EURASIP Journal on Applied Signal Processing 2004:12, 1841–1848c© 2004 Hindawi Publishing Corporation

Filtered-X Affine Projection Algorithms forActive Noise Control Using Volterra Filters

Alberto CariniInstitute of Science and Information Technologies, University of Urbino, 61029 Urbino, ItalyEmail: carini@sti.uniurb.it

Giovanni L. SicuranzaDepartment of Electrical, Electronic and Computer Engineering, University of Trieste, 34127 Trieste, ItalyEmail: sicuranza@univ.trieste.it

Received 1 September 2003; Revised 22 December 2003

We consider the use of adaptive Volterra filters, implemented in the form of multichannel filter banks, as nonlinear active noisecontrollers. In particular, we discuss the derivation of filtered-X affine projection algorithms for homogeneous quadratic filters.According to the multichannel approach, it is then easy to pass from these algorithms to those of a generic Volterra filter. It isshown in the paper that the AP technique offers better convergence and tracking capabilities than the classical LMS and NLMSalgorithms usually applied in nonlinear active noise controllers, with a limited complexity increase. This paper extends in twoways the content of a previous contribution published in Proc. IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing(NSIP ’03), Grado, Italy, June 2003. First of all, a general adaptation algorithm valid for any order L of affine projections ispresented. Secondly, a more complete set of experiments is reported. In particular, the effects of using multichannel filter bankswith a reduced number of channels are investigated and relevant results are shown.

Keywords and phrases: active noise control, adaptive Volterra filters, affine projection algorithms.

1. INTRODUCTION

Methods for active noise control are nowadays intensivelystudied and have already provided promising applicationsin vibration and acoustic noise control tasks. The initial ac-tivities originated in the field of control engineering [1, 2],while in recent years a signal processing approach has beensuccessfully applied. This approach strongly benefited of theadvances in electroacoustic transducers, flexible digital sig-nal processors, and efficient adaption algorithms [3, 4]. Thetechnique used in a single-channel active noise controller isbased on the destructive interference in a given location ofthe noise produced by a primary source and the interferingsignal generated by a secondary source.

Most of the studies presented in the literature refer to lin-ear models, while it is often recognized that nonlinear effectscan affect actual applications [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].Such effects may arise from the behavior of the noise sourcewhich rather than a stochastic process may be depicted as anonlinear deterministic noise process, sometimes of chaoticnature. Moreover, the primary path may exhibit a nonlinearbehavior thus motivating the use of a nonlinear controller.

Recently, a model for a nonlinear controller based onVolterra filters [15] has been presented in [9] with interest-

ing results. The model actually exploits the so-called diagonalrepresentation introduced in [16]. This representation allowsa truncated Volterra system to be described by the “diagonal”entries of its kernels. In fact, if the pth-order kernel is repre-sented as a sampled hypercube of the same order, the diag-onal representation implies the change of the Cartesian co-ordinates to coordinates that are aligned along the diagonalsof the hypercube. In this way, the Volterra filter can be rep-resented in the form of a filter bank, where each filter corre-sponds to a diagonal of the hypercube. This representation isparticularly useful for processing carrier-based input signals,since the frequency content of the output signal is directlyrelated to the frequency response of the diagonal elements ofthe kernel. It has been exploited in the derivation of efficientimplementations of Volterra filters for processing carrier-based input signals using fast convolution techniques [16].

A similar representation has been used in [17, 18] todefine the so-called simplified Volterra filters (SVF). Eventhough this model can be applied to kernels of arbitraryorder, the simplest example of a Volterra filter, that is, thehomogeneous quadratic filter, has been specifically consid-ered in [17, 18] with reference to the acoustic echo can-cellation problem. An SVF is still implemented as a filterbank, but here the stress is on the fact that, according to

1842 EURASIP Journal on Applied Signal Processing

the characteristics of the measured second-order kernel, it ispossible to use a reduced number of active channels. In fact,it has been noted that the relevance of the quadratic kernelelements is strongly decreasing moving far from the maindiagonal. Therefore, remarkable savings in implementationcomplexity can be achieved. It is worth noting that similarbehaviors have been observed also in other real-world non-linear systems.

The other relevant idea in [17, 18] is that, exploiting theSVF structure, it is possible to extend the affine projection(AP) adaptation algorithm, originally proposed by Ozeki andUmeda [19] for linear filters, to quadratic filters. It has beenshown that the AP technique offers also for quadratic filtersbetter convergence and tracking capabilities than the classicalLMS and NLMS.

Based on these premises, we present in this paper a cou-ple of novel filtered-X AP (F-X AP) algorithms for non-linear active noise control. The derivation of these algo-rithms is given according to the SVF model of a homoge-neous quadratic filter. Extensions to higher-order Volterrafilters can be obtained using the general diagonal represen-tation mentioned above. The advantages given by the AP al-gorithms can be appreciated when changes in the character-istics of the noise source or of the room acoustics occur. It isalso shown that the complexity increase with respect to theF-X LMS algorithm may be relatively small.

The outline of our paper is the following. In Section 2,the quadratic model used for the nonlinear active noise con-trol is briefly described. The derivation of the novel F-X APalgorithms for homogeneous quadratic filters is presented inSection 3. Results of extensive computer simulations are pre-sented in Section 4 for typical nonlinear situations and a fewconcluding remarks are given in Section 5.

2. MODELING THE NONLINEARACTIVE NOISE CONTROLLER

A single-channel acoustic noise controlling scheme is de-picted in Figure 1. The corresponding block diagram isshown in Figure 2. The noise source is sensed by a referencemicrophone and the primary path P consists of the acousticresponse from the reference microphone to the error micro-phone located at the canceling point. The signal to be atten-uated is marked as dp(n). The reference microphone collectsthe samples x(n) of the noise source and feeds them as inputto the adaptive controller. The controller is adapted accord-ing to the feedback signal e(n) coming from the error mi-crophone. The controller output y(n) generates an acousticsignal that, traveling through the secondary path S, gives asignal ds(n) which destructively interferes with the undesiredsignal dp(n). It is usually assumed that the secondary path islinear and time invariant, and that its impulse response s(n)has been obtained by separate estimation procedures. Then,the signal ds(n) is given as the linear convolution of s(n) withthe signal y(n) and thus the error signal can be expressed ase(n) = dp(n) + ds(n) = dp(n) + s(n) ∗ y(n), where ∗ in-dicates the operation of linear convolution. In the nonlinearsituation we are dealing with, the controller is described as

Adaptivecontroller Secondary path

e(n)y(n)x(n)

Referencemicrophone

Errormicrophone

Primary path

Noisesource

Figure 1: Single-channel adaptive controller.

F-X AP

Volterrafilter

y(n)S

ds(n)

x(n)P

dp(n)+

e(n)

Figure 2: F-X AP adaptive nonlinear controller.

FIR×z−1

FIR×z−1

FIR×z−1

FIR×x(n)

+y(n)

Figure 3: Simplified Volterra filter structure.

a Volterra filter implemented by a multichannel filter bank.The adaptation of the nonlinear filter is controlled by meansof the F-X AP algorithms introduced in the next Section.

3. SINGLE-CHANNEL F-X AP ALGORITHMS

We illustrate here the main steps leading to the derivation ofF-X AP algorithms for a homogeneous quadratic filter repre-sented as a filter bank according to the SVF implementationshown in Figure 3. The output y(n) in Figure 3 is obtained as

y(n) =M∑

i=1

yi(n), (1)

where M is the number of channels actually used, with M ≤N . N is the memory length of the quadratic filter and yi(n)is the output of the FIR filter in the generic channel i in

Filtered-X Affine Projection Algorithms 1843

Figure 3, given by the following equation:

yi(n) =N−i∑

k=0

h(k, k + i− 1)x(n− k)x(n− k − i + 1). (2)

Using a vector notation, (2) becomes

yi(n) = hTi (n)xi(n), (3)

where hi(n) is the vector formed with the N−i+1 coefficientsof the ith channel (1 ≤ i ≤M),

hi(n) = [h(0, i− 1) h(1, i) · · · h(N − i,N − 1)]T. (4)

The corresponding input vector xi(n), again formed withN−i + 1 entries, is defined as

xi(n) =

x(n)x(n− i + 1)x(n− 1)x(n− i)

...x(n−N + i)x(n−N + 1)

. (5)

If we now define two vectors of K =∑Mk=1(N−k+1) elements

h(n) = [hT1 (n) · · · hT

M(n)]T

,

x(n) = [xT1 (n) · · · xT

M(n)]T (6)

formed, respectively, with the vectors hi(n) and xi(n) relatedto single channels of the filter bank, then the output of thehomogeneous quadratic filter can be written as

y(n) = hT(n)x(n). (7)

While the F-X LMS algorithm minimizes, according to thestochastic gradient approximation, the single error at time n,

e(n) = dp(n) + ds(n) = dp(n) + s(n)∗ [hT(n)x(n)], (8)

the aim of an F-X AP algorithm of order L is to minimize thelast L F-X errors. The desired minimization can be obtainedby finding the minimum norm of the coefficient incrementsthat set to zero the last L a posteriori errors. More in detail,the a posteriori error at time n − j + 1, with j = 1, . . . ,L, isdefined as

ε(n− j + 1) = dp(n− j + 1) + s(n− j + 1)

∗ [hT(n + 1)x(n− j + 1)].

(9)

The set of constraints for an Lth order F-X AP algorithm iswritten in the following form:

dp(n) + s(n)∗ hT(n + 1)x(n) = 0,

dp(n− 1) + s(n− 1)∗ hT(n + 1)x(n− 1) = 0,

...

dp(n− L + 1) + s(n− L + 1)∗ hT(n + 1)x(n− L + 1) = 0.(10)

The function J to be minimized can then be defined as

J = δhT(n + 1)δh(n + 1)

+L∑

j=1

λj(dp(n− j + 1) + s(n− j + 1)

∗ hT(n + 1)x(n− j + 1)),

(11)

where

δh(n + 1) = h(n + 1)− h(n) (12)

and λj are Lagrange’s multipliers. By differentiating J withrespect to δh(n + 1), the following set of K equations is ob-tained

2δh(n + 1) = −L∑

j=1

λjs(n− j + 1)∗ x(n− j + 1), (13)

where the linearity of the convolution operation has been ex-ploited. Equation (13) can be also rewritten as

2δh(n + 1) = −G(n)Λ, (14)

where the K × L matrix G(n) is defined as

G(n)

=[s(n)∗x(n) s(n−1)∗x(n−1) · · · s(n−L+1)∗x(n−L+1)],

(15)

Λ = [λ1λ2 · · · λL]T. (16)

By premultiplying equation (14) by GT(n), the followingequation is obtained:

Λ = −(GT(n)G(n))−1

GT(n)2δh(n + 1). (17)

The L× 1 vector GT(n)2δh(n + 1) can be also written as

GT(n)2δh(n + 1)

= 2

{s(n)∗ xT(n)

}δh(n + 1){

s(n− 1)∗ xT(n− 1)}δh(n + 1)

...{s(n− L + 1)∗ xT(n− L + 1)

}δh(n + 1)

.

(18)

Apart from the factor 2, the jth element of this vector ( j =1, . . . ,L), can be written, after some manipulations, as

[s(n− j + 1)∗ xT(n− j + 1)

][h(n + 1)− h(n)

]

= s(n− j + 1)∗ [hT(n + 1)x(n− j + 1)]

− s(n− j + 1)∗ [hT(n)x(n− j + 1)]

= −dp(n− j + 1)− s(n− j + 1)

∗ [hT(n)x(n− j + 1)] = −ej(n).

(19)

In deriving this expression, the linearity of the convolu-tion operation and the constraints given by (10) have been

1844 EURASIP Journal on Applied Signal Processing

Initialization: hi = 0 ∀iy(n) =∑M

i=1 hTi (n)xi(n)

ej(n) = dp(n− j + 1) + s(n− j + 1)∗∑Mi=1 hT

i (n)xi(n− j + 1)

for j = 1, . . . ,L

e(n) = [e1(n) e2(n) · · · eL(n)]T

Gi(n) =[s(n)∗ xi(n) s(n−1)∗xi(n−1) · · · s(n−L+1)∗xi(n−L+1)]

hi(n + 1) = hi(n)− µiGi(n)(GT(n)G(n))−1e(n) for i = 1, . . . ,M

Algorithm 1: F-X AP adaptive algorithm of order L for an SVFusing the direct matrix inversion.

exploited. As a conclusion, (18) can be rewritten as

GT(n)2δh(n + 1) = −2e(n), (20)

where e(n) is the L × 1 vector of the F-X a priori estimationerrors,

e(n) = [e1(n) e2(n) · · · eL(n)]T. (21)

By combining (14), (17), and (20), the following relation isderived:

δh(n + 1) = −G(n)(

GT(n)G(n))−1

e(n). (22)

By splitting the vector δh(n + 1) in its components, that is,

δh(n + 1) = [δhT1 (n + 1) · · · δhT

M(n + 1)]T

, (23)

and accordingly partitioning the matrix G(n) in submatricesGi(n) of congruent dimensions, the following set of equa-tions is obtained:

δhi(n + 1) = −Gi(n)(

GT(n)G(n))−1

e(n) (24)

for i = 1, . . . ,M. As a consequence, the updating relations forthe coefficients of each branch are given by

hi(n + 1) = hi(n)− µiGi(n)(

GT(n)G(n))−1

e(n) (25)

for i = 1, . . . ,M, where µi is a parameter that controls boththe convergence rate and the stability of the F-X AP algo-rithm. The L × L matrix GT(n)G(n) represents an estimateof the filtered-X autocorrelation matrix of the signal formedwith products of couples of input samples, obtained usingthe last L input vectors. The computation of its inverse is re-quired at any time n.

Since this step is often a critical one, we can distinguishthe solution for the F-X AP algorithms of low orders, that is,with L = 2, 3 from that for greater orders. In fact, for L = 2, 3,even the direct inversion of the matrix is an affordable task.The only necessary care in order to avoid possible numericalinstabilities is to add a diagonal matrix δI , where δ is a smallpositive constant, to the matrix GT(n)G(n). The equationsemployed for updating the coefficients and filtering the inputsignal using SVFs are summarized in Algorithm 1.

A general and efficient solution which can be applied toany order L of affine projections is derived by resorting to asimpler and more stable estimate for the inverse of the matrixGT(n)G(n). We introduce the vectors

xi(n) = s(n)∗

x(n)x(n− i + 1)x(n− 1)x(n− i)

...x(n− L + 1)x(n− i− L + 2)

(26)

for i = 1, . . . ,M, and the M × L matrix

X(n) = [x1(n) x2(n) · · · xM(n)]T. (27)

Then a recursive approximation of the GT(n)G(n) matrix isgiven by

R(n) = λR(n− 1) + (1− λ)XT(n)X(n), (28)

where λ is a forgetting factor (0 < λ < 1) which determinesthe temporal memory length in the estimation of the auto-correlation matrix. The higher the forgetting factor, the moreinsensitive to noise is the estimate. In practice, λ is alwaystaken close to 1. The recursive estimate can now be used inthe coefficient updating equation (25), where the computa-tion of the inverse matrix is still required. To avoid such aninversion, it is convenient to directly update the inverse ma-trix R−1(n), as done for the recursive least square (RLS) algo-rithm. We define the following matrices:

R0(n) = λR(n− 1), (29)

Rl(n) = Rl−1(n) + (1− λ)xl(n)xTl (n) (30)

for l = 1, . . . ,M. Since, from (28),

R(n) = λR(n− 1) + (1− λ)[

x1(n)xT1 (n) + x2(n)xT

2 (n)

+ · · · + xM(n)xTM(n)

],

(31)

it immediately follows that R(n) = RM(n). By using the ma-trix inversion lemma [20], it is possible to derive from (30)the following updating rule:

R−1l (n) = R−1

l−1(n− 1)

− R−1l−1(n− 1)xl(n)xT

l (n)R−1l−1(n− 1)

1/(1− λ) + xTl (n)R−1

l−1(n− 1)xl(n).

(32)

These expressions can be written in a more compact form bydefining

kl(n) = R−1l−1(n− 1)xl(n)

1/(1− λ) + xTl (n)R−1

l−1(n− 1)xl(n), (33)

P(n) = R−1(n), and Pl(n) = R−1l (n). Therefore, P(n) =

PM(n) and P0(n) = (1/λ)P(n − 1). As a consequence, thefollowing recursive estimate for Pl(n) is derived:

Pl(n) = Pl−1(n)− kl(n)xTl (n)Pl−1(n). (34)

Filtered-X Affine Projection Algorithms 1845

Initialization: P(−1) = δI, hi = 0 ∀iy(n) =∑M

i=1 hTi (n)xi(n)

ej(n) = dp(n− j + 1) + s(n− j + 1)∗∑Mi=1 hT

i (n)xi(n− j + 1)

for j = 1, . . . ,L

e(n) = [e1(n) e2(n) · · · eL(n)]T

Gi(n) =[s(n)∗ xi(n) s(n− 1)∗ xi(n−1) · · · s(n−L+1)∗xi(n−L+1)]P0(n) = (1/λ)P(n− 1)

kl(n) = Pl−1(n)xl(n)/(1/(1− λ) + xTl (n)Pl−1(n)xl(n))

Pl(n) = Pl−1(n)− kl(n)xTl (n)Pl−1(n),

for l = 1, . . . ,M

P(n) = PM(n)

hi(n + 1) = hi(n)− µiGi(n)P(n)e(n)

for i = 1, . . . ,M

Algorithm 2: Filtered-X AP adaptive algorithm of order L for anSVF using the matrix inversion lemma.

Finally, by replacing in (25) (GT(n)G(n))−1 with the matrixP(n) = PM(n), the following updating expression for the ithbranch of an SVF is obtained:

hi(n + 1) = hi(n)− µiGi(n)P(n)e(n) (35)

for i = 1, . . . ,M. The corresponding updating algorithm isdescribed in Algorithm 2. Its complexity is given by O(ML2 +KL). Since O(K) = O(MN) and usually L < M, the numberof multiplications needed to implement SVFs equipped withthis adaptive AP algorithm is O(LMN), Therefore, its com-plexity is of the order of L times that of the correspondingLMS algorithm. More specifically, the complexity of the F-XAP algorithm for a quadratic filter with M = N is O(LN2)per sample, while the complexity of the F-X LMS algorithmis O(N2), as also reported in [9]. On the other hand, forhigh values of L, the F-X AP algorithms tend to behave asthe RLS algorithms with similar convergence rates and track-ing capabilities. However, the complexity of RLS algorithmsfor quadratic filters is O(N4) or O(N3) for their fast versions[15, page 271]. In addition, it is worth noting that often evensmall values of L, that is, L = 2, 3, are sufficient to obtainremarkable convergence improvements with respect to theF-X LMS algorithm. Moreover, while the F-X AP algorithmcan be applied to complete quadratic filters simply by settingM = N , using a small number of channels M often permits toobtain good adaptation performances with a reduced com-putational complexity, as shown in the next Section. In fact,with reference to these aspects, the implementation complex-ity O(LMN) indicates a sort of tradeoff between the numberL of APs used and the number of active channels M in thefilter bank realization.

Finally, it is worth noting that it is easy to pass from thealgorithm for a homogeneous second-order Volterra filter tothat of a generic Volterra filter. The SVF structure can becompleted with the branches associated with the linear termand the higher-order Volterra operators according to their di-

agonal representation. Each of these channels is then treatedby the algorithms of Tables 1 and 2 in a way similar to thechannels of the homogeneous second-order Volterra filter.

4. SIMULATION RESULTS

In this section, we present some simulation results obtainedwith the F-X AP algorithms of Tables 1 and 2.

In the first set of simulations, we consider the same ex-perimental conditions of [9, Section IV-A]. The source noiseis a logistic chaotic noise, that is, a second-order white andpredictable nonlinear process, generated with the recursivelaw,

ξ(i + 1) = 4ξ(i)(1− ξ(i)

), (36)

where ξ(0) is a real number between 0 and 1 different fromk/4 with k = 0, 1, . . . , 4. The nonlinear process is then nor-malized in order to have a unit signal power x(i) = ξ(i)/σξ .The primary and secondary paths are modeled with the fol-lowing FIR filters, respectively,

P(z) = z−5 − 0.3z−6 + 0.2z−7, (37)

S(z) = z−2 + 1.5z−3 − z−4. (38)

The system is identified with a second-order Volterra filterwith a linear part of memory length 10 and a quadratic partof memory length 10 and 10 diagonals (M = 10). Figures4 and 5 plot the ensemble average of the resulting meansquare error for 100 runs of the simulation system, using thedirect matrix inversion as in Algorithm 1 and the recursivetechnique in Algorithm 2, respectively. The four curves re-fer to different values of the affine projection order L. Theorder L = 1 corresponds to a normalized LMS adaptation al-gorithm, which is the same adaptation algorithm employedin [9] apart from the normalization. In the experiments ofFigure 4, the step size was equal to 0.005. In the experimentsof Figure 5, the λ factor was equal to 0.9 and the step sizevalue was set to 0.0009 in order to obtain the same conver-gence characteristics of the first set of experiments when theaffine projection order L equals 1. For higher orders of affineprojections, the improvement in the convergence behavior ofthe algorithm is evident. Moreover, the adaptation curves ofFigure 5 indicate a slight but steady reduction of the asymp-totic error for increasing values of L. This fact confirms thereliability of the recursive approximation leading to the algo-rithm of Algorithm 2.

In the second set of experiments, we simulate a suddenchange in the noise source and its propagation model and weinvestigate the ability of the F-X AP algorithm to track thenoise conditions. We employ the same experimental condi-tions of the first set of simulations, but after 100000 signalsamples we modify the primary path model according to thefollowing equation:

P(z) = z−5 + 0.3z−6 − 0.2z−7 (39)

1846 EURASIP Journal on Applied Signal Processing

0 0.5 1 1.5 2×105

Number of iterations

10−3

10−2

10−1

100

Mea

nsq

uar

eer

ror

L = 1L = 2

L = 3L = 4

Figure 4: Adaptation curves for different orders of affine projec-tions L using the method in Algorithm 1.

0 0.5 1 1.5 2×105

Number of iterations

10−3

10−2

10−1

100

Mea

nsq

uar

eer

ror

L = 1L = 2

L = 3

L = 4

Figure 5: Adaptation curves for different orders of affine projec-tions L using the method in Algorithm 2.

and we use as input signal x(i) = x2(i)/2, where x(i) isthe normalized logistic noise of the previous experiments.Figure 6 plots the resulting adaptation curves for different or-ders of affine projections when the algorithm of Algorithm 2is applied for the filter adaptation. Again, we can observe theimprovement in the convergence behavior determined by theAP algorithm.

In the last set of experiments, we investigate the effectsof modeling an active noise controller as a multichannelfilter bank with a reduced number of channels. The noisesource is the logistic chaotic noise of the first set of exper-iments. The primary and secondary paths are modeled asin (37), (38), respectively. The system is identified with anSVF with a linear part and a quadratic part, both of mem-ory length 10. Figures 7 and 8 plot the ensemble average

0 0.5 1 1.5 2×105

Number of iterations

10−3

10−2

10−1

100

101

Mea

nsq

uar

eer

ror

L = 1L = 2

L = 3L = 4

L = 1L = 2

L = 3L = 4

Figure 6: Adaptation curves with a sudden modification in thenoise conditions.

0 0.5 1 1.5 2×105

Number of iterations

10−3

10−2

10−1

100M

ean

squ

are

erro

r

lin

M = 4

M = 2

Figure 7: Adaptation curves with different number of channels andlinear primary path.

of the resulting mean square error for 100 runs of the sim-ulation system. The algorithm of Algorithm 2 was appliedwith an affine projection order L set to 4 and with λ = 0.9and µ = 0.0005.The curves in Figure 7 refer to the lin-ear controller and the quadratic controller with M = 2and M = 4. The convergence improvement with respectto the linear case can be easily appreciated, especially forM = 2. It has been experimentally verified that the curvesfor 4 ≤ M ≤ 10 converge with a progressively slower be-havior to about the residual error of the curve for M = 2.Therefore, using the full model, as done in [9], does notallow any improvement. It has been also observed that us-ing a number L of APs equal to 2 gives, as expected, thesame global performances, but with a reduced convergencespeed.

Filtered-X Affine Projection Algorithms 1847

0 0.5 1 1.5 2×105

Number of iterations

10−2

10−1

100

Mea

nsq

uar

eer

ror

M = 2

M = 4

M = 10M = 6

Figure 8: Adaptation curves with different number of channels andnonlinear primary path.

To complete this set of experiments, the primary path hasbeen then replaced by the following second-order Volterrafilter:

y(n) = x(n− 5)− 0.3x(n− 6) + 0.2x(n− 7)

+ 0.5x(n− 5)x(n− 5)− 0.1x(n− 6)x(n− 6)

+ 0.1x(n− 7)x(n− 7)− 0.2x(n− 5)x(n− 6)

+ 0.05x(n− 6)x(n− 7)− 0.02x(n− 7)x(n− 8)

+ 0.1x(n− 5)x(n− 7)− 0.02x(n− 6)x(n− 8)

+ 0.01x(n− 7)x(n− 9) + 0.5x(n− 5)x(n− 8)

− 0.1x(n− 6)x(n− 9) + 0.1x(n− 7)x(n− 10)

− 0.2x(n− 5)x(n− 9) + 0.05x(n− 6)x(n− 10)

− 0.02x(n− 7)x(n− 11) + 0.1x(n− 5)x(n− 10)

− 0.02x(n− 6)x(n− 11) + 0.01x(n− 7)x(n− 12)(40)

and all the simulations have been repeated with the sameparameters. The results obtained for different numbers ofbranches M of the quadratic part of the SVF are shown inFigure 8. Of course, the best approximation result is now thatfor M = 6, since in this case the SVF exactly corresponds tothe system to be modeled. We observe that when M = 2,the resulting SVF is inadequate to model the noise gener-ation system, while for M = 4 a better approximation isobtained. This case can be considered as a compromise interms of modeling accuracy, speed of convergence, and com-putational cost. From Figure 8, it can be noted again thatoverdimensioning the model using a complete second-orderVolterra filter, M = 10, does not offer particular advantages.In fact, this filter is able to model the noise generation sys-tem with slightly reduced accuracy and convergence speed atan increased computational cost with respect to the referencecase M = 6.

5. CONCLUSIONS

In practical applications, methods for active noise controlhave often to deal with nonlinear effects. In such environ-ments, nonlinear controllers based on Volterra filters im-plemented in the form of multichannel filter banks can beusefully exploited. One of the crucial aspects is the deriva-tion of efficient adaptation algorithms. Usually, the so-calledfiltered-X LMS or NLMS algorithms are used. In this paperwe proposed the use of the affine projection technique, andwe derived in detail the so-called filtered-X AP algorithms forhomogeneous quadratic filters. According to the multichan-nel approach, these derivations can be easily extended to ageneric Volterra filter. The extensive experiments we reportconfirm that the AP technique offers better convergence andtracking capabilities than the classical LMS and NLMS algo-rithms with a limited increase of the computational complex-ity.

ACKNOWLEDGMENT

This work has been partially supported by “Fondi RicercaScientifica 60%, Universita di Trieste.”

REFERENCES

[1] G. E. Warnaka, “Active attenuation of noise—the state of theart,” Noise Control Engineering, vol. 18, no. 3, pp. 100–110,1982.

[2] S. J. Elliot and P. A. Nelson, Active Control of Sound, AcademicPress, New York, NY, USA, 3rd edition, 1995.

[3] S. J. Elliott and P. A. Nelson, “Active noise control,” IEEESignal Processing Magazine, vol. 10, no. 4, pp. 12–35, 1993.

[4] S. M. Kuo and D. R. Morgan, Active Noise Control Systems: Al-gorithms and DSP Implementations, John Wiley & Sons, NewYork, NY, USA, 1996.

[5] D. Pavisic, L. Blondel, J.-P. Draye, G. Libert, and P. Chapelle,“Active noise control with dynamic recurrent neural net-works,” in Proc. European Symposium on Artificial Neural Net-works (ESANN ’95), pp. 45–50, Brussels, Belgium, April 1995.

[6] T. Matsuura, T. Hiei, H. Itoh, and K. Torikoshi, “Active noisecontrol by using prediction of time series data with a neuralnetwork,” in Proc. IEEE International Conference on Systems,Man and Cybernetics (SMC ’95), vol. 3, pp. 2070–2075, Van-couver, BC, Canada, October 1995.

[7] P. Strauch and B. Mulgrew, “Active control of nonlinear noiseprocesses in a linear duct,” IEEE Trans. Signal Processing, vol.46, no. 9, pp. 2404–2412, 1998.

[8] L. Tan and J. Jiang, “Filtered-X second-order Volterra adap-tive algorithms,” Electronics Letters, vol. 33, no. 8, pp. 671–672,1997.

[9] L. Tan and J. Jiang, “Adaptive Volterra filters for active controlof nonlinear noise processes,” IEEE Trans. Signal Processing,vol. 49, no. 8, pp. 1667–1676, 2001.

[10] M. Bouchard, B. Paillard, and C. T. L. Dinh, “Improved train-ing of neural networks for the nonlinear active control ofsound and vibration,” IEEE Transactions on Neural Networks,vol. 10, no. 2, pp. 391–401, 1999.

[11] C. A. Silva, J. M. Sousa, and J. M. G. Sa da Costa, “Active noisecontrol based on fuzzy models,” in Proc. 4th European Con-ference on Noise Control (EURONOISE ’01), pp. 1–14, Patras,Greece, January 2001.

[12] J. M. Conchinha, M. A. Botto, J. M. Sousa, and J. M. G. Sa daCosta, “The use of neural network models in an active noisecontrol applications,” in Proc. 4th European Conference on

1848 EURASIP Journal on Applied Signal Processing

Noise Control (EURONOISE ’01), pp. 1–13, Patras, Greece,January 2001.

[13] M. H. Costa, J. C. Bermudez, and N. J. Bershad, “Stochasticanalysis of the filtered-X LMS algorithm in systems with non-linear secondary paths,” IEEE Trans. Signal Processing, vol. 50,no. 6, pp. 1327–1342, 2002.

[14] O. J. Tobias and R. Seara, “Performance comparison of theFXLMS, nonlinear FXLMS and leaky FXLMS algorithms innonlinear active control applications,” in Proc. European Sig-nal Processing Conference (EUSIPCO ’02), pp. 1–4, Toulouse,France, September 2002.

[15] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Process-ing, John Wiley & Sons, New York, NY, USA, 2000.

[16] G. M. Raz and B. D. van Veen, “Baseband Volterra filters forimplementing carrier based nonlinearities,” IEEE Trans. Sig-nal Processing, vol. 46, no. 1, pp. 103–114, 1998.

[17] A. Fermo, A. Carini, and G. L. Sicuranza, “Simplified Volterrafilters for acoustic echo cancellation in GSM receivers,” inProc. European Signal Processing Conference (EUSIPCO ’00),Tampere, Finland, September 2000.

[18] A. Fermo, A. Carini, and G. L. Sicuranza, “Low-complexitynonlinear adaptive filters for acoustic echo cancellation inGSM hand-set receivers,” European Transactions on Telecom-munications, vol. 14, no. 2, pp. 161–169, 2003.

[19] K. Ozeki and T. Umeda, “An adaptive filtering algorithm usingan orthogonal projection to an affine subspace and its proper-ties,” Electronics and Communications in Japan, vol. 67-A, no.5, pp. 19–27, 1984.

[20] C. F. N. Cowan and P. M. Grant, Adaptive Filters, Prentice-Hall, Englewood Cliffs, NJ, USA, 1985.

Alberto Carini was born in Trieste, Italy,in 1967. He received the Laurea degree(summa cum laude) in electronic engineer-ing in 1994 and the “Dottorato di Ricerca”degree (Ph.D.) in information engineeringin 1998, both from the University of Trieste,Italy. He has received the Zoldan Award forthe best Laurea degree in electronic engi-neering at the University of Trieste duringthe academic year 1992–1993. In 1996 and1997, during his Ph.D. studies, he spent several months as a VisitingScholar at the University of Utah, Salt Lake City, USA. From 1997to 2003, he worked as a DSP engineer with Telit Mobile TerminalsSpA, Trieste, Italy, where he was leading the audio processing R&Dactivities. In 2003, he worked with Neonseven srl, Trieste, Italy, asaudio and DSP expert. From 2001 to 2004, he collaborated withthe University of Trieste as a Contract Professor of Digital SignalProcessing. Since 2004, he is an Associate Professor at the Informa-tion Science and Technology Institute (ISTI), University of Urbino,Urbino, Italy. His research interests include adaptive filtering, non-linear filtering, nonlinear equalization, acoustic echo cancellation,and active noise control.

Giovanni L. Sicuranza is Professor of sig-nal and image processing and Head of theImage Processing Laboratory at DEEI, Uni-versity of Trieste (Italy). His research in-terests include multidimensional digital fil-ters, polynomial filters, processing of im-ages and image sequences, image coding,and adaptive algorithms for echo cancella-tion and active noise control. He has pub-lished a number of papers in internationaljournals and conference proceedings. He contributed in chapters

of six books and is the Coeditor, with Professor Sanjit Mitra, Uni-versity of California at Santa Barbara, of the books Multidimen-sional Processing of Video Signals, (Kluwer Academic Publisher,1992), and Nonlinear Image Processing, (Academic Press, 2001). Heis the coauthor with Professor V. John Mathews, University of Utahat Salt Lake City, of the book Polynomial Signal Processing, (J. Wi-ley, 2000). Dr. Sicuranza has been a member of the technical com-mittees of numerous international conferences and Chairman ofEUSIPCO-96 and NSIP-03. He is an Associate Editor of “Multi-dimensional Systems and Signal Processing” and a Member of theEditorial Board of “Signal Processing” and “IEEE Signal Process-ing Magazine.” Dr. Sicuranza is currently the Awards Chairman ofthe Administrative Committee of EURASIP and a Member of theIMDSP Technical Committee of the IEEE Signal Processing Soci-ety. He has been one of the founders and the first Chairman of theNonlinear Signal and Image Processing (NSIP) Board of which heis still a Member.

Photograph © Turisme de Barcelona / J. Trullàs

Preliminary call for papers

The 2011 European Signal Processing Conference (EUSIPCO 2011) is thenineteenth in a series of conferences promoted by the European Association forSignal Processing (EURASIP, www.eurasip.org). This year edition will take placein Barcelona, capital city of Catalonia (Spain), and will be jointly organized by theCentre Tecnològic de Telecomunicacions de Catalunya (CTTC) and theUniversitat Politècnica de Catalunya (UPC).EUSIPCO 2011 will focus on key aspects of signal processing theory and

li ti li t d b l A t f b i i ill b b d lit

Organizing Committee

Honorary ChairMiguel A. Lagunas (CTTC)

General ChairAna I. Pérez Neira (UPC)

General Vice ChairCarles Antón Haro (CTTC)

Technical Program ChairXavier Mestre (CTTC)

Technical Program Co Chairsapplications as listed below. Acceptance of submissions will be based on quality,relevance and originality. Accepted papers will be published in the EUSIPCOproceedings and presented during the conference. Paper submissions, proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.

Areas of Interest

• Audio and electro acoustics.• Design, implementation, and applications of signal processing systems.

l d l d d

Technical Program Co ChairsJavier Hernando (UPC)Montserrat Pardàs (UPC)

Plenary TalksFerran Marqués (UPC)Yonina Eldar (Technion)

Special SessionsIgnacio Santamaría (Unversidadde Cantabria)Mats Bengtsson (KTH)

FinancesMontserrat Nájar (UPC)• Multimedia signal processing and coding.

• Image and multidimensional signal processing.• Signal detection and estimation.• Sensor array and multi channel signal processing.• Sensor fusion in networked systems.• Signal processing for communications.• Medical imaging and image analysis.• Non stationary, non linear and non Gaussian signal processing.

Submissions

Montserrat Nájar (UPC)

TutorialsDaniel P. Palomar(Hong Kong UST)Beatrice Pesquet Popescu (ENST)

PublicityStephan Pfletschinger (CTTC)Mònica Navarro (CTTC)

PublicationsAntonio Pascual (UPC)Carles Fernández (CTTC)

I d i l Li i & E hibiSubmissions

Procedures to submit a paper and proposals for special sessions and tutorials willbe detailed at www.eusipco2011.org. Submitted papers must be camera ready, nomore than 5 pages long, and conforming to the standard specified on theEUSIPCO 2011 web site. First authors who are registered students can participatein the best student paper competition.

Important Deadlines:

P l f i l i 15 D 2010

Industrial Liaison & ExhibitsAngeliki Alexiou(University of Piraeus)Albert Sitjà (CTTC)

International LiaisonJu Liu (Shandong University China)Jinhong Yuan (UNSW Australia)Tamas Sziranyi (SZTAKI Hungary)Rich Stern (CMU USA)Ricardo L. de Queiroz (UNB Brazil)

Webpage: www.eusipco2011.org

Proposals for special sessions 15 Dec 2010Proposals for tutorials 18 Feb 2011Electronic submission of full papers 21 Feb 2011Notification of acceptance 23 May 2011Submission of camera ready papers 6 Jun 2011