Implementation of a broadband duct ANC system using adaptive spatially feedforward structure

doi:10.1006/jsvi.2001.4019, available online at http://www.idealibrary.com on

IMPLEMENTATION OF A BROADBAND DUCT ANC

SYSTEM USING ADAPTIVE SPATIALLY FEEDFORWARD

STRUCTURE

M. R. BAI ANDP. ZEUNG

Department of Mechanical Engineering, National Chiao-¹ung ;niversity, 1001 ¹a-Hsueh Road, Hsin-Chu 30010, ¹aiwan, Republic of China. E-mail: [email protected]

(Received 26 April 2000, and in ,nal form 10 August 2001)

An adaptive spatiallyfeedforward algorithm is proposed for broadband attenuation of noise in ducts. Acoustic feedback generallyexists in this active noise control structure. Munjal and Eriksson (1988 Journal of Acoustical Society of America 84, 1086}1093) derived an ideal controller for the spatiallyfeedforward structure. The ideal controller can be partitioned into two parts. The "rst part represents a repetitive controller that can be implemented byan in"nite impulse response (IIR) "lter, whereas the second part represents the dynamics of transducer that can be implemented by a "nite impulse response (FIR) "lter. In the paper, the IIR "lter is merged with the original plant. The FIR "lter is adaptively updated bythe least-mean-square (LMS) algorithm to accommodate perturbations and uncertainties in the system. The proposed algorithm is implemented via a #oating point digital signal processor and compared with other commonlyused algorithms such as the Filtered-X LMS algorithm, the feedback neutralization algorithm, and the Filtered-U LMS algorithm. Experimental results show that the system has attained 15)7 dB maximal attenuation in the frequencyband 200}600 Hz.  2002 Elsevier Science Ltd.

1. INTRODUCTION

Active noise control (ANC) [1}3] techniques provide numerous advantages over conventional passive methods such as improved low-frequencyperformance, reduction of physical size and weight, zero back pressure programmable #exibility, and so forth. In ANC applications to date, feedforward control has been widelyused whenever a non-acoustical reference is available [4]. In practice, however, a non-acoustical reference required by feedforward control is usuallyunavailable. The spatiallyfeedforward control structure (Figure 1(a)) appears to be a more feasible approach in dealing with such situations, especiallywhen broadband attenuation is sought [5}7]. In the spatiallyfeedforward control structure, an upstream sensor is placed near the primarysource. The upstream sensor captures not onlythe primarynoise but also the signals from the downstream control source. The control structure is thus not actuallyfeedforward but only&&spatially'' feedforward.

An in-depth analysis of the spatially feedforward structure was conducted by Munjal and Eriksson [8]. Their investigation reveals that there exists an ideal controller for such an ANC structure. The ideal controller can be partitioned into two parts. The "rst part represents a repetitive controller that can be implemented byan in"nite impulse response (IIR) "lter, whereas the second part represents the dynamics of transducer that can be implemented bya "nite impulse response (FIR) "lter. In our method, a "xed IIR "lter is

Figure 1. The spatiallyfeedforward structure. (a) The ANC system of a duct; and (b) the equivalent circuit.

merged with the original plant. In addition, an FIR "lter (generallyof low order) is adaptivelyupdated bythe least-mean-square (LMS) algorithm to accommodate perturbations as well as uncertainties in the system. The proposed algorithm is implemented on the platform of a #oating point digital signal processor (DSP) and compared with other commonlyused algorithms such as the Filtered-X LMS (FXLMS) algorithm, the feedback neutralization algorithm, and the Filtered-U LMS (FULMS) algorithm. Experiments are carried out to validate the proposed ANC approach for attenuation of the random noise in a long duct. Experimental results show that the system attained 15)7 dB maximal attenuation within the frequencyband 200}600 Hz.

2. SPATIALLY FEEDFORWARD CONTROLLER FOR DUCT

The spatiallyfeedforward structure of a duct and its equivalent circuit are shown in Figure 1. Munjal and Eriksson [8] derived an ideal controller for achieving global noise cancellation downstream the control source in a "nite-length duct:

CGBC?J"!ZQ?_>



"

CCP,

where ZQ? is the acoustic impedance of the control source, >"c/S is the characteristic impedance of the duct, c is the sound speed, S is the cross-sectional area of the duct, k is the wave number, and lG is the distance between the upstream measurement microphone and the control source. In equation (1), C,!ZQ?/> is a function of the "nite impedance ZQ?

Figure 2. Two equivalent spatiallyfeedforward controllers. (a) Block diagram of zero spillover controller and (b) block diagram of the Roure's controller.

which depends onlyon the electro-mechanical constants of the control source. On the other hand, CP,e\IJG/(1!e\IJG) takes the form of the so-called repetitive controller [9].

The zero spillover controller [7] is shown in Figure 2(a), where w, y, z and u are exogenous noise, measurement, performance variable and control input, respectively; the transfer functions G1s are self-explanatoryfrom the subscripts:

C81.(j
)"_{GXU(j
)GWS(j
)!GXS(j
)GWU(j
)}GXU(j
) . (2) This controller requires the knowledge of the disturbance-related transfer functions GXU and

GWU that are generallyunavailable in practice. To resolve the problem, a more practical but

equivalent controller proposed byRoure [10] can be used. It is obtained bydividing the numerator and the denominator of equation (2) by GWU(j
):

G81.(j
)"_{GXU(j
)!GWS(j
) GXU(j
)/GWU(j
)}!GXU(j
)/GWU(j
)

"

!

H(j
)

H(j
)!H(j
)H(j
),C0MSPC(j
), (3)

where H(j
),GXU(j
)/GWU(j
) is the frequencyresponse function between the pressures measured at the performance microphone and the measurement microphone (Figure 3),

H(j
),GWS(j
) is the frequencyresponse function between the measurement microphone

and the control speaker, and H(j
),GXS(j
) is the frequencyresponse function between the performance microphone and the control speaker. It is noted that all functions of

Figure 3. The experimental set-up of a spatiallyfeedforward ANC system.

equation (3) are measurable. The block diagram of the Roure's controller is shown in Figure 2(b).

Next, we will prove the equivalence between the zero spillover controller and the ideal controller. Using Munjal and Eriksson's [8] notations, the transfer functions, GWU, GXU, GWS, and GXS can be identi"ed as GWU"pN/pQNG, GXU"pN/pQNG, GWS"p?/pQ?, and GXS"p?/pQ?, where pQNG is the acoustic pressure of the primarysource, pQ? is the acoustic pressure of the auxiliarysource, p? is the acoustic pressure of location 3 in Figure 1(b) with onlythe auxiliarysource on, and pN is the acoustic pressure of location 3 in Figure 1(b) with onlythe primarysource on, etc. One can then manipulate the zero spillover controller of equation (2) as follows: C81.(j
)"_{GXU(j
)GWS(j
)!GXS(j
)GWU(j
)}GXU(j
) " [pN/pQNG] [pN/pQNG][p?/pQ?]![p?/pQ?][pN/pQNG] " [C/ZQNG<R] [C/ZQNG<R][C/ZQ?<R]![C/ZQ?(CG#j>SG/ZQNG)/<R][CCG#j>SG#j>SGC/ZQ?/ZQNG<R] " ZQ?<R C!(CG#j>SG/ZQNG)(CCG#j>SG#j>SGC/ZQ?) & ZQ?<R ! j>SG<R "jZQ? >SG"!ZQ?>



Therefore in this paper, we chose the equivalent but simpler approach proposed byRoure [11] to implement the ideal controller for the following three reasons. First, as indicated by equation (4), the ideal controller is essentiallyof in"nite bandwidth and mayproduce excessive control output at high frequency. The excessive output beyond the control

bandwidth is likelyto saturate, and even destabilize the system. Second, errors exist in the identi"ed electro-mechanical}acoustical parameters of the system. Third, there is always some dynamics of the physical system that cannot be modelled by the simple lumped-parameter model.

3. ADAPTIVE SPATIALLY FEEDFORWARD ANC ALGORITHM 3.1. REVIEW OF SOME LMS-BASED ALGORITHMS

In what follows, a brief review of some widelyused LMS-based algorithms including the FXLMS algorithm, the feedback neutralization algorithm, and the FULMS algorithm will be given [12]. The block diagram of ANC system using the FXLMS algorithm is shown in Figure 4(a). The error signal is expressed as

e(n)"d(n)!y(n)"d(n)!s(n)*y(n)"d(n)!s(n)*[w2(n)xN(n)], (5) where s(n) is the impulse response of secondarypath S(z) at the time n, * denotes the linear convolution, w(n)"[w(n)w(n)2w*\(n)]2 is the coe$cient vector of the FIR "lter =(z),

x_{N (n)"[x(n)x(n!1)}2x (n!¸#1)]2 is the reference signal vector, and ¸ is the order of "lter = (z). The FXLMS method minimizes the instantaneous squared errorK(n)"e(n) and updates the coe$cient vector in the negative gradient direction with step size:

w(n#1)"w(n)!

2K(n)"w(n)#xN(n)e(n). (6) It should be noted that the e!ect of acoustic feedback is not considered in the FXLMS algorithm using feedforward structure.

The second approach that deals with the undesirable acoustic feedback problem is to use a feedback neutralization "lter within the controller. An ANC system using the FXLMS algorithm with feedback neutralization is illustrated in Figure 4(b). The feedback signal to the reference microphone is cancelled &&electronically'' byusing a feedback neutralization "lter FK (z) which models the transfer function from the loudspeaker input to the reference microphone output. Thus, the input signal x(n) is computed as

x(n)"u(n)!+\ K

fKKy(n!m!1), (7)

where u(n) is the signal from the reference microphone, fKK are the coe$cients of the Mth order feedback neutralization "lter FK (z), and y(n) is the cancelling signal. With the removal of acoustic feedback, the optimal controller can be found byordinaryFXLMS algorithm. Finally, ANC system using the FULMS algorithm is illustrated in Figure 4(c). Similar to the FXLMS "lter, the steepest-descent algorithm is used to search for the optimal controller which in FULMS is an IIR "lter. However, the search process could possiblybe trapped in local minima because the cost function in this case is not a quadratic function. The update formula of the "lter coe$cients in the FULMS algorithm is

w(n#1)"w(n)#[sL(n)*u(n)]e(n)"w(n)#u(n)e(n), (8) where  is the step size, sL is the estimated secondarypath, e(n) is the error signal,

u(n)"sL(n)*u(n),

u(n)"





Figure 4. Some commonlyused LMS-based ANC algorithms. (a) Block diagram of the FXLMS algorithm; (b) block diagram of the feedback neutralization algorithm; and (c) block diagram of the FULMS algorithm.

is a generalized reference vector, and

w(n)"





is an overall weight vector. Equation (8) can be further partitioned into two vector equations for adaptive "lters A(z) and B(z) representing the numerator term and the denominator term, respectively, of the IIR "lter in the FULMS algorithm.

a(n#1)"a(n)#x(n)e(n) and b(n#1)"b(n)#yL(n!1)e(n), (9) where x(n)"sL(n)*x(n) and yL(n!1)"sL(n)*y(n!1).

Figure 5. Block diagram of the adaptive spatiallyfeedforward algorithm.

3.2. ADAPTIVE SPATIALLY FEEDFORWARD ALGORITHM

In this section, an adaptive spatiallyfeedforward algorithm will be developed for the spatiallyfeedforward structure. The aforementioned Munjal's ideal controller is repeated here for convenience,

CGBC?J"!ZQ?_>



In viewing equation (10), we make a proposition that the ANC controller should take the form

C"CCP, (11) where C approximates the transducer dynamics (generally of low order FIR form) and

CP approximates the repetitive controller (generallyof IIR form), i.e., CP+e\IJG/ (1!e\IJG). The repetitive patterns in the frequencyresponse and the impulse response of the controller shall be seen in the experimental results. The repetitive peaks with nearly equal interval in the time domain and in the frequencydomain are actuallycreated by acoustic feedback in the spatiallyfeedfordward structure.

From the discussion in section 2, we are able to decompose Roure's controller into

C and CP because of its equivalence to Munjal's ideal controller. The idea of the adaptive

spatiallyfeedforward algorithm is shown in Figure 5. In the case where the temperature of the duct is not varied much, the repetitive peaks of the ideal controller are almost "xed. Thus, the transfer function, CP, is implemented bya "xed IIR "lter "tting the repetitive peaks in the frequencyresponse of Roure's controller. On the other hand, in order to accommodate system uncertainties and perturbations which frequently occur in practice, the transfer function, C, is realized byan FIR "lter adaptivelyupdated bya modi"ed FXLMS algorithm.

u"C*w, w"CP*w, (12, 13)

Note, however, that the "lter function S/(1!CPCF) in the update formula of equation (14) contains a term C which prevents direct implementation of the update algorithm. To alleviate the di$culty, the feedback term in equation (14) is neglected to get a simpli"ed and workable update law for the "lter weights

C(k#1)"C(k)!e(k)*(S*w), (15) where C and S are realized byFIR "lters, and &&*'' denotes linear convolution. Although the instant gradient estimate in the update law is somewhat biased, the latter experimental veri"cation shows that such a heuristic approach is e!ective for the spatiallyfeedfordward structure.

4. EXPERIMENTAL INVESTIGATION 4.1. EXPERIMENTAL RESULTS

A duct made of plywood shown in Figure 3 is used for verifying the proposed ANC method. The length of the duct is 440 cm and the cross-section is 25 cm;25 cm. There is 10 cm between the primarysource speaker and the measurement microphone. To reduce acoustic feedback, we use the backward control loudspeaker facing the open end of the duct. The distance between the measurement microphone and the control speaker is 235 cm to ensure causalityof the controller. The distance between the control speaker and the performance microphone is 110 cm. A TMS320C32 DSP equipped with four 16-bit analog IO channels is utilized to implement di!erent controllers. The sampling frequencyis chosen to be 2 kHz. Considering the cut-o! frequencyof the duct (&700 Hz) and the poor response of speaker at low frequency, we chose control bandwidth from 200 to 600 Hz.

In the following experiments, the FXLMS algorithm, the feedback neutralization algorithm, and the FULMS algorithm are compared for the spatiallyfeedfordward structure. The FXLMS is "rst examined byusing the reference signal from the upstream measurement microphone. As expected, this algorithm failed to attain noise attenuation (Figure 6(a)).

Next, we implement the feedback neutralization method that subtracts the signals from the feedback path to produce a non-contaminated reference input. What should be noted is that there is an inherent one sample delayin the feedback between F(z) and FK (z) because when the reference sample u(n) is measured, the output y(n) has not yet been computed. The experimental result of control performance obtained from this method is shown in Figure 6(b).

As a third approach, the FULMS algorithm is implemented byusing identical "lter order and step size to update the feedforward B(z) and the feedback "lters A(z). The experimental result of control performance is shown in Figure 6(c).

In addition to the commonlyused methods mentioned previously, an experiment was conducted for verifying the adaptive spatially feedfordward ANC algorithm proposed in this paper. First, we calculate the frequencyresponse of Roure's controller, as shown in Figure 7(a) via measurement of three frequencyresponses, i.e., H, H, and H in Figure 2(b). While measuring frequencyresponses H and H, the primaryspeaker is switched o! and the control speaker is switched on. Then, the control speaker is switched o! and the primaryspeaker is switched on to measure H. Second, we curve-"t the repetitive peaks of the frequencyresponse of Roure's controller in the control bandwidth to obtain an IIR "lter representation of CP in equation (11). The frequencyresponse regenerated bythe identi"ed model and the original frequencyresponse of the controller are compared in Figure 7(a). MATLAB functions, invfreqz tf 2sos are employed in this proceudre [13]. The function

Figure 6. Experimental results of some commonlyused LMS-based ANC algorithms. (a) The control performance of FXLMS implementation; (b) the control performance of feedback neutralization implementation; and (c) the control performance of FULMS implementation. **, Control o!; . . . , control on.

Figure 7. Design of spatiallyfeedforward controller. (a) The frequencyresponse of the IIR "lter: **, Roure's controller; } } } }, second order IIR "lters; (b) the frequencyresponse of the FIR "lter of the adaptive spatially feedforward controller; and (c) the impulse response of Roure's controller.

invfreqz converts the frequencyresponse of the IIR "lter into a transfer function which in

turn is converted byusing tf 2sos into second order transfer functions for DSP implementation. The result of this procedure is schematicallyshown in Figure 7(a). Third,

Figure 8. The control performance of the adaptive spatiallyfeedforward algorithm. **, Control o!; . . . , control on.

TABLE1

Comparison of ANC methods

Items FXLMS algorithm Feedback neutralization algorithm FULMS algorithm Adaptive spatially feedforward algorithm Control bandwidth (Hz) 200}600 200}600 200}600 200}600

Filter length of secondarypath 128 128 128 128

Filter length of feedback path 256

The feedforward "lter length 30

The feedback "lter length 30

The order of IIR "lter 20

Length of adaptive "lter 256 256 50

Scaling factor 0)5 0)25 0)5 0)5

Step size !1)0 !0)8 !0)3 !0)1

Total band attenuation (dB) 0)4 3)2 3)8 4)2

the FIR part of C is adaptivelyupdated byLMS algorithm with a reference signal that is the output of the IIR "lter. In the block diagram of Figure 5, the FIR "lter follows the IIR "lter and the cascaded "lter approximates the ideal controller C in equation (11). As shown in the results of Figures 7(a) and 7(c), the implemented controller exhibits the aforementioned repetitive patterns in the frequencyresponse and in the impulse response as well. Using the adaptive spatiallyfeedforward algorithm, the experimental result of control performance is obtained (Figure 8). The experimental results of the FXLMS algorithm, the feedback neutralization algorithm, the FULMS algorithm, and the adaptive spatiallyfeedforward algorithm are summarized in Table 1. It can be observed in the results that the adaptive spatiallyfeedforward algorithm yields the best performance in the total control bandwidth (4)2 dB). In addition, the adaptive spatiallyfeedforward algorithm requires fewer taps of the

"lter than the other methods. Although signi"cant attenuation in total band is obtained using adaptive spatiallyfeedfordward algorithm, the performance in the frequencyband near 200 and 500 Hz is not as good. This is due to the fact that the gain in that frequency region is not su$cientlyhigh to produce enough control output.

4.2. SOME IMPLEMENTATION ISSUES

First, the secondarypath and feedback path can be reliablymodelled o!-line via inverse fast Fourier transform and are realized byFIR "lters. Su$cient lengths are required to capture the fullydecayed impulse responses.

The adaptive spatiallyfeedforward algorithm approximates the physical nature of the transducer dynamics (FIR part) and the repetitive controller of the Munjal's ideal controller (IIR part) respectively. Because the lightly damped dynamics of the repetitive controller has been represented byan IIR "lter, onlyan FIR "lter of verylow order, e.g., 50 coe$cients is needed to represent the remaining dynamics of the controller. The frequency response of the FIR "lter can be predicted bydividing Roure's controller byIIR "lter. The LMS algorithm should converge to this "xed FIR "lter. Figure 7(b) illustrates the frequencyresponse of the FIR "lter that is nearlya constant gain.

Another implementation issue is regarding the e!ect of scaling on the convergence speed. From the following relation [12]:

0(( 1

pVY(¸#
), (16)

where is the step size, ¸ is the length of the adaptive"lter,
is the delayof the secondary path, and pVY is the power of the "ltered reference signal. The speed of convergence can be improved byhaving a greater step size bydecreasing pVY. This can be done in the DSP program bymultiplying the reference signal bya scaling factor, M, to decrease pVY and dividing the control output bythe same factor M.

5. CONCLUSIONS

In the paper, an adaptive spatiallyfeedforward algorithm has been developed. The ideal controller derived in section 2 via Roure's approach has proved that it can be partitioned into two parts: an FIR "lter and an IIR "lter. In the case where the temperature of the duct is not varied much, the repetitive peaks of the ideal controller are almost "xed. Thus, the IIR part representing a repetitive controller is implemented as a "xed controller, whereas the FIR part representing the transducer dynamics is implemented by an adaptive LMS "lter to accommodate perturbations and uncertainties in the system.

The proposed technique was compared with commonlyused methods such as the FXLMS algorithm, the feedback neutralization algorithm, and the FULMS algorithm by means of extensive experimental investigations. Among these ANC approaches, the proposed technique attains the best performance of noise attenuation (4)2 dB, total band), even in the presence of acoustic feedback.

In our experiments, the plant and the secondarypath are modelled o!-line. It is reported in Eriksson's work [14] that an adaptive controller with online plant modelling is capable of tracking plant variations. Future research will be focused on this particular aspect.

ACKNOWLEDGMENTS

The present work was supported bythe National Science Council in Taiwan, Republic of China, under the project number NSC 87-2212-E009-022.

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