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![Page 1: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/1.jpg)
Study on the Use of Error Term in Parallel-form Narrowband Feedback
Active Noise Control Systems
Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong
11th Dec, 2014
[email protected] Signal Processing Lab,
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore
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Contents
2
• Introduction on feedback ANC
• Parallel-form narrowband feedback ANC
• Theoretical analysis on the use of error
term
• Simulation results
• Conclusions and future works
![Page 3: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/3.jpg)
Introduction on feedback ANC
3
Active noise control: introduce an anti-noise to cancel the primary noise sourceFeedforward ANC
– Reference microphone: not desired due to feedback of the secondary source or physically constraints
– Non-acoustic sensors (e.g., tachometers)
Feedback ANC: internal model control (IMC)– Affected by frequency separation in primary noise;– Degraded by measurement noise and impulse noise;– Subject to secondary path estimation accuracy [5];
LMS
S(z)
ˆ( )S z
ˆ( )S z
f ( )x n
( )x n ( )e n
( )d n
( )y n
f ( )y n
a ( )y n
ˆ( )d n
Acoustic summation
W(z)
[5] V. L. Wang, W. S. Gan, A. W. H. Khong and S. M. Kuo, “Convergence Analysis of Narrowband Feedback ANC System With Imperfect Secondary Path Estimation,” IEEE Trans. Audio, Speech, Lang. Process., vol. 21, no. 11, pp. 2403-2411, Nov. 2013.
![Page 4: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/4.jpg)
Parallel-form narrowband FBANC
4
LMSJ
ˆ( )S z
f1( )x n
( )Jx n
( )d n( )y n
f ( )y n
a ( )y n
ˆ( )d n( )N z
LMS2
LMS1
ˆ( )S zˆ( )S zˆ( )S z
Signal grouping and generation
( )e n
2 ( )x n
1( )x n
f2 ( )x n
f ( )Jx n
ˆj
( )Je n
2 ( )e n
1( )e n
Acoustic summation
( )S z
1 ( )W z
( )JW z
2 ( )W z
1( )y n
2 ( )y n
( )Jy n
[7] T. W. Wang, W. S. Gan, and S. M. Kuo, “New feedback active noise control system with improved performance,” Proc. IEEE ICASSP, Florence, Italy, 2014, pp. 6712-6716.
ˆ( )S z
f1( )x n
( )d n( )y n
f ( )y n
a ( )y n
ˆ( )d n( )N z
LMS2
LMS1
ˆ( )S zˆ( )S z
Signal grouping and generation
( )e n
2 ( )x n
1( )x n
f2 ( )x n
ˆj
2 ( )e n
1( )e n
Acoustic summation
( )S z
1 ( )W z
2 ( )W z
1( )y n
2 ( )y n
20, 40, 60, 80, 100, 120, 140, 160 Hz
Even: 40, 80, 120, 160 Hz
Odd: 20, 60, 100, 140 Hz
0 1 2 3 4 5 6 7 8 9 10-35
-30
-25
-20
-15
-10
-5
0
Time, second
MS
E,
dB
Conventional IMC feedback ANC system
Proposed new feedback ANC system
Learning curves when the noise frequencies vary
0 1 2 3 4 5 6 7 8 9 10-35
-30
-25
-20
-15
-10
-5
0
Time, second
MS
E,
dB
Conventional IMC feedback ANC system
Proposed new feedback ANC system
Effect of impulsive noise
Compared to IMC based FBANC [7]:increase frequency separation;Improve convergence rate and noise reduction;More robust to impulsive noise.
![Page 5: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/5.jpg)
[9] C.-Y. Chang and S. M. Kuo, “Complete parallel narrowband active noise control systems,” IEEE Trans. Audio, Speech, Lang. Process., vol. 21, no. 9, pp. 1976–1986, Sep. 2013.
Parallel-form narrowband FBANC
5
To use single full-band error or individual narrowband error?
LMSJ
ˆ( )S z
1 ( )fx n
( )Jx n( )e n
( )d n( )y n
( )fy n
( )ay n
ˆ( )d n
( )S z
( )N z
( )JR z
1 ( )R z
2 ( )R zLMS2
LMS1
1 ( )W z
2 ( )W z
( )JW z
ˆ( )S zˆ( )S zˆ( )S z
Signal generation and grouping
( )e n
2 ( )x n
1( )x n
2 ( )fx n
( )Jfx n
ˆj
1( )e n
2 ( )e n
( )Je n
2nd order IIR delayless filter bank
[9]
Full-band error: 1 2 , 1, 2,...,
Narrowband error: 1 2 , 1, 2,...,
j j j
j j j jfj
jfn n n j J
n n n
e n
j Je n
w w x
w w x
22
2 2
2
1 11
,
where
0 1 determines the bandwidth of the bandpass filter
21 cos control the center frequency
jj
j
jj j
j
jj j
s
qp z
pR z
z q z p
p
fq p
f
![Page 6: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/6.jpg)
Theoretical analysis
6
Consider using single full-band error
Tones with different frequencies
1
1 2
2 2 ,
J
j j j i jf
j j j
i
j jf fj j
n n e n n
nn n n ne
w x x
w w x 1,
J
j ii i j
n e n
Taking expectation
1 2 2 2 ,j j j j j j j jn n w I R w P D
1,
1, 1,
TJ
j i ai jfi i j
J JT T
i jf ai jfi i j i i j
E d n y n n
E d n n E y n n
D x
x x
0
where
cos 2 cos 2 0, .i j i jE f n f n f f
disturbance
![Page 7: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/7.jpg)
Simulations
7
Simulation setup 1:•Assume perfect secondary path estimation, secondary path: s={1, -0.1, -0.2};
•Primary noise: 4 tones (80, 160, 240, 320) Hz at sampling frequency 2 kHz;
•Four frequency channels are considered in parallel, one tone in each channel, and thus two taps are enough for the adaptive filters;
•For narrowband errors, an IIR filter bank similar to [9] is used, with pj =0.99;
•Step size = 0.1;
![Page 8: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/8.jpg)
Result 1: Insignificance of D compared to P
8
1 1
1 1
1 1ˆ ˆ,
1 1
ˆ ˆ,1 , 2DPR1 = , DPR2 =
ˆ ˆ,1 ,2
n n
j jf j jfm m
j jn n
j jf j jfm m
j jj j
j j
d m x m m x m
n nn n
d m x m m x m
n nn n
n n
P D
D D
P P
0 1000 2000 3000 4000 5000 600010
-5
10-4
10-3
10-2
10-1
100
101
X: 1372Y: 0.01001
iteration (sample)
DP
R1
(a) correspond to first weight
0 1000 2000 3000 4000 5000 600010
-5
10-4
10-3
10-2
10-1
100
101
X: 1532Y: 0.01009
iteration (sample)
DP
R2
(b) correspond to second weight
Channel 1
Channel 2Channel 3
Channel 4
Current iteration
![Page 9: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/9.jpg)
Result 2: Weight adaptation
9
0 500 1000 1500 2000 2500 3000-0.5
0
0.5
Iteration number
w1,
0
0 500 1000 1500 2000 2500 3000-0.5
0
0.5
1
Iteration number
w1,
1
Full-band error, j = 0.1
Narrowband error, j = 0.1
0 500 1000 1500 2000 2500 30000
0.5
1
Iteration number
w2,
0
0 500 1000 1500 2000 2500 3000-0.5
0
0.5
Iteration number
w2,
1
0 500 1000 1500 2000 2500 3000-0.5
0
0.5
Iteration number
w3,
0
0 500 1000 1500 2000 2500 3000-0.2
0
0.2
Iteration numberw
3,1
0 500 1000 1500 2000 2500 3000-0.5
0
0.5
Iteration number
w4,
0
0 500 1000 1500 2000 2500 3000-0.5
0
0.5
Iteration number
w4,
1
Channel 1
Channel 2
Channel 3
Channel 4
![Page 10: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/10.jpg)
Result 3: Learning curves
10
Frequencies of the tones are: 80, 160, 240, and 320 Hz.
0 1000 2000 3000 4000 5000 6000-70
-60
-50
-40
-30
-20
-10
Iteration number
MS
E (
dB)
Full-band error, j = 0.1
Narrowband error, j = 0.1
Frequencies of the tones are: 40, 80, 120, and 160 Hz.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-70
-60
-50
-40
-30
-20
-10
Iteration number
MS
E (
dB)
Full-band error, j = 0.1
Narrowband error, j = 0.1
0 1000 2000 3000 4000 5000 6000-70
-60
-50
-40
-30
-20
-10
Iteration number
MS
E (
dB)
Full-band error, j = 0.1
Narrowband error, j = 0.1
Frequencies of the tones are: 100, 200, 300, and 400 Hz.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
-80
-60
-40
-20
0
20
40
60
Iteration number
MS
E (
dB)
Full-band error, j = 0.1
Narrowband error, j = 0.1
Frequencies of the tones vary from 50, 100, 150, and 200 to 100, 200, 300, and 400 Hz.
Using same step size, full-band error is better than narrowband error!
![Page 11: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/11.jpg)
Simulation 2: setup
11
• Assume perfect secondary path estimation, secondary path shown below;
• Primary noise: 4 tones with fundamental frequencies at 70, 80, 90, 100 Hz at sampling frequency 1.5 kHz;
• Four frequency channels are considered in parallel, one tone in each channel, and thus two taps are enough for the adaptive filters;
• For narrowband errors, IIR filter bank similar to [9] is used, with pj =0.95;
• Step size < 0.1* step size bound (theoretical);
0 100 200 300 400 500 600 700 800-20
-15
-10
-5
0
5
Frequency (Hz)
Mag
nitu
de (
dB)
0 100 200 300 400 500 600 700 800-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Frequency (Hz)
Pha
se (
* r
ad)
![Page 12: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/12.jpg)
More Results: different harmonics, small step size
12
Using a small enough step size, the converge performance between the two cases is quite close.
0 0.5 1 1.5 2 2.5 3 3.5 4-60
-50
-40
-30
-20
-10
0
10
Time (s)
MS
E (
dB)
70Hz
Full-band error, j=0.05
Narrowband error, j=0.05
0 0.5 1 1.5 2 2.5 3 3.5 4-60
-50
-40
-30
-20
-10
0
10
Time (s)
MS
E (
dB)
80Hz
Full-band error, j=0.05
Narrowband error, j=0.05
0 0.5 1 1.5 2 2.5 3 3.5 4-60
-50
-40
-30
-20
-10
0
10
Time (s)
MS
E (
dB)
90Hz
Full-band error, j=0.05
Narrowband error, j=0.05
0 0.5 1 1.5 2 2.5 3 3.5 4-60
-50
-40
-30
-20
-10
0
10
Time (s)
MS
E (
dB)
100Hz
Full-band error, j=0.05
Narrowband error, j=0.05
70, 140, 210, 280 Hz 80, 160, 240, 320 Hz
90, 180, 270, 360 Hz 100, 200, 300, 400 Hz
![Page 13: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/13.jpg)
Conclusions and Future work
13
We studied the use of error terms in parallel-form narrowband feedback active noise control:1.Using different error terms, the MSE (hence noise reduction) performance is not affected;
2.The convergence performance differs:• Using smaller enough step size (<0.1*bound), the performance of using
full-band error and narrowband error is quite close;
• When the step size is larger, use of full-band single error yields faster convergence (in most of the cases);
3.When there is additional disturbance in the error microphone, the narrowband error could be better as it rejects the disturbance.
4.Future work shall investigate the exact conditions for the use of error terms with respect to the secondary path, frequencies of the primary noise, step size, etc.
![Page 14: Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems Jianjun HE, Woon-Seng Gan, and Yong-Kim Chong 11 th Dec,](https://reader035.fdocuments.us/reader035/viewer/2022062802/56649ee45503460f94bf3515/html5/thumbnails/14.jpg)
More Results
14
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
70+140+210+280Hz
Full-band error, j=0.03
Narrowband error, j=0.03
0 1 2 3 4-60
-40
-20
0
20
Time (s)M
SE
(dB
)
70+140+210+280Hz
Full-band error, j=0.06
Narrowband error, j=0.06
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
70+140+210+280Hz
Full-band error, j=0.1
Narrowband error, j=0.1
0 1 2 3 4-1000
0
1000
2000
Time (s)
MS
E (
dB)
70+140+210+280Hz
Full-band error, j=0.12
Narrowband error, j=0.12
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.03
Narrowband error, j=0.03
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.06
Narrowband error, j=0.06
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.1
Narrowband error, j=0.1
0 1 2 3 4-500
0
500
1000
1500
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.12
Narrowband error, j=0.12
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.03
Narrowband error, j=0.03
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.06
Narrowband error, j=0.06
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.1
Narrowband error, j=0.1
0 1 2 3 4-100
-50
0
50
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.12
Narrowband error, j=0.12
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
100+200+300+400Hz
Full-band error, j=0.03
Narrowband error, j=0.03
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
100+200+300+400Hz
Full-band error, j=0.06
Narrowband error, j=0.06
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
100+200+300+400Hz
Full-band error, j=0.1
Narrowband error, j=0.1
0 1 2 3 4-60
-40
-20
0
20
Time (s)M
SE
(dB
)
100+200+300+400Hz
Full-band error, j=0.12
Narrowband error, j=0.12
Secondary path: 1 sample delay
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More Results
15
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
70+140+210+280Hz
Full-band error, j=0.06
Narrowband error, j=0.06
0 1 2 3 4-60
-40
-20
0
20
Time (s)M
SE
(dB
)
70+140+210+280Hz
Full-band error, j=0.07
Narrowband error, j=0.07
0 1 2 3 4-100
-50
0
50
Time (s)
MS
E (
dB)
70+140+210+280Hz
Full-band error, j=0.08
Narrowband error, j=0.08
0 1 2 3 4-100
0
100
200
Time (s)
MS
E (
dB)
70+140+210+280Hz
Full-band error, j=0.09
Narrowband error, j=0.09
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.04
Narrowband error, j=0.04
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.05
Narrowband error, j=0.05
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.06
Narrowband error, j=0.06
0 1 2 3 4-100
0
100
200
Time (s)
MS
E (
dB)
80+160+240+320Hz
Full-band error, j=0.07
Narrowband error, j=0.07
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.09
Narrowband error, j=0.09
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.1
Narrowband error, j=0.1
0 1 2 3 4-100
-50
0
50
100
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.11
Narrowband error, j=0.11
0 1 2 3 4-200
0
200
400
600
Time (s)
MS
E (
dB)
90+180+270+360Hz
Full-band error, j=0.12
Narrowband error, j=0.12
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
100+200+300+400Hz
Full-band error, j=0.04
Narrowband error, j=0.04
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
100+200+300+400Hz
Full-band error, j=0.05
Narrowband error, j=0.05
0 1 2 3 4-60
-40
-20
0
20
Time (s)
MS
E (
dB)
100+200+300+400Hz
Full-band error, j=0.06
Narrowband error, j=0.06
0 1 2 3 4-100
0
100
200
Time (s)M
SE
(dB
)
100+200+300+400Hz
Full-band error, j=0.07
Narrowband error, j=0.07
Actual secondary path