split filter structures for lms adaptive filtering

ELSEVIER Signal Processing 46 (1995) 255-266

SIGNAL PROCESSING

Split filter structures for LMS adaptive filtering

KC. Hoa,*, P.C. Chingb

a Department qf Electrical and Computer Engineering, Royal Militav College of Canada, Kingston, Ontario, Canada K?K SLO

b Department of Electronic Engineering. The Chinese University of Hong Kong, Shatin, N. T, Hong Kong

Received 19 September 1991; revised 20 December 1993, 31 October 1994 and 16 June 1995

Abstract

In this paper, the performance of a split adaptive filter in either a parallel or a serial form for non-white inputs is investigated. A parallel split structure is constructed by using two linear phase filters connected in parallel while for serial split it is configured as a cascade of two transversal filters. The adaptation characteristics of the well-known LMS algorithm has been shown to be governed by the eigenvalue spread of the input correlation matrix. By adopting the split structures, we illustrate that the eigenvalue ratios of the associated covariance matrices can be reduced thereby giving rise to a faster convergence speed. The parallel and serial split adaptive filters are examined for both joint process estimation and linear prediction. A new type of linear predictor is formed by combining both split methods and its performance for speech analysis is studied. Simulation results are included to validate the superiority of the proposed split filter structures in improving the rate of convergence for LMS adaptation.

In diesem Artikel wird die Leistungsfahigkeit eines split-adaptiven Filters in paralleler oder serieller Form fiir nicht-weiDe Eingangssignale untersucht. Eine parallele Split-Struktur wird durch Verwendung zweier parallelgeschal- teter linearphasiger Filter konstruiert, wahrend eine serielle Split-Struktur als Kaskade zweier Transversalfilter kon- figuriert wird. Es wurde gezeigt, dalj die Adaptionseigenschaften des bekannten LMS-Algorithmus durch die Eigenwert- verteilung der Eingangskorrelationsmatrix bestimmt sind. Wir zeigen, daB durch Verwendung der Split-Struktur die Eigenwertverhaltnisse der zugehorigen Kovarianzmatrizen verringert werden konnen, wodurch eine hohere Konver- genzgeschwindigkeit erreicht wird. Die parallelen und seriellen split-adaptiven Filter werden sowohl fur die Joint- Process-Schatzung als such fiir die lineare Pradiktion untersucht. Ein neuer Typ eines linearen Pradiktors wird durch Kombination beider Split-Methoden gebildet, und seine Leistungsftiigkeit fiir die Sprachanalyse wird untersucht. Simulationsergebnisse werden angegeben, urn die Uberlegenheit der vorgeschlagenen Split-Filterstrukturen hinsichtlich der Verbesserung der Konvergenzgeschwindigkeit der LMS-Adaption zu bestatigen.

R&urn&

Dans cet article, on examine les performances dun filtre adaptatif &pare dans une forme soit parallele soit serielle, fait pour les entrees non-blanches. Une structure paralltle stparte est contruite en utilisant 2 filtres a phase lineaire connectis en parallele tandis que, pour la separation sirielle, le filtre est fait dune cascade de deux filtres transversaux. Les caracteristiques d’adaptation de l’algorithme LMS sont gouvernees par la diffusion des valeurs propres de la matrice de

__- *Corresponding author

0165-1684/95/$9.50 0 1995 Elsevier Science B.V. All rights reserved

SSDI 0165-1684(95)00087-9

256 K.C. Ho, P.C. Ching J Signal Processing 46 (1995) 255-266

correlation dent& En adoptant des structures siparees, nous montrons que les rapports des valeurs propres des matrices de covariance associees peuvent 6tre rtduits, donnant lieu par la-meme a une vitesse de convergence accrue. Les filtres adaptatifs &pares serie et paralltle sont examines pour l’estimation de processus joints ainsi que pour la prediction lineaire. Un type nouveau de prtdicteur lineaire est formt en combinant les deux methodes de separation de filtre et ses performances sont itudites pour l’analyse de la parole. Des resultats de simulations sont inclus afin de valider la suptrioriti de la structure de filtres s&pares proposee en termes d’amtlioration du taux de convergence de l’adaptation LMS.

Keywords: LMS adaptive filter; Split filter structures; Convergence speed; Joint process estimation; Linear prediction

1. Introduction

The least-mean-square (LMS) algorithm is well- known for its simplicity. It has been extensively used in many applications such as noise cancella- tion, beamforming, channel equalization and others [l, 5, 10, 143. However, the convergence dynamics of the algorithm are very sensitive to the eigenvalue spread (the ratio of the maximum to the minimum eigenvalue) of the input autocorrelation matrix. In general, the larger the eigenvalue ratio, the slower the convergence rate.

An adaptive LMS filter can be realized in many configurations such as transversal, parallel, cascade and lattice forms, resulting in different properties and different dynamic convergence behaviour [lo, 21. Many of these realization methods can also be applied to IIR filtering [ll, 8,121. For LMS filter, although transversal ladder has the simplest form, it suffers from an exceptionally slow convergence rate when input data is highly correlated. The lattice filter, on the other hand, has a faster rate of adaptation but in the expense of a larger computational requirement. For most practical applications, an adaptive system is required to have a fast convergence speed in order to cope with non-sta- tionary signals and changing environments, while at the same time, the computational complexity should be kept as low as possible so that hardware implementation for real-time processing is feasible.

In this paper, two split filter structures, namely parallel and serial split, for LMS adaptive filtering are proposed. They can improve the convergence speed significantly by reducing the eigenvalue spread as well as maintain a low system complexity. The performances of the proposed adaptive structures have been analyzed and the results are

corroborated by computer simulations. The parallel split adaptive filter will first be described in Section 2, while Section 3 contains development and analysis of the serial split configuration. Com- parison of these two split techniques is given in Section 4. In Section 5, a new form of predictor is constructed by making use of both parallel and serial split method to further enhance the convergence rate. Lastly, conclusions are given in Section 6.

2. Parallel split LMS adaptive filter

Assuming that an unknown system with input x(k) and output y(k) is modelled by an adaptive filter W(z) as shown in Fig. 1. The adaptive filter can be configured (Fig. 2) as a sum of two linear phase subunits, namely P(z) and Q(z),

ZN-1

W(Z) = 1 wizei = $(P(z) + Q(Z)), i=O

N-l

P(Z) = igo pi(Z-’ - ZezN+’ +i), (1)

N-l

Q(Z) = igo qi(z-’ + zszN+l +i),

where P(z) is an antisymmetric and Q(z) is a symmetric linear phase digital filter. Let u,,i(k) = ~(k - i) - ~(k - 2N -t- 1 + i) and Uq,i(k) = ~(k - i) + x(k - 2N + 1 + i), the output of the two filters

are given by

N-l N-l

Z,(k) = C tJivp,i(k) and zp(k) = 1 qivq,i(k), i=O i=O

(2)

KC. Ho, P.C. Ching / Signal Processing 46 (1995) 255-266 257

x(k)

response

0)

_I

I I

W(z) ; e(k) error

Fig. 1. A parallel split adaptive filter.

Fig. 2. Hardware implementation of a parallel split-path adaptive filter for system modeling.

respectively, and the modelling error can be computed from

44 = YW - :(zP(w + Zq(W (3)

Since e(k) is linear with respect to pi and qi, the error surface is quadratic and has a unique minimum. I%nOtep = [pO,pl, . . . #N-l1’I;q = [qO,ql, . . . ,qN- IIT,

UP = [VP, o(k), VP, 1 (k), . *. , up, N - I MIT and uq = [%, o(k), uq, 1(k), . , Vq,N- 1 (k)lT, the LMS update equations

are as follows:

P(k + 1) = P(k) + Qb4~)~PW~ (4)

4(k + 1) = q(k) + 2cLq e(k) %(k), (5)

where pP and pq are step sizes of the two filter paths. The convergence behavior of an LMS filter update is known to be governed by the eigenvalue spread [9] of the input covariance matrix. It has been shown [6] that the eigenvalues of

258 K.C. Ho, P.C. Ching f Signal Processing 46 (1995) 255-266

f %(k)

- P(z) ~ /

x(k) 0 c e(k)

f

- Q(z) /

e,(k)

Fig. 3. A split-path adaptive prediction error filter.

R* = E[v,(k)~,(k)~] and R, = E[v,(k)~,(k)~] partition the eigenvalues of 2R = 2E[x(k)x(QT], where x(k) = [x(k), x(k - l), . . . ,x(k - 2N + l)lT is the regressor vector of W(z). There is a SO-50 chance that the eigenvalue ratios of both R, and R, are less than that of R, yielding a possible improvement in convergence. Since the trace of a matrix is equal to the sum of its eigenvalues, an intuitive choice of the two step sizes is

1 1

” = tr(Rp) and pq =-,

tr(RJ

where 1 < 1, and tr( *) is the trace operation. In practice, tr(RP) and tr(R,J are computed by moving averages of up(k)T up(k) and uq(k)T u,(k).

Following [14], since u,(k) and q(k) are uncorrelated, the total misadjustment is given by

M = M, + M, = 4 pp tr(Rp) + 4 pq tr(R&. (7)

Therefore, choosing the step sizes according to (7) not only improves the convergence speed, but also evens out the misadjustment of the two paths.

We have so far considered a general parallel split framework. In adaptive linear prediction, y(k) is zero, and po and 40 are fixed to unity. A schematic block diagram of the split-path predictor is depicted in Fig. 3 and it has two distinctive features. First of all, the outputs of the two filters are 90” out of phase because of their symmetric and antisymmetric linear phase characteristics. Thus, e,(k) and e,(k) are uncorrelated and independent adaptation

of the two paths is feasible. Secondly, the two filters include backward prediction error information because the sample x(k - 2N + 1) is also involved in the prediction process. This leads to a reduction in gradient noise. Consequently, updating the two filters separately enables a two-fold increase in convergence speed. Furthermore, the eigenvalue ratios of the associated matrices may be less than that of the parent filter, and further improvements in convergence is possible if different step sizes are used. Further details of the split-path adaptive predictor can be found in [6].

Extensive simulations have been carried out to evaluate the performance of the system. The unknown plant to be identified is

H(z) = 0.1 + 0.22-l + 0.62~-~ - 0.222-”

+ o.5z-4 + 0.42- 5. (8)

The noise was Gaussian, white with variance 0.1. The signal x(k) was generated by an AR(4) process as follows:

x(k) = 0.939x(k - 1) - O.O558x(k - 2)

+ 0.465x(k - 3) - 05184x(k - 4) + P(k),

(9)

where b(k) was Gaussian white with unity variance. The eigenvalue spreads of R, R, and R, were 154,3 1 and 45, respectively.

Fig. 4 compares the learning trajectories of the split and non-split models. The results were an

K.C. Ho, P.C. Ching 1 Signal Processing 46 (I 995) 255-266 259

0

I 6? SNR=-lOdf3

3 -1

I --_ split-path model

L 9

p

z \ i.,

p ,=0.0027, p q=O .0006

I i conventional model

$ _5- \,“\. j&J=0.0010

? \ :..

i \, ‘L.

ti - \ .Y,_, ‘\ ‘Y.*

E \ \ \-..... \ Lb\

‘Y .._..., w_

‘..? . ...__..., L. --‘I..

~‘~--Y.~. ‘.__“,~..~,..~,, ..-.-.-. ._.. -. . ..___. ._____ ,^, .._ -lO-

~--_~-r.,,,_____~__~~~~~~-~~,

0 1000 2000 3000 iterations

4000 5000

Fig. 4. Comparison of learning trajectories of the split and non-split systems.

0 1

--- split-path model pP=0.00048 , ~q=0.0~O06

conventional model

IJw=0.00010

I 8

2000 3000

iterations 00

Fig. 5. Comparison of adaptation characteristics of the split and non-split predictors

average of 2000 independent runs. The step sizes pP Fig. 5 gives the convergence dynamics of the split and c(~ were selected according to (6). It is notice- and non-split adaptive predictors. The input was able that the split configuration performed much given by (9) and the step sizes of the two systems were better and in this example, it achieved -9.5dB chosen so that they gave approximately the same mean-square error (MSE) at about 1500 iterations misadjustment. The split-path predictor was able to whereas 3200 iterations were needed for a trans- reduce the MSE to 0.5 dB at 300 iterations, which is versa1 filter. roughly 1000 iterations faster than its counterpart.

260 K.C. Ho, P.C. Ching / Signal Processing 46 (1995) 255-266

It is important to note that splitting of a parent filter may not necessarily lead to a decrease in eigenvalue spread. Suppose the input was changed to

x(k) = 0.600x(k - 1) - 0.7300x@ - 2)

+ 0.4440x(k - 3) - 0.5256x(k - 4)

+ P(k). (10)

The eigenvalue ratios of R, R, and R, were found to be 18.08, 18.08 and 3.17, respectively. Although the eigenvalue spread of Q(z) is decreased, the spread of P(z) is, however, identical to the original filter. For joint process estimation with this input, decompo- sing the filter into a parallel form will have very little improvement, if any at all. This can also be seen from the traces of Rp and R,, tr(Rp) = 11.87 and tr(R,J = 10.72. According to (6) we have pP x p’s and similar performance with the parent filter is observed. In practice, the ratio of tr(R&/tr(R,J can be used to indicate whether an adaptive filter should be split or not. If the ratio is close to unity, an FIR filter should be used, otherwise, a parallel split structure is recommended. The larger the ratio, the greater is the degree of improvement. Finally, it is worth to mention that because of a reduction in gradient noise, the split- path predictor can always double the convergence speed regardless of whether a smaller eigenvalue spread is obtained.

3. Serial split adaptive filter

Apart from parallel split, W(z) can also be constructed in cascade form [7,2]. This configuration is found to be useful in linear prediction of speech signal because it offers simple stability monitoring and is less sensitive to finite word length effect. It will be demonstrated that another merit of this structure is that it can provide a better convergence behavior. If the filter order of W(z) is 2N, where N = even, then W(z) can be separated into two parts:

W(z) = i&viz-i = A(z)C(z)

/N \/N \

(11)

x(k)

Fig. 6. A serial split adaptive system.

where ai and cj are the filter coefficients of A(z) and C(z). When N is odd, the order of A(z) is chosen to be N + 1 while that of C(z) is assigned to be N - 1 in order to eliminate the possibility of having com- plex filter coefficients. The parameter co is fixed to unity to keep the number of adapting parameters to 2N+l.

A typical serial split adaptive filter structure is shown in Fig. 6. Define vectors (z = [aI, a,, . . . , aNIT,

c = cw2 ,..., C~]T,X(k-l)=[x(k-l),x(k-2) ,..., x(k - N)lT and matrix X(k - 1) = [x(k - 2), x(k - 3), . . . , x(k - N - l)lT, the output error can be expressed as

e(k) = y(k) - x(k)ao - x(k - l)Ta

- cTx(k - l)ao - cTX(k - 1)~. (12)

The MSE, 5 = E[e(k)‘], is a function of order four in ao, (I and c together, and the error surface is not convex. There exist two minimum (due to the sym- metry of (I and c) and a proportionate number of saddle points [S]. After taking partial derivatives of e(k) with respect to the adapting parameters, the LMS adaptation are adk + 1) [ 1 a(k + 1)

c(k + 1)

= c(k) + 2pe(k) [x(k - 1) X(k - l)] 2::; . [ 1 (14)

K.C. Ho, P.C. Ching / Signal Processing 46 11995) 255-266 261

The algorithm (13) and (14) has O(N’) operations. If the step size /A is small, the gradient of ai- i(k) is roughly equal to that of ai(k - 1) and the same is true for the gradients of ci- i(k) and ci(k - 1). Fur- thermore, the gradient for adjusting cl(k) is readily available when the error e(k) is being computed. The algorithm thus requires only O(N) operations in contrast to LMS, which needs N additional multiplications and additions.

Because the error surface is not quadratic, the convergence behavior depends on the initial filter settings. The effective input to C(z) is a filtered version of x(k) by A(z). Thus the front-end subfilter will affect the correlation of the input to the sub- sequent subfilter and in turn will affect the overall convergence speed. This implies that the performance is dependent on the characteristics of both the input and the desired response of the adaptive units. In joint process estimation, there is no guarantee of any noticeable improvement over a conventional transversal filter.

However, if the serial split adaptive system is applied to linear prediction, a significant improvement in adaptation speed may be achieved. In this application, y(k) is set to zero and both a0 and co are fixed to unity. The adaptive algorithm is similar to (13) and (14) except a0 is not adapted. The major role of the two subunits is to successively whiten the correlated input sequence x(k) and the

output of the equivalent prediction filter {A(z)C(z)} should possess a uniform spectrum at steady state. The sequence produced by filtering x(k) through either A(z) and C(z) will be less correlated than the original input and therefore will give a faster convergence rate. It should be noted that due to the non-quadratic error surface, the adaptation behavior might be affected by the initial conditions.

Experiments have been conducted to study the convergence dynamics of the proposed serial split model. The input signal was generated recursively from

x(k) = 2.3588x@ - 1) - 2.6970x(k - 2)

+ 1.6456x@ - 3) - 0.5184x@ - 4) + /3(k)

(15)

and it was passed through the serial split predictor and the transversal adaptive predictor separately. The variable j?(k) was a zero mean Gaussian dis- tributed random sequence with unity power. The optimal prediction filter of (15) is

W(z)* = (1 - 1.55882-l + 0.81~-~)

x (1 - 0.8z-’ + 0.64~-~), (16)

The learning trajectories of the two adaptive models are depicted in Fig. 7. The step sizes were selected to keep the misadjustment roughly the

--- serial split model Jl=O.o008

. . . . . . . . . non-split model po.0002

1000 2000

iterations

,,..‘::

iv

301

Fig. 7. Comparison of convergence rate between the serial split and the transversal model.


same in both cases. To allow A(z) and C(z) converge to different values, they should have different initial values. Otherwise, they will remain locked up throughout adaptation as their gradients are essen- tially identical. A simple initialization method is to distribute zeros evenly on the z-plane. In our tests, they were initialized to

A(z) = 1 - z-i + z-2, C(z)= 1 +z-’ +z-2,

(17)

which corresponded to zero pairs at 1 L f 60” and 1 L + 120”. For the transversal model, W(z) is initialized to the product of the two in (17). From Fig. 7, it is noted that the serial split filter can adapt at a much faster speed. It took about 700 iterations to reach an MSE of value 0.5 dB whilst the transversal filter needed an exceedingly long time to converge.

Although in most of our simulations we experi- ence an improvement in convergence speed, there are cases where the convergence dynamics are getting worse, which is mainly because of wrong initial settings. The effect of different initial conditions is not precisely known at this point and there is no guarantee that the initialization method suggested will always provide an improvement in convergence behavior. Further investigation on this issue is undergoing.

4. Comparison of the two types of split filters

It has been demonstrated that both parallel and serial split model can achieve a speed-up in convergence by reducing the correlation of adaptive filter input. However, the mechanism for the two ap- proaches to accomplish the objective is quite different. For parallel split, there are two sets of eigenvalues, one for each path. The input correlations of the two subfilters can be smaller than that of the parent filter and will remain unchanged throughout adaptation. On the other hand, the serial split model modifies the correlation by filtering the in- coming signal through the other filter. Since the adaptive subunits are time-varying, the degree of decorrelation varies until steady state is reached.

In joint process estimation, the parallel split model is preferred. The partition of eigenvalues in a parallel system allows different step sizes to be

used for the two paths so as to increase the convergence speed. For serial split, the filtering characteristics of the two subunits depends on the desired response and a reduction in correlation is not assured.

For linear prediction, both split methods can attain a better performance. The parallel split method incorporates the backward prediction to the adaptation to diminish gradient noise so that larger step sizes can be chosen to further increase the adaptation rate. In serial split model, the role of the two cascades is to flatten the input spectrum. As long as the two subfilters are properly initialized, the input to the two subunits will be whitened and thus giving a faster convergence.

Simulations were performed to compare the adaptation characteristics of the two split predictors. The AR process given in (15) with an eigenvalue spread of 6432 and the one in (9) having an eigenvalue spread of 154 were to be whitened by the two models. The initial conditions for the two subunits of the serial split model were set according to (17). The step sizes of both systems were assigned to some predefined values to maintain the same excess MSE. For the input process as given by (15), Fig. 8 indicates that the serial split method outper- formed the other and reached an MSE value of 2 dB at about 200 iterations, whilst the parallel split predictor took 1100 iterations instead. However, when the input process of (9) was used, the parallel split model gave a better performance and this is illustrated in Fig. 9.

These observations can be explained as follows. When the input signal has a large eigenvalue spread, partition of the eigenvalues could not in general lessen the ratio. In this case, the serial model will perform better because it can continu- ously decrease the correlation. On the other hand, when the input process has a relatively small eigenvalue spread, the serial split predictor could hardly reduce the correlation any further. Whereas par- titioning the correlation matrix can effectively decrease the eigenvalue spread. This merit together with its capability in reducing gradient noise makes the parallel split predictor more attractive. Hence, we conclude that a serial predictor is more suitable for input having larger eigenvalue spread while a parallel predictor is preferred when the input eigenvalue ratio is relatively small.

K.C. Ho, P.C. Ching / Signal Processing 46 (1995) 255-266 263

--_ serial split model ~=0.0008

. . . . . . . . parallel split model po.0002

I . .I, 1.. , ‘. , . .

0 500 1000 1500 2

iterations

Fig. 8. Comparison of convergence rate between the parallel and serial split models with large eigenvalue spread input

.-_ parallel split model

/~~=0.00048 , p,=o.Oooo6

5 - ij

k I\ ......... serial split model 1: \;

~,=0.00013 Lz cu :, ,_t

K !L

$ i

% - f.,

E \... I ,; . . . . . . ...‘.‘!., C~,

.. “.“-x”.; ,,, o _ \‘,‘, p r ‘..’ >;;;yL.&;Q..L&&.

J v-q,y .,..._..._.. ;..

\ / ._----~~~.~~&&.(+/-

I I 0 1000 2000 3000 4000 501

iterations

30

Fig. 9. Comparison of convergence rate between the parallel and serial split models with small eigenvalue spread input.

Regarding complexity, the parallel split method is better because it requires extra additions only, whereas for serial split, additional multiplication are needed.

5. A new split predictor

We have separately examined the parallel split and the serial split adaptive techniques. Both con-

figurations have proven to be more advantageous than their respective non-split counterparts in linear prediction. This motivates the idea of merging the two types of split predictors together to form a new one that incorporates all the advant- ages.

The simplest way to combine the two methods is, first of all, parallel split an adaptive predictor W(z) with independent adjustment of


the two filter paths. This gives a two-fold increase in adaptation speed by reducing the gradient noise. Serial split is then applied to the two linear phase subfilters. A further improvement in convergence speed may be acquired because the associated input correlations of the serial split adaptive subunits are getting smaller as adaptation goes on. Notice that one can assign different step sizes to the two split-paths to achieve additional gain in convergence rate. The block diagram of this new model of order 2N - 1 is shown in Fig. 10,

where N/2

PA(Z) = 1 pa,i(Zmi + Z-N+i)p

i=O

(N/Z) - 1

PC(Z) = C Pe,i(Zmi + ZpN+l +‘), N = even; i=O

(18) W- 1W

PA(Z) = 1 Pa,i(Zmi + ZmN+’ +i)p

i=O (N- I)/2

PC(Z) = 1 pc,i(Z-’ + Z-N+i), A’ = odd. i=O

Fig. 10. The combined model of a parallel and a serial split predictor.

6 I\

63 I I\ _--_ z p,,

combined model ----.--.. parallel split model

I I 1 ,

0 200 400 600 600 1

iterations

Fig. 11. Comparison of convergence rate for a parallel split, a serial split and a combined predictor.

K.C. Ho, P.C. Ching / Signal Processing 46 (1995) 255-266 265

Table 1 Comparison of complexity in the application of joint process estimation

+ * + Relative complexity (* = +)

LMS PS_LMS SS-LMS Gradient lattice RLS FTF [4]

2M+2 2Mf2 0 1 3M + 2.5 2M + 2.5 0 1.25 2.5M + 2.5 2.5M + 2.5 0 1.25 lOA!+ llMf3 2M+l >5 G(M’) Wf’) O(M) G(M) -1M -7M 2 3.5

Table 2 Comparison of complexity in the application of linear prediction

+ * + Relative complexity (* = +)

LMS PS_LMS SS-LMS PSS_LMS PSSKLMS (2ndOC) Gradient lattice Cl] RLS FTF [4]

2M+l 2M 0 1 3M + 0.5 2M + 0.5 0 1.25 2.5M + 0.5 2.5M - 0.5 0 1.25 3.5M + 1 2.5M 0 1.5 4M+3 3M 0 1.75 6M 6M M >3

O(M’) O(M’) O(M) O(M) -5M -5M 2 - 2.5

Note: PS-LMS: parallel split LMS. SS-LMS: serial split LMS. RLS: recursive least square. FTF: fast transversal filter. PSS_LMS: parallel serial split LMS. PSS_LMS (2ndOC): parallel serial split LMS with second-order cascades.

Simulation tests have been conducted to examine the convergence dynamics of this prediction model. The input sequence generated by (15) was to be identified separately by this new predictor, the parallel split model in Fig. 3 and the serial split model in Fig. 6. The step sizes for the new system and the serial split model were both set at 0.0008 and that of the parallel system was chosen to be 0.0002 in order to produce the same misadjustment. The averaged squared output errors of the three models, E[~(/c)~], are depicted in Fig. 11. The combined structure had the best performance and the MSE was reduced to a value of 1 dB at about 80 iterations, which is 200 iterations less than the serial

split approach. Other experiments were also performed and they verified the superiority of this combined split prediction model.

To further study the applicability of the new predictor, we have used it for speech analysis. Be- sides the improvement in adaptation rate, it was illustrated in [3] that when the adaptive subunits in Fig. 10 are constructed as cascade of second-order sections, the filter parameters will be equivalent to the line spectral pair (LSP) coefficients, which are particularly efficient for low bit rate transmission [13]. It is found that the performance of the new predictor for small prediction order is close to the gradient lattice filter [l], and requires less


computations. Implementation details of the new speech analysis system can be found in [3].

6. Conclusions

Two split filter structure, viz. the parallel and the serial split, for LMS adaptive filtering have been studied. In both methods, the correlation of the associated inputs are reduced which lead to a better convergence characteristics. Applications of the two split models to joint process estimation and linear prediction have been investigated. It is shown that parallel split is suitable for input signal with small eigenvalue spread while serial split is useful for input signal having large eigenvalue ratio. A new form of predictor by combining the two models is constructed. If the cascades are of second- order sections, its coefficients become the LSP parameters. The new predictor can provide a convergence speed close to the gradient lattice filter but with a simpler system complexity.

Tables 1 and 2 show the complexity of the LMS algorithm, the split LMS algorithms and some other adaptation methods for joint process estimation and linear prediction. The amount of actual extra cost in computation will depend on practical implementation, however, it will never exceed 25% of the total requirement of the LMS algorithm, assuming that the cost of addition and multiplication is identical.

Acknowledgements

The authors are grateful to the anonymous re- viewers for their many constructive comments and suggestions on the original manuscript. One of the authors K.C. Ho would like to thank the Croucher Foundation for the award of a studentship to sup- port the work of this project. This project is also

partially supported by a research grant awarded by the Hong Kong Research Grant Council.

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split filter structures for lms adaptive filtering

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