Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems...

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

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

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.

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.

[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

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

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;

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

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

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)




0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-70

-60

-50

-40

-30

-20

-10

Iteration number

MS

E (

dB)



0 1000 2000 3000 4000 5000 6000-70

-60

-50

-40

-30

-20

-10

Iteration number

MS

E (

dB)




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)



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!

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)

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



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



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



70, 140, 210, 280 Hz 80, 160, 240, 320 Hz

90, 180, 270, 360 Hz 100, 200, 300, 400 Hz

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.

More Results

14

0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

70+140+210+280Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)M

SE

(dB

)

70+140+210+280Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

70+140+210+280Hz



0 1 2 3 4-1000

0

1000

2000

Time (s)

MS

E (

dB)

70+140+210+280Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-500

0

500

1000

1500

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-100

-50

0

50

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

100+200+300+400Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

100+200+300+400Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

100+200+300+400Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)M

SE

(dB

)

100+200+300+400Hz



Secondary path: 1 sample delay

More Results

15

0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

70+140+210+280Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)M

SE

(dB

)

70+140+210+280Hz



0 1 2 3 4-100

-50

0

50

Time (s)

MS

E (

dB)

70+140+210+280Hz



0 1 2 3 4-100

0

100

200

Time (s)

MS

E (

dB)

70+140+210+280Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-100

0

100

200

Time (s)

MS

E (

dB)

80+160+240+320Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-100

-50

0

50

100

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-200

0

200

400

600

Time (s)

MS

E (

dB)

90+180+270+360Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

100+200+300+400Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

100+200+300+400Hz



0 1 2 3 4-60

-40

-20

0

20

Time (s)

MS

E (

dB)

100+200+300+400Hz



0 1 2 3 4-100

0

100

200

Time (s)M

SE

(dB

)

100+200+300+400Hz



Actual secondary path

Study on the Use of Error Term in Parallel- form Narrowband Feedback Active Noise Control Systems...

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