322 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 1, JANUARY 1995

The e, = € ( e ) which realizes the optimum leads to (31), and then (36) leads to (32).

Taking the supremum over zn, in ( 3 3 , we obtain 2 , = Ch,, (0,) and the probability of the hest path arriving at e, at time n is

l o g ~ , ( e , ) = q n ( e n ) + hen(O,)

which leads to (33). Comments: 0 The storage requirements of this algorithm are still

reduced: for smoothing it will be N state pointers (as in Viterbi algorithm) and A’ vectors.

0 We can model a signal whose law is a mixture of linear models: each element of the mixture will be indexed by a pair ( e , e’) and Cec , (u ) will be (with some abuse of notation)

where p ( e. e ’ ) is the probability of the transition from e to e’. As before, functions K(z) will be quadratic forms zTVez and 0 ( y ) is a matrix product 0 y . If we drop indices in the ( e , e ’ ) -dependent linear model (Fe,,, Q e , , , H , , , , Re , , ) , we have to identify the likelihood (27) to

-(z’ - F X ) ~ Q - ’ .r’ - F x ) / 2 - (y’ - H z ’ ) ~ R - ’ ( ~ ’ - H x ’ ) / 2 ((

and this leads to

H ~ R - ’ H + Q-’) = 2ATUe:,,,A + 2L:t

F ~ Q - ‘ F = 2ue, , - 2ve

F ~ Q - ’ + Q - ~ F = U , , ~ A + A ~ U , , ~

R-’ = 0.

With those notation the process may be described in the following way: starting with a discrete state (e, z), the process jumps to another one e’ with probability p ( e , e ’ ) , and a new state x’ is chosen with the dynamics of (Fe ,! , Q e e , ) and an observation y’ is then produced with z‘ and ( H e , , , R e e t ) (cf (1)). Setting e = e’ in the equations above (steady state), we see the principal restriction of this mode: the matrices R - ’ H ( = 0) and P+F*PI’(= A ) are independent of e.

REFERENCES

[ I ] L. R. Rabiner and B. H. Juang, “Introduction to hidden Markov models,” ZEEE ASSP Mag., Jan. 1986.

[2] T. Kailath, “A view of three decades of linear filtering theory,” ZEEE Trans. Inform. Theory, vol. IT-20, no. 2, Mar. 1974.

[3] A. Gelb, Applied Oprimal Estimation. Boston, MA: M.I.T. Press, 1989. [4] B. Jamison, “Reciprocal processes,” Zeit. Wahrsch., vol. 30, pp. 65-86,

1974. [5] A. J. Krener, R. Frezza, and B. C. Levy, “Gaussian reciprocal processes

and self-adjoint differential equations of second order,” Stochastics and Stochastic Reps., vol. 34, pp. 29-56, 1991.

[6] B. C. Levy, R. Frezza, and A. J. Krener, “Modeling and estimation of discrete-time Gaussian reciprocal processes,’’ IEEE Trans. Automat. Contr., vol. 35, no. 5, pp. 1013-1023, 1991.

[7] A. J. Krener, “Reciprocal diffusions and stochastic differential equations of second order,” Srochasrics, vol. 24, pp. 393424, 1988.

[8] B. C. Levy, A. J. Krener, “Dynamics and Kinematics of Reciprocal Diffusions,” J. Math. Phys., vol. 34, pp. 1846-1975, May 1993.

[9] M. Mariton, Jump Linear Systems in Automatic Control. New York: Marcel Dekker, 1990.

[ I O ] G. D. Forney, “The Viterbi algorithm,” Proc. ZEEE, vol. 61, no. 3, Mar. 1973.

Echo Canceler Performance Analysis in Four-Wire Loop Systems with Correlated AR Subscriber Signals

John Homer, Member, ZEEE, and Iven M. Y. Mareels, Senior Member, ZEEE

Abstract-By using a simple example we illustrate the effect of auto- correlation and cross correlation of subscriber signals on the achievable performance of adaptive echo cancelers in a four-wire telephone network.

Index Terms-LMS adaptation, FIR filter, bias, averaging, convergence rate.

I. INTRODUCTION

Adaptive echo cancelers are used in four-wire loop telephone net- works to suppress the effects of echoes. The commonly used double echo-canceler (DEC) system with an LMS adaptive FIR echo canceler placed at each end of the four-wire loop-Fig. 1-typically performs well. However, poor performance, such as slow or inadequate echo cancelation, and bad behavior, such as signal bursting, has been observed [l]. Such behavior has been linked to the presence of correlation within and between the subscriber signals [l], [ 2 ] . One approach is to quantify this link so that line-coding schemes can be used confidently to control the subscriber signal correlation levels and enhance the performance of the DEC system [3].

In this light, various authors [2], [3] have determined bounds on the correlation levels of the subscriber signals within which good echo cancelation is guaranteed. The bounds, however, only identify sufficient conditions for good performance. The aim of this communication is to derive an explicit equation relating asymptotic performance of the DEC system to the cross-correlation and auto- correlation levels of the subscriber signals. Although a number of simplifying assumptions are made, the results indicate the necessity of reducing signal correlation levels if good echo cancelation is to be achieved.

Consider the DEC system of Fig. 1, with subscribers 1 and 2, at either end of the network, sending signals SI and s2 and receiving

Manuscript received October 27, 1993; revised May 20, 1994. The activ- ities of the Cooperative Research Centre for Robust and Adaptive Systems (CRCRASys) are funded by the Australian Commonwealth Government under the Cooperative Research Centres Program.

J. Homer is with the Department of Electrical and Computer Engineering, University of Queensland, St. Lucia, Qld. 4067, Australia.

I. M. Y. Mareels is with CRCRASys and the Department of Systems Engineering, Research School of Information Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia.

IEEE Log Number 9406686.

0018-9448/95$04.00 0 1995 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. I , JANUARY 1995 323 ,

local j ; local 2-wireline ; central 4-wire loop facility I 2-wueline

// Y1.k

Y2k

Fig. 1. Double echo canceler four-wire loop network.

signals y2 and y ~ , respectively. Echoes in such a four-wire loop network occur as a result of part of the received signal y l or ya leaking through the hybrid at the receiver's end and into the opposite transmission channel. In Fig. I , the echo path passing through the hybrid nearest subscriber 1 or 2 is parametrized by H1 or 02,

respectively. Echo cancelers, parametrized by el and &, sit inside the four-wire loop and next to the hybrids. Considering the end of the four-wire loop near subscriber 1 , the echo canceler samples the signal y2 in the receiving channel of the loop and outputs a signal r l , which supposedly replicates the echo passing through the neighboring hybrid. The replica, T I , is then subtracted from the echo containing signal IC1 = e l + SI in the transmitting channel.

The performance of an adaptive echo canceler is measured by its ability to produce an accurate replica of, and hence suppress, the echo passing through the neighboring hybrid. This ability is adversely affected by the tendency of the echo canceler to interpret any part of the transmitting channel signal (e.g.. ([>I), that is correlated with the receiving channel signal (e.g., y ~ ) , as being part of the echo. This results in the DEC system performing poorly when the subscriber signals are cross-correlated and, because of the inherent feedback structure of the four-wire loop, when the subscriber signals show broad autocorrelation functions.

We consider a simplified model (an example) of a DEC four- wire loop system in which the subscriber signals are modeled as first-order filtered white noise. After developing the equations which describe the dynamics of the DEC system, averaging theory is utilized to describe the adaptive system's behavior. The simplicity of the example allows us to obtain explicit analytic expressions quantifying the effect of cross correlation and autocorrelation on the achievable performance.

The communication is organized as follows. In the next section we introduce our example, and state the assumptions made on the signals entering the loop. Initially, the subscriber signals are assumed to be of equal power. Then we obtain an approximate system description, average out the time variations due to the subscribers' signals, and analyze the simplified system. A discussion of the adaptive system's performance follows. Finally, we briefly explore the case in which the subscriber signals are of unequal power.

11. SYSTEM DESCRIPTION Consider the DEC network shown in Fig. 1. All signals are

sampled. At sampling instant k subscriber 1 (2) sends signal SI. I, ( s z , ~ ) while receiving signal y ~ , k ( y 1 . k ) . The echo e l , k ( e 2 . k )

resulting from signal leakage through the hybrid HI (H2) is generated by a single delay-time-invariant FIR filter with tap coefficient H I (02).

The echo canceler EC1 (ECZ) produces the replica T I , k ( r 2 , I C ) of

+ S2k+e2,k

the echo vi? a single delay FIR filter with adaptively adjusted tap coefficient 0 1 , k ( O Z . k ) . The LMS algorithm is used for adaptation.

We make the following assumptions regarding the four-wire loop network.

Assumption I: Each echo path is attenuating, that is

Assumption 2: There is no transmission delay in the channels of the four-wire loop.

Remark: 1) Assumption 2 is made to simplify the mathematics. An extension of the work to include transmission delay within the four-wire loop channels would be straightforward.

The received signals are given by

Yl, k + l = S i . k + i + 61. k + i Y 2 . k Y I , o = 2/10

Y2. k+1 = s 2 . k + l + 0 2 . k+l Yd. k Y Z , 0 Y20. (2)

8,. k = 8 , -6'. k is the ith residual echo parameter or parameter error. Application of the LMS algorithm (to minimize the mean of the

squared error (echo containing) terms, yf, k , i E (1, 2)) leads to the following update equations for the tap coefficients in the echo cancelers:

0t , k+l = o t > h + {"Y,, k + l Y ~ . k

The term p is the update stepsize. It is a small positive constant. The initial estimates are set to zero, which corresponds to assuming that there is no echo.

Rewriting (3), in terms of the residual echo parameter, gives

H l . k + l = 8 1 , k - ~ ~ y t . k + l Y 2 , k 81,O = @ I

0 2 . k + l = 0 2 . k - P Y 2 . k + l Y l , k 0 2 . 0 = 82. (4)

Equations ( 2 ) and (4) describe the dynamics of the adaptive DEC system driven by external signals s l , k , sa, k . It is a nonlinear time- varying fourth-order recursion with driving signals SI, k and s ~ , k .

In order to characterize explicitly the influence of cross correlation between subscribers signals and the autocorrelation of the subscriber signals on the echo cancelers' ultimate performance we assume that the subscriber signals SI. l i . sa, k may be described as

2 112

2 112

S I . ~ ~ + I = 0 3 1 , k + (1 - a ) wli s 1 , o = 0

( 5 ) 5'2, k+1 = a S 2 , I; + (1 - ( I ) Tr2, k SZ, 0 = 0

where la1 < 1 and 111. k and 112, k are wide-sense-stationary signals and such that the following assumptions hold for i , j = { 1, a}, i # j .

324 IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 41, NO. I , JANUARY 1995

Assumption 3: Bounded signals

( 1 2 8 , k ( < bx v k .

Assumption 4: Zero-mean property

Assumption 5: Well-defined autocorrelation properties

Assumption 6: Well-defined cross-correlation properties

V m , I , M , where JpI 5 1.

The constant C is positive and independent of the integers m, 1, and Af in the above inequalities.

Remarks: 2) The numerator (1 - u2)' / ' in (5) is included so that the signals s1 k, SL k , V I , k r up k have the same power (12 norm).

3) The signals cl, are deterministic in the sense that Assumptions 4-6 do not refer to a probability distribution or to convergence in a probabilistic sense.

The wide-sense stationarity of VI k and V L , k implies that SI, 1. and sz k are also wide-sense-stationary signals. Because of the stability of the filter involved, the subscriber signals s t , k enjoy properties similar to the signals v , k . More specifically, for some positive constant S only depending on a, W , and V

The above inequalities embody all the assumptions on the subscriber signals we need to derive our results.

Equation (8) indicates that the signal parameter a is a measure of the broadness of the autocorrelation and cross-correlation functions of the subscriber signals, e.g., the lag 1 = 13 dB at which the autocorrelation function drops to 3 dB of the zero-lag level is 13 dB = log(2)/log(l/u). On the other hand, (9) indicates that the signal parameter p is a measure of the degree of cross correlation between the subscriber signals.

111. ANALYSIS

Equations (2), (4), and ( 5 ) completely describe the DEC system of interest, which we are now going to analyze subject to Assumptions 1-6 .

The residual echo parameters are essentially slowly time-varying, as y is small (see (4)). The received signals are of a mixed time-scale nature, containing both fast time variations due to the driving signals and slow time variations due to the adaptation mechanism. It is these time-scale separation properties that make analysis tractable.

First we find approximately how the signal y k = (yl. k-, y2. k ) de- pends on the slowly time-varying parameter error 8 k = (81, k , OZ. k ) .

Then we are in a position to use averaging as presented in [4]. In this fashion we obtain a simpler system, the solutions of which, subject to certain conditions, can be made to approximate, arbitrarily closely, those of the original adaptive system (2), (4), and (5).

Let us introduce the function L = ( 2 1 . 2 2 ) + yk(:) = -

( Y l . k ( Z L Yz. k(Z)) via - -

! / l , k + l ( z l * : 2 ) = S l , k + l + z l g 2 , k ( z l - 2'2) y1.0 = y l , O - ! / 2 , k + l ( Z l r 1 ^ 2 ) = S z . k + 1 + ~ 2 ~ ~ , k ( - l , 2 2 ) g2.0 =!/2.0.(10)

By construction, the function jJk ( 2 ) is well-defined and analytic in z E R2 on the domain / ~ I : Z I < 1.

Comparing (10) with (2) it can be seen that j j k ( 6 k ) is a good approximation for Yk. Indeed, ~ ~ / i ( ~ n - ) - y k l = ~ ( p ) for as long as 8, remains in the domain D : I O l , k 8 2 . k I < 0 < 1. (0 is some constant independent of p but possibly dependent on the system parameters.) Because of Assumption 1 and the zero initial parameter estimates of (3), 8, belongs to D at least on a time scale of l / p (i.e.? for all k E [o. L / p ) some L > 0). Replacing yk in (4) by g k ( 8 k ) then yields

on a time scale 1//i. Equation (1 1 ) is in a formLsuit_able for averaging. It is a nonlinear

time-dependent recursion in 8 k . 6 k is clearly slowly time-varying for p small. The time dependence in this recursion enters via the variable yk and may be averaged out to obtain

- - - - 81, k + l = 81. k - p ~ ~ ~ ' y l y % ( l ) ( @ k )

0 2 . k+l = HZ, k - p A L 1 i j Z j 1 ( 1 ) ( 6 1 k )

01.0 = 81

0 2 . 0 = 0 2 - -

(12)

where e, = ( e l , k , % , k ) and

A U ~ ~ B , ( ~ ) ( Z ) = lim vZ, k ( z ) ~ J , k - l ( z ) ,

M-l+m

k=m M - 30

V > 1 i , j E (1, 2) (13)

and gt, k ( ~ ) is as defined by (10). This limit exists (on the domain Izl 22 1 < 0 < I), is independent of m and, moreover, the limiting value is approached uniformly in m , by virtue of Assumptions 4-6.

According to av_eraging theory [4], the solution 3, to (12) ap- proximates 0 k = (el, k , 6'2. k ) (solution to (4)) for sufficiently small

It follows directly from (7)-(9) and the definition of jJk ( 2 ) (see (10)) with l z l zz l < 0 < 1 that 6 ( p ) = O(,,&).

Notice that we also have the approximation (because vk(z) is analytic in z in D )

I B k ( e k ) - yk( = ~(fi), on a time scale 1 / p . (15)

Furthermore [4], if the averaged system has an asymptotically stable equilibrium within the domain D then the approximations can be

325

extended to hold on an infinite time scale

Remark: 4) The strength of the simplifications due to the exploita- tion of time-scale separation properties can be gleaned from (16). Arbitrarily accurate information about the actual behavior of the DEC system as described by (2) , (4), and ( 5 ) which is four-dimensional, nonlinear, and time-varying can be gleaned from a linear second-order system (10) and a nonlinear time-invariant second-order system (12).

Through the application of the averaging approximation results above and analysis of the averaged system equations of (12) we obtain the following result.

Theorem 1: Consider the DEC system described by (2), (4), and (5) . Let Assumptions 1-6 be satisfied. Let 8 E (max(a’. 0:. 0;). 1). Then there exists a positive constant p * ( O , a. p ) such that for all positive p < p*

l e k - 8kl = O(&), vli 2 0

I & ( G k ) - ykl = O(JTT), vli 2 0. (17)

Here $1, is defined in (12) and yk(;) is defined in (10) while yk are the variables of the DEC system.

nentially fast (with rate A) to

and

Furthermore, the averaged parameters 81, k. gz, k converge expo-

- - - A - ( I + 0,’) + d(1 + f 1 2 ) 2 -3azPz 01.k. 0 2 . b + 0 1 2 , s = (18)

2ap

as x k + 0.

attraction) can be estimated as (see bottom of page) The convergence rate X (uniform estimate over the domain of

Proof: See the Appendix, Subsection A. Equations (17) and (18) indicate that the parameter estimate 0,

converges exponentially fast (with rate A) to an O( J77) neighborhood of ( 0 1 $812. S . 8 2 $312. *) . Let us interpret the result of Theorem 1:

1) When the external subscriber signals are “white,” i.e., n = 0, and regardless of the cross correlation, even when the external subscriber signals are identical (perfectly correlated), the DEC system operates optimally in that the echoes are fully canceled. Indeed, in this case the parameter errors converge to a JF: small neighborhood of zero exponentially fast (e, = ( 8 1 2 , ~ , sl2, * ) = (0. 0)) . The transients are govemed by a convergence rate overbounded by X <

2) In the case in which the external subscriber signals are uncor- related, i.e., p = 0, the DEC system operates optimally in that the echoes are fully suppressed. In this situation the parameter errors converge to a fi small neighborhood of zero exponentially fast. The transients are governed by a convergence rate overbounded by X < l-pV2(1--a2)/[(l+O)(1+aZO)]. Notice thattheconvergence rate (or at least its bound) is adversely affected by the presence of the autocorrelation coefficient a.

3) In the case in which the subscriber signals are correlated and not “white,” i.e., np # 0, the DEC system no longer can achieve full echo cancelation. The parameter error is biased. The bias grows with both the autocorrelation and cross correlation of the subscriber signals. This is the subject of Fig. 2. At the same time the transients become longer.

1 - p\,’Z/(l + e).

I -0.5 0 0.5 1

P Fig. 2. Plot of e 1 2 . with cross-correlation factor p and autocorrelation factor a.

as given in (18) for correlated subscriber signals ( 5 )

1 , I

-0.5 I -’: 500 ld00 1500 2’300 25bO 3000 3;OO 4dOO

-0.5- J -

-’O 500 Id00 1500 2000 2 i O O 3dOO 3500 4000

time, k Fig. 3. Plot of 0 1 . k . 0 2 , k over time with 0 1 , o = 0.8, 02 .0 = -0.7, p = 0.002, and signal parameters a = 0.8, p = 0.65.

- -

4) Many simulations have been carried out on the single-delay DEC system with correlated first-order autoregressive subscriber signals. The results agree with Theorem 1 and the above discussion. Figs. 3 and 4 provide an illustratio? by showing the time evolution of the residua! echo parameters ( B l . k , 0 2 . k ) and residual echoes (01, kyz. k-I, 62, k y ~ , & I ) , respectively, for the case in which the subscriber signals ( 5 ) have an autocorrelation factor a = 0.8 and a cross-correlation factor p = 0.665. p = 0.002 and V = 1 for this simulation. As indicated in Fig. 3, both residual echo parameters converge approximately to 8 1 2 , = -0.368, while Fig. 4 indicates incomplete asymptotic echo suppression.

A. Subscriber Signals of Unequal Power

The previous section considered subscriber signals which were of equal power and described by the first-order autoregressive (AR1)

326 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. I , JANUARY 1995

2, I

I 1000 2000 3000 4000 5000

-0 1000 moo 3000 4000 5000

time, k Fig. 4. p = 0.002, and signal parameters U = 0.8, p = 0.65.

Plot of residual echoes over time with 0 1 . 0 = 0.8, 02 o = -0.7,

processes of (5). In this subsection we briefly explore the case in which subscriber signals are ARI processes of unequal power. In particular, we consider when Assumptions 5 and 6 are replaced by the following.

Assumption 7:

where b; > 0.

Assumption 8:

C

where I . J = (1. 2}, / # J .

replaced by Consequently, the subscriber signal inequalities of (8) and (9) are

The following theorem quantifies the effect, on asymptotic echo- canceler performance, of subscriber signals having unequal power.

Theorem 2: Consider the DEC system described by (2), (5), as well as by Assumptions 14, 7, 8, and

i l . k + l = 81, k - $y13 k + l y 2 . k

82. k+l = 62. k - -Y2. k+lyl , k

6 1 . 0 = 81 2

(21) P

8 2 , 0 = 82- 17;

Let 8 E ( a 2 , 1). Suppose

(22) v2 v 1’71 15 -I811 + 2 1 8 2 1 < 2& < 1.

while 6 k and yn are the variables of the DEC system

nentially fast (with rate A) to Furthermore, the averaged parameters $1 k . 82 converge expo-

- - 3 1; - ( l+a”+J( l+nL)”-4n2y2 61 k - 0 1 = - 1; 2trp

as xn -+ o

as X~ -+ 0. (25)

The convergence rate X (uniform estimate over the domain of attraction) can be estimated as (see bottom of page)

Pro$ The proof follows along the same lines as the proof of Theorem 1.

Remurks: 5) The condition of (22) imposes a constraint on the ratio of

the power of the subscriber signals. This condition guarantees that (31.0, 32, o ) lies in an invariant subset of the domain of attraction of the equilibrium ($1, s , 6% s ) of (25). However, because (22) is only a sufficient condition, then its violation does not necessarily mean ($1, k , 82. k ) will fail to converge to ($1. S. $2, S ) . On the other hand, violation of (22) will increase the chances of unstable behavior, such as bursting, in the original system. Consider, for example, the case

T:L/b’, >> 1 and a , p # 0, such that $ I , , $ = AI >> 1.

The width of the stability domain

in the region of the equilibrium (31, s, 32. ,?) = ( M , 1/M) is O( l /M). Recall that the averaged system only approximates the original system within an error O(fi). Thus if &I is sufficiently large, then stability problems become a real possibility.

6) As indicated by (21), the DEC system considered in Theorem 2 is different from that of Theorem 1 , not only because the subscriber signals have unequal power, but also because the adaptation constant of each echo canceler has been normalized with respect to the power of the far-end subscriber signal. This normalization was included to enable quantitative estimation of the domain of attraction of the equilibrium of (25).It is important to note, however, that this normalization does not affect the location of the equilibrium of the averaged system or, equivalently, the location of the asymptotjc O(fi) ball to which the original system parameter vector 8 k

converges. This is because the equation - -

z ~ ~ l ~ l $ ~ ( ~ ) ( O k ) = A i ’ - , - yLyl(l)(0h) = 0

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 1 , JANUARY 1995 327

the solution of which identifies the equilibria of the averaged system (24), is independent of any normalization of p . Consequently, assum- ing p is sufficiently small and $0 = (ol , e,) lies sufficiently close to e, = ( e ~ , s r 8 2 , s) so that $0 is within the domain of attraction of e,, then the averaged parameter vector e k of the DEC system described by (2), (3, as well as Assumptions 1 4 , 7, 8 and (4)-in which p is not normalized-will also converge to the equilibrium 8, of Theorem 2.

The result of Theorem 2 indicates that when the subscriber -signals are of unequal power, the adverse effect of correlation within and between the subscriber signals on asymptotic echo-canceler performance will be accentuated at one end of the four-wire loop and diminished at the other end. In particular, in order to achieve sufficiently good asymptotic performance of both echo cancelers within the four-wire loop DEC system, the autocorrelation and cross correlation of the subscriber signals should be reduced to smaller levels as the ratio of the power of the subscriber signals increases (decreases) above (below) unity.

- -

IV. CONCLUSION

We illustrated a technique for the analysis of an adaptive DEC system. The main result indicates that the cross-correlation and the autocorrelation length of the subscriber signals should be small in order to obtain good asymptotic performance in reasonable time. The need for minimal autocorrelation and cross correlation of the subscriber signals is accentuated when the subscriber signals are of unequal power.

The technique is applicable to DEC systems of a more complicated structure and subscriber signals with general spectra, but this leads necessarily to more involved algebra.

APPENDIX

A. Proof of Theorem I The averaged system (12) has equilibria given by the solutions to

We, therefore, begin by determining an explicit expression for these averaged terms of (13). This is achieved by combining (10) and (5) with Assumptions 4-6 to construct a Lyapunov equation, the solution of which depends explicitly on the average (13)-see Subsection B. The constructed Lyapunov equation was solved by using MAPLE V to obtain

A~, lyz ( l ) ( z ) 2 2 - 172 apz1 22 + a2z-1z2 + z1z; + 2apz122 + 21 + aZz2 + up -

(1 - z f z i ) ( l - a22122) (AI)

- 1.r2 a p z f z ; + a2z;z1 + zzz: + 2apzlz2 + 2 2 + a2z1 + u p

('42)

From (Al) and (A2) we conclude that the averaged system (12) has two finite equilibria e, = ( ~ I Z , ~ , 812,s) and e, = (el2 ,, e12 given by (via MAPLE V)

AVylyZ(l)(Z)

- (1 - %:z;)(l- aZz122)

- - ( I + a') + J( l+ $12 - 4a2pZ 812 s =

2aP

It is easily verified that for all la( < 1 and all Ipl < 1, et, = (H12. (,, g12, 1 , ) lies outside the domain D of interest, indeed

(812. > 1. Consequently, this equilibrium will be ignored. On the other hand, 8, = ( 8 1 ~ , ~ , e12. . ) E D. This is because, from the definition of 0 in Theorem 1

la( < dG < 1 (A41

- which guarantees le12 , I < & < 1. We, therefore, consider only 8,.

From the Jacobian evaluated at 8, we obtain that the equilibrium 0, is (locally) asymptotically stable. A domain of attraction may be estimated as follows.

-

First we rewrite the system as follows:

[81. k + l t $2, k + l ]

Here

V2(1 + a ' ) ffk ( l - B 1 , k e 2 , k ) ( l - a 2 8 , k 8 2 . k ) '

On the domain E c D, a k is positive and bounded above by V2(1 + a2)/[(1 - 0 ) ( 1 - a 2 0 ) ] . Let fi be sufficiently small (and positive), such that

As the initial conditions are such that they belong to E , we conclude by induction that the domain E is invariant and that the trajectories converge to the invariant subset F = {g1, . = 32, . } n E. Indeed

since

-+ 0 as k -+ CO. (A6)

We now consider the dynamics restricted to the invariant set F . In F the dynamics are governed by the recursion

328 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 1, JANUARY 1995

which can be rewritten as

\ ,

On the domain lqjkl < fi we have the estimate

Thus with 1 ~ 0 1 < fi, ok converges exponentially to 3 1 2 , s f

B. Maple Procedures to Solve a Lyapunov Equation

The following Maple procedures were used to find the expres- sions for the averaged terms Av,l,2(l) ( z ) and z). These procedures allow one to solve a Lyapunov equation of the type P = A P A T + BWBT where W = W T 2 0.

1 . with (linalg) ; 1. tenp:=proc(a) local i, j,k,l,n,aa; 2. n:=coldim(a) ; 3. aa:=matrix(nA2,nA2) ; 4. for i to n do

for j to n do

for k to n do

for 1 to n do

5. aa[(i-l)*n+k, (j-l)*n+ll :=a[i, jl*a[k,l] 6. od; od; od; od; 7. aa 8. end; 1 . colstack: =proc (b) local i, j , n; 2. n:=coldim(b) ; 3. aa:=matrix(nA2,1) ; 4. for i to n do

for j to n do

5 . aa[(i-l)*n+j,ll:=b[i,jl 6. od;od; 7. aa 8. end; 1. makemat:=proc(p) local n,i, j,a; 2. n:=sqrt (rowdim(p) ) ;

3. a:=matrix(n,n); 4. for i to n do

f o r j to n do

5 . a[i,jl :=p[(i-l)*n+j,l] 6. od;od; 7. a 8. end; 1. lyap:=proc(a,b,v) local d,c,f; 2. d:=add(scalarmul(tenp(a) , -

1) , array (identity, 1. .coldim(a) “2,l. .coldim(a) -2) ) ;

3. c:=multiply(multiply(b,v) ,transpose(b) ) ;

5. makemat (linsolve (d, f) ) 6. end;

4. f : =colstack (c) ;

The Lyapunov equation relevant to our system can be constructed by combining (2) and (5) in conjunction with Assumptions 4-6 as follows:

Let

XI = (!/I h y2 k SI k 32 k I 7

U & = (711 k I12 k)T.

Then

; ~ k + l ; c ~ + ~ = A.r:kxzAT + A.rkt,;BT + Bvk.rEAT + Bukv: BT

where

Define M-Cm ~ 1

k = m

Assumption 5 implies that ILf-CT7-I

liin l/M xkt,,’ = 0(4 x 2 zero matrix). M-CxZ

k = m

Combining this with Assumption 6 leads to

P,,,+I = dP,,AT + BWBT

where

For n2, zf, 2.; < 1 and 7’1, k , 19, h bounded (Assumption 3) we have X k bounded. This leads to the existence of the following limit and, consequently, the Lyapunov equation:

lini pI,+l = P = A P A ~ + B W B ~ .

The averaged term At’yly2(,)(i) is obtained from the solution P of this Lyapunov equation via

nl-oci

with

c = ( 0 21 a 0 )

c2 = (0 1 0 0 ) .

The term A u ~ ~ ~ ~ ( ~ ) ( z ) is similarly obtained.

REFERENCES

[ 11 W. A. Sethares, C. R. Johnson, Jr., and C. Rohrs, “Bursting in adaptive hybrids,” IEEE Trans. Commun., vol. 37, no. 8, pp. 791-799, Aug. 1989.

[2] I. M. Mareels and R. K. Bod, “Performance oriented analysis of a double hybrid adaptive echo cancelling system,” J. Math. in Estimation and Contr., vol. 2, no. 1, pp. 71-94, 1991.

[3] J. Homer, I. M. Y. Mareels, and R. R. Bitmead, “Signal dependent performance of echo cancellers,” in Proc. Inc. Symp. on Adaptive Systems and Signal Processing (ACASP’92), 1992, pp. 621-626.

[4] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dy- namical Sysrems (Applied Mathematical Sciences 59). New York: Springer-Verlag, 1985.