ANALYSIS AND DESIGN OF ONE-BIT NOISE-SHAPING QUANTIZER USING VARIABLE STRUCTURE CONTROL APPROACH

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1 266 Aian Jounal of Contol, Vol. 7, No. 3, pp , Septembe 2005 ANALYSIS AND DESIGN OF ONE-BI NOISE-SHAPING QUANIZER USING VARIABLE SRUCURE CONROL APPROACH Jwuheng Hu and Shiang-Hwua Yu ABSRAC hi pape offe a taightfowad analyi of a one-bit noie-haping quantize baed on vaiable tuctue contol method. he method ue the concept of equivalent contol to acce the bound on the tate ignal when the liding condition i eached. Fo a table loop filte, the global tability can be poved to exit by uing the Lyanpunov tability theoy. Fo an untable loop filte, thi pape popoe a tate containt appoach to enue that the liding condition i achieved. Simulation example and an expeimental cicuit contucted uing off-the-helf electonic component ae given to demontate the coectne of the theoetical analyi. KeyWod: Noie haping, vaiable tuctue contol quantization, igma delta modulation. I. INRODUCION Signal quantization i an aea that i elated to many engineeing application. Among vaiou quantization cheme, igma-delta modulation i a well-known technique ued to alte the fequency ditibution of quantization noie via the feedback connection of a quantize with a filte. hough feedback, the quantize output tack the input ignal while quantization noie i uppeed within the fequency band of inteet. hi technique ha vaiou pactical ue in data conveion [] and powe conveion [2]. he hitoy of igma-delta modulation goe back to at leat 952, when de Jage [3] invented the delta modulato fo coding analog ignal uing negative feedback. he delta modulato i a imple feedback ytem with a one-bit quantize in the fowad path and an integato in the feedback path. A it name implie, the delta modulato geneate a binay ignal, which caie infomation about the diffeentiation of the input ignal. In 962, Inoe, Yauda and Muakami [4] modified the delta modulato by adding an integato a a pe-filte to obtain a modulato which diectly caie infomation about the input magnitude. By educing the block diagam, one can obeve that thi modulato i equivalent to one with a ingle integation in Manucipt eceived Augut 5, 2003; accepted Januay 0, he autho ae with Depatment of Electical and Contol Engineeing, National Chiao ung Univeity, 00 a Hueh Road, Hinchu 300, aiwan, R.O.C. ( jhu@cn.nctu. edu.tw). the fowad path of the feedback loop. hi modulato wa a pototype fo the vaiou igma-delta modulato of today. Becaue it ue a one-bit quantize in the feedback loop, the igma-delta modulato i eentially a nonlinea dynamic ytem. By fa the mot popula method fo analyzing a igma-delta modulato i baed on the linea ytem method, which model the quantize a a ouce of additive noie and yield an output noie pectum detemined by the magnitude epone of the loop enitivity function unde the white-noie aumption concening the quantization eo. Accodingly, the enitivity function i alo called the noie tanfe function fo igma-delta modulato. Howeve, thi i a minome, ince the whitenoie aumption i incoect [5], and the o-called noie tanfe function fail to accuately pedict the quantization noie pectum. In effect, the linea ytem method cannot explain many of the chaacteitic of the igma-delta modulato, epecially it tability. Vaiou method have been ued to deive the tability condition, including, fo example, nom analyi [6], the decibing function method [7,8] and geometical analyi [9-]. Howeve, all of thee method have deficiencie o limitation and, thu, can only achieve limited ucce. Unlike othe analyi method, the theoy of liding mode [2,3], which i a contol technique that ue a witching cheme to tabilize a ytem o to tack a ignal, eem to not have attacted much attention depite it potential uefulne in developing a igoou theoy of igma-delta modulato. he theoy of liding mode wa fit applied by Sia-Ramiez [4] to analyze

2 J.S. Hu and S.H. Yu: Analyi and Deign of One-Bit Noie-Shaping Quantize 267 the tability of the delta and igma-delta modulato. he fit-onde cae wa mainly conideed. Zounto and John [5] popoed vaiable-tuctue igma-delta modulato baed on the vaiable tuctue contol with the liding mode; they achieved tability by adapting the loop filte. Altenatively, the igma delta modulato can eentially be viewed a a elay feedback ytem (RFS) with a fat elay witch [6]. Advanced theoie and analyi fo RFS have been invetigated in ecent yea with epect to tability and limit cycle behavio [6-8]. he peviou eeach wok on vaiable tuctue contol and RFS povide a good amount of infomation to help one undetand the delta-igma modulato. Howeve, fo enginee, a clea deign guideline deived fom the theoy i needed, epecially fo high-ode loop filte. Unfotunately, little eeach (baed on vaiable tuctue o RFS) ha focued on thi iue [9,20]. he pupoe of thi wok i to intoduce a taightfowad analyi of one-bit noie-haping quantize baed on the well-known vaiable tuctue feedback contol famewok. he elative degee of the loop filte conideed i eticted to one, and all the zeo ae table. It i hown that if the loop filte i alo table, global contaction of the tate tajectoy can be enued. When the loop filte contain untable pole (e.g., on the imaginay axi), a popoed tate containt i applied to allow the tate tajectoy to each the liding uface. In pactice, all the tate have atuation limit (i.e., ae bounded by the upply voltage). Although the liding condition can till be maintained when ome tate ae atuated, the noie-haping quantization popety i detoyed becaue of non-lineaity. hi wok alo peent an appoach to etimating the tate magnitude in a pactical implementation uing a zeo-th ode holde. wo example ae given to illutate the deign method. One of them wa implemented uing off-the-helf electonic component. II. QUANIZAION AND NOISE SHAPING A geneal quantization poblem can be fomulated a Y () H () R() H () E(), n whee R() i the ignal to be quantized, E() the quantized eo and Y () the quantized eult. H () and H n () ae fequency epone of the ignal and noie haping function. Intuitively, if E() i made to be white, then electing H n () i equivalent to haping the quantization noie. Fo ingle-bit quantization, Y () i a bi-level ignal, and H () and H n () atify N2 ( ) D( ) Y () R() E(), () N() N() whee N (), N 2 (), and D() ae polynomial with deg(d) > deg(n 2 ). Equation () can be ewitten a N () N () E R Y D () D () 2 () () () G () R() G () Y (). (2) 2 By connecting E() and Y () with a ymmetic bi-level quantize, we can ealize the one-bit noie-haping quantize hown in Fig.. Fom the contol ytem pepective, the objective of the feedback implementation hown in Fig. i to make e a mall a poible by mean of a elay feedback. Let G and G 2 be tictly pope tanfe function, witten a and n n2 bn bn2 b0 () n n n2 an an2 a0 G G n n2 dn dn2 d0 2 () n n n2 an an2 a0 (3). (4) he input ignal i aumed to be bounded by ( ). Conide the minimal tate pace epeentation of the quantize hown in Fig. (Eq. (2)) a d X AX Bu B, e CX. (5) dt he ability to keep the output e at (o aound) zeo (the o-called liding condition) by uing output elay feedback depend on the ode of the liding et [2]. hi degee i defined a the mallet intege m uch that CA m B 0. It ha been hown that fo m 2, the ability to maintain e = 0 i vey limited (ee [6,7]). heefoe, fo the quantize, the election of the noie ill haping function i then eticted to deg(d()) deg(n ()) = o that 0 (ee Eq. ()). hi in tun mean that the elative degee of G () i (i.e., b n 0 in Eq. (3)). III. MAINAINING SLIDING CONDIION he behavio of the feedback ytem unde the liding condition can be analyzed uing the equivalent contol technique. In ode to maintain tate tajectoy on the liding uface e = 0, the equivalent contol of Eq. (5) mut be [2] u G 2 () G () Fig.. A geneal one-bit noie-haping quantize, whee G and G 2 epeent the noie and ignal haping function, the bounded input ignal, and y the output binay equence. e y

3 268 Aian Jounal of Contol, Vol. 7, No. 3, Septembe 2005 ueq CAX. (6) A a eult the tate tajectoy unde the liding condition i govened by the following dynamic: d BC BC X I AX I B dt. (7) he following fact can be deived fom tandad contol theoy [22]. Fact. he chaacteitic equation of the tate tanition matix of Eq. (7) i n n2 n n2 0 ( b b b ) 0. Fact 2. he tanfe function fom input to a ignal = CAX i BC BC G () CAI I A I B, (8) which i not minimal with at leat a pole-zeo cancellation at = 0. Fact i tue ince the matix (IBC/) i ingula. Fact how u that to have the liding condition, the zeo of G () mut be table. hi guaantee a bounded tate tajectoy when liding occu. Fact 2 enue that G () i table, and that it l -nom exit, that i, G ( ) ht ( ) dt, whee h(t) i the impule epone of 0 G (). he tatic witching contol feedback of Figue can be witten a u y U ign( e), U 0. (9) It i then neceay to find out the magnitude of U in ode to maintain the liding condition. Conide the enegy function Ve 2 e 2. he deivative of V e i d V e U ign() e ign() e e, dt whee = CAX. (0) heefoe, to tap e on the liding uface (e = 0), a conevative etimate of the minimum magnitude U mut be U () whee denote the maximum amplitude of. he ignal i actually the output of Eq. (7), and it magnitude when the tate ae tapped on the liding uface i etimated a G ( ), (2) whee denote the l -nom. he following lemma ummaize the analyi. Lemma. he liding condition e = 0 of Eq. (5) unde the feedback in Eq.(9) can be maintained if the following condition ae atified: () he elative degee of G () i, and all the zeo ae table. (2) ( ) U G. Note that Lemma ay nothing about the tability of the feedback ytem unde abitay initial condition. Once U i detemined and atifie Lemma, the tate tajectoy will tay inide a bounded et called the liding egion: n S X X R, U and e 0. (3) Anothe impotant iue egading pactical implementation i enuing that all the tate tay within thei atuation limit. If they do not, then even though the liding condition can till be guaanteed, the lineaity of the opeation will be lot and Eq. () and (2) will no longe hold. Checking to ee whethe o not the atuation limit have been exceeded can be done by invetigating the l -nom of the tanfe function fo each tate of the liding egion (Eq. (3)). IV. CONVERGENCE O SLIDING SURFACE It i impotant to know the et of initial condition that guaantee the liding condition. Although in pactice one can alway ue technique like eetting to foce the initial condition to tay within the liding egion (Eq. (3)), knowing the tability bounday can povide help in deigning the ytem o even in educing the amount of cicuity. If G () i table, then it i poible to aive at condition which enue liding globally. In thi pape, a pactical appoach i taken by containing the tate to enue liding condition. 4. Stable open loop filte Fo a table open loop filte, it i poible to etablih a global contaction egion of the ytem (Fig. ). Howeve, the pupoe i to make the tate tajectoy hit and tay on the liding uface. When G () atifie condition (.) in Lemma and i table (all pole ae on the open LHP), G () i alo tictly poitive eal (SPR). he Kalman-Yacubovich lemma [23] tate that P and Q exit and ae both poitive definite and ymmetic uch that A P + PA = Q and PB = C. (4) Define V 0.5X PX ; we have

4 J.S. Hu and S.H. Yu: Analyi and Deign of One-Bit Noie-Shaping Quantize 269 d V X QX X PBu X PB. dt 2 Fom Eq. (5) we have d V X QX ( ) U e X PB. (5) dt 2 If we let Q = W W and PB = W q, then Eq. (5) can be ewitten a d V ( ) WX q ( WX q ) dt 2 2 q q ( ) U e. (6) Define the following thee et: n 2 y X X R, e CX q q U, (7) n X X R, WX q 2 q, (8) n X X R, X PX, 0. (9) It can be hown that fo a finite, i finite ince W i non-ingula. We can etablih the invaiant contaction egion of the tate X in the following lemma. Lemma 2. If the tanfe function between u and e in Eq. (5) i tictly poitive eal, then i an invaiant contaction egion of the tate unde the feedback in Eq. (9), whee Min y. Poof. Fit, fo evey X y, we have (fom Eq. (6)) d V ( ) WX q ( WX q) 0. dt 2 Similaly, fo evey X, we have 0 dt d V. i defined uch that fo evey X, we have V 0 but thee exit X uch that V 0. heefoe, X will ente and tay in. he contaction egion tate nothing about enteing the liding uface. If we let e(0) > 0, then we have u(0) = U /. If no witching occu, then the teady tate epone in Fig. become t U e () t CA B g2 ( t) ( ) d. (20) 0 If the ign of both CA B and ae the ame, then the epone i le than o equal to zeo if U G 2( ). (2) CA B A a eult, witching mut occu. Simila analyi can be applied when e(0) < 0. he tajectoy will tay on the liding uface if it hit the liding egion (Eq. (3)). Note that Eq. (2) i a ufficient condition conideing a geneal bounded input. If = 0, fo example, witching will happen when the ign of both CA B and ae the ame, egadle of the magnitude of U. A imple geometical intepetation i given below to illutate the tability condition. Fit, obeve that,. o ee thi, conide the following inequalitie. Fo 0, WX q WX q ( ) q WX q q q q q, and fo 0, WX q WX q ( ) q WX q q q q q. Figue 2 depict vaiou et (Eq. (7)-(9)), howing the liding uface and the liding egion fo a 2 nd ode ytem. he illutation how that the liding uface within the invaiant et i alo the liding egion. 4.2 Containing the tate Fo a moe geneal cae whee the open loop filte could be untable, a tate containt i popoed in thi pape to enue the liding condition. Fom Eq. (0) and (), the enegy function V e can be made to deceae if the magnitude of CAX i bounded. Aume that the bound i S (liding egion) x q x 2 x q e=cx=0 (liding uface) U Fig. 2. Geometic intepetation of Lemma 2. y x U

5 270 Aian Jounal of Contol, Vol. 7, No. 3, Septembe 2005 poible and i equal to. he tate tajectoy will eventually hit and tay on the liding uface if U. (22) In addition, Fact and 2 ae applied if doe not each the bound when on the liding uface. heefoe, uing the concept of equivalent contol (Eq. (6)-(8)), we have G ( ). (23) One poibility of impoing the limit on i to make a phyical ignal in the implementation. Fo example, conide the canonical fom of Eq. (5): dx X dt a0 a an2 an bn dn bn2 dn2 u, (24) b d b0 d0 e[ 0 0] X x. he ignal i actually the tate x 2. V. PRACICAL DESIGN ISSUES In eality, howeve, it i impoible fo a witch to act infinitely quickly a aumed in peviou ection. One of the application of a igma-delta modulato i analog-todigital conveion, whee a ample-and-hold function i added (ee Fig. 3). Now the witching action of Eq. (9) become u() y U ign((( e k) )), ( k) k, U 0. (25) u G() e ZOH y Decimation Fig. 3. An A/D conveion uing igma-delta modulation whee i the ampling peiod. Let e(k) = e(k); fom Eq. (5), K ek ( ) ek ( ) CX ( ) d ( k) K ( CAX ) duign e( k ) ( k) CAX( k) Uign( e( k)), (26) whee i of ode, i.e., = O(). heefoe, ek ( ) ek ( ) ek ( ) ign e( k ) CAX( k ) U e( k) (27) Fo a ufficiently mall, it i poible to have a U value that i lage than the optimal etimate obtained with Eq. (3), which make Eq. (27) le than zeo when e(k ) 0. A a eult, we have ek ( ) ek ( ) when e(k ) 0. In othe wod, e(ki tapped in a bounded egion (o-called chatteing). Refeing to Eq. (26), the wot-cae incement of e(ti et ( ) ek ( ) Max ( ) U ( k) k 2 U, ( k ) t k. Since e(k) i deceaing, the bound of e(t) can be etimated a et ( ) 2U. (28) Equation (28) not only give an etimate of the chatteing amplitude, but alo eveal that the amplitude i popotional to the amplitude of the witch. In othe wod, a good choice of U i vey impotant fo minimizing the extent of the chatteing that affect the S/N atio of the conveion. VI. SIMULAION AND EXPERIMEN Example. A 4 th ode table loop filte i conideed (G () in Eq. (2)): N() G () D () ( 6.28) (29)

6 J.S. Hu and S.H. Yu: Analyi and Deign of One-Bit Noie-Shaping Quantize 27 N () i the denominato of a 3 d ode Buttewoth lowpa filte with a cutoff fequency at 20 KHz. he input ignal i aumed to be bounded by ( ), and the input channel i the ame a the feedback (i.e., N () = N 2 () in Eq. () o B = B in Eq. (5)). It can be veified that when B = B, the tanfe function G () = 0 (Eq. (5)). Futhemoe, we have =, CA B = and G 2 () = G () = heefoe, the lowe bound of feedback paamete U ued to maintain the liding condition i (Eq. ()) U. (30) he ame lowe bound can alo be computed to enue witching (Eq. (2)). A imulation wa conducted uing digital implementation (Fig. 3) with a ampling fequency of 000 KHz (50 time the bandwidth of 20 KHz). he gain U wa et at 2. With Eq. (28), the eo e(t) wa bounded a 6 et ( ) 2U 4 0. hi value wa then ued a the etimate of the quantization eo bound (Eq. ()). Figue 4 how the pectum of the imulation eult when the input ignal i a unit magnitude inuoidal wave of KHz. Note that the quantization eo i puhed into a high fequency ange, and that the theoetical pediction i quite accuate. Example 2. In thi example we conide implementing a 4 th ode loop filte a N() G () D () ( ) 4 (33) and N2 ( ).00 G2 (). (34) 4 D () Figue 5 how the chematic cicuit of the igmadelta modulato. he compaato bandwidth limitation i equivalent to a MHz ampling fequency a depicted in Fig. 3. he cicuit implementation hown in Fig. 5 give the following tate-pace epeentation, whee the tate ae the voltage of the opeational amplifie: dx X dt u, e[ 0 0 0] X x. he coeponding paamete ae computed a (35) = 98530, = 0, and G () = Fom Fig. 5, the maximum amplitude of the input ignal i eticted to the cicuit upply voltage, VDD = 5 Volt (i.e., 5 ). Conideing again a bounded input efeence ignal, fom Eq. () and (2), we have 6 U If thi lowe bound i ued, the feedback contol law i (Eq. (9)) u = 3.09 ign(e). (36) Fig. 4. Spectum of the binay equence fo the KHz inuoidal efeence ignal (thin line) in Example and a theoetical pediction of the quantization eo pectum (the thick line, epeent G ( j) in Eq. (27)). Fig. 5. Cicuity of the 4 th -ode igma-delta modulato in Example 2 (CMP: compaato; VDD: 5 Volt).

7 272 Aian Jounal of Contol, Vol. 7, No. 3, Septembe 2005 Secondly, ince the loop filte i not table, it i neceay to contain the vaiable = CAX (ee Section 4.2). Fom Eq. (23) and (35), we have = x 2 VDD x (37) he bound of x 3 and x 4 on the liding uface can alo be computed a x and x (38) he eult of Eq. (36) to (38) how that the tate ignal do not exceed the upply voltage limit. Figue 5 how a implified implementation in which the witching contol gain (Eq. (36)) and the tate containt (Eq. (37)) ae u = 5.0 ign(e) and x 2 5.0, epectively. (39) A a eult, the eo e (o x ) i bounded by (Eq. (28)): Fig. 6. State ignal (teady tate) of the imulation in Example 2, whee the input i a KHz inuoidal ignal with a 5-volt amplitude and the initial condition i X [ ] ; (a) x ; (b) x 2 ; (c) x 3 ; (d) x 4. et ( ) x( t) 2U (40) A imulation that aumed all the component in Fig. 5 ae ideal and that ued the contol law (Eq. (39)) and a MHz ampling fequency wa conducted and the time equence of each tate i plotted in Fig. 6. In the imulation, a non-zeo initial condition of X = [ ] wa applied (not on the liding uface). It i hown that the ytem i table, and that all the tate ignal tay within the atuation limit of 5 volt. Figue 7 how the ignal of x 2 at the initial tage. Notice that it hit the atuation limit initially but become table aftewad. he cicuit hown in Fig. 5 wa contucted uing off-the-helf electonic component (i.e., an opeational amplifie). Figue 8 how the tate ignal obtained in the expeiment. It veifie that the ytem i table, a pedicted, and that all it tate tay within the atuation limit. Howeve, notice that the ignal ae moothe than in the imulation (Fig. 6) becaue the electonic component wee not ideal and could not act infinitely quickly. Fig. 7. he ignal of x2 in Example 2 at the initial tage (hitting the atuation limit (5V) but becoming table aftewad). VII. CONCLUSION An analyi of one-bit noie-haping quantize uing vaiable tuctue contol method ha been peented. he method ue the concept of equivalent contol to obtain the bound on the tate ignal when the liding condition i eached. Fo a table loop filte, global tability can be poved uing the Lyanpunov tability theoy. Fo an untable loop filte, thi pape ha popoed a tate containt appoach to enuing that the liding condition exit. Simulation example and an expeimental cicuit uing off-the-helf electonic component have been ued to demontate the theoetical analyi. Fig. 8. State ignal of the expeimental cicuit (C to C4 epeent tate x to x 4 ; PK-PK mean peak-to-peak).

8 J.S. Hu and S.H. Yu: Analyi and Deign of One-Bit Noie-Shaping Quantize 273 ACKNOWLEDGEMENS hi wok wa uppoted by the National Science Council of aiwan unde gant numbe NSC E and by the Minity of Education, aiwan, unde gant numbe 9--FA REFERENCES. Adam, R.W., P.F. Feguon, A. Ganean, S. Vincelette, A. Volpe, and R. Libet, heoy and Pactical Implementation of a Fifth-Ode Sigma-Delta A/D Convete, J. Audio Eng. Soc., Vol. 39, pp (99). 2. Silva, J.F., PWM Audio Powe Amplifie: Sigma Delta Veu Sliding Mode Contol, Poc. IEEE Int. Conf. Electon. Cicuit Syt., Liboa, Potugal, Vol., pp (998). 3. de Jage, F., Delta Modulation A Method of PCM anmiion Uing the One Unit Code, Philip Re. Rept., Vol. 7, pp (952). 4. Inoe, H., Y. Yauda, and J. Muakami, A elemeteing Sytem by Code Modulation - Modulation, IRE an. Space Electon. elemety, Vol. SE-8, pp (962). 5. Gay, R.M., Quantization Noie in A/D Convete, Delta-Sigma Data Convete: heoy, Deign and Simulation, S.R. Nowothy, R. Scheie and G.C. eme, Ed., IEEE Pe (997). 6. Scheie, R. and Y. Yang, Stability et fo Single- Bit Sigma-delta Modulato with Second-Ode FIR Noie anfe Function, Poc. IEEE Int. Symp. Cicuit Syt., San Diego, CA, U.S.A., pp (992). 7. Baid, R.. and.s. Fiez, Stability Analyi of High-Ode Delta-Sigma Modulation fo ADC, IEEE an. Cicuit Syt.-II, Vol. 4, No., pp (994). 8. van Engelen, J. and R. van de Plache, New Stability Citeia fo the Deign of Lowpa Sigma-Delta Modulato, Poc. Int. Symp. Low Powe Electon. Deign, Monteey, CA, U.S.A., pp. 4-8 (997). 9. Wang, H., A Geometic View of Modulation, IEEE an. Cicuit Syt.-II, Vol. 39, No. 2, pp (992). 0. Hein, S. and A. Zakho, On the Stability of Sigma Delta Modulato, IEEE an. Signal Poce, Vol. 4, No. 7, pp (993).. Faell, R. and O. Feely, Bounding the Integato Output of Second-Ode Sigma-Delta Modulato, IEEE an. Cicuit Syt.-II, Vol. 45, No. 6, pp (998). 2. Utkin, V.I., Sliding Mode in Contol Optimization, Spinge-Velag, Belin, Gemany (992). 3. Hung, J.Y., W. Gao, and J.C. Hung, Vaiable Stuctue Contol: A Suvey, IEEE an. Ind. Electon., Vol. 40, No., pp (993). 4. Sia-Ramiez, H., Analog Signal Encoding in Delta Modulation Cicuit Uing the heoy of Vaiable Stuctue Sytem, Poc. 27 th IEEE Conf. Deci. Cont., Autin, X, U.S.A., Vol. 2, pp (988). 5. Zounto,. and D.A. John, Vaiable-Stuctue Compenation of Delta-Sigma Modulato: Stability and Pefomance, IEEE an. Cicuit Syt.-I, Vol. 49, No., pp. 4-5 (2002). 6. Johanon, K.H., A.E. Baabanov, and K.J. Åtöm, Limit Cycle with Chatteing in Relay Feedback Sytem, IEEE an. Automat. Cont., Vol. 47, No. 9 (2002). 7. Johanon, K.H., Relay Feedback and Multivaiable Contol, Doctoal Dietation, Depatment of Automatic Contol, Lund Intitute of echnology, Sweden (997). 8. Gonçalve, J.M., A. Megetki, and M.A. Dahleh, Global Stability of Relay Feedback Sytem, IEEE an. Automat. Cont., Vol. 46, No. 4 (200). 9. Plekhanov, S., I.A. Shkolnikov, and Y.B. Shteel, High Ode Sigma-Delta Modulato Deign via Sliding Mode Contol, Poc. Ame. Cont. Conf., Denve, Coloado, U.S.A. (2003). 20. Yu, S.-H. and J.-S. Hu, Analyi and Deign of a Single-Bit Noie-Shaping Quantize, Poc. Ame. Cont. Conf., Denve, Coloado, U.S.A. (2003). 2. Fidman, L.M. and A. Levant, Highe Ode Sliding Mode a a Natual Phenomenon in Contol heoy, Robut Contol via Vaiable Stuctue & Lyanpunov echnique, F. Gaofalo and L. Glielmo, Ed., Spinge-Velag, New Yok, Vol. 27, Lectue Note Cont. Infom. Sci., pp (996). 22. Kailath,., Linea Sytem, Pentice Hall, Englewood Cliff, NJ, U.S.A. (980). 23. Vidyaagaa, M., Nonlinea Sytem Analyi, Pentice-Hall, Inc., Englewood Cliff, NJ, U.S.A. (978).

9 274 Aian Jounal of Contol, Vol. 7, No. 3, Septembe 2005 Jwuheng Hu wa bon in aipei, aiwan in 962. He eceived the B.S. degee fom the Depatment of Mechanical Engineeing, National aiwan Univeity, aiwan, in 984, and the M.S. and Ph.D. degee fom the Depatment of Mechanical Engineeing, Univeity of Califonia at Bekeley, in 988 and 990, epectively. He i cuently a Pofeo in the Depatment of Electical and Contol Engineeing, National Chiao-ung Univeity, aiwan, R.O.C. Hi cuent eeach inteet include micophone aay ignal poceing, active noie contol, embedded ytem deign, and obotic. Shiang-Hwua Yu eceived the M.S. and Ph.D. degee in electical and contol engineeing fom National Chiao-ung Univeity, in 995 and 200, epectively. In 2004, he joined the faculty of National Sun Yat-Sen Univeity, whee he i cuently an aitant pofeo in the depatment of electical engineeing. Hi eeach inteet ae in the aea of elay feedback contol, igma-delta modulation and mixed-ignal integated-cicuit deign.