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DYNAMIC RANGE, NOISE AND LINEARITY OPTIMIZATION OF CONTINUOUS-TIME OTA-C FILTERS S. Koziel1, A. Ramachandran2, S. Szczepanski1, E. Sánchez-Sinencio2 1

Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, 80-952 Gdansk, Poland, e-mail: [email protected]. 2 Department of Electrical Engineering Texas A&M University, College Station, TX 77843, USA following system of differential equations (xi denote internal node voltages, ui and uo are input and output voltage, respectively): ª ¦n G1l xl (t ) º ª G u (t )  C u (t ) º b1 i « l1 » « b1 i (1) » »« Tx t « # # » « n » «¬¦l 1 Gnl  xl (t ) »¼ «¬Gbn u i (t )  Cbn u i (t )»¼ n (2) u o (t ) Go1 ¦ Gcl  xl (t )  Gd ui (t )

ABSTRACT A general framework for performance optimization of continuous-time OTA-C filters is presented. Efficient procedures for evaluating nonlinear distortion and noise valid for any filter of arbitrary order are developed based on matrix description of a general OTA-C filter model. A systematic optimization procedure using equivalence transformations is discussed. An application example of the proposed approach to optimal block sequencing and gain distribution of 8th order cascade Butterworth filter is given. Theoretical results are verified using transistor-level simulation with CADENCE..



where matrix T is defined as ªCb1  ¦n C1 j j 1 « «  C12 T « # « «  C1n ¬

1. INTRODUCTION There has been a growing interest displayed in the design of continuous-time filters based on the transconductancecapacitor (OTA-C) technique for more than two decades [1], [2]. The operational transconductance amplifiers offer a higher bandwidth than their voltage-mode counterparts, can be easily tuned electronically, and have a better suitability for operating in reduced supply environment [3], [4]. Although OTA-C filters offer excellent high frequency performance, their other properties in terms of low power consumption, low parasitics effects, noise, linearity and dynamic range, etc., still need improvements [5], [6]. In this paper, a framework for performance optimization of OTA-C filters is described. The presented approach uses an algebraic description of a general OTA-C filter model that leads to both differential system describing time dynamics of the filter and compact formulas for noise evaluation. This allows us to perform fast transient and noise analysis of any OTA-C filter and use it in a filter optimization system.

º » » "  C2n » % # » n " Cbn  ¦ j 1 C nj » ¼ "

 C12 n

Cb 2  ¦ j 1 C 2 j #  C2n

 C1n

(3)

and Gkl, Gbk, Gcl, Gd and Go, k,l=1,2,...,n are in general nonlinear functions of their input variables. We also used the vector notation for node voltages: T T (4) x (t ) >x1 t " xn t @ , x (t ) >x1 t " x n t @ If all filter transconductors are linear, i.e. Gkl(y)=gkly, Gbk(y)=gbky, Gcl(y)=gcly, Gd(y)=gdy and Go(y)=goy, k,l=1,2,...,n, we can define the following matrices:



C1n xn

Cb1

 C12 x2

xn-1

Cn-1,n

Cbn

C11

Cnn

 Gc1



Gbn x1

-Go

Gcn

xn

G11 G21

G1n x2



Gn1

xn



x1



Consider a general model of OTA-C filter shown in Fig.1, which includes all possible OTA-C filter structures as particular cases [7]. The structure in Fig.1 contains n internal nodes denoted as xi, i=1,...,n, an active network consisting of n input transconductors Gbi, the set of internal feedback and feedforward transconductors Gij, an output summer consisting of transconductors Gci and Go as well as a feedforward transconductor Gd, and passive network with capacitors Cbi, and capacitors Cij. The filter structure in Fig.1 can be described by the



Gb1



in

2. NONLINEAR ANALYSIS OF OTA-C FILTERS

Gn-1,n

xn-1

Gn,n

Gd

This work was supported in part by the State Scientific Research Committee, Poland, under Grant 4T11B01625

0-7803-8715-5/04/$20.00 ©2004 IEEE.



l 1

Figure 1. General structure of OTA-C filter

41

out

G

ª g11 " g1n º « # % # » « » «¬ g n1 " g nn »¼

d dº ª " gbn  Cbn » B « gb1  Cb1 dt dt ¼ ¬ C

>c1

"

cn @,

D

(cf. Fig.1) then one has also take into consideration the noise of transconductors Gci, Gd and Go. The current-to-voltage transfer function from output node to itself is H0=g0-1. Let us define the matrices n n n S w >S w .ij @ , S f >S f .ij @ , S wb >S wb .i @i 1 ,

T

(5)

d

with ci=-gci/go, i=1,2,...,n and d=-gd/go. This is a special case which allows us to rewrite (1), (2) as follows: (6) Tx (t ) Gx (t )  Bu i (t ), u o (t ) Cx (t )  Du i (t ) or, in the domain of Laplace transform: Uo(s)=CX+DUi(s) (7) sTX=GX+BUi(s), in which X is the Laplace transform of the vector x. System (7) is the state variable matrix description of the general OTA-C filter model introduced in [7]. Turning back to the general case note that if the matrix T is invertible, equation (1) can be reformulated as x t

§ ª n G  x (t ) º G u (t )  C u (t ) · º¸ ¨ « ¦l 1 1l l b1 i » ª« b1 i »¸ 1 ¨ »« T « # # »¸ ¨« » « n Gbn u i (t )  C bn u i (t )»¼ ¸ ¨« ¬  G x t ( ) » ¦ nl l ¼ ©¬ l 1 ¹

i, j 1

S fb

>S @

S wd

S wd ,

n

fb .i i

i, j 1

, S wc 1

>S wc .i @in 1 ,

S fd ,

S fd

S wo

S fc

S wo ,

>S @

(12)

n fc .i i 1

S fo

S fo

representing the white noise (subscript w) and 1/f noise (subscript f) of transconductors gij, gbi, gci, gd and go, respectively. Let G [| g ij |]in, j 1 , B [| g bi |]in 1 , C [| g ci |]in 1 , D | g d | , O | g o | . Denote by D the Hadamard product of two matrices, i.e. for A [aij ]in, j 1 and B [bij ]in, j 1 we have A D B [aij bij ]in, j 1 (the same definition holds, with obvious

(8)

changes for nu1, 1un and 1u1 matrices). The output noise 2 spectrum u no  jZ of the filter can be then calculated as

>

@

T 2  jZ | H cv  jZ | 2 PG , S w , S f Z ˜ Iˆ  PB , S wb , S fb Z  u no

The above assumption is very natural. In particular, it is satisfied if every internal node of the filter has a grounded capacitor. The problem of invertibility of matrix T has been thoroughly addressed in [7]. Denote the vector on the right-hand side of (8) by f(u(t),x(t)). Then we have (9) x t f u t , x t , Now we can endow (9) with a proper initial condition (e.g. x(0)=0) and solve it numerically to get the transient response of the filter in Fig.1. Having the transient response we can easily perform the Fourier transform in order to evaluate THD or other nonlinearity measure. It should be noted that unlike the approaches based on Volterra series representation [8] or harmonic injection method [9], the above method is not restricted to handle weak nonlinearities only.

>

@

 H PC , S wc , S fc ˜ Iˆ  PD , S wd , S fd  P O , S wo , S fo 2 0

(13)

2

where |Hcv(jZ)| =Hcv(jZ) D Hcv(-jZ) and the function P is defined as: P(A,B,C)(Z)=A D (B+(2S/Z)A D C), and T 2 Iˆ >1 " 1@ is nu1 vector. In general, u no Z is a rational function of Z. Formula (13) allows us to calculate the output noise spectrum of any OTA-C filter. In order to get the output noise voltage one needs to integrate (13) over the suitable frequency range. Equivalent input noise spectrum can be obtained by dividing (13) by the square of the transfer function of the filter given by H(s)=C(sT-G)-1B+D (cf. (7)). 4. PERFORMANCE OPTIMIZATION The nonlinear distortion and noise evaluation tools presented in Sections 2 and 3 are well suited to be used in computer-aided design and optimization of OTA-C filters. The optimization methodology which takes advantage of the matrix description of the OTA-C filter model in Fig.1 is the following. Let P and Q be two diagonal nun matrices with positive elements, i.e. Q=diag{q1,...,qn} (14) P=diag{p1,...,pn}, We also assume that the matrix T is diagonal (i.e. the filter contains only grounded capacitors). Then the transfer function formula (linear case) can be rewritten as 1 1 Hs CsT  G B  D CPP1 sT  G Q1QB D (15)

3. NOISE ANALYSIS OF OTA-C FILTERS The output noise of any OTA-C filter is a combination of the noise contributions of its all transconductors. The noise in CMOS amplifier with transconductance gm can be described in terms of an equivalent input referred noise voltage with spectral density Sn(f) modeled as [10]: (10) Sn(f)=Sw/gm+Sf/f where both Sw and Sf depend on amplifier topology. Since each transconductor injects its noise current (its spectrum density equals Sn˜gm2) into one of the internal nodes of the filter, its contribution to the output noise spectrum is determined by the current-to-voltage transfer function from the respective node to the output of the filter. It follows from (7) that the vector Hcv(s) of current-to-voltage transfer functions Hi(s) from i-th node to the output of the filter is (11) Hcv(s)=[H1(s) ... Hn(s)]=C(sT-G)-1 where T, G and C are given by (3) and (5), respectively. The elements of G and C are, in general, first order terms in the Taylor series expansion of the filter transconductor transfer functions. If non-trivial output summer is present

CPsQTP QGP QB D Let us define the following matrices T QTP , G QGP , C CP , B QB , D D (16) It is seen from (15) that the equivalence transformation (16) leads to the matrices T , G , C , B and D , and define a new filter, which has the same topology and transfer function but different (re-scaled) element values, and, usually, different performance parameters. The task is now to find the matrices P and Q so that the filter performance 1

42

parameter(s) of interest is(are) optimized. Thus, we came up with a clear optimization procedure: 1. Take an initial realization of the transfer function H(s) given by the set of matrices T, G, B, C and D; 2. Use the matrix elements of P and Q, i.e. p1,...,pn, q1,...,qn, as optimization variables and optimize the target function F=F(T,G,B,C,D;P,Q), which may be dynamic range, integrated noise, THD, etc. Note that Q, P and T need not be diagonal, however, they cannot be arbitrary invertible matrices as well, because matrix T obtained as a result of tranformation (16) has to be symmetric, positively definite with positive diagonal and non-positive non-diagonal entries in order to define an OTA-C filter (see [7] for details). Usually, we have some design constraints such as the maximum value of total capacitance of the filter, maximum power consumption of the filter which depends on transconductance value, allowable capacitance ratio, i.e. the ratio of maximum to minimum capacitance values in the filter, and so on. Some or all of these constraints have to be taken into account in the optimization process. In general, constraints can be written as follows j=1,...,Nc (17) mjdcj(T,G,B,C,D;P,Q)dMj, where Nc is the number of constraints, cj is the j constraint function (e.g. total capacitance of the filter) which is dependent on matrices describing the filter and optimization variables, while mj and Mj are minimum and maximum values of cj (which may be finite or infinite). The optimization itself can be carried out using any available numerical procedure embedded into the optimization system. Choice of the optimization procedure depends on the complexity of the problem and constraints. For the rest of this section we discuss a representative example: performance optimization of OTA-C filters in cascade realization. The problem is to find the optimal pole-zero pairing, optimal cascading sequence, and optimal gain distribution so that the parameter of interest is optimized. There is a rich literature (e.g. [11],[12]) discussing that problem and its solutions (usually quite complex procedures which are impracticable for high-order filters, or just rules of thumb) in more or less general setting and usually for some specific performance parameters. In practice, the only way to find truly optimal solution of the general problem is exhaustive search through all possible cascade sequencing and performing parameter optimization for each of them. Fortunately, using the procedures described in the previous sections we can afford it because these tools are very fast. For the sake of illustration we shall consider performance optimization of 8th order Butterworth filter in cascade realization (Fig.2) with biquads shown in Fig.3. We consider two variants of the filter: variant I with differential pair transconductors (Fig.4) implemented in standard 0.35Pm CMOS process, and variant II with linearized OTAs [13] (Fig.5) implemented in 0.5Pm technology. 3dB cutoff frequency of the filter equals 8MHz for both

variants. Capacitance values are: C11=0.51pF, C12=1.96pF, C21=0.60pF, C22=1.66pF, C31=0.90pF, C32=1.11pF, C41=2.56pF, C42=0.39pF, where the first index refers to the biquad number (e.g. Ci1 is C1 in biquad Hi and so on). in

H1(s)

H2(s)

H3(s)

out

H4(s)

Figure 2. Block diagram of 8th order cascade filter C1 +

in

-

C2 +

gb + -

-

+

gm -

+

-

+ gm + -

gm + -

C1

out

C2

Figure 3. Fully differential biquad used in the filter in Fig.2 VDD=1.25V M4

M3 outin+

CMFB

out+ in-

M2

M1

Ibias VSS=-1.25V

Figure 4. Simple differential-pair OTA Vdd=+1.65V M15

M17

M16

Ibias

M10

M9 R

Vbias1

R

CMFB2

M18

M19 Vbias1

VCMR

out+ M21

outVbias2

M5

M6

in+

M7 M3

Vbias2

M8

M23

in-

M4

M1 M22

M20

M2

CMFB1

CMFB1 M11

M12

M13

M24

M14

Vss=-1.65V

Figure 5. Linearized OTA circuit [13]

In the optimization process we assume two degrees of freedom. The first one is biquad sequencing (which is equivalent to pole-zero pairing for Butterworth filter is an all-pole one). The second are biquad gains, which will be denoted as Ki, i=1,2,3,4. Gains are adjusted by changing transconductance gb of input transconductor; the value of transconductance gm=100PA/V is fixed; gb[70.7PA/V,141.4PA/V], which allows us to change Ki in the range [2-1/2,21/2] ([-3dB,+3dB]). We assumed unity gain setting for the whole filter, i.e. (K1+K2+K3+K4=0dB). In terms of the general OTA-C filter model, the matrices corresponding to the filter in Fig.2 are G

ªG1 «G « 2 « 02 « ¬ 02

02 G1 G2 02

02 02 G1 G2

02 º G 0 2 »» with 1 02 » » 02 G1 ¼

ª gm  gm º , G2 «g 0 »¼ ¬ m

ª0 g m º «0 0 » ¬ ¼

- 2u2 zero matrix

T=diag{C11, C12, C21, C22, C31, C32, C41, C42} (18) C=[0 ... 0 1], D=0 B=[gm 0 ... 0]T, We assume that initial value of input tranconductance gb is equal to gm for each biquad. Setting gains as above is equivalent to using the transforms (16) with P, Q as in (14) with q1=q2=K1, q3=q4=K1K2, q5=q6=K1K2K3, q7=q8=1, and

43

pi=1/qi, i=1,...,8. Thus, we actually have a constrained optimization problem with three independent variables. As mention before, it is possible to implement more general transform of the form (16) if matrices Q and P are not diagonal which allow us to modify also filter topology. Actually, biquad permutation in the considered problem is implemented with non-diagonal Q and P (they are block matrices made of 2u2 identity matrices). In order to perform both biquad permutation and gain adjustment we use composite transforms (16), however, we omit the details. The optimization was carried out using the software written in C, implementing both nonlinearity and noise evaluation procedures of Sections 2 and 3, and numerical optimization routines. OTA nonlinearity was represented, for the purpose of solving equation (9), by spline interpolation of its tabularized transfer function. Both nonlinearity and noise parameters of transconductors depend on linear transconductance, which was modeled using polynomial approximation of appropriate coefficients. Optimization was carried out three times, each time for different objective: noise (goal: minimization of input noise integrated over 3dB bandwidth), linearity (goal: minimization of THD for input signal level 0.15Vpp [0.4Vpp for variant II] at frequency 1MHz) and dynamic range (goal: maximization of DR at THD=-40dB [-55dB for variant II] with input signal frequency 1MHz). We treat the filter with biquad sequencing H1H2H3H4 and Ki=0dB, i=1,2,3,4, as the reference. Table 1 shows target function values for the reference filter. Tables 2 and 3 show optimization results for the Filter in Fig.2 in variant I (simple OTA) and II (linearized OTA [13]), respectively. Note that there is a very good agreement between theoretical and transistor-level simulation results. We can observe that optimal linearity,

dynamic and noise performance are obtained for different biquad sequencing and gain distributions. Note also that optimal biquad sequencing does not depend on OTA used for filter implementation, which is not the case for gain distribution if target parameter involves linearity of the circuit. It should be emphasized that the optimization process including exhaustive search through all 24 biquad permutations and numerical optimization of target function involving multiple THD evaluations (note that in case of DR it is necessary to perform nested nonlinear optimization in order to find input signal level corresponding to required THD value) is fully automated and very fast. For example, transient analysis of the filter in Fig.2 using OTA macromodeling and integration of eqn. (9) involves less then 0.04s of CPU time regardless filter variant vs. 16.2s (variant I) and 17.2s (variant II) for transistor-level CADENCE simulation, with almost no loss of accuracy). 5. CONCLUSIONS A framework for performance optimization of continuous-time OTA-C filters is presented based on matrix description of a general OTA-C filter model. Efficiency and accuracy of the approach is shown by finding optimal block sequencing and gain distribution for example filter and comparing the theoretical results to CADENCE simulation. 6. REFERENCES [1] R.L. Geiger, E. Sánchez-Sinencio, ”Active filter design using operational transconductance amplifiers: A tutorial,” IEEE Circuit and Devices Mag., Vol.1, pp.20-32, 1985. [2] T. Deliyannis, Y. Sun, and J. K. Fidler, Continuous-time active filter design, CRC Press, USA, 1999. [4] B. Nauta, Analog CMOS filters for very high frequencies, Kluwer Academic Publishers, 1993. [5] Y. Sun (Editor), Design of high frequency integrated analogue filters, The Institution of Electrical Engineers, London, 2002. [6] Y. Palaskas, Y. Tsividis, „Dynamic Range Optimization of Weakly Nonlinear, Fully Balanced, Gm-C Filters With Power Dissipation Constraints”, IEEE Trans. Circuits Syst.-II, Vol.50, No.10, pp.714-727, 2003. [7] S. Koziel, S. Szczepanski, R. Schaumann, „General Approach to Continuous-Time Gm-C Filters”, Int. J. Circuit Theory Appl., Vol.31, pp.361-383, 2003. [8] P. Wambacq, W. Sansen, Distortion Analysis of Analog Integrated Circuits, Kluwer Academic Publishers, 1998. [9] J. Mahattanakul, C. Bunyakate, „Harmonic injection method: a novel method for harmonic distortion analysis” in Proc. Int. Symp. Circuits Syst. ISCAS, Vol.3, pp.85-88, 2001. [10] A. Brambilla, G. Espinosa, F. Montecchi, E. SánchezSinecio, ”Noise optimization in operational transconductance amplifier filters,” in Proc. Int. Symp. Circuits Syst. ISCAS, Vol.1, pp. 118 -121, 1989. [11] G.S. Moschytz, P. Horn, Active Filter Design Handbook, John Wiley & Sons, 1981. [12] K. Su, Analog Filters, Kluwer Academic Publishers, 2002. [13] S. Szczepanski, S. Koziel, „Linearized CMOS OTA using Active-Error Feedforward Technique”, to appear, ISCAS’2004.

Table 1. Reference filter parameters (values in [ ] from simulation) Filter Target function realization THD [dB] DR [dB] Noise [PV] Variant I 432 [437] -31.5 [-32.4] 40.7 [41.1] Variant II 1170 [1066] -46.5 [-46.4] 42.3 [42.9] Table 2. Optimization results for the filter in Fig.2 - variant I (values in parentheses refer to CADENCE simulation results) Optimal configuration Target Biquad Biquad gains [dB] Target param. Improv. over param. sequence K1 K2 K3 K4 value reference [dB] Noise H3H4H2H1 3.0 3.0 -3.0 -3.0 169 [172] PV 8.2 [8.1] THD H3H1H2H4 -3.0 -2.1 2.1 3.0 -37.7 [-38] dB 6.2 [5.6] DR H4H3H2H1 1.8 -1.4 -0.9 0.5 47.4 [47.5] dB 6.7 [6.4] Table 3. Optimization results for the filter in Fig.2 - variant II (values in parentheses refer to CADENCE simulation results) Optimal configuration Target Biquad Biquad gains [dB] Target param. Improv. over param. sequence K1 K2 K3 K4 value reference [dB] Noise H3H4H2H1 3.0 3.0 -3.0 -3.0 458 [533] uV 8.1 [6.1] THD H3H1H2H4 -1.7 -1.3 0.0 3.0 -54.8 [-53.3]dB 8.3 [6.9] DR H4H3H2H1 2.4 -1.5 -1.0 0.1 48.9 [48.4] dB 6.6 [5.5]

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