A moving zone of quiet for narrowband noise in a one-dimensional duct using virtual sensing

Cornelis D. Petersen,∗ Anthony C. Zander,† Ben S. Cazzolato,‡ and Colin H. Hansen§

Active Noise and Vibration Control GroupSchool of Mechanical EngineeringThe University of Adelaide, SA 5005, Australia

(Dated: December 5, 2006)

A frequent problem in active noise control is that the zone of quiet created at the error sensortends to be very small. This means that the error sensor generally needs to be located close toan observer’s ear, which might not always be a convenient or feasible solution. Virtual sensingis a method that can move the zone of quiet away from the error sensor to a desired locationthat is spatially fixed. This method has been investigated previously, and has shown potentialto improve the performance of an active noise control system. However, it is very likely thatthe desired location of the zone of quiet is not spatially fixed. An active noise control systemincorporating a virtual sensing method thus has to be able to create a moving zone of quiet thattracks the observer’s ears. This paper presents a method for creating a moving zone of quiet basedon the LMS virtual microphone technique. To illustrate the proposed method, it is implementedin an acoustic duct and narrowband control results are presented. These results show that amoving zone of quiet was effectively created inside the duct for narrowband noise.

I. INTRODUCTION

The aim of a local active noise control system is tocreate a zone of quiet at a specific location, for instanceat the passenger’s ear inside a vehicle cabin. Generally,the greatest noise reduction is achieved at the error sen-sor location, and the created zone of quiet tends to bevery small. Elliott et al.1 have shown both analyticallyand experimentally that in a diffuse sound field, the zoneof quiet in which the noise is reduced by 10 dB or moretypically has the shape of a sphere with a diameter ofone-tenth of an acoustic wavelength. This means thatthe error sensor usually has to be placed close to an ob-server’s ear, which might not always be a convenient orfeasible solution. Virtual sensing2–5 is a method that hasbeen developed in order to overcome these problems thatare often encountered in local active noise control sys-tems. This method requires a non-intrusive sensor whichis placed remotely from the desired location of maximumnoise attenuation. The non-intrusive sensor is used toprovide an estimate of the pressure at the desired loca-tion, which is spatially fixed. The estimated pressurecan then be minimized by a local active noise controlsystem such that the zone of quiet is moved away fromthe physical location of the transducers to the spatiallyfixed desired location of maximum attenuation, such asa person’s ear.

The concept of virtual sensing has been shown to im-prove the performance of a local active noise control sys-tem2–20. However, it is very likely that the desired loca-

∗Electronic address: cornelis.petersen@mecheng.adelaide.edu.

au†Electronic address: anthony.zander@mecheng.adelaide.edu.au‡Electronic address: ben.cazzolato@mecheng.adelaide.edu.au§Electronic address: colin.hansen@mecheng.adelaide.edu.au

tion of maximum attenuation is not spatially fixed. Forinstance, a passenger inside a vehicle cabin will movetheir head, thereby changing the desired location of thezone of quiet. This means that an active noise controlsystem incorporating a virtual sensing method has to beable to create a moving zone of quiet that tracks thepassenger’s head. Although active noise control at a spa-tially fixed virtual location has been investigated previ-ously by a number of authors, the concept of creatinga zone of quiet at a moving virtual location based onvirtual sensing has not been investigated to the authors’knowledge.

Elliott and David2 were the first to suggest a virtualsensing method called the virtual microphone arrange-ment. An important assumption made in this methodis that the primary pressures at the physical and virtualmicrophones are equal. Furthermore, a preliminary sys-tem identification step is required in which models of thetransfer paths between the control source and the physi-cal and virtual microphones are estimated. The assump-tion of equal primary pressures and knowledge of thesetransfer paths allow the pressure at the virtual micro-phone to be estimated. The virtual microphone arrange-ment has been thoroughly investigated by a number ofauthors3,10.

A virtual sensing method called the remote microphonetechnique was suggested by both Popovich11, and Roureand Albarrazin4. The remote microphone technique re-quires the estimation of two transfer paths and one filter,which are usually estimated in a preliminary system iden-tification step. The two transfer paths are the secondarytransfer paths also needed in the virtual microphone ar-rangement2. However, an additional filter needs to beestimated in the remote microphone technique, that com-putes an estimate of the primary pressure at the virtualmicrophone from the primary pressure at the physical mi-crophone. The virtual microphone arrangement assumes

Moving zone of quiet inside duct 1

this filter to be unity, and is therefore a simplified versionof the remote microphone technique.

Cazzolato5 suggested a virtual sensing method basedon forward difference prediction techniques. In this ap-proach, the pressure at the virtual microphone is esti-mated by summing the weighted pressures from a num-ber of microphones in an array. The weights for eachof the elements in the microphone array are determinedusing forward difference prediction techniques. Kestell12and Munn13 investigated the forward difference predic-tion approach for simple sound fields, namely an acous-tic duct and a free field. Experiments showed that alinear prediction method, which uses a two-microphonearray, proved to be better in practice than a quadraticprediction method, which uses a three-microphone ar-ray14–16. The reason for this was the high sensitivityof the quadratic prediction method to short wavelengthextraneous noise. In an effort to overcome this prob-lem, higher-order virtual microphone arrays, which werethought to be able to spatially filter out the extrane-ous noise, were investigated13. A higher number of mi-crophones than the order of the prediction algorithm isused in this method, which results in an over-constrainedproblem that is solved by a least squares approximation.Unfortunately, the accuracy of these higher-order predic-tion algorithms was shown to be very much affected bythe phase and sensitivity mismatches and relative po-sition errors between the microphones in the array17.These mismatches and position errors are generally un-avoidable, especially if the number of microphones usedis large.

Instead of using forward difference prediction tech-niques to determine the weights for each of the elementsin the microphone array, Cazzolato18 suggested that theoptimal values for the weights could be determined us-ing the LMS algorithm21. A similar method was alsosuggested by Gawron and Schaaf19, who applied it tolocal active noise control inside a car cabin. The adap-tive LMS virtual microphone technique involves placinga microphone at the virtual location, after which the mi-crophone weights are adapted by the LMS algorithm soas to optimally predict the sound pressure at this loca-tion. After the weights have converged, the microphoneis removed from the virtual location and the weights arefixed to their optimal value. The adaptive LMS virtualmicrophone technique was theoretically investigated foran acoustic duct13. The results indicated that the adap-tive LMS algorithm was able to completely compensatefor relative position errors and sensitivity mismatches be-tween the microphone elements, and partly compensatefor phase mismatches between the microphone elements.The adaptive LMS virtual microphone technique provedto outperform the forward difference prediction techniquein real-time control experiments conducted in an acousticduct13,20.

The virtual sensing methods discussed so far were alldeveloped with the aim to create a zone of quiet at aspatially fixed virtual microphone. The primary aim of

this paper is to present a method for creating a zone ofquiet at a moving virtual microphone inside an acous-tic duct based on the adaptive LMS virtual microphonetechnique18. In previous work13,18, this technique hasonly been applied to the case of using one spatially fixedvirtual microphone, and it is therefore initially extendedhere to the case of using multiple spatially fixed virtualmicrophones. The analysis is based on a state-spacemodel of the active noise control system under consid-eration, because this allows an easy extension to the caseof a moving virtual microphone instead of a spatiallyfixed virtual microphone. The developed moving virtualmicrophone method is then combined with the filtered-xLMS algorithm22. To illustrate the proposed method, itis implemented in a one-dimensional acoustic duct withthe aim to create a moving zone of quiet for narrowbandnoise. Experimental results are presented and comparedto experimental results for narrowband active noise con-trol at a spatially fixed virtual microphone, and a spa-tially fixed physical microphone.

In a practical application, an important issue thatwould need to be addressed is how to determine the de-sired location of maximum attenuation, i.e. the movingvirtual location. This could for instance be done usinga 3D head tracking system based on camera vision, or ahead tracking system based on ultrasonic position sensingsuch as the Logitechr head tracker. However, the issueof determining the desired location of the zone of quietis beyond the scope of this paper, and the focus hereis on the development of a method for creating a zoneof quiet at a moving virtual location inside an acousticduct, while assuming that an exact determination of thislocation is available.

In Section II, the acoustic duct arrangement used toillustrate the proposed method is introduced. In Sec-tion III, the LMS virtual microphone technique is ex-tended to include multiple spatially fixed virtual micro-phones, given a state-space model of the considered ac-tive noise control system. In Section IV, the theory dis-cussed in Section III is extended to develop a movingvirtual sensing algorithm that computes an estimate ofthe error signal at a moving virtual microphone. In Sec-tionV, it is explained how the moving virtual sensingalgorithm is combined with the filtered-x LMS algorithmto create a zone of quiet at a moving virtual microphone.In Section VI, the proposed algorithm is implemented inreal-time on an acoustic duct arrangement, and narrow-band control results are presented.

II. PROBLEM DEFINITION

A. Acoustic duct arrangement

Fig. 1 shows a schematic diagram of the one-dimensional acoustic duct arrangement considered here.The rigid-walled acoustic duct has a length L, a primarysource located at xp, and a control source located at xs.

Moving zone of quiet inside duct 2

x

v n( )

x nv( )x1 xM-2 xM-1 xMxs xp

moving virtualsensing algorithm

e nv( )

êv( )n

x n( )u n( )

x nv( )

ADC & anti-aliasing filter

filtered-xLMS algorithm

ep( )n

DAC &reconstruc-tion filter

DAC &reconstruc-tion filter

targetzone

FIG. 1: Schematic diagram of a the one-dimensionalacoustic duct arrangement under consideration.

The primary source is excited by the disturbance sourcesignal x(n), and the control source by the control sig-nal u(n). Both signals are first passed through a DA-converter and a reconstruction filter before being sent totheir respective sources. It is assumed here that the dis-turbance source signal x(n) is a stationary process, andthat the primary sound field created inside the acousticduct is therefore stationary as well.

The aim of the active noise control system is to min-imize the virtual error signal ev(n) at a moving virtualmicrophone located inside the acoustic duct, indicated bythe black dot in Fig. 1. This moving virtual microphonetracks the desired location of the zone of quiet definedby the moving virtual location xv(n) = xM + v(n), withv(n) the moving virtual distance shown in Fig. 1. It willbe assumed that the moving virtual microphone stayswithin a confined region inside the acoustic duct calledthe target zone, which has been indicated in Fig. 1 by thegrey box. The virtual error signal at the moving virtualmicrophone is given by

ev(n) = dv(n) + yv(n), (1)

where dv(n) is the virtual primary disturbance that needsto be attenuated, and yv(n) is the virtual secondary dis-turbance created by the control source, such that

yv(n) = Gvu(n)u(n), (2)

with Gvu(n) the linear time-varying virtual secondarytransfer path. This transfer path is time-varying due tothe movement of the virtual microphone.

The virtual primary disturbance dv(n) is a non-stationary process due to the movement of the virtualmicrophone, even though it is assumed that the primarysound field inside the acoustic duct is stationary. Whenthe primary disturbance is non-stationary, the commonapproach in active noise control is to use an adaptive con-trol algorithm. Adaptive control algorithms are able to

track the changes in the statistical properties of the pri-mary disturbance and adjust the controller accordingly22.Here, the filtered-x LMS algorithm22 is used to adapt anFIR control filter that generates the control signal u(n)in Fig. 1. A feedforward control approach is adopted asit is assumed that a feedforward reference signal x(n) isavailable for control purposes. As illustrated in Fig. 1,this signal is equal to the disturbance source signal thatexcites the primary source.

B. Implementing the filtered-x LMS algorithm

If the virtual error signal ev(n) was directly measuredby a physical microphone, and the virtual location wasspatially fixed, the filtered-x LMS algorithm could be im-plemented as22

w(n + 1) = w(n)− µrv(n)ev(n), (3)

where µ is the convergence coefficient, w(n) ∈ RI is avector containing the filter coefficients

w(n) =[

w0(n) w1(n) · · · wI−1(n)]T

, (4)

and ev(n) is the virtual error signal directly measuredby the spatially fixed physical microphone. The vectorrv(n) ∈ RI in Eq. (3) is given by

rv(n) =[

rv(n) rv(n− 1) . . . rv(n− I + 1)]T

, (5)

where rv(n) = Gvux(n) is the virtual filtered-referencesignal generated by filtering the reference signal x(n)with the linear time-invariant virtual secondary transferpath Gvu. However, the virtual error signal ev(n) is notdirectly measured in the active noise control system con-sidered here, and the virtual microphone is not spatiallyfixed but moving, such that the virtual secondary trans-fer path is linear time-varying, as indicated in Eq. (2). Toimplement the filtered-x LMS algorithm, these two issuesthus need to be addressed.

1. Moving virtual sensing algorithm

As indicated by Eq. (3), the filtered-x LMS algorithmneeds the feedback information contained in the virtualerror signal ev(n) to update the control filter coefficientsw(n). This signal is not directly measured by a physicalmicrophone in the problem considered here. A movingvirtual sensing method thus needs to be developed that isable to compute an estimate ev(n) of the virtual error sig-nal at the moving virtual location xv(n). As illustratedin Fig. 1, the moving virtual sensing algorithm uses in-formation about the sound field provided by a physicalmicrophone array, and knowledge of the moving virtuallocation xv(n) to compute an estimate ev(n) of the vir-tual error signal. The physical microphone array, indi-cated by the grey dots in Fig. 1, measures the physical

Moving zone of quiet inside duct 3

error signals

ep(n) =[

e1(n) e2(n) · · · eMp(n)

]T, (6)

at the spatially fixed physical locations

x =[

x1 x2 · · · xM

]T. (7)

The physical error signals ep(n) are first passed throughan anti-aliasing filter and an AD-converter before beingused by the moving virtual sensing algorithm to com-pute an estimate ev(n) of the virtual error signal. Asillustrated in Fig. 1, instead of the unavailable virtual er-ror signal ev(n), the filtered-x LMS algorithm uses thisestimate to update the control filter coefficients.

2. Generating the virtual filtered-reference signal

The second issue that needs to be addressed to im-plement the filtered-x LMS algorithm in Eq. (3) is thatthe virtual microphone is moving. For a spatially fixedvirtual microphone, the virtual secondary transfer pathGvu is linear time-invariant, and the virtual filtered-reference signal can be generated as usual, such thatrv(n) = Gvux(n). However, for a moving virtual mi-crophone, the virtual secondary transfer path is lineartime-varying, and the virtual filtered-reference signals arethen generated as

rv(n) = Gvu(n)x(n). (8)

To generate the virtual filtered-reference signal, a modelof the time-varying virtual secondary transfer pathGvu(n) is thus needed in a practical implementation forevery sample n. This is generally not possible in a prac-tical situation, and a method for generating estimates ofthe virtual filtered-reference signals is therefore presentedin Section V.

C. State-space model of system

In this section, a state-space model23 of the acousticduct arrangement in Fig. 1 is introduced. This state-space system models the transfer paths between the inputsignals into the primary and control sources, and the out-put signals at the physical microphones located at x, themoving virtual microphone located at xv(n), and a num-ber of spatially fixed virtual microphones located withinthe target zone at

xv =[

xv1 xv2 · · · xvMv

]T. (9)

The state-space system that models these transfer pathsis given by

z(n + 1) = Az(n) + Buu(n) + Bxx(n)ep(n) = Cpz(n) + vp(n)

ev(n) = Cvz(n) + vv(n)ev(n) = Cv(n)z(n) + vv(n), (10)

ep( )n

ev( )n

Ho

ev( )n

weights

S

+_

Mp

ev( )n

Mv

Mv

Mv

ˆ

FIG. 2: Block diagram of the LMS virtual microphonetechnique using Mp physical microphones, and Mv

spatially fixed virtual microphones.

with z(n) ∈ RN the state of the plant, u(n) the controlsignal, x(n) the disturbance source signal, ep(n) ∈ RMp

the physical error signals at the spatially fixed physicalmicrophones, ev(n) ∈ RMv the virtual error signals atthe virtual microphones spatially fixed at xv within thetarget zone, with

ev(n) =[

ev1(n) ev2(n) · · · evMv(n)

]T, (11)

and ev(n) the virtual error signal at the moving virtuallocation xv(n). The signals vp(n) ∈ RMp , vv(n) ∈ RMv ,and vv(n) in Eq. (10) are the measurement noise signalson the physical microphones, spatially fixed virtual mi-crophones, and moving virtual microphone, respectively.The state-space matrices in Eq. (10) are real-valued andof appropriate dimensions.

One may wonder why measurement noise on the vir-tual microphones has been included in Eq. (10). The rea-son is that physical microphones are usually located atthe virtual locations in a preliminary identification stageof the plant, and the measurement noise on these micro-phones thus has to be included in the analysis.

III. SPATIALLY FIXED VIRTUAL MICROPHONE

The LMS virtual microphone technique18 can be usedto compute an estimate of the virtual error signals ev(n)at the spatially fixed virtual locations xv in Eq. (9), whichare located throughout the target zone shown in Fig. 1.However, this algorithm18 has only been derived for thecase of one spatially fixed virtual microphone. In thissection, this algorithm is therefore extended to includemultiple spatially fixed virtual microphones.

A. LMS virtual microphone technique

A block diagram of the LMS virtual microphone tech-nique using multiple spatially fixed virtual microphonesis shown in Fig. 2.

An estimate ˆev(n) of the virtual error signals is cal-culated by summing the weighted physical error signalsep(n), which can be expressed as

ˆev(n) = HTep(n), (12)

Moving zone of quiet inside duct 4

where the matrix H ∈ RMp×Mv contains the physicalmicrophone weights given by

H =[h1 h2 · · · hMv

]T, (13)

with hi ∈ RMp the weights used to compute the estimateevi(n) = hT

i ep(n). The physical error signals ep(n) ∈RMp in Eq. (12) are defined as

ep(n) = dp(n) + yp(n), (14)

with dp(n) the physical primary disturbances, and yp(n)the physical secondary disturbances. The aim of the filterproblem illustrated in Fig. 2 is to find a set of real weightsH that minimize the virtual output error εv(n) given by

εv(n) = ev(n)− ˆev(n). (15)

These weights will be called the optimal physical micro-phone weights, and are derived in the following.

B. Modified LMS virtual microphone technique

Using Eq. (14), the estimated virtual error signal inEq. (12) can also be written as

ˆev(n) = HT (dp(n) + yp(n)) = ˆdv(n) + ˆyv(n), (16)

where ˆdv(n) and ˆyv(n) are the estimated virtual primaryand secondary disturbances, respectively. Eq. (16) illus-trates that the microphone weights H are applied to boththe primary and secondary disturbances at the physi-cal microphones. The underlying assumption made byprevious researchers13 is therefore that the optimal mi-crophone weights for the estimation of both dv(n) andyv(n) are equal, which might not always be true. Forexample, for active noise control in the near-field of thesecondary source, the spatial characteristics of the pri-mary and secondary fields can be very different2. Thisdifference is accounted for in the virtual microphone ar-rangement2, where it is assumed that the primary fieldchanges relatively little between the physical and virtualmicrophones, such that the primary disturbances at thetwo microphones are assumed equal. The secondary dis-turbances at the physical and virtual microphones are,however, not assumed equal due to the near-field prop-erties of the secondary source. For the LMS virtual mi-crophone technique, the described situation will result indifferent optimal microphone weights for the primary andsecondary sound fields. In conclusion, it is thus impor-tant to find optimal microphone weights for the estima-tion of both the primary and secondary disturbances atthe virtual microphone. If these weights are equal, theLMS virtual microphone technique can be implementedas illustrated in Fig. 2. If these weights are not equal,the virtual sensing algorithm needs to separate the phys-ical error signals into their primary and secondary com-ponents. The optimal weights for each component can

Hx

u n( )Gpu

ep( )n

ev( )n

ev( )nS

+_

Mp

S

+

yp( )n

Mp

+ Mv

Mv

Mv

H Hu x-

ev( )nˆ

FIG. 3: Block diagram of the modified LMS virtualmicrophone technique using Mp physical microphones,

and Mv spatially fixed virtual microphones.

then be applied to obtain optimal estimates of dv(n) andyv(n), which can then be superimposed to get an optimalestimate ˆev(n) of the virtual error signal. The separationof the physical error signals ep(n) into their primary andsecondary components can be achieved as illustrated inFig. 3.

The estimated virtual error signal in this figure is givenby

ˆev(n) = HTx ep(n) + (Hu −Hx)TGpuu(n), (17)

where Hx ∈ RMp×Mv are the weights for the primaryfield, Hu ∈ RMp×Mv the weights for the secondary field,and Gpu the physical secondary transfer path matrixbetween the control source and the Mp physical micro-phones. It can be seen that the block diagram in Fig. 3reduces to the block diagram in Fig. 2 if the weights forthe primary and secondary field are equal, such thatHx = Hu.

C. Optimal microphone weights

In this section, an optimal solution for the physical mi-crophone weights is derived given the state-space modelin Eq. (10). By setting x(n) = 0 in this equation, optimalweights for the secondary field can be derived. Similarly,optimal weights for the primary field can be derived bysetting u(n) = 0. First, consider the case where the dis-turbance source signal x(n) is set to zero. An estimateof the virtual secondary disturbances yv(n) is now com-puted as

ˆyv(n) = HTuyp(n). (18)

From Eqs (10) and (18), a state-space system that modelsthe virtual output error εv(n) = yv(n) − ˆyv(n) is thengiven by

z(n + 1) = Az(n) + Buu(n)

εv(n) = Cεz(n) + vv(n)−HTuvp(n), (19)

where the matrix Cε ∈ RMv×N is defined as

Cε = Cv −HTuCp. (20)

Moving zone of quiet inside duct 5

The optimal weights Huo ∈ RMv×Mp are now defined asthe weights that minimize the cost function Jε defined as

Jε = tr E[εv(n)εv(n)T

]. (21)

It is assumed that the signal u(n) in Eq. (19) is a zero-mean white and stationary random process during iden-tification of the weights. Furthermore, the measurementnoise signals vp(n) and vv(n) are assumed to be zero-mean white and stationary random processes that areuncorrelated to u(n), such that the following covariancematrices can be defined

E

u(n)

vp(n)vv(n)z(0)

[u(k) vp(k)T vv(k)T z(0)T 1

]=

Qu 0 0 0 00 Rp Rpv 0 00 RT

pv Rv 0 00 0 0 Π0 0

δnk,

(22)with δnk the Kronecker delta function23, such that δnk =1 if n = k, z(0) ∈ RN the initial state, Rp the covari-ance matrix of the measurement noise on the physicalmicrophones, Rv the covariance matrix of the measure-ment noise on the spatially fixed virtual microphones,and Rpv the cross-covariance matrix between the mea-surement noise on the physical and spatially fixed virtualmicrophones. In Appendix A, it is shown that the costfunction in Eq. (21) can then be written as

Jε = tr(HT

uRuHu − 2HTu pu + cu

), (23)

with

Ru = E[yp(n)yp(n)T] = CpΠuCTp + Rp, (24)

pu = E[yp(n)yv(n)T] = CpΠuCTv + Rpv, (25)

cu = E[yv(n)yv(n)T] = CvΠuCTv + Rv, (26)

where Πu > 0 is the solution to the discrete-time Lya-punov equation23

Πu = AΠuAT + BuQuBTu . (27)

The optimal weights can now be found by differentiat-ing the cost function Jε in Eq. (23) with respect to theweights Hu, and setting all of the resulting derivatives tozero, which results in22

Huo = R−1u pu. (28)

Similarly, optimal weights Hxo = R−1x px for the primary

field can be computed, with

Rx = E[dp(n)dp(n)T] = CpΠxCTp + Rp, (29)

px = E[dp(n)dv(n)T] = CpΠxCTv + Rpv, (30)

yv( )nˆyp( )n

y nv( )

hu( )nev( )n

weights

S

+_

Mp

FIG. 4: Block diagram for deriving the optimaltime-varying weights for the secondary field.

where Πx > 0 is the solution to the discrete-time Lya-punov equation23

Πx = AΠxAT + BxQxBTx . (31)

The above derivations show that the weights for the pri-mary and secondary field are equal if Πu = Πx.

IV. MOVING VIRTUAL MICROPHONE

In this section, the LMS virtual microphone techniqueintroduced in Section III is extended to a moving virtualsensing algorithm. This algorithm can be used in theacoustic duct arrangement shown in Fig. 1 to estimatethe virtual error signal at the moving virtual locationxv(n). The LMS moving virtual microphone technique isderived in Section IVA assuming that the matrix Cv(n)in Eq. (10) is known at every sample n. This will notbe the case in a practical situation, and a more practicalimplementation of the developed moving virtual sensingalgorithm is therefore presented in Section IVB. This im-plementation is based on linear spatial interpolation be-tween the estimated virtual error signals ˆev(n) at the spa-tially fixed virtual microphones located within the targetzone at xv.

A. LMS moving virtual microphone technique

In this section, the analysis of the LMS virtual mi-crophone technique presented in Section III is extendedto develop the LMS moving virtual microphone tech-nique. Because the virtual microphone is now moving,a time-varying optimal solution for the physical micro-phone weights is derived given the state-space model inEq. (10). Here, time-varying optimal weights for the sec-ondary field are derived by setting x(n) = 0 in Eq. (10).A block diagram of the optimal filter problem for thiscase is shown in Fig. 4.

An estimate of the virtual secondary disturbance yv(n)at the moving virtual location xv(n) is now computed as

yv(n) = hu(n)Typ(n), (32)

with hu(n) ∈ RMp the time-varying weights for the sec-ondary field. Using Eqs (10) and (32), a state-spacesystem that models the virtual output error εv(n) =

Moving zone of quiet inside duct 6

yv(n)− yv(n) is now given by

z(n + 1) = Az(n) + Buu(n)

εv(n) = Cε(n)z(n) + vv(n)− hu(n)Tvp(n), (33)

where the matrix Cε(n) ∈ R1×N is defined as

Cε(n) = Cv(n)− hu(n)TCp. (34)

The time-varying optimal weights huo(n) ∈ RMp are de-fined as the weights that minimize the cost function Jε(n)at every sample n, with

Jε(n) = E[εv(n)2

]. (35)

Following the derivations presented in Section IIIC, it canbe shown that this cost function can be written, similarto Eq. (23), as

Jε(n) = hu(n)TRuhu(n)− 2hu(n)Tpu(n) + cu(n), (36)

with Ru as defined in Eq. (24), and

pu(n) = E[yp(n)yv(n)T] = CpΠuCv(n)T + Rpv, (37)

cu(n) = E[yv(n)yv(n)T] = CvΠuCv(n)T + Rv, (38)

where Πu > 0 is the solution to the discrete-time Lya-punov equation in Eq. (27), Rv is the covariance of themeasurement noise on the moving virtual microphone,and Rpv the cross-covariance matrix between the mea-surement noise on the physical microphones and the mov-ing virtual microphone, such that

Rpv = E[vp(n)vv(n)], Rv = E[vv(n)2]. (39)

The time-varying optimal weights can now be found bydifferentiating the cost function Jε(n) in Eq. (36) with re-spect to the weights hu(n), and setting all of the resultingderivatives to zero at every sample n, which results in22

huo(n) = R−1u pu(n). (40)

Similarly, time-varying optimal weights for the primaryfield can be computed as hxo(n) = R−1

x px(n), with Rx

as defined in Eq. (29), and

px(n) = E[dp(n)dv(n)] = CpΠxCv(n)T + Rpv, (41)

where Πx > 0 is the solution to the discrete-time Lya-punov equation in Eq. (31).

To implement the derived moving virtual sensing al-gorithm, the cross-covariance vectors pu(n) and px(n)in Eqs (37) and (41) need to be known at every samplen. This is generally not possible in practice, and a morepractical implementation of the developed moving vir-tual sensing algorithm is therefore presented in the nextsection.

e nv( )

ev( )nS

+_

Mv

inter-polation

êv( )n

Mv

xv( )n

xv

ev( )nˆLMS virtualmicrophonetechnique

Mp

ep( )n

u( )n

FIG. 5: Block diagram of the practical implementationof the LMS moving virtual microphone technique basedon spatial interpolation, using Mp physical microphones,Mv virtual microphones spatially fixed at xv, and one

moving virtual microphone located at xv(n).

B. Practical implementation using spatial interpolation

The assumption that the moving virtual location xv(n)stays within the target zone shown in Fig. 1 is now used toarrive at a more practical implementation of the movingvirtual sensing algorithm derived in the previous section.A block diagram of the proposed implementation basedon spatial interpolation is shown in Fig. 5.

The approach taken here is to first compute an esti-mate ˆev(n) of the virtual error signals at each of the Mv

spatially fixed virtual microphones located at xv definedin Eq. (9). These spatially fixed virtual microphones arepositioned throughout the target zone in Fig. 1, and theestimate of the virtual error signals at each of these loca-tions can be computed using the LMS virtual microphonetechnique described in Section III. As shown in Fig. 5, anestimate ev(n) of the virtual error signal at the movingvirtual location xv(n) is then computed using linear spa-tial interpolation between the error signals ˆev(n). Fig. 5shows that the information needed to compute the lin-ear spatial interpolation are the estimated virtual errorsignals ˆev(n), the spatially fixed virtual locations xv atwhich these virtual error signals are estimated, and themoving virtual location xv(n), which is assumed knownhere.

It can be shown that an equivalent estimate of thevirtual error signal can be computed as

ev(n) = hxo(n)Tep(n) + (huo(n)− hxo(n))TGpuu(n),(42)

where the weights huo(n) ∈ RMp and hxo(n) ∈ RMp areestimates of the optimal time-varying weights defined inthe previous section. These estimates are obtained usinglinear spatial interpolation between the weights Huo andHxo derived in Section IIIC for the spatially fixed virtuallocations xv. As an example, the mth component hm

uo(n)of huo(n) is computed as

hmuo(n) =

xv(n)− xv(i+1)

xvi − xv(i+1)H(m,i)

uo +xvi − xv(n)xvi − xv(i+1)

H(m,i+1)uo ,

(43)with the moving virtual location xv(n) such that xvi ≤xv(n) ≤ xv(i+1), and with xvi the spatially fixed virtual

Moving zone of quiet inside duct 7

locations defined in Eq. (9). If the weights for the primaryand secondary field are equal, Eq. (44) reduces to

ev(n) = ho(n)Tep(n), (44)

with ho(n) = huo(n) = hxo(n). This is the implementa-tion used in the experiments presented in Section VI.

V. FILTERED-X LMS ALGORITHM

A. Implementation

As discussed previously, to implement the filtered-xLMS algorithm in Eq. (3), the virtual filtered-referencesignals rv(n) = Gvu(n)x(n) need to be generated, withGvu(n) the time-varying virtual secondary transfer path.In a practical situation, this transfer path will not beavailable for every sample n, and a method for generat-ing an estimate rv(n) of the virtual filter-reference signalneeds to be developed. Here, an estimate of the virtualfiltered-reference signal is generated as

rv(n) = huo(n)Trp(n), (45)

with huo(n) the time-varying optimal weights definedin the previous section, and rp(n) ∈ RMp the vectorof filtered-reference signals for the physical microphones,which is generated as

rp(n) = Gpux(n), (46)

with Gpu the physical secondary transfer path betweenthe control source and the physical microphones. Substi-tuting Eq. (46) into Eq. (45) gives

rv(n) = Gvu(n)x(n), (47)

with Gvu(n) = huo(n)TGpu. Effectively, this methodthus provides an estimate of the time-varying virtual sec-ondary transfer path Gvu(n). The filtered-x LMS algo-rithm can now be implemented as

w(n + 1) = w(n)− µrv(n)ev(n), (48)

with ev(n) the estimated virtual error signal at the mov-ing virtual location xv(n), and rv(n) ∈ RI given by

rv(n) =[

rv(n) rv(n− 1) . . . rv(n− I + 1)]T

. (49)

This is the implementation used in the experiments pre-sented in Section VI.

B. Tracking

The aim of the filtered-x LMS algorithm in Eq. (48)is to track the changes in the statistical properties ofthe estimated virtual primary disturbance at the mov-ing virtual location xv(n), and adjust the control filter

coefficients w(n) accordingly. The amount and speed oftracking needed is dependent on both the temporal rateof change of the moving virtual location, and the spa-tial rate of change of the relative magnitude and phasebetween the primary and secondary sound field over thetarget zone. This spatial rate of change determines howmuch the filter coefficients need to be adjusted over thespatial region through which the virtual microphone ismoving. This information, together with the temporalrate of change of the moving virtual location, determinesthe amount and speed of the tracking that is needed tosuccessfully create a moving zone of quiet. However. asstated by Haykin24, the tracking details of a time-varyingsystem are very problem specific. As a result, the track-ing behaviour of the suggested algorithm will be problemspecific as well, and general statements on how fast thevirtual microphone can be moved cannot be made.

As discussed by Haykin24, the convergence rate andthe tracking capability of an adaptive algorithm are gen-erally two different properties. Whereas convergence isa transient phenomenon, tracking is a steady-state phe-nomenon. For an adaptive algorithm to exercise its track-ing capability, it must therefore first pass from the tran-sient mode to the steady-state mode24. In the exper-iments presented in the next section, the performanceobtained with the proposed method was measured afterthe filtered-x LMS algorithm in Eq. (48) had passed fromthe transient mode to the steady-state mode. Thus, inthe experiments presented in the next section, it is thetracking capability of the filtered-x LMS algorithm thatwas investigated.

VI. ACOUSTIC DUCT EXPERIMENTS

The algorithms introduced in the previous sectionswere implemented on the acoustic duct arrangementshown in Fig. 1. The experimental arrangement is nowdescribed in more detail, after which experimental resultsare presented and discussed.

A. Experimental arrangement

The rectangular acoustic duct in Fig. 1 that was usedin the real-time experiments was of length L = 4.830 m,width 0.205 m, and height 0.205 m. A loudspeaker lo-cated at xp = 4.730 m was used as a primary source, andanother loudspeaker located at xs = 0.500 m as a controlsource. The primary loudspeaker was excited by a tonaldisturbance source signal x(n). As shown in Fig. 1, thissignal was also used as a feedforward reference signal inthe filtered-x LMS algorithm. A physical microphone ar-ray consisting of two electret microphones was located atx = [1.425 1.475]m. A traversing microphone mountedon a cable wrapped around pulleys at each end of theduct was located inside the duct. This cable was woundonto a 0.150 m diameter spool, which was mounted on the

Moving zone of quiet inside duct 8

shaft of a DC servo-motor encoder unit. This encoderprovided dual track TTL output signals of 500 pulsesper revolution, which enabled accurate position controlof the traversing microphone. In a preliminary identifi-cation procedure, the traversing microphone was placedat a number of spatially fixed virtual locations xv in or-der to determine optimal microphone weights for theselocations. Furthermore, the traversing microphone wasposition controlled to measure the primary and controlledsound pressure at the moving virtual location xv(n). Toimplement the developed algorithms in real-time, a host-target software program called xPC Targetr was used.A sampling frequency of fs = 4kHz was employed in thereal-time experiments.

B. Experimental results

The filtered-x LMS algorithm discussed in Section Vwas implemented in the acoustic duct arrangement tocreate a moving zone of quiet inside the acoustic duct.Only I = 2 filter coefficients were used in the experimentspresented here, since the duct was excited by a tonal ex-citation signal22. For broadband noise, more filter coeffi-cients need to be used. The aim of the experiments wasto estimate and minimize the virtual error signal ev(n) ata moving virtual location xv(n) = x2+v(n) that changedsinusoidally with time, with v(n) the moving virtual dis-tance. The expression governing the desired position ofthe virtual microphone is given by

v(n) = 0.070 + 0.050 sin(

2πn

Tvfs

), (50)

where Tv is the period of the sinusoidally time-varyingmoving virtual distance v(n) in Fig. 1. The virtual mi-crophone is thus moving sinusoidally between a virtualdistance bounded by 0.020 m and 0.120 m. The per-formance at the moving virtual distance was measuredfor two excitation frequencies f of 213 Hz and 249Hz.These frequencies correspond to the sixth and seventhresonance frequencies of the acoustic duct. For these ex-citation frequencies, the performance at the moving vir-tual distance v(n) was measured for three different val-ues of Tv in Eq. (50) given by 10 s, 5 s, and 2.5 s. Themaximum amplitude of the sinusoidally time-varying ve-locity of the moving virtual microphone is thus given by5cm × 2π/2.5 s ≈ 12.6cm/s. This was considered to berepresentative of the likely motion of a head in the in-tended applications. In total, six experiments were thusconducted in order to measure the performance of theimplemented algorithms for various speeds of the movingvirtual microphone, and various spatial characteristics ofthe sound field through which the virtual microphone ismoving.

0 0.05 0.1 0.15−3

−2

−1

0

1

2

3

4

virtual distance (m)

optim

al m

icro

phon

e w

eigh

ts

FIG. 6: Optimal physical microphone weights ho

plotted against fixed virtual distance v.

1. Preliminary identification procedure

In a preliminary identification procedure, the optimalmicrophone weights Huo for the secondary field givenin Eq. (28) were determined for Mv = 16 spatially fixedvirtual locations xv = x2 + v, with v given by

v =[

0.000 0.010 0.020 . . . 0.150]

m. (51)

These spatially fixed virtual distances v were evenly po-sitioned throughout the target zone located within a vir-tual distance range of 0.000–0.150 m. The acoustic ductwas excited with band-pass filtered white noise in thefrequency range of 50–500 Hz while determining the op-timal microphone weights. For each of the spatially fixedvirtual distances in Eq. (51), the optimal weights werecalculated using Eq. (28) based on 30 s of data obtainedfrom the two physical microphones and the traversingmicrophone positioned at the spatially fixed virtual dis-tance of interest. The results of this procedure are shownin Fig. 6, where the optimal weights for the secondarysound field have been plotted against v.

As discussed in Section III, it is necessary to verifyif the identified optimal weights for the primary andsecondary sound fields are equal. Optimal microphoneweights Hxo for the primary sound field were determinedin a similar way, and it was observed that the result-ing weights for both cases were almost identical. TheLMS virtual microphone technique as shown in Fig. 2 wastherefore implemented in the acoustic duct.

Using the optimal microphone weights shown in Fig. 6,the LMS virtual microphone technique was implementedin the acoustic duct for a number of spatially fixed virtualdistances. The tonal attenuations achieved at these vir-tual distances were measured for excitation frequenciesof 213 Hz and 249 Hz. The results are shown in Fig. 7,which confirms previous experimental results presentedby other researchers13,20, and indicates that good perfor-mance can be achieved at these fixed virtual distances

Moving zone of quiet inside duct 9

0 0.05 0.130

35

40

45

50

55

60

virtual distance (m)

atte

nuat

ion

(dB

)

FIG. 7: Average tonal attenuation achieved at anumber of fixed virtual distances. f = 213 Hz

f = 249Hz.

when using the LMS virtual microphone technique.

2. Relative spatial change of primary and secondary soundfields over target zone

In Section V, it was discussed that the amount andspeed of the tracking needed from the filtered-x LMSalgorithm to successfully create a moving zone of quietis dependent on the spatial rate of change of the relativemagnitude and phase between the primary and secondarysound field over the target zone, and the temporal rate ofchange of the moving virtual location xv(n). The mea-sured spatial rate of change of the relative magnitude andphase between the primary and secondary sound fieldinside the acoustic duct over the target zone has beenplotted in Fig. 8, for the excitation frequencies of 213 Hzand 249Hz. In this figure, the relative magnitude andphase between the primary and secondary sound fieldshave been normalized against the relative magnitude andphase at a virtual distance of v = 0.000 m.

For an excitation frequency of 213 Hz, the relative mag-nitude and phase change by about 0.58 dB and 4.2◦ overthe target zone, respectively. For an excitation frequencyof 249Hz, the relative magnitude and phase change byabout 0.77 dB and 13.0◦ over the target zone, respec-tively. Thus, when the virtual location is moving throughthe target zone located between a virtual distance vbounded by 0.000 m and 0.150 m, the filtered-x LMS al-gorithm has to account for these changes in the relativemagnitude and phase between the primary and secondarysound field by adjusting the filter coefficients, such thattracking of the non-stationarities in the estimated virtualprimary disturbance is achieved. In the experimentalresults presented next, the performance at the movingvirtual location was measured after convergence of thefiltered-x LMS algorithm, such that the tracking capa-

0 0.05 0.1 0.150

0.5

1

rela

tive

mag

nitu

de (

dB)

0 0.05 0.1 0.150

5

10

15

virtual distance (m)

rela

tive

phas

e (°

)

FIG. 8: Spatial change of relative magnitude and phasebetween primary and secondary sound fields plotted

over target zone inside duct. f = 213 Hzf = 249 Hz.

bility of the adaptive algorithm was investigated.

3. Performance at moving virtual location

To illustrate the increase in local control performancethat can be obtained at a moving virtual location whenusing the suggested method, the performance at the mov-ing virtual location xv(n) = x2 + v(n) was also mea-sured for the case of active noise control at a spatiallyfixed physical microphone located at v = 0.000 m, andactive noise control at a spatially fixed virtual micro-phone located at v = 0.020 m. For the spatially fixedphysical microphone, the physical error signal was di-rectly measured by the physical microphone located atx2 = 1.475 m, and was minimized using the standardformulation of the filtered-x LMS algorithm22. For thespatially fixed virtual microphone, the virtual error sig-nal at v = 0.020 m was estimated using the LMS virtualmicrophone technique described in Section III. The esti-mate was then minimized using the filtered-x LMS algo-rithm. Unlike the currently proposed method, both ofthese active noise control systems cannot account for thefact that the desired location of the zone of quiet is notspatially fixed.

The performance of the three active noise control sys-tems that each employ a different sensing method wasmeasured at the moving virtual location xv(n) = x2 +v(n), where v(n) is defined by Eq. (50) with Tv givenby 10 s, 5 s, and 2.5 s, respectively. The results of thesemeasurements are illustrated in Fig. 9, where the aver-age tonal attenuation measured at the moving virtuallocation has been plotted against time for excitation fre-quencies of 213 Hz and 249 Hz. In Fig. 9, the dashed lineis the average tonal attenuation measured using active

Moving zone of quiet inside duct 10

noise control at the spatially fixed physical microphonelocated at v = 0.000 m, the dash-dotted line the aver-age tonal attenuation measured using active noise con-trol at the spatially fixed virtual microphone located atv = 0.020 m, and the solid grey line the average tonalattenuation measured using active noise control at themoving virtual microphone that tracks the moving vir-tual distance v(n). Each of these lines was generated byaveraging the results of 30 data-sets of 10 s measured withthe traversing microphone, which was position controlledto track the moving virtual distance v(n). The resultingposition of the traversing microphone as measured by theencoder is plotted against time in the bottom parts of thesubfigures in Fig. 9. Furthermore, the average tonal at-tenuations in decibels were low-pass filtered in order toprevent noisy plots in Fig. 9.

Fig. 9 indicates that using a spatially fixed virtual mi-crophone gives better performance at the moving virtuallocation than using a spatially fixed physical microphone.However, the best results are obtained when using amoving virtual microphone that tracks the desired lo-cation of the zone of quiet defined by the moving virtualdistance v(n). When the moving virtual distance is atv = 0.020 m in Fig. 9, the system that uses a moving vir-tual microphone gives similar results to the system thatuses a spatially fixed virtual microphone, indicating thatthe filtered-x LMS algorithm is able to provide sufficienttracking of the non-stationarities in the estimated virtualprimary disturbance. When the moving virtual distanceis not at v = 0.020 m, however, Fig. 9 shows that theactive noise control system that uses a moving virtualmicrophone provides the best performance. The resultsin this figure indicate that a moving zone of quiet haseffectively been created at the moving virtual location inthe acoustic duct, for all of the six narrowband experi-ments that were conducted. When using a moving virtualmicrophone, the average tonal attenuation at the movingvirtual location does not fall below 42 dB for all values ofTv and an excitation frequency of 213 Hz. For the spa-tially fixed physical and virtual microphones, the averagetonal attenuation at this frequency reduces to 24 dB and25 dB, respectively, at a virtual distance of v = 0.120 m.For an excitation frequency of 249 Hz, the average tonalattenuation at the moving virtual location does not fallbelow 36 dB for all values of Tv when using a moving vir-tual microphone. For the spatially fixed physical and vir-tual microphones, the average tonal attenuation at thisfrequency reduces to 17 dB and 18 dB, respectively, at avirtual distance of v = 0.120 m. These results indicatethat for the narrowband acoustic duct experiments pre-sented here, the filtered-x LMS algorithm is able to pro-vide sufficient tracking of the non-stationarities in the es-timated virtual primary disturbance. As a result, a mov-ing zone of quiet is effectively created inside the acousticduct for narrowband noise.

VII. CONCLUSIONS

The LMS virtual microphone technique is a virtualsensing method that can be used to estimate the errorsignal at a virtual microphone that is spatially fixed. Inthis paper, this technique has been extended to a mov-ing virtual sensing algorithm that is able to estimatethe error signal at a moving virtual microphone whichis tracking the desired location of the zone of quiet. Theproposed algorithm can be combined with the filtered-x LMS algorithm to create a moving zone of quiet. Thefiltered-x LMS algorithm can provide the tracking neededto account for the non-stationarities in the primary dis-turbance, which are caused by the movement of the vir-tual microphone. A practical application of the suggestedmethod is the creation of a moving zone of quiet thattracks a person’s head. Here, the developed algorithmwas implemented in a one-dimensional acoustic duct, andresults of narrowband control experiments were presentedto validate the proposed method. These results showedthat the proposed algorithm was able to create a mov-ing zone of quiet inside an acoustic duct for narrowbandnoise. This resulted in an increase in local control perfor-mance compared to using a spatially fixed virtual micro-phone or a spatially fixed physical microphone. The ex-perimental results indicate that the developed algorithmhas the potential to improve the scope of successful lo-cal active noise control applications. The proposed algo-rithm can also be used for broadband active noise control,and in three-dimensional sound fields. Ongoing researchis currently being conducted to investigate if the filtered-x LMS algorithm can provide the necessary tracking forbroadband noise, and to analyse the performance of thesuggested method in a three-dimensional sound field.

Acknowledgments

The author gratefully acknowledges the University ofAdelaide for providing an ASI scholarship, and the Aus-tralian Research Council for supporting this research.Thanks also to the editor, Dr Kenneth A. Cunefare, forhis constructive comments on the original manuscript.

1 S. J. Elliott, P. Joseph, A. J. Bullmore and P. A. Nel-son. “Active cancellation at a point in a pure tone diffusesound field.” Journal of Sound and Vibration 120, 183–189(1988).

2 S. J. Elliott and A. David. “A virtual microphone arrange-ment for local active sound control.” Proceedings of the1st International Confererence on Motion and VibrationControl, Yokohama (1992), pp. 1027–1031.

3 J. Garcia-Bonito, S. J. Elliott and C. C. Boucher. “Gen-eration of zones of quiet using a virtual microphone ar-rangement.” Journal of the Acoustical Society of America101(6), 3498–3516 (1997).

4 A. Roure and A. Albarrazin. “The remote microphonetechnique for active noise control.” Proceedings of Active’99 (1999), pp. 1233–1244.

Moving zone of quiet inside duct 11

0 2 4 6 8 10

20

40

60

atte

nuat

ion

(dB

)

0 2 4 6 8 100

0.05

0.1

0.15

time (s)

virt

ual d

ista

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(m)

(a) f = 213Hz and Tv = 10 s

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0.1

0.15

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virt

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(m)

(c) f = 213 Hz and Tv = 5 s

0 2 4 6 8 10

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(dB

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0 2 4 6 8 100

0.05

0.1

0.15

time (s)

virt

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ista

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(m)

(d) f = 249Hz and Tv = 5 s

0 2 4 6 8 10

20

40

60

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(dB

)

0 2 4 6 8 100

0.05

0.1

0.15

time (s)

virt

ual d

ista

nce

(m)

(e) f = 213Hz and Tv = 2.5 s

0 2 4 6 8 10

20

40

60

atte

nuat

ion

(dB

)

0 2 4 6 8 100

0.05

0.1

0.15

time (s)

virt

ual d

ista

nce

(m)

(f) f = 249Hz and Tv = 2.5 s

FIG. 9: (Bottom) Moving virtual distance v(n) plotted against time. (Top) Average tonal attenuation at movingvirtual distance plotted against time for active noise control at: physical microphone spatially fixed at

v = 0.000 m; virtual microphone spatially fixed at v = 0.020 m; and moving virtual microphone at v(n).

Moving zone of quiet inside duct 12

5 B. S. Cazzolato. Sensing systems for active control of soundtransmission into cavities. Ph.D. thesis, Department ofMechanical Engineering, The University of Adelaide, SA5005 (1999).

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7 B. Rafaely, S. J. Elliott and J. Garcia-Bonito. “Broad-band performance of an active headrest.” Journal of theAcoustical Society of America 106(2), 787–793 (1999).

8 M. Pawelczyk. “Noise control in the active headrest basedon estimated residual signals at virtual microphones.” Pro-ceedings of the Tenth International Congress on Sound andVibration (2003), pp. 251–258.

9 J. L. Nielsen, A. Sæbø, G. Ottesen, T. A. Reinen andS. Sørsdal. “A local active noise control system for lo-comotive drivers.” Proceedings of Inter-Noise ’00, Nice,France (2000), pp. 1–6.

10 J. Garcia-Bonito, S. J. Elliott and C. C. Boucher. “A vir-tual microphone arrangement in a practical active head-rest.” Proceedings of Inter-Noise ’96, Liverpool (1996),pp. 1115–1120.

11 S. R. Popovich. “Active acoustic control in remote re-gions.” US Patent No. 5,701,350 (1997).

12 C. D. Kestell. Active control of sound in a small single en-gine aircraft cabin with virtual error sensors. Ph.D. thesis,Department of Mechanical Engineering, The University ofAdelaide, SA 5005 (2000).

13 J. M. Munn. Virtual sensors for active noise control. Ph.D.thesis, Department of Mechanical Engineering, The Uni-versity of Adelaide, SA 5005 (2003).

14 C. D. Kestell, C. H. Hansen and B. S. Cazzolato. “Ac-tive noise control in a free field with virtual error sensors.”Journal of the Acoustical Society of America 109(1), 232–243 (2000).

15 C. D. Kestell, C. H. Hansen and B. S. Cazzolato. “Activenoise control with virtual sensors in a long narrow duct.”International Journal of Acoustics and Vibration 5(2), 1–14 (2000).

16 J. M. Munn, C. D. Kestell, B. S. Cazzolato and C. H.Hansen. “Real-time feedforward control using virtual errorsensors in a long narrow duct.” Proceedings of the AnnualAustralian Acoustical Society Conference (2001).

17 J. M. Munn, B. S. Cazzolato, C. H. Hansen and C. D.Kestell. “Higher-order virtual sensing for remote ac-tive noise control.” Proceedings of Active ’02, ISVR,Southampton, UK (2002), pp. 377–386.

18 B. S. Cazzolato. “An adaptive LMS virtual micro-phone.” Proceedings of Active ’02, ISVR, Southampton,UK (2002), pp. 105–116.

19 H. J. Gawron and K. Schaaf. “Interior car noise: Activecancellation of harmonics using virtual microphones.” Pro-ceedings of the 2nd International Conference on VehicleComfort: Ergonomic, Vibrational and Thermal Aspects,Bologna, Italy (1992), pp. 739–748.

20 J. M. Munn, B. S. Cazzolato and C. H. Hansen. “Virtualsensing: Open loop vs adaptive LMS.” Proceedings of theAnnual Australian Acoustical Society Conference (2002),pp. 24–33.

21 B. Widrow and S. D. Stearns. Adaptive Signal Processing.Prentice-Hall signal processing series. Prentice-Hall, Inc.

(1985).22 S. J. Elliott. Signal Processing for Active Control. Aca-

demic Press, 1st edition (2001).23 T. Kailath, A. H. Sayed and B. Hassibi. Linear estimation.

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Saddle River, New Jersey, 4th edition (2002).

APPENDIX A: DERIVATION OF COST FUNCTION

Eqs (23) and (27) can be derived as follows. First, thestate covariance matrix Πu(n) is defined as

Πu(n) = E[z(n)z(n)T]. (A1)

From Eq. (10), the state covariance matrix satisfies therecursion

Πu(n + 1) = AΠu(n)AT + AE[z(n)u(n)T]BTu +

BuE[u(n)z(n)T]ATu + BuQuBT

u .(A2)

It can be shown, using Eqs (10) and (22), that the currentstate z(n) is uncorrelated to the current and past inputs{u(k), k = 1, . . . , n}23, such that

E[z(n)u(n)T] = 0. (A3)

This can be seen by deriving the following expressionfrom Eq. (10)

z(n) = Anz(0) +n∑

m=1

An−mBuu(m− 1). (A4)

The state z(n) is thus a linear combination of the initialstate z(0) and the past inputs {u(k), k = 1, . . . , n − 1}.From Eq. (22), the input u(n) is uncorrelated to all ofthese variables, thereby arriving at Eq. (A3). Eq. (A2)now reduces to

Πu(n + 1) = AΠu(n)AT + BuQuBTu . (A5)

When the state z(n) reaches its mean steady state value,the state covariance matrix Πu(n + 1) = Πu(n) = Πu inEq. (A5), and solving the discrete-time Lyapunov equa-tion in Eq. (27) thus gives the steady state solution Πu.

The expression for the cost function Jε given inEq. (23) is derived next. Using a similar reasoning thatwas used to derive Eq. (A3), it can be shown that thecurrent state z(n) is uncorrelated to the current and pastmeasurement noise signals {vp(k), vv(k), k = 1, . . . , n}23,such that

E[z(n)vp(n)T] = 0, E[z(n)vv(n)T] = 0. (A6)

Using Eqs (22), (A3) and (A6), the cost function inEq. (21) can now be written as defined in Eq. (23).

Moving zone of quiet inside duct 13