Convergence analysis of the multiple-channel filtered-U recursive LMS algorithm for active noise...

*Corresponding author.E-mail address: [email protected] (C. Mosquera)

Signal Processing 80 (2000) 849}856

Convergence analysis of the multiple-channel "ltered-Urecursive LMS algorithm for active noise control

Carlos Mosquera*, Fernando PeH rez-GonzaH lez

ETSI Telecomunicacio& n, Universidad de Vigo, 36200 Vigo, Spain

Received 13 July 1999; received in revised form 23 November 1999

Abstract

The active noise control in large-dimension ducts usually requires the use of a multiple-channel adaptive system withseveral error sensors, secondary sources and reference sensors. Here we present the convergence analysis of themultiple-channel "ltered-U recursive LMS algorithm, which employs adaptive IIR "lters to improve the cancellationcapabilities with respect to those of the corresponding FIR case, namely, the multiple-channel "ltered-X LMS.A positive-de"nite condition is found to be su$cient to ensure convergence. Such a condition includes also the e!ects ofthe error path modeling mismatch, extending and generalizing previous results existing in the literature. ( 2000Elsevier Science B.V. All rights reserved.

Zusammenfassung

Die aktive RauschunterdruK ckung in langen akustischen AusbreitungskanaK len macht den Einsatz eines mehrkanaligenadaptiven Systems mit mehreren Fehlersensoren, sekundaK ren Quellen und Referenzsensoren noK tig. Wir praK sentieren hierdie Konvergenzanalyse des mehrkanaligen `Filtered-Ua rekursiven LMS Algorithmus, der adaptive IIR Filter zurVerbesserung der RauschunterdruK ckungseigenschaften benutzt, im Gegensatz zu dem entsprechenden FIR Fall, der alsmehrkanaliger `Filtered-Xa LMS Algorithmus bekannt ist. Es wird eine positiv de"nite Bedingung gefunden, diehinreichend fuK r die Sicherstellung der Konvergenz ist. Solch eine Bedingung beinhaltet ebenfalls die E!ekte einerFehlmodellierung des Fehlerpfades und erweitert und verallgemeinert damit fruK here, in der Literatur existierendeErgebnisse. ( 2000 Elsevier Science B.V. All rights reserved.

Re2 sume2

Le contro( le actif de bruit dans les voies de communication de grande dimension requiert l'utilisation d'un systemeadaptatif multi-canaux ayant plusieurs capteurs d'erreur, sources secondaires et capteurs de reH feH rence. Nous preH sentonsici une analyse de la convergence de l'algorithme LMS reH cursif U-"ltreH multi-canaux, qui emploie des "ltres IIRadaptatifs pour ameH liorer la capaciteH de suppression vis-a-vis des "ltres FIR, nommeHment le LMS X-"ltreH multi-canaux.Une condition de positiviteH est trouveH e e( tre su$sante pour assurer la convergence. Une telle condition inclue aussi lese!ets du malajustement de la modeH lisation du chemin d'erreur, ce qui eH tend et geH neH ralise des reH sultats anteH rieurementpublieH s. ( 2000 Elsevier Science B.V. All rights reserved.

Keywords: Active noise control; Adaptive algorithms; Convergence; IIR; MIMO

0165-1684/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 1 6 8 - 1

Fig. 1. Structure of a multiple-channel acoustic ANC system.

Fig. 2. Block diagram of the adaptive structure in the single-channel case.

Fig. 3. Block diagram where S(z) has been splitted into two"lters.

1. Introduction

The active noise control (ANC) problem hasreceived a great deal of attention in the last fewyears. Although an old idea, it has not been untilrecently that the development of digital signal pro-cessors capabilities has boosted the use of activeways of reducing the levels of noise at low frequen-cies, for which passive methods cannot cope withthe required levels of attenuation [2]. To illustratethe application, consider the structure of a mul-tiple-channel acoustic ANC system as that shownin Fig. 1. There are J reference sensors, K second-ary canceling sources and M error sensors. Thenoise measured by the J microphones is activelycanceled by the K secondary loudspeakers, whichare controlled by adaptive "lters to minimizethe sum of the squares of the M error sensor out-puts. The single-channel case corresponds toJ"K"M"1, for which di!erent types of algo-rithms have been presented throughout the pastyears. From those, the most widely used are the"ltered-X LMS (FxLMS) and the "ltered-U recur-sive (FuRLMS) algorithms, which update adaptiveFIR and IIR "lters, respectively. Fig. 2 representsthe control structure in the single-channel case. S(z)represents the combined e!ects of the loudspeaker,error microphone, conversion "lters and acousticpath followed by the error signal before being cap-tured by the error microphone. Strictly speaking,this so-called error "lter should be split into two

"lters, one in the estimated output path, S2(z), and

another in the error path, S1(z), as in Fig. 3. S

1(z)

represents the reconstruction "lter, the loudspeakerresponse and the residual transfer function from theloudspeaker output to the cancellation point. S

2(z)

models the common path followed by the desiredand estimated signal, including the error micro-phone and antialiasing "lter. The main contribu-tion of S

1(z) is a delay for the frequency range under

consideration, delay which is to be compensated byan at least equal acoustic delay in the desired trans-fer function H(z); this is the so-called causality con-straint of broadband active cancellation systems[2]. The residual contribution of S

1(z) in addition

to the delay is usually included in the identi"cationprocess, i.e., the transfer function to be identi"edmodi"es to accommodate this residual term [13].Thus, S(z) in Fig. 2 includes S

2(z) and the result of

extracting S1(z) from the lower branch, and it is the

"lter which a!ects directly the convergence proper-ties, as shown later on.

The FxLMS algorithm "lters the input u(n) by anestimation SK (z) of the error path "lter before enter-ing the LMS algorithm. Its convergence is guaran-teed for a su$ciently small adaptation size if

ReGS(e+u)SK (e+u)H'0, u3[0,2p). (1)

850 C. Mosquera, F. Pe& rez-Gonza& lez / Signal Processing 80 (2000) 849}856

Fig. 4. Block diagram of the adaptive structure in the multiple-channel case.

This property, "rst presented in [6], and sub-sequently developed in many other references, statesthe importance of the modeling of the error pathtransfer function in phase terms. This result wasindependently extended to the FuRLMS algorithmin [7,13]. In this case, the estimated output y( (n) isalso "ltered before entering the adaptation equation.The positive real-part condition is now as follows:

ReGS(e+u)

SK (e+u)A(e+u)H'0, u3[0,2p), (2)

where A(z) denotes the poles of the unknown trans-fer function. The FuRLMS algorithm was "rst de-rived by Eriksson et al. [1] as an extension of theFxLMS algorithm to the zero-pole case, to dealbetter with the feedback from the canceling loud-speaker to the reference microphone.

The single-channel con"guration, well suited forboth narrowband and broadband cancellation innarrow ducts, happens to be clearly insu$cient formore complicated enclosures or large dimensionducts. In fact, the number of secondary sourcesneeded to achieve perfect narrowband noise cancel-lation in an enclosure is the same as the number ofexcited acoustic modes, number proportional tothe volume of the enclosure [2]. This is why mul-tiple-channel versions of Filtered LMS algorithmshave been developed. Thus, the results exposedabove regarding the convergence of the single chan-nel FxLMS case have been extended to the Mul-tiple-channel FxLMS algorithm in [12], althoughfor a unique reference, i.e., J"1 in Fig. 1.

We present here a general analysis of multiplechannel algorithms. The multiple-input/multiple-output FuRLMS (MIMO FuRLMS) algorithm isdescribed in the next section, and its convergenceanalyzed subsequently, extending the positive real-part results of the single-channel FuRLMS [7,13]and the multiple-channel FxLMS algorithms [12].The MIMO FuRLMS was "rst presented in [3,5],as an extension of the adaptation of multiple FIR"lters, which result computationally intensive forlightly damped systems, and which cannot elimin-ate feedback in general. For MIMO systems actingin the frequency domain see [10,11]. Successfulcancellation results were reported in those works,although analytical convergence results have re-mained unpublished. We consider an arbitrary

number of actuators, dealing with the multiple-reference case. It should be pointed out that severalreferences are a must when the noise sources aredistributed inside the enclosure, and not located atone single point as in the single-duct case [2]. Andalthough at the early stages of active multiple-channel cancellation the combination of severalsingle input}single output controllers was sugges-ted, cross-connection topologies proved to be bet-ter [5]. This is due to the struggle among individualcontrollers to minimize residual noise at one loca-tion without regard to other locations.

Standard notation is employed: vectors andmatrices are boldface lower-case and upper-caseletters, respectively; the matrix transpose and theHermitian are denoted by ( ) )5 and ( ) )H; I

pis the

p]p identity matrix; E[ ) ] is the mathematical ex-pectation; trM ) N is the trace operator.

2. The multiple-channel 5ltered-U recursive LMSalgorithm

Fig. 4 shows the block diagram in the multiplechannel case. A "lter C(z) has been introduced inthe path error to help achieve convergence. Themission of this "lter is to smooth out the negativee!ects of the poles of the unknown system on theconvergence of the algorithm, as usually done inclassical adaptive IIR "ltering [9]. Its e!ects will bemore clearly understood after deriving the conver-gence conditions.

The di!erent channels of the estimated outputcan be expressed as

y(k(n)"

J+j/1

bkj

(n)uj(n)#

K+i/1

aki(n)y(

i(n),

k"1,2,K (3)

C. Mosquera, F. Pe& rez-Gonza& lez / Signal Processing 80 (2000) 849}856 851

with

uj(n),[u

j(n),2, u

j(n!N)]5, j"1,2, J (4)

and

y(k(n),[y(

k(n!1),2,y(

k(n!N)]5, k"1,2, K.

(5)

The di!erent inputs and outputs can be stackedtogether in vector form

u(n),[u1(n),2,u(

J(n)]5, (6)

y( (n),[y(1(n),2,y(

K(n)]5. (7)

Note that we are assuming Nth-order "lters. Theabove equations can be combined into a single-matrix form expression:

y( (n)"U5(n)h(n), (8)

where the regressor matrix is a P]K block diag-onal matrix de"ned as

U(n)"A/(n) 0 2 0

0 /(n) 2 0

2 2 2 2

0 2 0 /(n)B (9)

with

/(n),[/5y((n) /5

u(n)]5

,[ y( 51(n),2, y( 5

K(n),u5

1(n),2,u5

J(n)]5. (10)

The number of rows P of U(n) is equal toK(KN#J(N#1)). The vector of adaptive coe$-cients is

h(n),[h51(n),2, h5

K(n)]5, (11)

which comprises the coe$cients hk(n), k"1,2, K

corresponding to all the relations between the J in-puts and K outputs with the output k:

hk(n),[h5

ka(n) h5

kb(n)]5

,[a5k1

(n),2,a5kK

(n),b5k1

(n),2,b5kJ

(n)]5,

k"1,2,K, (12)

aki(n),[a

ki,1(n),2, a

ki,N(n)]5, i"1,2,K, (13)

bkj

(n),[bkj,0

(n),2, bkj,N

(n)]5, i"1,2, J. (14)

The MIMO FuRLMS adaptive equation is givenby

h(n#1)"h(n)#kU&(n)e(n), (15)

where the "ltered regressor U&(n) is obtained as

follows:

U&(n)"[SK (z)U5(n)]5

"ASK11

(z)/(n) 2 SKM1

(z)/(n)

2 2 2

SK1K

(z)/(n) 2 SKMK

(z)/(n)B. (16)

SKkm

(z) is the (k,m)th component of the M]Kmatrix SK (z) in z, an estimation of the error pathmatrix S(z). The algorithm does not have accessto the error, but a "ltered version of ite(n)"[e

1(n),2,e

M(n)] given by

e(n)"C(z)S(z)[d(n)!y( (n)]

"C(z)S(z)(H(z)!HK (z))u(n), (17)

where C(z) denotes a transfer function M]Mmatrix, which is to be chosen by the designer toimprove the convergence properties of the algo-rithm, following the corresponding single-channelcase algorithms [9]. The step size is not constrainedto be the same for the M error measurements.Di!erent adaptation constants can be readily in-cluded in the matrix C(z).

Note the use of a mixed notation in the transferdomain and the time domain, following the con-vention in the literature of adaptive IIR "ltering. Inthis notation z~1 denotes the unit delay operator,so that for any sequence u[n] one has

z~1u[n]"u[n!1] for all n.

This notation allows to write H(z) and HK (z) in a waythat encompasses all the true and adaptive coe$-cients:

H(z)"(IK!A

0(z))~1B(z),A~1(z)B(z). (18)

The feedback matrix A0(z) is

A0(z)"A

A11

(z) A12

(z) 2 A1K

(z)

2 2 2 2

AK1

(z) 2 AK,K~1

(z) AKK

(z)B,(19)


where Aki(z)"a0

ki,1z~1#2a0

ki,Nz~N contains the

true coe$cients relating the ith output with thekth output. The feedforward coe$cients aregrouped into the transfer functions B

kj(z)"

b0kj,0

#b0kj,1

z~1#2b0kj,N

z~N, which form thematrix B(z):

B(z)"AB11

(z) B12

(z) 2 B1J

(z)

2 2 2 2

BK1

(z) 2 BK,J~1

(z) BKJ

(z)B. (20)

The same applies for the adaptive MIMO "lterHK (z), whose coe$cients are simultaneously adaptedas follows:

aki(n#1)"a

ki(n)#k

M+

m/1

[SKmk

(z)y(i(n)]e

m(n),

i"1,2,K, (21)

bkj

(n#1)"bkj

(n)#kM+

m/1

[SKmk

(z)uj(n)]e

m(n),

j"1,2,J (22)

for k"1,2,K. This set of simultaneous equations,when stacked together, form the adaptive matrixequation (15).

3. Convergence analysis

The asymptotic behavior of the algorithm can beobtained by examining the di!erential equationassociated to the algorithm for slow adaptation,i.e., with a su$ciently small step size. This is theso-called ordinary di!erential equation (ODE)method, "rst presented by Ljung in [4]. Distur-bance terms independent of the input u(n) do notcontribute to the di!erential equation, so that glo-bal convergence in the noise-free case will implyglobal convergence in mean in the noisy case, underslow adaptation conditions.

The associated di!erential equation to (15) be-comes

dh

dt"E[U

&(n)e(n)], (23)

where

e(n)"C(z)S(z)(d(n)!y( (n))"C(z)S(z)e(n). (24)

In the following derivations the error term e(n) willbe expressed as a function of the error in the coe$-cients

hI "h0!h (25)

with h0 the true coe$cients, and h the frozen coe$-cients in (11).

The un"ltered error e(n) in Fig. 4 can be manipu-lated in the following way:

e(n)"(H(z)!HK (z))u(n)

"U5d(n)h0

a#U5

u(n)h0

b!U5

y((n)h

a!U5

u(n)h

b

(26)

where we have used the following decompositions:

ha"[h5

1a,2,h5

Ka]5, (27)

hb"[h5

1b,2,h5

Kb]5, (28)

h0a"[h05

1a,2,h05

Ka]5, (29)

h0b"[h05

1b,2,h05

Kb]5, (30)

and

Uu(n)"A

/u(n) 0 2 0

0 /u(n) 2 0

2 2 2 2

0 2 0 /u(n)B . (31)

The input term /u(n) is de"ned in (10). The same

applies for the output matrices Uy((n), U

d(n) and the

terms /y((n) and

/d(n)"[d 5

1(n),2, d 5

K(n)]5. (32)

With the appropriate manipulations in (26), we canexpress the un"ltered error as

e(n)"d(n)!y( (n)

"(U5d(n)!U5

y((n))h0

a#U5

y((n)hI

a

#U5u(n)hI

b. (33)

By using a more compact form

e(n)"A~1(z)U5(n)hI , (34)


where A(z) has been de"ned in (18), the di!erentialequation (23) can be expressed as

dh

dt"E[U

&(n)e(n)]

"E[(SK (z)U5(n))5C(z)S(z)A~1(z)U(n)5hI ] (35)

or

dhIdt

"!RhI , (36)

where

R"E[(SK (z)U5(n))5C(z)S(z)A~1(z)U5(n)] (37)

is a P]P matrix, such that if R#R5 is positivede"nite, then the equation is stable, guaranteeingconvergence. This leads us to state the followingtheorem.

Theorem 1. The multiple-channel FuRLMS algo-rithm converges for a suzciently low step size underpersistent excitation conditions if the following posit-ive-dexnite condition is verixed for all u3[0,2p):

A~H(e+u)SH(e+u)CH(e+u)SK (e+u)

#SK H(e+u)C(e+u)S(e+u)A~1(e+u)'0. (38)

Proof. Let d be any column vector of P elements.We have

d5(R#R5)d

"E[(SK (z)U5(n)d)5C(z)S(z)A~1(z)U5(n)d]

#E[(C(z)S(z)A~1(z)U5(n)d)5SK (z)U5(n)d]

"E[(SK (z)a(n))5C(z)S(z)A~1(z)a(n)]

#E[(C(z)S(z)A~1(z)a(n))5SK (z)a(n)], (39)

where a(n)"U5(n)d is a K]1 vector random pro-cess. The expectation terms can be written as innerproducts if we make them equal to

trME[(SK (z)a(n))(C(z)S(z)A~1(z)a(n))5]N

# trME[(C(z)S(z)A~1(z)a(n))(SK (z)a(n))5]N (40)

or equivalently

trG1

2pPp

~p

SK (e+u)Sa(e+u)A~H(e+u)SH(e+u)CH(e+u) duH# trG

1

2pPp

~p

C(e+u)S(e+u)A~1(e+u)Sa(e+u)

]SK H(e+u) duH (41)

with Sa (e+u) the K]K power spectrum matrix ofthe process a(n). Since the matrix product is com-mutative under the trace operator, then

d5(R#R5)d

"

1

2pPp

~p

trMSa(e+u)[A~H(e+u)SH(e+u)CH(e+u)SK (e+u)

#SK H(e+u)C(e+u)S(e+u)A~1(e+u)]Ndu. (42)

Now, the trace of the product of two symmetricpositive-de"nite matrices is always positive, andzero only in the case that one of the matrices isidentically zero [12]. Then, under persistent excita-tion conditions which guarantee that Sa (e+u)'0for all u [9], we have that d5(R#R5)d'0 if thefollowing condition is veri"ed:

A~H(e+u)SH(e+u)CH(e+u)SK (e+u)

#SK H(e+u)C(e+u)S(e+u)A~1(e+u)'0 (43)

for all u3[0,2p), which concludes the proof.

3.1. Discussion of the results

Condition (38) states a su$cient condition forguaranteeing the convergence of the MIMOFuRLMS algorithm for a su$ciently small stepsize.Nonetheless, note that the ODE analysis does notallow to precise how small the stepsize must be.Simpli"ed conditions can be derived for particularcases, achieving the results already presented in theliterature:f MIMO FxLMS. Let A(z)"I

Kand C(z)"I

Min

(38). Then, the convergence of the MIMOFxLMS algorithm is guaranteed provided that

SH(e+u)SK (e+u)#SK H(e+u)S(e+u)'0 (44)


condition already established in [12] for thesingle reference case, i.e., J"1 in Fig. 1.

f Single-channel FxLMS. The previous conditionin the single-channel case can be expressed as

ReGS(e+u)SK (e+u)H'0, (45)

the well-known positive real-part conditionwhich determines that the phase of the error pathtransfer function must be matched with an errornot higher than 903 [2].

f Single-channel FuRLMS. In this case (38) canalso be rephrased as the real part condition de-rived in [7,13] (in this latter reference withC(z)"1):

ReGC(e+u)S(e+u)SK (e+u)A(e+u)H'0. (46)

The general multivariable form of the conver-gence condition (38) is not easy to interpret.Generally speaking, we need to makeSK H(e+u)C(e+u)S(e+u)A~1(e+u) equal to P(e+u), withP(e+u) a K]K matrix such that PH(e+u)#P(e+u)'0. For instance, if we choose P(e+u)"I

K,

then condition (38) is veri"ed if

SK H(e+u)C(e+u)S(e+u)"A(e+u).

Even further, for the square case, i.e., M"K, andprovided that S(e+u) and SK (e+u) are non-singular forall frequencies, then a su$cient condition for hav-ing convergence is

C(z)"SK ~H(z)A(z)S~1(z). (47)

In order to further exploit condition (38) it wouldbe desirable to come up with a synthesis method todesign both C(z) and SK (z) robustly, i.e., such thatcondition (38) is veri"ed for a set of possible A(z)and S(z). In a real system environmental conditionsare variable, and as a result, the transfer functionsinvolved A(z) and S(z) are not "xed. In [8] somerobust methods are detailed to "nd SK (z) ful"lling(45) for an uncertainty set of error path transferfunctions, since no perfect identi"cation of S(z) canbe performed in a real environment, due to un-modeled e!ects, variable conditions, etc. The exten-sion of these methods to the multivariable caserequires to verify the positive de"niteness for all

frequency and all A(z) and S(z) in the uncertaintysets, which amounts to working with the character-istic polynomial of expression (38) in order to verifythat all its roots are positive for every frequencyand every element in the set. We will not pursuefurther this analysis here given the complexity in-volved.

Note that no constraint is imposed on the stabil-ity of the transfer functions, other than the usualassumption that the system under identi"cation isstable, which amounts to saying that A(z) is min-imum-phase. There is no reason why the error pathtransfer function should be minimum-phase; in fact,as experimental results show, S(z) does usually havezeros outside the unit circle [2]. Therefore, thepositive real conditions expressed above in thesingle-channel cases are not equivalent to a strictlypositive real (SPR) property, of common use in thecontrol "eld [9]. On the other side, the role playedby C(z) is clearly unveiled in condition (46). If SK (e+u)is to approximate S(e+u), C(e+u) should compensatefor the e!ects of the poles of the system in A(e+u), asit happens in adaptive IIR algorithms without "l-tered error [9].

Despite the fact that for the sake of clarity noexplicit feedback has been considered in the blockdiagrams shown in this paper, this can be readilyincluded in our analysis. Now the new function tobe identi"ed changes into

H(z)(IJ#F(z)H(z))~1 (48)

with F(z) a J]K transfer matrix from the cancelingloudspeakers to the reference microphones. All theconsiderations made above apply for the new trans-fer function (48). The details for the single-channelcase are given in [7,13].

4. Conclusions

A convergence analysis of the multiple-channelFuRLMS algorithm has been presented based onthe ODE method. A general de"nite positive con-dition was shown to be su$cient to guaranteeconvergence, condition which boils down to theconditions already presented in the literature forsimpli"ed versions, namely, single channel or all-zero cases.


Acknowledgements

We would like to thank Dr. A.K. Wang for herhelpful comments and suggestions.

References

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