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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 12, DECEMBER 1993

Adaptive Switched-Capacitor Filters Based on the LMS Algorithm

Ulrich Menzi and George S. Moschytz, Fellow, IEEE

929

Abstract-The implementation of adaptive FIR filters based on the LMS algorithm using SC circuits is described. The filters consist of the basic elements: delay element, summing circuit, integrator, and multiplier. In an error analysis, the influence of nonideal effects of SC networks on the behavior of a given filter is investigated. It is shown that the nonidealities in the FIR filter part of the circuit can be eliminated by an additional constant tap element, whereas the main error source in the adaptation part are the multiplier- and integrator offset-errors, which can be compensated using special offset-free circuits. Using the proposed offset-compensation schemes, the accuracy of a switched-capacitor adaptive filter is mainly determined by the nonlinearity errors of the multipliers.

I. INTROLXJCTI~N

I N spite of the growing number and diversity of applications for adaptive filters (e.g., prediction, equalization,

interference cancellation, system identification [l]), and of the many known filter structures and adaptive algorithms, the adaptive finite impulse response (FIR) filter based on the least-mean-square (LMS) algorithm is still among the most commonly used. The reasons for this are, among others, its simplicity, its computational efficiency, and its lack of numerical problems when implemented with finite-precision arithmetic. In most of the applications of adaptive FIR filters based on the LMS algorithm, the realization is all digital, although some other approaches can be found in the literature, e.g., a mixed-mode analog-digital realization in [2], and an all- analog approach based on CCD technology in [3]. However, in those applications in which low power, low supply voltage, small chip-size, and low cost have a high priority, none of the above solutions are very satisfactory. It is the purpose of this paper to show that in such cases switched capacitor (SC) networks, implemented with standard MOS technology, and combining analog building blocks with digital logic circuits on the same chip, can readily satisfy the above requirements. To facilitate this approach, software tools for the automatic design of all, or parts, of such adaptive filters have been developed [4]. These tools take into account the fact that some of the nonideal effects of SC circuits will deteriorate the performance of the adaptive filter considerably unless they are compensated for PI.

In this paper the realization of general adaptive FIR filters based on the LMS algorithm is presented and SC circuits

Manuscript received May 18, 1992; revised April 30, 1993. This paper was recommended by Associate Editor D.J. Allstot.

The authors are with the Institute for Signal and Information Processing, ETH Zurich, 8092 Zurich, Switzerland.

IEEE Log Number 92 11958.

Id(n) Fig. 1. Adaptive FIR filter.

that can be implemented by MOS technology are described. The paper describes the influence of the nonidealities of MOS devices on the behavior of adaptive filters and how most of the effects can be eliminated by appropriate compensation methods. As a result, we believe that a potential application of the adaptive SC filters described is that of adaptive equalizers for data transmission.

II. A BFCIEF SUMMARY OFTHE LMS ALGORITHM

The configuration of an adaptive filter based on the LMS algorithm (AFLMS) of order (M-l) is shown in Fig. 1. This structure can be separated into two circuit parts, namely, the FIR jilter part and the adaptation part. In the FIR filter part the input signal u(n) is processed in each sample period n, according to the equation

c&l) = wTu(n) (1) where the tap-weight vector w and the tap-input vector u(n) are defined as follows:

MT := [WI,. . . , w&f] u’(n) := [u(n), . . . ) u(n - M + l)].

Referring to Fig. 1, d^(n) is the estimation of the desired response d(n) and e(n) is the estimation error. We assume u(n) and d(n) to be jointly weakly stationary and to have zero mean. According to Wiener filter theory this filter operates in its optimum mode; when the index of performance J(w), defined as the mean-squared error (MSE), is minimum. This can be represented as follows:

MSE = E[e’(n)] = J(w) = ai - 2pTw + wTRw (2)

where

R := E[u(~~)u~(n)] (3) 1057-7122/93$03.00 0 1993 IEEE

930 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS ,VOL.40,NO. 12,DECEMBER 1993

p := E[u(n]d(n)] (4) 0; := E[d2(n)]. (5)

J(w) is a second-order function with a unique minimum. This minimum can be found by differentiating J(w) with respect to w and setting the result equal to zero. This leads to

Jmin = J(wo) = 02 - pTRP1p (6) w. = R-‘p. (7)

Jmin is called the Wiener error, wo the Wiener solution. Often, the normalized MSE is used, namely,

Jmin e[dB] := lolog-. ff2

(8)

In the adaptation circuit, the tap-weights of the FIR filter are adjusted such that their expected value converges to the optimum solution MO. This adjustment is performed in each sample period n according to the well-known LMS algorithm [l]

w(n + 1) = w(n) +$7+X). [d(n) - J(n)] (9)

where # is the convergence factor.’ The dynamic behavior of the adaptive filter is described by E[J(n)] which denotes the expectation of the MSE J(n) at time n. E[J(n)] is called the average mean-squared error (AMSE). According to [l] the AMSE after complete conversion E[J(w)] is as follows:

‘[J(m)] = Jmi,+E[Je,(m)] = Jmi,+ PtJmintr[R] (I()) 2-pttr[R]

where tr[.] denotes the trace operator, and JeX(n) is the so-called excess mean-squared error, which is defined as

Jex(TL) := J(n) - Jmin. (11)

III. AN APPROACH TO THE DESIGN OF SC ADAPTIVE FILTERS

Our approach to the SC realization of an adaptive FIR filter based on the LMS algorithm (as shown in Fig. 1) is illustrated in Fig. 2. It shows that the adaptation algorithm in (9) can be realized by an array of noninverting discrete- time integrators. Using SC integrators, the scaled step size ~1 is realized by a capacitor ratio, i.e., the inverse of the normalized time constant l/r, of the integrator. Thus, this structure consists only of the four discrete-time basic elements delay element (II), summing circuit (+), integrator (s), and four-quadrant multiplier (x). The first three are standard SC circuits and therefore easy to realize. However, the realization of the four-quadrant multiplier is more difficult. The SC multipliers described in the literature (e.g., [6]) are slow, need a large amount of hardware, and are therefore not suitable for applications requiring many multipliers and high processing speed. For this reason, we use continuous-time MOS four- quadrant analog multipliers (e.g., [7], [ll]). Such multipliers behave like discrete-time building blocks when driven with analog sampled input signals, as is the case in our application.

’ The index t is used here to make a distinction between the theoretical step size pt and the step size p in an SC implementation of an adaptive filter, which must be scaled due to multiplier scaling factors (see Section V).

d(n) Fig. 2. SC realization of an adaptive filter.

The advantages of these building blocks are their relatively small size and low power consumption.

In order to examine the influence of nonideal effects of SC-based adaptive filters, we proceed as follows:

1) Derive linear discrete-time error models for the four basic elements which include the most important nonideal effects except for noise and nonlinear distortion (Section IV).

2) Analyze the influence of the pertinent nonidealities on the behavior of the AFLMS (Sections 5.1-5.3), taking the linear discrete-time models of the SC circuits into account.

3) Derive models for the four nonideal basic elements, but now with respect to noise. Analyze the influence of noise on the behavior of the AFLMS (Section 5.4), taking the noise models into account.

4) Examine the influence of multiplier nonidealities by simulating a simple first-order AFLMS in a system identification application. Compare with the influence of the linear errors found in Section V (section VI).

5) Based on the comparison of the different error sources and on their effect on performance, develop compensation methods with which acceptable accuracy of SC adaptive filters based on the LMS algorithm can be attained (Section VII).

IV. LINEAR MODELS OF THE BASIC ELEMENTS

4.1. Linear Model of the Three Basic SC Elements

The models for the three basic elements implemented in SC technology can be derived in the following way: According to [8], the nonideal effects stray capacitance (of capacitors, lines, and switches), offset errors due to clock feedthrough and pn- junction leakage current, capacitor-mismatch, dc offset voltage of opamps, and finite gain of opamps are considered. The nonideal effects: finite on-resistance of switches, finite opamp- bandwidth, finite opamp-output-resistance, and finite slew rate can be neglected, by obeying generally established design rules (e.g., [8]) concerning the upper limit of the clock frequency. Simple SC realizations of a delay element, a summing circuit, and a noninverting, discrete-time integrator can be found in the literature (e.g., [6], [8], [9]). Analyzing these SC circuits with the relevant nonidealities mentioned above, linear discrete-

MENZI AND MOSCHYTZ: ADAPTIVE SWITCHED CAPACITOR FILTERS BASED ON LMS ALGORITHM 931

(4

Cc) Fig. 3. Discrete-time models of three SC basic elements. (a) Delay element.

(b) Summing circuit. (c) Integrator.

time models of the three SC basic elements can be derived. These are depicted in Fig. 3, where the symbols (D), (+), (Sk,) and (qz) denote an ideal delay element, an ideal summer, a weighting error, and an offset error, respectively. The symbols (1;) and (CL) denote, respectively, the leakage error and the capacitor ratio of the integrator. Note that errors due to capacitor mismatch are incorporated in the weighting errors.

Analytically, the models can be described as follows:

delay element: ~,,~(n) = (1 + 6kD)Xin(n - 1) + 40 (12)

Fig. 4. Model of the multiplier.

of the output signal (( 1 + S/G,) denotes the factor by which the real scaling-factor deviates from the standard scaling-factor); and S(Xi,r(n), Xi,a(n)) is the multiplier error:

S(zinl(n), xin2(n)) = QOO + QOlZinZ(n)

+ Q102in1(n) + Q02$r12(n)

+ Q205%1 (n) + a12~inl (n>x?*2(n)

+ a2lx&l(n)2in2(n) + a032%2(n)

+ a302;n, (n) + . . . (15)

with craa the offset-error, aa1 and ~~10 the feedthrough errors, and all remaining errors denoted as nonlinear errors.

Ignoring nonlinear errors, our model of the multiplier, depicted in Fig. 4, consists of the ideal multiplier (x), a scaling- factor m,, a weighting-error (S/cm), linear input feedthrough (.fzl and fz2), and an output offset-error (qm). This linear discrete-time model has the following input-output relationship:

&d(n) = m,(l + ~kn)~inl(n)~in2(n>

+.fzlxinl(n) + fac2Zin2(n) + qrn. (16)

summing circuit: zoUt (n) = (1 + ak,l)xinl(n)

f(1 + Sks2)zin2(n) + 4s (13) V. ANALYSIS OF THE LINEAR NONIDEAL FILTER

integrator: 5,,t(n) = (1 + S~J)~,,~(TZ - 1)

+p-l(l + Skl)Xi,(72 - 1) + p--141. (14)

These expressions are used in Section V to evaluate the effect of nonidealities on the performance of LMS-based adaptive filters.

4.2. Linear Model of the Multiplier

For the four-quadrant multiplier of the AFLMS we use a continuous-time MOS four-quadrant analog multiplier, which behaves like a discrete-time circuit when driven by sampled analog input signals, as in our application.

A nonideal multiplier can be described by the following relationship:

z3,t(~) = (1+ hn) msZinl(n)5;,2(n) + h(xinl(n), xin2(n))

where Xinr(n), Zina(n), and x0”,(n) denote the input and output signals of the multiplier; m, is the standard scaling factor of the multiplier, which is required to prevent overflow

By replacing the ideal elements in Fig. 2 by their nonideal linear models (see Figs. 3 and 4), we are able to carry out an error analysis of the AFLMS and to evaluate the influence of the different nonidealities. This will provide answers to the following questions:

1)

2)

3)

Which are the optimum tap-weights (=Wiener solution) of the nonideal FIR filter in the minimum mean-squared error (MSE) sense? What is the minimum MSE in this case? (See Section 5.1.) To which tap-weights does the nonideal adaptation circuit converge? (See Section 5.2.) How large is the increase of the AMSE due to nonideal effects? (See Section 5.3.)

5.1. Nonidealities in the FIR Filter Part Here we investigate the FIR filter part only, i.e., we deter-

mine the optimum tap-weights (=Wiener solution) in the case of nonidealities. First, however, we show that an additional tap with the constant value A leads to an improvement in the performance of an adaptive FIR filter when offset errors

932 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS , VOL. 40, NO. 12, DECEMBER 1993

Fig. 5. Model of the adaptive FIR filter.

occur.2 From the principle of orthogonality [l] we know that the FIR filter operates in its optimum mode (in the minimum MSE sense) when each tap element n(1z - i + 1) and the estimation error e(n) = d(n) - d^(n) are orthogonal to each other. Since we assume that d(n) has zero mean, while d^(n) has nonzero mean due to offset in the FIR filter circuit, the mean of the error e(n) is not zero. Thus, the error e(n) is not orthogonal to a constant value A (i.e., E[Ae(n)] # 0). Extending the filter by an additional tap with the constant value A, however, causes the error e(n) to be orthogonal to this new tap element which means that e(n) has zero mean. Thus’ the extension of the adaptive filter by an additional tap element with constant value A is a simple, yet powerful method of compensating all the offset errors in the FIR part of an adaptive filter. This is shown next within the context of our SC-based adaptive filters.

Consider the model of an adaptive filter of (A4 - 1)th order with an additional constant tap element A, as depicted in Fig. 5. This model is characterized as follows:

1)

2)

3)

Referring to Fig. 2, only the Zth tap element (1 = 0 . . . AJ), and the additional tap with the constant value A, is shown. The weighting errors and offset errors of the delay elements are drawn outside the delay line. This applies, if we assume that the Zth offset error (1 = 1, . . . , M) is the offset error of the Zth delay element plus the sum of the offset errors of all previous delay elements, and the Zth weighting error is the weighting error of the Zth delay element times the product of the weighting errors of all previous delay elements. The leakage error 1; is omitted in the model of the integrator since it is negligibly small.

21n analogy to the method used in [12] for the compensation of biased signals.

4) We assume that the convergence factor p is realized by the inverse of the normalized integrator time constant l/r.

5) The factor a has been introduced to prevent overflow of the elements of the tap-weight vector.

6) No multiplier is needed for the tap element with the constant value A, since the multiplication with a constant value can be realized by the capacitor ratio in the appropriate branch of the summing circuit and in the integrator. Nevertheless, our model of the four-quadrant multiplier is also used for the constant tap element in order to obtain a uniform treatment for each tap.

We define the following vectors and matrices:

u; := [A, u(n), . . . , u(n - M + 1)]

w;(n) := [WA(n), wl(n), . “, w&f(n)]

!7: := [QdO, ml, . . . > aafl T .-

.- [qto, al,. . ., cmfl

2 := Lfwo, .fwl,~~~,fd41

f: := [fuo, fd,~~~,.fuM]

lT := [l, 1,. . . ) 11.

Kd. Kt, and K, are (A4 + 1) x (kZ + l)-diagonal matrices with (I+Skdr), (l+S&), (1+,&l) (Z = O...M) asdiagonal elements.

For the investigation of the nonidealities in the FIR filter part, the tap-weight vector is assumed to be fixed, i.e., independent of the sample period n. The output of the filter is therefore

d(n) = [(w~Km,KtKs + fzKdKs)(uA(n) + q& + qpI‘$f,) + lTK&

+ 4s - f:KK,-‘f,$ (17)

By defining converted tap-weight and tap-input vectors

w: := [w~Kdm,K~K, -I- fTKdK,]i (18)

7&(n) := ?&A(n) + Kilqd + K;‘rn,‘K~‘f, (19)

qtot := [lTKsq, + qs - .f:KK,-‘f,li (20) a

(17) can be rewritten as follows: ^

d(n) = w:%(n) + qtot. (21)

Taking the expected value of the squared error gives

&z(w) = E[(d(n). - +c(n) - qtot)“]. (22)

By defining the term d,(n) as

d,(n) := d(n) - qtot (23)

(22) can be transformed into a closed-form expression, which is exactly the same as in the ideal case of (2), namely,

Jc(wc) = -w&(n) - +c(4)21 (24) + Jc(wc> = 02, - 2p3+ + w:R,w, (25)

MENZI AND MOSCHYTZz ADAPTIVE SWITCHED CAPACITOR FILTERS BASED ON LMS ALGORITHM 933

where

R, := E[u,(n)u~(n)]

PC := ~[~c(~)&(~)l

a;, := E[&(n)] = 0; - q;&.

Thus it can be concluded that all modeled errors in the FIR f the adaptive filter are compensated, on the one hand

by the additional constant tap element A, and on the other by the LMS algorithm itself.

5.2. Nonidealities in the Adaptation Part In analogy to the ideal case, the converted tap-weight vector wcu, for which J=(m,) is minimum, can now readily- be Here we show how the adaptation part of the LMS based

determined by differentiating JJw,) with respect to w, and adaptive filter behaves when all nonidealities in the FIR

setting the result equal to zero: filter-part and in the adaptation-part according to Fig. 5 are considered. For this analysis we define the following vectors

Rcwco = P,. P-3 and matrices:

The vector equation (26) can be solved for the unknown f: := [fee, fel, . . . 1 f&r1 vector W,O. For the representation of the result, the vector q; := ['ho, 'hl,"','hM] wca is separated into a part with the converted tap weight of the tap A (W&a), and a part with the remaining converted

4: := [qio, qi1,. . * ,mfl. tap-weights (wFUo = [w,rc, . . . , w,Ma]): F,, K,, and K; are (n/l + 1) x (M + l)-diagonal matrices

& = [%AO, WclO,“’ ,&MO] = [%AO, w:uO]* with fal, m,(l + Slc,[), (1 + 6Zc;l) (1 = 0. . . M) as diagonal elements. In this case, the adaptation equation, according to

The solution of the vector equation (26), which denotes the (9), can be written as

optimum tap-weight vector for the case of nonidealities in the WA(‘n + 1) = WA(n) + paKdm,K,Ki(ZLA(n) FIR filter part, is obtained as

+ &l’ld) wcuo = R-lp (27) . (d(n) - d++44 - Qtot)

and + ~Fa&Ki(%t(~) + K&d

W,lOCl + . . . + 'WcMOcM + qtot (28)

+ paKif,(d(n) - w:(~)~c(~) - qtot) 'WcAO = -

A + co

where

c1 = 4dl + m;‘(l + 6ktl)-1fwl for 1 = 0, 1,. . . , M 1 + C&l >

and R and p are, respectively, the correlation matrix and the cross-correlation vector in the ideal case ((3), (4)). The “real”

+ /&q, + ,w.

By making the following substitutions:

M := paKdm,K,Ki F,, := F,rn;‘K;‘a-’ f,, := KT1rn,‘K;‘fe

(33)

(34) (35) (36)

optimum tap-weight vector WA0 can now be found by the Q mc := Kilm-lK;la-lq

conversion equation (18): .- K,-‘1*L91K;‘K;la”‘lq (37)

Q;c .- z c (38)

wA0 = aK~lm,lK~lK,‘wco - rn,‘K,-‘f,. (29) the nonideal adaptation equation (33) can be rewritten as

The minimum MSE can then be calculated as follows: wA(n+ 1) ='UIA(~)+M(UA(~)+Kd'Qd+f,,)

J =min = Jc(wco) = 02, - PTWCO. (30) . (d(n) - a+44 - 4tot) +MF,c(u~(n) +K&)

Inserting wee into (26) gives the simple result: + M(q,c f 4. (39) J cmin = 0; - pTR-l P (31) Taking the expectation of both sides of (39), making the

+ Jcmin = Jmin. (32) assumption that w,(n) is independent of uA(n) according

The results presented in (27)-(32) can be interpreted as fol- to the “fundamental assumption” [l] and assuming that the

IOWS: 1) Jcmin is the same as in the ideal case, i.e., it algorithm has converged ( + E[wA(n+l)] = E[WA(n)]), we obtain

is independent of all nonidealities in the FIR filter part. 2) Compared to the ideal case, the optimum tap-weight vector E[(uA(n) +K,-'% + f,,)(d(n) - Qtot)] WA0 is scaled by the diagonal matrix aK:‘rn;’ Ktpl KL1 and shifted by the vector m;lKtlf,. It is shown in Section 5.2

+Fcdb~(~) +K,&] +Qmc +!I';,

that the LMS algorithm, implemented in the adaptation part, = E[(uA(n)+ K,-'qd +f,,) .'&n)]

exactly converges to this scaled and shifted tap-weight vector, . -q4~)1. (40) as long as the adaptation part is ideal. 3) All offset errors in With the definitions the FIR filter part are compensated by the additional constant tap element A. The value of the coefficient of this tap element Rcf := E[(uA(n)+ f&d +'f,,)&)]

depends on the nonidealities in the FIR filter part ((28)). PC, := E[('11A(n) +K;lQd+f,,)(d(n)- qtot)]


(40) can be rewritten as follows:

-f+4c~)l = R,,‘P,, + R&m, + qic

+FaJquA(n) + K;lqdl). (41)

It can readily be shown that L,

R,,‘p,, = R,‘p, = wco. (42)

Moreover we find that3

F&[uA(n)+ K&] = F,&&. (43)

Thus the (converted) tap-weights to which the nonideal adaptation algorithm converges are as follows:

+J,(~xJ)] = wco + R,,‘(qm + q;c + F&&d. (44)

The term Fa,Kilqd can be neglected, since it is a second- order error term:

E[wc(~)l z wco + R,,‘(q,, + qi,). (45)

The convergence point of the actual tap-weight vector can be found by (45) and (18):

E[wA(cs)] = ~K;%L,~K;~K,~

++o + R;l(qmc + q;,)] - mi%%. (46)

With (37), (38), and neglecting second-order error terms, we find from (46):

.[wco + a-lms -lR;‘(q, + qi)] - m,‘K,-‘fu. (47)

The results presented in (44)-(47) can be interpreted as follows: 1) Assuming that the adaptation part is ideal, i.e., qm, qi, and F, are zero, the adaptation algorithm converges to the optimum solution, i.e., to the solution that was found previously in Section 5.1 when only the nonidealities in the FIR filter part were considered (see (29)). 2) Considering all nonidealities (in the FIR filter and the adaptation part), the algorithm does not converge to the optimum solution. The main error sources responsible are the offset errors of the integrators and adaptation multipliers, whereas the feedthrough error of the adaptation multiplier only leads to a second-order error (see (44)). 3) The convergence factor ,LL (implemented by the capacitor ratio of the integrators) has to be scaled due to the scaling factor m, occurring in the multipliers in both the FIR filter and the adaptation part:

p = mg2pt. (48)

3With the assumption that fao = 0, since the multiplication with the constant tap element A is realized by a capacitor ratio and not by a four- quadrant multiplier.

Explanation for Nonoptimum Convergence: The reason that the LMS algorithm does not converge to the optimum solution can be explained as follows: In the ideal case, the input signal of the Ith integrator (i.e., ,~w(n-l+l)e(n)) has zero mean after convergence, i.e., the condition for optimum filter operation (E[u(n - 1-t l)e(n)] = 0) is fulfilled.4 In the nonideal case, however, the input of the Zth integrator is p~(n, - 2 + l)e(n) plus an offset error PQ. In order to prevent the integrator from going into saturation, the expectation of this input signal has to be zero. This is provided for by the LMS algorithm itself which causes E[zl(n - 1+ l)e(n)] to be -41. Thus, on the one hand the input signal of the Zth integrator has zero mean, while on the other the filter does not operate in its optimum mode.

5.3. Increase of the Average MSE (AMSE) Due to Adaptation Offsets

As mentioned above, the adaptation offsets cause the tap- weight vector not to converge to the optimum solution. In this section we shall determine the increase of the ASME (E[J(co)]) due to this effect. For this purpose we define the tap-weight error vector

CA(n) := w(‘Iz) -WA,, (49)

where WA0 is the optimum tap-weight vector of the (nonideal) Wiener filter according to (29). According to the model in Fig. 5, and assuming that, except for adaptation offsets and multiplier scaling factors, the influence of all other nonidealities is negligible, we find the following recursive equation for the tap-weight error vector c(n):

cA(n+ l)=[I- pm&A(n)uz(n)]‘cA(n)

+~m~.[am,‘,A(n)d(~)-‘11A(n)~TA(n)wAO]

+ P(Qm + %I. (50)

Note that I denotes the M x M-unity matrix. With definition (49), J(n) can be represented by the so-called quadratic form [l]:

J(n) = Jmin + 03 -2m;C~(TX)RACA(TA) (51)

where RA is defined as follows:

RA := E[uA(Tz)u~(TI)]. (52)

With (11) we have

-Lx(n) = a -2m~C;(TL)R&&L)

+ -qJex(n)l = a -2m~~[C~(7t)RACA(?Z)]. (53)

By defining the weight-error correlation matrix KA(~) as

KA(~) := E[CA(+:(n)] (54)

we find from (53)

-qJex(n)l = a- 2mS tr[RA KA(TX)]. (55)

It has been shown (e.g., [l]) that a recursive equation for the time evolution of KA(TL) can be found. By the evaluation

4Assuming ergodic processes generating the input signals of the adaptive filter.

MEN21 AND MOSCHYTZ: iDAFTIVE SWITCHED CAPACITOR FILTERS BASED ON LMS ALGORITHM 93s

of this recursive equation for n --t cc we find for EIJex(oo)] (56), which can be found at the bottom of the page.

Thus we find that

E[J(m)] = Jmin + E[J~~“‘(~)] + E[J&(W)] (57)

where E[$‘(cc)] is the additional AMSE caused by*adaptation offsets, as shown in (58), which is at the bottom of the page.

These results can be interpreted as follows: 1) According to (57), the AMSE after convergence is composed of three distinct terms: Wiener error (Jmi,), the ideal adaptation error (E[J:d,““*(m)]), d an an adaptation error caused by the adaptation offsets (E[J&(oo)]). 2) According to (58), E[J,&(cc)] has the following characteristics:

1) E[J,O, (oc)] is large for a high filter order (M - 1) and is nearly independent of the convergence factor pt.

2) E[J,&( cc)] increases with increasing offset errors. 3) JY[J~~(c~)] is large for small scaling factors a. 4) The dependence of E[Jzx( cc)] on the offset errors and

the scaling factor a is quadratic, whereas the dependence of EIJix(oo)] on the filter order (M - 1) is linear.

5.4. Analysis of Noise Effects

Due to various physical effects, noise occurs in the MOS switches, opamps, and multipliers [8], [lo]. As in the previous sections, models for the four nonideal basic elements, but now with respect to noise, are therefore derived in what follows.

5.4. I. Noise Models for the SC Basic Elements: In [lo] the noise analysis of SC circuits is described, where the models used for the switches and opamps are as follows:

Switch: Here we distinguish between the “on” and “off’ state

1)

2)

- of the switch: In the “off’ state the MOS switch is considered as an ideal open circuit. In the “on” state the MOS switch can be modeled by its on-resistance R,, and the noise source u,,~ with the constant two-sided spectral power density

S,, = 2/&R,,

connected in series (Fig. 6(a)). Here k denotes the Boltzmann constant (k = 1.38 x 1O-23 J/K) and 0 is the absolute temperature.

% (a) @I

Fig. 6. (a) Noise model of an MOS switch in the on-state. (b) Noise model of an opamp.

Fig, 7. Discrete-time noise models, (a) Delay element. (b) Summing circuit. (c) Integrator.

Opamp: The noisy opamp can be modeled by an ideal opamp and the noise source u,n with the constant two-sided spectral power density

So, = 2lcOR,,

connected with the positive input node (Fig. 6(b)>.’ Here R,, denotes the noise-equivalent resistance of the opamp.

Using these noise models, the three SC basic elements of the AFLMS can be analyzed according to the method described in [lo] and discrete-time noise models derived from the results. This procedure is described in detail in [5]. Here only the resulting noise models are presented (Fig. 7). In these models, the symbols (D), (+), and (us) denote an ideal delay element, an ideal summer, and a source with a discrete-time independent white noise voltage. The analytical calculation of the variance of these noise voltage sources is given in [5].

5 Note that the l/f component of the opamp has been omitted in our model, since its contribution to the broadband noise variance, which is the significant value in our noise analysis, can be neglected.

2am2mT2(q, + qJTRil(q, + q;) - $am2mY2(q, + dT(q, + d I

2 - pt WA] (56)

2a-2m;2(q, + qi)TRil(q, + qJ - $am2mY2(q, + dT(qm + a) w:xw1 = 1

2 - pt ~[RA] (58)

936 IEEETRANSACTIONSONCIRCUITSANDSYSTEMS-I:FUNDAMENTALTH~ORYANDAPPLICATIONS,VOL.~~,NO. 12,DECEMBER 1993

Fig. 8. Noise-model of the multiplier.

Analytically, the models can be described as follows: delay element:

Z,ut(TZ) = xin(n - l) + uV(n) (59)

summing circuit:

lout = xinl(n) h 2in2(n) + u.9(n)

integrator:

(60)

xi,t(n) = Zint(n - 1) + pzin(n - 1) + UII(~ - 1) (61)

2,,t(nj = xint(n) + UIA(~). (62)

5.4.2. Noise Models of the Multiplier: As mentioned ear- lier, the multiplier used in our AFLMS is a continuous-time circuit. In analogy to the SC basic elements, a discrete- time noise model (Fig. 8) can be derived, since the noise is considered only in the discrete time instances n. In this model, the symbols (x), (+), (m,), and (uM(~)) denote an ideal multiplier, an ideal summer, a scaling-factor, and a source with a discrete-time independent white noise voltage. The variance of the noise voltage source is given by the specific multiplier circuit.

Analytically, the model can be described as follows:

X,,t(TL) = ma~in1(~)2in2(n) + uA4(n)* (63)

In analogy to Sections 5.2 and 5.3 an analysis of the AFLMS (shown in Fig. 2) with these noise models can be carried out. This procedure is described in detail in [5]. The expression for the increase of the AMSE due to noise effects after conversion (E[J&(oo)]) is shown in (64), at the bottom of the page, where a$, ai, u&, afl, and ~9~ denote the variances of the discrete-time noise voltage sources in the noise models according to Figs. 7 and 8.

5.5’. Summary of the Main Theoretical Results

The error analysis presented in this section can be summa- rized as follows.

1) The nonidealities in the FIR filter part (modeled in Fig. 5) have no influence on the performance of the adaptive filter based on the LMS algorithm (AFLMS) if a tap element with a constant value is added to the filter.

First-order adaptive

filter 4

1 44 1 Fig. 9. Adaptive filter in a first-order system identification application.

2) The nonidealities in the adaptation part (modeled in Fig. 5) have no influence on the quality of the AFLMS, except for the offset errors occurring in the multipliers and integrators.

3) The offset errors in the adaptation part prevent the tap- weight vector from converging to the optimum solution.

4) After convergence, the average MSE (AMSE) is generally larger than for the ideal case.

5) The theoretical convergence factor has to be scaled due to the multiplier scaling factor.

6) Noise errors in both the FIR filter and the adaptation part lead to an increase of the AMSE after convergence.

VI. SIMULATION OF A FIRST-ORDER ADAPTIVE FILTER

6.1. Adaptive System Identijcation

The computer simulation of a simple first-order (two-tap) system-identification problem (Fig. 9) using the LMS algorithm was used a) to test the theoretical results in this paper, and b) to examine the influence of noise and nonlinear multiplier errors on the behavior of the filter. In this example, both the first-order adaptive filter (with an additional constant tap element of value 0.5, a convergence factor pt = 0.24, and a scaling factor a = 1) and the system to be identified (i.e., “plant”), with the two tap-weights as1 = as2 = 0.707 and the (white) plant noise of variance 0.004V2, are fed with a white noise input signal of variance 0.25V2. Since the error in this configuration is the difference between the output of the “plant” and of the adaptive filter, the LMS algorithm forces the adaptive filter to imitate the “plant.” Note that in this application the plant noise is equal to the Wiener error [51. Typical values of the modeled parameters in both the FIR filter and the adaptation part are as follows:

1) Delay Element: 6kD = -0.001, QD = Cl.015 V. 2) Summing Circuit: 6k,t = -0.01, qs = 0.06 V. 3) Integrator: 61c1 = -0.01, qI = 0.016 V.

A4

Ma$ + CT; + ~m;u$(l - 1)~; l=l 1

+’ tr [Rv](2m3& + ,urechMg& + prechai) + MpreCh~;2(~& + g,“,) a2 2 - prech tr[Rv] 1 (64)

MENZI AND MOSCHYTZ: ADAPTIVE SWITCHED CAPACITOR FILTERS BASED ON LMS ALGORITHM 937 ’

Fig. 10. Simulated learning curves of an adaptive filter in a first-order system-identification problem with nonidealities in the FIR filter part. Upper curve: Without constant tap element A. Lower curve: With constant tap element A.

4) Multiplier: m, = 0.667, 6lc, = -0.01, fZ1 = fZ2 = 0.003, qm = 0.01 v.

6.2. Verijication of the Results

Case I: Ideal Case: In the ideal case, theory and simulation result in the following values:

E[J(oo)] = 0.00443 v2 E[wT,(oo)] = [0, 1.061, 1.0611 V.

Case 2: Nonidealities Only in the FIR Filter Part: In the first experiment, only nonidealities in the FIR filter part are considered, whereas the adaptation part is ideal. The lower learning curve in Fig. 10 represents this case. The numerical results of the theoretical calculation (with (10) and (29)) and the simulation are as follows:

Theory Simulation

E[J(co)] = o.o0443v* 0.00442V’

%JTA(~)I = [-0.372, 1.059, 1.0581 V [-0.371, 1.057, 1.0571 v

A close agreement between simulation and theory is evident. As a comparison, the upper learning curve in Fig. 10 shows the case where the additional constant tap element is omitted. E[J(co)] is then 0.01592V2. Case 3: Nonidealities in Both the FIR Filter and the Adaptation Part: In the second experiment the nonidealities in both the FIR filter and the adaptation part are considered. The numerical results of the theoretical calculation (with (47) for the calculation of E[wz(oo)] and (57) and (58) for the calculation of E[J(oo)]) and the simulation are as follows:

Theory Simulation

pt = 0.1: E[J(~)l = 0.0188 v2 0.0190 v2 eJTA(~)l = [-0.249, 1.296, 1.3621 [-0.256, 1.302, 1.3021 v

pt = 0.24: &)] = 0.0202 v2 0.0199 v2 %4(c=)l= [-0.259, 1.311, 1.3251 [-0.256, 1.302, 1.3021 v V

Again a close agreement between simulation and theory is apparent.

C&e 4: Simulation of Noise Effects: As shown in [5], the typical variances of the discrete-time noise voltage sources in the noise-models of a first-order AFLMS, realized with simple SC circuits, are as follows:

1) Delay Element: (T$ = 12.8 nV2. 2) Summing Circuit: ai = 36.5 nV2. 3) Integrator: 04~ = p2 . 9.8 nV2; a;* = 7.7 nV2. The variance of the multiplier noise voltage-source may

typically be

UK .= 90 nV2.

Using these values, the theoretical calculation and the simulation provide the results below for the AMSE (E[J(oo)]). The convergence point of the tap-weight vector (E[w~(w)])~ for our first-order AFLMS in the given system-identification application with zero plant noise7 is also given:

Theory Simulation

pt = 0.1 E[J(co)] = 258 nV* 252 nV2 E[&(ca)] = [1.061, l.O61]V

pt = 0.24 E[J(co)] .= 309 nV2 300 nV2 E[w~(co)] = [1.061, 1.0611 V

These results can be interpreted as follows: 1) Since the plant noise is zero in our simulation, the value of E[J(co)] is caused by circuit noise only. Thus we here denote E[J(oo)] as E[J,“,(co)]. H owever, this value is much smaller than the increase of the AMSE due to offset errors. Moreover, it is nearly independent of pt. 2) The convergence point of the tap-weight vector is unaffected by noise. 3) A close agreement between theory and simulation is evident.

In the analysis of the previous section, multiplier nonlinearities have been omitted. These effects are considered next.

6.3. Simulation of Multiplier Nonlinearity

6.3.1. Introduction: Every term in (15) with 2inr and/or Zins having an exponent larger than unity is referred to as a nonlinear multiplier error term. For the quantitative determination of the nonlinear behavior of a multiplier circuit, the total harmonic distortion (THD) is normally used [ 111. For the measurement of the THD with reference to the input node 1, a sinusoidal signal with maximum amplitude is applied to input node 1, and the maximum dc voltage is applied to the input node 2. In this configuration, the measured THD of the output signal is denoted as the THD with reference to input node 1 (= kr). The THD with reference to input node 2 (= k2) is defined in the same way. Typical THD values of currently available CMOS multipliers are 0.4%-l%.

6Note that we have omitted the constant tap element in this simulation since it has no influence on noise.

‘The plant noise is zero in this simulation, since the increase of the AMSE due to noise is much smaller than the plant noise in the simulation of section 6.2. Zero plant noise means that the Ah4SE, after convergence, is zero in the ideal case.


In the following simulation, the influence of multiplier nonlinearity on the behavior of the AFLMS is investigated. For this purpose the coefficients a;; of nonlinear terms up to the third order in (15) have been chosen such that distinct THD values Ici and ka are given as a result.

6.3.2. Simulation Results: In the following, we present the results of the simulation of our first-order AFLMS in the given system-identification application with different values for the multiplier THD Li and lcs and the convergence factor pt. For the same reason as in the noise simulation, the plant noise was set to zero, meaning that the resulting AMSE is caused exclusively by multiplier nonlinearities. We therefore denote E[Jl(~)l as -w,“,lWl~ and consider the three following cases: Case I: Nonlinear terms only in the FIRJilterpart: pt = 0.1 (with constant tap element A):

h,z = 0.25% 0.5% 1% w,“,‘(~)l= 2.2 /iv2 7.8 pV2 31 pv2 E[wA (==)I =

[jyi$y [j?.g] [jy){{

pLt = 0.24 (with constant tap element A):

0.25% 0.5% 1% 2.5 pv2 8.7 pv2 35 pv2

124 fiV2

E[“A(m)] =

pCLt = 0.24 (without constant tap element A):

@“A (=J)] =

Case 2: Nonlinear terms only in the adaptation part: ,ut = 0.24 (with constant tap element A):

kl,z E[J,“,‘(co)] 1

0.25% 0.5% 0 0 1.5 pv 4.8 fiV2

E[wA(m)] = [;;Ah] [:1%] $g, $i;,.

Additional simulations showed that the influence of nonlinearities in the adaptation part are independent of the convergence factor pt, and independent of the existence of a constant tap element A. Moreover, it can be shown that E[JF!(cc)] is mainly caused by nonlinear terms with even exponents, which in the mean behave as offset errors. Case 3: Nonlinear terms in both the FIR jilter- and the adaptation part: pt = 0.24 (with constant tap element A):

E[J(m)l = Jmin + EIJAd,““‘(m)l + EIJ&(CQ)l

+E[J,‘,(m)l + -f&J,“,41 (65)

where 1) Jmin is the Wiener error,

2%

2) E[J$“‘(~)] is the ideal adaptation error, 3) E[J&( co)] is the error due to offset terms, 4) E[J&(m)] is the error due to noise, 5) E[J,n,“(co)] is th e error due to multiplier nonlinearity. In order to compare the different error terms, the results

of another simulation of our first-order AFLMS in the given system-identification application is presented. The plant noise was again set to zero, meaning that Jmin and E[JLF’(oc)] disappear and the AMSE, after convergence, is exclusively caused by circuit nonidealities. In Fig. 12, four learning curves are renresented:

1000

800

600

400

1%

Fig. 11. Increase of the AMSE in function of multiplier nonlinearities for a 2-tap and a IO-tap AFLMS.

The results of this simulation can be interpreted as follows: 1) Nonlinear terms in both the FIR filter and the adaptation part lead to a considerable increase of the ASME, which is at least an order of magnitude larger than the increase of the AMSE due to noise. 2) By using a constant tap element, the increase of the AMSE due to nonlinearity in the FIR filter part can be drastically reduced. The reason for this is that many nonlinear terms have nonzero mean and therefore behave, in the mean, as offset errors. 3) The increase of the AMSE due to nonlinearity in the adaptation part is mainly caused by nonlinear terms with even exponents, which behave, in the mean, as offset errors. 4) The convergence factor pt has almost no influence on E[J,n,” (oc)]. 5) The convergence-point of the tap-weight vector is almost unaffected by nonlinear terms.

In Fig. 11 the AMSE increase due to nonlinearities according to Case 3 is presented graphically for both short (2 taps) and long (10 taps) filters. Note that the increase of the AMSE is proportional to the filter order and quadratic with respect to the multiplier nonlinearity.

6.4. Comparison of the, Influence of Different Error Sources

According to our error analysis, the average MSE, after convergence, of an adaptive filter based on the LMS algorithm (AFLMS) is composed of five different terms:

I 1) Learning curve 1 applies when all nonidealities consid-

ered in this paper are taken into account, and no constant tap element is used. The AMSE after convergence, and

MENZI AND MOSCHYTZ: ADAPTIVE SWITCHED CAPACITOR FILTERS BASED ON LMS ALGORITHM 939 a

E[Jldl [v*l

0.010

E

:

I.OE-4 .A

3

l.M-6 4

109 - 300

Fig. 12. Comparison of the influence of different error terms.

its normalized value

. are in this case

E[J(cm)] = 0.0320 V2 + ea = -8.9 dB.

2) Learning curve 2 applies when all nonidealities considered in this paper are taken into account and a constant tap element is used. The AMSE after convergence, and its normalized value, are in this case

E[J(cm)] = 0.0157 v2 + E, = -12.0 dB.

3) Learning curve 3 applies when all nonidealities considered in this paper are taken into account, without the offset errors in the adaptation part, and a constant tap element is used. The AMSE after convergence, and its normalized value, are in this case

E[J(co)] = 14.0 /Lvs -+ E, = -42.5 dB.

4) Learning curve 4 applies when all nonidealities considered in this paper are taken into account, with the exception of the offset errors in the adaptation part and the multiplier nonlinearities, and a constant tap element is used. The AMSE after convergence, and its normalized value, are in this case

E[J(m)] = 495 nV2 + 6, = -57 dB.

These curves indicate that the error terms of an adaptive SC filter based on the LMS algorithm (AFLMS) caused by circuit nonidealities are clearly dominated by offset errors in both the FIR filter and the adaptation part. Therefore, the quality of an SC AFLMS can be significantly improved, if it is possible to compensate for these effects. Such compensation methods will be discussed in what follows.

VII. COMPENSATION OF ADAPTATION OFFSETS

As mentioned above, the compensation of offset errors in the FIR filter part can readily be accomplished by using an additional constant tap element, whereby some specific features of the LMS algorithm are utilized. By contrast, the compensation of the adaptation offsets has to be carried out

Fig. 13. Circuit for the realization of the adaptation algorithm.

on the circuit level. For the description of the compensation procedure we consider the adaptation formula of the Zth tap weight. According to the model in Fig. 5, and assuming that, except for the adaptation offsets and multiplier scaling factors, all nonidealities are neglected, we find

wz(n+l) = wl(n)+P m,u(n-Z+l)e(n)+p(q,l+qir). (66)

A possible circuit model for the realization of (66) with a two- phase switched capacitor network is shown in Fig. 13. Two different offset voltages occur, namely, the multiplier offset V, = qml and the offset voltage of the integrator (K, = qil). In what follows, we show how these offset errors can be compensated for.

7.1. Compensation of the Multiplier Offset

The operation of the circuit in the uncompensated case (Fig. 13) is as follows:

1) In phase 1, the charge Qc, on the capacitor Ci is

Qc, = G[m,u(n - 1 + l)e(n) + Vm].

2) In phase 2, Qc, is recharged to the integrator capacitor by connecting Ci between ground and the inverting opamp input. If the integrator is ideal (--+ V,, = 0), Ci is discharged completely. Note that, in this phase, the multiplier is not used.

For compensation we exploit the fact that the multiplier is not used in phase 2. The compensated circuit is shown in Fig. 14. The operation of the circuit is as follows:

1) In phase 1 the behavior of the circuit is the same as in the uncompensated case, i.e.,

&cl = G[m,u(n - Z + l)e(n) + Vm].

2) In phase 2 both multiplier inputs are connected to ground. Qc, is recharged to the integrator capacitor by connecting Ci between the multiplier output and the noninverting opamp input. In this case Qc, is not recharged completely. Due to the multiplier offset, which appears at the multiplier output in phase 2, a residual charge

QT = GVm is stored on C1 at the end of phase 2. Thus the charge, moved to the integrator capacitor Cl, is

Qint = Qc, - QT = CI~,U(~ - 1 + l)e(n)

which is exactly the same as in the ideal case.


I I

Fig. 14. Realization of the adaptation algorithm with multiplier offset compensation.

We conclude that the influence of the multiplier offset can be completely eliminated by the circuit depicted in Fig. 14. The additional hardware cost is negligible, namely, 2 switches per tap. Fig. 15. Integrator CFI compensation.

7.2. Compensation of the Integrator Offset

The offset error in an integrator is caused by the opamp offset-voltage and the clock feedthrough (CFT). 7.2.1. Compensation of the Opamp Offset-Voltage: For the compensation of the opamp offset voltage we use offset- free SC integrators, of which several alternatives have been published in the literature. For our purpose, we selected the compensated circuit of Nagaraj [15], since it is very simple and has only two phases. 7.2.2. Compensation of the Clock Feedthrough (CFT) Offset Error: The only significant remaining offset error in the integrator is the one caused by clock feedthrough (CFT). A method for the elimination of such an effect was found by Huang and Moschytz [ 161. The basic idea is the following: In contrast to opamp offset voltages, which are different in each integrator circuit, the offset error terms caused by CFT can be assumed to be the same in all the integrators, at least to a first order. This is because the circuit structure of the integrators on a given chip is the same; the switches all have the same size and are all driven by the same clocksignal. We therefore denote this error term as a “common-mode” offset. Based on this assumption, the circuit in Fig. 15 can be used for CFT compensation. Assuming a common-mode offset term denoted by V&T, the analysis of the upper circuit of Fig. 15 results in the following difference equation:

wo(n + 1) = wo(72) - Zwo(n) + V&T. (67)

Because Cl/C1 = p, we find

‘wO(n + 1) = wO(n>(l - p) + VCFT. (68)

According to (68), WO(~) can be interpreted as a geometrical series with the factor (1 - p). Since this factor is smaller than 1, for n -+ co this series converges to

VCFT wrJ(cm) = -.

CL (69)

According to Fig. 15, the output of the upper circuit, WC,(~), is connected with an additional input-capacitor (Cl) in every integrator of the AFLMS. Therefore the update formula of the lth tap weight is now:

Wl(n+l) = WZ(n)+~(e(n)u(,-l+l)-WO(n))+VCFT. (70)

Taking the expectation of both sides of (70), using the independence of wl (n) from w(n - I + 1) according to the independence theory [l], and assuming that the algorithm has converged (-+ E[wl(n + l)] = E[wl(n)]), we obtain

E[e(n)u(n - 1 + l)] = -VCFT + ,UWO(OO).

From (69) and (71) we find

(71)

E[(e(n)u(n - 1+ l)] = 0. (72)

From (72) we deduce that the principle of orthogonality is valid. Therefore, the filter is operating in its optimum mode, meaning that the CFT effect is compensated for.

We have found this compensation method to be clearly verified by the simulation of a linear model of the AFLMS.

Thus we conclude that the influence of all offset errors occurring in an adaptive SC filter based on the LMS algorithm (AFLMS) can be compensated for. Using these compensation methods, the accuracy of an SC AFLMS is determined by multiplier nonlinearities, according to Section 6.3. Assuming a multiplier THD of 0.5% for both input nodes, a first-order AFLMS can reach a normalized average MSE of -42 dB (curve 3 in Fig. 12).

7.3. Comparison with Digital Realization

The error analysis of a digitally realized AFLMS is carried out in [17]. In digital filters, an increase of the average MSE occurs due to the finite precision of digital networks. In [17] it is shown that the accuracy can be improved if more bits are used for the filter’s coefficients than for the data. It is shown in [5] that, in the given example (first-order system identification), the resolution of the coefficients must be at least 12 bits if the digital adaptive filter is to be more accurate than its SC counterpart. It can therefore be concluded that the SC realization of adaptive filters based on the LMS algorithm is advantageous in those applications, in which the errors of the algorithm (Wiener error and excess mean- squared error) is considerably larger than the errors caused by the circuit nonidealities. In such cases an SC realization has the advantage, over a digital realization, of lower power consumption and lower circuit complexity.

MEN21 AND MOSCHYTZ ADAPTIVE SWITCHED CAPACITOR FILTERS BASED ON LMS ALGORITHM 941

VIII. HARDWARE IMPLEMENTATION OF AN ADAPTIVE SC FILTER

A first-order adaptive SC filter was built with dis&ete components. In order to get by with only two-phase circuits, the following circuits were chosen for the SC basic elements: Gillingham delay element [ 131, Haug summing circuit [ 141 and Nagaraj integrator [15]. The circuit of the whole filter is shown in Fig.. 16. Note that the filter is completely offset-compensated, thereby comprising all the compensation schemes described in this paper (i.e. additional tap element A, as well as multiplier and integrator offset compensation). In Fig. 16, opamp 1 belongs to the (Gillingham) delay element, opamp 2 to the (Haug) summing circuit, opamp 3 and 4 to the (Nagaraj) integrators of the tap-weights w1 and 2~2, opamp 5 to the (Nagaraj) integrator of the constant tap element, and opamp 6 to the circuit for CFT compensation (see Section 7.2.2). In order to test this filter, we used the system-identification application described in Section 6.1. In what follows, the results of the measurement, the simulation of the ideal AFLMS, and the theoretical calculation for different values of the plant noise variance (which is equal to the Wiener error Jmin), are listed8

Theory Simulation Measurement

Jmin =OV*: E(J(m)] = 0 v2 0 v* 12 pvz Jmin = 0.00114 v2 0.00109 v* 0.001 v2

E[J(~)l = 0.00114 va

Jmin = f/)gga 0.00456 V2 0.00454 v2 0.004 vz

Jmin = 0.01 V2 fE’ly$2= 0.0114 v2 0.0112 vs.

We found that the results of the measurements were entirely independent of offset voltages, which were adjustable to distinct values in our bread-board circuit. The measured learning curves for different Wiener errors are shown in Fig. 17. In Fig. 18 the measured learning curve (solid line) is shown together with the simulated learning curve (dotted line) of the corre- sponding ideal filter, with a Wiener error Jmin = 0.004 V2. Very close agreement between measurement and simulation can clearly be seen in both experiments. It can be concluded that our SC AFLMS, built with discrete components, fits the theoretical AFLMS very well for Wiener errors larger than errors caused by circuit nonidealities (in our case 12 pV2).

In [4] software tools for the automatic generation of chip layouts are described that permit the design of the adaptive SC filters presented in this paper as CMOS integrated circuit chips. Realizations of circuit chips for the assembly of filter-tap and adaptation modules are presently in preparation.

IX. CONCLUSION

The SC realization of an adaptive FIR filter based on the LMS algorithm has been presented. An error analysis shows that all linear errors in the FIR filter part can be eliminated by an additional constant tap element. However,

8Note that in our breadboard circuit the input signal variance was 1 V2

)I

Fig. 16. Completely offset-compensated first-order adaptive SC filter.

I.OE-5 L yy-w-M+*d ‘I _*(__~_,. 0 200

Fig. 17. Measurement of the first-order SC AFLMS in a system-identification application. 1: Jmin = 0.01 V2.2: Jmin = 0.004 V2.3: Jmin = 0.001 V2. 4: Jmin = 0 V2.

IO 50 90

Fig. 18. Measurement (solid line) and simulation (dotted line) with Jmin = 0.004 V2.

offset errors, which occur in the multipliers and integrators of the adaptation part, lead to a deterioration in’ the filter performance. By using compensation schemes for both the multiplier and the integrator offset voltages, these errors can also be eliminated. The accuracy of the compensated adaptive filter is determined by the multiplier nonlinearity. Simulations of a two-tap adaptive filter show that a normalized average MSE of better than -40 dB can be obtained, if a multipher THD of 0.5% is assumed for both input nodes. Compared to their digital counterparts, the accuracy of adaptive SC filters is lower, if the coefficient bit length of the digital realization is


sufficiently long. Thus, the realization of adaptive filters based on the LMS algorithm is advantageous in those- applications, where the errors of the algorithm (Wiener error and excess mean-squared error) is larger than the errors caused by the circuit nonidealities. In such cases the SC realization has the advantage of lower power consumption and lower circuit complexity.

HI

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REFERENCES

S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: Prentice- Hall, 1986. Y. Hague, “An adaptive transversal filter,” in Proc. IEEEICASSP, 1983, pp. 1208-1211. T. Enomoto, et al., “Monolithic analog adaptive equalizer integrated circuit for wide-band digital communication networks,” IEEE J. Solid State Circuits, vol. SC-17, no. 6, Dec. 1982. A. Muralt, P. Zbinden, and G. S. Moschytz, “CAD tools for the synthesis and layout of SC files and networks,” Analog Integrated Circuits and Signal Processing, vol. 3, 1993, pp. 229-242. U. Menzi, “Switched-capacitor circuits for estimation algorithms used in adaptive filters and MFSK receivers” (in German), Ph.D dissertation 9600, ETH, Zurich, Switzerland, 1991. P. Allen and E. Sanchez-Sinencio, Switched Capacitor Circuits. New York: Van Nostrand Reinhold, 1984. Z. Wang, Current-Mode Analog Integrated Circuits and Linearization Techniques in CMOS Technology, series in microelectronics. Zurich: Hartung-Gorre Verlag, 1990. R. Gregorian and G.C. Temes, Analog MOS Integrated Circuits. New York: Wiley, 1986, ch. 7. K. Nagaraj, “Switched capacitor delay circuit that is insensitive to capacitor mismatch and stray capacitance,” Electron. Lett., vol. 20, pp. 663-664, 1984. B. Furrer, “Noise in switched-capacitor filters” (in German), Ph.D. dissertation 7284, ETH, Zurich, 1983. J. Pena-Finol, et al., “A MOS four-quadrant analog multiplier using quarter-square technique,” IEEE J. Solid-State Circuits, vol. SC-22, Dec. 1987. B. Widrow, et al., “Adaptive noise cancelling: Principles and applications,” Proc. IEEE, vol. 63, no. 12, Dec. 1975. R. Gillingham, “Stray-free switched capacitor unit-delay circuit,” Elec- tron. Left., pp. 663-664, 1984. K. Haug, G. C. Temes, and K. Martin, “Improved offset compensation schemes for SC circuits,” in Proc. IEEE ISCAS, Montreal, 1984, pp. 1054. K. Nagaraj, et al., “Switched-capacitor integrator with reduced sensitivity to amplifier gain,” Electron. Lett., pp. 1103-l 105, 1986. Q. Huang and G. S. Moschytz, “Multiplierless analog LMS adaptive FIR filters,” to be published. C. Caraiscos and L. Bede, “A roundoff error analysis of the LMS adaptive algorithm,” IEEE Trans. Acoustics, Speech, Signal Process., vol. ASSP-32, no. 1, Feb. 1984.

Ulrich Menzi was born in Beme, Switzerland, in 1959. He received the M.S. and Ph.D. degrees in electrical engineering from the Swiss Federal In- stitute of Technology, Zurich, Switzerland, in 1984 and 1992, respectively.

From 1984 to 1992 he was at the Institute of Signal and Information Processing, Swiss Federal Institute of Technology, working in the field of switched capacitor networks. In 1992 he joined Ascom-Ericsson-Transmission, Beme, where he is engaged in HDSL data transmission.

George S. Moschytz ( -F’OO) received the E.E. Diploma in 1958 and the Ph.D. degree (on optical scanning of code-addressed envelopes) in 1962, both from the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland.

From 1960 to 1962 he was with RCA Laborato- ries in Zurich where he worked on envelope-delay measurement techniques for the transmission of color TV signals and on conversion techniques for NTSC, PAL, and SECAM TV signals. From 1963 to 1972 he was with Bell Labs, Holmdel, NJ, where

he developed and later supervised methods of designing hybrid-integrated active RC filters, phase-locked loops and oscillators, as well as silicon- integrated logic circuits and modulators for use in data transmission equipment and modems. Since 1973 he has been Professor for Network Theory and Signal Processing and Director of the Laboratory for Signal and Information Processing, Swiss Federal Institute of Technology (ETH), Zurich. He has authored many papers in the field of network theory, active and switched- capacitor filter and network design, and sensitivity theory, and holds several patents in these areas. He is the author of Linear Integrated Networks: Fundamentals (New York: Van Nostrand Reinhold, 1974), Linear Integrated Networks: Design (New York, Van Nostrand Reinhold, 1975), the coauthor of Active Filter Design Handbook (New York: Wiley, 1981), and editor of MOS Switched-Capacitor Filters: Analysis and Design (New York, IEEE Press, 1984). His present interests are digital, switched-capacitor, and adaptive filters, neural networks for signal processing, and the application of signal processing techniques to medical problems (electromyography and hearing aids).

Dr. Moschytz is President of the IEEE Swiss Chapter on Digital Commu- nication Systems and a member of the Swiss Electrotechnical Society. From 1981 to 1982 he was President of the Swiss Section of the IEEE. He is also the Swiss representative on the Commission for the Development of European Science and Technology (CODEST) in Brussels. He is an elected member of the Swiss Academy of Engineering Sciences, winner of Best Paper Award (for a paper on active filter design using tantalum thin-film technology), and a member of the Eta Kappa Nu Honor Society.

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