On reduced-order filter design for uncertain cascaded 2-1 sigma-delta modulators

Transcription

1 On reduced-order filter design for uncertain cascaded - siga-delta odulators Chun-Chen Lin Yung-Shan Chou Dept. of Electrical Engineering akang University aipei aiwan R.O.C. Venkataraanan Balakrishnan School of Electrical and Coputer Engineering Purdue University West Lafayette IN 4797 USA Abstract In this paper we present a new robust atching filter design ethod for uncertain - cascaded siga-delta odulators. his ethod addresses a well known liitation of - infinity loop shaping techniques that they yield filters of high order (equal to the su of the plant order and the order of the weighting function) thus increasing the coplexity of circuit ipleentation. In contrast the new ethod yields filters whose order is equal to the plant order independent of the weighting function. We copare the new ethod with other existing fixedorder designs and establish its efficacy. Keywords uncertainty cascaded siga-delta odulator reduced-order filter -infinity loop shaping linear atrix inequality. I. INRODUCION Siga-delta (ΣΔ) odulators [] are iportant devices which have found widespread application in high-speed analog-to-digital (A/D) conversion for odern digital signal processing. Cascaded siga-delta odulators are preferred over single loop odulators as they offer greater stability. In order to have a higher signal-to-noise ratio (SNR) cascaded siga-delta odulators rely heavily on accurate atching of analog stages with a digital filter to prevent leakage of quantiation noise. owever perfect atching is ipossible in practice due to the liited accuracy of the ipleentation technologies as well as paraeter variations. echniques that do not explicitly account for this often yield designs that perfor poorly []. hus in recent years considerable effort has been devoted to the study of robust atching filters under the fraework of odel atching [ 3 4]. he basic idea is to recast the filter design proble as a proble of iniiing the worst-case value over all possible uncertainties of a easure of a certain odel isatch (we will present details in Section II). he specific isatch easure that has been ost often used is the nor which easures the peak value of the isatch over all frequencies. o achieve higher-order noise shaping a weighting function was introduced in []; the uncertainties were of the polytopic type and the proble becae that of the iniiation of the nor of the weighted atching error over a linearied polytopic odel. While the introduction of weighting functions is useful in shaping the noise transfer function (NF) so as to increase the SNR it also increases the order of the filter which in turn leads to increased coplexity of circuit ipleentation. o alleviate this proble two fixed-order designs have been proposed [3 4]. In [3] the central polynoial linear atrix inequality (LMI) ethod has been eployed. In particular the order of the resulting infinite ipulse response (IIR) filter is independent of that of the introduced weighting function. Moreover the filter order can be chosen to be any positive integer. In contrast to the atheatical approach in [ 3] a design based ore on engineering insight was presented in [4]. A low-frequency linearied odel of a - cascaded odulator was derived and again a fixed-order (but finite ipulse response (FIR)) filter design was presented based on a foral optiiation ethod. In this paper we revisit the weighted nor iniiation forulation in [] and directly address the issue of high filter order. Our ain contribution is a new reducedorder filter design procedure that yields filters whose order is equal to the plant order independent of the weighting function. We show that this approach yields filters whose perforance copares favorably with those presented in [3 4]. he rest of this work is organied as follows. In Section II an uncertain cascaded - ΣΔ odulator is briefly described and the proble forulation is presented. In Section III the proposed reduced-order IIR filter design is provided. Section IV shows the siulation results with coparison to soe of the existing fixed-order works. Section V is the conclusion. U ( ) ( ) E 3 ( ) Figure. A cascaded - odulator with -bit quantiers E E Y Y y F F Y ICSSE

2 II. CASCADED - SIGMA-DELA MODULAOR he linear odel for a cascaded - ΣΔ odulator is presented in Fig. E and E are the quantiation noises of the first stage and the second stage respectively []. he first stage quantiation noise E is extracted and re-quantied at the second-stage. Accordingly the NF fro E to output Y is NF ( ) F ( ) SF ( ) () NF SF 3 he noise effect fro E is less significant and hence can be neglected [ 3 4]. Ideally the E ter can be copletely eliinated fro the odulator output Y by a atching filter NF ( ) F( ) SF ( ) (3) owever perfect cancellation of E is not possible in practice owing to non-ideal analog coponents. Recall that two coon sources of analog iperfections in a ΣΔ odulator are finite aplifier gain and isatch in capacitor values. hese factors can be odeled as paraetric uncertainties in the gains and poles of the integrators. herefore a non-ideal integrator can be odeled as [ 3 4] δ a i ( ) i 3 (4) ( δb ) δ a [) and δ b [) are the paraeter deviations fro the noinal values. ere we assue the integrators and 3 NF can be described by are ideal and thus the uncertain ( δb) + ( δb) NF ( ) (5) ( δb + δa) and SF. Our work is to iniie the effect of quantiation noise E on the output Y in the signal band. his can be forulated as a weighted nor iniiation proble in W( ) ( NF ( ) F( ) ) (6) F( ) W( ) is a weighting function that is eployed to shape the NF for E. For later use we assue that the transfer functions W( ) F ( ) and NF ( ) have the following state-space realiations: : W W W W () W C I A B + D (7) F C I A B + D (8) : F F F F NF : C I A B + D (9) δb + δa + δ a A B C ( ) D δb () In order to take the uncertainties δ a and δ b into account in proble (6) we assue the uncertain atrices ( A B C D ) of NF belong to the following uncertainty polytope: Ω ( A B C D) ( A B C D) αi i ( A B C D ) αi αi i () ere giving the values of α i i with αi and α+ α + + α produces an eleent of Ω. hen it is readily verified the state-space realiation of the weighted atching error W( ) ( NF F ) denoted by ( ): C I A B + D is given by AM BM CW BM D W A B ( ) : αi αi AW BW i C D i CM DMCW DM D W () A B M BM AF BF M C D C F M D D F A C (3) nf nw np np and AF R AW R A R B R np C R D R. his recasts the design proble (6) to that of finding a filter of for (8) via the solution of the following optiiation proble: in γ (4) A B C D F F F F subject to C I A B + D < γ. Referring to (6) it is known that the conventional -infinity loop-shaping technique derives a full-order filter nf np + nw. In the following section we present a reduced-order filter design ethod and the filter order nf is reduced to be np. III. MAIN RESUL In this section we present the ain result. he following lea is useful in the developent. Lea []. For all ( A B C D) belonging to Ω the condition C I A B + D < γ holds if there exist a atrix G and atrices P P ( i ) satisfying G+ G P GA GB I C D. > i (5) A G C P () i B G D γ I We now present heore that states that there exists a robust atching filter with order equal to that of the first stage of the ICSSE

3 cascaded odulator if certain atrix inequality constraints (6) are satisfied. heore : Assue nw np. For all vertices () () () ( i i i ) A B C D ( i ) there exists a suboptial filter (8) of order nf np to proble (4) if optiiation proble (6) is feasible for i. In the case the filter is given by C A B D F : Q I M Q M Q + Q (7) Proof: We will invoke Lea to derive the solvability condition for the filter with order nf np. o proceed partition the atrices G and P as follows: P P P 3 G G G3 P P P P G G G G P3 P P 33 G3 G3 G 33 (8) all the subatrices P have the diension np np. Without loss of generality is assued to be nonsingular. Define λ α Inf α is a scalar paraeter. Under the constraint G 3 λg apply congruence transforation J diagii ( ) to (5) (i.e. ultiplying (5) on the left by J and on the right by J ) diag I G I and define P P G G G G P G G G G P (9) g g g we obtain () i.e. G + G P I P Pg P3 > () Pg Pg P33 γ I G + G Pg 3 G3 + G3 P3 5 GA 6 G AF G G G B C + G B C + G A () G B C + λg B C + G A 7 W F W 3 W 8 GB DW G BFDW G3BW G G Pg i G λ G Pg 5 GA 6 G AF G G 7 GB CW GBC F W GAW 8 GB DW GBD F W GB W 33 G33 G33 P33 35 G3A 36 λgaf G G 37 3 W F W 33 W 38 G3B DW + λgbfdw + G33BW 45 C 46 CG F G 47 D CW DFCW 48 D DW DFDW Now define new variables as follows: M G N G S G () QA G AFG G QB G BF QC CFG G QD DF We obtain the atrix constraints in proble (6). Furtherore if (6) is feasible it iplies the positive definiteness of the () block of (6) i.e. M + M > () It follows that G + G > () subject to in λ ( P ) g g ( Pg ) 3 g 33 ( P33 ) 3333 M N S QA QB QC QD i ( i) γ (6) G + G P M + S P G + G P G A Q G B C + Q C + G A G B D + Q D + G B > g A W B W 3 W W B W 3 W M + M Pg N + Mλ Pg SA QA SB CW QBCW NAW SB DW QBDW NBW G33 G33 P33 G3A λqa G3B CW + λqbcw + G33 AW G3B DW + λqbdw + G33BW I C QC D CW QDCW D DW QDDW P Pg P3 Pg Pg P33 γ I ICSSE

4 ence both M and G are invertible. It follows fro (8) and () that the digital filter is given by F( F) F + F QC( M QA) QB + QD ( ) + F C I A B D C A B D QG G I G QG G G Q + Q Q I M Q M Q Q C A B D (4) With the defined change of variables we obtain the synthesis condition given in (6) and the filter recovery procedure shown in (4). In order to confir the correctness of the results we shall further verify that the atrices P and G in (8) can be recovered fro any solution of proble (6). Specifically we need to show that the atrices P G 3 can be recovered since the atrices P 3 33 G 3 33 were obtained as part of the solution. For this purpose we recall that M G G G N G G G S G G G (5) M N S can be deterined when (6) is feasible. With the solution and let be a given nonsingular atrix X we obtain G and G via the following forulas: G M X X N G X S. hen it is easily found that G X G. With a prior deterined paraeter λ we iediately obtain G 3 λg. Next we can obtain P by reversing (9) i.e. and P P X g P X P X g P X P g. his copletes the proof. Reark. he proposed design has the following advantages. First heore provides a new solvability condition for deriving robust atching filters for uncertain cascaded odulators. In particular the order of the proposed filters is independent of that of the weighting functions. his overcoes the well-known liitation of the state-space -infinity loop shaping ethod the resulting filters are of the sae order as the plant plus weighting functions. Second when λ in (6) is specified proble (6) reduces to a linear objective iniiation proble over LMI constraints which can be efficiently solved by existing software [8]. Reark. While the weighting functions play an iportant role in arriving at a design with good SNR it is difficult to deterine their order a priori as this will be applicationdependent. heore was derived under the assuption that nf nw np ( np is the order of the first stage of the odulator) and the iposed constraint G3 λ G ( α I nf ) G α R. In the exceptional case that a candidate weighting function W has order n less than np a new weighting function can be fored by ultiplying W with the delay ( np n ) eleents to fulfill the requireent nw np without altering its agnitude response; see e.g. Section IV. On the contrary if W has order greater than np then the proposed ethod ay be extended by replacing λ with a n npatrix. For siplicity λ can be chosen to be λ α I np np ( n np ). IV. SIMULAION Nonlinear siulations are carried out and validated with MALAB/SIMULINK [8][9] for a cascaded - ΣΔ odulator with -bit quantier. Specifically the odulator of this experient is aied at applying to an audio syste. he experiental paraeters are set up as follows. he signal bandwidth (BW) is 5K. A 8K sinusoidal wave is used to perfor a standard test. he oversapling ratio (OSR) is chosen to be 64. he sapling frequency f s is 3.M and the nuber of tie points used for FF is he following paragraphs consist of two parts. In Part A we consider three weighting functions each of which has order less than or equal to or greater than the plant order ( np ). We will nuerically verify that the resulting (reduced-order) filters obtained by applying heore and Reark have order the sae as that of the plant independent of that of the introduced weighting functions. In Part B we copare the best filter obtained in Part A with soe of the existing fixedorder filter designs. A. Filter Design with Weights of Different Order Our work is to iniie the effect of the leaky quantiation noise E on the output Y in the signal band. o achieve the goal it s iportant to design the digital filter such that the agnitudes of NF is relatively sall in the frequency range [5K] i.e. we want the NF be a high-pass one. As far as the noise effect beyond the signal band is concerned it can be reduced by a subsequent deciation filter []. In discretetie doain the cut-off frequency of the desired NF can be coputed by the following forula [7 pp. 54]: 3 BW 5 π π.4965 (rad/s). (6) 6 fs 3. Accordingly three low-pass weighting functions (7) (8) and (9) are considered. W (7).9 W (8) W3 (9) o apply heore an augented W ( ) is given by W a.9 (3) ICSSE

5 ABLE I. GIVEN DAA & OUCOME Case Paraeter A B C 3.6 λ.6.6 λ.6.35 λ.35 Weighting function Filter Perforance SNR (db) Eq. Order index (3) + F ( ) w (8) F ( ) w (9) 3 F ( ).455. γ γ γ as alluded to in Reark. As shown in Fig. the Bode plots for these weights (7) (8) (9) and (3) are low-pass and have cut-off frequency around.49 (rad/s). Afterward we suppose that the uncertainties in the gain and pole of the integrator ( ) are within the ranges δ a. and.[3 4]. By apping the uncertain paraeters δ b δ a and b δ to ( A B C D ) of () these uncertain atrices can be described by a four-vertex polytope i.e. 4. By using heore with 4 and the weighting functions (8)-(3) three filters of the sae order as that of the plant were derived. able shows the results and the searched paraeters λ i ( i 3 ) which were coputed by the function finsearch of MALAB. As shown in Fig. 3 the design with W 3 yields filter F which perfors better than that with W and W in ters of lower agnitudes for the NF at lowfrequencies especially for frequency interval [..49] (rad/s). his iplies the lowest noise power in the signal band by using filter F. Specifically using filter F results in the best SNR value 9.46 db when a dbfs input signal is given. Fro the above siulation results we conclude that the proposed ethod can be used to design filters whose order is equal to the plant order even if the order of the eployed weighting functions is different. B. Coparison with Existing Methods We copare the perforance of the proposed filter F with the following filters: Method A [5]: Ft ; Method B []: F ( ) + ; n Method C [3]: Fc ; Method D [4]: F ( ) o It should be noted that ethod A derived a full-order -infinity filter F t for proble (4) without using weighting functions. he Bode plots of the NF atched by all filters are provided in Fig. 4. As expected F which eployed a weighting function in the design produces lower agnitude than F t does at low-frequencies. Siilarly F outperfors the other filters in ters of the lower agnitudes for the NF in a large portion of the frequency interval [.49] (rad/s). With a dbfs input signal the resulting SNR values for the odulators atched with filters F t F n F c F o and F are and 9.46 db respectively. In Fig. 5 it can be seen that the proposed filter F achieves the best SNR perforance for different input levels. V. CONCLUSION In this work we have studied the synthesis proble of robust atching filters for uncertain - cascaded siga-delta odulators. A new design ethod which involves iniiing the worst case nor of a certain weighted atching error over linearied polytopic odel has been presented. In particular the ethod overcoes a liitation of the well known -infinity loop shaping techniques in ters of filter order i.e. the filter derived by the proposed ethod has order independent of the weighting function. his reduces the coplexity of circuit ipleentation. he siulation results deonstrate the effectiveness of the proposed ethod. Finally the proposed ethod is general which is applicable to the other cases. REFERENCES [] S.R. Norsworthy R. Schreirer and G.C. ees Delta-Siga Data Converters: heory Design and Siulation IEEE Press 997. [] F.W. Yang and M. Gani An approach for robust calibration of cascaded siga-delta odulators IEEE rans. Circuits Systes I vol. 55 no. pp Mar. 8. [3] J. McKernan M. Gani D. enrion and F.W. Yang Robust filter design for uncertain - siga-delta odulators via the central polynoial ethod IEEE Signal Processing Lett. vol. 5 pp [4] J. McKernan M. Gani D. enrion and F.W. Yang Optial lowfrequency filter design for uncertain - siga-delta odulators IEEE Signal Process. Lett. vol. 6 pp ICSSE

6 [5].J. Gao J. La L.. Xie and C.. Wang New approach to ixed / filtering for polytopic discrete-tie systes IEEE rans. Signal Process. vol. 53 no. 8 pp [6] K. Zhou and J. C. Doyle Essentials of Robust Controlrentice all 998. (Chapter 6) [7] D. Johns and K. Martin Analog Integrated Circuit Design Wiley 997. [8] P. Gahinet A. Neirovski A. J. Laub and M. Chilali Manual of LMI Control oolbox Math Works Inc 995. [9] R. Schreier. he Delta-Siga oolbox Version 7. Available: Magnitude (db) W W a Magnitude (db) F t F n F c - W - F o - W 3 - F Noralied Frequency f Figure. Bode plots of weighting functions. plots of weighting functions. plots of weighting functions Noralied Frequency f Figure 4. Bode plots of the uncertain NF atched by all filters; paraeter deviations δ aδ b. in i (i3) Magnitude (db) F w SNR (db) F t F n F c - F w - F Noralied Frequency f F o F Input Level (dbfs) Figure 3. Bode plots of the uncertain NF atched by filters; paraeter deviations δ aδ b. in i (i3). Figure 5. SNR perforance vs. input aplitude; paraeter deviations. ; paraeter deviations δ aδ b. in i (i3). ICSSE