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MULTICHANNEL ACTIVE NOISE CONTROL ALGORITHMS USING INVERSE FILTERS

F. Yu, M. Bouchard School of Information Technology and Engineering University of Ottawa 161 Louis-Pasteur, P.O. Box 450, Station A Ottawa (Ontario), K1N 6N5 e-mail : bouchard@site.uottawa.ca

ABSTRACT

In this paper, a simple inverse structure for multichannel active noise control (ANC) is introduced. This structure, combined with adaptive FIR filters and filtered-x LMS–based algorithms, can produce both a reduction of the computational load and an increase of the convergence speed, compared to the standard multichannel filtered-x LMS algorithm or its fast exact realizations. The proposed structure is based on the use of delayed non-causal models of the inverse plant between some error sensors (typically microphones in ANC systems) and some actuators (typically loudspeakers). Simulations using models of a realistic acoustic plant and inverse plant were performed, and the convergence gain that can be achieved with the proposed structure was demonstrated, while reducing the computational load of the control algorithm.

In the proposed approach, the FIR controller is split in two parts: multichannel predictors and multichannel delayed non-causal models of the inverse plant between some error sensors (typically microphones in ANC systems) and some actuators (typically loudspeakers). Figure 2 shows the structure of the proposed approach using the delayed non-causal filter, for the simplified monochannel case. The approach of splitting the controller in two parts is not new [5], and these two parts are sometimes referred to as the regulator and the compensator. However, the approach in [5] did not use adaptive FIR filters as controllers and delayed non-causal models of the inverse plant.

R e fe re n c e s ig n a l D is tu r b a n c e + P la n t + Σ

A d a p t iv e f ilt e r P la n t m odel

1. INTRODUCTION

For active noise control (ANC) systems, a common approach is to use adaptive FIR filters trained with the filtered-x LMS algorithm [1], for both feedforward systems and Internal Model Control (IMC) feedback systems, in monochannel or multichannel systems. Figure 1 shows the structure of a feedforward ANC system using the filtered-x LMS algorithm, for the simplified monochannel case. The widespread use of the filtered-x LMS algorithm is mainly due to the simplicity and the low computational load of the algorithm. Variations of the algorithm called the modified filtered-x LMS algorithms have also been published [2]-[4], which can achieve a faster convergence speed by using a larger step size in the algorithm. Recently, fast exact realizations of the filtered-x LMS and the modified filtered-x LMS algorithms have been published [2]. In most cases, these fast realizations can reduce the computational complexity of the filtered-x LMS and modified filtered-x LMS algorithms for multichannel systems. Many algorithms that can achieve faster convergence than the multichannel filtered-x LMS or the modified filtered-x LMS algorithms for ANC systems have also been published over the years. However, these algorithms can only provide increased convergence speed at the cost of increasing the computational load, compared to the multichannel filtered-x LMS algorithm or its fast exact realizations. In this paper, a simple multichannel algorithm that can both reduce the computational load and increase the convergence speed compared to the multichannel filtered-x LMS algorithm or its fast exact realizations is introduced, using an inverse structure and filtered-x LMS-based algorithms.

R e fe re n c e s ig n a l

x

Figure 1 : structure of the standard filtered-x LMS algorithm

D is tu rb a n c e In v e rs e p la n t m o d e l P la n t m odel P la n t + +

A d a p tiv e f ilte r In v e rs e p la n t m o d e l

Σ

x

c o m b in e d

Figure 2 : structure using the delayed non-causal inverse filter With the proposed inverse structure using adaptive FIR filters, there are two benefits of using delayed non-causal filters modeling the inverse plant of an ANC system. One benefit is that the combination of the delayed non-causal models and the models of the direct plant becomes approximately pure delays (approximately because models are never perfect). Using these pure delay operations can eliminate some costly convolutions. The second benefit of using the delayed non-causal filters is due to the fact that the convergence speed of filtered-x LMS-based algorithms is related to the eigenvalue spread in the correlation matrix of the filtered reference signals in the algorithms. This eigenvalue spread that can be reduced using the proposed structure, because the resulting combination of the delayed noncausal models and the models of the direct plant will have flat, uniform frequency responses, thus eliminating the eigenvalue spread caused by filtering with the models of the direct plant in the standard filtered-x LMS structure (as shown in Fig. 1).

2

In Section 2 of this paper, the multichannel algorithms that result from the use of the filtered-x LMS and the modified filtered-x LMS [2] algorithms with the proposed inverse structure are described. In Section 3, the computational load of the different algorithms described in this paper will be compared to lowcomputational fast exact realizations of the multichannel filteredx LMS and modified filtered-x LMS ANC algorithms. Simulations using a realistic acoustic plant and inverse plant measured on a headphone were performed to show the convergence gain that can be achieved with the proposed inverse structure, while reducing the computational load of the control algorithm. These simulation results will be described in Section 4.

z (n ) value at time n of the signal obtained by filtering the i, j,k

reference signal xi (n ) with the filter g (see (2)) j,k

v ′ (n ) value at time n of the filtered reference signals, i.e. i, k ,k

the signals obtained by filtering the z ( n ) signals i, j, k ′ with the direct plant models h d ′ (n) k ′′ ( n ) ek

j,k

(see (1))

2. DESCRIPTION OF THE INVERSE ALGORITHMS

2.1 The multichannel inverse filtered-x LMS algorithm

As previously mentioned in the paper, it is possible to both reduce the computational load and increase the convergence speed of the filtered-x LMS-based algorithms used for the training of adaptive FIR filters in ANC systems, by using noncausal models of the inverse plant. This scheme can be applied to the multichannel filtered-x LMS and the modified filtered-x LMS algorithms, and therefore the resulting algorithms have been called the multichannel inverse filtered-x LMS (IFX) and inverse modified filtered-x LMS (IMFX) algorithms. To describe explicitly those algorithms, the following notation is defined: I J K L M P number of reference sensors in a feedforward ANC system number of actuators in an ANC system number of error sensors in an ANC system length of the FIR adaptive filters length of the non-adaptive FIR filter models of the direct acoustic plant length of the non-adaptive FIR filter models of the inverse acoustic plant value at time n of the primary sound field (disturbance signal) at the k th error sensor value at time n of the signal measured by the k th error sensor xi ( n )

tk (n)

value at time n of the estimate of d (n ) for the k modified structure [2] value at time n of the error computed in the modified structure [2]

w (n ) value at time n of the l i,k ,l

th

coefficient in the adaptive

FIR filter linking xi (n ) and t k ( n )

h

j, k , m

value of the m

th

coefficient in the non-adaptive FIR

filter models of the direct plant between y j (n ) and e (n) . k

g

j,k , p

value of the p

th

coefficient in the non-adaptive FIR

filter modeling the inverse plant between e ( n) and k

y j (n ) .

h ? h =? , h ,Lh ? j, k , M ? ? ? j, k ,1 j, k ,2 T (1) T (2) T (3)

j,k

g

j,k

? =? g , g ,Lg ? j , k ,2 j, k , P ? ? j, k ,1 ?

d (n ) k

w

i,k

(n) = ? (n ), w ( n ), L w (n )? w ? i , k ,2 i, k , L ? ? i , k ,1 ? T

e (n) k

x i (n ) = ? x (n ), x i (n ? 1), L x i (n ? L + 1)? ? ? ? i ?

′ ( n ) = ? x (n ), x (n ? 1), xi ? i ? i

(4)

T

value at time n of the i

th

reference signal in a

feedforward ANC system (from a reference sensor) estimated value of the disturbance signal d k ( n + D ) at time n, where D is the delay required to make causal (see (2)). t k ( n ) is obtained by filtering j,k the reference signals xi ( n ) with the adaptive filters wi,k (n) (see (3)).

L xi (n ? P + 1)???

T

(5) T

g

′′ ( n ) = ? x (n ), x (n ? 1), L x (n ? P ? M + 2)? xi i i ? ? ? i ? t ( n ), t ( n ? 1), L t ( n ? P + 1)? t (n) = ? ? ? k k k ? ?k

(6)

(7) T (8)

y j (n )

value at time n of the j th actuator signal

y (n ), y j (n ? 1), L y j (n ? M + 1)? y j (n) = ? ? ? ? ? j

3

z i,j,k z (n) = ? ( n ), z (n ? 1), L z ( n ? M + 1)? ? ? i, j, k i, j, k ? ? i, j, k T where " * " is the convolution operator and only computed once. Since h

j =1

∑ T g j,k ′ * h j,k

J

is

(9) v i,k ′,k T v ′ ( n ), v ′ ( n ? 1), L v ′ (n ? L + 1)? (n ) = ? ? ? i k k i k k i k k , , , , , , ? ? (10)

and g are models of the j,k j,k direct and inverse plant, their combination can also be estimated by pure delays (the same delay required to make g causal). j,k Therefore, it is possible to simplify equations (14)-(17) and use (18), to have a simplified IFX algorithm:

w i,k ( n + 1) = w i,k ( n) ? ? xi ( n ? D )e ( n ) k

In this notation for ANC systems, feedforward controllers have been implicitly assumed because of the use of reference signals xi ( n ) . However, it is well known that feedback ANC systems using the Internal Model Control (IMC) approach behave like feedforward controllers, and that the same adaptive filtering algorithms as in feedforward systems can be used for those feedback systems. The error signals ek ( n ) measured by the error sensors are the sum of the disturbance signals and the contribution of the actuators at the error sensors, as described by (11) if the models h

(18) causal.

where D is the delay required to make g

j,k

2.2 The multichannel inverse modified filtered-x LMS algorithm

The inverse modified filtered-x LMS (IMFX) algorithm combines a modified filtered-x LMS algorithm [2] with the inverse structure using the models of the inverse plant. The algorithm can be described by equations (12)-(13), (17) and (19)(21):

j,k

h

of the direct plant are exact :

e (n ) = d ( n) + k k

j =1

∑

J

T j,k

y j (n)

(11)

The inverse filtered-x LMS (IFX) algorithm is described by equations (12)-(16). In these equations, k and k ′ are both used as an index for the error sensor signals or their prediction: t ( n ) = ∑ w ( n ) T xi ( n ) k i,k i =1 y j (n ) = I (12)

d ′ ( n) = e (n ) ? k k

j =1

I

∑

J

h

T j,k

y j (n)

(19)

e ′′ ( n ) = d ′ (n ) + k k

i =1k ′ =1 (n ) ? ?

∑ ∑

K

w

i,k ′

(n ) T v

i,k ′,k

(n)

(20)

k =1

∑

K

g

t (n) j,k k T ′ (n ) xi

T

w (13)

i,k ′

( n + 1) = w

i,k ′

k =1

∑

K

v

i,k ′,k

(n )e ′′ ( n ) k

(21).

Again, since h (14)

z

i, j, k

(n) = g

j,k

and g are models of the direct and inverse j,k j,k plant, their combination in equation (17) can be estimated by pure delays. Moreover, the combination of h and g in j,k j,k equations (13),(19) can also be simplified to pure delays. The resulting simplified IMFX algorithm is described by equations (12),(13) and by (22)-(24): ′ ( n) = e (n ) ? t (n ? D ) dk k k e ′′ ( n ) = d ′ (n ) + k k (22). (23)

v

i, k ′, k

(n) =

j =1

∑

J

h

T j,k

z

i,j,k ′

(n )

(15)

w

i,k ′

( n + 1) = w

i,k ′

(n ) ? ?

k =1

∑

K

v

i,k ′,k

(n )e ( n) k

(16)

where ? is a scalar convergence gain. Note that the g in (14) can be convoluted with the h

j,k

filters

filters in (15), and these j,k convolutions only need to be computed once. Therefore, equations (14)-(15) can be combined in a single equation, which will significantly reduce the number of computations of the algorithm: T ? J ? ? ? T ′′ (n ) v ′ (n) = ? ∑ g ′ * h ? xi j,k ? i, k , k ? j =1 j,k ? ?

i =1

∑ wi,k (n)T xi (n ? D )

I

w ( n + 1) = w ( n ) ? ? x i ( n ? D )e ′′ ( n ) i,k i,k k

(24).

3. COMPUTATIONAL LOAD

The computational load of the simplified IFX and simplified IMFX algorithms was compared with the computational load of the fast exact versions of the filtered-x LMS and the modified filtered-x LMS found in [2]. Table 1 shows the result for various numbers of reference sensors, actuators, error sensors, and filter lengths. It is clear from Table 1 that the simplified IFX algorithm has a lower computational load than the fast exact filtered-x LMS, and the simplified IMFX algorithm has a much lower

(17),

4

computational load than the fast exact modified filtered-x LMS. Combining this fact with the expected increase of convergence speed that the IFX and IMFX algorithms can produce, the use of these inverse algorithms does seem to be an interesting option. The convergence of these algorithms is discussed in the next section.

I J K L1 L2 M P fast exact fx-lms 1,622 3,246 6,923 8,030 10,940 34,670 41,180 93,260 1,110 2,222 5,259 5,214 7,868 20,334 26,588 76,620 854 1,710 4,427 3,806 6,332 13,166 19,292 68,300 fast exact modified fx-lms 3,158 9,390 22,283 32,606 51,900 188,270 297,180 1,168,460 1,878 5,294 12,939 17,502 28,348 97,134 154,588 614,220 1,238 3,246 8,267 9,950 16,572 51,566 83,292 337,100 simplif. IFX simplif. IMFX

Amplitude 10 5 0 -5 -10 -15 15

45

75

105

135

165

195 225 Samples

1 1 2 1 2 1 2 10 1 1 2 1 2 1 2 10 1 1 2 1 2 1 2 10

1 2 3 4 4 10 10 10 1 2 3 4 4 10 10 10 1 2 3 4 4 10 10 10

1 2 2 4 4 10 10 10 1 2 2 4 4 10 10 10 1 2 2 4 4 10 10 10

300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250 250

256 256 256 256 256 256 256 256 128 128 128 128 128 128 128 128 64 64 64 64 64 64 64 64

256 256 256 256 256 256 256 256 128 128 128 128 128 128 128 128 64 64 64 64 64 64 64 64

757 1,007 2,026 2,526 3,538 4,538 6,100 7,100 8,100 10,100 30,610 33,110 35,610 40,610 75,610 100,610 629 879 1,514 2,014 2,770 3,770 4,052 5,052 6,052 8,052 17,810 20,310 22,810 27,810 62,810 87,810 565 815 1,258 1,758 2,386 3,386 3,028 4,028 5,028 7,028 11,410 13,910 16,410 21,410 56,410 81,410

Figure 4 : non-causal inverse model of headphone acoustic plant

0 -5 -10 -15 -20 0 2000 4000 6000 8000 Iterations

A

B

C

-25 Convergence (dB)

Figure 5 : convergence curves of A) IFX/IMFX algorithms B) modified filtered-x LMS algorithm C) filtered-x LMS algorithm

5. CONCLUSION

In this paper, an inverse structure was introduced for the use of adaptive FIR filters in ANC systems. Multichannel adaptive FIR filter learning algorithms based on the filtered-x LMS algorithm were introduced for this inverse structure. It was shown in simulations with a realistic acoustical plant that some versions of the introduced algorithms can achieve both a reduction of the computational load and an increase of the convergence speed, compared to standard algorithms for ANC such as the multichannel filtered-x LMS algorithm, the modified filtered-x LMS algorithm or their fast exact realizations. Some theoretical work on the effect of plant model and inverse plant model errors would be of interest.

Table 1 : Computational load of fast exact filtered-x LMS, fast exact modified filtered-x LMS and simplified IFX/IMFX algorithms

4. SIMULATIONS

Simulations were performed using an acoustic plant measured on a headphone, as shown by the impulse response in Fig. 3. The inverse impulse response of this plant is shown in Fig. 4. It can be seen that a delay of 10-15 samples is required to make the inverse filter causal. The convergence speed of the filtered-x LMS algorithm, the modified filtered-x LMS algorithm, the IFX/IMFX algorithms and their simplified versions appears in Fig. 5, for a 20 dB SNR on the model of the plant and the inverse model of the plant. All the algorithms using the inverse filter produced a better performance, as expected, and the simplification using a pure delay did not affect the performance of the simplified inverse algorithms in this case. It was found that for a SNR of more than 10 dB on the model of the plant and the inverse model of the plant, the algorithms using the inverse filter produced a faster convergence speed.

Amplitude

0.25 0.05 -0.15 -0.35 0 25 50 75 100 125 150 175 200 225 250 Samples

6. REFERENCES

[1] S.M. Kuo and D.R. Morgan, Active noise control systems : algorithms and DSP implementations, New-York : J. Wiley & Sons, 1996 [2] S. Douglas, "Fast, exact filtered-x LMS and LMS algorithms for multichannel active noise control", Proc. ICASSP-97, Munich, Germany, 1997, pp.399-402 [3] I.S. Kim, H.S. Na, K.J. Kim and Y. Park, "Constraint filtered-x and filtered-u least-mean-square algorithms for the active control of noise in ducts", J. Acoust. Soc. Am., vol. 95, pp. 3379-3389, 1994 [4] M. Rupp and A.H. Sayed, "Robust FxLMS algorithms with improved convergence performance", IEEE Trans. Signal Processing, vol. 6, pp.78-85, 1998 [5] E.F. Berkman and E.K. Bender, "Perspectives on active noise and vibration control", Sound and vibration, vol. 31, pp. 80-94, 1997

Figure 3 : impulse response of the headphone acoustic plant

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