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Signal Processing

Signal Processing 91 (2011) 98–113

0165-16

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/sigpro

A new adaptive recursive RLS-based fast-array IIR filterfor active noise and vibration control systems

Allahyar Montazeri n, Javad Poshtan

Faculty of Electrical Engineering, Iran University of Science and Technology, Narmak 16846, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 29 June 2008

Received in revised form

2 June 2010

Accepted 10 June 2010Available online 16 June 2010

Keywords:

Adaptive IIR filters

Active noise and vibration control

Stability

Fast array algorithm

Recursive least square

84/$ - see front matter & 2010 Elsevier B.V. A

016/j.sigpro.2010.06.013

esponding author. Tel.: +98 21 44222782.

ail address: [email protected] (A. Montaz

a b s t r a c t

Infinite impulse response filters have not been used extensively in active noise and

vibration control applications. The problems are mainly due to the multimodal error

surface and instability of adaptive IIR filters used in such applications. Considering

these, in this paper a new adaptive recursive RLS-based fast-array IIR filter for active

noise and vibration control applications is proposed. At first an RLS-based adaptive IIR

filter with computational complexity of order O(n2) is derived, and a sufficient condition

for its stability is proposed by applying passivity theorem on the equivalent feedback

representation of this adaptive algorithm. In the second step, to reduce the

computational complexity of the algorithm to the order of O(n) as well as to improve

its numerical stability, a fast array implementation of this adaptive IIR filter is derived.

This is accomplished by extending the existing results of fast-array implementation of

adaptive FIR filters to adaptive IIR filters. Comparison of the performance of the fast-

array algorithm with that of Erikson’s FuLMS and SHARF algorithms confirms that the

proposed algorithm has faster convergence rate and ability to reach a lower minimum

mean square error which is of great importance in active noise and vibration control

applications.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

The use of active noise and vibration control (ANVC)systems as a complement of passive methods for cancel-ing unwanted low frequency noise and vibration signalshave widely been investigated in various applications bymany researchers. For a thorough review of the subject aninterested reader can refer to the books [1,3,13], and thesurvey papers [2,14]. A large group of adaptive algorithmsused in ANVC applications use FIR filters as the controller,whose parameters are adapted using stochastic gradientdescent algorithms (such as LMS algorithms, Newton-LMS, etc.) or recursive least-squares (RLS) techniques. Oneof the main problems with the use of filtered-LMS familyalgorithms, in which either the input reference signal or

ll rights reserved.

eri).

the error signal is filtered by a model of secondary pathbefore being applied to the LMS algorithm, is theirs slowconvergence rate especially in a multi-channel case. Infact, this phenomenon is often due to the dependence ofthe convergence rate of the algorithm to the differencebetween the minimum and maximum values of auto-spectrum of the regression vector. This difference isdetermined by the auto-spectrum of the reference signaland the dynamics of the secondary path model, and cancause very small convergence rate for broadband noise orvibration signals [3]. The convergence rate can be madeindependent of the auto-spectrum of the regressionvector by using the Newton-LMS algorithm [4]. In thisway, the convergence rate depends only on the step size,and can be chosen such that a fast convergence isobtained. However, since the update of filter coefficientsin Newton-LMS algorithm is based on the estimates of theinverse of the auto-correlation of regressor and cross-correlation between regressor and disturbance, the

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A. Montazeri, J. Poshtan / Signal Processing 91 (2011) 98–113 99

computational complexity of the algorithm is too high(O(n3) where n is the number of control filter weights) formost practical applications of active noise and vibrationcontrol problems. One way to reduce the computationalcost of the Newton-LMS algorithm is to pre-compute theinverse of the auto-correlation matrix of the regressionvector and use it as a constant matrix in the adaptationalgorithms [1]. Such an algorithm would not, however,be able to accommodate the significant changes in thestatistical properties of the reference signal. Another wayto approximate the Newton-LMS update direction is touse the affine projection algorithm (APA) [5]. Fastimplementation of affine projection algorithm in ANCapplications is proposed by [6,7]. Fast affine projectionalgorithm can provide a good trade-off between theconvergence speed and computational complexity of theNewton-LMS algorithm; however, its convergence speedis lower than that of the recursive-least-squares algorithm[8]. Although the RLS algorithm is generally derived froma rather different perspective, the update equation inNewton-LMS algorithm has many similarities with thatin the RLS algorithm. The use of RLS-based algorithms inANVC applications with FIR filter structure has becomemore and more common [9–11]. This is due to the factthat, in contrary to the stochastic gradient descentalgorithm, the convergence behavior of RLS-type algo-rithms is quite independent of the statistics of theincident noise or vibration signal. The computationalcomplexity of RLS-type algorithms is of order O(n2), wheren is the length of the control filter. One of the mainproblems in using plain RLS-type algorithms in ANVCapplications is that they suffer from numerical instabilitydue to finite precision computations. To overcome thisdifficulty it has been shown that array-based RLS filteringmethods (such as the QR and inverse QR algorithms) willbe more reliable in finite precision implementations [12].In order to reduce the computational complexity of RLS-type algorithms used in ANVC applications to the order ofO(n) floating point operations (flops) per sampling instant,a fast transversal filter (FTF) is proposed in [10]. Thenumerical stability of this algorithm is improved byusing QR decompositions and lattice structures in [11].However, the comparison study in [8] shows that theperformance of FTF implementation of RLS algorithmsused in ANVC applications is reduced in comparison withthe original RLS algorithm. Array methods [4,12] arepowerful algorithmic variants that are theoreticallyequivalent to the recursive least-squares algorithms, andhence unlike FTF implementations, an array RLS algorithmwill exhibit the same performance as the originalalgorithm. Besides, computations in the array form areperformed in more compact and reliable manners. Byexploiting the structure of data in the regression vector, afast array implementation of RLS algorithm for adaptingFIR filters is suggested in [12].

Despite the stability and convex performance surfaceof algorithms used to adapt FIR filters in ANVC applica-tions, there are many situations in which the use of IIRfilters is of interest. Using IIR filters in ANVC applicationswas first proposed in [15], in which FuLMS algorithm isused for adaptation of filter weights. One of the main

problems with IIR-based adaptive filters in ANVC applica-tions is their stability analysis. The stability analysis ofFuLMS algorithm, because of using IIR filters, is muchmore complicated than FxLMS algorithm. Some sufficientconditions for stability and convergence of FuLMS algo-rithm using the ODE method of Ljung [17], is derived in[16], and these results are generalized to MIMO case in[18]. It is shown in [19] that the perfect cancellationassumption of the analysis performed in [16] is notnecessary and it can be relaxed in some sense. On theother hand, FvLMS is proposed in [28] to improve thegradient estimate of FuLMS algorithm. Lattice IIR filterswhich are recently introduced to ANC systems in [31,32]take the advantages of stability of these filters. However,all of these algorithms suffer from slow adaptation rate(slower than FxLMS algorithm), and cannot ensureconvergence to the global minimum of performancesurface. These drawbacks were the main motivation forusing equation error method in [29] to derive analgorithm for which global convergence is assured.Nevertheless, due to equation-error nature of the algo-rithm in [29], large deviation from global minimum mayoccur when measurement noise exists or the order of thecontrol filter is not sufficient. Considering these problems,a Steiglitz–Mcbride type adaptive IIR algorithm is pro-posed in [30], but the stability of the algorithm is assumedbeforehand, and there is no guarantee that the poles of IIRfilter in this algorithm do not move outside the unit circle.Other than the ODE method, another approach lessaddressed in the literature for the stability analysis ofadaptive IIR filters is based on hyperstability theory. Thistheory was proposed for the first time in [20], and is usedto analyze the stability of nonlinear and time-varyingclosed-loop systems. Although hyperstability theory rootsin control system community (for example the reader canrefer to [21]), its application in signal processing commu-nity has also been reported with the name of HARF andSHARF algorithms in [22,23]. The use of SHARF and HARFalgorithms in ANVC applications is reported in [24–26];however, their design and analysis are not based on arigorous theory, and a large number of challenging pointsneed to be further investigated. This is effectively due tothe existence of the secondary-path transfer function inANVC systems in comparison with the conventionaloutput error identification problems.

Considering this review, the contribution of this paper istwo-fold: First is to introduce a new fast array RLS-typeadaptive IIR filter with computational complexity of orderO(n), which is numerically stable for real-time implementa-tions. The developed array algorithm for IIR adaptive filtersis an extension of the algorithms proposed in the literaturefor FIR filters. It is shown by numerical simulations that theproposed algorithm has faster convergence rate and lowerminimum mean square error in comparison with classicadaptive IIR algorithms used in ANVC applications. Second,the design and stability analysis of the proposed algorithmis based on Popov hyperstability theory, by which theproblem is transformed to stability analysis of a nonlinearfeedback control system. In fact, it is shown that undercertain SPR conditions the global convergence of theproposed adaptive IIR filter is guaranteed. It is worthwhile

A. Montazeri, J. Poshtan / Signal Processing 91 (2011) 98–113100

to mention that the proposed approach, in contrast tostatistical ODE method used in the literature, is establishedbased on quite a deterministic framework, and hence nostatistical hypothesis need to be considered in the deriva-tion and analysis of the algorithm. This fact is examinednumerically under different spectra of the incident noise.After this introduction in Section 2 an RLS-type adaptive IIRalgorithm is synthesized by transforming the ANVCproblem into an output error identification problem. Theanalysis of the stability of the proposed algorithm andderiving conditions to guarantee the global convergence ofthe algorithm is addressed in Section 3. Since thecomputational complexity of the proposed algorithm is oforder O(n2), a fast array implementation of this adaptive IIRfilter is developed in Section 4. The performance of thealgorithm from the viewpoints of speed convergence andachievable minimum mean square error are compared withFuLMS and SHARF IIR adaptive algorithms in Section 5.

2. Adaptive IIR filter for ANVC systems

2.1. Synthesis of the algorithm

To derive the algorithm, we start with the blockdiagram of a typical ANVC system in Fig. 1. The aim isto adapt the coefficients of IIR filter W(q, n) in (1) suchthat the sum of squares of a posteriori errors {e(i)} in errormicrophone is minimized:

Wðq,nÞ ¼b0ðnÞþ b1ðnÞq

�1þ � � � þ bnBðnÞq�nB

1þ a1ðnÞq�1þ � � � þ anAðnÞq�nA

: ð1Þ

The regularized weighted performance index consid-ered for minimization is the sum of the squared a

posteriori errors as

JðnÞ ¼ ln1ð0Þðw�wð0ÞÞT ðw�wð0ÞÞþ

Xn

i ¼ 1

ln�i1 ðiÞe

2ðiÞ,

eðiÞ ¼ duðiÞþSðqÞ½xðiÞT wðnÞ�: ð2Þ

where d0(i) is the disturbance to be canceled at time i,S(q)is the transfer function of secondary path, x(i) and wðnÞdenote the regression vector and the vector of the filtercoefficients, respectively, defined by

wðnÞ ¼ ½a1ðnÞ,a2ðnÞ,. . .,anAðnÞ,b0ðnÞ,b1ðnÞ,b2ðnÞ,. . .,bnB

ðnÞ�T ,

xðiÞ ¼ ½�uði�1Þ,. . .,�uði�nAÞ,. . .,xðiÞ,. . .,xði�nBÞ�T :

Fig. 1. A typical active noise and vibration control system with IIR

control filter.

In (2), S(q)[.] means filtering of the signal in the bracketby the secondary path, and a posteriori error e(n) isobtained from Fig. 1 if the updated filter weights at time n

(i.e. wðnÞ) are used for calculating the control signal. Onthe other hand a priori error e(n), is obtained when thecurrent filter weights at time n (i.e. wðn�1Þ) are used togenerate the control signal. In order to be able to combinethe transfer function S(q) with the regression vector x(i) in(2), it is necessary to assume the filter weights arechanged slowly. Then (2) can be approximated as follows:

eðiÞ ¼ duðiÞþxTf ðiÞwðnÞ, ð3Þ

where eðiÞ is the approximation of e(i) in (2) when slowadaptation assumption of the filter weights holds. Besides,xf(i) is the filtered regression vector whose elements arethe current and past samples of the measured referencesignal filtered with the estimated transfer function ofsecondary path, and past samples of the controlleroutputs when this filtered measured reference signal isused as the input:

xf ðiÞ ¼ ½�uf ði�1Þ,. . .,�uf ði�nAÞ,. . .,xf ðiÞ,. . .,xf ði�nBÞ�T ,

xf ðiÞ ¼ SðqÞxðiÞ, uf ðiÞ ¼Wðq,iÞxf ðiÞ: ð4Þ

The difference of uf(n) in (4) with uuf ðnÞ in the blockdiagram of Fig. 2 is that, in (4), the updated filter weightsat time n are used to generate uf(n), while in generatinguuf ðnÞ current weights are applied.

The use of approximation in (3) is equivalent tothe assumption of interchangeability of W(q,n) and S(q)in Fig. 1. Hence, the block diagram of Fig. 1 can betransformed to its equivalent form in Fig. 2. In practice,since only an estimate of S(q) will be available for theadaptive algorithm, these two figures will not remainequivalent. This point is considered and will be furtherdiscussed in the proof of Theorem 1. The last stepfor derivation of the algorithm is to transform theblock diagram of Fig. 2 to an output-error identificationproblem. Since in practice the transfer functions D(q) andS(q) are non-minimum phase, by following the approachpresented in [19], the block diagram of Fig. 2 can bewritten in the decomposed scheme shown in Fig. 3. InFig. 3, Di(q), Do(q), Si(q), So(q) are the inner and outerfactors of D(q) and S(q) as follows:

DðqÞ ¼DiðqÞDoðqÞ,

SðqÞ ¼ SiðqÞSoðqÞ:

Fig. 2. Equivalent block diagram of Fig. 1.

l1ðnÞþl2ðnÞxf ðnþ1ÞQ ðnÞxf ðnþ1Þ

Fig. 3. Block diagram of Fig. 2 in the decomposed form.

Fig. 4. Equivalent representation required to transform Fig. 3 to an output-error problem.

Fig. 5. Output-error identification problem equivalent to the block

diagram of Fig. 3.


Then by going from xf(n) to d0(n) in Fig. 3 in the reversedirection of arrows, the block diagram of Fig. 4(a) and itsequivalent representation is obtained. Besides, it isassumed in Fig. 4(b) that the equivalent noise is stated by

mu1ðnÞ ¼m1ðnÞþxðnÞ�PðqÞD�1o ðqÞD

*i ðqÞm2ðnÞ,

xðnÞ ¼ SiðqÞ½S*i ðqÞPðqÞD

*i ðqÞ��DiðqÞsðnÞ;

where Gn(q) means the conjugate of the transfer functionG(q). This relation results from the input-outputequivalence of block diagrams of Figs. 4(a) and (b). Inthe light of the analysis performed in [19], it can be shownthat x(n) is independent from the measured referencesignal x(n), and hence can be considered as ameasurement noise at the output of the plant. Byconsidering the transfer function from xf(n) to d

0

(n) inFig. 4 (b) as

BðqÞ

AðqÞ¼ ½S*

i ðqÞPðqÞD*i ðqÞ�þD�1

o ðqÞS�1o ðqÞ ¼

b0þb1q�1þ � � � þbnBq�nB

1þa1q�1þ � � � þanAq�nA

,

ð5Þ

the block diagram of Fig. 2 can be transformed to anoutput error identification problem as shown in Fig. 5. Byfollowing the approach presented in [21] for an adaptiveoutput error identification problem, the algorithmrequired for adapting the weights of an IIR filter inANVC applications will be derived. For this purpose, it is

assumed that an estimation of secondary-path transferfunction SðqÞ is available. The proposed algorithm toupdate the coefficients of the adaptive IIR filter in Fig. 1 attime n can be formulated as follows:

xf ðnÞ ¼ ½�uf ðn�1Þ,. . .,�uf ðn�nAÞ,xf ðnÞ,. . .,xf ðn�nBÞ�T ,

xf ðiÞ ¼ SðqÞxðiÞ,uf ðiÞ ¼Wðq,iÞxf ðiÞ, ð6Þ

kðnþ1Þ ¼Q ðnÞxf ðnþ1Þ

T, ð7Þ


Q ðnþ1Þ ¼1

l1ðnÞQ ðnÞ�

l2ðnÞQ ðnÞxf ðnþ1ÞxTf ðnþ1ÞQ ðnÞ

l1ðnÞþl2ðnÞxTf ðnþ1ÞQ ðnÞxf ðnþ1Þ

" #,

ð8Þ

eðnþ1Þ ¼ duðnþ1ÞþSðqÞ½xT ðnþ1ÞwðnÞ�, ð9Þ

v0ðnþ1Þ ¼ eðnþ1ÞþXnH

i ¼ 1

hieðnþ1�iÞ, ð10Þ

eðnþ1Þ ¼eðnþ1Þ

1þ xTf ðnþ1ÞQ ðnÞxf ðnþ1Þ

, ð11Þ

wðnþ1Þ ¼ wðnÞ�kðnþ1Þv0ðnþ1Þ: ð12Þ

Here, two weighting sequences 0ol1(n)r1 and0rl2(n)o2 determine how adaptation gain changes intime, and are especially useful for non-stationaryenvironments. Specifically, the effect of l1(n)o1 is tointroduce increasingly weaker weighting on the old data,while the effect of l2(n)40 is to change the weights onthe recent samples. Moreover, eðnÞ and e(n) are a priori

and a posteriori errors, and v0(n) is filtered a priori error. In(10), hi is the coefficient of an FIR filter used to filter a

priori and a posteriori errors. This filter can be designed forexample by the method proposed in [33], however,because of the effect of this filter on the performance ofthe proposed adaptive filter, in practice, some kinds oftrials and errors will be required to tune the coefficientsproperly. The role of this filter is described in the nextsection in more detail. The order of numerator anddenominator of the IIR filter W(q, n) may be determinedfrom (5). In fact, by having an estimation of the primary,secondary, and detection paths, and inserting them in (5),a lower bound for nA and nB will be determined. If thedegrees of numerator and denominator of the adaptive IIRfilter are selected to be lower than what is required, sinceperfect cancellation assumption will not be satisfied, theresidual error remaining after the convergence of thealgorithm will be increased.

Remark 1. In the derivation of the proposed algorithm, inorder to be able to interchange the adaptive filter and thesecondary path, it is assumed that the adaptive filtercoefficients are changing slowly. However, since in theRLS-based algorithms the change of filter coefficients,especially at the few starting samples is fast, thisassumption may be violated in practical applications. Inthis case, the proposed algorithm can be used in themodified filtered-x form with some minor modifications[34,35]. In the examples that will be studied in Section 4,although the convergence is fast, since the magnitude ofthe signals due to the nature of examples are small, theslowly varying assumption of the adaptive filter weightsis still true, and the algorithm remains stable. Besides,proper initialization of the adaptation gain matrix Q(n) isimportant for this purpose.

2.2. Stability analysis

To prove the stability of the algorithm proposed by(6)–(12) for the adaptation of filter weights in Fig. 1, at

first it is necessary to show the relation between eðnÞ ande(n) in Figs. 1 and 2. By comparing Figs. 1 and 2,

eðnþ1Þ ¼ eðnþ1Þþu00f ðnþ1Þ�uuf ðnþ1Þ: ð13Þ

Assuming slow adaptation condition holds thedifferenceu00f ðnþ1Þ�uuf ðnþ1Þ in (13) will be negligible,and hence in practice eðnþ1Þ is replaced by the measure-ment of e(n+1) from the error microphone. With thisassumption, it is required just to prove that

limn-1

eðnþ1Þ-0, ð14Þ

and by using (13), this will lead the actual errormicrophone signal e(n+1) to converge asymptotically tozero. Considering the stated comment, the sufficientconditions for the stability of the algorithm synthesizedin Section 2 is proved by the following theorem.

Theorem 1. The algorithm proposed by (6)–(12) to update

the coefficients of the adaptive IIR filter in Fig. 1 is

asymptotically stable, and will converge to the desired filter

weights if the transfer function HðqÞSðqÞ=AðqÞSðqÞ�l2=2 is

strictly positive real where A(q) is the denominator of the

transfer function defined in (5), and HðqÞ ¼ 1þPnH

i ¼ 1 hiq�i is

a stable FIR filter of order nH. Moreover, maxnl2ðnÞrl2o2 .

Proof. To show that the filter weights of the adaptive IIRfilter in Fig. 1 converges to the desired values in a stablemanner and minimizes the regularized performance index(1), it is equivalent to prove (as shown in Section 2) thatthe estimated parameters vector wðnÞ of controller W(q, n)in Fig. 5 converges to the values for which the a priori

estimation error eðnþ1Þ tends asymptotically to zero. Thiswill be true if we prove

limn-1

eðnþ1Þ ¼ 0: ð15Þ

By defining the a posteriori estimation error in Fig. 5 as

eðnþ1Þ ¼ duðnþ1Þþuf ðnþ1Þ and considering (5) we will

get

eðnþ1Þ ¼ d1ðnþ1Þþmu1ðnþ1Þþuf ðnþ1Þ,

eðnþ1Þ ¼ �A*ðqÞd1ðnÞþBðqÞxf ðnþ1Þþmu1ðnþ1Þþuf ðnþ1Þ,

ð16Þ

where

AðqÞ ¼ 1þq�1A*ðqÞ, A*ðqÞ ¼ a1þa2q�1þ � � � þanAq�nAþ1,

uf ðnþ1Þ ¼Wðq,nþ1Þxf ðnþ1Þ:

By adding and subtracting the term An(q)(uf(n)+m01(n))

to the right-hand side of (16) we obtain

eðnþ1Þ ¼ �A*ðqÞeðnÞþA*ðqÞmu1ðnÞþBðqÞxf ðnþ1ÞþA*ðqÞuf ðnÞ

þmu1ðnþ1ÞþWðq,nþ1Þxf ðnþ1Þ: ð17Þ

Since in the proposed algorithm SðqÞ is used to update

wðnÞ, it is useful to represent (17) in terms of xf ðnÞ. Noting

that the relation between xf ðnÞ and xf(n) can be stated by

xf ðnÞ ¼SðqÞ

SðqÞxf ðnÞ, ð18Þ


we can replace xf(n) in (17) in terms of xf ðnÞ. Considering

slow adaptation assumption, (17) will be equal to

AðqÞeðnþ1Þ ¼ AðqÞmu1ðnþ1ÞþSðqÞ

SðqÞðBðqÞxf ðnþ1Þ

þA*ðqÞuf ðnÞÞþSðqÞ

SðqÞWðq,nþ1Þxf ðnþ1Þ: ð19Þ

Noting A*ðqÞuf ðnÞþBðqÞxf ðnÞ ¼ �xTf ðnÞw where W is the

vector of optimal value of filter weights stated by (5),

and the definition of the a posteriori error, we may rewrite

(19) as

eðnþ1Þ ¼mu1ðnþ1Þ�1

AðqÞ

SðqÞ

SðqÞðxT

f ðnþ1Þ ~wðnþ1ÞÞ, ð20Þ

where ~wðnþ1Þ ¼w�wðnþ1Þ. By subtracting both sides of

(12) from w and then multiplying by �xTf ðnþ1Þ, we may

have

xTf ðnþ1Þ ~wðnþ1Þ ¼ xT

f ðnþ1Þ ~wðnÞþ xTf ðnþ1ÞQðnÞxf ðnþ1Þvðnþ1Þ:

ð21Þ

Considering the relation between vðnþ1Þ and eðnþ1Þ as

vðnþ1Þ ¼HðqÞeðnþ1Þ; ð22Þ

and noting the update equation of Q(n) from (8), theproposed IIR adaptive algorithm can be stated as anonlinear time-varying feedback system with lineartime-invariant part in the feedforward path and anonlinear time-varying part in the feedback path byputting (20)–(22) together. The block diagram of thissystem after loop transformation is shown in Fig. 6.According to the popov hyperstability theory, this system isstable when the feedback system is passive and thefeedforward system is strictly passive. The proof forthe passivity of the feedback path is similar to what isproved in [21], and is not repeated here. To make thefeedforward path strictly passive, a stable polynomialHðqÞ ¼ 1þ

PnH

i ¼ 1 hiq�i with an order less than that of A(q),

Fig. 6. Feedback representation of the proposed algorithm.

should be designed so that the feedforward path remainsstrictly passive &.

3. Fast RLS array implementation

3.1. Preliminary results

The update Eq. (12) requires the gain vector k(n) inorder to compute wðnÞ. In turn, the evaluation of k(n)requires the matrix Q(n), and updating Q(n) needs O(n2)operations per iteration. Besides, the computation of thedenominator of k(n) also requires O(n2) operations. Sincethese update steps are the main computational bottleneckin the algorithm, the effort is to develop a time-update fork(n) directly based on k(n�1). In order to implement afast array form of the algorithm proposed in Section 2, thefirst thing to be noted is the structure of data in theregression vector xf ðnÞ. By splitting the regression vectorinto samples of the previous outputs of the filter x1f ðnÞ,and the samples of the reference signal x2f ðnÞ, the shiftstructure of the regressor for two successive samplingtimes will be captured by noting the following relation:

½�uf ðnÞ,xT1f ðnÞ,xf ðnþ1Þ,xT

2f ðnÞ�

¼ ½xT1f ðnþ1Þ,�uf ðn�nAÞx

T2f ðnþ1Þ,xf ðn�nBÞ�, ð23Þ

where

x1f ðnÞ ¼ ½�uf ðn�1Þ,. . .,�uf ðn�nAÞ�T ,

x2f ðnÞ ¼ ½xf ðnÞ,. . .,xf ðn�nBÞ�T : ð24Þ

Now assuming that l1 and l2 are constant andindependent of time index n, let us define

gðnþ1Þ ¼l�1

1

1þðl2=l1ÞxTf ðnþ1ÞQ ðnÞxf ðnþ1Þ

, ð25Þ

kðnþ1Þ ¼l�1

1 Q ðnÞxf ðnþ1Þ

1þðl2=l1ÞxTf ðnþ1ÞQ ðnÞxf ðnþ1Þ

, ð26Þ

kuðnþ1Þ ¼l2

l1kðnþ1Þ, ð27Þ

guðnþ1Þ ¼ l1gðnþ1Þ: ð28Þ

Then we will get

kuðnþ1Þgu�1ðnþ1Þ ¼l2

l21

Q ðnÞxf ðnþ1Þ, ð29Þ

and

gu�1ðnþ1Þ ¼ 1þ

l2

l1xT

f ðnþ1ÞQ ðnÞxf ðnþ1Þ: ð30Þ

By partitioning k0(n) and the adaptation gain matrixQ ðnÞ as

Q ðnÞ ¼Q11ðnÞnA�nA

Q12ðnÞnA�ðnBþ1Þ

Q21ðnÞðnBþ1Þ�nAQ22ðnÞðnBþ1Þ�ðnBþ1Þ

" #, ð31Þ

and using the following lemma, we may write the left-hand side of (29) and (30) in a more suitable form for anarray implementation.


Lemma 1. The time-update equation for the left-hand side

of can be rewritten in the following forms:

ku1ðnþ1Þgu�1ðnþ1Þ

0


0

266664

377775¼

0

ku1ðnÞgu�1ðnÞ

0

ku2ðnÞgu�1ðnÞ

266664

377775þl�1

1 dQðnÞ

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775,

ð32Þ

gu�1ðnþ1Þ ¼ gu�1

ðnÞþ

�uuf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775

T

dQ ðnÞ

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775, ð33Þ

where

dQ ðnÞ ¼l2

l1

Q11ðnÞ 0 Q12ðnÞ 0

0 0 0 0


0 0 0 0

26664

37775

0BBB@

�

0 0 0 0

0 Q11ðn�1Þ 0 Q12ðn�1Þ

0 0 0 0

0 Q21ðn�1Þ 0 Q22ðn�1Þ

266664

377775

1CCCCA:

Proof. The major difference between the proof of thislemma and the one stated in [12] is the type of appliedpartitioning and augmentation. To derive time-updateequations (32), we start from (29). By using the parti-tioned form of Q(n) in (31) and the definition of x1f ðnÞ andx2f ðnÞ in (24), both sides of (29) can be rewritten inaugmented forms as


0


0

266664

377775

¼l2

l21

Q11ðnÞx1f ðnþ1ÞþQ12ðnÞx2f ðnþ1Þ

0

Q21ðnÞx1f ðnþ1ÞþQ22ðnÞx2f ðnþ1Þ

0

26664

37775

¼l2

l21


0 0 0 0


0 0 0 0

26664

37775

x1f ðnþ1Þ

�uf ðn�nAÞ

x2f ðnþ1Þ

xf ðn�nBÞ

266664

377775, ð34Þ

where

kuðnþ1Þ ¼ku1ðnþ1ÞnA�1

ku2ðnþ1ÞðnBþ1Þ�1

" #:

Using the shift structure of the regressor in (23), (34)

can be written as


0


0

266664

377775¼

l2

l21


0 0 0 0


0 0 0 0

26664

37775�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775:

ð35Þ

By writing (29) for time index n, and augmenting it with

a rather different manner we will have

0

ku1ðnÞgu�1ðnÞ

0

ku2ðnÞgu�1ðnÞ

266664

377775

¼l2

l21

0

Q11ðn�1Þx1f ðnÞþQ12ðn�1Þx2f ðnÞ

0

Q21ðn�1Þx1f ðnÞþQ22ðn�1Þx2f ðnÞ

266664

377775

¼l2

l21

0 0 0 0

0 Q11ðn�1Þ 0 Q12ðn�1Þ

0 0 0 0

0 Q21ðn�1Þ 0 Q22ðn�1Þ

266664

377775�uf ðnÞ

x1f ðnþ1Þ

xf ðnþ1Þ

x2f ðnþ1Þ

266664

377775: ð36Þ

Then, subtracting (36) from (35) and defining dQ(n) as

dQ ðnÞ ¼l2

l1


0 0 0 0


0 0 0 0

26664

37775�

0 0 0 0

0 Q11ðn�1Þ 0 Q12ðn�1Þ

0 0 0 0

0 Q21ðn�1Þ 0 Q22ðn�1Þ

266664

377775

0BBBB@

1CCCCA

ð37Þ

yields


0


0

266664

377775¼

0

ku1ðnÞgu�1ðnÞ

0

ku2ðnÞgu�1ðnÞ

266664

377775þl�1

1 dQðnÞ

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775:

ð38Þ

This proves the update relation (32). For derivation of

(33) the same approach is followed, starting from (30). By

rewriting (30) for two consecutive indices and subtracting

them we will get


ðnÞþl2

l1ðxT

f ðnþ1ÞQ ðnÞxf ðnþ1Þ

�xTf ðnÞQ ðn�1Þxf ðnÞÞ: ð39Þ

Next, it is desired to combine the two terms in the

parentheses of the right-hand side of (39). For this purpose,

each term is dealt with individually. For the first term,

xTf ðnþ1ÞQ ðnÞxf ðnþ1Þ ¼ ½xT

1f ðnþ1Þ�uf ðn�nAÞxT2f ðnþ1Þ xf ðn�nBÞ�

�


0 0 0 0


0 0 0 0

26664

37775

x1f ðnþ1Þ

�uf ðn�nAÞ

x2f ðnþ1Þ

xf ðn�nBÞ

266664

377775,

and by using (31), the expression above can be rewritten as

xTf ðnþ1ÞQ ðnÞxf ðnþ1Þ ¼ ½�uuf ðnÞx

T1f ðnÞxf ðnþ1ÞxT

2f ðnÞ�

�


0 0 0 0


0 0 0 0

26664

37775�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775:

ð40Þ

For the second term,

xTf ðnÞQ ðn�1Þxf ðnÞ

1 That is, J can be decomposed to J= I�S [4].


¼ ½�uf ðnÞxT1f ðnÞxf ðnþ1ÞxT

2f ðnÞ�

�

0 0 0 0

0 Q11ðn�1Þ 0 Q12ðn�1Þ

0 0 0 0

0 Q21ðn�1Þ 0 Q22ðn�1Þ

266664

377775�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775: ð41Þ

Then by replacing (41) and (40) into (39) it will take the

following form:

gu�1ðnþ1Þ

¼ gu�1ðnÞþ �uf ðnÞxT1f ðnÞxf ðnþ1ÞxT

2f ðnÞh i l2

l1

�


0 0 0 0


0 0 0 0

26664

37775�

0 0 0 0

0 Q11ðn�1Þ 0 Q12ðn�1Þ

0 0 0 0

0 Q21ðn�1Þ 0 Q22ðn�1Þ

266664

377775

0BBBB@

1CCCCA

�

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775:

By using the definition of dQ(n), we will get the update

Eq. (33) &.

Since the vector kðnÞ is needed for updating thecoefficients of the IIR filter, it is required to recoverk1(n) and k2(n) from the time-update (32), having thetime-update of gu�1

ðnÞ.Eqs. (32) and (33) in lemma1 show that the update of

the gain vector kuðnÞgu�1ðnÞ as well as gu�1

ðnÞ depends onlyon the value of dQ(n), and hence for fast implementationof the algorithm it is required to compute the time-updateof dQ(n) with an order of O(n) operations. For this purposewe start by the initialization of the algorithm with thefollowing parameters:

kuð0Þ ¼ kð0Þ ¼ 0,gð0Þ ¼ l�11 ,guð0Þ ¼ 1,Q ð0Þ ¼P�1

¼P�1

11 P�112

P�121 P�1

22

" #,Qð�1Þ ¼ l1Q ð0Þ,

P�111 ¼ Z

l21 � � � 0

^ & 0

0 0 lnAþ11

2664

3775, P�1

22 ¼ ZlnAþ2

1 � � � 0

^ & 0

0 0 lnAþnBþ21

2664

3775:

P�112 ¼P�1

21 ¼ 0: ð42Þ

In this case, dQ(0) will be initialized by a matrix ofM+2 by M+2 with M=nA+nB+1, whose rank is four andhas two positive and two negative eigenvalues. Thismatrix can be for example as follows:

dQ ð0Þ ¼ Zl21l2

1 � � � 0 0 � � � 0

^ & ^ ^ & ^

0 � � � �lnA

1 0 � � � 0

0 � � � 0 lnA

1 � � � 0

^ & ^ ^ & ^

0 � � � 0 0 � � � �lM1

26666666664

37777777775: ð43Þ

Since dQ(0) is a matrix with rank four it can be factoredas

dQ ð0Þ ¼ l21L0S0L

T

0 , L0 ¼

ffiffiffiffiffiffiffiffiZl2

q1 0 0 0

^ ^ ^ ^

0 lnA=21 0 0

0 0 lnA=21 0

^ ^ ^ ^

0 0 0 lM=21

266666666664

377777777775

,

S0 ¼

1 0 0 0

0 �1 0 0

0 0 1 0

0 0 0 �1

26664

37775, ð44Þ

where L0 is a matrix with the size of (M+2)�4 and S0 isthe initial signature matrix. In Theorem 2, it will be shownthat if we start from the initial condition above and use

the array algorithm proposed in next section, then dQ(n)

can be similarly factorized as dQ ðnÞ ¼ l21LnSnL

T

n with Sn=S0

for all n=1,2,... .

3.2. Fast array algorithm

Using the initializations in (44), it is possible toassume that dQ(n) in (32) and (33) can be factorized asl2

1LnSnLT

n by the statement that will be proved in Theorem2 (this is true for n=0 based on the initialization (44)).Considering this assumption the following equations willbe resulted:


0


0

266664

377775¼

0

ku1ðnÞgu�1ðnÞ

0

ku2ðnÞgu�1ðnÞ

266664

377775þl1LnSnL

T

n

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775,

ð45Þ


ðnÞþ

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775

T

l21LnSnL

T

n

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775:

ð46Þ

A closer inspection of the Eqs. (45) and (46) revealsthat they are in inner-product and norm-preservingmatrix form, respectively [4], and hence there is aJ-unitary1 transformation Y such that

A B

D E

� �H¼

C 0

F Z

� �ð47Þ

and

HI 0

0 S

� �HT¼

I 0

0 S

� �


with

A¼ gu�1=2ðnÞ,S¼ Sn, E¼ Ln, B¼

�uf ðnÞ

x1f ðnÞ

xf ðnþ1Þ

x2f ðnÞ

266664

377775

T

l1Ln,

D¼

0

ku1ðnÞgu�1=2ðnÞ

0

ku2ðnÞgu�1=2ðnÞ

266664

377775, ð48Þ

ZSnZT ¼

0 0 0 0

0 ku1ðnÞku1TðnÞgu�1

ðnÞ 0 ku1ðnÞku2TðnÞgu�1

ðnÞ

0 0 0 0

0 ku2ðnÞku1TðnÞgu�1

ðnÞ 0 ku2ðnÞku2TðnÞgu�1

ðnÞ

266664

377775þLnSnLn�

ku1ðnþ1Þku1Tðnþ1Þgu�1

ðnþ1Þ 0 ku1ðnþ1Þku2Tðnþ1Þgu�1

ðnþ1Þ 0

0 0 0 0

ku2ðnþ1Þku1Tðnþ1Þgu�1

ðnþ1Þ 0 ku2ðnþ1Þku2Tðnþ1Þgu�1

ðnþ1Þ 0

0 0 0 0

266664

377775,

ð51Þ

and

C ¼ gu�1=2ðnþ1Þ, F ¼

ku1ðnþ1Þgu�1=2ðnþ1Þ

0

ku2ðnþ1Þgu�1=2ðnþ1Þ

0

266664

377775: ð49Þ

From Theorem 2 it can be proved that (in Theorem 2)

Z ¼ffiffiffiffiffiffiffiffil�1

1

qLnþ1 and this will result in an array algorithm for

updating kuðnÞgu�1ðnÞ, gu�1ðnÞ, and Ln.

Theorem 2. If dQ(n) can be factored as l21LnSnL

T

n at time n,

where Ln is a matrix of rank four and dimension (M+2)�4,and Sn is a diagonal signature matrix with +1 and �1 in the

principal diagonal, then the following properties hold for

dQ(n+1) at time n+1:

1.
dQ(n+1) can be factored as l21Lnþ1Snþ1L
T

nþ1 where Lnþ1

is a matrix with rank four and the dimension (M+2)�4,and relates toLn with an array algorithm.

2.
The signature matrix Sn+1 will remain unchanged and
coincides with Sn.

Proof. Since the proof is to some extent similar to what isgiven in [12], here just the main different parts arestressed. In order to extract the state-space fast array formof the proposed RLS-based algorithm, it is good to startfrom the array form (47) with unknown right-hand side,and known A, B, D, E matrices as given in (48). By squaringboth the sides of (47) and using (45) and (46), it can beeasily seen that

X ¼ gu�ð1=2Þðnþ1Þ and Y ¼

ku1ðnþ1Þgu�ð1=2Þðnþ1Þ

0


0

266664

377775:

But deriving an expression for the term Z, needs more

contemplation. By equating the left and right-hand sides

of the (2,2) block matrices, when squaring (47), we

will get

0

ku1ðnÞgu�ð1=2ÞðnÞ

0


266664

377775

0


0


266664

377775

T

þLnSnLn ¼ YYTþZSnZT :

ð50Þ

Substituting for Y in (50) and doing necessary multi-

plication yields

where

LnSnLn ¼l2

l31


0 0 0 0


0 0 0 0

26664

37775�

0 0 0 0

0 Q11ðn�1Þ 0 Q12ðn�1Þ

0 0 0 0

0 Q21ðn�1Þ 0 Q22ðn�1Þ

266664

377775

0BBBB@

1CCCCA:

ð52Þ

In order to be able to collect the expression above, first

it is required to write (8) in terms of ku1ðnÞ and ku2ðnÞ. For

this purpose, by using the partitioned from of Q(n) and

splitting xf ðnÞ into x1f ðnÞ and x2f ðnÞ, (8) can be written as

Q ðnþ1Þ ¼ l�11 Q ðnÞ�

l21

l2

ku1ðnþ1Þ

ku2ðnþ1Þ

" #ku1ðnþ1Þ

ku2ðnþ1Þ

" #T

gu�1ðnþ1Þ,

ð53Þ

Q11ðnþ1Þ Q12ðnþ1Þ


" #

¼ l�11

Q11ðnÞ Q12ðnÞ

Q21ðnÞ Q22ðnÞ

" #

�l2

1

l2

ku1ðnþ1Þku1Tðnþ1Þ ku1ðnþ1Þku2

Tðnþ1Þ


Tðnþ1Þ

" #gu�1ðnþ1Þ:

ð54Þ

Multiplying both the sides of (54) by l2=l21yields

l2

l21



" #

¼l2

l31

Q11ðnÞ Q12ðnÞ

Q21ðnÞ Q22ðnÞ

" #

�ku1ðnþ1Þku1

Tðnþ1Þ ku1ðnþ1Þku2

Tðnþ1Þ


Tðnþ1Þ

" #gu�1ðnþ1Þ:

ð55Þ

By comparing (55) with (51) and (52), expression (51)

can be collected as

ZSnZT ¼l2

l21

Q11ðnþ1Þ 0 Q12ðnþ1Þ 0

0 0 0 0

Q21ðnþ1Þ 0 Q22ðnþ1Þ 0

0 0 0 0

26664

37775

0BBB@

Table 1Computational complexity of the algorithm in RLS form and Fast array

form.

Steps RLS form Fast array implementation

1 ns+M –

2 2M2+M+2 ns+M

3 6M2+M+1 25(M+3)

4 M 2(M+3)

5 nH M

6 M nH

7 M 4

8 – M

Total 8M2+6M+ns+nH+3 30M+ns+nH+85


�

0 0 0 0

0 Q11ðnÞ 0 Q12ðnÞ

0 0 0 0

0 Q21ðnÞ 0 Q22ðnÞ

266664

377775

1CCCCA: ð56Þ

By noting the definition of dQ(n+1), (56) can be written

as

ZSnZT ¼1

l1dQ ðnþ1Þ: ð57Þ

Expression (57) tells that dQ(n+1), as for dQ(n), can be

factorized by a matrix of rank four and the signature

matrix Sn which is remained unchanged. As a result it can

be deduced that Z ¼ffiffiffiffiffiffiffiffil�1

1

qLnþ1, and Sn+ 1=Sn. &

The algorithm proposed in section II by (6)–(12) can besummarized in the array form following the steps below:

1.
Initialize the algorithm with the following para-meters:
L0 ¼

ffiffiffiffiffiffiffiffiZl2

q1 0 0 0

^ ^ ^ ^

0 lnA=21 0 0

0 0 lnA=21 0

^ ^ ^ ^

0 0 0 lM=21

266666666664

377777777775

,

S¼

1 0 0 0

0 �1 0 0

0 0 1 0

0 0 0 �1

26664

37775: ð58Þ

By measuring the new samples of the reference signal
2. x(n+1), update the regression vector in the pre-arraywith these new calculated samples:
xf ðnþ1Þ ¼ SðqÞxðnþ1Þ, uf ðnÞ ¼Wðq,nÞxf ðnÞ: ð59Þ

Find a J-unitary (J= I�S) transformation matrix Yn
3. such that the first element of the first row of the post-array matrix is positive and its last four elements areequal to zero:
gu�ð1=2ÞðnÞ ½�uf ðnÞxT1f ðnÞxf ðnþ1ÞxT

2f ðnÞ�Ln

0


0


266664

377775 l�1

1 Ln

266666664

377777775Hn

¼

gu�ð1=2Þðnþ1Þ 0 0 0 0� �


0


0

266664

377775

ffiffiffiffiffiffiffiffil�1

1

qLnþ1

266666664

377777775:

ð60Þ

Extract the required elements of the first column of
4. post-array matrix to calculate k0(n+1), g0(n+1), andsubsequently k(n+1) and g(n+1), for updating the
coefficients of the filter:

kuðnþ1Þ ¼ku1ðnþ1Þgu�ð1=2Þ

ðnþ1Þ


" #ðgu�ð1=2Þðnþ1ÞÞ,

ð61Þ

guðnþ1Þ ¼ ðgu�ð1=2Þðnþ1ÞÞðgu�ð1=2Þ

ðnþ1ÞÞ, ð62Þ

kðnþ1Þ ¼l1

l2kuðnþ1Þ, ð63Þ

gðnþ1Þ ¼1

l1guðnþ1Þ: ð64Þ

Measure the a priori residual error:
5.
eðnþ1Þ ¼ duðnþ1ÞþSðqÞ½xT ðnþ1ÞwðnÞ�: ð65Þ

Calculate filtered a priori error used by algorithm:
6.
v0ðnþ1Þ ¼ eðnþ1ÞþXnH

j ¼ 1

hjeðnþ1�jÞ: ð66Þ

Calculate a posteriori residual error by
7.
eðnþ1Þ ¼ 1�g�1ðnþ1Þ�l1

l2:gðnþ1Þ

� �eðnþ1Þ: ð67Þ

Update the coefficient vector with
8.
wðnþ1Þ ¼ wðnÞ�kðnþ1Þv0ðnþ1Þ: ð68Þ

Repeat steps 2–9 until the algorithm converges to the
9. optimal weights of the IIR filter.
The computational complexity of both algorithms(number of multiplications in each iteration) are shownand compared in Table 1. It is assumed that the secondarypath is modelled with an FIR filter of order ns, and M is thenumber of coefficients of the IIR control filter.

4. Simulation results

The capabilities and performance of the proposedalgorithm is evaluated here by computer numericalsimulations. For this purpose, two numerical examplesselected from the literature are studied here. Althoughthese examples may seem simple, they represent themain characteristics of the adaptive IIR filters encoun-tered in real applications. The first example introduces theperformance surface in which the global minimum is

Fig. 7. Power spectral density of w(n) for three values of parameter a. Fig. 8. Nyquist plot of HðqÞ=AðqÞ�1=2 for example 1.


placed at a point which is hard to reach using gradientalgorithms, while the performance surface for the secondexample has two local minima, and it is very probablethat gradient algorithms converge to the local optimum.To illustrate that the convergence rate of the proposedalgorithm (unlike the filtered-LMS type algorithms) is notaffected by statistical properties of the reference signal,three kinds of the input signal are assumed as the incidentunwanted noise. These signals are generated by passing azero-mean unit-variance white noise through a filter withthe following transfer function:

HðqÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffi1�a2p

1�aq�1, ð69Þ

where a is a real-valued constant in the range �1 to +1.The power spectral densities of the incident noise s(n) forvalues of a=0, 0.5, 0.75 are plotted in Fig. 7. We note thata=0 corresponds to the case where s(n) is white. As aincreases from 0 to 1, s(n) becomes more colored and forvalues of a close to 1, most of its energy is concentratedaround o=0. The performance of the proposed algorithmfor these three signals is compared with FuLMS, and alsowith FuLMS with SHARF smoothing filter algorithms usingthe following examples. In all simulations the results areplotted after ensemble averaging of 100 differentrealizations of the incident unwanted signal.

Example 1 [27]: This example without any delays inthe primary path is studied first in [27] in a systemidentification framework. The primary-path transfer func-tion is assumed to be

PðqÞ ¼0:05q�6

1�1:75q�1þ0:81q�2, ð70Þ

and the secondary path, and detection-path transferfunctions are selected FIR filters defined by D(q)=q�1

S(q)=0.5q�3+0.35q�4+0.3q�5. Since the aim is to showthe performance of the algorithm from speed of conver-gence, as well as achievable minimum mean square errorviewpoint, the estimated secondary path is assumed to beequal to the real one. For this example the primary-pathtransfer function has two complex conjugate poles with

the amplitude of 0.9, and the local minimum in the errorsurface is placed where it is hard reach by the algorithms.The smoothing filter H(q) is chosen an FIR filter with twocoefficients h0=1 and h1=�0.9, so that the positive realcondition stated in Theorem 1 is satisfied. In Fig. 8, theNyquist plot of HðqÞ=AðqÞ�1=2 shows that its trajectorylies completely in the right half plane. The controller W(q)is selected an IIR filter with a sixth-order numerator and aseventh-order denominator. In fact, by inverting thetransfer functions of the detection and secondary pathsand multiplying them by the transfer function of theprimary path, an idea about the order of the controller canbe determined. In this case, if perfect cancellation isachievable, W(q) should have sixth-order numerator andeighth-order denominator. However, in order to have amore realistic simulation, and to show the performance ofthe algorithm when perfect cancellation is not achievable,the order of denominator is chosen equal to seven. Thealgorithm is initialized by the gain matrix with the valueof Q(1)=30I, and constant weighting sequences l1=1 andl2=1. This corresponds to the decreasing adaptation gainwhich results in a good convergence of the filter weightstowards the proper values. The values of these twoparameters have high influence on the performance ofthe algorithm and must be tuned for each problemcarefully. For example, by choosing l1=1 and l2=0 (i.e.constant adaptation gain) and the same initial value forthe gain matrix, the slowly varying assumption based onwhich the algorithm is developed will be violated, and thealgorithm will become unstable. By decreasing the matrixgain, the algorithm will converge, but with slowerconvergence time than before. Hence, by suitableselection of these two parameters, the algorithm hasmore flexibility and its performance will be increased.For more information and some measures regarding theselection of these two parameters an interested readercan refer to [21]. The performance of the proposedalgorithm is compared with FuLMS, and FuLMS withSHARF smoothing filter. To obtain the highestconvergence rate for these algorithms, the maximumvalues of step size is selected so that any further increase

Fig. 9. Comparing the performance of three algorithms.

Fig. 10. Convergence behavior of two coefficients of IIR controller.


will cause instability of the algorithms. The error signalfor three algorithms after 100 ensemble averaging isshown in Fig. 9. As can be clearly seen, both the FuLMSand the FuLMS with SHARF smoothing filter fail toconverge to the global minimum of the performancesurface, whereas the proposed algorithm converges veryfast to the global minimum. The convergence behavior oftwo filter weights for these three algorithms is plotted in

Fig. 10. As can be seen from Figs. 9 and 10, since theweights of IIR filter for FuLMS and FuLMS with SHARFsmoothing filter have not converged to the correct values,no obvious reduction in the error signal is observed. Theperformance of the proposed algorithm in front of threetypes of input signals mentioned in Fig. 7 is alsoevaluated. The initial condition for all runs are the sameas for a=0 (Q(1)=30I, l1=1, l2=1). The error signals for

Fig. 11. Convergence behavior of error signal for three different unwanted disturbances.

Fig. 12. Nyquist plot of HðqÞ=AðqÞ�1=2 for example 2.


different values of a are plotted in Fig. 11. As can be seenin this figure, since no statistical assumption is used inderivation of the algorithm the error signal and minimummean square error are not affected.

Example 2 [24,30]: This example without any delays inthe primary and secondary paths is studied first in[24,30]. For the second simulation example, the pri-mary-path transfer function is assumed to be

PðqÞ ¼0:05q�3�0:4q�4

1�1:1314q�1þ0:25q�2, ð71Þ

and the detection path and the secondary path transferfunctions are FIR filters defined by D(q)=q�1,S(q)=q�1+0.7q�2+0.6q�3, respectively. Similar to theprevious example, the estimated secondary path transferfunction in the algorithm is chosen to be equal to itsactual one. The primary-path transfer function has twopoles at 0.8303 and 0.3011, and the error surface for thisproblem differs from that of the previous problem in thatit is bimodal (has two error surface minima). Thesmoothing filter H(q) is chosen an FIR filter with twocoefficients h0=1 and h1=�0.85, so that the positive realcondition stated in Theorem 1 is satisfied. The Nyquistplot of HðqÞ=AðqÞ�1=2 shown in Fig. 12 confirms that thestrict positive real condition is satisfied.

The controller W(q) is selected an IIR filter with afourth-order numerator and a fifth-order denominator.Here, similar to the previous example, in order to have asimulation close to what is happening in real applications,the order of the denominator is selected one degree lowerthan what is necessary for perfect cancellation. The initialvalues of the gain matrix is chosen Q(1)=7I, and theweighting sequences l1 and l2, are fixed equal to one. Theerror signals for three algorithms as well as the conver-

gence behavior of two coefficients of the filter are plottedin Figs. 13 and 14. As can be seen in these figures,although both FuLMS and FuLMS with SHARF smoothingfilter exhibit convergence to local optimum, the proposedalgorithm converges to the global optimum with a veryfaster convergence rate. The performances of theproposed algorithm in case of three types of unwanteddisturbances are plotted in Fig. 15. In this example, thevalue of initial gain matrix for a=0, 0.5, 0.75 is selected tobe Q(1)= 7I for all cases, and no noticeable changes in theconvergence rate, and excess in the mean square error isobserved. However, if this is the case, it can be easilycompensated by changing the value of the initial gainmatrix.

Fig. 13. Comparing the performance of three algorithms.

Fig. 14. Convergence behavior of two coefficients of IIR controller.


5. Conclusions

This paper deals with the design and stability analysisof a new fast array adaptive IIR filter in active noise andvibration control applications. Since the proposedalgorithm is derived in a quite deterministic framework,its convergence behavior does not depend on the spectralcontent of the incident noise or vibration. The designand analysis of the proposed algorithm is based onconversion of the original problem to an output-erroridentification problem and then its global convergence is

proved using the hyperstabilty theory. The fast arrayform of the algorithm reduces its computational complex-ity to the order of O(n). Besides, because of itsmatrix nature, it has good numerical stability againstround-off and finite precision errors which is a necessityin real-time implementation of ANVC algorithms.The convergence speed as well as the achievableminimum mean square error of the proposed algorithmis compared by numerical simulations with the wellknown IIR adaptive filters in ANVC applications. Theexamples are chosen from the literature to approve

Fig. 15. Convergence behavior of error signal for three different unwanted disturbances.


the superiority of the proposed algorithm in com-parison with both the FuLMS and FuLMS with SHARFsmoothing filter algorithms previously studied in theliterature.

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