Using SIP techniques to verify the trade-off between SNR and information capacity of a sigma delta modulator

Transcription

1 Audo Engnrng Socty Convnton Ppr 6697 Prsntd t th 0th Convnton 006 y 0 3 Prs, Frnc hs convnton ppr hs bn rproducd from th uthor's dvnc mnuscrpt, wthout dtng, corrctons, or consdrton by th Rvw Bord. h AES ts no rsponsblty for th contnts. Addtonl pprs my b obtnd by sndng rqust nd rmttnc to Audo Engnrng Socty, 60 Est 4 nd Strt, Nw Yor, Nw Yor , USA; lso s All rghts rsrvd. Rproducton of ths ppr, or ny porton throf, s not prmttd wthout drct prmsson from th Journl of th Audo Engnrng Socty. Usng SIP tchnqus to vrfy th trd-off btwn SNR nd nformton cpcty of sgm dlt modultor Chrlott Yu-Fn Ho, Joshu D. Rss nd Bngo Wng-Kun Lng 3 Dprtmnt of Elctronc Engnrng, Qun ry, Unvrsty of London, l End Rod, London, E 4NS, Untd Kngdom. c.ho@qmul.c.u Dprtmnt of Elctronc Engnrng, Qun ry, Unvrsty of London, l End Rod, London, E 4NS, Untd Kngdom. Josh.rss@qmul.c.u 3 Dprtmnt of Elctronc Engnrng, Kng s Collg London, Strnd, London, WCR LS, Untd Kngdom. wng-un.lng@cl.c.u ABSRAC h Grzon-Crvn nos shpng thorm stts tht th dl nformton cpcty of sgm dlt modultor dsgn s chvd f nd only f th nos trnsfr functon (NF) s mnml phs. In ths ppr, t s found tht thr s trd-off btwn th sgnl-to-nos rto (SNR) nd th nformton cpcty of th nos shpd chnnl. In ordr to vrfy ths rsult, loop fltrs stsfyng nd not stsfyng th mnml phs condton of th NF r dsgnd v sm-nfnt progrmmng (SIP) tchnqus nd solvd usng dul prmtrzton. Numrcl smulton rsults show tht th dsgn wth mnml phs NF chvs nr th dl nformton cpcty of th nos shpd chnnl, but th SNR s low. On th othr hnd, th dsgn wth nonmnml phs NF chvs postv vlu of th nformton cpcty of th nos shpd chnnl, but th SNR s hgh. Rsults r lso provdd whch compr th SIP dsgn tchnqu wth Buttrworth nd Chbyshv structurs nd dl thortcl SDs, nd vlut th prformnc n trms of SNR nd vrty of nformton thortc msurs whch cptur nos shpng qults.. INRODUCION It s wll nown tht contnuous-tm sgnl cn b smpld nto dscrt-tm sgnl nd chvd prfct rconstructon from th corrspondng dscrt-tm sgnl v n dl lowpss fltrng f th smplng rt s hghr thn twc of th bndwdth of th corrspondng contnuous-tm sgnl. hs smplng rt s clld th Nyqust

2 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty smplng rt. If th smplng rt s hghr thn th Nyqust smplng rt, t s clld n ovrsmplng. In ths cs, th sgnl s concntrtd t vry nrrow bnd nd th bndwdth of th sgnl s nvrsly proportonl to th ovrsmplng rto (). If quntzton s ppld on th corrspondng dscrttm sgnl nd ssumng tht th quntzton nos s vnly dstrbutd n th whol frquncy bnd, thn th SNR cn b mprovd by ncrsng th nd th numbr of bts of th quntzr v n dl lowpss fltrng of th corrspondng quntzd dscrt-tm sgnl. h SNR cn b furthr mprovd by pplyng nos shpng tchnqus v propr dsgn of loop fltr. Bcus of th ovrsmplng nd nos shpng tchnqus, sgm dlt modultors (SDs) cn chv vry hgh SNR vn for vry cors quntzton stps[]. As rsult, SDs wth good prformnc domnt th hgh rsoluton, low frquncy nd of th A/D nd D/A convrtr mrt[]. hy r prtculrly wllsutd for udo pplctons, whr thr low cost nd hgh prformnc for nput sgnls blow 50Hz ms thm spclly pplng. h most common loop fltr dsgn mthods r bsd on Chbyshv structurs [3] or Buttrworth structurs[4]. Howvr, n ordr to chv good SNR, th NF should b clos to zro nd th SF should b clos to on n th sgnl bnd, nd th NF should b clos to on nd th SF should b clos to zro n th nos bnd. Snc ths chrctrstcs r dfnd n th frquncy domn, contnuous constrnts should b cpturd n th dsgn. o dl wth ths, w hv rcntly formultd th dsgn problm s sm-nfnt progrmmng problm[5, 6] nd solvd th problm v dul prmtrzton[7]. Numrcl smulton rsults show tht th SD dsgnd usng th SIP pproch cn chv vry hgh SNR. Howvr, thr s nothr ndx for vlutng th prformnc of SDs, whch s bsd on th nformton cpcty of th nos shpd chnnl. It ws rportd n th Grzon-Crvn nos shpng thorm[8] tht mxmum nformton cpcty of th nos shpd chnnl s chvd f nd only f th NF s mnml phs. Hnc, to chv good prformnc bsd on th nformton cpcty of th nos shpd chnnl, th NF should b mnml phs. In ths ppr, loop fltrs stsfyng th mnml phs condton of th NF r dsgnd. h gol of ths ppr s to clrfy th SIP dsgn tchnqu nd how t my b usd for mnmum phs NF dsgn, nd to vlut ts prformnc nd compr t wth othr dsgn mthods nd wth thortcl lmts. h comprson btwn mnmum phs nd non-mnmum phs dsgns llows us to dmonstrt nd nlyz th trd-off btwn nformton cpcty of n SD nd ts SNR prformnc. In Scton of ths ppr, w rvw th formulton of SD dsgn by Sm-Infnt Progrmmng, nd ts soluton usng dul prmrrzton s prsntd n Scton 3. hough, ths In Scton of ths ppr, th dsgn of th loop fltr stsfyng th mnml phs condton of th NF s formultd s n SIP problm. h SIP problm s solvd v dul prmtrzton nd th mplmntton ssus r rvwd n Scton 3. hough Scton 3 rprsnts rvw of th stt of th rt, t should b notd tht [6] dscrbs th formulton of th SIP problm nd th justfcton for dul prmtrzton but not ts mplmntton. h mplmntton of dul prmtrzton ws dscrbd n [7] but hr th SIP problm s rphrsd n th contxt of ths ppr nd mphss s plcd on ssus concrnng prctcl mplmntton of dul prmrrzton s ppld to SD dsgn. Scton 4 s focusd on vluton of SD dsgns nd comprson wth thory. In Scton 4., vrous dsgns r comprd n trms of thr SNR prformnc. In Scton 4., nformton thortc msurs r dvsd nd mplmntd whch ttmpt to cptur th ffctvnss of nos shpng, nd how closly th SDs prform to thortcl lmts s suggstd by th Grzon-Crvn nos shpng thorm. hs prformnc ndxs nclud th rto of th totl powr of th shpd quntzton nos to tht of th unshpd quntzton nos, nd th msur of th totl loss of chnnl nformton cpcty ftr nos shpng. Scton 4.3 dscusss th trd-off btwn th SNR nd th nformton cpcty of th nos shpd chnnl. A justfcton for ths trd-off s md, nd numrcl smulton rsults r llustrtd to show tht thr s trd-off btwn th SNR nd th nformton cpcty of th nos shpd chnnl. Fnlly, n Scton 5, th rsults r summrzd nd commnt on futur drctons of rsrch.. FORULAION OF SIP-BASED DESIGN OF AN SD WIH INIAL PHASE NF h formulton of th dsgn of n SD s n SIP problm ws shown n [5, 6]. Howvr, n [5, 6], th dsgn ws bsd on mnmzng th rppl nrgy of NF nd SF n th sgnl bnd, subjct to th H constrnts on ths rppls s wll s th stblty condton of both th SF nd NF. As rsult, ths dsgn chvs vry hgh SNR. Howvr, th nformton cpcty of th nos shpd chnnl ws not consdrd. AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg of

3 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty h dsgn of th loop fltr s dvdd nto two prts. h frst prt s to formult th dsgn of dnomntor coffcnts of th loop fltr s contnuous constrnd optmzton problm whr th objctv s to mnmz th nrgy of th dnomntor trnsfr functon (xcludng th DC pols) n th sgnl bnd, subjct to th mnml phs condton of th NF. h scond prt s to formult th dsgn of numrtor coffcnts s contnuous constrnd optmzton problm whr th objctv s to mnmz th nrgy of th numrtor trnsfr functon (xcludng th dly lmnts) n th nos bnd, subjct to th contnuous constrnt dfnd by th stblty condton of both th NF nd th SF. h rsons for choosng ths cost functons nd constrnts wll b dscussd n th followng prgrphs. W ssum tht th loop fltr s rtonl nd cusl fltr wth unt smpl dly n th numrtor trnsfr functon nd thr my b som DC pols n th dnomntor trnsfr functon, tht s: [ ] R( + ( η ) x ) 0 ω,, () N whr nd N r th numbrs of roots of th polynoml of n th numrtor nd dnomntor trnsfr functons of th loop fltr (xcludng th DC pols nd pur dly lmnts), rspctvly, r s th numbr of DC pols (possbly zro), nd n, b m for n=,,, N nd m = 0,,, r th fltr coffcnts. By groupng th fltr coffcnts n th numrtor nd dnomntor s x [ b,, b ] b 0 nd x, ], rspctvly, whr th [, N suprscrpt dnots th trnspos oprtor, nd dfnng η [ ] nd,,, jnω η,,,, thn N [ ] ( η) xb H =. () j r ( ω ) ( + ( η ) x ) h sgnl trnsfr functon nd nos trnsfr functon of th SD cn b xprssd s SF rspctvly. ( ) H ω =, NF = + H + H.. Dtrmnton of dnomntor coffcnts N (3) B P ; =, (4) whr s th ovrsmplng rto. For SDs hvng good SNR, th mgntud of th SF should b pproxmtly qul to nd tht of th NF should b pproxmtly qul to 0 for ll frquncs n th sgnl bnd. hs holds f + ( η ) x 0 ω N BP Hnc, w dfn cost functon s + ( η ) x = x Q x + b x BP nd N (5) + p, (6) whr Q (R( η )R( η ) + Im( η )Im( η )) N N N N BP BP ( η ω ) b R ( ) n whch Q N p BP (7), (8) s postv dfnt mtrx. o cptur th mnml phs condton of th NF n th dsgn, H (ω) should b stbl. hs mpls tht [ ] R( + ( η ) x ) 0 ω, N. (9) Dnot A R(( η N) ) nd c. hn th dsgn of th dnomntor coffcnts cn b formultd s th followng SIP problm: Problm (P ) mn x Qx + b x + p x subjct to A x + c 0 ω, [ ] (0) h SIP problm cn b solvd by dul prmtrzton[7], whch gurnts th globl optml soluton nd stsfs th contnuous constrnt. Dnot th pssbnd of th loop fltr, or th sgnl bnd, s AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 3 of

4 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty.. Dtrmnton of numrtor coffcnts hough th chrctrstcs of th NF nd SF for frquncs n th sgnl bnd r cpturd n th dsgn, th corrspondng chrctrstcs n th nos bnd lso nd to b cpturd n th dsgn. h stblty of ths two trnsfr functons nd th frquncy chrctrstcs of th loop fltr should b consdrd s wll. For SDs hvng good SNR, th mgntud of th SF should b pproxmtly qul to 0 for ll frquncs n th nos bnd. hs mpls tht th rppl nrgy of th loop fltr n th nos bnd should b smll. Howvr, x s obtnd from solvng th problm P nd r s nown from th dsgn spcfctons, so th dnomntor trnsfr functon dos not nd to b consdrd hr. hus, to chv ths gol, th rppl nrgy of th numrtor trnsfr functon should b smll. h objctv of th optmzton problm s to mnmz th rppl nrgy of th numrtor trnsfr functon n th nos bnd subjct to th stblty condton of th NF nd SF. h cost functon cn b formultd s: ( η ) xb. B S h stblty condton of th NF nd SF s r R( ( η ) xb + ( ) ( + ( ηn ) x ) 0 ω [, ], whch s quvlnt to r ( η ) xb + R( ) ( + ( ηn ) x ) 0 ω [, ], whr η [ cos ω, cos ω,, cos( + ) ω]. Hnc, th optmzton problm cn b rprsntd s th followng SIP problm: Problm (P ) mn xb Qbxb xb, () subjct to A x + c 0, ω, whr b b b [ ] ( R( η ) R( η ) Im( η ) Im( ( ω ) ) Q η ) b + B S Not tht problm P dos not dpnd on th numrtor coffcnts. hus th globl optml soluton of problm P cn b obtnd v dul prmtrzton mthod[7]. h dnomntor coffcnts r obtnd from solvng problm P, nd thus th globl optml soluton of problm P cn thn b obtnd ndpndntly. Hnc, n ths formulton, trtv dsgn of th numrtor nd dnomntor coffcnts s vodd. 3. IPLEENAION OF DUAL PARAEERIZAION FOR SOLVING SIP PROBLES hr r mny xstng mthods for solvng SIP problms. For dscrtzton mthods[9], th contnuous constrnts r dscrtzd, rsultng n fnt numbr of convx nd qudrtc constrnts ftr dscrtzton. h problm thn bcoms qudrtc progrmmng problm nd cn b solvd ffcntly v mny xstng solvrs, such s tlb, tc. Howvr, ths mthod dos not gurnt tht th soluton obtnd would stsfy th corrspondng contnuous constrnts. In ths scton, w rvw th dul prmtrzton mthod for solvng SIP problms[7]. Snc th orgnl problm conssts of contnuous constrnts, whch r dffcult to solv, w trnsform th orgnl problm nto n quvlnt fnt dmnsonl nonlnr progrmmng problm v squnc of rgulr convx progrms nd solv th fnt dmnsonl nonlnr progrmmng problm v th dul prmtrzton mthod. hs mthod s gurntd to obtn globlly optml soluton tht stsfs th contnuous constrnt f soluton xsts. For th dtls, pls rfr to [7] nd [0]. Bsd on ths thory, th corrspondng fnt dmnsonl dul problms of P nd P r: Problm (PDP ) nd mn x, μτ, x Q x c μ ( ) = ( ) = Qx + b + A μ = 0 subjct to μ 0, ω Δ for =,,, () nd ( η ( ) A ( ω ω, b ) ) jω r ( ) ( + ( ηn ( ) ) c b R ω) x. AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 4 of

5 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty Problm (PDP ) mn xb, μτ, x Q x c μ subjct to b b b ( b ) = ( ( ω )) Qx+ A μ = 0 b b = μ 0, ω Δ for =,,,, (3) whr μ nd ω r, rspctvly, th dscrt multplrs nd th dscrt frquncs, n whch τ [ ω, ω,, ] nd s th dmnson of x. ω h mplmntton procdurs [7] of th dul prmtrzton mthod r summrzd n th followng lgorthm. Frst w ntlz prmtrs n th lgorthm (Stp ), thn comput locl optml soluton by solvng fnt dmnsonl nonlnr progrmmng problm (Stps -4). Fnlly, th globl optml soluton s computd v locl srch for th fnt dul problm (Stp 5). Algorthm Stp. Intlzton Arbtrrly choos n ntl guss of th fltr 0 S coffcnts x, whr S s th numbr of fltr coffcnts to b dtrmnd. For th problm P, S = N, whl for th problm P, S = +. Choos smll postv numbr ε > 0 whch dfns th ccptbl rror on th constrnts. Choos mnmum trton numbr N to prvnt th lgorthm from trmntng prmturly. Choos squnc of ntl fnt prmtrzton sts Δ = { ω j : j =,,, } tht stsfs th mtrc d Δ, Δ mx mnϖ ω, ( ) 0 ω Δ ω Δ whr ϖ nd ω r frquncs. St th tmporrly fnt prmtrzton sts th mpty st, nd th trton ndx = 0. Stp. Dtrmn th st of dscrt frquncs Incrmnt th trton ndx by, tht s, = +. Fnd pont n th ntl fnt prmtrzton sts ϖ Δ such tht th constrnts rch thr mxmum vlu t ϖ for ll th ponts n, tht s Δ g ( x, ϖ ) = mx g ( x, ω), mx ω Δ mx whr g (, ϖ ) ll th lmnts n th constrnt vctor ( ) E 0 mx x s th mxmum lmnt mong g x,ϖ. to If th constrnts mt th spcfctons t ϖ, tht s, g ( x mx, ϖ ) < ε, thn st th fnt prmtrzton sts Z to th prvous tmporrly fnt prmtrzton sts, tht s Z = E. If th trton ndx rchs th mnmum trton numbr N, tht s, N', thn go to Stp 5. Othrws, st th currnt vlus of th fltr coffcnts x, th multplrs μ nd th tmporrly fnt prmtrzton sts E to thr prvous vlus, tht s (, ) ( x μ = x, μ ) nd E = E, nd rpt Stp gn. Othrws, st ϖ nd th prvous tmporrly fnt prmtrzton sts s th currnt fnt prmtrzton sts, tht s, Z { ϖ } =. E Stp 3. Solvng th problm PDP( Z ) Solv th problm PDP( Z ) x, μ ) PDP( ) to obtn soluton for (, whr th problm Z s th fnt dmnsonl problm of PDP subjct to th fnt prmtrzton sts, tht s Z = ω. Z { } Stp 4. St th currnt tmporrly fnt prmtrzton sts s subst of th currnt fnt prmtrzton sts, tht s E, wth no mor Z thn S + ponts such tht th soluton of th problm PDP( E ) s n th form ( μ ). hn go to x, Stp gn. Stp 5. Locl srch for th fnt dul problm Suppos th currnt fnt prmtrzton sts Z hs ponts ϖ, ϖ,, ϖ. Strtng from ( x, μ, τ ), whr x nd μ r dfnd prvously nd τ = [ ϖ, ϖ, s th tupl formd by th, ϖ ] ponts n Z, fnd locl mnmum (, for th x μ, τ ) problm PDP, whr th problm s th PDP dscrtzton vrson subjct to th ntl prmtrzton sts Δ. hn x s tn s th soluton for th problm P. Solvng th problm PDP( Z ) n Stp 3 nd fndng th locl mnmum n Stp 5 cn b ccomplshd by usng xstng optmzton solvrs. h bov AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 5 of

6 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty lgorthm cn b summrzd by th flowchrt dpctd n Fgur. Intlzton Incrmnt trton ndx Dtrmn frquncy whch mxmzs constrnt functon Error s ccptbl? Ys Dtrmn frquncy whch mxmzs constrnt functon St currnt prmtrzton st, fltr coffcnts & multplrs s prvous vlus No Ys Fnd locl mnmum x, μ, τ ( ) x s tn s th soluton No Itrton ndx rchs N Includ ths frquncy n fnt prmtrzton st Solv problm PDP(Z ) & choos subst of fnt prmtrzton st wth no mor thn S+ ponts Fgur. Flowchrt showng th mplmntton of th dul prmtrzton mthod. 4. PERFORANCE EVALUAION o vlut th prformnc of SDs dsgnd usng Sm-Infnt Progrmmng, s wll s th trd-off btwn th SNR nd th nformton cpcty of nos shpd chnnl, SDs wr dsgnd usng SIP pprochs wth nd wthout mnml phs NF, nd usng Buttrworth nd Chbyshv fltr dsgn tchnqus. Dsgn of Chbyshv structurs[3] ws ccomplshd v th functon synthssnf n th dlt-sgm mtlb toolbox[], nd dsgn of Buttrworth structurs[4] ws ccomplshd v mposng th mxmlly flt condton on th dsgn. Whr possbl, dsgns wr lso comprd wth thortcl lmts. Ech SD, unlss othrws notd, hd n ovrsmplng rto =64 (smplng frquncy 64x44.Hz), on DC pol, r=, zro ntl condtons, nd th numbr of roots n th numrtor nd dnomntor trnsfr functons r both 4, =N=4 (s Eq. ()),.. fltr ordr 4+=5. h quntzr ws sngl bt wth th dcson boundry t zro nd th sturton lvl t on. 4.. SNR-bsd Prformnc Evluton Frst, w compr th prformncs of ch SD dsgn n trms of ts sgnl-to-nos rto. h thortcl lmt of th sgnl-to-nos rto my b stmtd by []: SNR = 0log ( σ ) 0log ( σ ) stmtd 0 x 0 N 0log + ( 0N + 0) log N R (4) Hr, N rprsnts th fltr ordr, σ x nd σ r, rspctvly, th powr of th nput sgnl nd th powr of th quntzton nos. If snusodl nput wth mgntud U s mployd, thn σ. For x = U / th quntzr wth dynmcl rng btwn - nd, thn σ =, whr L s th numbr of bts of L 3( ) th quntzr. In th cs of ffth ordr SD wth 64 tms, th thortcl lmt of SNR gvn n (4) rducs to: SNRstmtd 0 0 L ( ) = 0log U + 0log (5) Fgur shows th rltonshp btwn th SNRs nd th nput mgntuds, whr th ntl condtons of th bov SDs r zro, th quntzton rgon of th quntzr s btwn - nd, L =, R = 64, nd N = 5. h thortcl lmt of SNR ws found from (6) nd th msurd SNRs wr computd usng th dlt-sgm mtlb toolbox [3, ]. h SD dsgnd v th SIP pproch wthout th mnml phs condton of th NF consstntly outprforms th SDs dsgnd v th Chbyshv AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 6 of

7 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty structur, th Buttrworth structur nd th SIP pproch wth th mnml phs condton of th NF by pproxmtly 3.75dB, 3.04dB nd 9.4dB, rspctvly, n thr corrspondng stbl rgons. h non-mnml phs SD dsgnd v th SIP pproch lso hs hghr dynmc rng, wth stbl bhvor up to nputs of pproxmtly 0.68, comprd to 0.66 for th Chbyshv structur nd mnml phs SIP dsgn, nd 0.60 for th Buttrworth structur. hs s surprsng rsult, snc mor nos shpng s oftn thought to rsult n trd-off wth dynmc rng, nd snc no ttmpt ws md to dsgn for stbl bhvor t hgh mpltuds. Fgur 3. Rltonshps btwn SNRs nd th numbr of bts of th quntzr. Fgur 4 shows th rltonshps btwn th SNRs nd th s, whr th ntl condtons of SDs r zro, th quntzton rgon of th quntzr s btwn - nd, L =, N = 5, U ws st to 0.3 n ordr to gurnt stblty nd thortcl lmt of SNR wr computd from (4). h non-mnml phs SIP dsgn outprforms th optmzd Chbyshv dsgn by t lst 4.48dB, nd shows ncrsd comprtv prformnc wth ncrsd. Fgur. Rltonshps btwn SNRs nd nput mgntuds. Fgur 3 dpcts th rltonshp btwn th SNR nd th numbr of bts n th quntzr. Intl condtons of ll SDs r zro, th thortcl lmt of SNR s gvn by (5), nd th SNRs of th vrous SDs r clcultd v th dlt-sgm mtlb toolbox[]. As bfor, ffth ordr SD wth =64 ws usd for ll dsgns. W chos n nput mpltud of U=0.44 bcus ths gurnts stblty for ll th smultd dsgns. In lmost ll css, th SD dsgnd v th non-mnmum phs SIP dsgn outprforms th SDs dsgnd v th Chbyshv structur, th Buttrworth structur nd th SIP pproch wth mnml phs NF by 3.58dB,.85dB nd 9.37dB, rspctvly. Improvmnts ovr Chbyshv nd Buttrworth structurs r prtculrly notcbl for low numbr of bts. Fgur 4. Rltonshp btwn SNRs nd s. Fnlly, Fgur 5 shows th rltonshps btwn th SNRs nd th fltr ordrs, whr U = 0.3, L =, nd =64. h Chbyshv dsgn nd SIP dsgn wthout th mnml phs condton of th NF show nrly dntcl prformnc t fltr ordrs of 3 nd 7, but th SIP dsgn wthout th mnml phs condton of th NF shows up to 4.7dB mprovmnt whn th fltr ordr N=5. hs s sgnfcnt, snc ths typ of fltr ordr s commonly usd, prtculrly n udo pplctons, bcus t AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 7 of

8 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty rprsnts bst comproms btwn nos shpng nd stblty. mgntud nd mng th pproxmtons concrnng quntzton rror mor nccurt. Nvrthlss Eq. (4) s nown to produc n uppr bound on th SNR tht cn b xtrmly ccurt for unshpd, or frst nd scond ordr fltrs. Furthrmor, th xpctd (6L+)dB mprovmnt wth doublng th nd pproxmtly 6dB mprovmnt wth ddng bt to th quntzr r obsrvd n ll dsgns. Fgur 5. Rltonshp btwn SNRs nd ordrs of fltrs. surmnt of th SNR llows us to stmt th ffctv rsoluton of th dsgn. hs s gvn by th R ff SNR.76 =. (6) 6.0 Not tht ths s not qut th Effctv Numbr of Bts, whch s mor ccurtly gvn by rplcng SNR wth th SINAD n th bov formul[]. For comprson, rsults for th SNR nd ffctv rsoluton for th vrous dsgns r gvn n bl, whr =64, th fltr ordr s 5, nd bt quntsr s usd. In ll rsults dpctd n Fgurs -5, though th SD dsgnd usng th SIP pproch wth non-mnml phs NF outprforms othr mthods, t dos not com clos to th thortcl lmt of SNR. h thortcl lmt of SNR gvn n (4) ms svrl ssumptons, most notbly th ssumpton of unform dstrbuton of th quntzton rror. Its ssumptons lso ntroduc sgnfcnt pproxmtons for hgh ordr SDs. hs s shown drmtclly n Fgur 5. h thortcl lmt of SNR s 70.08dB hghr thn tht of th SIP dsgn wth non-mnml phs NF whn N=7, nd nthr smultd dsgn shows th prdctd 08dB mprovmnt from N=3 to N=7. Anothr dffculty n ths cs s th nblty to stmt vry hgh sgnl-to-nos rtos, whr fnt prcson rsults n srous undrstmts of SNRs bov bout 30dBs. Fnlly, th uthors now of no rportd SD dsgn whch chvs clos to th thortcl lmt suggstd by (4) for hgh ordr (N>) fltr. hs s prtly bcus of th dffcults n crtng dl fltrs. But t s lso bcus such dsgns r unstbl, thus lmtng th nput 4.. Informton hortc nd Nos Shpng trcs In [8], Grzon nd Crvn consdrd th sgm dlt modultor s trnsmsson chnnl, nd usd nformton thortc consdrtons to show tht for spcfd NF, th nformton cpcty of th nos shpd chnnl cnnot xcd tht of th non-nos shpd chnnl. In othr words, th nos shpng fltr wth th smllst possbl output rror powr s th fltr tht lvs th nformton cpcty of th chnnl unltrd (mxmum).hs mpls tht, G log ( ) 0 NF ω d ω (7) whr qulty s obtnd (mxmzd us of th nformton cpcty of th nos shpd chnnl) for mnmum phs fltrs. hs rsult my b rphrsd s sttng tht th rs bov nd blow th 0-dB ln (on dcbl plot of th sgnl spctrum vs lnr frquncy) wll b qul for ny optml (.., mnmum-phs) nos shpr. In [3], ths rsult ws usd to show tht, f th NF s spcfd just ovr th sgnl bnd, thn th rto of th totl powr of th shpd quntzton nos to tht of th unshpd quntzton nos s mnmzd by sttng th NF to constnt or flt shp ovr th rmnng bnds. Such n dl NF s dpctd n Fgur 6. NF mgntud (db) Frquncy No nos shpng Constrnd to -0dB n pssbnd Constrnd to -70dB n pssbnd Fgur 6. NFs of SDs wth dl fltrs for =8. Hnc, th dl NFs cn b xprssd n th form of: 60 AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 8 of

9 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty NF A ω / = B / < ω <, (8) ( ) ( ) whr NF ω = NF ω, n whch th suprscrpt dnots th complx conjugt. Bsd on th nos shpng thorm[8], A 0 nd G = 0 f th NF s mnml phs. From (7) nd (8), w hv tht, for mnmum phs NF, 0 G = log A+ log B. (9) = log A+ log B = 0 hs mpls tht ( ) log A= log B A= B. (0) Eq. (0) mpls tht, for comprson purposs w should consdr two typs of dl fltr. h frst llows drct comprson of th SIP dsgn, whch mposs constrnd NF n th sgnl bnd, wth th dl fltr wth th sm constrnt. Snc th loop fltr dsgnd v th SIP pproch wthout th mnml phs condton of th NF cn chv th 5 bound A = on th NF n th sgnl bnd, w hv tht dlly, B =.840. For th scond typ of dl fltr, w consdr n dl constrnt on th NF n th pssbnd. Snc th NF should b clos to zro n th sgnl bnd, whl th NF should b clos to on n th nos bnd, A = 0 nd B = Informton Cpcty trcs h frst mtrc w mployd for comprson s bsd on th nformton cpcty of th nos shpd chnnl, nd dfnd by Eq. (7). For th frst dl (non-zro) NF, by dfnton G = 0. For th scond dl NF, G s ngtv nfnty bcus A = 0. Howvr, for lowpss SD, w r prmrly concrnd wth lowpss prformnc. ht s, w would l to msur how wll th nformton cpcty hs bn utlzd n th sgnl bnd. hrfor, t s mor prctcl to comput ths mtrc n th sgnl bnd only, tht s: / G = log NF( ) ω. () / h vlu of G s lwys lss thn zro bcus NF n th sgnl bnd. It s ngtv nfnty for th SD wth th scond dl NF bcus A=0. bl lsts how th vrous SD dsgns prform n trms of G nd G. h vlu of G for th SD dsgnd v n SIP pproch wth th mnml phs condton of NF s stll nonzro bcus thr s DC pol on th corrspondng loop fltr trnsfr functon. For ths rson, non of th dsgns chv G=0, though th mnmum phs dsgn outprforms th non-mnmum phs dsgn, s xpctd. h mnmum phs fltr ctully prforms poorly n trms of G. hs s most lly du to ovrshoot wthn th pssbnd (s Fgur 7). Intrstngly, on ths mtrc, ll othr smultd fltrs ctully outprform th dl fltr wth non-zro NF, wth th non-mnmum phs SD prformng bst. hs s bcus th dl fltr ssums constnt NF wthn th sgnl bnd st to th constrnt 5 A = , whrs th othr fltrs oftn hv NFs tht r mostly wll-blow A n th sgnl bnd Nos Shpng trcs It s mportnt to vlut th nos shpng chrctrstcs of th loop fltr. o chv ths gol, th rto of th totl powr of th shpd quntzton nos to tht of th unshpd quntzton nos s vlutd[3]. Assum tht th unshpd quntzton nos powr spctrl dnsty s flt n th frquncy spctrum wth th mgntud dnotd s S, thn th rto of th totl powr of th shpd quntzton nos to tht of th unshpd quntzton nos s: K = S NF = = S S NF S NF. () Not tht ths quton s lttl bt dffrnt from tht n [3] bcus th orgnl formul hd typng rror. For th SD wth th frst dl NF (constrnd, non-zro), w hv, from Eq. (0), AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 9 of

10 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty K = NF = A+ Bd 0 ω. (3) ( ) B + B = R For th SD wth th scond dl NF (A=0, B=), w hv: K = NF. (4) = = As n Scton 4.., w r prmrly concrnd wth th nos shpng chrctrstcs n th low pss rgon. hus t s mor prctcl to comput ths mtrc n th sgnl bnd only, tht s: K = = / / S ( ) / NF ω d / / / S NF ω (5) As wth G, K s lwys lss thn zro bcus NF n th sgnl bnd, nd ngtv nfnty (on dcbl scl) for th SD wth zro NF n th sgnl bnd bcus A=0. h rsults r shown n bl, whr vlus for K nd K r gvn on dcbl scl. From ths rsults, w cn s tht th rto of th totl nos powr of th shpd quntzton nos to tht of th unshpd quntzton nos s smllst for th Chbyshv dsgn nd th mnml phs SD dsgnd v th SIP pproch. On th othr hnd, th Buttrworth structur prforms worst wth th hghst totl nos powr. Howvr, f w only consdr th rto of th powr of th shpd quntzton nos to tht of th unshpd quntzton nos n th sgnl bnd only, K, thn th SIP pproch wthout th mnml phs prforms bst, whl mnml phs dsgn prforms worst. o compr th nos shpng chrctrstcs, th NFs of th bov SDs r plottd n Fgur 7. It cn b sn tht th SD dsgnd v th SIP pproch wthout th mnml phs condton of th NF hs th bst nos shpng chrctrstcs. hr r on vrg, pproxmtly 9.5dB, 5.dB nd.7db mprovmnts ovr th frquncy spctrum 0-0Hz comprd to th SDs dsgnd v th Chbyshv structur, th Buttrworth structur nd th SIP pproch wth th mnml phs condton of NF, rspctvly. Howvr, for th SD dsgnd wth th SIP pproch wth th mnml phs condton of th NF, thr s srous ovrshoot n th NF spctrum. hs s bcus th H constrnts on th NF spctrum, whch bound th NF throughout th sgnl bnd, hv bn rmovd n th dsgn procdur n ordr to ntroduc nw mnml phs constrnts. Hnc, t ccounts for th wors prformncs of K nd G rd-off btwn SNR nd nformton cpcty of nos shpd chnnl h sgnl-to-nos rto rflcts th rconstructon rror of th nlog-to-dgtl convrson. Hgh SNR corrsponds to low rconstructon rror. In ordr to chv hgh SNR, SF should b clos to on nd NF should b clos to zro n th sgnl bnd. As rsult, th frquncy rspons of th loop fltr should b nfnty n th sgnl bnd. Howvr, f th NF s mnml phs, ths mpls tht th loop fltr s stbl nd th rgon of convrgnc of th loop fltr trnsfr functon ncluds th unt crcl. In whch cs, th frquncy rspons of th loop fltr s wll dfnd for ll frquncs, whch contrdcts th proprty of th frquncy rspons of th loop fltr hvng hgh SNR prformnc. As rsult, thr s trd-off btwn th SNR nd th nformton cpcty of nos shpd chnnl. hs grs wth th rsults dscrbd n Sctons 4. nd CONCLUSION In ths ppr, optml SD dsgns bsd on smnfnt progrmmng, wth mnml nd non-mnml phs NFs wr comprd wth Buttrworth structurs, Chbyshv structurs nd thortcl dsgns. h non-mnml phs SD dsgnd usng SIP dmonstrtd hgh SNR nd hgh nformton thortc prformnc n th sgnl bnd, whrs th mnml phs dsgn hd gnrlly poor prformnc n th sgnl bnd. Yt ths mnml phs dsgn hd hgh prformnc ovr th ntr spctrum nd cm closr to th lmts suggstd by th Grzon-Crvn nos shpng thorm thn ny of th othr dsgns. hs suggsts trd-off btwn th SNR nd th nformton cpcty of chnnl. It my b prtly ccountd for by th xplnton gvn n Scton 4.3, nd lso prtly du to ovrshoot of th NF n th sgnl bnd of th mnmum phs dsgn. Furthr wor n ths r would nvolv prformnc nlyss of dsgns wthout DC pols, dsgns ncorportng th mnml phs constrnt whl prvntng ovrshoot, nd comprson wth othr optmzd dsgn mthods. AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg 0 of

11 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty SD dsgn mthods SNR (db) R ff (bts) G (bts) G (bts) K (db) K (db) SIP pproch wthout mnml phs NF SIP pproch wth mnml phs NF Chbyshv structur Buttrworth structur Idl fltr wth nonzro NF n sgnl bnd Idl fltr wth zro NF n sgnl bnd bl. Vrous prformnc ndcs of SDs dsgnd v SIP pprochs, Chbyshv structur nd Buttrworth structur. Idl SNRs ssumd -bt, 64 tms ovrsmpld, ffth ordr SD. Fgur 7. NF of SDs dsgnd v SIP pprochs, Chbyshv structur nd Buttrworth structur. AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg of

12 Ho, Rss nd Lng rd-off btwn SNR nd nformton cpcty 6. ACKNOWLEDGEENS hs wor ws supportd by rsrch grnt from Qun ry, Unvrsty of London. 7. REFERENCES [] P.. Azz, S. H. V., nd J. V. Spgl, "An ovrvw of sgm-dlt convrtrs: how -bt ADC chvs mor thn 6-bt rsoluton," IEEE Sgnl Procssng gzn, pp. 6-84, 996. [] W. Kstr, "Whch ADC Archtctur Is Rght for Your Applcton?," Anlog Dlogu, vol. 39, pp. -8, 005. [3] S. Norsworthy, R. Schrr, nd G. ms, Dlt-Sgm Dt Convrtrs: IEEE Prss, 997. [4] D. Rfmn nd E. Jnssn, "Sgnl procssng for Drct Strm Dgtl: A tutorl for dgtl Sgm Dlt modulton nd -bt dgtl udo procssng," Phlps Rsrch, Endhovn, Wht Ppr 8 Dcmbr 00 [5] C. Y.-F. Ho, B. W.-K. Lng, J. D. Rss, Y.- Q. Lu, nd K.-L. o, "Dsgn of Intrpoltv sgm dlt modultors v sm-nfnt progrmmng," to ppr n IEEE rnsctons on Sgnl Procssng, 005. [6] C. Y.-F. Ho, B. W.-K. Lng, nd J. D. Rss, "Dsgn of Intrpoltv Sgm Dlt odultors v Sm-nfnt Progrmmng Approch," Procdngs of th Advncd A/D nd D/A Convrson chnqus nd hr Applctons (ADDA), Lmrc, 005. [7] C. Y.-F. Ho, B. W.-K. Lng, Y. Q. Lu, P. K. S. m, nd K. L. o, "Effcnt lgorthm for solvng sm-nfnt progrmmng problms nd thr pplctons to nonunform fltr bn dsgns," to ppr n IEEE rnsctons on Sgnl Procssng, 006. [8]. Grzon nd P. G. Crvn, "Optml Nos Shpng nd Dthr of Dgtl Sgnls," Procdngs of th 87th AES Convnton, Nw Yor, Nw Yor, USA, 989. [9] H. H. Dm, S. Nordbo, C. A., nd K. L. o, "Frquncy domn dsgn for dgtl Lgurr ntwors," IEEE rnsctons on Crcuts nd Systms I: Fundmntl hory nd Applctons, vol. 47, pp , 000. [0]. S. Bzrr nd C.. Shtty, Nonlnr progrmmng: thory nd lgorthm. Chchstr, Nw Yor: Wly, 979. [] R. Schrr, "h dlt-sgm modultors toolbox vrson," 6.0 d: Anlog Dvcs Inc., dfl.do?objctid=9 []. UNIPR, IALEL, INFINEON tchnologs, ENSERB nd INESC Porto, "DYNAD: thods nd drft stndrds for th DYNmc chrctrzton nd tstng of Anlogu to Dgtl convrtrs." [3] R. Nwroc, G. J.., nd. B. Sndlr, "Informton-thortc constrnts on nos shpng ntwors wth mnmum nos powr gn," Intrntonl Journl of Crcut hory nd Applctons, vol. 7, pp , 999. AES 0th Convnton, Prs, Frnc, 006 y 0 3 Pg of