AUAT -- 0.02nm/"C, slightly higher than values we find in the literature for silica-germania fibres [5]. In Fig. 5 we can see the effect on the intensity transfer function. The next FFP resonance is centred in the bandpass grating, so the device filtering is now centred at 1526.400nm, with the same BW as before, I70MHz. In this way, we can tune the grating bandpass over the different FFP free spectral ranges which are only limited by the grating thermal stability. As it can be heated up to a few hundred degrees, the filter structure can be tuned over a range of several hundred gigahertz. Alternatively, a stress tuning mechanism can be used, although it has not been reported here.

5

I

l

t

II

0.8

U

l

I

0

!A

A

I

i' I

,

I

I

1

Fig. 5 Filter response showing eflect on intensity transfer function

room temperature

The second stage is the fine tuning, which consists of tuning the FFP resonance along the selected (by the FG) free spectral range by means of a tuning voltage. By applying voltages between 0 and 12V, the FFP resonances have been smoothly shifted as expected. Summary and conclusions: We have presented the first experimental demonstration of a novel continuously tunable ultrahigh optical Wter. This structure combines the high selectivity of the FFP and the bandpass nature of the fibre grating. The structure operation has been presented and two tuning stages have also been addressed, achieving a maximum tuning range limited by the thermal stability of the grating. Acknowledgments: This work has been funded by Spanish CICYT TIC95-0859-CO2, the Valecian Regional Government (FPI grant) and Pirelli Cavi S.P.A.

0 IEE 1997 Electronics Letters Online No: 19907444

R.I. Laming (Optoelectronics Centre, University of Southampton, Southampton, SO9 5NH, United Kingdom)

References

2 3

4 5

'Dense wavelength division multiplexing networks:

principles and applications', ZEEE J. Sel. Areas Conzmun., 1990, 8, pp. 948-964 CAPMANY, J., LAMING, R.I., and PAYNE, D.N.: 'Novel highly selective and tunable optical bandpass filter using a fibre grating and a Fabry-Perot', Microw. Opt. Technol. Lett., 1994, pp. 499-501 COLE, M J., LOH, w.W., LAMMING, R.I., ZERVAS, M.N., and BARCELOS, s.: 'Moving fibreiphase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask', Electron. Lett., 1995, 31, pp. 1488-1489 KOGELNIK, H.: 'Filter response of nonuniform almost-periodic structures', B.T.S.J., 1976, 55, pp. 109-126 LAUZON, J., THIBAULT, s., MARTIN, J., and OZJELLETTE,F.: 'Implementation and characterisation of fibre Bragg gratings linearly chirped by a temperature gradient', Opt. Lett., 1994, 19, pp. 2027-2029

ELECTRONICS LETTERS

Indexing terms: Adaptivefilters, Volterra series

The authors propose filtered-X second-order Volterra adaptive algorithms based on a multichannel structure application to active noise control. The developed algorithms can be used as alternatives in the case where the standard filtered-X LMS algorithm does not perform well. Introduction: The filtered-X least mean square (LMS) algorithm has been successfully applied in the area of active noise and vibration control [l]. In the case where the standard filtered-X LMS algorithm does not perform well, a fdtered-X nonlinear adaptive algorithm can be used as an alternative. Recently, adaptive nonlinear filters such as adaptive second-order Volterra fiters [2, 31 using the LMS and recursive least squares (RLS) algorithms are investigated. The investigations have dealt with the improvement of convergence speed and reduction of computational complexity for the developed algorithms. However, the applications of the adaptive Volterra filters to active noise control have not been addressed. Therefore in this Letter, we use the adaptive secondorder Volterra filters for the applications of active noise control and present the filtered-X second-order Volterra adaptive algorithms based on a multichannel structure.

10th April 1997

Vol. 33

N-1

N-1 N-1

i=O

i=o

j=1

where x(n) and y(n) represent the filter input and output, respectively; h,(i) and hl(ij) the linear and quadratic filter coefticients, respectively. If we rewrite eqn. 1 as N-l

N-1

i=O

i=O

N-2

z=o we can realise this second-order Volterra filter by FIR multichannels. The relationship between the filter output and N+1 channel inputs is expressed as

I 1 February 1997

B. Ortega, J. Capmany, and D. Pastor (Departamento de Comunicaciones, ETSI Telecomunicacion, Camino de Vera s/n, 46071 Valencia, Spain)

BRACKET, C.A.:

Li-Zhe Tan and Jean Jiang

Multichannel structure and algorithms: Consider a second-order Volterra filter described by the input-output relationship

- _ _ - A T = 13.5"C

1

Filtered-X second-order Volterra adaptive algorithms

N-1

3=0

where H,(z) = ZEihh,(z]zl,H&) = C:$'h,(i)z-', Y(z) is the ztransform of the filter output, X(z) and X,(z) are the z-transforms of the correspondingchannel inputs (x(n)and x2,(n)x(n)x(n-j),j = 0, 1, ..., N-1). Hence, we transform the nonlinear second-order Volterra filter to a multichannel input linear filter. Using the multichannel structure, we now propose a codiguration of the filtered-X second-order Volterra adaptive algorithm for the active noise control system illustrated in Fig. 1. As shown in Fig. 1, d(n) is the primary disturbance obtained from the output of the nonlinear primary path with the reference signal x(n) as input and e(n) is the residue noise given by

e(n) = d ( n ) - s(n)* y(n) (4) where s(n) is the impulse response of the secondary path S(z) (from the adaptive filter output to the error microphone output) and * denotes linear convolution. Notice that y(n) is the output of the adaptive second-order Volterra filter expressed as y(n) = HT(n)X(n), where the input vector X(n) and corresponding adaptive coefficient vector H(n) at time index n are defined as X ( n ) = [ z ( n ) , z (n l), ..., z ( n - N + l), z 2 ( n ) , z 2 (n l),...,z ( n ) z ( n- N + 1)IT (5)

No. 8

671

and

H ( n ) = [h1(O;n),h1(l;n) ,...,h l ( N

l;n),

-

hzo(0;n ) ,h2o(l;n),...,h2,N-1(0;n)IT (6) By modifying the filtered-X LMS algorithm [4], we obtain the filtered-X second-order Volterra LMS algorithm with the coefficient adaptation as follows:

+

+

H ( n 1) = H ( n ) p e ( n ) U ( n ) (7) where p is a diagonal matrix with pL for the fEst N diagonal entries and pn for the rest of the diagonal entries, and U(n) is the filtered signal vector shown below U ( n )= [u(n),u(n - l), ...)u(n - N l),

+

( 1 4 . 3 5 ~ ~We ) . choose the memory span of the adaptive secondorder Volterra filter as N = 9 so that the number of filter coefficients is 54(N(N+3)/2). We set the step sizes as pL = 0.01 and pe = 0.001 for the filtered-X second-order Volterra LMS, and the forgetting factor as 1 = 0.995 for the fdtered-X second-order Volterra RLS. As a comparison, we also use the standard filtered-X LMS algorithm with a fiter order of 54 for the adaptive FIR filter, and set the step size as 0.0002. The squared residue errors are plotted in Fig. 2 for 3500 iterations. As illustrated in Fig. 2, the standard filtered-X LMS performs poorly, while both developed algorithms can perform well to reduce the primary disturbance. Specially, the filtered-X second-order Volterra RLS algorithm achieves the significantly improved performance in terms of convergence speed.

...,~

UZO(~),U~ OI), (~

, ~ - i ( n ) ](8) ~ Note that the dimension of X(rz), H(n) and U(n) is (N(N+3)/2) x 1. Based on the multichannel structure, the N+l independent elements of U(n) can be obtained from U(n) = s^(n)* X(n), where X(n> = [x(n),x,o(n),x,,(~), X2.H-1 @>iTN.F!, XI, U ( 4 = [u(n),U2o(n),u,,(n), ..., u2N-l (n)]&+,, s^(n)is the lmpulse response of the secondary path estimate S(z). Note that u(n) = s"(n)* x(n) and U&) = s^(n)*{x(n)x(n-j)},j = 0,1, ..., N-1. The implementation is also depicted in Fig. 1.

20

I

-80

I

1

---2

nonlinear primary path

G

0

I 1000

2000

3000

number of iterations

Fig. 2 Performance comparison for filtered-X adaptive algorithms

(i) standard filtered-X LMS (ii) filtered-X second-order Volterra LMS (iii) filtered-X second-order Volterra RLS

e hi

I

'

-

1

I

Conclusion: We have developed the filtered-X second-order Volterra adaptive algorithms based on a multichannel structure for active noise control. The simulation results demonstrate that the developed algorithms outperform the standard fitered-X LMS algorithm when the primary path has a quadratic nonlinearity. The ftltered-X second-order Volterra RLS algorithm converges very fast at the cost of an increase in the computational complexity.

I

LMS o r RLS

376111 Fig. 1 Active noise control system using filtered-X second-order Volterra adaptive algorithm

Similar to the derivation in [4],we propose the filtered-X second-order Volterra RLS algorithm as following

e(.)

= d ( n ) - s ( n ) * y(n) = d(n)- s(n) * { ~ ~ ( n ) ~ ( n ) }

(9) X-lP(n - l ) U ( n ) k(n)= 1+ X-lUT(n)P(n - l ) U ( n ) H(n

+ 1) = H ( n ) + k ( n ) e ( n )

24 February 1997 Electronics Letters Online No: 19970477 Li-Zhe Tan and Jean Jiang (Interactive College of Technology, 5303 New Peachtree Road, Atlanta, Georgia 30341, USA)

References ELLIOTT, s.J., and NELSON, P.A.: 'Active noise control', IEEE Sig. Process. Mag., 1993, pp. 12-35 2 MATHEWS, v.J.: 'Adaptive polynomial filters', IEEE Sig. Process. Mag., 1991, pp. 10-26 3 TAN, L.z., and HANG, J.: 'Adaptive second-order Volterra delay filter', Electron. Lett., 1996, 32, (9), pp. 807-809 4 KUO, s.M., and MORGAN, D;.: Active noise control systems-

1

(10) (11)

P ( n ) = X-lP(n - 1) - X - ' k ( n ) U T ( n ) P ( n - 1) (12) where h (0 e h 5 1) is the forgetting factor, k(n) the gain vector and U(n) defined in eqn. 8. Note that e(n) is the residual noise measured by the error sensor; and the coefficient vector in eqn. 11 is shifted by one sample as compared with the one in the standard RLS since the coefficient vector is required before the next sample arrives in order to generate the adaptive filter output y(n) in eqn. 9.

Simulation: In the following simulation, we use the zero-mean Gaussian noise with a variance of 1.0 as the input signal and obtain the primary disturbance d(n) with a quadratic nonlinearity by the following procedure. We first filter the input x(n) using a delayed bandpass filter given by C,(z) = ~-~/(1-0.22~) to generate p(n), where p(n) = c,(n)* x(n);we then obtain q(n) using p(n) and a quadratic function in the following expression q(n) = p(n)+0.5p2(n);we finally filter q(n) via another delayed bandpass filter represented by C,(z) = ~ ~ / ( 1 4 . 2to5 give ~ ~ )the primary disturbance d(n) = c,jn)* q(n). Notice that c , ( ~ )and c&) are the impulse responses of C,(z) and C,(z), respectively. The secondarypath transfer function is assumed to be S(z) = ~ - ~ / ( 1 + . 3 9(a ) delayed bandpass) while the secondary path estimate is S(z) = z2/

672

0 IEE 1997

algorithms and DSP implementations' (John Wiley and Sons, New York, 1996)

Modified Moose estimator for tracking highly manoeuvring targets Chi-Min Liu and Kuo-Guan Wu Indexing terms: Target tracking, Kalman filters

The Moose's adaptive state estimator has proved successful in tracking a manoeuvring submarine 111. However, its application to track highly manoeuvring aircraft will encounter problems in complexity and tracking accuracy [2, 31. The authors present a new algorithm that extends the Moose estimator to solve its problems. The new algorithm is verified through Monte-Carlo simulation and shows a better performance than the origmal Moose algorithm.

ELECTRONICS LETTERS

10th April 1997

Vol. 33

No. 8