UNDERSTANDING SIGMA DELTA MODULATION: The Solved and Unsolved Issues

Transcription

1 UNDERSTANDING SIGMA DELTA MODULATION: Th Solvd and Unsolvd Issus By Joshua D. Riss, AES Mmbr Sigma dlta modulation is th most popular form of analog-to-digital convrsion usd in audio applications. It is also commonly usd in D/A convrtrs, sampl-rat convrtrs, and digital powr amplifirs. In this tutorial th thory bhind th opration of sigma dlta modulation is introducd and xplaind. W xplain how prformanc is assssd and rsolv som discrpancis btwn thortical and xprimntal rsults. W discuss th issus of usag, such as limit cycls, idl tons, harmonic distortion, nois modulation, dad zons, and stability. W charactriz th currnt stat of knowldg concrning ths issus and look at what ar th most significant problms that still nd to b rsolvd. Finally, practical xampls ar givn to illustrat th concpts prsntd. INTRODUCTION Sigma dlta modulation (SDM) is prhaps bst undrstood by comparison with traditional puls-cod modulation (PCM). A PCM convrtr typically sampls an input signal at th Nyuist fruncy and producs an N-bit rprsntation of th original signal. This tchniu, howvr, ruirs uantization to N lvls. Whthr implmntd using succssiv approximation rgistrs, piplind convrtrs, or othr tchnius, high rsolution is difficult to obtain in PCM convrsion du to th nd to accuratly rprsnt many uantization lvls and th subsunt circuit complxity. This is th motivation for sigma dlta modulation, a form of puls-dnsity modulation, which xploits ovrsampling and sophisticatd filtr dsign in ordr to mploy a low-bit uantizr with high ffctiv rsolution. In this tutorial, w will considr common dsigns of sigma dlta modulators as usd for analog-to-digital convrsion. Th basic principl is th sam for SDMs mployd in D/A or sampl-rat convrsion. W will rstrict th analysis to asynchronous, discrt-tim dsigns. Howvr, ths ar by far th most common dsigns and includ most fdforward, fdback, and multistag implmntations. W will xplain th thory of opration, mphasizing signal-to-nois ratio stimation and comparison with PCM convrsion. W will also introduc th linar modl, which assists in undrstanding filtr dsign and nois shaping principls. Sinc sigma dlta modulation is highly nonlinar, thr ar various phnomna that cannot b xplaind using this tchniu, such as instability and limit cycls. Th litratur on ths phnomna can b confusing, so w attmpt to giv a clar dfinition of th trms and clarify th currnt stat of undrstanding. Finally, w introduc svral stat-ofth-art tchnius that can b usd to dal with ths unwantd phnomna. THE LINEAR MODEL AND PULSE-CODE MODULATION Th thory of uantization is wllstablishd (s [] and rfrncs thrin). Th allowd valus in th Fig.. Transfr charactristics for a 3-bit uantizr and V= output signal, aftr uantization, ar calld uantization lvls, whras th distanc btwn two succssiv lvls, is calld th uantization stp siz,. For a uantizr with b bits covring th rang from + to -, thr ar b uantization lvls, and th width of ach uantization stp is b = /( - ) () This is dpictd in Fig. for a 3-bit uantizr. Th rounding, or midrisr, uantizr assigns ach input sampl x(n) to th narst uantization lvl. Th uantization rror is simply th diffrnc btwn th input and output to th J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary 49

2 uantizr, =Q(x)-x. It can asily b sn that th uantization rror (n) is always boundd by / ( n) / () Sinc uantization is a highly nonlinar procss, th xact ffct of uantization on th signal contnt and th natur of uantization nois may b difficult to masur. For this rason, svral assumptions ar oftn mad.. Th rror sunc, (n), is a stationary, random procss.. Th rror sunc is uncorrlatd with itslf and with th input sunc x(n). 3. Th probability-dnsity function of th rror is uniform ovr th rang of uantization rror. Such assumptions ar known to b, in gnral, untru. Howvr, thy ar a rasonabl approximation for largamplitud, tim-varying input signals whn b is larg and succssiv uantization rror valus ar not highly corrlatd. Furthrmor, as w shall s, rsults obtaind through th us of this approximation yild accurat stimats of th signal-to-(uantization)- nois ratio, or SNR. Ths assumptions allow us to rprsnt uantization as th introduction of an additiv whit-nois sourc. This is dpictd in Fig.. As w shall s, this modl nabls in-dpth undrstanding of th signal and nois in uantization systms. Mor xact modls xist that includ gain trms applid to th signal and uantization nois, as in [], but th modl dpictd hr is sufficint for analysis. Th assumption that th uantization rror is uniformly distributd ovr a uantization stp givs / / p ( ) = (3) 0 > / Sinc th rror is whit nois, th powr spctral dnsity of th nois will also hav a uniform distribution within th limits of th Nyuist band. Th probability-dnsity function and powr spctral-dnsity function ar dpictd in Figs. 3A and B rspctivly. If th sampling rat satisfis th sampling thorm, i.., th signal is sampld at last twic th highst fruncy in th input signal, f s >f B, thn uantization is th only rror in th Fig.. Th linar modl of uantization Fig. 3. (a) Probability-dnsity function and (b) powr spctral-dnsity function for th uantization rror undr th linar modl. A/D convrsion procss (jittr and othr ffcts ar not considrd hr). Using th assumption of uniform distribution, th avrag uantization nois is givn by (4) and th uantization nois powr is givn by E p( ) d = E{ } = d = {( ) } = = / d = / p ( ) d = / / = 0 (5) From (), w gt (6) To find th SNR, w also nd to stimat th signal powr. Now assum w ar uantizing a sinusoidal signal of amplitud A, x(t)=acos(πt/t). Th avrag powr of th signal is thus = x E{( ) } = E{ } = T b b T 0 = = 3 ( - ) 3 x x x ( Acos( πt / T)) dt = A (7) From (6) and (7), th signal-to-nois ratio may now b givn by, Fig. 4. SNR as a function of th numbr of bits in th uantizr for a PCM-ncodd signal, sampld at th Nyuist fruncy. 50 J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary

3 AES is now a Scitopia.org partnr on hundrd fifty yars of contnt fiftn ladrs in scinc & tchnology rsarch thr million documnts on gatway to it all Sarchscitopia.org to find uality contnt from ladrs in scinc and tchnology rsarch. Scitopia.org gnrats rlvant and focusd rsults with no Intrnt nois. From pr-rviwd journal articls and tchnical confrnc paprs to patnts and mor, scitopia.org is a rsarchrs havn on arth. sarch Intgrating Trustd Scinc + Tchnology Rsarch Scitopia.org was foundd by: Acoustical Socity of Amrica Amrican Gophysical Union Amrican Institut of Physics Amrican Physical Socity Amrican Socity of Civil Enginrs Amrican Socity of Mchanical Enginrs Amrican Vacuum Socity ECS IEEE Institut of Aronautics and Astronautics Institut of Physics Publishing Optical Socity of Amrica Socity of Automotiv Enginrs Socity for Industrial and Applid Mathmatics SPIE

4 SNR(dB)= b x 3A 0log 0log = 0 0 0log 0 A b (8) Thus th signal-to-nois ratio incrass by approximatly 6 db for vry bit in th uantizr. Using this formula, an audio signal ncodd onto CD (a 6-bit format) using PCM, has a maximum SNR of Also not from (7) that th SNR is linarly rlatd to th signal strngth in dcibls. In Fig. 4, th SNR is givn as a function of th numbr of bits in th uantizr for two PCM ncodd signals, sampld at th Nyuist fruncy. E. (8) is usd to prdict th SNR, and th simulatd SNR is masurd dirctly in th tim domain from signal varianc and uantization-rror varianc. Th input signals ar -khz sinusoids, whr it is assumd that th sampling rat is 44. khz, with full rang amplitud A= and with small amplitud A=0.. It can asily b sn, for a highbit uantizr, that th signal-to-nois ratio is indd givn by E. (8). Th only significant rror is for a low numbr of bits du to th approximation first introducd in E. (6). E. (8) also givs a mthod by which th prformanc of sigma-dlta modulators may b compard with Nyuist-rat PCM convrtrs. By invrting this formula for a full-scal input signal and incorporating all th nois and distortion into th signal-tonois-and-distortion ratio (SINAD), w hav th masurabl ffctiv numbr of bits of a uantization, ENOB = SINAD (9) NOISE SHAPING AND OVERSAMPLING Lt s now assum that th signal is ovrsampld. That is, rathr than acuiring th signal at th Nyuist rat, f B, th actual sampling rat is f s = r+ f B. Th ovrsampling ratio is OSR= r =f s /f B. Thus, th uantization nois is sprad ovr a largr fruncy rang yt w ar still primarily concrnd with nois blow th Nyuist fruncy. Th in-band uantization nois powr Fig. 5. Distribution of uantization nois for Nyuist rat sampling, 4 tims, and 8 tims ovrsampling can b found by intgrating th powr spctral dnsity ovr th passband, (0) whr S ( f) = / f s is th powr spctral dnsity of th (unshapd) uantization nois. Most of th nois powr is now locatd outsid of th signal band. As dpictd in Fig. 5, th uantizationnois powr within th band of intrst has dcrasd by a factor OSR. Th signal powr occurs ovr th signal band only, so it rmains unchangd and is givn by E. (7). Th signal-tonois ratio may now b givn by, SNR(dB)= 0log = S f df n ( ) = f f s fb fb = / OSR x r + 0log 0 0 0log A b+ 3. 0r+. 76 () Thus for vry doubling of th ovrsampling ratio, th SNR improvs by 3 db. Th 6-dB improvmnt with ach bit in th uantizr rmains, so w can say that doubling th ovrsampling ratio incrass th ffctiv numbr of bits by half a bit. B Ovrsampling givs us a mans to rduc th ruird numbr of bits in th uantizr. Howvr, this alon is not sufficint. According to E. (), to achiv a CD-uality rcording (ENOB=6, f s =44,00 khz) with an 8-bit uantizr w would nd an ovrsampling ratio of 6, or sampl rat of approximatly.89 GHz, which is unfasibl. STF and NTF undr th linar modl Th abov ovrsampling systm prforms no nois shaping. Considr a filtr placd in front of th uantizr (known as th loop filtr), and th output of uantization is fd back and subtractd from th input signal, as shown in Fig. 6. W now hav a systm that may b rprsntd by transfr functions applid to both th input signal and th uantization nois. In th Z domain, th output may b rprsntd as Yz () = STFzXz () () + NTFzEz () () () whr STF is th signal transfr function and NTF is th nois transfr function. To find ths valus, not that th input to th loop filtr is X(z)- E(z) so that Y(z)=H(z)[X(z)-Y(z)]+E(z). Fig. 6. Rprsntation of a sigma dlta modulator using th linar modl 5 J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary

5 Rarranging trms, w hav, UNDERSTANDING SIGMA DELTA MODULATION: Th Solvd and Unsolvd Issus And thus, Yz ()[ + Hz ()] = HzXz () () + Ez () (3) Hz () STF () z = Hz (), + NTF () z = + Hz () (4) So th linar modl allows us to find th ffcts on th signal and nois du to any choic of filtr function. This filtring and fdback, combind with ovrsampling, ar th ssntial lmnts of sigma dlta modulation. Th primary goal of sigma dlta modulator dsign is to choos a filtr rsulting in high stability and fw artifacts, such that ovr th passband, STFz ( ), NTFz ( ) 0 (5) If such is th cas, thn th nois has bn shapd away from th passband and th signal passs unchangd. First-ordr sigma dlta modulation A/D convrtr A first-ordr SDM has a singl intgrator in th loop filtr. Th simplst dsign has no additional gain trms and may b givn in th tim domain as, un ( + ) = xn ( ) yn ( ) + un ( ) (6) which is dpictd in th block diagram Fig. 7A). Rcalling that, =Q-u and dscribing th prvious tim stp, w hav Fig. 8. Normalizd powr spctral dnsity for puls-cod modulation and for firstordr SDM (no ovrsampling). Actual valus ar found by multiplying by / f. yn ( ) = xn ( ) + ( n) ( n ) (7) A Z-domain block diagram is givn by Fig. 7B, and th corrsponding uation is Yz ( ) = Xzz ( ) + Ez ( )( z ) (8) Thus th signal transfr function is givn by z -. Th signal is unaffctd and only dlayd by on sampl. Th nois transfr function is -z -, which pushs th nois to high fruncis. Using trigonomtric idntitis, NTF ( f ) = jπf / fs jπf / fs ( )( ) = 4sin ( π f / f s ) (9) Unlik ovrsampld PCM, which has unity NTF, th nois shaping in sigma dlta modulation implis a nonconstant nois powr, givn in th basband by, fb n = ( ) ( ) fb (0) whr, again, S ( f) = / f is th s powr spctral dnsity of th unshapd uantization nois. Th total nois powr,, rmains unchangd, but now th nois has bn pushd up to th high fruncis. This is dpictd in Fig. 8, which givs th powr spctral dnsity for first-ordr SDM as compard with PCM. Assuming a high OSR, f f s B and using a Taylor sris xpansion, sin(x)=x-x 3 /3!+ x 5 /5!, E. (0) can b solvd to giv = n S f NTF f df fs f π f f B B s f 4 sin( / ) π s π 3 OSR 3 () s Th signal-to-nois ratio may now b givn by, 3r 3 x SNR(dB)=0log 0 π 0log A+ 60. b r () Th ffct of first-ordr nois shaping is vidnt. W now gt an improvmnt of 9 db for ach doubling of th ovrsampling ratio, rathr than th 3-dB improvmnt that occurs without nois shaping. Fig. 7. (a) first-ordr sigma dlta modulator givn by its block diagram and altrnativly (b), by its z-transform block diagram with th uantizr approximatd by a nois sourc. SNR for high-ordr sigma dlta modulators This tchniu can b xtndd to highr-ordr filtrs. Th transfr J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary 53

6 function of a gnric Nth-ordr SDM is givn by N Yz () = Xzz () + Ez ()( z ) = STFzXz () () + NTFz () Ez () (3) Thus, th nois transfr function is givn by NTF f j f / f s ( ) = ( π ) N (4) Using an intgral formula, this givs fb n ( ) ( ) fb f = S f NTF f df = fb s fb [ sin( π f / f )] N π N + ( N + ) OSR (5) whr th approximat uality was found by using a Taylor sris xpansion on th sin trms and kping only th first nonzro trms. Compard with th first-ordr SDM, this provids mor supprssion of th uantization nois ovr th low fruncis and mor amplification of th nois outsid th signal band. E. (5) can b usd to find th gnral formula for th SNR of an idal Nth-ordr SDM, SNR(dB)=0log 0log 0 s N df N+ ( N + ) ( ) 0log A+ 60. b log ( N + ) 9.94N ( N+ ) r (6) Thus w s a larg improvmnt with incrasing SDM ordr. For a scond-ordr SDM (N=), thr is a 5-dB improvmnt in th SNR with ach doubling of th ovrsampling ratio. In gnral, for an Nth-ordr SDM, thr is a 3(N+) db improvmnt in th SNR with ach doubling of th ovrsampling ratio, and a 6-dB improvmnt with ach additional bit in th uantizr. Thus, us of high-ordr SDMs and a high ovrsampling ratio offrs a much bttr SNR than that obtaind by simply incrasing th numbr of bits. Of cours, this is an approximation. It dpnds on th cofficints of th modulator, on th approximations usd in th drivation, and othr factors. Nvrthlss, it provids an 0 π N x + r Fig. 9. SNR as a function of r, th log of th ovrsampling ratio. In ach cas, 7 sampls and a 6-bit uantizr wr usd, and th input signal had a fruncy 0f s / 7. uppr limit on th SNR. Fig. 9 dpicts th SNR for PCM ncoding, a firstordr SDM, and a scond-ordr SDM. In ach cas 7 sampls and a 6-bit uantizr wr usd, and th input signal had a fruncy 0f s / 7. Unlik in th PCM simulation for Fig. 4, th uantization rror and input signal sampls ar no longr tim alignd, so th signal and nois powr wr calculatd in th fruncy domain. Thr is strong agrmnt btwn thory and simulation, with th diffrncs bing attributabl to th assumption of high ovrsampling ratio (for th Taylor sris truncation), th difficulty in accurat masurmnt of SNR, particularly at larg valus, and th assumptions mntiond arlir, particularly that of uniform PDF ovr th rang / to +/. Nvrthlss, th 3-dB, 9-dB, and 5-dB incrass for doubling th OSR hav bn confirmd, and thr is rasonabl agrmnt throughout. ISSUES IN SIGMA DELTA MODULATION As rportd in [3, 4], 64-tims-ovrsampld -bit A/D convrtrs, using a fifth-ordr SDM hav bn dsignd and achiv an SNR ovr 0 dbs. Yt from E. (6), w find that SNR(dB) 0log 0 A (7) To dat, no on has dsignd a sigma dlta modulator with such high prformanc. As an xampl, considr Fig. 0. Implmntation of a ralistic fifth-ordr fdforward SDM 54 J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary

7 Fig.. Estimats of th signal-to-nois ratio as a function of input signal amplitud for an idal SDM, drivd from thory, and for a practical SDM, both drivd from thory and stimatd from simulation. In all cass, th SDM was assumd to b -bit, fifth-ordr, with an ovrsampling ratio of 64. a -bit, arbitrary-ordr fdforward SDM that may b rprsntd as, yn ( ) = sgn( c s( n)) s ( n+ ) = s ( n) + x( n) y( n) s ( n+ ) = s ( n) + s ( n)... s ( n+ ) = s ( n) + s ( n) (8) N N N x is th input signal and y th output bitstram, as bfor, and u = c s is th input to th uantizr, whr c is a cofficint vctor dtrmining noisshaping charactristics. A practical fifth-ordr implmntation of this dsign, Fig. 0, is dscribd in [5], and is intndd to b usd for analog-todigital convrsion in audio applications (w will rturn to this SDM latr in this sction to dmonstrat othr phnomna). Th SDM is lowpass, has a cornr fruncy of 80 khz for a sampl rat of 64 x 44. khz, and is givn by th cofficints, c = [ , , , , ] (9) Fig. dpicts th thortical SNR as a function of input signal amplitud for an idal fifth-ordr SDM, and both thortical and simulatd SNR stimats for th SDM givn by E. (8) and (9). In both cass, th SDM was assumd to b bit with an ovrsampling ratio of 64. Th thortical SNR is computd dirctly from th signal powr and in-band nois powr. In-band nois powr is found from th intgral givn by E. (0), which is numrically intgratd for th practical SDM and may b found from E. (6) for th idal SDM. Not that th practical dsign has an SNR almost 80 db lss than that of th idal dsign, and that simulation shows a dramatic drop in prformanc for input amplitud gratr than Th archtypal Nth-ordr SDM with NTF (-z - ) N is highly unstabl; hnc loop filtrs with lss-aggrssiv nois shaping ar usd. But vn this lssaggrssiv SDM bcoms unstabl with larg input valus. This instability problm is not xplaind by th linar modl. In fact, thr ar a host of issus in sigma dlta modulation that ar causd by fdback around a highly nonlinar uantizr. In this sction w will look at th rlatd causs of ths unwantd bhaviors and highlight th currnt stat of undrstanding and rsarch. Limit cycls Considr a singl-bit first-ordr SDM as givn by E. (6) with constant input x=.5, and an initial condition, say, u(n)=0.>0 un ( + ) = =. 4 un ( + ) = =. un ( + 3 ) = =. 6 un+ ( 4) = =. (30) Thus, th input to th uantizr rpats with a priod of 4 itrations and th uantizr producs a rpating output bitstram +,-,+,+ Not that th avrag valu is [3*(+)+*(-)]/4, which is th sam as th input. This rlationship can b asily shown for a first-ordr SDM sinc rpatd application of E. (6), lads to un ( + k) = k i= 0 k xn ( + i) yn ( + i) + un ( ) i= 0 (3) If w assum constant input and that u rpats aftr k cycls, w hav k kx = y ( n + i) i= 0 (3) which implis that avrag output is ual to th input signal. Th occurrnc of a rpating sunc in th output bitstram is known as a limit cycl. It poss problms in th signal procssing of th output. Mor sriously, whn SDMs ar applid in audio applications, limit cycls can rsult in audibl artifacts. For instanc, in th cas just mntiond, th input was purly DC, yt th limit cycl rsults in a suar wav with 75% duty cycl and fruncy f s /4 at th output. Th thory of limit cycls in low- Fig.. Plot of th input to th uantizr as a function of th itrat for a fifth-ordr SDM. Th uantizr input, and hnc th output bitstram, ntr a limit cycl around itrat 300 and again at approximatly J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary 55

8 ordr (first- and scond-ordr) sigma dlta modulators is wll undrstood. Yt limit cycls can xist in high-ordr SDMs oprating undr normal conditions. Rturning to th SDM dscribd in th introduction at th bginning of this sction, with an input of 0.7 and initial conditions s=0, it xhibits a short-trm limit cycl. This can b sn in a tim sris of th input to th uantizr, as dpictd in Fig.. This xampl illustrats both th xistnc of limit cycls in high-ordr dsigns and th fact that limit cycls may also appar for finit duration. It is only in rcnt yars that th thory of limit cycls has bn xtndd to charactriz bhavior in highr-ordr SDMs [5-0]. Rfman s Thorm shows that, for most SDM dsigns, DC input implis that th output bitstram is priodic if and only if th stat-spac variabls (a vctor dscribing th currnt stat of th systm) ar priodic. Using this condition it was possibl to find all limit cycls that may occur for a givn sigma dlta modulator and th st of initial conditions that may gnrat thm. This allowd a dscription of snsitivity to limit cycls for diffrnt SDM dsigns and th as by which various tchnius may b usd to brak out of a limit cycl. Othr significant rcnt rsults includ analysis of limitcycl bhavior with nonconstant, priodic input and dvlopmnt of limit-cycl dtction and rmoval tchnius. Dsign of SDMs to avoid limit cycls is accomplishd ithr by using mor complx nois shaping structurs [] or through th addition of dithr or a control [] in ordr to supprss limit cycls in an xisting dsign. Qustions lingr concrning how problmatic limit cycls ar whn th input has a small amount of nois, such as with analog implmntations of SDMs whn th limit cycl is of finit duration or whn th input is not constant but th ovrsampling ratio is larg nough such that th input appars narly constant ovr a short duration. Th framwork stablishd in [5] may b usd to addrss ths issus. Idl tons Thr is littl thortical undrstanding of idl tons, vn for th simplst Fig. 3. Illustration of th dfinitions usd to distinguish a limit cycl (lft) from an idl ton (right). A limit cycl consists of a finit numbr of discrt paks in th fruncy spctrum; an idl ton is a pak in th fruncy spctrum, but suprimposd on a nois background. low-ordr sigma dlta modulators. Prhaps th most significant is arly work by Candy [3] that showd that in a first-ordr SDM th idl ton is simply an alias of an ovrton of a suar wav that is discrtly sampld. Thus it is suspctd that a similar phnomnon may account for idl tons in highr-ordr SDMs. Th idlton phnomnon in highr-ordr SDMs has bn dscribd in [4, 5], and xprimntal vidnc of idlton bhavior in a scond-ordr bandpass SDM was rportd in [6]. It is important to distinguish btwn limit cycls and th rlatd phnomnon of idl tons. Th rptitiv pattrns that xist in th output bitstram ar rfrrd to as idl pattrns or limit cycls. Whras an idl ton is rprsntd by a discrt pak in th fruncy spctrum of th output of a SDM, but suprimposd on a background of nois (s Fig. 3). In this cas thr is no uniu sris of rpating bits. This distinction is oftn blurrd, and altrnativ dfinitions ar somtims usd. Kozak and Kal [7] do not distinguish btwn limit cycls and idl tons pr s, but instad rfr to idl tons as priodic pattrns with constant input and harmonic tons as priodic pattrns rsulting from sinusoidal input. Bourdopoulos [8] rportd idl tons occurring with both constant and sinusoidal input for a third-ordr SDM. Th lattr w would rfr to as harmonic distortion. H also linkd th gnration of idl tons to th xistnc of almost rpating pattrns. Ldzius [9] usd this link to infr that sinc a linar modl of an SDM cannot account for limit cycls, it must not b abl to account for idl tons ithr. This link to limit cycls (xactly priodic suncs) has also bn suggstd by Angus [0] and Fig. 4. Powr spctrum of a 64 tims ovrsampld fifth-ordr SDM with constant input of amplitud 0.7, dpicting idl tons. 56 J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary

9 AssistAnt Profssor of Audio nginring tchnology Tach 4 crdit hours pr yar. Advis studnts with acadmic progrss and mntor studnts in rsarch and othr class-rlatd activitis. Engag in activitis to support th mission and vision of Blmont Univrsity and th Collg. Engag in scholarly activity and profssional dvlopmnt. Participat in dpartmntal work as appropriat. For additional information about th position and to complt th onlin application, visit Rviw of applications will bgin immdiatly. Blmont Univrsity is an ual opportunity /affirmativ action mployr undr all applicabl civil rights laws. Womn and minoritis ar ncouragd to apply. Locatd nar Nashvill s Music Row, th Mik Curb Collg of Entrtainmnt and Music Businss at Blmont Univrsity nrolls 00+ majors and combins classroom xprinc with ral-world applications. Facilitis fatur stat-of-th-art classrooms and rcording studios, including th award-winning Ocan Way Nashvill studios, Historic RCA Studio B, and th Robrt E. Mulloy Studnt Studios in th Cntr for Music Businss.

10 Jsprs []. Yt rcnt work by th authors has shown that, although ths tons ar rlatd to th input in much th sam way as for limit cycls, thy ar not asily accountd for within th currnt thory of limit cycls [9]. Howvr tmpting, th assumption that idl tons rsult from limit cycls has so far lft svral phnomna unxplaind. First, th limit cycl producs a spctrum with discrt paks of comparabl amplitud occurring at multipls of th limit-cycl fruncy [9]. Thus, for a limit cycl with a short priod, all paks would b out of th audibl band. For a limit cycl with sufficintly long priod to produc audibl tons, almost all harmonics would b prsnt and significant. This is in dirct contrast to idl tons, which ar known to produc a rlativly small numbr of paks and th highr harmonics ar of small amplitud. Furthrmor, limit cycls ar highly snsitiv to initial conditions and input. An infinitsimal chang in th input will compltly rmov a limit cycl, and a chang in initial conditions will dstabiliz it. Idl tons, on th othr hand, may b obsrvd rgardlss of initial conditions and prsist as th input is changd. Finally, it has bn shown that limit cycls xist only if th input is constant or priodic [6] whras thr ar no known similar constraints on idl tons. Th phnomnon of idl tons is illustratd in Fig. 4, which dpicts th powr spctrum of th fifth-ordr SDM dscribd in th prvious sction on limit cycls, with constant input of 0.7. Th strongst idl ton occurs at khz, or xactly 3/0 of th sampling fruncy of 64 x 44. khz. Th nxt strongst is th first harmonic of this ton. This is also vidnc of a simpl rlationship obsrvd in []. In that papr th authors notd that th fruncy of th fundamntal idl ton, rfrrd to as f FIT, is proportionally rlatd to th amplitud of th input signal, f = A f FIT DC s (33) Hr, w rfr to th amplitud as rlatd to full scal. An input of amplitud 0.7 is 7/0 of th full scal from - and +, or 3/0 of th full scal on a rvrs x-axis, from + to -. Sinc 7f s /0 is abov f s /, w would xpct Fig. 5. Powr spctrum of a 64 tims ovrsampld fifth-ordr SDM with sinusoidal input of fruncy 5 khz and amplitud 0.7, dpicting harmonic distortion. th fruncy of th idl ton to b 3f s /0, as obsrvd. Th othr significant idl tons ar du to harmonics and aliasing of th fundamntal ton. Although th fundamntal idl ton is typically far outsid th audibl rang, highr-ordr harmonics of this ton may alias back to lowr fruncis. It is important to not that th rlationship givn by E. (33), though asily obsrvd, has no known thortical justification. Nor dos thr yt xist any thory that will stimat th amplituds of th idl tons. This is prhaps mor rlvant sinc th dsignr is concrnd with unwantd tons that ar significantly abov th nois floor. Furthrmor, th fundamntal and its harmonics may not account for all obsrvd tons. Thus a mor dtaild undrstanding of idl tons rmains a significant challng. Proposd solutions to idl tons and harmonic distortion ar th addition of dithr or th us of chaotic SDMs [3]. Ths sam solutions ar ffctiv in liminating limit-cycl bhavior. Thus, it is blivd that ths ar, at last in part, dynamical systms phnomna, but not a dirct consunc of limit cycls. Nonlinar dynamics tchnius may thrfor b succssful in dvising a paralll analytical approach to th undrstanding of idl tons. Harmonic distortion As with idl tons, thr ar multipl dfinitions of harmonic distortion in th litratur. Harmonic distortion, though also apparing as undsird tons, occurs whn sinusoidal input is applid. It may b dfind as th prsnc of harmonics (signals whos fruncis ar intgr multipls of th input signal) and othr spurs in th output spctrum that wr not prsnt in th input signal. In this cas w ar concrnd both with paks that ar du to unwantd harmonics or aliasing of th input signal and thos that bar no apparnt rlationship to th input fruncy. Kozak and Kal [7] rfr to harmonic tons as priodic pattrns rsulting from sinusoidal input (ths would still b considrd limit cycls undr our prvious dfinition). Exprimntal vidnc of harmonic distortion was givn in [4] and [5]. Thortical rsults hav typically bn limitd to approximat analysis of scond-ordr or third-ordr harmonics [6] or low-ordr modulators [3, 7]. Thy ar also spcific to harmonic distortion that appars as a rsult of circuit imprfctions, as opposd to th inhrnt distortion componnts in high-ordr SDMs. Aliasing of th distortion componnts back into th passband was dscribd in [8], though no thortical approach was providd. Harmonic distortion is dpictd in th powr spctrum shown in Fig. 5, which is idntical to Fig. 4 xcpt now th constant input of 0.7 has bn rplacd with sinusoidal input of 58 J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary

11 Figur 6. Th first two conditional momnts of rror (avrag uantization nois and avrag uantization nois amplitud 0.7. Th distortions ar far mor numrous than th idl tons producd from constant input and ar spcially intriguing bcaus thy show no obvious rlationship to th input fruncy or amplitud. In both cass, fortunatly, th tons ar far outsid th audio band (though this is not always th cas). Harmonic distortion, vn though it is a srious issu, is not wll undrstood. It is known that it is mor problmatic in low-ordr dsigns, and th sam tchnius that ar usd to rmov idl tons may b applid hr. Th origins of harmonic distortion li in th fact that, in a low-bit rprsntation, sinusoidal signals ar rprsntd in th output bitstram as clos to a suar wav. Sinc suar wavs hav strong harmonics, ths harmonics appar in th output spctrum. Howvr, no gnral thory xists allowing on to analytically driv th dominant harmonics and thir amplituds or to dtrmin th suscptibility of diffrnt filtr dsigns to harmonic distortion. Nois modulation W bgin by looking at dithr distributions and th rsultant uantization rror in PCM systms. Th approach usd is slightly simplr though lss rigorous than that in []. Th analysis in this sction is prsntd primarily bcaus th analysis of SDM systms can b mad via an xtnsion of this approach. Rcall that an infinit midrisr uantizr has th input-output charactristic Qw ( ) = w/ + /, whr w is th input to th uantizr. If th input to a PCM systm, x, is fd dirctly into th uantizr, thn th total rror btwn input and output is simply = x x / + / (34) Thus th mth momnt of th rror for a givn x is m p ( ) = ( x x / + / ) (35) Undr such circumstancs, all rror momnts ar dpndnt on th input valu x. In particular, th scond momnt, th uantization nois powr, dpnds on th signal. This is known as nois modulation. It can b prcivd in th uantization of audio signals and is gnrally undsirabl. Howvr, if random nois with uniform probability distribution from / to +/, othrwis known as rctangular PDF (RPDF) dithr of siz Last Significant Bit (LSB), is applid immdiatly bfor uantization thn th PDF of th input to th uantizr has th form, pw ( ) = / x / < w x+ / 0 othrwis (36) Th input rangs ovr LSB, so th output can assum only possibl valus. If w dfin y = x / + / x / + /, thn th rror has th distribution, = ( y) p = y y p = y (37) and hnc, using E. (35), th first rror momnt bcoms indpndnt of th input. p ( ) = ( y) y y( y) = 0 (38) Not that this was also th rsult of th uniform distribution assumption in th linar modl, E. (4), only now th addition of dithr maks th assumption fully valid. In fact, it can b shown that nth-ordr dithr will mak th first n rror momnts indpndnt of th input. Triangular (TPDF) dithr of width LSBs, which can b gnratd by summing two rctangular PDF dithrs of width LSB, will rndr both th first- and scond-ordr momnts of th rror indpndnt of th input signal. That is, th uantization nois will hav a constant avrag of zro and a constant (nonzro) powr indpndnt of th input signal charactristics. Th dithring forcs th uantization nois to los its cohrnc with th original input signal, but has th drawback of raising th avrag spctral nois floor. Th ffct of dithr on th conditional momnts of rror is dpictd in Fig. 6. This shows th first two conditional momnts of rror (avrag uantization nois and avrag uantization nois powr) as a function of th systm input for a PCM systm without dithr, with RPDF dithr and with TPDF dithr. This PCM systm uss th sam 3-bit midrisr uantizr as dpictd in Fig.. Th rsults wr found from simulation of 00,000 data points with a constant input to th systm and a random numbr gnrator usd to crat th dithr signal. With RPDF dithr, th avrag uantization nois rmains fixd at zro, rgardlss of th input signal. With TPDF dithr both th avrag uantization nois and th avrag uantization nois powr ar constant, thus th nois modulation has bn rmovd. Howvr, du to th finit bits in th uantizr, th rsults do not hold for th lowst and highst uantization lvls with RPDF dithr, and do not hold for th lowst two and highst two uantization lvls with TPDF dithr. To dat, th thory of nois J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary 59

12 rlatd to limit cycls in that th output is an ndlssly rpating bitstram. A formal dfinition would b that a dad zon is a continuous rang of SDM input, such that, for givn initial conditions, th sam output bitstram would b producd. If on considrs a nonzro initial condition, thn E. (4) bcoms k k i k i uk ( ) = cu( 0) + cx+ ( ) ( c) (46) i = 0 Fig. 7. Avrag output as a function of DC input for first- and scond-ordr SDMs with intgrator lak modulation has not yt bn xtndd from PCM to SDM. Though th us of dithr to prvnt nois modulation in puls-cod modulation is wll undrstood, thr ar many intriguing ustions that rmain whn dithr is applid to a sigma dlta modulation systm. Th fdback loop affcts th distribution of th input to th uantizr in complicatd ways. Many rsarchrs hav suggstd that this lads to a form of slf-dithring, but corrlations rmain btwn th uantization nois and th input signal. In addition, onbit sigma dlta modulation is fruntly usd, yt RPDF dithr of LSB or TPDF dithr of LSBs, as mntiond abov, will caus th uantizr to b ovrloadd [9], and no amount of dithr in a on bit systm will produc constant nois powr [30]. Dad zons For crtain input signals, th input may not b proprly ncodd by th SDM. That is, thr is a rang of input for which th sigma dlta modulator may produc th sam avrag output valu. This rang is known as a dad zon. Considr a first-ordr SDM with a lossy intgrator, such that E. (6) is rplacd by un ( ) = cun ( ) + xn ( ) yn ( ) (39) whr c<. For x<<, y(0)=+, this yilds, u() = cu(0)+x-y(0) = x-<0 u() = cu()+x-y() = (+c)x+(-c)>0 u(3) = cu()+x-y() = (+c+c )x-(-c+c )<0 (40) which givs th output bitstram y(n)=+,-,+,-, Th gnral formula for th input to th uantizr bcoms, k uk ( ) = cx+ ( ) ( c) i = 0 (4) Whn x is ual to 0, and c<, thn this producs th limit cycl +,-. In ordr to brak out of this limit cycl for nonzro x, at som point w must hav, for odd k, i uk ( ) > 0 cx> ( c) or for vn k, i k i k i = 0 k i uk ( ) < 0 cx< ( c) i = 0 k i = 0 k i = 0 (4) (43) In th limit of larg k, ths formulas bcom x x and c < c > + c + c (44) That is, all input in th rang c c x (45) + c < < + c will not brak out of a limit cycl, and thus will hav no ffct on th output. Th rgion of input dfind by E. (45) is known as a dad zon, and th phnomnon is somtims rfrrd to as a thrshold ffct. It is strongly i i which in th limit of larg k still rducs to E. (4). Thus, this dad zon is indpndnt of initial conditions. It should b mntiond that in a firstordr SDM, dad zons occur for vry limit cycl. Sinc th limit cycls xist with any rational input, this can b rphrasd as stating that dad zons xist in th nighborhood of vry rational input. Similar phnomna xist in scond-ordr SDMs. Fig. 7 dpicts th avrag output of a sigma dlta modulator as a function of DC input. It can b sn that both firstand scond-ordr SDMs may xhibit dad zons. Dad zons ar familiar to th SDM community and ar mntiond in many dsign txtbooks [3]. Thr was som xtnsiv analysis of dad zons in firstand scond-ordr SDMs in th work of Fly [3-36]. Howvr, sh did not rfr to thm as such, and instad discussd th staircas structur of th SDM output as a function of input, which charactrizs th dad zons. To dat, no on who has charactrizd th dad zons that can xist in high-ordr SDMs nor fully charactrizd th possibl rlationships btwn dad zons, initial conditions, and input. Howvr, though this sms tractabl, thr is littl vidnc that dad zons ar a srious issu in highordr SDM dsigns. Stability In our arlir introduction to th issus in SDM, w statd that th primary rason that idal SNRs ar not achivd with high-ordr SDMs is that many dsigns ar highly unstabl. Thr has bn much rsarch into stability issus in SDMs, but many ssntial ustions rmain unsolvd. At its cor, on would lik to driv 60 J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary

13 valus of constant input such that, for crtain initial conditions, th magnitud of th uantizr input will divrg toward infinity. A similar ustion is, givn initial conditions and constant input, dtrmin if this lads to stabl bhavior. For th first-ordr SDM, it is asy to show that it is stabl for all input -<x< and to find th rlatd bitstram bhavior. Rturning to th tim-domain xprssion givn by E. (6), assum x is constant and -<x<. Th following rlationships show that a ngativ u will incras until it is positiv, and a positiv u will dcras until it is ngativ. u(n)<0 u(n+) = u(n)+x+>u(n) u(n) 0 u(n+) = u(n)+x-<u(n) (47) Thus it is oscillating btwn positiv and ngativ valus. Now assum that at last on bit flip has occurrd (i.., th transint bhavior has passd and w ar not starting from arbitrary initial conditions). From E. (47), th maximum valu of u occurs whn th prvious valu is just blow zro and th minimum valu occurs whn th prvious valu is ual to zro un ( ) 0 un ( + ) x+ < un ( ) = 0 un ( + ) = x (48) Thus u is limitd to th rang [-+x;+x]. Th first-ordr SDM can b itratd to giv, whn th output bits all hav th sam sign, un ( + m) = un ( ) + mx ( y) (49) Assum that thr ar xactly n + positiv output bits in a row. Thn w hav u() un ( ) = u( ) + [ n ]( x ) 0 (50) Sinc th maximum valu of u is x+, w hav that th maximum numbr of output bits occurs whn, x + + [ n + ]( x ) 0 (5) so th maximum numbr of positiv output bits is givn by th largst n + such that n + x (5) similarly, th maximum numbr of ngativ output bits is givn by th Fig. 8. Th absolut magnitud of th uantizr input in a fifth-ordr SDM as a function of itrat for various constant input magnituds. At x=0.7, th uantizr input is stabl, at x=0.9 th uantizr input is unstabl and oscillating with xponntially incrasing priod and amplitud, and at x=. th uantizr input is unstabl but nonoscillating. largst n - such that n (53) + x Th standard scond-ordr SDM has similar stability proprtis in that stability is indpndnt of initial conditions, but proof of bounddnss is not trivial. A linar programming approach is usd by Farrll and Fly [37]. Thy assum that thr hav bn som numbr n - itrations with ngativ output. From this, thy idntify th maximum valus of th stat-spac variabls for th first positiv output bit. This valu is usd to idntify th maximum numbr of positiv output bits, which rsults in n +. Thy thn find th maximum numbr of ngativ output bits that rsult from th n + positiv bits. This nw valu of n - is strictly lss than n + and hnc th oscillations ar boundd. This succssfully finds th bounds on th scondordr SDM and may b xtndd to scond-ordr SDMs with laky or chaotic intgrators, and thir rsults bar strong agrmnt with simulation. To th bst of our knowldg, thr is no succssful analytical approach to stability in high-ordr SDMs (ordr gratr than ). Thr ar svral altrnativ approachs to stability in scond-ordr SDMs, som prliminary work on third-ordr dsigns, and only sktchd approachs to stability in highr-ordr SDMs. Thus th ustion bcoms, Can any xisting approachs b xtndd to highrordr SDMs? Of cours, thr is th rlatd ustion of whthr xisting approachs ar corrct. Risbo [38] discussd stability of SDMs in dtail, primarily from a nonlinar dynamics prspctiv. But, with th xcption of first-ordr SDMs, h did not attmpt a mthod for its dtrmination. A computational approach to finding th invariant sts, which consist of initial conditions giving ris to stabl bhavior, is drivd by Schrir [39 4]. Although nithr analytical nor rigorous, it is significant bcaus sourc cod is availabl and bcaus rsults ar providd that may b confirmd or dnid by othr mthods. Hin and Zakhor s approach [4] is to us th limit cycls as a masur of stability. Thir mthod is not rigorous, in that it postulats that th limit cycls hav a convrgnt bound on th stat-spac variabls, and that this is also th bound for nonlimit cycl bhavior. Wang [43] convrtd a third-ordr modulator to a continuous-tim systm by looking at th vctor fild uations. By considring only boundary points, h is abl to convrt th 3-dimnsional flow into a -dimnsional rturn map. Fixd points of this map thn yild insight into stability of th SDM. Zhang [44, 45] usd a modl of th uantizr to stimat th stability of a third-ordr SDM. Th linarization implis that important phnomna hav bn J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary 6

14 omittd. Furthrmor, thr is littl comparison of th rsults with simulation. Anothr work by Zhang [46] bars a strong rsmblanc to th linar programming approach of Fly. Stinr and Yang [47, 48] us a transformation that dcoupls th stat-spac variabls, xcpt through thir intraction in th uantization function. Thy suggst how this may b usd to tackl stability but thr is littl analysis. This approach has bn xpandd by Wong [49] to dal with practical high-ordr SDMs. Wong provids simulatd rsults for many high-ordr SDMs, but his analysis dos not confirm simulation. Mladnov t al. [50] also us a rlatd transformation and hav shown promising rsults on simpl but highordr SDMs. Th author and collaborators ar currntly invstigating th potntial of this tchniu. Now w will show that th statspac variabls ar always unboundd for u >, and that th stat-spac variabls oscillat btwn positiv and ngativ valus for u <. Not that this oscillation dos not guarant stabl bhavior, but an undrstanding of th oscillations may lad to an undrstanding of stability. Rturning to th arbitrary SDM givn by E. (8), it is asy to s that s ( n) > 0 s ( n+ ) > s ( n) i i+ i+ s ( n) < 0 s ( n+ ) < s ( n ) i i+ i + (54) If x>, x y( n) > 0, rgardlss of c. Thrfor, s ( n+ ) > s ( n),.g., s always incrass. Hnc, for som k, s ( k)> 0. At which point s will incras, and at som point it will bcom positiv, and so on. This implis that, at som point all statspac variabls will incras. Similarly, if x<-, at som point, all stat-spac variabls will dcras. Thus, for any fdforward SDM in this form, th bounds ar always <=. Assum 0<x<, Assum y(n)>0. So s ( n+ ) n n = s ( ) + x < s ( ) (55) So s dcrass. s may still incras, but vntually s bcoms lss than 0. Thn s starts to dcras, and so on. ( n Evntually s ) N < 0, and y(n) flips to -. Now, th sam procdur happns again, but with ach variabl incrasing. This givs oscillation. Th problm coms whn ach oscillation taks longr than th prvious on. An xampl of this oscillation is dpictd in Fig. 8. For x=0.9, th systm is unstabl but still oscillating btwn - and + output, though th oscillations ar xponntially incrasing in both amplitud and priod. CONCLUSION In this tutorial w hav discussd mthods of opration, dsign, and us of sigma dlta modulators. Although w considrd th dsign of a sigma dlta modulator as usd for analog-to-digital convrsion, th rsults drivd hr could asily b gnralizd. Th dscriptions apply ually for SDMs usd in othr applications, such as D/A convrtrs or Class D amplifirs. Th SNR formulas in th linar modl and puls-cod modulation sction and th sction on nois shaping and ovrsampling can also b modifid for diffrnt input signals or filtr dsigns. Th xampls in th sction on SDM issus could apply to multibit SDMs as wll. W v idntifid svral ky issus in sigma dlta modulation: limit cycls, idl tons, dad zons, harmonic distortion, nois modulation, and stability. To som xtnt limit cycls may b considrd a mostly solvd problm, whras for ach of th othr problms th issus ar undrstood for low-ordr dsign but th thory is not yt stablishd for th high-ordr dsigns. Dad zons hav bn ffctivly charactrizd for lowordr dsigns, but thy hav not bn rportd to b problmatic in highordr or commrcial dsigns. Nois modulation is wll undrstood for PCM, yt thr is no wll stablishd thory for vn low-ordr sigma dlta modulators. Idl tons and harmonic distortion, though not wll undrstood, ar clarly rlatd phnomna. With idl tons in particular, wll-dfind and simpl rlationships btwn th input signal and th fruncis of th tons hav bn obsrvd that do not yt hav a thortical basis. Nois modulation, idl tons, harmonic distortion, and limit cycls may b dalt with, at last in part, through th application of dithr. Howvr, dithr is lss ffctiv whn usd with a low bit uantizr, which is also whn ths issus ar most srious. Furthrmor, dithr is not hlpful in daling with stability issus, and will actually dcras th stabl rang of a sigma dlta modulator. A bttr undrstanding of stability (and of othr issus) is ndd so that robust, highprformanc implmntations may b dvlopd. Thr hav bn many promising rcnt rsults that may lad toward a thory of stability in sigma dlta modulators; this rmains an activ ara of invstigation for th author. REFERENCES [] S. P. Lipshitz, R. A. Wannamakr, and J. Vandrkooy, Quantization and Dithr: A Thortical Survy, J. Audio Eng. Soc., vol. 40, pp (99 May). [] S. H. Ardalan and J. J. Paulos, An Analysis of Nonlinar Bhavior in Dlta Sigma Modulators, IEEE Trans. on Circuits and Systms I: Fundamntal Thory and Applications, vol. 34, pp (987). [3] D. Rfman and E. Janssn, Signal Procssing for Dirct Stram Digital: A Tutorial for Digital Sigma Dlta Modulation and -Bit Digital Audio Procssing, Philips Rsarch, Eindhovn, Whit Papr 8 (00 Dc.). [4] D. Rfman and P. 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16 Sigma Dlta Modulators, Int. J. Circ. Thory and Appl., vol. 5, pp (997). [36] O. Fly, Nonlinar Dynamics of Sigma Dlta Modulators and Phas- Lockd Loops, Qun Mary Univrsity of London, London, UK, invitd talk (00 Jan.). [37] R. Farrll and O. Fly, Bounding th Intgrator Outputs of Scond Ordr Sigma Dlta Modulators, IEEE Trans. on Circuits and Systms, Part II: Analog and Digital Signal Procssing, vol. 45, pp (998). [38] L. Risbo, Sigma Dlta Modulators Stability Analysis and Optimization, PhD Thsis, Elctronics Institut, Tchnical Univrsity of Dnmark, 994, pp. 79, (994) < publications/phdthsis.html>. [39] R. Schrir, M. Goodson, and B. Zhang, An Algorithm for Computing Convx Positivly Invariant Sts for Dlta-Sigma Modulators, IEEE Trans. on Circuits and Systms I: Fundamntal Thory and Applications, vol. 44, pp (997). [40] B. Zhang, M. Goodson, and R. Schrir, Invariant Sts for Gnral Scond-Ordr Lowpaws Dlta-Sigma Modulators with DC Inputs, Proc. ISCAS, pp. 4 (994). [4] M. Goodson, B. Zhang, and R. Schrir, Proving Stability of Dlta- Sigma Modulator Using Invariant Sts, Proc. ISCAS, pp (995). [4] S. Hin and A. Zakhor, On th Stability of Sigma Dlta Modulators, IEEE Trans. Signal Procssing, vol. 4, pp (993). [43] H. Wang, On th Stability of Third-Ordr Sigma Dlta Modulation, Proc. ISCAS, pp (993). [44] J. Zhang, P. V. Brnnan, D. Jiang, E. Vinogradova, and P. D. Smith, Stabl Boundaris of a Third- Ordr Sigma Dlta Modulator, Proc. Southwst Symp. on Mixd-Signal Dsign, pp (003). [45] J. Zhang, P. V. Brnnan, D. Jiang, E. Vinogradova, and P. D. Smith, Stability Analysis of a Sigma Dlta Modulator, Proc. Int. Symp. on Circuits and Systms, ISCAS, pp. I-96 I-964 (003). [46] J. Zhang, P. V. Brnnan, P. D. Smith, and E. Vinogradova, Bounding Attraction Aras of a Third-Ordr Sigma Dlta Modulator, Proc. Int. Conf. on Communications, Circuits and Systms, ICCCAS, pp (004). [47] P. Stinr and W. Yang, A Framwork for Analysis of High-Ordr Sigma Dlta Modulators, Circuits and Systms II: Analog and Digital Signal Procssing, IEEE Trans., pp. 0 (997). [48] P. Stinr and W. Yang, Stability of High Ordr Sigma Joshua D. Riss was born in 97, and is a Lcturr with th Cntr for Digital Music and th Digital Signal Procssing group in th Elctronic Enginring dpartmnt at Qun Mary, Univrsity of London. H has bachlor s dgrs in both physics and mathmatics, and rcivd his Ph.D. in physics from th Gorgia Institut of Tchnology, spcializing in th analysis of chaotic tim sris. In Jun of 000, h accptd a rsarch position in th Audio Signal Procssing rsarch lab at King s Collg, London, and movd to Qun Mary in 00. H mad th transition from chaos thory to audio procssing through his work on sigma dlta modulators, which has lad to a nomination for a bst papr award from th IEEE, as wll as a UK patnt. His work also includs significant contributions to th filds of THE AUTHOR Dlta Modulators, Proc. IEEE Int. Symp. on Circuits and Systms, ISCAS 96, pp (996). [49] N. Wong and T.-S. Tung-Sang Ng, DC Stability Analysis of High- Ordr, Lowpass Sigma-Dlta Modulators With Distinct Unit Circl NTF Zros, IEEE Trans. On Circuits And Systms-II: Analog And Digital Signal Procssing, vol. 50 (003). [50] V. Mladnov, H. Hgt, and A. V. Rormund, On th Stability Analysis of High Ordr Sigma-Dlta Modulators, Analog Intgratd Circuits and Signal Procssing, vol. 36 (003). music rtrival and procssing, audio ffcts, satllit navigation, and nonlinar dynamics. Riss is vry activ in th Audio Enginring Socity, including bing a mmbr of th Rviw Board for th Journal and vic chair of th Tchnical Committ on High- Rsolution Audio. H was on th Organizing Committ of DAFx003 and MIREX005 and was rcntly program chair for th 005 Intrnational Confrnc on Music Information Rtrival and gnral chair of th 007 AES Intrnational Confrnc on High Rsolution Audio. As coordinator of th EASAIER projct, h lads an intrnational consortium of svn partnrs working to improv accss to sound archivs in musums, libraris, and cultural hritag institutions. 64 J. Audio Eng. Soc., Vol. 56, No. /, 008 January/Fbruary