PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

A method for detecting convergence completion of adaptive filter
cancelling feedback path appeared in active noise control systems
Kensaku FUJII1; Mitsuji MUMEYASU2
1

Kodaway Laboratory, Japan
2

Kansai University, Japan

ABSTRACT
We propose a method for detecting the convergence completion of the feedback control filter working as an
adaptive filter. Generally, the feedback path is manually estimated before starting active noise control. Thisestimation is practically desired to be automatically completed. We first introduce a step size control method
capable of completing the estimation with a designed error even where the power of disturbance fluctuates.
We subsequently propose a method for detecting the convergence completion of the adaptive filter. In
practical use, since the impulse response of the feedback path and its power gain are unknown, we cannot
directly monitor the difference between the impulse response and the coefficients of the adaptive filter. For
solving this problem, we apply two adaptive filters with two different step sizes to estimating the feedback
path. Actually, we have already presented a recursive filter expression illustrating the structure of thenormalized least mean square algorithm. According to the expression, the square error between thecoefficients of the two adaptive filters converges on the designed error multiplied by the square of thedifference between the two step sizes. We finally verify using computer simulations that the proposed method
can successfully detect the convergence completion.
Keywords: Feedback, Path, Cancellation

1. INTRODUCTION
Feedforward type active noise control system (1) detecting a primary noise with a microphone,inevitably forms a feedback path, which is generally cancelled by the feedback control filter connectedin parallel. The simultaneous equations method (2-4) is proposed as an algorithm not requiring thecancellation unless howling occurs. The feedback path, however, is p ractically desired to be
sufficiently cancelled, because the cancellation error degrades the noise reduction effect (5). The
feedback control filter used for the cancellation is generally operated as an adaptive filter before
starting the noise control, whose coefficients are manually fixed after the feedback path is estimatedwith a sufficiently small error (6). In practical use, the operation is desired to be automatically stopped
after the estimation error decreases to a designed level.
The automatic stop operation requires a method for detecting that the estimation error decreased toless than the designed level. Three methods for the detection are accordingly proposed. One is themethod of tracking the transition of the gradient vector used for updating the coefficients of theadaptive filter (7). Another is the method of using the cross-correlation between the output of theadaptive filter and the remainder subtracted the previously mentioned output from the output of the
feedback path (8). The other is the method of continuing to update the coefficients for a prescribedtime (9). These methods, however, cannot be applied to practical systems in which the power of the
primary noise arriving at the microphone is supposed to be unknown or to fluctuate. The methods also
cannot guarantee the estimation error to decrease to the designed level.
The automatic stop operation moreover requires defining the upper limit of the estimation error notdegrading the noise reduction effect. The upper limit depends on the shape of the impulse response of
the acoustic paths formed in the system. As such examples, two upper limits are shown by using
simulations. One is --10 dB derived by using the practical impulse responses whose direct sound
1
2

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kodawayfujii@m.ieice.org
muneyasu@kansai-u.ac.jp

component is considerably large in comparison to the reflection components (10). The other is --20 dB
shown by using artificial impulse responses whose components consist of exponentially decayed
random numbers (11). In practical use, the coefficients of the adaptive filter need to be estimated withthe designed error even where the power of the primary noise is unknown or fluctuates.
In this study, we first present a method capable of steadily reducing the estimation error to thedesigned level under the fluctuation of the primary noise power. This method is characterized byapplying the block implementation normalized least mean square (NLMS) algorithm (12, 13) to theevaluation of the short time average power of the primary noise disturbing the estimation of the
feedback path. The power is then evaluated by substituting the short time square sum of the difference
between the outputs of the adaptive filter and the microphone for the real one. The square sum isaccordingly overestimated by the estimation error.
The overestimated power is next applied to the step size control method proposed in (14, 15). The
step size is then estimated under the desired one. This underestimated step size is actually preferable,
which operates so as to reduce the estimation error to smaller than the designed level. Applying theunderestimated step size to the adaptive algorithm can decrease the estimation error to less than t he
designed level even under the power fluctuation of the primary noise.
The remained issue is the derivation of the method for detecting that the estimation error decreased
to less than the designed level. A method for detecting the decrease is proposed (16). The method uses
two adaptive filters, and applies two different step sizes to them. Then, the square sum of thedifference between the coefficient vectors of the two adaptive filters converges on the designed
estimation error multiplied by the square of the difference of the two step sizes, which is used as thedetection threshold. The convergence of the adaptive filter can be judged by detecting that the squaresum of the difference decreases to less than the threshold.
In practical systems, the power gain of the feedback path is supposed to be unknown. The method
proposed in (16) accordingly normalizes the square sum of the difference with the power of either
adaptive filter, which is approximate to that of the feedback path after the convergence. Unf ortunately,some more simulation results show that the method cannot detect where the power gain of the feedback
path is not unity. In this study, we show that non-normalized square sum is suitable for the detection,
which is verified by computer simulation.

2. STEP SIZE CONTROLIn this study, the block implementation normalized least mean square (NLMS) algorithm (14, 15)
formulated byሺ௡ାଵሻ௃బ

ࡴ௡ାଵ ൌ ࡴ௡ ൅ ߤ௡ሺ௡ାଵሻ௃బ෍ ݁௝ ࢄ௝ ൙ ෍ ࢄ௝் ࢄ௝ ǡ௝ୀ௡௃బ ାଵ(1)௝ୀ௡௃బ ାଵis applied to the estimation of the feedback path for evaluating the short time average power of the
primary noise, where ࡴ௡ is the coefficient vector given to the adaptive filter at the ݊th block, ߤ௡
is the step size, ݆ is the sample time index, ݁௝ is the difference between the outputs of the adaptive
filter and the microphone, ‫ܬ‬଴ is the block length, ࢄ௝ is the vector of the reference signal provided
to the adaptive filter and the feedback path. Accordingly, the short time average power of the
primary noise can be approximately evaluated byሺ௡ାଵሻ௃బܳ௡ ൌ෍ ݁௝ଶǤ

(2)௝ୀ௡௃బ ାଵThe short time average power is then overestimated by the estimation error. The estimation error,
however, decreases along with the convergence of the adaptive filter, and finally to a negligible
extent (14).
The step size is subsequently calculated (14, 15) by using(3)ߤ௡ ൌ ʹ‫ܥ‬଴ ܲ௡ ‫ܬ‬଴Τሺܳ௡ ‫ ܫ‬൅ ܲ௡ ‫ܥ‬଴ ሻǤ
The estimation error decreases to the designed level ‫ܥ‬଴ by applying the step size to Eq. (1), where‫ ܫ‬is the number pf taps of the adaptive filter and6899ሺ௡ାଵሻ௃బ෍ ࢄ௝் ࢄ௝ Ǥ

ܲ௡ ൌ

(4)

௝ୀ௡௃బ ାଵ3. CONVERGENCE DETECTION METHOD
The impulse response of the feedback path is supposed to be unknown in practical system s. We
cannot accordingly calculate the transition of the estimation error between the impulse response
samples and the coefficients of the adaptive filter. We then require a method capable of indirectly
detecting that the estimation error decreased to the designed level. In this paper, we next introduce
the detection method proposed in (16). The method is characterized by applying two adaptive filtersto the estimation of the feedback path.
The first order recursive filter expression of the NLMS algorithm (17) is used for the derivationof the threshold applied to the detection of the convergence of the adaptive filter. In the derivation
process, the vector expression of the NLMS algorithm formulated byࡴ௝ାଵ ൌ ࡴ௝ ൅ ߤ݁௝ ࢄ௝ Τࢄ௝் ࢄ௝(5)is resolved into the element expression,‫ܪ‬௝ାଵ ሺ݉ሻ ൌ ‫ܪ‬௝ ሺ݉ሻ ൅ ߤ݁௝ ‫ݔ‬௝ ሺ݉ሻΤܲ௝ ǡ(6)where ‫ܪ‬௝ ሺ݉ሻ is the ݉th element of the coefficient vector given to the adaptive filter at ݆th sample
time, ߤ is the step size, ‫ݔ‬௝ ሺ݉ሻ is the ݉th element of the reference signal vector andܲ௝ ൌ ࢄ௝் ࢄ௝ Ǥ
(7)
The element expression can be rearranged to the first order recursive filter expression (17).
ூିଵ‫ܪ‬௝ାଵ ሺ݉ሻ ൌ ‫ܪ‬௝ ሺ݉ሻߙ௝ ሺ݉ሻ ൅ ݄ሺ݉ሻሼͳ െ ߙ௝ ሺ݉ሻሽ ൅ ߤ෍ ሼο௝ ሺ݅ሻ‫ݔ‬௝ ሺ݅ሻ ൅ ݊௝ ሽ‫ݔ‬௝ ሺ݉ሻ൘ܲ௝ ǡ(8)௜ୀ଴ሺ௜ஷ௠ሻ

where ݄ሺ݉ሻ is the ݉th impulse response sample of the feedback path,ߙ௝ ሺ݉ሻ ൌ ͳ െ ߤ‫ݔ‬௝ଶ ሺ݉ሻൗܲ௝ ǡ(9)ο࢐ ሺ݅ሻ ൌ ݄ሺ݅ሻ െ ‫ܪ‬௝ ሺ݅ሻǡ(10)and ݊௝ is the primary noise disturbing the estimation. In this expression, ݄ሺ݉ሻ is supposed to beinvariable until the estimation is completed, which can be regarded as a direct current. According to(18, 19), the transition of the estimation error can be described by substitutingூିଵതതതതതതതതߙఫ ሺ݉ሻ ൌ ෍ ߙ௝ ሺ݉ሻ൘‫ ܫ‬ൌ ͳ െ ߤΤ‫ ܫ‬൏ ͳ

(11)

௠ୀ଴

for ߙ௝ ሺ݉ሻ. Then, the transition of the estimation error can be calculated as the step response of theതതതതതതതത
first order low pass filter whose recursive coefficient is ߙఫ ሺ݉ሻ.
The filter expression shown in Eq. (8) can be easily developed to the block implementation type,(12)‫ܪ‬௡ାଵ ሺ݉ሻ ൌ ‫ܪ‬௡ ሺ݉ሻߙ௡ ሺ݉ሻ ൅ ݄ሺ݉ሻሼͳ െ ߙ௡ ሺ݉ሻሽ ൅ ‫ܩ‬௡ ሺ݉ሻǡwhereሺ௡ାଵሻ௃బ෍ ‫ݔ‬௝ଶ ሺ݉ሻ൘ܲ௡ ǡߙ௡ ሺ݉ሻ ൌ ͳ െ ߤ௡

(13)௝ୀ௡௃బ ାଵሺ௡ାଵሻ௃బ

‫ܩ‬௡ ሺ݉ሻ ൌ ߤ௡෍ூିଵ෍ ሼο௡ ሺ݅ሻ‫ݔ‬௝ ሺ݅ሻ ൅ ݊௝ ሽ‫ݔ‬௝ ሺ݉ሻ൘ܲ௝ ǡ(14)௝ୀ௡௃బ ାଵ ௜ୀ଴ሺ௜ஷ௠ሻ

(15)ο௡ ሺ݅ሻ ൌ ݄ሺ݅ሻ െ ‫ܪ‬௡ ሺ݅ሻǤIt should be noted here that the white noise is used as the reference signal for the estimation of
the feedback path. Then, the transition of the estimation error calculated by Eq. (12) is equal to that6900

provided by Eq. (5) (12, 13). Equation (12) indicates that the adaptive filter coefficient ‫ܪ‬௡ାଵ ሺ݉ሻ isestimated as the step response generated by applying ݉th element of the impulse response of the
feedback path to the first order recursive filter. As seen from the expression, the coefficient‫ܪ‬௡ାଵ ሺ݉ሻ fluctuates around ݄ሺ݉ሻ after the convergence, asሺ௡ାଵሻ௃బ ‫ܪ‬௡ାଵ ሺ݉ሻ ൌ ݄ሺ݉ሻ ൅ ߤ௡

௡՜ஶ

ூିଵ

෍ ሼߜ௡ ሺ݅ሻ‫ݔ‬௝ ሺ݅ሻ ൅ ݊௝ ሽ‫ݔ‬௝ ሺ݉ሻ൘ܲ௝ ǡ

෍

(16)

௝ୀ௡௃బ ାଵ ௜ୀ଴ሺ௜ஷ௠ሻ

where ߜ௡ ሺ݅ሻ denotes the estimation error added to the coefficient ‫ܪ‬௡ ሺ݅ሻ. The second term of Eq.
(16) shows that the coefficient ‫ܪ‬௡ାଵ ሺ݉ሻ fluctuates around to ݄ሺ݉ሻ, which expresses the designed
estimation error. The fluctuation equal to the estimation error can be moreover expressed asሺ௡ାଵሻ௃బ

ߜ௡ାଵ ሺ݉ሻ ൌ ߤ௡ூିଵ෍ ሼߜ௡ ሺ݅ሻ‫ݔ‬௝ ሺ݅ሻ ൅ ݊௝ ሽ‫ݔ‬௝ ሺ݉ሻ൘ܲ௝ Ǥ

෍

(17)

௝ୀ௡௃బ ାଵ ௜ୀ଴ሺ௜ஷ௠ሻ

In practical use, the power of the reference signal is supposed to be less than that of the primarynoise so as not to be perceived. ߜ௡ ሺ݅ሻ is then expected to be negligible in comparison with ݊௝ after
the convergence. Equation (17) can be accordingly approximated toሺ௡ାଵሻ௃బ

ூିଵ෍෍ߜ௡ାଵ ሺ݉ሻ ൎ ߤ௡݊௝ ‫ݔ‬௝ ሺ݉ሻ൘ܲ௝ Ǥ(18)௝ୀ௡௃బ ାଵ ௜ୀ଴ሺ௜ஷ௠ሻ

We next apply another step size ܽߤ௡ to either adaptive filter, whereܽ൏ͳ
Then, the fluctuation of the coefficient of the either adaptive filter can be expressed byᇱሺ݉ሻ ൎ ܽߤ௡ߜ௡ାଵሺ௡ାଵሻ௃బூିଵ෍෍݊௝ ‫ݔ‬௝ ሺ݉ሻ൘ܲ௝ ൌ ܽߜ௡ାଵ ሺ݉ሻǤ

(19)

(20)

௝ୀ௡௃బ ାଵ ௜ୀ଴ሺ௜ஷ௠ሻ

It should be noted here thatᇱሺ݉ሻ ൌ ሺͳ െ ܽሻߤ௡݀௡ାଵ ሺ݉ሻ ൌ ߜ௡ାଵ ሺ݉ሻ െ ߜ௡ାଵሺ௡ାଵሻ௃బூିଵ෍෍݊௝ ‫ݔ‬௝ ሺ݉ሻ൘ܲ௝ ൌ ሺͳ െ ܽሻߜ௡ାଵ ሺ݉ሻ(21)௝ୀ௡௃బ ାଵ ௜ୀ଴ሺ௜ஷ௠ሻ

is equal to the difference of the coefficients of the two adaptive filters. The square sum of thedifferences can be accordingly evaluated by,(22)‫ܦ‬௡ ൌ ሺࡴ௔௡ െ ࡴ௕௡ ሻ் ሺࡴ௔௡ െ ࡴ௕௡ ሻǡ௔௕
where ࡴ௡ and ࡴ௡ denote the coefficient vectors of the adaptive filters, respectively. Equations(21) and (22) show the square sum ‫ܦ‬௡ converges on the ሺͳ െ ܽሻଶ times of the designed estimation
error ‫ܥ‬଴. The convergence of the adaptive filter can be judged by monitoring the transition of ‫ܦ‬௡ .

4. SIMULATION RESULTS
Figure 1 shows the impulse response of the feedback path applied to simulations, whose samples
are given the series of exponentially decayed regular numbers, and whose power gain is adjusted to 0
dB. The other conditions are shown below.
• Reference signal and Primary noise: white noise• Power ratio of reference signal to primary noise: -6 dB and 0 dB
• Block length ‫ܬ‬଴: 1000
• Designed estimation error ‫ܥ‬଴: 0.001 (-30 dB)• Step size ratio ܽ: 0.9

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Figure 1 – Impulse response of feedback path used for simulations
Under the conditions, the difference ‫ܦ‬௡ାଵ ሺ݉ሻ gives the relation,(23)݀௡ାଵ ሺ݉ሻ ൌ ሺͳ െ ܽሻߜ௡ାଵ ሺ݉ሻ ൌ ͲǤͳߜ௡ାଵ ሺ݉ሻ
Equation (23) states that the square sum of the differences, ‫ܦ‬௡ , decreases to 20 dB less than thedesigned estimation error ‫ܥ‬଴. In the conditions shown above, the designed estimation error ‫ܥ‬଴ is-30 dB, which states that the convergence of the adaptive filter can be judged by detecting that ‫ܦ‬௡decreased to less than -50 dB.
Figure 2 shows the transition of the estimation errors,(24)‫ܦ‬௡௔ ൌ ሺࢎ െ ࡴ௔௡ ሻ் ሺࢎ െ ࡴ௔௡ ሻǡ(25)‫ܦ‬௡௕ ൌ ሺࢎ െ ࡴ௕௡ ሻࢀ ሺࢎ െ ࡴ௕௡ ሻǡand the square sum of the differences,(26)‫ܦ‬௡௖ ൌ ሺࡴ௔௡ െ ࡴ௕௡ ሻ் ሺࡴ௔௡ െ ࡴ௕௡ ሻǡ
where (a), (b) and (c) show the transitions of the estimation errors and the square sum of thedifferences, ‫ܦ‬௡௔ , ‫ܦ‬௡௕ and ‫ܦ‬௡௖ , respectively. In this result, the square sum of the differences, ‫ܦ‬௡௖ ,increases from less than -60 dB, which is caused by initializing the coefficients of the adaptive filters
to zero. The square sum of the differences, ‫ܦ‬௡௖ , subsequently decreases to less than -50 dB again.
This result shows that the convergence with the designed estimation error, -30 dB, can be judged bydetecting that the square sum of the differences, ‫ܦ‬௡௖ , decreases to less than -50 dB.

Figure 2 – Transition of the estimation errors and the square sum of the differences (the power gain: 0 dB)
Figure 3 shows the transitions of the estimation errors and the square sum of the differences
calculated by reducing the power gain of the feedback path to -10 dB. In this result, the estimation
errors decrease to less than the designed level, -30 dB, and consequently the convergence of theadaptive filter can be judged by detecting that the square sum of the differences decrease s to thethreshold, -50 dB. On the other hand, normalizing the estimation errors and the square sum of thedifferences with the power gain of the feedback path increases the transitions by 10 dB.
Consequently, the estimation errors and the square sum of the differences do not decrease to lessthan the designed level, -30 dB, and to less than the detection threshold, -50 dB, respectively.

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Figure 3 – Transition of the estimation errors and the square sum of the differences (the power gain: -10 dB)
Figure 4 also shows the transition of the estimation errors and the square sum of the differences
calculated by increasing the power gain of the feedback path to 10 dB. Similarly, normalizing theestimation errors and the square sum of the differences with the power gain decrease s the transitions
by 10 dB. Consequently, the estimation errors decrease to less than -40 dB, which is 10 dB less thanthe designed level, -30 dB. The square sum of the differences also decreases to less than -60 dB,
which is 10 dB less than the detection threshold -50 dB. The two results shown in Figs. 3 and 4
demonstrate that the normalization presented in (16) is unnecessary for the detection where thepower gain of the feedback path is not unity.

Figure 4 – Transition of the estimation errors and the square sum of the differences (the power gain: 10 dB)
Figure 5 shows the transitions of the estimation errors and the square sum of the differences
calculated by changing the power ratio of the reference signal to the primary noise from 0 dB to -6
dB at the indicated point, where the power gain of the feedback path is 10 dB. This result shows thatthe estimation error decreases steadily to the designed level, -30 dB, though the power ratio changes
at the point and that the convergence of the adaptive filter can be judged by detecting that the squaresum of the differences deceases to less than -50 dB.

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Figure 5 – Transition of the estimation errors and the square sum of the differences calculated changing thepower ratio of the reference signal to the primary noise at the indicated point (the power gain: 10 dB)

5. SUMMARY
In this paper, we have proposed the method capable of detecting the convergence of thecoefficients of the adaptive filter with the designed estimation error, and then have shown that themethod successfully works. In the near future, we will apply the proposed method to re-estimating
the feedback path under active noise control.

REFERENCES
1. Elliott S J, Nelson P A. Active Noise Control. IEEE Signal Processing Magazine. 1993; 10(4): 12-35.
2. Fujii K, Muneyasu M, Ohga J. Active noise control systems by using the simultaneous equations
method without estimation of the error path filter coefficients. IEICE Trans Fundamentals. 1999;
J82-A(3): 299-305.
3. Fujii K, Hashimoto S, Muneyasu M. Application of a frequency domain processing technique to thesimultaneous equations method. IEICE Trans Fundamentals. 2003; E86-A(8): 2020-2027.
4. Fujii K, et al. Verification of simultaneous equations method by an experimental active noise control
system. Acoustical Science & Technology. 2006; 27(10): 270-277.
6. Fujii K, Muneyasu M. A method for re-estimating feedback path under active noise control. Proc
ICSV24; 23-27 July 2017; London UK 2017. #730.
7. Kuo S J, Morgan D R. Active Noise Control System. New York, USA: Wiley-Interscience Publication:
1996.
8. Nagayasu K. Stabilization of adaptive control on active noise control. Proc 1993 Autumn Meeting
Acoust Soc of Jpn; September 1993; Toyama, Japan 1993. p. 515-6.
9. Fujii K, Ohga J. A method of estimating the pre-fixed filter coefficients requisite for active noise control
systems. IEICE Trans Fundamentals. 1996; J79-A(8): 1369-1377.
10. Fujii K, Ohga J. Equation for brief evaluation of convergence rate of the NLMS algorithm. IEICE Trans
Fundamentals. 1993; E76-A(12): 2048-2051.
11. Fujii K, et al. Method for identifying feedback path in feedforward type noise control system. J Acoust
Soc Jpn, 2014; 70(4): 162-171.
12. Iwamatsu Y, Fujii K, Muneyasu M. Frequency domain method to estimate the coefficients of feedback
control filter for active noise control systems. Acoust Sci & Tech, 2006; 27(5): 264-269.
13. Fujii K, Ohga J. A method of continuing to adjust adaptive filter coefficients in voiceless noise terms
available for acoustic echo canceller systems. IEICE Trans. Fundamentals, 1995; J78-A(11): 1404-1409.
14. Fujii K, Ohga J. Convergence time reduction provided by a block length control method applied to the“summational” NLMS algorithm. IEICE Trans Fundamentals, 1997; J80-A(1): 27-35.
15. Fujii K, Oka R, Saitoh N, Muneyasu M. A Step size control method steadily reducing estimation error
even during double-talk term. IEICE Trans. Fundamentals, 2009; J92-A(11): 874-887.
16. Yoshioka T, et al. A step size control method improving estimation speed in double talk term. IEICE
Trans Fundamentals, 2013; E96-A(7): 1543-1551.
17. Fujii K, Muneyasu M. Proposition of a method for detecting the convergence of coefficients ofpre-estimating filter applied to active noise control systems. Proc ICSV25; 8-12 July 2018; Hiroshima,
Japan 2018. #465.
18. Fujii K, Sakai Y, Masuda H, Ohga J. A new expression and stability condition of learning identification
algorithm. IEICE Trans Fundamentals, 1990; J73-A(11): 1769-1775.
19. Fujii K, Sakai Y, Ohga J. Influence of time constant on coefficients estimation error derived from a low
pass filter expression for learning identification algorithm. IEICE Trans Fundamentals, 1991; J74-A(4):635-643.
20. Fujii K, Sakai Y, Ohga J. Formulation of estimation error transition and processing cost reduction
methods derived from low pass filter expression of learning identification algorithm, IEICE Trans
Fundamentals, 1991; J74-A(7): 966-973.

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