IEEE TRANSACTIONS ON ACOUSTICS. SPEECH, AND SIGNAL PROCESSING. VOL. 38, NO. 2. FEBRUARY 1990 253

Control-Theoretic Design of the LMS and the Sign Algorithms in Nonstationary Environments

c. P. KWONG, SENIOR MEMBER, IEEE

Abstract-The feedback structure of the LMS algorithm proposed by Widrow et al. [l] is reexamined from a control system design view- point. The minimization of the misadjustments due to gradient noise and lag can then be recasted as the disturbance rejection and tracking R ( S ) ky-w problems in control. A frequency-domain approach to the latter prob- lems is presented, which has the advantages of transparency, ease of computation, and generality compared with the time-domain approach previously used. With the same set of assumptions of white input and a Markovian plant, it is shown that the optimum step size obtained by the present approach is identical to that obtained in [I]. Applying the same approach, the optimum step size of a simplified version of the LMS algorithm-the sign algorithm-is derived for the case when the

~ i ~ . 1, A feedback control

Disturbance VIS)

I

Measurement Noise w ( s )

system with disturbance and measurement noise.

- - plant is slowly varying and the input signals are Gaussian.

We can write (1.1) as

I. INTRODUCTION Y = S V - T W + T R . (1.4) N control literature, the most widely studied problem I is the analysis and design of feedback control. Fig. 1

shows a very typical situation that illustrates the formu- lation and the main philosophy of tackling many real problems.

For simplicity, we assume that the controlled process, usually called the plant, and the controller are linear time- invariant, continuous systems modeled by their respective transfer functions P ( s ) and C ( s ) . The transformed com- mand and output signals are denoted as, respectively, R (s) and Y (s) as shown in Fig. 1. In a tracking problem, the output is required to follow the command closely and rapidly. However, to realize this goal, we will face prac- tical difficulties such as the presence of measurement noise. There may also be disturbance affecting the system performance. (The plant uncertainty, another form of de- terioration, is also commonly modeled as disturbance.) In Fig. 1, both the measurement noise and the disturbance are included in the feedback control model.

The output Y ( s ) can be readily written as (omitting the understood variable s)

(1.1) Y = ( V - P C W + PCR)( 1 + PC)-I.

Let

S = (1 + PC)-’ (1.2) and

T = PC( 1 + P C ) - ’ . (1.3)

Manuscript received May 3, 1988; revised April 4, 1989. The author is with the Department of Information Engineering, Chinese

IEEE Log Number 8932774. University of Hong Kong, Shatin, N.T., Hong Kong.

Observe that both S and T should be “small” to reduce the effects of V and W, which constitutes a commonly called disturbance rejection problem. S , the sensitivity function, can be made small by increasing the loop gain PC, which also results in an approximately unity (trans- mission function) T. It is seen that a unity transmission function will take the command directly to the output, and hence large loop gain can reduce the disturbance and at the same time solve the tracking problem. However, the measurement noise will then appear without attenuation at the system output. In fact, it is easily seen that S and T are subject to the constraint

S + T = l . (1.5)

It is therefore obvious that, besides the well-known fact that the loop gain cannot be arbitrarily large because it causes instability, its relationship with noise attenuation must be taken into consideration.

Thus, it is not surprising that the above tradeoff prob- lem has such a predominant position in the development of feedback control. The study of this problem has a long history and it is impossible to survey here even briefly the different approaches and solutions. However, the follow- ing approach, known as the statistical design method, has direct relevance to our present investigation.

The statistical design method [2] has its origin as Wiener’s optimal filtering theory [3]. In this method, the disturbance and the measurement noise are modeled as independent random processes with power spectra, re- spectively, !€)v( o) and cPw( o). Define an error

E = S V - T U r . (1.6)

0096-35 18/90/0200-0253$01 .OO O 1990 IEEE

254 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING. VOL. 38. NO. 2. FEBRUARY 1990

Then E is also a random process and its power spectrum can be proved to be

As a design criterion the integral of + P E ( ~ ) over all fre- quencies

which is equivalent to the mean square value of the error, is minimized. Given a particular form of the controller, the minimization leads to an optimization of the control- ler's parameters.

In Section 11, we will show how the optimization of the Step size in the LMS algorithm for adaptive identification can be rephased as a statistical design problem for control systems. A design criterion, similar to (1.8), is used. Be- cause of the existence of a simple computing algorithm, it is shown that the optimization of the step size can be more easily handled than the previously used time-domain approach [l]. The optimum step size for the LMS algo- rithm is derived for white input using the present design criterion, and the same optimum step size is obtained. However, another expression for the optimum step size is also derived by using less restrictive assumptions. In Sec- tion 111, the basic approach is applied to the sign algo- rithm (SA) which is a simplified version of the LMS al- gorithm. Although the analysis in this section is highly simplified, it does provide useful preliminary results to- ward a more complete theory. A comparison of the per- formance of the LMS and the sign algorithms is then per- formed using the obtained results.

11. STATISTICAL DESIGN OF THE LMS ALGORITHM A feedback control structure (Fig. 3) has been proposed

in [ l ] for the analysis of the LMS algorithm

in identifying an unknown time-varying FIR plant (Fig. 2) . In [l] and the present study, it is assumed that the input data ak and the measurement noise uk are samples of stationary random processes. The variances of ak and uk are, respectively, a: and U:. Moreover, the measure- ment noise U, is also white and is independent of ak. How- ever, the desired response dk = gk * a, + uk is nonsta- tionary due to the time variation of the plant weight vector gk. Hence, the identification problem has commonly been referred to as nonstationary. To simplify our analysis, we have assumed that ak is white, so that it is sufficient to consider only one of the decoupled single-input single- output feedback systems. In Fig. 3, gk is the ith compo- nent of the optimal weight vector gk at the kth instant; ck is the ith component of the filter's weight vector at the kth

Unit Celays

Noise uk

Fig. 2. Adaptive identification of an unknown time-varying plant

- I ---i

Fig. 3. Feedback control structure for the analysis of the LMS algorithm.

instant; the subscripts i are omitted for convenience. The feedback structure of Fig. 3 models the relationship be- tween the true gradient Vk and the tracking error, and how this error is caused by the propagation of the gradient noise vI. Note that the feedback model is essentially the same (with gk replaced by g ) for the stationary case when the plant weight vector is fixed. The model is time invariant for both cases since the autocorrelation matrix of ak is constant even as gk varies.

The gradient noise in the LMS algorithm is seen to be

v, = b, + V , = 2rkak + V,. ( 2 . 2 )

Taking the z-transform of the variables in Fig. 3, we ob- tain a feedback control system (Fig. 4) for the LMS al- gorithm which has a structure similar to Fig. l . The two governing transfer functions are denoted as

2a:K 2 - 1

L ( z ) = -

and

(2.3)

K D ( z ) = -

2 - 1 (2.4)

Clearly, G ( z ) and C ( z ) , the transformed plant weight and the transformed filter's weight, can be regarded as, respectively, the transformed command and output sig- nals in control terminology. Furthermore, the trans- formed gradient noise, V ( z ) , can be considered as dis- turbance, which is coupled to the system output via D ( 2 ) . It is thus easily conceived that the performance analysis of the LMS algorithm may be formulated in the language of control. In particular, the statistical design principle described in Section I may be employed.

KWONG: CONTROL-THEORETIC DESIGN OF LMS 255

G ( z ) E ' z ) ;

- > C ( Z ) 2oiK 1

2-1

1 " 2lr --*

= - I T(e'Q)12+G(e") d Q Fig. 4, Transform-domain representation of Fig. 3.

The output C ( z ) can be readily written as

L D l + L l + L e = - G + - V.

From the theory of discrete systems, we realize immedi- ately that ck is bounded for bounded gk and vk, if and only if L / ( 1 + L ) and D/( 1 + L ) have their poles lying in- side the unit circle. The required boundedness, or the bounded-input bounded-output (BIBO) stability of the system, is then achieved if the root of the polynomial z - 1 + 2a:K lies inside the unit circle, or

-

which is a well-known result [ 13. It is also obvious from Fig. 4 that the only adjustable

parameter, the step size K, plays a key role in the system performance. K now appears as the open-loop gain of the control system, and at the same time the gain of the trans- mission from the disturbance to the system output. Whereas a large open-loop gain will generally result in a fast tracking from control theory, as the gain for the dis- turbance, a larger value of K means a higher degree of deterioration on the system performance due to U,. Hence, the tradeoff involved in the choice of the step size is clear. We use, in the following, the statistical design approach to optimize such choice.

The error signal E ( z ) is given by

E = G - C

1 D l + L l + L

G - - V = TG - SV (2.7) =-

where we have used (2.5) and defined

(2.10)

In practice, the gradient noise vk is nonstationary and de- pendent on gk [cf. (2.2)], but may be considered approx- imately stationary and independent on gk when ck I gk. In (2.10), + ' G ( ~ ) and +,,(z) are, respectively, the power spectra of g, and vk; I' is the boundary of the unit circle in the z-plane; and 0 = UT, where Ts is the sampling period. Then, one way of applying the statistical approach of control system design [e.g., the minimization of (1.8)] to the adaptive algorithm is to minimize the mean square error ME given by (2.10). From (2.10) we see that ME can be computed numerically for a range of the step size K given the power spectra of gk and uk. Then the optimum value of K can be determined in principle from the plot of ME against K. From the properties of +G and of stationary gk and uk, we can write

+&) = PG(Z) P G ( z - ' ) (2.11)

+ V ( 4 = P v ( z ) Pv(Z-7 (2.12)

and

where PG(z) and P V ( z ) are ratios of algebraic polyno- mials. Then the contour integrals in (2.10) can be easily computed by an simple algorithm using the coefficients of the transfer functions T ( z ) PG(z) and S(z) Pv(z) [41. This method of computation will be illustrated for both the LMS and the sign algorithms.

It is interesting to obtain I T( e j* ) l 2 and I S( e/* ) l 2 in (2.10) and see how they vary with the value of the step size K. Fig. 5 shows some of these plots from which we observe that T is a high-pass filter, and S is a low-pass filter. That means MG, which accounts for the tracking error, will be small if the variation of gk is slow (so that +G is a low-pass spectrum of narrow bandwidth). On the

and

other hand, -the high-frequency components of the dis- turbance will be attenuated by S. From Fig. 5 we can also see that increasing K will reduce the magnitude of 1 T( e / * ) 1' at low frequencies, but increase the bandwidth of I S( e/*) 1 2 . Thus, the need of trading off the tracking

- - 2 - 1 (2.8) 1 T=-

l + L z - l + 2 u : K

- K and disturbance rejection performances-by optimizing k is once again seen in the frequency domain context.

Using the same set of assumptions used in [l] , we de- rive in the following the optimum step size for the LMS algorithm using the approach just developed. Although

- D S=- 1 + L z - 1 + 20$' (2.9)

Note that T and S in (2.8) and (2.9) are not the same as in (1.3) and (1.2).

256 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. 38. NO. 2 . FEBRUARY 1990

0 = 1

Fig. 5 . Plots of 1 T(e ’* ) I’and I S(e’’)12 for (a) K = 0.05, (b) K = 0.1, (c) K = 0.2, and (d) K = 0.3.

the same value of optimum K will be obtained, which im- plies the equivalence of the time-domain approach (used in [I] and in other previous studies) and the frequency approach (used in the present study), we stress that the later approach has its advantages of transparency, ease of computation, and generality (for instance, we will use this approach to deal with the sign algorithm in Section 111).

As in [ l ] , we assume that gk is a first-order Markov process with memory constant a. It is well known that gk can be generated by passing a white noise with variance a 2 to a first-order filter with transfer function z / (z - a). Then

a2 ( 2 . 1 3 ) 2 2-’ z - az- ’ - CY @ G ( Z ) = - ~

and hence we can write

with reference to (2 .11) . Furthermore, we have

T ( z ) pG(z>

- ( z 2 - 2 ) . - 2’ + (2a iK - - l ) ~ + (a! - 2 a ~ : K ) ’

( 2 . 1 5 )

By using the algorithm derived in [ 4 , Table I11 in its Ap- pendix] for the contour integrations in ( 2 . l o ) , which in- volves evaluating the ratio of two determinants, we obtain the mean square tracking error

2a2 ( 1 - a!r,,)(l + ar,, + a + r,,)

MG = ( 2 . 1 6 )

where we define

rp = 1 - 2u:K. ( 2 . 1 7 )

For the mean square disturbance error, if we make the gk, the gradient noise (2 .2) reduces

vk = 2ukak. ( 2 . 1 8 )

Then, from the basic assumptions that uk and ak are white and independent, vk is a white process with constant power spectrum

@v = E [ v i ] = 4u;a:. (2 .19 )

Using the same algorithm for evaluating M G , we obtain

approximation ck to

( 2 . 2 0 )

By plotting MG and MV against K using (2 .16) and (2 .20) , K can then be chosen such that both errors are satisfac- torily small. Another criterion is to minimize MG + My.

If we assume that a P 1 so that gk is slowly varying, (2 .16) reduces to

( 2 . 2 1 ) . uL - - U L

( 1 - r p ) ( 1 + r,,) MG =

4u:K( 1 - a : K ) ’

A further assumption of small step size, such that 1 - U ~ K I 1 , gives

( 2 . 2 2 )

Differentiating (2.22) with respect to K and setting the result to zero gives the optimum step size K ,

K , =(T. (2 .23 ) 2% @U

Note that the result is identical to that obtained in [ I ] for the minimization of “misadjustments due to lag and gra- dient noise” using the same assumptions a! 1 and 1 - a:K I 1. However, as mentioned above, the present fre- quency-domain approach has several advantages over the time-domain approach used in [ I ] .

In obtaining K,, the assumption 1 - a;K I 1 can be removed by using the sum of (2 .20) and (2.21) rather than the approximate expression (2 .22) . I Proceeding in this way, that is, differentiating the sum of (2.20) and (2.21) with respect to K and setting the result to zero, we obtain an equation for the optimum K

4 a i a t K i + 2a2a:K, - u2 = 0 . ( 2 . 2 4 )

The positive solution is

a J a 2 a ; + 40; - a2a, 4a, a:

K , = ( 2 . 2 5 )

For comparison, we plot in Fig. 6 the optimum step sizes obtained by (2 .23) and (2 .25) . It is seen that, for the given uu and S I N , the difference between the two optimum step sizes increases as U increases.

‘The author is grateful to an anonymous reviewer who suggested this approach.

1'

KWONG: CONTROL-THEORETIC DESIGN OF LMS

a

Fig. 6. Comparison of the optimum step sizes obtained by (2.23) and (2.25).

111. STATISTICAL DESIGN OF THE SA To reduce the complexity of the LMS algorithm is one

of the recent research topics in the area of adaptive filter- ing [5]-[8]. The main idea is to quantize the input data or the modeling error. The latter approach leads to the so- called sign algorithm (SA).

There has been no study on the design of the SA in nonstationary environments. However, we will show in the following that the statistical design approach can be used to obtain approximate design equations for slowly varying identification problems.

The adaptation equation of the SA is

which is different from that of the LMS algorithm. Fur- thermore, the true gradient and the gradient noise for the two algorithms are also different from each other because of the two distinct error criteria being used (the least mean- square criterion in the LMS and the minimum mean-ab- solute criterion in the SA [9]). Hence, Fig. 3 does not directly apply to the SA. A modification is proposed in the following so that the SA can also be analyzed within a control framework. The key idea is to derive a relation- ship between the gradient vectors of the two algorithms based on some assumptions.

The gradient vector for the SA is given by the deriva- tive of the mean-absolute error E 1 dk - c - ak 1 with re- spect to c [9]

where dk = g ak + uk is the desired response. For the LMS a1gorithm;the gradient vector is given by the deriv- ative of the mean-square error E [ (dk - c - a$] with respect to c [ 13 :

257

A simple relationship between VsA and V L M s can be es- tablished if dk and ak are zero mean and jointly Gaussian. Based on a lemma proved in [ 101, we can write, with the said assumption,

r

(3.4)

where rk = dk - c square error given by [l]

ak. However, ~ [ r i l is the mean-

~ [ r ; ] = a: + a:(c - g ) ( C - g ) . ( 3 . 5 )

Hence, (3.2) can be written as

VLMS . (3.6) vSA = & J a; + uf(c - g ) - (c - 8 )

If we make a further assumption that c is closed to g'so that U; >> a:(c - g) * ( c - g ) , then (3.6) simplifies to

(3.7)

Note that the above relationship between Vs, and VLMs is derived using a fixed optimal weight vector for sim- plicity (in other words, dk is assumed stationary), other- wise (3.4) cannot be applied and the analysis will become very difficult. The assumptions used to attain (3.7) are quite strong. Nevertheless, this simple relationship will be useful to the analysis and design of the SA when the unknown plant is varying so slowly that ck = gk and dk is almost stationary are both valid approximations. It is hoped that the present simplified study could contribute to the construction of a more complete analysis.

By using (3.7), a control system structure similar to Fig. 4 can be constructed for the SA. This is shown in Fig.7. The governing transfer function L is given by

(3.8)

(3.3)

and the transfer function D is the same as that for the LMS algorithm. Accordingly, the transfer functions T and S for the statistical design of the SA are given by

and

s = - - D - K (3.10) l + L z - l + P a f K

where we have defined P = ( - ) / U , , .

Similar to the BIBO stability criterion (2.6) for the LMS algorithm, the SA is BIBO stable (under the Gaussian as-

258 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING. VOL. 38, NO. 2. FEBRUARY 1990

I

Fig. 7 . Feedback control model for the analysis of the SA.

sumptions of dk and ak) if the root of the polynomial t - 1 + PuiK lies inside the unit circle, or

r 2 - - > > K O . J; 2 (3.11)

This is compared to the sufficient condition for the con- vergence of the SA derived in [ 1 1 1 (with uk being Gauss- ian )

r dr2 > K > 0. (3.12)

The gradient noise in the SA is seen to be

vk = sgn (rk)ak + VLMS. (3.13)

By using the approximation ck = &, we may neglect the last term on the right-hand side of (3.13). Furthermore, the term sgn ( r k ) is assumed to be white and independent on ak. Then vk is a white process with constant power spectrum

9v = E[v:] = U?. (3.14)

uJ3 = 1 SIN - 30 dB

(I = 0.99

0 - 0.01

It will be shown by computer simulation that (3.14) is a valid approximation. Given (3.14), MY is calculated as before and is given by

(3.15) K

2P(1 - fPuiK)' MV =

3F1 n o s 0 1 0 1 5 0 2 0 2 5 c K~~~

(b) Flg. 8. Plots of MF for (a) the SA, and (b) the LMS algonthm.

Using the same assumptions in Section I1 that gk is a first-order Markov process with memory constant a z 1 , we obtain the mean square tracking error

It is interesting to compare the performances of the SA and the LMS algorithm. For simplicity, we further as- sume, as for obtaining (2.22), that K in (3.15) and (3.16) is so small that 1 - (Pu:K/2) z 1 . Then we can write

(3.17)

3

Differentiate (3.17) with respect to K and setting the result to zero gives the optimum step size

which is compared to the K* of the LMS algorithm (2.23). If we substitute (3.18) into (3.17), and (2.23) into

(2.22), we obtain an interesting ratio:

which is slightly larger than 1 . That means the SA has a performance comparable to the LMS algorithm if the plant to be identified is slowly varying.

KWONG: CONTROL-THEORETIC DESIGN OF LMS 259

gorithm has been derived. An expression that relates the performances of the LMS and the sign algorithms has also been obtained. Computer simulation has been used to support the theoretical results derived for the SA. The proposed approach of analysis and design could be easily extended to a variant of the sign algorithm-the dual sign algorithm [6]-of which the convergence performances have been shown to be superior to that of the sign algo- rithm.

I I aa = I

s/N - 30 dB a = 0.99

b ‘\ ! .

a = 0.01

i, t, t ‘, \ -,

Fig. 9. Comparison of the theoretical and simulation results for the SA

As an illustration, we plot, for the two algorithms, ME = MG + MY (using the expressions without the small step size assumption, i.e., (2.20)-(2.21) and (3.15)-(3.16) for accuracy) against the respective step sizes using the same set of data (Fig. 8). We see that the optimum KSA is about 0.0084 and the corresponding ME is 4.4 X On the other hand, the optimum KLMs is about 0.132 and the cor- responding ME is 3.66 x The ratio of the two ME’s is quite close to that given by (3.19) (which involves the small step size approximation).

The main results derived in this section for the SA, given by (3.15) and (3.16), are checked by computer sim- ulation based on the same set of assumptions on the input data, measurement noise, and the characteristics of the time-varying plant. In the simulation tests, 30 OOO sam- ples are used to calculate the mean square error ME for each given value of the step size K. The result is plotted in Fig. 9 together with the theoretical curve for compari- son. It is seen from the closeness of the two curves that the theory proposed here does provide valuable informa- tion for the analysis and design of the SA.

IV. CONCLUSIONS We have presented in this paper a control-theoretic

viewpoint to the design of gradient-type algorithms for the adaptive identification of time-varying systems. Spe- cifically, a frequency-domain method is proposed for the optimization of the step sizes of the LMS and the sign algorithms, which has several advantages over the previ- ously used time-domain approach. Furthermore, an im- proved estimate of the optimum step size of the LMS al-

ACKNOWLEDGMENT The author is grateful to the anonymous reviewers for

their constructive criticism, especially to the one who pointed out a major mistake which appeared in the first manuscript.

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[7] V. J. Mathews and S. H. Cho, “Improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 450- 454, Apr. 1987.

[8] W. A. Sethares and C. R. Johnson, Jr., “A comparison of two quan- tized state adaptive algorithms,” IEEE Trans. Acoust., Speech, Sig- nal Processing, vol. 37, pp. 138-143, Jan. 1989.

[9] A. Gersho, “Adaptive filtering with binary reinforcement,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 191-199, Mar. 1984.

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[ I l l T. A. C. M. Claasen and W. F. G. Mecklenbriiuker. “Comparison of the convergence of two algorithms for adaptive FIR digital filters,” IEEE Trans. Circuits Sysr., vol. CAS-28, pp. 510-518, June 1981.

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C. P. Kwong (S’81-M’82-SM’89) was born in Hong Kong. He received the M.Sc. degree in communications in 1977 from the Loughborough University of Technology, England, and the Ph.D. degree in control engineering in 1982 from the Chinese University of Hong Kong, Hong Kong.

In 1982 he joined the Department of Electronic Engineering of the Hong Kong Polytechnic. From 1984 to 1989 he was a Lecturer in the Department of Electronics of the Chinese University of Hong Kong. He is now a Senior Lecturer in the Depart-

ment of Information Engineering at the same university. His research in- terests are in the areas of signal processing and control theory.