A multichannel hierarchical approach to adaptive Volterra filters employing filtered-X affine...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005 1463

A Multichannel Hierarchical Approach to AdaptiveVolterra Filters Employing Filtered-X Affine

Projection AlgorithmsGiovanni L. Sicuranza, Senior Member, IEEE, and Alberto Carini, Member, IEEE

Abstract—It is shown in this paper how the use of a recently in-troduced algebra, called V-vector algebra, can directly lead to theimplementation of Volterra filters of any order in the form ofa multichannel filterbank. Each channel in this approach is mod-eled as a finite impulse response (FIR) filter, and the channels arehierarchically arranged according to the number of the filter coef-ficients. In such a way, it is also possible to devise models of reducedcomplexity by cutting the less relevant channels.

This model is then used to derive efficient adaptation algorithmsin the context of nonlinear active noise control. In particular, it isshown how the affine projection (AP) algorithms used in the linearcase can be extended to a Volterra filter of any order . The deriva-tion of the so-called Filtered-X AP algorithms for nonlinear ac-tive noise controllers is easily obtained using the elements of theV-vector algebra. These algorithms can efficiently replace the stan-dard LMS and NLMS algorithms usually applied in this field, espe-cially when, in practical applications, a reduced-complexity multi-channel structure can be exploited.

Index Terms—Adaptive Volterra filters, affine-projection algo-rithms, multichannel filterbank, nonlinear active noise control,V-vector algebra.

I. INTRODUCTION

VOLTERRA filters [1] have been thoroughly studied in thecontext of nonlinear filters because of their nice properties

that permit the derivation of a theoretical framework, includingthe well-known linear filters as a particular case. According tothese properties and, in particular, to the linearity of the filteroutput with respect to the filter coefficients, adaptation rules forVolterra filters can be obtained by extending classical algorithmsexploited for linear filters. This extension is often based on amultichannel approach [2]–[4] in which the Volterra filter is re-alized by means of a linear filterbank where each filter processesa product of samples of the input signal.

The multichannel structure has been more recently exploitedin [5] to implement the so-called simplified Volterra filters(SVFs). Even though this model can be in principle applied tokernels of arbitrary order, the simplest example of a Volterrafilter, i.e., the homogeneous quadratic filter, has been specif-ically considered in [5] with reference to the acoustic echo

Manuscript received October 15, 2003; revised April 13, 2004. This workwas suppoted in part by Grant MIUR PRIN 2004092314. The associate editorcoordinating the review of this manuscript and approving it for publication wasDr. Kenneth E. Barner.

G. L. Sicuranza is with the Department of Electrical Electronic andComputer Engineering, University of Trieste, 34127 Trieste, Italy (email:[email protected]).

A. Carini is with the Information Science and Technology Institute, Univer-sity of Urbino, 61029 Urbino, Italy (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2005.843705

cancellation problem. The stress in [5] is on the fact that,according to the characteristics of the measured second-orderkernel, it is possible to use a reduced number of active channels.In fact, it has been noted that the relevance of the quadratickernel elements is strongly decreasing moving far from the maindiagonal. Therefore, remarkable savings in implementationcomplexity can be achieved. This is an important aspect sinceone of the main drawbacks of Volterra filters is their implemen-tation complexity. It is worth noting that similar behaviors havebeen observed in other real-world nonlinear systems, as well.The argument of reduced number of active channels has beenalso presented in [6] with reference to the so-called bandedregular Volterra kernels. As far as the adaptation algorithm isconcerned, in [5], an updating rule based on the so-called affineprojection (AP) algorithm has been devised extending it fromthe linear case, as originally proposed by Ozeki and Umeda[7], to quadratic filters. The significant aspect of this class ofalgorithms is the ability to offer, in the presence of correlatedsignals, convergence rates higher than those of LMS algorithmsand tracking capabilities better than those of RLS algorithms[8]. In fact, while least mean squares (LMS) and normalizedLMS (NLMS) adaptive algorithms minimize, according to thestochastic approximation, the error between the conditioningsignal and the actual output signal at time , AP algorithms tryto minimize the last a posteriori errors at time ,

. As a consequence, AP algorithms behave as theLMS algorithms for while tending to behave as the RLSalgorithms for an increasing number of errors considered.On the other hand, even when the input signal is white, thepresence of products of samples in the input signal to a Volterrafilter introduces correlation among its entries. This is the reasonwhy AP techniques can offer, for this class of filters, betterconvergence and tracking capabilities than the classical LMSand NLMS algorithms. According to the structure of SVFs,these performances are obtained with a limited increment ofthe computational complexity.

A further example of application of the multichannel struc-ture can be found in [9], where adaptive Volterra filters are usedto implement nonlinear active noise controllers. In recent years,intensive studies have been carried out in the field of vibrationand acoustic noise control with promising results [10], [11].The technique used in a single-channel active noise controller isbased on the destructive interference in a given location of thenoise produced by a primary source and the interefering signalgenerated by a secondary source. However, most of the studiespresented in the literature refer to linear models, whereas it is

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1464 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

often recognized that nonlinear effects can be present in the ac-tual applications [9], [12], [13]. Such effects may arise from thebehavior of the noise source, which, rather than being a sto-chastic process may be depicted as a nonlinear deterministicnoise process, sometimes of chaotic nature. Moreover, the pri-mary path may exhibit a nonlinear behavior thus further moti-vating the use of a nonlinear controller. The multichannel modelfor nonlinear controllers presented in [9] actually exploits theso-called diagonal representation introduced in [14]. This rep-resentation allows a truncated Volterra system of any order tobe described by the diagonal entries of its kernels. In fact, if the

th-order kernel is represented as a sampled hypercube of thesame order , the diagonal representation implies the change ofthe Cartesian coordinates to coordinates that are aligned alongthe diagonals of the hypercube. A diagonal is defined here asany line that is parallel to the main diagonal of the hypercube.Using this coordinate system, the Volterra filter can be repre-sented in the form of a filterbank where each filter correspondsto a diagonal of the hypercube. This representation is partic-ularly useful since the frequency content of the output signalis directly related to the frequency response of the diagonal el-ements of the kernel. It has been exploited to derive efficientimplementations of Volterra filters for processing carrier-basedinput signals using fast convolution techniques [14].

It is worth noting that the SVF filters implement the samediagonal structure as in the diagonal coordinate representation,but the stress here is on the fact that, according to the character-istics of the measured second-order kernel, it is possible to ar-range the channels of the filterbank in a hierarchical sequence.Moving far from the main diagonal, less relevant terms are en-countered, and thus, the use of a reduced number of diagonalsas active channels is possible. As a consequence, a reductionin computational complexity can be achieved without compro-mising the whole approximation performance.

This hierachical arrangement is characteristic of the main el-ements (the so-called V-vectors) of the novel algebra proposedin [15] and [16]. This algebra, which is called V-vector algebra,constitutes a formal basis for the development of adaptation al-gorithms for Volterra filters as an extension of adaptive tech-niques used for linear filters. It is naturally organized to dealwith linear and nonlinear multichannel structures. It can alsobe viewed as an actual implementation of the diagonal coordi-nate representation system mentioned above, where the chan-nels are arranged according to the length of the correspondingdiagonal. In other words, the diagonal of the sampled hypercubeis sequentially arranged according to the increasing distance ofeach diagonal from the main one. Therefore, nonlinear systemscharacterized by fading terms far from the main diagonal canbe approximated using only the first filter channels. This resultis achieved without any change in the coordinate system, whichremains as the Cartesian coordinate system that is usually em-ployed to represent Volterra filters.

The aim of this paper is to describe a unified approach for thederivation of adaptation algorithms using the AP technique forVolterra filters of any order . Such a derivation is easily de-scribed if the V-vector algebra previously mentioned is suitablyexploited. It does not require the change to the diagonal coor-dinate system and permits us to describe the adaptation algo-

rithm for Volterra filters in a very compact way as the vector-ma-trix representation of the standard algebra does for linear fil-ters. Moreover, it automatically arranges the channels of themultichannel filterbank in a hierarchical sequence. The corre-sponding order is based on the distance of the elements of theVolterra kernels involved in each channel from those involvedin the main diagonal. For systems characterized by kernel ele-ments that fade when moving far from the main diagonal, thisorder permits us to obtain compact approximations by keepingonly the first channels of the whole filterbank. Accordingly, theAP algorithm developed in this paper is characterized by a lim-ited complexity increase with respect to the standard LMS andNLMS algorithms. In fact, for a homogeneous Volterra filter oforder , the complexity per sample is incremented by a factor

, where is the number of the a posteriori errors minimizedby the AP algorithm, as shown in Section IV. It should be alsonoted that in general, even small values of , i.e., , 3, and4, are sufficient to obtain remarkable improvements with respectto the LMS and NLMS algorithms, which indeed correspond tothe case with . This fact makes it unnecessary to resortto the development of fast versions of the AP algorithms as, forexample, those described in [17]–[19].

The AP algorithms for Volterra filters of any order willbe derived in the general case of the so-called Filtered-X algo-rithms. Such algorithms are used in the context of the activenoise control, and the difference with respect to the standardalgorithms consists of filtering the input signal with the im-pulse response of the so-called secondary path [9]. Therefore,another contribution of this paper is the derivation of Filtered-XAP algorithms for nonlinear active noise controllers. These al-gorithms can efficiently replace the standard LMS and NLMSalgorithms usually applied in this field, especially when the con-siderations on the reduced hierarchical multichannel structurecan be applied. The first example of a Filtered-X AP algorithmapplied to quadratic filters is encouraging in such a direction, asshown in [20].

As a final comment, it should be noted that the standardadaptive filters equipped with an AP algorithm are includedin the Filtered-X model since the usual adaptation algorithmscan easily be derived by simply assuming, as the impulseresponse of the secondary path, the unitary impulse . Asa consequence, all the derivations shown in this paper remainvalid for Volterra filters used as standard adaptive filters.

The paper is organized as follows. In Section II, the basicelements of the V-vector algebra are recalled, together withthe definition of some new operations that are essential for thederivation of AP algorithms. In Section III, the hierarchical mul-tichannel representation of Volterra filters is described in de-tail, and in Section IV, the general Filtered-X AP algorithm forVolterra filters of any order is derived. Simulation results aredescribed and commented on in Section V. Finally, concludingremarks are reported in Section VI.

II. BASIC ELEMENTS OF V-VECTOR ALGEBRA

In this section, the basic elements of the V-vector algebraare briefly revisited together with the main operations betweenV-vectors and V-matrices. Only the definitions essential to the

SICURANZA AND CARINI: MULTICHANNEL HIERARCHICAL APPROACH TO ADAPTIVE VOLTERRA FILTERS 1465

successive derivations are reported here. A complete account onthe V-vector algebra can be found in [15] and [16]. A couple ofnew definitions required in the next section are also introducedin this section.

A. Basic Definitions

Here, the basic definitions related to the V-vector algebra aresummarized.

Definition 1: A V-vector is a nonrectangular matrix inwhich the number of elements in each row does not increasegoing from the top to the bottom of the matrix.

The elements of a V-vector are identified by acouple of indexes , , where the first index indicates the rowand the second indicates the column. The diagonal bracketsemphasize the nonrectangular structure of the matrix. Anexample is given in Fig. 1.

Definition 2: The first column of a V-vector is the arrayformed with the first elements in each row of the nonrectangularmatrix.

Definition 3: The last column of a V-vector is the arrayformed with the last elements in each row of the nonrectangularmatrix.

The shape of a V-vector is identified by its type, i.e., the setof integers that indicate the numberof rows and the number of elements in each row of the V-vector.The type of the V-vector in Fig. 1 is the set of numbers (3,2,1).The type of a V-vector will be indicated with a calligraphicletter, e.g., , , etc. It is worth noting that the role of the typeof a V-vector is the same of the length of a vector of the standardalgebra. Moreover, a V-vector of type (1,1, , 1), i.e., with allthe integers equal to one, is a vector of length .

Definition 4: A V-matrix of type isa V-vector of type whose elements are again V-vectors(subV-vectors) of type .

The type of a V-matrix plays the role of the dimensions of amatrix of the standard algebra. The structure of a V-matrix is il-lustrated in Fig. 2. The elements of a V-matrixare identified by two pairs of indexes, where the first pair indi-cates the subV-vector, whereas the second one identifies theelement in the subV-vector. A V-matrix of type

is a matrix of dimensions specified by the numberof ones between parentheses.

Other elements corresponding to those of the linear algebra,such as Identity and Transposed V-matrices, are then defined.

Definition 5: The Identity V-matrix is a V-matrixof type with the elements equal to one when thecouple of indexes is equal to and equal to zero elsewhere.

Definition 6: The Transposed V-matrix of a V-matrixof type whose elements are is a V-ma-

trix of type whose elements are .

It is worth noting that a V-vector can be also seen as aV-matrix of type , where the subV-vectors coincide withthe single elements of the V-vector. Then, according to the

previous definition, the transposed V-vector is defined

Fig. 1. V-vector of type (3,2,1).

as a V-matrix of type whose entries still are . We willmake use of this property in the following definition of innerproduct of two V-vectors so that a notation similar to that usedin the usual algebra can be applied.

B. Basic Operations Between V-Vectors and V-Matrices

Here, the fundamental operations between V-vectors andV-matrices are reported.

Sum of Two V-Vectors: Let and indicate twoV-vectors of type , whose elements are and , respec-

tively; then, their sum is a V-vector of the same type whoseelements are given by .

Sum of Two V-Matrices: Let and indi-cate two V-matrices of type , whose elements are

and , respectively; then, their sum is a V-matrixof the same type whose elements are given by

.

Inner Product of Two V-Vectors: Let and indi-cate two V-vectors of type , whose elements are and ,

respectively; then, the inner product

is a scalar given by .

Product of a V-Matrix and a V-Vector: Let indi-cate a V-matrix of type whose elements are , and

let indicate a V-vector of type whose elements are ;

then, the product is a V-vector of type

, whose elements are given by .

Product of Two V-Matrices: Let indicate aV-matrix of type , whose elements are , and let

indicate a V-matrix of type , whose elements

are ; then, the product is a V-ma-

trix of type whose elements are given by

.It is now possible to introduce the definition of the inverse

V-matrix.Definition 7: The Inverse of a V-matrix of type

is the V-matrix of the same type, which, pre-

or post-multiplied by , gives the identity V-matrix.Two further operations used in the development of the AP

algorithm are those of derivative of a V-vector and convolutionof a finite-extent sequence with a V-vector.

Definition 8: The derivative of a V-vector of type

, whose elements are , with respect to a V-vectorof type , whose elements are , is the V-vector of typeformed with the derivatives of the single elements ,

i.e., .


Fig. 2. V-matrix of type (3,2,1) � (3,2,1).

Definition 9: The convolution of a finite-extent se-quence with a V-vector , formed with theelements of a finite-extent input sequence, is the V-vectorformed with the convolved elements of the V-vector, i.e.,

.

III. MULTICHANNEL HIERARCHICAL REPRESENTATION OF

VOLTERRA FILTERS

Discrete-time causal polynomial filters [1] are described bythe input–output relationship

(1)

where the function is a polynomial of order in the vari-ables within the parentheses. The linear filter is a particular caseof polynomial filters since the relationship in (1) becomes linearif for all . With reference to (1), polyno-mial filters can be classified into recursive and nonrecursive fil-ters. Recursive filters are characterized by (possibly nonlinear)feedback terms in their input–output relationships and, as infi-nite impulse response linear filters, possess an infinite memory.Nonrecursive polynomial filters, which are commonly calledVolterra filters, are characterized by input–output relationshipsthat result from a double truncation of the Volterra series, i.e.,a memory truncation, by limiting the memory of the filters to afinite number of terms in each kernel, and an order truncationby limiting the number of Volterra kernels

(2)

Without any loss in generality, (2) exploits the so-called tri-angular representation of the Volterra kernels [1].

In the multichannel approach, the truncated Volterra filteris implemented as a linear filterbank using a set of FIR fil-ters whose coefficients are the elements of the Volterra kernels.The input signals to such channels are the corresponding prod-ucts of input samples. According to the diagonal representa-tion [9], [14] or the SVF scheme [5], the channels of the fil-terbank are formed by the diagonals of the Volterra kernels. TheV-vector algebra is the most efficient tool we have at our dis-posal to describe this realization in a compact form avoidingany change of the coordinate system. Moreover, it allows usto arrange the filter channels in a hierarchical sequence, i.e.,according to the increasing distance from the main diagonal.In fact, the products of input samples and the correspondingfilter coefficients of any Volterra kernel can be ordered in aV decreasing structure, as shown in [15], by applying a suit-able filter-order-memory-length recursion. As a consequence,the rows of the V-vectors correspond to the diagonals of the di-agonal representation. The last rows of the V-vector representa-tion can be simply discarded in the frequent case when the rele-vance of the elements of the Volterra kernels decreases movingfar from the main diagonal.

In the following, we will use a simple measure of the dis-tance of a diagonal of a Volterra kernel from the main one and,thus, derive an algorithm to implement the filter-order-memory-length recursion given in [15].

A. Multichannel Representation of Homogenous VolterraFilters

Let us first consider the generic homogeneous Volterrafilter characterized by the th-order Volterra kernel

. The first row of the correspondingV-vector is formed with the coefficients of the maindiagonal of the th-order hypercube representing thekernel, i.e., the coefficients marked with the indices

.Then, the successive rows are formed with all the coefficientsthat lay on the other diagonals. A simple parameter that isderived from the indices of the coefficients belonging to each


diagonal, is used to indicate the distance from the main diag-onal, i.e.,

P (3)

The parameter P is actually used as a label that is asso-ciated with each diagonal of the kernel hypercube, whichallows us to list all the diagonals depending on the increasingdistance from the main one and, at the same time, the de-creasing number of coefficients of the corresponding FIR filtersin the multichannel implementation. The above mentionedfilter-order-memory-length recursion can be implemented ac-cording to the following pseudo-code, where the memory-lengthrecursion is nested inside the filter-order recursion.

---Begin filter-order recursion---for

for

for

---Begin memory-length recursion---

for

end---End memory-length recursion---

end

endend---End filter-order recursion---

The external for cycles on the current indicesP P 2 (the filter-order recursion) designate the

first elements of each row of the V-vector. The internalfor cycles on the current indices (the memory-lengthrecursion) designate the further elements laying on eachrow of the V-vector. In practice, with reference to thegeometrical interpretation of a Volterra kernel, the filter-orderrecursion allows us to move on the hypersurface ,within the constraints , dueto the triangular representation. Once an element has beenselected by the filter-order recursion and a value for P hasbeen computed, the memory-length recursion allows us to

collect all the elements for which the same value of P ismeasured. The vectors sequentially return all the indices

that univocally identify the coefficients ofthe th-order Volterra kernel . Then,the rows of the V-vector of the products of input samples

are identified in the samesequence by the same indices . The number of rows inthese V-vectors is given by the number of combinations withrepetitions of objects taken at at the time, i.e.,

(4)

and the number of elements in each row is equal to the cor-responding value of computed in the recursions describedabove.

We are now in a position to represent a homogeneousVolterra filter of any order as a multichannel filterbank usingV-vectors. In this case, if the V-vector of the products of inputsamples at time is denoted as , then its rows arethe inputs to the channels of the filterbank. The correspondingV-vector formed with the coefficients of the Volterra kernel isdenoted as , and its rows contain the coefficients of thekernel diagonals. It is worth noting that both the V-vectors areof type , where is the numberof rows in the V-vectors, i.e., the number of kernel diagonals,and are the corresponding lengthes, i.e.,the number of coefficients in each diagonal. As a consequence,the input–output relationship of the multichannel filterbank issimply described by the equations

(5)

which imply the use of a set of FIR filters defined by the rowsof . The block diagram of the multichannel filterbankis depicted in Fig. 3. The block contains a delay line ofelements and an array of multipliers to compute the products ofthe input samples. The block contains an adder tree.

As mentioned before, the FIR filters that do not contain rele-vant coefficients can be cut off.

1) Example: Let us consider now, as an example, a cubicVolterra kernel with memory length equal to three sam-ples, i.e., . According to the previous algorithm, thereare six diagonals, which are defined on the basis of the followingindices that identify the coefficients of the given kernel

(6)

The elements of the corresponding V-vector formed with theinput samples , , and are shownin Fig. 4.

The filterbank that implements the input–output relationshipis shown in Fig. 5. In this simple example, the first FIR filterhas three taps , and , the second


Fig. 3. Multichannel filterbank.

Fig. 4. Input data V-vector of an order 3, memory length 3, homogeneousVolterra filter.

and third FIR filters have two taps and, respectively, and the last three FIR filters

have only one tap and , respec-tively.

B. Multichannel Representation of a Complete Volterra Filter

The multichannel representation of a complete Volterra filterof any order can be derived by first building the V-vectorsrelated to each kernel and then by rearranging all their rowsin a unique V descendant structure according to the respec-tive length . In the case where all the kernels have the samememory length , this arrangement coincides with that de-rived by simply considering the distance P. The first row inthis structure is formed by the coefficients of the linear termthat, by definition, is assumed to be characterized by a distance

. Then, the rearrangement is done by ordering in thefinal V-vector the rows of the single V-vectors according to thevalues of the distance p, , and of the index . Anexample is given in Fig. 6, where the input data to a completeVolterra filter of order 3 and memory length 3 are shown. Therows of the V-vectors of the linear, quadratic, and cubic termsare interleaved according to the values of , , and . The se-quence of the values labeling the rows from the top to the bottomis , , , , , , ,

, , and . Using arrangements of this typefor the input samples and for the Volterra filter coefficients, it ispossible to directly apply the AP algorithm described in the nextsection to a Volterra filter of any order and memory length

.

IV. DERIVATION OF THE AP ALGORITHM USING THE

V-VECTOR ALGEBRA

In this section, we present the derivation of the AP algorithmfor a Volterra filter of any order and memory length . Thederivation of the AP algorithm for Volterra filters is done in thecontext of the so-called Filtered-X algorithms used in the fieldof the active noise control [9]. We first briefly describe howa single-channel active noise control problem is usually mod-eled. The corresponding block diagram is shown in Fig. 7. Thenoise source is sensed by a reference microphone, and the pri-mary path consists of the acoustic response from the referencemicrophone to the error microphone located at the cancelingpoint. The signal to be attenuated is marked . The refer-ence microphone collects the samples of the noise sourceand feeds them as input to the adaptive controller. The controlleris adapted according to the feedback signal coming fromthe error microphone. The controller output generates anacoustic signal that, traveling through the secondary path ,gives a signal that destructively interferes with the unde-sired signal . It is usually assumed that the secondary pathis linear and time-invariant and that its impulse responsehas been obtained by separate estimation procedures. Then, thesignal is given as the linear convolution of withthe signal , and thus, the error signal can be expressed as

, where indicatesthe operation of linear convolution.

In the nonlinear situation, the controller is described as a trun-cated Volterra filter implemented as a filterbank according to thestructure implied by the V-vector representations of both theinput data and the filter coefficients, as schematically shown inthe example in Fig. 3. Therefore, the output from the filterbankis given, as in (5), by

(7)

where and are V-vectors of type

. While the Filtered-X LMSalgorithm minimizes according to the stochastic gradient ap-proximation the single error at time , the aim of the Filtered-XAP algorithm of order is to minimize the last a posteriori


Fig. 5. Multichannel filterbank implementation of the cubic Volterra filter of the Example.

Fig. 6. Input data V-vector of a complete Volterra filter of order 3 and memorylength 3.

Fig. 7. Filtered-X AP adaptive nonlinear controller.

errors. This minimization can be performed by finding the min-imum norm of the coefficient increments that set to zero theseerrors. More in detail, the a posteriori error at time ,with , is defined as

(8)

where the symbol denotes the convolution operation. The setof constraints for an th-order Filtered-X AP algorithm is ex-plicitly written in the following form:

...

(9)The function to be minimized can then be defined as

(10)

where

(11)

and are Lagrange’s multipliers.

By differentiating with respect to , the fol-lowing set of equations is obtained:

(12)

where the linearity of the convolution operation has beenexploited. To rewrite the previous equation in a morecompact form, let us define the V-vector of type


, which actually is a column vector formedwith the entries , , so that

...(13)

Let us also define the V-matrix suchthat

...

(14)

It can be noted that is a column of V-matricesof type . Using such definitions, (12) becomes

(15)

Premultiplying (15) with the V-matrix , thefollowing equation is obtained:

(16)

It can be also noted that according to (14)

...

(17)

The elements of the V-vector on the right side of (17) arescalars that can be interpreted, apart from a minus sign, as the setof the a priori errors for . In fact, the genericelement in the row can be rewritten as

(18)

where the constraints on the a posteriori errors have been takeninto account. Therefore, the following equation is derived:

(19)

where is the V-vector of type , i.e., a column vector,of the last a priori errors.

By using (15), (16), and (19), the following equation can bederived after some simple manipulations:

(20)

Finally, the updating equation becomes

(21)

In this equation, the step size has been introduced in orderto control the speed of convergence of the algorithm. It can be

noted that the V-matrix is of typeand thus is actually a matrix of dimensions so

that the inversion required is an affordable task since oftenassumes in practice small values, i.e., , 3, and 4. If wethen split the V-vector in its rows, the followingadaptation rule can be easily derived for each channel of thefilterbank:

(22)where a different adaptation constant is used for eachchannel.

A more general solution for the computation of the inverse

of the matrix has been recentlyproposed in [21], where a recursive estimate of this matrix isused. The algorithm is based on the direct update of the inversematrix, as done for RLS algorithms. This technique applies toany order of APs.

It is worth noting that the model described above for the ac-tive noise control clearly includes the usual situations in whichadaptive filters are used, i.e., when the input signal is not filtered.In this case, in fact, it is simply required that we use, as the im-pulse response of the secondary path, the unitary impulse .As a consequence, the updating rules reported above can beused within this condition for the adaptation of standard adap-tive Volterra filters.

The good convergence and tracking behaviors of these AP al-gorithm have been experimentally verified, and a few simulationresults are reported in the next section. Even though a theoretical


study of these properties is not available at this time, the generalgood behavior of the AP algorithms can be inferred by the factthat for , they correspond to LMS algorithms, while forincreasing values of , they tend to behave as RLS algorithms.Therefore, their faster convergence and tracking capabilities areexplained. In this respect, it should also be taken into accountthat the inputs to the multichannel filterbank implementing theVolterra filter are given in the form of products of input samplesexhibiting correlations among them, even when the input signalis white.

The operations required for the computation of

and and those involved in the updating (21) are all

of the order of , if , as it frequentlyhappens. Therefore, the overall computational complexity peroutput sample is given by , whereis the total number of coefficients involved in the Volterra filter.It clearly appears that this complexity is times that of theLMS/NLMS algorithms .

V. SIMULATION RESULTS

In order to validate the Filtered-X AP algorithm derived inthe previous section, in this section, we report the results of afew simulations.

To test the convergence characteristics of the Filtered-X APalgorithm, we assume, as an experimental environment, thesame conditions described in [9, Sec. IV-A]. A logistic chaoticnoise, i.e., a second-order white and predictable nonlinearprocess, is used as source noise. The noise is generated by theequation

(23)

where is a real number between 0 and 1, which, here, ischosen to be equal to 0.9. The nonlinear process is then normal-ized in order to have a unit signal power . Theprimary and secondary paths are modeled with the followingFIR filters, respectively:

(24)

(25)

The system is identified with a second-order Volterra filterwith a linear part of memory length 10 and a quadratic part ofmemory length 10 and ten diagonals . In our tests, the

direct inversion of the V-matrix ,which is actually an matrix, is applied at any iteration.Even though this step is often a critical one, for the Filtered-XAP algorithms of low orders, i.e., , 3, and 4, the matrixinversion is an affordable task. Moreover, in order to avoid pos-sible numerical instabilities, a diagonal matrix can be addedto the matrix, where is a small positive constant. Fig. 8plots the ensamble average of the resulting mean square errorfor 100 runs of the simulation system with adaptation constants

for both the linear and the quadratic parts of theVolterra filter. The four curves refer to different values of theAP order . As previously mentioned, the order corre-sponds to a normalized LMS adaptation algorithm. Apart from

Fig. 8. Adaptation curves for different orders of affine projection L.

Fig. 9. Adaptation curves with a sudden modification in the noise conditions.

a normalization factor, it is the same adaptation algorithm ap-plied in [9]. It is evident from the figure how the convergencebehavior of the algorithm improves with the order of the APs.

To test the tracking characteristics of the Filtered-X AP algo-rithm, a sudden change in the noise source and its propagationmodel is applied. Initially, the experimental conditions are thesame described above. After 100 000 samples, the primary pathmodel is modified according to the following equation:

(26)

The input signal is modified to , where is thenormalized logistic noise of (23) with . Fig. 9 plotsthe resulting adaptation curves for different orders of the affineprojections. We can observe the improvement in the trackingbehavior determined by the AP algorithm.

Finally, we investigate the effects of modeling an active noisecontroller as a multichannel filterbank with a reduced numberof channels. The noise source is the logistic chaotic noise of the


Fig. 10. Adaptation curves with different number of channels.

first set of experiments, and the primary path is modeled by thefollowing second-order Volterra filter:

(27)

The secondary path is the linear filter of (25). The system isidentified with a multichannel filterbank with a linear part anda quadratic part both of memory length equal to 10. Fig. 10plots the ensamble average of the resulting mean square errorfor 100 runs of the simulation system. Here, the affine projec-tion order has been set equal to 4, and the adaptation constantsare equal to 0.0015. The different curves refer to different num-bers of channels of the quadratic part. It can be clearly ap-preciated that two channels are not adequate to model the givensystem, whereas four channels remarkably improve the approx-imation at a reduced computational complexity. Finally, modelswith six and ten channels give, in practice, the same adapta-tion error. These configurations can be actually considered tobe equivalent in terms of convergence speed and approximationaccuracy. However, the configuration with should beclearly preferred since it offers a reduced computational com-plexity in comparison to the complete multichannel structure.

VI. CONCLUSIONS

In this paper, a Volterra filter of order is realized as a mul-tichannel filterbank. This kind of implementation can be effi-ciently described by resorting to the so-called V-vector algebra.The fundamental elements of this recently presented algebra arereviewed, together with the basic operations involving V-vec-tors and V-matrices. Then, the multichannel representation ofhomogeneous and complete Volterra filters is derived by usingthis specific algebra. The single channels are arranged accordingto the decreasing length of the FIR filters involved so that it ispossible to reduce the computational complexity when the lastand less important channels can be deleted.

It is also shown in the paper how this algebra allows the directderivation of AP adaptive algorithms for Volterra filters of anyorder . These algorithms are derived with reference to theirFiltered-X versions used in the context of the active noise con-trol. In such a way, it is possible to design and implement an effi-cient nonlinear active noise controller of reasonable complexity.

REFERENCES

[1] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Pro-cessing. New York: Wiley, 2000.

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[9] L. Tan and J. Jiang, “Adaptive Volterra filters for active noise controlof nonlinear processes,” IEEE Trans. Signal Process., vol. 49, pp.1667–1676, Aug. 2001.

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Giovanni L. Sicuranza (SM’85) is Professor ofsignal and image processing and head of the ImageProcessing Laboratory at the Department of Elet-trotecnica Elettronica Informatica, University ofTrieste, Trieste, Italy. His research interests includemultidimensional digital filters, polynomial andVolterra filters, processing of images and imagesequences, image coding, and adaptive algorithmsfor echo cancellation and active noise control. Hehas published a number of papers in internationaljournals and conference proceedings. He is the

co-editor, with Prof. S. Mitra, University of California at Santa Barbara, of thebooks Multidimensional Processing of Video Signals (Boston, MA: Kluwer,1992) and Nonlinear Image Processing (New York: Academic, 2001). He isthe co-author, with Prof V. J. Mathews, University of Utah, Salt Lake City, ofthe book Polynomial Signal Processing (New York: Wiley, 2000).

Dr. Sicuranza was Chairman of the VIII European Signal Processing Confer-ence EUSIPCO-96 (Trieste, Italy, 1996) and the Sixth Workshop on NonlinearSignal and Image Processing, NSIP-03, (Grado, Italy, 2003). He is an Asso-ciate Editor of Multidimensional Systems and Signal Processing and a memberof the editorial boards of Signal Processing and the IEEE SIGNAL PROCESSING

MAGAZINE. He has served as a project manager of ESPRIT and COST researchprojects funded by the European Commission and as a consultant to several in-dustries. He was a member of the Administrative Committee of the EuropeanAssociation for Signal, Speech, and Image Processing (EURASIP) and is cur-rently a member of the IMDSP Technical Committee of the IEEE Signal Pro-cessing Society. He was one of the founders and the first Chairman of the Non-linear Signal and Image Processing (NSIP) Board.

Alberto Carini (M’99) was born in Trieste, Italy, in1967. He received the Laurea degree (Summa CumLaude and Print Honor) in electronic engineering in1994 and the “Dottorato di Ricerca” (Ph.D.) degree ininformation engineering in 1998, both from the Uni-versity of Trieste, Trieste, Italy.

In 1996 and 1997, during his Ph.D. studies, hespent several months as a Visiting Scholar at theUniversity of Utah, Salt Lake City. From 1997 to2003, he was a DSP Engineer with Telit MobileTerminals s.p.a., Trieste, where he lead audio pro-

cessing R&D activities. In 2003, he was with Neonseven s.r.l., Trieste, as anaudio and DSP expert. From 2001 to 2004, he collaborated with the Universityof Trieste as a contract professor of digital signal processing. Since 2004, hehas been an Associate Professor with the Information Science and TechnologyInstitute (ISTI), University of Urbino, Urbino, Italy. His research interestsinclude adaptive filtering, nonlinear filtering, nonlinear equalization, acousticecho cancellation, and active noise control.

Prof. Carini received the Zoldan award for the best Laurea degree in electronicengineering at the University of Trieste during 1992–1993.

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