th oigil ol. Pfly, th ll ol hyiclly wll thticlly otit. Thi th ol ittl fficit. Th hot of thticl oc to i ll ol fo th fil of wht w cll ol O Ructio OR. Th o fo th i tht th ol i uc i iz y o tchiqu. Th ol i u to ly t, ig i y hyicl lw utio. Soti th tho uily cll Ruc O olig., which houl th ccout fo th t of olig i uch wy tht th i ol ll ight wy. I y c it i y wll oil to uc th iz of ol. W ly w tht thticl ul of ictiztio y l to ol which too lg h too uch til fo th qui ciio. Alo th fil outut of th ol h o flxiility i th qui ccucy. I lictio ll o i oft iil. Thi flxiility gi oo fo ll oxitio ft tho. Futho, w oft oly itt i cti tt of th ol, fo itc oly i th outut u to cti iut. Thi, whil th ol coti ifotio of iut tht io. ol O Ructio ti to quicly ctu th til ftu of tuctu. Thi tht i ly tg of th oc, th ot ic oti of th oigil ol ut ly t i th ll oxitio. At cti ot th oc of uctio i to. At tht oit ll cy oti of th oigil ol ut ctu with ufficit ciio. Th o uctio of li ti iit yt i li i lot ll fil of lcticl giig. Th u of o uc ol fo tt iultio of colx yt i lot i th utilizig full o ol. Thi i u to th fct tht th low o tf fuctio c lyz o ily. Thfo, o uctio lgoith t tchiqu i th itgt cicuit couity fo lyi, oxitio iultio of ol iig fo itcoct lctogtic tuctu lyi. Cotiol uctio tho ill i littu fo ucig th o of lg-cl li IO yt i fqucy oi [ ]. So of th xitig tho o th glol oti of h oo i th

3 Dig of Digitl IIR Filt with th Atg 9 littu ogt which th fqucy oi tho uig th cotiu-fctio xio of th clicl P oxitio wll ow fo thi couttiol ilicity tchig of fw iitil ti-ot. How, tility of uch uc-ol i ot gut though th oigil yt i tl. Futh, o ix tho h uggt y coiig th lgoith of two ifft uctio tho [5 7]. I it of hig l uctio tho, o lwy gi th tifctoy ult fo ll th yt. I lt c, c out i ol o uctio o coct of oit ol ttio o oit ig lu ttio, Routh oxitio, tility-qutio tho. A ltti och uggt y u of ch i to u th tho logig to th co ctgoy tio o to oti tt uc-ol xt th licility of th tho. I thi coctio th wo of Shh [8], Ch, t l [9], [], Pl, t l [] ut, t l [], woth tioig. S K Bhgt [3] oo o w ix tho fo th ilifictio of igl-iut igl-outut yt of Stility ig tho, uch, g- coicl xio. I th oo tho, ix tho i oo fo th uctio of li h igitl IIR filt. Th oti filt tf fuctio gi ig cifictio i cot i to tf fuctio i oi fo lt it i to ol o uctio tg. Th oito olyoil of uc o ol i oti y uig th fcto iiio tho uto olyoil i oti y oo tho. Th uc o ol oti y th oo tho will th tility of th oigil yt. Th oo tho i xli i th followig ctio. Digitl IIR Filt Scifictio A i th c of log filt, ith th gitu /o th h ly o i cifi fo th ig of igitl filt fo ot lictio. I o itutio, th uit l o o th t o y cifi. I ot cticl lictio, th ol of itt i th lot of lizl oxitio to gi gitu o cifictio. Th h o of th ig filt c coct y ccig it with ll ctio. I thi, w tict ou tttio to th gitu oxitio ol oly. I th c of IIR filt ig, th ot coo ctic i to cot th igitl filt cifictio i to log low ototy filt cifictio, to ti th log low filt tf fuctio H tig th cifictio, th to tfo it ito th i igitl filt tf fuctio z. Thi och h wily u fo y o uch th xti tl ill, yil clo-fo olutio thy highly c. Th gitu o cifictio of igitl filt i th i th to gi with o cctl tolc. I itio, titio i cifi tw th th to to it th gitu to o off oothly.

4 9 K. Rh, A. Nilu.uuy Coi th low filt with th gitu of, i th fi y, w qui tht th gitu oxit uity with o ±δ, i.., - δ δ, fo I th to, fi y π, w qui tht th gitu oxit zo with o of δ, i.., δ, fo π. Th fquci ctily cll th g fqucy th to g fqucy. Th liit of th tolc i th to, δ δ uully cll th il lu. Not tht th fqucy o of igitl filt i fuctio of. A ult, th igitl filt cifictio gi oly fo th g π. Digitl filt cifictio oft gi i t of th lo fuctio, A - log, i B. H th il α th iiu to ttutio α gi i B, i.., th lo cifictio of igitl filt gi y α -log - δ B α -log δ B 3 Th to g fquci, i ot lictio, cifi i Hz, log with th lig t of th igitl filt. Sic ll filt ig tchiqu lo i t of oliz gul fquci, th cifi citicl fquci to oliz fo cific filt ig lgoith c li. Lt F T, F P F S ot ctily th lig fqucy, th to g fquci ctily. Ω πfp P FT FT Ω S πfs 5 FT FT Fo th ig of IIR igitl low filt z o th coio of log low filt H, th filt o of H i fit tit fo it cifictio. Th o of utt woth low filt i oti fo th qutio log / 6 log / Wh, Ω ε Ω A Th ultt IIR filt tf fuctio i Z oi i u fo uctio of IIR filt colxity. Sttt of Pol Coi th o li h igitl IIR filt ot i Z oi,

5 Dig of Digitl IIR Filt with th Atg 93 X z A A z A z A z 7 z B z Y z B B z B z. B z Whil lyig th oo ol o uctio tchiqu, th li tfotio tchiqu ZS i li to oti th tf fuctio i oi. N 8 D. [ i ] xq i,,. wh,,.q,,,.,,,,.. -. Th cooig th < o uc o i of th fo N 9 D. wh [ i ] xq i,,.,,.q,,, -.,,,.. -, I li tfotioz- i th li t qutio 9 to fo th uc o li h IIR filt. I thi, th oti uc ol ti th iott chctitic of th oigil igitl IIR filt yt oxit it o cloly oil fo th ty of iut. Poo tho Th uc o oti y th oo tho coit of two t. I fit t, th uc o oito olyoil i oti y uig th tho ci i []. Th uto olyoil i oti futh i co t o th ultt oito olyoil. St Otiig th oito olyoil fo th gi yt tf fuctio Th th o oigil yt gi i qutio 8 i qut to th th o uc ol t y th qutio 9

6 9 K. Rh, A. Nilu.uuy.. O co ultilyig gig th qutio By qutig th cofficit of th ow of o oth i i th qutio ilifyig th ltio, th uc o oito olyoil i oti, D..... St Coi th gi yt tf fuctio gi i 8.. B B B B A A A K ; 3 Wh, K, A i i ; i,,,3 B ;,,,3 Lt th tf fuctio of th oxitig low-o yt,. q q D D D D C C C K wh q Th followig ltio houl tifi cloly oil fo 5.. q q C C C B B B B D D D D A A A. u u l l l l 6 Wh, u q 7 Th qutio ouc th ow of c witt, u u L L L 8 u u L L L 9 To tify th o qutio,

7 Dig of Digitl IIR Filt with th Atg 95 L L L u ; if u If u<, th th o gt y th low o ol i, ε O coig th qutio, w gt x i x x ixi i x L fo x,, 3..u y i x y lil yi i y l 3 fo y,, 3.. l Fo th qutio 3, th uc o ol uto cofficit c oti. Filly th uc o ol i i th fo of q C C C q K D D. D D Th tf fuctio oti i qutio i cot i to z oi y uig th i li tfotio. Dig Exl Coi th igitl IIR filt with th tf fuctio i ict fo Hz i qutio 5 th gl tuctu of lttic-l IIR filt i how i fig.. 3 z z z H z 3.567z.65z z z z 3 z.567z.65z z.375z z.567z.65z X f N f N- f f f N g N - N - g z - Z - g - Z - C C C C N N- C Figu : Stuctu of Lttic-l IIR filt oh N th o.

8 96 K. Rh, A. Nilu.uuy Lt,.5833z.375z.667 z 8 3 z.567z.65z.333 By lyig th li tfotio ZS, th tf fuctio z co i gi y, By lyig th t- tt i th oo tho, th uc o oito olyoil i oti, D With hl of oito olyoil D, th uto olyoil i oti fo th oo tho, N O lyig i li tf fuctio, th uc o tf fuctio i z oi i oti,.5z.3z.73 z.5z.6z.56 H 3 Th ult w co i oth cotiuou wll i ict oi how i fig. fig.3 ctily. Th o ix J tw th oigil uc o filt i oti.93. Figu : Coio of St o.

9 Dig of Digitl IIR Filt with th Atg 97 Figu 3: Coio of oigil uc o IIR filt i ict itt. Th clo of th high o filt with uc o IIR filt i fqucy oi i how i fig.. Th uc o IIR filt clo with oigil IIR filt lo th clo loo tility. Th uc o IIR filt h th ol t i i. Th i o ol t ight hlf of th -l hc th uc o yt i tl. Th tility citio i hc o with th hl of Nyquit lot i how i fig.5. Figu : Fqucy o of oigil uc o IIR filt.

98 K. Rh, A. Nilu.uuy Figu 5: Nyquit lot of oigil uc o IIR filt. Th uc o tf fuctio z H i lo c tt,.5.3 z.73 z H z 33.5.6 z.56 z Th ict fo of uc o IIR filt c t i uc o Lttic-l IIR filt fo o coio.

10 98 K. Rh, A. Nilu.uuy Figu 5: Nyquit lot of oigil uc o IIR filt. Th uc o tf fuctio z H i lo c tt,.5.3 z.73 z H z z.56 z Th ict fo of uc o IIR filt c t i uc o Lttic-l IIR filt fo o coio. Coi th ict fo of Ltticl IIR filt i th fo of z.5.3 z.73 z H z 3 z.5.6 z.56 z Fo th qutio 3, th lttic-l filt co-fficit oti.5,.6,.56,.5,.3.73.th cofficit c i oti with th qutio of, C i i C i ;, -,.. 35 i Th lu of t i oti fo th o qutio.. Th uc o tuctu of lttic-l IIR filt i how i fig.6. 36

11 Dig of Digitl IIR Filt with th Atg 99 f f x.56. g - - g Z - Z - C.5 C.37 C.3 y Figu 6: Stuctu of Ruc o Lttic-l IIR filt. Cocluio Thi oo th ol ol o uctio fo th ig of IIR filt. Th uc o ol ult i uctio i u of ly uit oth coot i th filt ig tg. Th uc o ol h clo with ct to th high o yt i thi foc chctitic. Th oo tho i thticlly il lo li to th lg cl yt with uttil it o oth cotiol tho. Th iultio ult how th goo oxitio tility citio of th uc o ol. Rfc [] R. P, J. Pl, A. K. Pt, ultiil yt oxitio uig olyoil iti, Joul of th Ititutio of Egi, ol. 76, , 995. [] Y. Bititz U. Sh, iil P ol uctio fo ultiil yt, Joul of Dyic Syt, ut Cotol, ol. 6, o., , 98. [3] C. F. Ch, ol uctio of ultiil cotol yt y of tix cotiu fctio, Ittiol Joul of Cotol, ol., o.,. 5 38, 97. []. R. Clf. Hly, Cotiu fctio ol uctio tchiqu fo ultiil yt, Pocig of th Ititutio of Elcticl Egi, ol., o. 5, , 97. [5] R. P, P ty ol o uctio fo ultiil yt uig outh oxitio, Cout & Elcticl Egiig, ol. 6, o. 6,. 5 59,.

12 K. Rh, A. Nilu.uuy [6] R. P, A. K. ittl, S. P. Sh, A ix tho fo th uctio of ulti-il yt, Joul of th Ititutio of Egi, ol. 85,. 77 8, 5. [7] L. Shih Y.Wi, A ix tho fo ultiil yt uctio, IEEE Tctio o Autotic Cotol, ol., o. 3,. 9 3, 975. [8] Y Shh, ol Ructio uig th Routh Stility Citio th P Aoxitio Tchiqu, Ittiol Joul of Cotol, ol, o 3, 975, [9] T C Ch, C Y Chg K W H, ol Ructio uig Stility qutio tho th P Aoxitio tho, Joul of Fli Ititut, ol 39, 98, [] T C Ch, C Y Chg K W H, ol Ructio uig Stility qutio tho Cotiu Fctio tho, Ittiol Joul of Cotol, ol 3, o, 98, 8-9. [] J Pl, A K Sih N K Sih, Ruc-o olig uig Pol Clutig Ti-ot tchig, Joul of Th Ititutio of Egi Ii, Pt El, ol 76, 995, -6. [] D K ut, S K Bhgt J P Twi, A ix tho fo th Silifictio of Li Dyic Syt, i Pocig of Ittiol Cofc o Cout Alictio i Elcticl Egiig Rct Ac, IIT, Roo, Fuy -3,, [3] S K Bhgt, J P Twi T Sii, So ix tho fo th Silifictio of High-o Sigl-iut Sigl-outut Syt, IE I Joul.EL, Vol 85,.-3, St. [] K.Rh, A.Nilu.uuy, O uctio y o iiiztio tchiqu, Pocig of Ittiol Cofc o Coutig, Couictio Ntwoig 8, -6, Dc 8.

OSXX1608C1A. Features. Outline Dimension. Applications. Absolute Maximum Rating (Ta=25 ) Directivity. Electrical -Optical Characteristics (Ta=25 )

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