Nonlinear echo cancelling using look-up tables and Volterra systems

L. Weruaga-Prieto A.R. Figueiras-Vidal

Indexing t m : Echo enncelling, Look-up tables, Volterra systems, FIRfilters

Abstract: Two classical approaches to nonlinear echo cancellation are to use look-up tables com- bined with FIR filters and Volterra systems; their practical interest is higher when adapted with LMS-type algorithms, because results are easily implementable and offer good tracking capabil- ities. A complete analysis of look-up-table-based schemes is presented, and a new one is i n t r cdud which offers near optimal characteristics but without requiring variable steps: in practice a piecewise-constant adaption algorithm is enough. The theoretical equivalence of these schemes with those based on Volterra kernels is established; a comparison of operational characteristic is then made. The important option of applying growing Volterra structures is proposed for the first time, and its possibilities are verified (as well as the pre- vious analysis) by means of simulation examples. The paper ends with suggestions to be explored to obtain more efficient nonlinear echo cancellers.

1 Introduction

A classical problem in full-duplex data transmission is the presence of electrical echoes as an effect of the imper- fect balance provided by the hybrid coils in making the 412 and 214 wire transitions. The standard scheme to alle- viate this difficulty is to apply an echo canceller, as Fig. 1

Fig.

canceller

t I ' t ' transmitted symbols

- x(k) Scheme of an echo canceller in a subscriber Imp

shows: a system which replicates the echo path, creating an estimate y(k) of the echo z(k) which, substracted from this, reduces the perturbation to a clearly lower residual echo e(k).

When considering low rate transmissions, such as those used in telephone-network modems, insertion of an

0 IEE, 1994 Paper 1427K (ES), 6rst received 17th February and in rcvisod form 11th July 1994 The authors a n with GPSS/DSSR, ETSI Telccom - UPM, Ciudad Universitaria s/n, 28040, Madrid, Spain

IEE Pr0c.-Vis. Image Signal Process., Vol. 141, No. 6, December 1994

adaptive linear system as the echo canceller is enough to obtain sufficient reduction of the echo level; References 1-3 describe these cases. However, when the transmission rate increases to hundred of kilobits per second, such as in high-speed digital subscriber loops (HDSL) 141, non- linear effects become important: they are mainly due to hybrid saturation, pulse asymmetries, and nonlinearities in A/D and D/A convertors. Since typically around 60dB echo reduction is needed (to obtain a signal-to- echo ratio of 20 dB from a received signal 50 dB below the transmission level and a standard 10 dB trans-hybrid losses), we must take care of these nonlinear effects, as first pointed out in Reference 5.

A series of nonlinear schemes has been proposed for nonlinear echo cancellation; among them are:

(a) those using a (linear) transversal filter combined with a look-up table (LUT), the latter to model the non- h e a r effects 16-93 ;

(b) schemes based in classical nonlinear structures, such as Hammerstein and Wiener filters [lo] or Volterra kernels [ll-131; in principle, we will consider here the second, which is completely general and accepts a finite- order exact implementation for echo cancelling in data transmission ;

(c) one other possibility is to use neural networks [l4, 151, with an architecture allowing a reasonable compromise between speed and degree of cancellation.

In this paper, we present an extensive study of the first two approaches, together with their analyses, and a com- parative discussion. Section 2 is dedicated to LUT based schemes, including, in addition, the complete analysis of the optimal LMS structure and a suboptimal switched- LMS step approach. Section 3 covers the use of Volterra- based echo cancellers and discusses their relative performances with respect to the cases of Section 2, and even conditions for equivalence. Finally, general conclu- sions are drawn and some suggestions are made for further work following up the discussions in this paper.

2 Look-up-table-based schemes

2.1 Previous work Look-up-table-based (LUT) cancellers have been pro- posed as a possible way of cancelling nonlinear echoes [6-91. Considering the transmitted data as pvalued symbols and the echo depending on the last n transmit- ted data, the echo can take p" forms; an adaptive memory of this size is prohibitive.

This work has been supported by CICYT grant TIC 92 0800-CO5-01.

357

To reduce this complexity, in Reference 6 the nonlin- earity is modelled by an m-length nonlinearity cascaded with a k-length linear dispersive element, proposing an echo canceller of k small memories of p" coefficients: the overall number of taps is reduced to kp". In Reference 7, a greater reduction of the number of taps is obtained, assuming that the nonlinear component of the echo is typically of shorter duration than the overall echo. The echo canceller proposed is composed of an n-length transversal filter plus a memory of p" locations, with the m first taps of the transversal filter overlapped with the look-up-table. The resulting filter-coefficient number is n + p"; this is a reasonable value.

The following Sections analyse different forms of the latter scheme.

22 LMS-type algorithms The look-up-table-based echo canceller analysed in this Section is shown in Fig. 2; it consists of two blocks: a

zlk) + elk)

Fig. 2 being discusred

Structure of the l o o k - u p t o b ~ p l ~ - F I R - b ~ e d echo canceller

(linar) FIR filter, and a nonlinear filter based on an adaptive memory. A shift register of length n is necessary to store the last n transmitted symbols, which are the input to the above canceller elements. This shift register, at time k, contains the vector x(k) = [x(k), x(k - l) , .... x(k - n + l ) ] . In our analysis, we will assume a plevel, independent and identically distributed symbol sequence.

The FIR filter is an n-tap transversal filter with coeffi- cients s,(k) = [sl0(k), s,,(k), .... s,,,-,], and its input is x(k). The LUT uses only the m lint components of x(k) (i.e. the last m transmitted symbols), having p" memory locations, each representing the canceller nonlinear output for each one of the possible m-length pvalued input sequences. The memory contains the vector s,(k) = [smo(k), s,,(k), .... smp- l(k)]. In this case, the LUT overlaps with the m first FIR coefficients.

We defme the following vectors: x(k) = [x'(k), ~ " ( k ) ] , x'(k) = [x(k), x(k - I), .... x(k - m + l ) ] , and x"(k) = [x(k - m), .... . j k - n + l ) ] . In the same way, the FIR filter, s'k), is diwded into an overlapped part with the LUT, containing d'k) = [ sdk ) , s,,(k), .... sla- l(k)], and a nonoverlapped part, corresponding to the vector d'k) = Csdk), .... SI"- I(k11.

The linear and nonlinear canceller outputs are, respectively,

y,(k) = s,(k)x(k)' = sl(k)x'(k)' + G(k)x"(k)' ( 1 ) ydk) = s,(kMk)' (2)

where 4 k ) is an all-zero pa-length vector except one posi- tion which contains a '1' :

(3) 44 = [o,o, 0, .... 0, i , o , .... 01

358

This position is one-to-one determined by one of the p" possible LUT input sequences.

Likewise, the echo can be expressed as the sum of a linear and a nonlinear part :

(4) where b, and b, represent the linear and nonlinear echo paths, respectively. We will not consider the noise term in our analysis: essentially, its effect will be only to increase the final mean-square error (MSE) by the power of that noise.

z(k) = zAk) + z,(k) = h, x(k)' + h, 4k)'

22.1 Constant-step algorithms: In Reference 7, the authors suggest adapting the overall structure using an LMS with two constant steps: a for the FIR and f i for the LUT; the proposed adaptive algorithm is

(5 ) (6)

s,(k + 1 ) = sdk) + c*e (k )

s,(k + 1 ) = d k ) + Be(kMk) The resulting steps were obtained heuristically, after developing a cumbersome theoretical analysis. There is an interaction between the LUT and the first m FIR coef- ficients: since the LUT can also cancel the linear echo, it interacts with the FIR, competing in order to cancel the linear echo. This prevents us obtaining an analytical solution; but it is possible to think that a nonoverlapped scheme, i.e. the LUT and only the last n - m FIR coeffi- cients, will serve to cancel the echo. The corresponding adaptive algorithm becomes

(7) (8)

where a" and B are the constant adaption steps of the nonoverlapped FIR and the LUT, respectively.

An analysis of the algorithm along these lines, sug- gested in Reference 8, is as follows: the output of the non- overlapped canceller is

s;'(k + 1 ) = s;(k) + a"e(k)x"(k)

s,(k + 1 ) = SAk) + Be(kMk)

y(k) = $(k)x"(k)' + s,(kMk)'

z(k) = b;lr.(k)' + K,4k)'

(9)

(10)

On the other hand, the echo can be written as

where N,4k)' represents the nonlinear plus the m first symbols linear linear echoes:

The resulting error between the echo and the canceller Nm4k)' = bm4k)'+ &x'(k)' ( 1 1 )

output is

4) = z(N - Ak) = brx"(k)' + Nm 4k)'

- {S;(k)x"(k)' + SAkMk)'} (12) The canceller coefficients are adapted by means of the LMS algorithm (eqns. 7 and 8). The optimisation study of the canceller convergence is included in Reference 8: under the usual hypotheses, and maximising the speed reduction of the mean-square error (MSE), by making zero the derivatives of the MSE evolution, we obtain the step values a" and /3 which offer the maximum speed of convergence :

(13)

(14)

1 a" = u:(n - m + p")

P" /g=- n - m + p a

IEE hoc.-Vis. Image Signal Process., Vol. 141, No. 6, December 1994

and the maximum reduction rate of the MSE

222 Three-variable step algorithm: In real cases, the echo is usually quasilinear, and the nonoverlapped scheme does not take advantage of this characteristic, since the linear echo depending on the last m transmitted symbols is cancelled by a nonlinear structure. To do better, we will consider the overlapped scheme, but with a very important difference in the adaptive algorithm, the use of three variable steps: one for the overlapped FIR part, one for the nonoverlapped FIR and one for the LUT.

(16)

(17)

In this case, we can write

y(k) = sAk)x'(k)= + S;'(k).r''(k)= + s,(k).(k)T

z(k) = h;x'(k)= + k;x"(k)= + k,4k)'

and the echo is

The residual echo results:

e(k) = {hi - sAk)}x'(k)= + {hl - S;'(k)}x"(k)=

+ {k" - sn(k))4k)T (18) The overall structure is adapted with an LMS of three variable steps

S;(k + 1) = sxk) + a'(k)e(k)x'(k)

S;'(k + 1) = S;'(k) + a"(k)e(kw(k) s,(k + 1) = sdk) -k B(k)e(k)a(k)

(19)

(20)

(21) mote that the whole FIR is not adapted using the same LMS step; the overlapped part is adapted with step a'(k), and the nonoverlapped part with a"(k).]

The analytical study of this case is contained in Refer- ence 9: it follows the equivalent method corresponding to the previous case. The evolution of the optimal three variable steps is given by

{ 1 - a'(k) - Zb(k)}a'(k) 1 - 44

a'(k + 1) =

(1 - a"(k)}a"(k) 1 - 4 k )

a"(k + 1) =

4 k ) = ma"(k) + (n - m)a"'(k)

+ pmb2(k) + 2ma'(k)b(k) (25)

a'(k) = a'(k)ui (26)

a"(k) = a"(k)o;f (27) b(k) = B W P - (28)

where we use the normalised steps

When the otpimal steps are used, the canceller reaches its maximum speed of convergence

E{e2(k + l)} = (1 - ma"(k) - (n - m)a"'(k) - p"bz(k)

The first question which arises from the theoretical result is: what are the initial values of the optimal steps? These values depend on the character of the echo: this means that, assuming all-zero initial canceller coefficients, a"(0) is proportional to the ratio between the energy of the

- 2m'(k)b(k))E{e2(k)} (29)

nonoverlapped linearecho part and the overall echo energy, a'(0) is proportional to the ratio between the overlapped linear echo and the overall echo energy and b(0) is proportional to the ratio between the nonlinear echo and the overall echo energy. For the algorithm to work successfully, knowledge of these echo characteristics is a necessary condition. This drawback, added to the high computational load of this LMS algorithm, make the optimal scheme of limited use in practical applica- tions.

223 First series of simulation results: In this section, we will present some representative simulation results which serve to give an idea about (relative) performances of the above schemes. The conditions of the simulation are :

(a) the transmitted symbols are antipodal binary, { - 1, 1); then, p = 2;

(b) the echo lengths are n = 30 and m = 5 ; (c) all the LUT and FIR coefficients have zero initial

values; (d) the echo has been modelled with an n-length FIR

plus a p" size LUT; the coefficients have been generated as random Gaussian numbers, in the FIR case according to a N(0,0.1825) and in the LUT case with a N(0, lo-') (note that the nonlinear echo power, as in real cases, is about 40 dB below that of the linear echo);

(e) there is an additive uncorrelated white Gaussian noise of variance We apply three different cancellers:

(i) a constant-step LMS overlapped LUT + FIR (that of Reference 7 ) with a = 0.0161 and /3 = 0.516;

(ii) a constant-step LMS nonoverlapped LUT + FIR with a" = 0.0175 and /3 = 0.561;

(iii) a three-optimal-variable-step LMS LUT + FIR, with the steps calculated by eqns. 22-25.

Fig. 3 shows the MSE evolution for the three cancellers (each curve has been obtained by averaging the square error of 20 simulation runs and smoothing it with a 20- point-length moving-average window): it can be seen that both constant-LMS-step schemes have similar per-

100,

k

Fig. 3 Euolutwn of the 4ume error (SE) (i) mnstant-step LMS ovcrlappd look-up table + FIR

(U) mnstant-stcp LMS nonoverlapped look-up table + FIR (iii) thrct-variablestcp LMS look-up tabk + FIR mmpudins to thc simulation case in Section 22.3: x(k)= ( -1 , l ) (p = 2),n = 30.m = 5 Tbc echo path is modclled by a look-uptable +FIR system of paramcm N(0, 0.1825) and N(0, IO-'), mpcztively

IEE Proc.-Vis. Image Signal Process., Vol. 141, No. 6, December I994 359

formances (though the nonoverlapped one is slightly better); the three-variable-step scheme clearly out- performs the others owing to its ability to cancel the linear echo in the first part of the adaptation.

Fig. 4 shows the evolution of the three variable steps; note that, initially, the steps take values b(0) s 0 and

r

'0 \ 200 400 600 800 1000

Fig. 4 a'(k): nonoverlappd FIR part step d ( k ) : ovcrlappd FIR part s k p Hk): lookpuptable step mrresponding Io the simulation case in w o n 22.3: dk) = ( - I , I} (p - 2), n = 30, m = 5 The echo p t b is modclled by a look-up tabk + FIR system of parameters N(0, 0.1825)and N(O,IO-'L rcspMivcly

Evolution of the three variable steps for a quasilinear echo

a'(0) e a"(0) = l/n in order to cancel primarily the linear echo, of greater magnitude, and the LUT remains ked. The overall scheme looks like an adaptive FIR with an optimal-step LMS. Later, the steps evolve abruptly to the values

a'(k) + 0 (30)

1 a"(k) and b(k) + -

n-rn+p"'

i.e. the canceller tends towards the nonoverlapped scheme with constant steps (as a consequence, it can be said that, when a constant-step LMS algorithm is used, the nonoverlapped scheme is the optimum structure, the interaction between the LUT and the overlapped part of the FIR being a drawback for better performance of the canceller).

2 2 . 4 Suboptimal switched LMS scheme: From the above, a straightforward and efficient mehod of avoiding the use of the computationally complex variable-step algorithm is to use a two-mode sequential scheme:

(i) in the 'linear mode', the period along which the echo is essentially linear, only the transversal filter (n coefficients) is updated, using

1 a' = an = - n

(the look-up table remains fixed);

b = O (33) (ii) the 'nonlinear mode' begins when the 'linear mode'

finishes; now the rn 6rst FIR coefficients stay ked, with the values reached at the end of the linear mode, and the

following values are used for the adaption steps:

a' = 0 (34)

1 n-m+p"

a" = b = - (35)

To use this procedure, it is necessary to detect when to switch from the 'linear' to the 'nonlinear' mode. A good way of detecting the end of the 'linear' mode is as follows; during the 'linear' mode, the MSE evolves according to

E{e2(k + l)} = 1 - - E{ez(k)} ( 3 and the expected value of the variable

is a more-or-less constant value for all k,

(38) However, when the linear echo has been nearly cancelled, the MSE does not evolve according to eqn. 36, and tends towards the nonlinear echo power; E{r(k)} then begins to increase exponentially. Detection of this situation, mon- itoring an estimation of E{e(k)], gives us the time to switch to the nonlinear mode. The evolution of the MSE in the nonlinear mode corresponds to that of the non- overlapped scheme with constant steps.

2.3 More simulation results In this Section, we will present more simulation results about the switched and optimal schemes. The conditions of the experiment are the same than in Section 2.2.3. We apply:

(i) a three-optimal-variable-step LMS LUT + FIR,

(ii)a switched LMS LUT + FIR(Section 2.2.4). with the steps calculated using eqns. 22-25;

Fig. 5 shows the results of the simulations; we can con- clude that the opimal LMS scheme (i) and the suboptimal switched scheme (ii) offer very similar performances.

2.4 General discussion of look-up-table-based

The results in Sections 2.2.3 and 2.3 serve to support the ideas of our theoretical analysis. We can conclude that:

(a) both constant-step schemes, overlapped and non- overlapped, offer similar performances, though this latter scheme is slightly better (as the evolution of the three- variable steps predicted);

(b) the switched scheme is the LMS scheme which offers better agreement between cancellation and compu- tational load, and also does not need any a priori infor- mation about the characteristics of the echo.

cancellers

3 Volterra-based schemes

Volterra series have been widely used in nonlinear system identification 111, 131 and, in particular, in nonlinear echo cancellation for data transmission [12] and for data

IEE Proc-Vis. Image S i g d Process., Vol. 141, No. 6, December I994 360

equalisation [13]. Although LMS algorithms have been studied for this structure, we will adapt the formulation to make easier comparisons with the LUT-based schemes.

' O 0 h lo-$\

l@@ "0 200 400 600 800 1000

Fig. 5 Euolution of the squme error (SE) (i) three-variable-step LMS look-up tabk + FIR

(U) switched-LMS look-up table + FIR corresponding to the simulation w e m Sed00 2.3

Tbe mho path is modelled by a look-up table + FIR of pararnctcrs N(O,O.l825) and N(0, IO-*), mpCaivcly

*(g= ( - 1 , l ) (J = Z)," = 33, m = 5

3.1 LMS-orthogonalised- Volterra-plus- FIR-filter

Since we are faced with the same problem as previously, it is reasonable to propose the structure of an ortho- gonalised Volterra-plus-FIR canceller: it is composed by an (n - +length FIR, which takes care of the part of the linear echo depending on the last (n - m) transmitted symbols, in parallel with an orthogonalised Volterra filter, dedicated to the nonlinear echo and the other part of the linear echo. The advantages of using orthogonal polynomials as adaptive filters are well known, and the basic concepts of this orthogonal-polynomial theory can be found in References 16 and 17, while an example of polynomial filter in the sense suggested here is given in Reference 18.

The canceller input is stored in two vectors: an n-length vector uik) = [P('){x(k)}, P"){x(k - l)}, . . . , P"){x(k - n + l ) } ] including the linear inputs, and a (p" - m)-length vector U, = [Q(O)(k), Q(2)(k), . . . , Q("("-'))(k)], where e'@) is the vector whlch contains the ith-degree orthonormal products of x(k), ..., x(k - m + 1). In the same way, the filter coefficients are stored in two vectors: qdk) and q,(k).

scheme

This scheme is adapted using the LMS algorithm

4l(k + 1) = qXk) + Pl(&oe()al(k)

q,(k + 1) = q,(k) + PAkMkbAk)

(39)

(40) Under the usual assumptions

(i) q,(k) and rkk) are (approximately) independent random vectors; the same can be said about qAk) and oAk) ;

and, since the components of U,@) and u,(k) are sta- tistically orthonormal, i.e.

(ii) e(k) is uncorrelated with ukk) and udk);

E{ul(k)Tul(k)} = I , (41) I E E Pr0c.-Vis. I?nage Signal Process., Vol. 141, No. 4 December 1994

E{~Ak)~on(k ) I = I F - m (42)

E{%(k)TuXk)) = o p - , x . (43) it is easy to v e e that the same analysis applied for the LUT + FIR canceller can be applied here. Therefore, we conclude :

Constant-step L M S : The optimal constant steps are

and the optimal speed of convergence

Variable-step L M S : The optimal variable steps are computed by the recursive equations

(47)

where

n(k) n p W + (P" - m)&) (48)

npdk) + (p" - m)k(k) = 1 (49)

verifying the equation

The initial step values depend on the energy of the linear and the nonlinear components of the echo. The optimal MSE evolution is

E{e2(k + 1)}

= { 1 - nPX4 - @" - m)P"(k))E{e2(k)l (50) Identically to the LUT case, the Volterra-plus-FIR variable-steps scheme suffers an abrupt transition in quasilinear problems : therefore, the switching algorithm can be applied in just the same way as previously. 32 Equivalence with look-up-table-based schemes In Appendix 6, we discuss whether the two LMS can- cellers (LUT + FIR and orthogonalised Voltena + FIR) can be considered equivalent; the necessary and sufEcient requirement for both cancellers to be exactly equivalent adaptive structures is the verification of

This is fully supported by simulation results.

3.3 Practical considerations The above schemes are appropriate when there is no detailed information about the nonlinear echo com- ponents: since they are general, they will provide an ade- quate solution.

If an LMS algorithm is to be used, both schemes being equivalent (under the hypothesis of selecting the adequate parameters), the reason for choosing which canceller to apply will be a question of computational load: in this sense, the nonlinear part of the LUT + FIR canceller requires one memory access and one multiplication (to adapt the LUT) per iteration, while the Volterra + FIR structure needs 2p"' products (p'" to calculate the output

361

and p" to update coefficients) plus the orthonormalisa- tion operations. Consequently, the LUT + FIR is prefer- able.

It is true, on the other hand, that, when the size of the transmission constellation and the length of the nonlinear echo component are large, p" becomes a high value, and both schemes could be impractical. Under this situation, the only alternative is to use any additional information one can obtain about the nonlinear echo component in order to reduce its degree of freedom below p".

To this end, the previous schemes clearly offer differ- ent possibilities: a LUT + FIR is a rigid scheme, and does not usually allow complexity reduction (with some trivial exceptions e.g. when modelling even or odd func- tions, its complexity can be reduced by a half). However, the FIR + Volterra structure is more flexible. It is pos- sible to use a part, P < pm, of its kernels in a practical implementation; this reduction will be useful if the resulting scheme matches the nonlinear echo (the know- ledge of which will be needed to make the reduction). Note, on the other hand, that the necessary changes are simple: for example, when applying an LMS algorithm, the (optimal) adaptation step must be inversely pro- portional to the number P of kernels selected.

A practical approach is to include progressively more kernels, when necessary, starting from an initial structure. This will be convenient when the information about the nonlinear echo component is insufficient for an initial decision on kernels selection. It is possible to start with a linear scheme, and to introduce nonlinear kernels if the degree of cancellation obtained is not enough; this method is adequate when the increasing strategy is so; for example, when including higher-order polynomials and the echo is quasilinear and the higher-order effects become less and less important, a situation which is fre- quent in the practice.

A simulation will serve to illustrate this idea. The simulation conditions are:

(i) the constellation is the 2B1Q line code { - 3, - 1, 1, 3}, recommended for HDSL;

(ii) n = 30, m = 5; (iii) the echo plant has been modelled with an FIR

plus a truncated Volterra model such that the quadratic- term energy is 40 dB below that of the linear terms, that of cubic terms is 3 dB below that of the quadratic ones, that of the fourth-order terms is 3 dB below that of the cubic ones, and that of the fifth-order terms is 3 dB below that of the fourth-order ones; the parameters have been obtained with normal distributions;

(iv) all initial coefficients are zero.

Three cancellers are used. The first assumes a general nonlinearity, the LUT plus FIR scheme; the second uses a true-order (5) truncated Volterra-plus-FIR scheme; and the third does not use any assumption about the echo, incorporating the idea of growing of its structure: a growing-Volterra-plus-FIR scheme. The growing Vol- terra filter initially has only linear terms, and, in the adaption process, it includes more kernels (quadratic, cubic etc.) if the degree of cancellation obtained is not enough, a situation which is detected by means of the switch algorithm of Section 2.2.4 adapted to the present filter configuration. Fig. 6 shows the evolution of the MSE of the three cancellers for 20 simulation runs.

Clearly, both Volterra + FIR schemes outperform the LUT + FIR, and the reason is obvious: both schemes have a lower number of coefficients than the LUT + FIR scheme. However, more important is the successful result

362

of the growing-based scheme: it does not use assump tions about the echo nature and is faster than the trun- cated Volterra-based canceller.

100 lc2bi "*C! 500 1000 1500 2000 2600

Fig. 6 (i) Switched look-uptabk + FIR scheme; (ii) Switched orthogonalised Voltma + FIR (iii) Growing ortbogooalipd Voltcrra + FIR Corresponding to the simulation casc in Section 3.3 x(k) = (-3, - 1 . 1 , 3 ] (p = 4Ln = 30.m = 5 The echo path is modclkd by a 6flhorda Vollcrra system; dcuila arc m th main text

Euolution ofthe squme error (SE)

The subject of deciding how to grow the Volterra network when the practical situation is not so simple needs further research; however, to use growing tech- niques based on neural-network growing schemes is an obvious option.

4 Conclusions and further work

This paper has presented an analysis and discussion of two alternatives for nonlinear echo cancellation in high- speed data transmission: look-up table (LUT) and Volterra-based schemes in parallel with an FIR.

The LUT-plus-FIR schemes using LMS adaption are interesting approaches, because they are computationally moderate and offer high tracking capabilities. Here, after studying the optimal selection of both constant and variable adaption steps, we introduce a suboptimal piecewise-linear algorithm, useful for the very frequent cases of quasilinear environments.

The analogous Volterra-based canceller family has performances identical to those above, although it requires a larger computational load. Additionally, these schemes include the possibility of being implemented in growing architectures: more study is needed of develop ment of ad hoc growing methods.

Research in this direction can be directed to the use of other nonlinear systems. A good alternative is neural net- works, nonlinear devices with high capabilities and flex- ibility. Although the first attempts at using neural networks on nonlinear echo cancellation have not been very promising [14, 151, it seems clear that the architec- tures selected were inadequate for the application. However, others can be useful; in particular, the Cerebel- lar model articulation controller (CMAC) [18] is inter- esting because of its similarity to a reduced version of a LUT and its high convergence speed and computational simplicity.

IEE Proc.-Vis. Image Signal Process, Vol. 141, No. 6, December 1994

The idea of increasing advantages by combining differ- ent nonlinear schemes is, of course, a very promising research direction.

5 References

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2 FALCONER, D.D.: 'Adaptive reference echo cancellation', IEEE Tram., 1982, COM-30, (9), pp. 2083-2094

3 MESSERSCHMITT, D.G.: 'Echo cancellation in speech and data transmission', IEEE J . Sel. Areas Conunun, 1984,s (2), pp. 283-297

4 Special issue on high speed digital subscriber lines; IEEE 1. Sel. Areas Commun., 1991.9, (6)

5 AGAZZI, O., HODGES, D.A., and MESSERSCHMIm, D.G.: 'Nonlinear echo cancellation of data signals', IEEE Trans., 1982, COM-30, (11). pp. 2421-2433

6 SMITH, MJ., COWAN, C.F.N., and ADAMS, P.F.: 'Nonlinear echo cancellers based on transpose distributed arithemetic', IEEE Trans., 1988, CAS% (l), pp. 618

7 YAMAZAKI, K, ALY, S., and FALCONER, D.D.: 'Convergence behaviour of a jointly-adaptive transversal and memory-bad echo canceller', IEE Proc. F , 1991,138, (4), pp. 361-370

8 WERUAGA-PRIETO, L., CID-SUEIRO, J., and FIGUEIRAS VIDAL, A.R.: 'Analysis of a look-up table plus separate transversal filter for adaptive nonlinear echo cancelled. First COST 229 WG.2 workshop on Adaptive algorithms, Bayona, Spain, 1991, pp. 142-151

9 WERUAGA-PRIETO, L., CID-SUEIRO, J., and RGUEIRAS- VIDAL, A.R.: 'Optimal variable step LMS look-uptable plus trans- versal filter nonlinear echo cancelled. Proceedings of IEEE ICASSP 92, San Francisco, USA, 1992, vol. 4, pp. 229-232

10 CASAR-CORREDERA, J.R., GARCfA-OTERO, M., and RGUEIRASVIDAL, A.R: 'Data echo nonlinear cancellation'. Proceedings of IEEE ICASSP '85, Tampa, USA, 1985, vol. 3, pp. 1245-1248

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neuronales en cancelaci6n ews' . Actas VI Symp. Capitulo EspaF,ol de la URSI, Spain, 1992, vol. 2, pp. 989-993

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15 WERUAGA-PRIETO, L., and FIGUEIRAS-VIDAL, A.R.: 'Redes

8

The outputs of the nonoverlapped LUT + FIR- and orthogonalised Volterra + FIR-based echo cancellers are, respectively,

Appendix: Equivalence of LMS LUT- and Volterra-based cancellers

~ d k ) = sXk)x"(k)' + s,(k)a(k)'

~ d k ) = qX&Oo)' + q.(kbAk)'

(52)

(53) Vector oXk) is an orthonormalised version of x(k). Since x(k) is zero mean and assuming that E{x(k)2} = 1 (scaling),

44 = oXk) (54) I E E Roc.-Vis. Image Signal Process., Vol. 141, No. 6, December 1994

Now, the difference between both canceller outputs results (using the standard notation):

Let us assume that both schemes represent the same non- linear system at time k, i.e. the difference between both canceller outputs (eqn. 55) is zero: it is not W c d t to sec that the nonoverlapped linear parts in both cancellers must necessarily be identical:

{s;(k) - d ( k ) } m ) ' = 0 (56)

sAkWk)' - d'W0)' - qm(k)om(k)' = 0 (57)

4Yk) = qXk) (58)

and the same for both overlapped parts:

As eqn. 56 is true for every t('(k), we can write

Both cancellers are operationally identical at time k + 1 if, and only if, y,(k + 1) and ydk + 1) are equal for any input vector; if this is true, we will have

Y'(k + 1) - Ydk + 1) = {S;(k) - d(k)}of(k + 1)'

+ b''e(k) - PIe(k)J4(k)o;(k + 1)'

+ sAk + 1Wk + 1)' - M k ) + Pre(k)ddk)}

x o;(k + 1)' - Iq,(k) + P" e(k)o"(k)b"(k + 1)'

= O (59) Starting from eqn. 59, we can obtain the necessary and sufficient conditions for the equivalence of the two schemes. According to eqn. 58, eqn. 59 becomes

(a" - p,)e(k)o;(k)o;(k + 1)' + s,(k + 1Wk + 1)' - {dXk) + PI e(k)dXk)bJ;(k + 1)' - {q,W + Pn e(k)o"(k)bAk + 1)' = 0 (0)

From eqns. 7 and 39, considering eqn. 58, it can be seen that, except for the trivial case edk) = e,,(k) = 0, the only way to obtain d ( k + 1) = 4 ( k + 1) is

a" = p1 (61)

s,(k + 1Wk + 1)' - {d'k) + PI e(kY(k)}r;(k + I)'} then eqn. 60 results:

- {q,(k) + pDe(kb,(k)}oAk + 1)' = 0 (62) Now, there are two possible cases:

(i)'If 4 k ) # 4 k + l ) , then s,(k + 1Wk + 1)' = s,,(k)a(k + 1)' (63)

because the memory location invoked by 4 k + 1) was not updated at time k.

Given the equality between both cancellers at time k for any input, and, in particular, selecting the input vector at k + 1, eqn. 57 can be reh t ten as

s,(k)cl(k + 1)' - dXk)d;(k + 1)' - q,(k)oAk + 1)' = 0

(64)

(65)

and, using eqns. 63 and 64, eqn. 62 results:

p1 a;(k)a;(k + 1)' + p,, u"(k)o,,(k + 1)' = 0

(ii) The memory location has been updated according to eqn. 7, and eqn. 62 becomes

{s,(k) + Be(k).(k)}4k)' - {dud + PI e(k)dXkMO)'

- {qAk) + ~,e(k)o.(k)14k)' 0 (66) 363

Using eqn. 57 in the above expression [and noting that u(k)a(k)T = 11, this results in

(67) The condition defined by eqn. 62 has been reduced to verification of eqns. 65 and 67, but the dependence of the new conditions on the input vectors can be suppressed by noting that, if

(68)

PI 4 W W T + P" M 4 ~ A k Y = B

CuXk), aAk)I f CoXk + 11, u,(k + 1)l

[IJXk), o,(k)lCoXk + l), U"(k + 1)lT = 0

then

(69) and that

The vector [ox& u,(k)] contains the p'" possible ortho- normal polynomials, and therefore precisely fulfils the above conditions (eqns. 6S70).

Given eqns. 68 and 69, eqn. 65 is only verified assuming

P" = PI (71)

In the same way, and given eqn. 70, eqn. 67 is verified when

B - = PI(= P A P"

Therefore eqns. 61, 71 and 72 allow us to say that the necessary and sufficient condition for both cancellers being exactly equivalent adaptive structures is the verifi- cation of

It is easy to see that, in the general case, for E{x(k)'} = U:, the above expression is modified by multiplying the adaption step a" appropriately:

364 IEE Pr0c.-Vis. Image Signal Process., Vol. 141, No. 6, December 1994