LOW SENSITIVITY DESIGN OF MULTIPLIERLESS ELLIPTIC IIR DIGITAL FILTERS Vlentn Ilev Anzov Dertment of Telecommunctons, Techncl Unversty of Sof Techncl Unversty, Dertment of Telecommunctons, 1756 Sof, Bulgr, e-ml: v@tu-sof.bg In ths er n mroved method of desgn of low senstvty multlerless elltc IIR dgtl flters s roosed. The desgn technque s bsed on the senstvty nlyss nd selecton of roer llss sectons. The llss sectons used to relze the llss structures re wth mnmzed senstvtes. Usng the symbolc nlyss, the exressons for hse senstvtes to multler coeffcents for the selected llss sectons hve been derved. The mltude senstvtes of the obtned relztons hve been nvestgted. The comrson between the resulted dgtl multlerless IIR flters hs been mde for dfferent coeffcents wordlength. The results re confrmed exermentlly. Keywords: IIR dgtl flters, llss sectons, low senstvty, multlerless 1. INTRODUCTION In VLSI mlementtons multler s very costly. Therefore t s very ttrctve to reresent the multlctons s seres of shft nd dd oertons. In ths wy n multlerless relztons the overll flter comlexty s sgnfcntly reduced. The desgn of multlerless IIR dgtl flters s very dffcult becuse of the hgh senstvty of ther frequency resonse to the multler coeffcents. Therefore, t s very mortnt to mnmze the frequency resonses senstvty. When the flter senstvty s low t s ossble to shorten coeffcents wordlength wthout voltng the flter secfctons. It s well known tht the flter structure consstng of two llss subflters n rllel s hvng very low coeffcent senstvty (eseclly n the ssbnd). Ths structure s lso very economcl t s relzng two n th - order trnsfer functons usng only n multlers. In [1][][3], t hs been shown tht the rllel-llss-flter-structure my hve smll number of shft nd dd oertons f the trnsfer functon (TF) s derved by the blner trnsformton from n elltc mnml Q-fctors (EMQF) nlog rototye. Becuse of the roertes of EMQF flters [4], (n+1)/ multler coeffcents cn be mlemented wthout quntzton. Then the remnng (n-1)/ less senstve coeffcents re quntzed usng cnonc sgn dgt (CSD) code. Ths clss of flters hs ll coeffcents mlemented wth smll quntzton error. The urose of ths contrbuton s to modfy ths roch nd to ly t to desgn of nrrowbnd lowss low senstvty multlerless IIR flters. The llss sectons used to relzed the llss structures re wth mnmzed senstvtes for trnsfer functon coeffcents vlues corresondng to oles ner z 1 [5][6]. The frequency resonses of the multlerless relzton so obtned nd the hse senstvty to multler coeffcents re comred wth those of multlerless IIR flter mlemented usng llss sectons selected by Mlć nd Lutovc [1][].
. IIR DIGITAL FILTERS DESIGN USING AN EMQF ANALOG PROTOTYPE The desgn method suggested by Mlć nd Lutovc [1][] strts wth hse resonse senstvty nlyss of the llss sectons tht mlements the desred dgtl IIR flter to ts multler coeffcents. The (n+1)/ most senstve coeffcents cn be mlemented s exct vlues wth mnml number of shft nd dd oertons by n rorte desgn of EMQF trnsfer functon. Then the remnng (n-1)/ less senstve coeffcents re quntzed usng cnonc sgn dgt (CSD) code. It s shown n [1][] tht n IIR flter, f derved from n EMQF nlog rototye, cn be desgned to hve n ll second order sectons of the llss networks one common coeffcent whose vlue deends only on the frequency t whch the flter ttenuton s nd my be djusted ccordng to the redetermned number of shft nd dd oertons. In ths wy, the drect control s estblshed over the vlues of (n-1)/ multlcton constnts nd they cn be mlemented s exct vlues, elmntng thus the nfluence of ther hgh senstvtes. In ths contrbuton, multlerless elltc flter desgn s bsed on the TF H formed by the blner trnsformton from n EMQF nlog rototye [1][3][4]. An mortnt roerty of EMQF flter s tht ts oles re lced on crcle wth the center t the orgn of the s lne. The rdus of the crcle s Ω, where the normlzed stobnd edge frequency s Ω nd for s j Ω the flter hs the ttenuton of. The relton between f (ssbnd edge frequency of the dgtl flter) nd f (stobnd edge frequency of the dgtl flter) nd the stobnd edge frequency of the nlog rototye Ω hs been estblshed n [3]. For gven f nd f, Ω s estmted from the exresson Ω. (1) The frequency where dgtl flter hs the ttenuton of corresonds n the nlog flter domn to the frequency Ω, nd cn be determned by the relton tg π f. () The trnsfer functon of n odd order IIR flter H cn be resented s sum of two llss functons H nd H b. The llss subflters H nd H b re usully relzed s cscde connecton of frst nd second order sectons. In order to obtn nrrowbnd lowss low senstvty multlerless IIR flter relztons, we need llss sectons hvng low senstvtes when relzng oles ner z 1. Such sectons hve been roosed nd nvestgted n [5][6]. We shll cll them ST1 (Fg. 1) (the frst order) nd STA (Fg. 1b) (the second order secton) nd they relze the followng trnsfer functons: (1 ) + z 1 d (1 d)(1 c) z + z H ST 1( z), H ST A ( z). (3) 1 (1 ) z 1 (1 d)(1 c) z + (1 d) z
) b) Fg. 1 Low senstvty llss sectons: ) frst order (ST1); b) second order (STA) For comrson the sme desgn rocedure wll be led for the oulr lttce wve sectons selected by Mlć nd Lutovc n [1]. The TFs of these sectons (clled GM-sectons) re: H b + z 1( z) GM bz, 1 (1 1) z + z H GM ( z). (4) 1 (1 1) z 1z The coeffcents of the dgtl llss sectons hve been derved from the corresondng frst nd second order nlog rototyes Eq.(5) by blner trnsformton s + B s A 0 s + B S( s), S( s) s + B0 s. (5) + A s + B If both the relton between the oles oston nd the coeffcents of the TF nd Eq.(1), Eq.() re used, the followng exressons for coeffcents n llss sectons cn be obtned: 1 B0 b 1 b. (6) 1 + B0 A B 1 1 r. (7) A + B + 1 1 1 + B B B 0 1 + B0 c B 1 + B A d A + B +1 where r s the comlex ole rdus. 1 tg πf. (8) tg πf 1 +. (9) πf f 1 r tg c. (10) tg π d, (11) 3. SENSITIVITIES INVESTIGATIONS We hve nvestgted the worst-cse (WS) mltude resonse senstvtes of the obtned elltc IIR flters (STAST1, GM1GM) for two dfferent TFs orders (3
nd 5) usng the ckge PANDA [7]. The results for the WS senstvtes vs. frequency re gven n Fg. (thrd order TF) nd Fg. b (ffth order TF). The IIR flters obtned usng llss sectons ST1 nd STA re mrked s STAST1. Resectvely, the flter relztons wth GM-sectons re mrked s GM1GM. It s clerly seen tht IIR flters STAST1 hve mny tmes lower senstvtes wthn the entre frequency rnge, but the dfference s consderbly stronger n the stobnd nd eseclly t mxml onts. The WS senstvty nlyss n Fg. demonstrtes tht dfference between senstvtes of both relztons (STAST1 nd GM1GM) for ffth-order TF becomes even more sgnfcnt. ) b) Fg. Worst-cse senstvtes of dfferent IIR flter relztons: ) thrd order; b) ffth order 4. EXPERIMENTS We desgned nd nvestgted two sets of flters wth secfctons s gven n Exmles 1,. In both cses frst, the desgn rocedure s led for llss secton ST1 nd STA (wth TFs gven n Eq.(3)). Then, for comrson, the sme rocedure s used for GM-sectons (wth TFs gven n Eq.(4)). Usng the symbolc nlyss, the exressons for hse senstvtes to multler coeffcents for the selected sectons hve been derved. Fgs. 3, 4, 5 nd 6 dsly the hse resonse senstvtes to coeffcents of the corresondng frst or second-order sectons of the both exmles. Exmle1 The flter secfctons re s follows: F 0.006, F 0. 0, A 0. db, A 0dB. The flter coeffcents cn be dvded nto two grous tht wll be consdered sertely. The frst grou ncludes the most senstvty coeffcents (tht s - GM1GM nd c - STAST1) tht cn be controlled by TFs, nd the coeffcents 1 (GM1GM) or d (STAST1) reresentng the ole r closest to the unt crcle. The second grou ncludes the remnng coeffcents b (GM1GM) nd (STAST1). For gven secfctons, the rnge of ermtted vlues of or c s determned from F nd F ccordng to [1]. Then, ll vlues n the whole rnge re quntzed n CSDcode. The vlues of or c wth the smllest quntzton error s chosen. From Eq.(8) nd Eq.(10) the frequency f 3 db s evluted. Keeng the vlues of f 3 db unchnged, the
trnston bndwdth s to be djusted to rovde the sutble vlue for 1 or d ccordng to [1][3]. At the end ll remnng coeffcents re quntzed wth dfferent coeffcents wordlength. Tble I gves the results for ll coeffcents. Exmle The flter secfctons re: F 0.006, F 0. 01, A 0. 1dB, A 0 db. The secfctons re fulflled wth ffth order TF flter. The second order sectons ' '' hve common coeffcent (GM1GM) or c ' c' ' (STAST1). The coeffcents of the obtned multlerless IIR flters re gven n Tble I. We hve nvestgted the frequency resonses of the multlerless relztons so obtned. It s clerly seen from Fgs. 7 nd 8 tht the shes of the ttenuton chrcterstcs of the multlerless flters obtned ccordng to our roch (STAST1) do not dffer from the del even though coeffcents of the second grou re quntzed to 3bts. TABLE I MULTIPLIERLESS REPRESENTATION OF THE FILTER COEFFICIENTS STAST1 GMGM1 d 5 7 1 4 6 c 10 9 1 4 8 + + b 3 STAST1 GMGM1 d ' 8 10 + 1' 7 9 + c ', c' ' 11 ', ' ' 0 1 d '' 6 7 + 1 ' ' 4 6 5 6 + b 4 + Fg. 3 Phse resonse senstvty of 3 th order GMGM1 TF Fg. 4 Phse resonse senstvty of 3 th order STAST1 TF Fg. 5 Phse resonse senstvty of 5 th order GMGM1 TF Fg. 6 Phse resonse senstvty of 5 th order STAST1 TF
Fg. 7 Attenuton chrcterstcs Fg. 8 Attenuton chrcterstcs of 3 th order TFs of 5 th order TFs 5. CONCLUSIONS In ths er modfcton of the desgn method ntlly suggested by Mlć nd Lutovc hs been roosed. The hse senstvtes to multler coeffcents for the selected llss sectons hve been derved. The hlf most senstve coeffcents hve been reresented wth smll number of shfters nd dders. The remnng coeffcents re quntzed wth dfferent coeffcents wordlength. Consequently, the obtned multlerless IIR relztons re wth smll quntzton error nd low comlexty. It s shown tht the senstvtes of the obtned flters cn be consderbly decresed f llss sectons used re wth mnmzed senstvtes. It s frmly demonstrted (through exerments) tht flters desgned ccordng to our roch re styng wthn the gven secfctons even fter very severe quntzton of the multler coeffcents whle the mgntudes of the reference flters (s desgn n [1][]) re totlly destroyed. 6. REFERENCES [1] Mlc L, M. Lutovc, Desgn of multlerless elltc IIR flters wth smll quntzton error, IEEE Trns. Sgnl Processng, vol.47,.469-479, no., Feb.1999 [] Lutovc M, L. Mlc, Desgn of comuttonlly effcent elltc IIR flters wth reducednumber of shft-nd-dd oertons n multlers, IEEE Trns. Sgnl Processng, vol.45,.4-430, Oct.1997 [3] Mlc L, M. Lutovc, Low-comlexty lttce wve dgtl flters, Proceedngs of 13 th Euro Conf. ECCTD, Budest, Hungry,.1141-1146, Aug.1999 [4] Rbrenovc D, M. Lutovc, Elltc flters wth mnml Q-fctors, Electron. Lett, vol 30, no.3,. 06-07, Feb.1994. [5] Stoynov G, H. Clusert, A comrtve study of frst order dgtl ll-ss structures, Frequenz, vol. 48, No 9/10, Set./Oct. 1994. 1-6 [6] Stoynov, G., V. Dmtrov, Senstvty nd nose nlyss of second-order dgtl ll-ss sectons (n Bulg.). Proc. 3-th Sc. Conf. "Communcton, electroncs nd comuter systems'93", Sof, Bulgr, My 1993, vol. 1,. 104-115. [7] Sugno H, A. Nshhr, Frequency-domn smultor of dgtl networks from the structurl descrton, Trns.of the IEICE of Jn, Vol. E73, No.11,.1804-1806, Nov.1990.