Cai et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
Time-domain acoustic contrast control design with response differential constraint in personal audio systems Yefeng Cai, Ming Wu, Li Liu, and Jun Yanga) State Key Laboratory of Acoustics and the Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China [email protected], [email protected], [email protected], [email protected]
Abstract: The acoustic contrast control (ACC) approach is applied to reproduce the focused sound in personal audio systems utilizing an array of loudspeakers. A time-domain design of ACC is developed here for broadband input signals, where a response differential term is introduced to control the frequency response. Based on experimental results in an anechoic chamber, the proposed method demonstrates the potential capability to provide excellent acoustic contrast over the continuous frequency and maintain a flat frequency response. Furthermore, compared with the previous method, fewer parameters need to be tuned in the proposed method. C 2014 Acoustical Society of America V
PACS numbers: 43.60.Fg, 43.60.Dh, 43.38.Hz [CG] Date Received: February 25, 2014 Date Accepted: April 21, 2014
1. Introduction Personal audio systems can provide individuals with a private listening space without disturbing other people by utilizing a set of loudspeakers.1 Generally, two different kinds of zones are created by the personal audio systems—the bright zone, a region where a desired sound pressure level is reproduced, and the dark zone, where the sound pressure level needs to be attenuated. For this purpose, Choi and Kim2,3 used the acoustic contrast as a measure of performance and proposed the acoustic contrast control (ACC) approach to maximize this contrast. The acoustic contrast is defined as the ratio of acoustic energy between the bright zone and the dark zone. The performance of the ACC method has been investigated in various applications.4–7 However, the traditional ACC approach is usually designed at a set of discrete control frequencies and transformed into the time domain using the inverse discrete Fourier transform.3 The resulting design cannot avoid the causality problem, and it is impossible to obtain satisfactory acoustic contrast over a broad range of frequencies, especially when the filter length is short. To tackle these problems, Elliott and Cheer8 first tried to design ACC in the time domain and presented a broadband acoustic contrast control (BACC) method. Although the contrast problem can be partially alleviated by the BACC approach, the frequency response in the bright zone cannot be controlled; this may result in intolerable distortion. The response variation (RV) term has been introduced in the BACC-RV method in our previous work to overcome this problem.9 The improved method can obtain good contrast over a broad range of frequencies and provide a flat frequency response in the bright zone. To achieve excellent results, three parameters have to be tuned in the BACC-RV method, which might be a potential problem in practice.
a)
Author to whom correspondence should be addressed.
EL252 J. Acoust. Soc. Am. 135 (6), June 2014
C 2014 Acoustical Society of America V
Cai et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
In this letter, the response differential (RD) term instead of the RV term is introduced to manipulate the frequency response in the bright zone. Compared with the BACC-RV method, the proposed BACC-RD method only needs to tune two parameters and still can yield a good effect. The mathematical formulation of the BACC-RV and BACC-RD methods will be presented in detail in the following sections. The performance of BACC-RD method will be evaluated by experimental results and compared with the ACC and BACC-RV approaches. 2. BACC-RV method The structure of BACC-RV is illustrated in Fig. 1, where each of the L loudspeakers is driven by the output of a finite impulse response (FIR) filter wl ðnÞ. All of the filters have the same length M. Let hBlk ðnÞ denote the impulse response between the lth loudspeaker and kth control point in the bright zone, and I denotes the length of hBlk ðnÞ. Therefore the sampled output yBk ðnÞ at the kth control point in the bright zone can be expressed as yBk ðnÞ ¼
L X I 1 X
hBlk ðiÞ
l¼1 i¼0
M 1 X
wl ðmÞxðn m iÞ;
(1)
m¼0
where xðnÞ is the input signal to the system. To investigate the frequency response of the system, xðnÞ is assumed to be the Dirac delta function in the following derivation. Equation (1) can be equivalently rewritten in a concise way as yBk ðnÞ ¼ wT rBk ðnÞ;
(2)
where the ML 1 coefficient vector w is defined as w ¼ ½w1 ð0Þ; …; w1 ðM 1Þ; …; wL ð0Þ; …; wL ðM 1ÞT ;
(3)
and the filtered signal vector rBk ðnÞ is given by rBk ðnÞ ¼ ½hB1k ðnÞ; …; hB1k ðn M þ 1Þ; …; hBLk ðnÞ; …; hBLk ðn M þ 1ÞT :
(4)
It should be noted that in this case, yBk ðnÞ is the global impulse response with the length ðM þ I 1Þ. Therefore the average acoustic energy eB in the bright zone can be described by eB ¼
K MþI X X2 k¼1
y2Bk ðnÞ=K ¼ wT RB w;
(5)
n¼0
where is the number of control points in the bright zone, and RB PMþI P K 2 ¼ K rBk ðnÞrTBk ðnÞ=K is the normalized correlation matrix in the bright zone. k¼1 n¼0
Fig. 1. Structure of BACC-RV method.
J. Acoust. Soc. Am. 135 (6), June 2014
Cai et al.: Time-domain acoustic contrast control design EL253
Cai et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
The average acoustic energy eD in the dark zone can be similarly written as eD ¼ wT RD w;
(6)
where RD is the normalized correlation matrix in the dark zone. To maximize the acoustic contrast in the time domain, the BACC method leads to the following optimization problem:8 max w
wT RB w ; wT RD w þ dwT w
(7)
where d is a regularization parameter and can improve the robustness of the system. The design of the BACC method would try to reserve the signal at frequency with the highest acoustic contrast observed in the frequency domain and filter out the signal at other frequencies to maximize the acoustic contrast in the time domain. Therefore it can easily result in poor frequency response in the bright zone.9 To deal with this problem, the RV term is introduced by the BACC-RV method to control the flatness of the frequency response. The frequency response pBk ðf Þ at the kth control point in the bright zone is expressed as pBk ðf Þ ¼
MþI X2
yBk ðnÞej2pf nTs ¼ wT sBk ðf Þ;
(8)
n¼0
where f is the analog frequency, Ts is the sampling period, and sBk ðf Þ is ML 1 vector given by sBk ðf Þ ¼ ½rBk ð0Þ; …; rBk ðM þ I 2Þ½1; ej2pf Ts ; …; ej2pf ðI þM2ÞTs T :
(9)
The RV term10 is defined to measure the response variation over the frequency range of interest in the bright zone, and its formulation is written as RV ¼
K X J 1 X jpBk ðfj Þ pBk ðfref Þj2 ; JK k¼1 j¼1
(10)
where J is the number of discrete frequency bins in the frequency range of interest, and fref is the reference frequency. To achieve a flat response, the value of RV should be as small as possible. As a result, the BACC-RV method aims to solve the problem8 max w
wT RB w ; bwT RD w þ ð1 bÞRV þ dwT w
(11)
where b is a weight factor the value of which is between 0 and 1. It provides a tradeoff between the acoustic contrast in the time domain and the flatness of the frequency response in the bright zone.
3. Proposed BACC-RD method It can be learned from Eqs. (10) and (11) that three parameters need to be tuned in the BACC-RV approach, namely the weight factor, the regularization parameter, and the reference frequency. The performance of acoustic contrast is sensitive to the reference frequency. However, there is no rule to follow in selecting the reference frequency. To address this problem, a RD term is used instead of the RV term to measure the fluctuation of the frequency response. The RD term is defined as the mean square of the
EL254 J. Acoust. Soc. Am. 135 (6), June 2014
Cai et al.: Time-domain acoustic contrast control design
Cai et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
first-order differential of the frequency response in the bright zone in which the reference frequency parameter is not used, and its formulation is given as RD ¼
K X J 1 X 1 jpBk ðfjþ1 Þ pBk ðfj Þj2 : ðJ 1ÞK k¼1 j¼1
(12)
Define 2
1
6 6 0 V¼6 6 ⯗ 4 0
1
0
1
1
⯗ 0
⯗
0
3
7 07 7; ⯗ ⯗7 5 1 1
2 6 Sk ¼ 6 4
sTBk ðf1 Þ ⯗ sTBk ðfJ Þ
3 7 7: 5
By using Eqs. (8) and (13), Eq. (12) can be expressed as ( ) n o K K X X 1 1 H H T ðVSk wÞ ðVSk wÞ ¼ w < ðVSk Þ ðVSk Þ w; RD¼ ðJ 1ÞK k¼1 ðJ 1ÞK k¼1
(13)
(14)
where
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
Time-domain acoustic contrast control design with response differential constraint in personal audio systems Yefeng Cai, Ming Wu, Li Liu, and Jun Yanga) State Key Laboratory of Acoustics and the Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China [email protected], [email protected], [email protected], [email protected]
Abstract: The acoustic contrast control (ACC) approach is applied to reproduce the focused sound in personal audio systems utilizing an array of loudspeakers. A time-domain design of ACC is developed here for broadband input signals, where a response differential term is introduced to control the frequency response. Based on experimental results in an anechoic chamber, the proposed method demonstrates the potential capability to provide excellent acoustic contrast over the continuous frequency and maintain a flat frequency response. Furthermore, compared with the previous method, fewer parameters need to be tuned in the proposed method. C 2014 Acoustical Society of America V
PACS numbers: 43.60.Fg, 43.60.Dh, 43.38.Hz [CG] Date Received: February 25, 2014 Date Accepted: April 21, 2014
1. Introduction Personal audio systems can provide individuals with a private listening space without disturbing other people by utilizing a set of loudspeakers.1 Generally, two different kinds of zones are created by the personal audio systems—the bright zone, a region where a desired sound pressure level is reproduced, and the dark zone, where the sound pressure level needs to be attenuated. For this purpose, Choi and Kim2,3 used the acoustic contrast as a measure of performance and proposed the acoustic contrast control (ACC) approach to maximize this contrast. The acoustic contrast is defined as the ratio of acoustic energy between the bright zone and the dark zone. The performance of the ACC method has been investigated in various applications.4–7 However, the traditional ACC approach is usually designed at a set of discrete control frequencies and transformed into the time domain using the inverse discrete Fourier transform.3 The resulting design cannot avoid the causality problem, and it is impossible to obtain satisfactory acoustic contrast over a broad range of frequencies, especially when the filter length is short. To tackle these problems, Elliott and Cheer8 first tried to design ACC in the time domain and presented a broadband acoustic contrast control (BACC) method. Although the contrast problem can be partially alleviated by the BACC approach, the frequency response in the bright zone cannot be controlled; this may result in intolerable distortion. The response variation (RV) term has been introduced in the BACC-RV method in our previous work to overcome this problem.9 The improved method can obtain good contrast over a broad range of frequencies and provide a flat frequency response in the bright zone. To achieve excellent results, three parameters have to be tuned in the BACC-RV method, which might be a potential problem in practice.
a)
Author to whom correspondence should be addressed.
EL252 J. Acoust. Soc. Am. 135 (6), June 2014
C 2014 Acoustical Society of America V
Cai et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
In this letter, the response differential (RD) term instead of the RV term is introduced to manipulate the frequency response in the bright zone. Compared with the BACC-RV method, the proposed BACC-RD method only needs to tune two parameters and still can yield a good effect. The mathematical formulation of the BACC-RV and BACC-RD methods will be presented in detail in the following sections. The performance of BACC-RD method will be evaluated by experimental results and compared with the ACC and BACC-RV approaches. 2. BACC-RV method The structure of BACC-RV is illustrated in Fig. 1, where each of the L loudspeakers is driven by the output of a finite impulse response (FIR) filter wl ðnÞ. All of the filters have the same length M. Let hBlk ðnÞ denote the impulse response between the lth loudspeaker and kth control point in the bright zone, and I denotes the length of hBlk ðnÞ. Therefore the sampled output yBk ðnÞ at the kth control point in the bright zone can be expressed as yBk ðnÞ ¼
L X I 1 X
hBlk ðiÞ
l¼1 i¼0
M 1 X
wl ðmÞxðn m iÞ;
(1)
m¼0
where xðnÞ is the input signal to the system. To investigate the frequency response of the system, xðnÞ is assumed to be the Dirac delta function in the following derivation. Equation (1) can be equivalently rewritten in a concise way as yBk ðnÞ ¼ wT rBk ðnÞ;
(2)
where the ML 1 coefficient vector w is defined as w ¼ ½w1 ð0Þ; …; w1 ðM 1Þ; …; wL ð0Þ; …; wL ðM 1ÞT ;
(3)
and the filtered signal vector rBk ðnÞ is given by rBk ðnÞ ¼ ½hB1k ðnÞ; …; hB1k ðn M þ 1Þ; …; hBLk ðnÞ; …; hBLk ðn M þ 1ÞT :
(4)
It should be noted that in this case, yBk ðnÞ is the global impulse response with the length ðM þ I 1Þ. Therefore the average acoustic energy eB in the bright zone can be described by eB ¼
K MþI X X2 k¼1
y2Bk ðnÞ=K ¼ wT RB w;
(5)
n¼0
where is the number of control points in the bright zone, and RB PMþI P K 2 ¼ K rBk ðnÞrTBk ðnÞ=K is the normalized correlation matrix in the bright zone. k¼1 n¼0
Fig. 1. Structure of BACC-RV method.
J. Acoust. Soc. Am. 135 (6), June 2014
Cai et al.: Time-domain acoustic contrast control design EL253
Cai et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
The average acoustic energy eD in the dark zone can be similarly written as eD ¼ wT RD w;
(6)
where RD is the normalized correlation matrix in the dark zone. To maximize the acoustic contrast in the time domain, the BACC method leads to the following optimization problem:8 max w
wT RB w ; wT RD w þ dwT w
(7)
where d is a regularization parameter and can improve the robustness of the system. The design of the BACC method would try to reserve the signal at frequency with the highest acoustic contrast observed in the frequency domain and filter out the signal at other frequencies to maximize the acoustic contrast in the time domain. Therefore it can easily result in poor frequency response in the bright zone.9 To deal with this problem, the RV term is introduced by the BACC-RV method to control the flatness of the frequency response. The frequency response pBk ðf Þ at the kth control point in the bright zone is expressed as pBk ðf Þ ¼
MþI X2
yBk ðnÞej2pf nTs ¼ wT sBk ðf Þ;
(8)
n¼0
where f is the analog frequency, Ts is the sampling period, and sBk ðf Þ is ML 1 vector given by sBk ðf Þ ¼ ½rBk ð0Þ; …; rBk ðM þ I 2Þ½1; ej2pf Ts ; …; ej2pf ðI þM2ÞTs T :
(9)
The RV term10 is defined to measure the response variation over the frequency range of interest in the bright zone, and its formulation is written as RV ¼
K X J 1 X jpBk ðfj Þ pBk ðfref Þj2 ; JK k¼1 j¼1
(10)
where J is the number of discrete frequency bins in the frequency range of interest, and fref is the reference frequency. To achieve a flat response, the value of RV should be as small as possible. As a result, the BACC-RV method aims to solve the problem8 max w
wT RB w ; bwT RD w þ ð1 bÞRV þ dwT w
(11)
where b is a weight factor the value of which is between 0 and 1. It provides a tradeoff between the acoustic contrast in the time domain and the flatness of the frequency response in the bright zone.
3. Proposed BACC-RD method It can be learned from Eqs. (10) and (11) that three parameters need to be tuned in the BACC-RV approach, namely the weight factor, the regularization parameter, and the reference frequency. The performance of acoustic contrast is sensitive to the reference frequency. However, there is no rule to follow in selecting the reference frequency. To address this problem, a RD term is used instead of the RV term to measure the fluctuation of the frequency response. The RD term is defined as the mean square of the
EL254 J. Acoust. Soc. Am. 135 (6), June 2014
Cai et al.: Time-domain acoustic contrast control design
Cai et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4874236]
Published Online 13 May 2014
first-order differential of the frequency response in the bright zone in which the reference frequency parameter is not used, and its formulation is given as RD ¼
K X J 1 X 1 jpBk ðfjþ1 Þ pBk ðfj Þj2 : ðJ 1ÞK k¼1 j¼1
(12)
Define 2
1
6 6 0 V¼6 6 ⯗ 4 0
1
0
1
1
⯗ 0
⯗
0
3
7 07 7; ⯗ ⯗7 5 1 1
2 6 Sk ¼ 6 4
sTBk ðf1 Þ ⯗ sTBk ðfJ Þ
3 7 7: 5
By using Eqs. (8) and (13), Eq. (12) can be expressed as ( ) n o K K X X 1 1 H H T ðVSk wÞ ðVSk wÞ ¼ w < ðVSk Þ ðVSk Þ w; RD¼ ðJ 1ÞK k¼1 ðJ 1ÞK k¼1
(13)
(14)
where
