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Localization of moving periodic impulsive source in a noisy environment
ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 22 (2008) 753–759 www.elsevier.com/locate/jnlabr/ymssp
Localization of moving periodic impulsive source in a noisy environment Jong-Hoon Jeon, Yang-Hann Kim Center for Noise and Vibration Control (NOVIC), School of Mechanical, Aerospace & Systems Engineering, KAIST (Korea Advanced Institute of Science and Technology), 373-1 Science Town, Daejon-shi 305-701, Republic of Korea Received 21 June 2007; accepted 2 September 2007 Available online 8 September 2007
Abstract Sound generated from a machine often carries information on the condition of the machine. For example, the faults in the rotating part of a machine generate impulsive sound. This means that by visualizing the radiation pattern from a moving machine, the condition of the machine: where the fault is, can be examined. The moving frame acoustic holography method [H.-S. Kwon, Y.-H. Kim, Moving frame technique for planar acoustic holography, Journal of the Acoustical Society of America 103(4) (1998) 1734–1741.] enables us to visualize sound field from a moving source of pure tone or narrow band signal. This method cannot be applied directly, however, to the moving periodic impulsive sound because the impulsive signal is obviously different from the pure tone or narrow band signal. In this paper, we propose a means to use the method to localize moving periodic impulsive source that is embedded by noise. Moving periodic impulsive signal can be regarded as the combination of moving several discrete pure tones. This enables us to visualize the sound field at each frequency component. We also found that the signal to noise ratio can be readily improved by averaging several holograms. This is because periodic impulsive noise has several discrete frequency components. r 2007 Elsevier Ltd. All rights reserved. Keywords: Periodic impulsive noise; Source identification; Moving frame acoustic holography; Moving source; Noise
1. Motivation and objective The moving frame acoustic holography method [1,2] is a novel way to visualize the sound field generated by the moving sound source. As shown in Fig. 1, the method assumes the plane that has a sound source and is moving with a constant speed u to the positive x direction. In fact, the method can be extended to the case the speed u is not constant [3]. A microphone array is assumed to be on the ground. The method also presumes that there is a coordinate, which is moving with the source plane (the hologram coordinate). Let us denote Pf h0 ðxm ; ym ; zm ; f Þ to be the sound pressure measured by a fixed microphone array at the position (xm, ym, zm) due to the moving source having pure tone fh0. The origin of hologram coordinate is set to pass (xm, 0, 0) on the fixed coordinate when t ¼ 0. Then the method says that Pf h0 ðxm ; ym ; zm ; f Þ can be expressed by the sound Corresponding author. Tel.: +82 42 869 3025; fax: +82 42 869 8220.
E-mail addresses: [email protected] (J.-H. Jeon), [email protected] (Y.-H. Kim). 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.09.001
ARTICLE IN PRESS J.-H. Jeon, Y.-H. Kim / Mechanical Systems and Signal Processing 22 (2008) 753–759
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Fig. 1. Pictorial expression of the moving sound source and fixed microphone array.
pressure with respect to the hologram coordinate in the wave number domain: Df ^ Ph ðkx ; ym ; zm ; f h0 Þ, (1) u where Df means frequency resolution of the measured signal and P^ h ðkx ; ym ; zm ; f h0 Þ means x directional wave number transform of the sound pressure on the hologram plane for the frequency fh0. kx means x directional wave number, and has relation with the frequency: Pf h0 ðxm ; ym ; zm ; f Þ ¼
2pðf f h0 Þ . (2) u Therefore the method predicts the sound pressure on the hologram plane by using the inverse spatial Fourier transform: Z 1 1 ^ Ph ðxh ; ym ; zm ; f h0 Þ ¼ Ph ðkx ; ym ; zm ; f h0 Þejkx xh dkx . (3) 2p 1 kx ¼
Once we have the hologram, we can localize the sound source by predicting the sound field on the source plane: Ph(xh,ym,0;fh0) [4,5]. However, the method is limited to the sound source that emits pure tone or narrow band signal. We have attempted to expand the method to localize the moving sound source that is generating periodic impulsive sound, especially in a noisy environment. In this situation, two major problems make it difficult to use the method directly. The first one is that the periodic impulsive signal is obviously different from the pure tone signal. Furthermore, the method does not consider the noise effect. The first problem can be readily solved if we find the relation between moving periodic impulsive signals and the moving pure tone signals. This motivates us to model and analyze the moving periodic impulsive signal first. 2. Signal modeling of moving periodic impulsive sound Suppose that we have a source that generates an impulse at time t ¼ t0. Its period is DT, and the rmi is the distance between the sound source and the microphone at (xm, ym, zm) when the source radiates ith impulsive sound. Then the sound pressure measured at the microphone positioning at (xm, ym, zm) can be written as 1 X P0 1 pðxm ; ym ; zm ; tÞ ¼ d t t0 þ ði 1ÞDT þ rmi (4) þ nðxm ; ym ; zm ; tÞ, c r i¼1 mi where P0 means the monopole amplitude assuming that the impulsive sound has spherical wave fronts, and n(xm, ym, zm;t) is the noise at the position (xm, ym, zm) and time t, and d( ) means Dirac delta function that models the impulse and c is the speed of sound. Fig. 2(a) shows the moving periodic impulsive signal with white noise that is uncorrelated with the periodic impulsive signal. The signal-to-noise ratio, which is defined
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Fig. 2. Moving periodic impulsive signal with white noise in (a) time domain and (b) frequency domain (signal-to-noise ratio is 0 dB).
by the ratio between the variance of the impulsive signal and noise, is 0 dB. The impulse period is 0.1 s and Mach number is 0.01. The Fourier transform of p(xm, ym, zm;t) can be written as Pðxm ; ym ; zm ; f Þ ¼
1 X P0 i¼1
rmi
ej2pf ðt0 þði1ÞDTþð1=cÞrmi Þ þ Nðxm ; ym ; zm ; f Þ,
(5)
where N(xm, ym, zm; f) is the Fourier transform of n(xm, ym, zm;t). Fig. 2(b) illustrates Eq. (5). The fluctuating part represents the noise, and the envelope part is due to the moving periodic impulsive signal. Since the nonmoving periodic impulsive signal can be expressed as the combination of discrete pure tone signals: fhi ¼ i/DT, where i ¼ 1,2,3,y [6], the moving periodic impulsive signal with noise can be expressed by the combination of moving pure tones with noise. That is Pðxm ; ym ; zm ; f Þ ¼
1 X
Pf hi ðxm ; ym ; zm ; f Þ þ Nðxm ; ym ; zm ; f Þ.
(6)
i¼1
This makes it possible to extract Pf hi ðxm ; ym ; zm ; f Þ: the frequency component of each pure tone frequency. In other words, if we design a band pass filter which extracts Pf hi ðxm ; ym ; zm ; f Þ, then we can localize the moving periodic impulsive source. However, it is noteworthy that we may fail to localize the sound source if the noise within the band is significant. This problem motivates us to find the way to reduce the effect of noise, which is normally not likely possible in most case unless we have enough set of data to be averaged. 3. Moving periodic impulsive source localization The first step to apply the moving frame acoustic holography method to localizing the periodic impulsive source is, therefore to design the band pass filters. These enable us to separate each envelope part of the signal (Fig. 2(b)). Because we can regard the moving periodic impulsive signal as the combination of several moving pure tones, we can readily separate each band signal unless the two nearby bands overlap each other: the side band overlapping problem [1]. Let fhi be the ith pure tone component of periodic impulsive source and let us call this ith center frequency: fhi ¼ i/DT. The Dopplerized signal of ith center frequency have the frequency band between 1/(1+2M)fhi and 1/(12M)fhi [1], where M is Mach number. Then the filter passing the ith band can be written as ( i 1 i 1 1; 1þ2M DT of o 12M DT ; Bi ðf Þ ¼ (7) 0 otherwise:
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Within the frequency range that does not have the side band overlapping, that is, for the center frequency satisfying the following equation: 1 2M 1 . 4M DT Eq. (7) gives us the Dopplerized pure tone embedded by the band pass filtered noise: f hi o
(8)
Bi ðf ÞPðxm ; ym ; zm ; f Þ ¼ Pf hi ðxm ; ym ; zm ; f Þ þ Bi ðf ÞNðxm ; ym ; zm ; f Þ.
(9)
Fig. 3 illustrates the band pass filtering process especially for the third band. By using Eq. (1), Eq. (9) can be rewritten as Df ^ Ph ðkx ; ym ; zm ; f hi Þ þ Bi ðf ÞNðxm ; ym ; zm ; f Þ, (10) u where kx ¼ 2p(ffhi)/u in this case. Inserting Eq. (10) into Eq. (3) brings us the hologram with spatially distributed noise Nh(xm, ym, zm; fhi): Z u 1 1 Bi ðf ÞPðxm ; ym ; zm ; f Þejkx xh dkx ¼ Ph ðxh ; ym ; zm ; f hi Þ þ N h ðxh ; ym ; zm ; f hi Þ, (11) Df 2p 1 Bi ðf ÞPðxm ; ym ; zm ; f Þ ¼
where Nh(xm, ym, zm; fhi) can be written as Z u 1 1 N h ðxh ; ym ; zm ; f hi Þ ¼ Bi ðf ÞN h ðxm ; ym ; zm ; f Þejkx xh dkx . Df 2p 1
(12)
By using the conventional acoustic holography process [4,5], we can obtain the sound field on the source plane embedded by noise. To reduce the effect of noise, we can use distinctive characteristics of periodic impulsive signals and the noise. First, we can obtain several holograms for center frequencies fh1, fh2, y . Furthermore, whereas the sound field due to periodic impulsive sources has its maximum value at the source position, noise uncorrelated with the periodic impulsive signals is randomly distributed. These characteristics motivate us to average holograms: PN a ðxh ; ym ; 0Þ ¼
Na 1 X Ph ðxh ; ym ; 0; f hi Þ þ N h ðxh ; ym ; 0; f hi Þ , N a i¼1
(13)
where PN a ðxh ; ym ; 0Þ is the averaged hologram and Na is the number of hologram we can obtain. The effect of noise can be reduced through the average process, so that we can obtain the averaged hologram on the source
Fig. 3. The band pass filtering process for the third band.
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plane for high Na: PN a ðxh ; ym ; 0Þ
Na 1 X Ph ðxh ; ym ; 0; f hi Þ. N a i¼1
(14)
It is noteworthy that the sound pressure at a single frequency has spherical wave front. Thus Eq. (14) can be rewritten as PN a ðxh ; ym ; 0Þ
Na 1 P0 X ej2pf hi ðt0 þR=cÞ , N a R i¼1
(15)
where R is the distance between the sound source and (xh, ym, 0) on the hologram plane. Through the above equation, we can find that the averaged hologram has its maximum magnitude at the source position (RE0), which enables us to localize the moving periodic impulsive source. Therefore, we can summarize as follows. The periodic impulsive signal that is moving is found to be extracted by designing appropriate filter in the frequency domain. This is the solution of the first problem. The second problem that is related with noise, is found to be easily solved by averaging the hologram data that are normally available. Next attempt is to see whether this proposed means can practically work. 4. Example and conclusion A simple example: the single source case shows the effect of hologram average. The position of source is (Dxh, Dyh, zm) ¼ (5, 3, 0)m with respect to the hologram coordinate in Fig. 1, and the period of impulse is 0.1 s. The sound source is moving along the x-axis with the speed 3.4 m/s. The impulsive signal is imbedded by the white noise whose signal-to-noise ratio is 0 dB. Fig. 4(a and b) shows the moving frame acoustic holography results at f ¼ 120 Hz with and without noise, respectively. When there is no noise, the position of source can be easily found (Fig. 4(a)). However the effect of noise degrades the capability to detect the position of source (Fig. 4(b)). Fig. 5(a and b) shows the hologram average result within the frequency 120–240 Hz with and without noise. As shown in the figures, the effect of noise is considerably decreased compared with those without averaging process, so that we can find the position of source well. Fig. 5(c) shows the spatially averaged error [7] which shows the difference between the average result with and without noise. It can be seen that the error decreases as number of average increases. We can, therefore, conclude the following. We have proposed a method to localize the moving periodic impulsive source in a noisy environment. It was verified that the sound field from that moving periodic impulsive noise could be visualized through the moving frame acoustic holography method by using the band
Fig. 4. Hologram at f ¼ 120 Hz (a) when there is no noise and (b) when there exists noise.
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Fig. 5. Hologram average results between f ¼ 120 and 240 Hz (a) without noise, and (b) with noise of S/N ratio 0 dB, and (c) spatially averaged error with respect to number of averages.
pass filter. Also we have proposed a method to average holograms from several pure tone components to remove the effect of background noise. The method has been verified mathematically, and the feasibility of the proposed method was studied by the simulation. This method can be used to monitor whether or not there are lose parts on moving vehicles, including high-speed train. It can be also applied to monitor the assembly quality of products. Acknowledgment This study was supported by Korea Railroad Research Institute and the Ministry of Education and Human Resources Development of Korea _Brain Korea 21 project.
References [1] H.-S. Kwon, Y.-H. Kim, Moving frame technique for planar acoustic holography, Journal of the Acoustical Society of America 103 (4) (1998) 1734–1741. [2] S.-H. Park, Y.-H. Kim, An improved moving frame acoustic holography for coherent band-limited noise, Journal of the Acoustical Society of America 104 (6) (1998) 3179–3189. [3] S.-H. Park, Y.-H. Kim, Visualization of pass-by noise by means of moving frame acoustic holography, Journal of the Acoustical Society of America 110 (5) (2001) 2326–2339.
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[4] J.D. Maynard, E.G. Williams, Y. Lee, Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH, Journal of the Acoustical Society of America 78 (4) (1985) 1395–1413. [5] W.A. Veronesi, J.D. Maynard, Nearfield acoustic holography (NAH): II. Holographic reconstruction algorithms and computer implementation, Journal of the Acoustical Society of America 81 (5) (1987) 1307–1322. [6] A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1999. [7] Sea-Moon Kim, Yang-Hann Kim, Characteristics and errors of four acoustic holographies, Transactions of the Korean Society of Mechanical Engineers 19 (4) (1995) 950–967.
Localization of moving periodic impulsive source in a noisy environment Jong-Hoon Jeon, Yang-Hann Kim Center for Noise and Vibration Control (NOVIC), School of Mechanical, Aerospace & Systems Engineering, KAIST (Korea Advanced Institute of Science and Technology), 373-1 Science Town, Daejon-shi 305-701, Republic of Korea Received 21 June 2007; accepted 2 September 2007 Available online 8 September 2007
Abstract Sound generated from a machine often carries information on the condition of the machine. For example, the faults in the rotating part of a machine generate impulsive sound. This means that by visualizing the radiation pattern from a moving machine, the condition of the machine: where the fault is, can be examined. The moving frame acoustic holography method [H.-S. Kwon, Y.-H. Kim, Moving frame technique for planar acoustic holography, Journal of the Acoustical Society of America 103(4) (1998) 1734–1741.] enables us to visualize sound field from a moving source of pure tone or narrow band signal. This method cannot be applied directly, however, to the moving periodic impulsive sound because the impulsive signal is obviously different from the pure tone or narrow band signal. In this paper, we propose a means to use the method to localize moving periodic impulsive source that is embedded by noise. Moving periodic impulsive signal can be regarded as the combination of moving several discrete pure tones. This enables us to visualize the sound field at each frequency component. We also found that the signal to noise ratio can be readily improved by averaging several holograms. This is because periodic impulsive noise has several discrete frequency components. r 2007 Elsevier Ltd. All rights reserved. Keywords: Periodic impulsive noise; Source identification; Moving frame acoustic holography; Moving source; Noise
1. Motivation and objective The moving frame acoustic holography method [1,2] is a novel way to visualize the sound field generated by the moving sound source. As shown in Fig. 1, the method assumes the plane that has a sound source and is moving with a constant speed u to the positive x direction. In fact, the method can be extended to the case the speed u is not constant [3]. A microphone array is assumed to be on the ground. The method also presumes that there is a coordinate, which is moving with the source plane (the hologram coordinate). Let us denote Pf h0 ðxm ; ym ; zm ; f Þ to be the sound pressure measured by a fixed microphone array at the position (xm, ym, zm) due to the moving source having pure tone fh0. The origin of hologram coordinate is set to pass (xm, 0, 0) on the fixed coordinate when t ¼ 0. Then the method says that Pf h0 ðxm ; ym ; zm ; f Þ can be expressed by the sound Corresponding author. Tel.: +82 42 869 3025; fax: +82 42 869 8220.
E-mail addresses: [email protected] (J.-H. Jeon), [email protected] (Y.-H. Kim). 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.09.001
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754
Fig. 1. Pictorial expression of the moving sound source and fixed microphone array.
pressure with respect to the hologram coordinate in the wave number domain: Df ^ Ph ðkx ; ym ; zm ; f h0 Þ, (1) u where Df means frequency resolution of the measured signal and P^ h ðkx ; ym ; zm ; f h0 Þ means x directional wave number transform of the sound pressure on the hologram plane for the frequency fh0. kx means x directional wave number, and has relation with the frequency: Pf h0 ðxm ; ym ; zm ; f Þ ¼
2pðf f h0 Þ . (2) u Therefore the method predicts the sound pressure on the hologram plane by using the inverse spatial Fourier transform: Z 1 1 ^ Ph ðxh ; ym ; zm ; f h0 Þ ¼ Ph ðkx ; ym ; zm ; f h0 Þejkx xh dkx . (3) 2p 1 kx ¼
Once we have the hologram, we can localize the sound source by predicting the sound field on the source plane: Ph(xh,ym,0;fh0) [4,5]. However, the method is limited to the sound source that emits pure tone or narrow band signal. We have attempted to expand the method to localize the moving sound source that is generating periodic impulsive sound, especially in a noisy environment. In this situation, two major problems make it difficult to use the method directly. The first one is that the periodic impulsive signal is obviously different from the pure tone signal. Furthermore, the method does not consider the noise effect. The first problem can be readily solved if we find the relation between moving periodic impulsive signals and the moving pure tone signals. This motivates us to model and analyze the moving periodic impulsive signal first. 2. Signal modeling of moving periodic impulsive sound Suppose that we have a source that generates an impulse at time t ¼ t0. Its period is DT, and the rmi is the distance between the sound source and the microphone at (xm, ym, zm) when the source radiates ith impulsive sound. Then the sound pressure measured at the microphone positioning at (xm, ym, zm) can be written as 1 X P0 1 pðxm ; ym ; zm ; tÞ ¼ d t t0 þ ði 1ÞDT þ rmi (4) þ nðxm ; ym ; zm ; tÞ, c r i¼1 mi where P0 means the monopole amplitude assuming that the impulsive sound has spherical wave fronts, and n(xm, ym, zm;t) is the noise at the position (xm, ym, zm) and time t, and d( ) means Dirac delta function that models the impulse and c is the speed of sound. Fig. 2(a) shows the moving periodic impulsive signal with white noise that is uncorrelated with the periodic impulsive signal. The signal-to-noise ratio, which is defined
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Fig. 2. Moving periodic impulsive signal with white noise in (a) time domain and (b) frequency domain (signal-to-noise ratio is 0 dB).
by the ratio between the variance of the impulsive signal and noise, is 0 dB. The impulse period is 0.1 s and Mach number is 0.01. The Fourier transform of p(xm, ym, zm;t) can be written as Pðxm ; ym ; zm ; f Þ ¼
1 X P0 i¼1
rmi
ej2pf ðt0 þði1ÞDTþð1=cÞrmi Þ þ Nðxm ; ym ; zm ; f Þ,
(5)
where N(xm, ym, zm; f) is the Fourier transform of n(xm, ym, zm;t). Fig. 2(b) illustrates Eq. (5). The fluctuating part represents the noise, and the envelope part is due to the moving periodic impulsive signal. Since the nonmoving periodic impulsive signal can be expressed as the combination of discrete pure tone signals: fhi ¼ i/DT, where i ¼ 1,2,3,y [6], the moving periodic impulsive signal with noise can be expressed by the combination of moving pure tones with noise. That is Pðxm ; ym ; zm ; f Þ ¼
1 X
Pf hi ðxm ; ym ; zm ; f Þ þ Nðxm ; ym ; zm ; f Þ.
(6)
i¼1
This makes it possible to extract Pf hi ðxm ; ym ; zm ; f Þ: the frequency component of each pure tone frequency. In other words, if we design a band pass filter which extracts Pf hi ðxm ; ym ; zm ; f Þ, then we can localize the moving periodic impulsive source. However, it is noteworthy that we may fail to localize the sound source if the noise within the band is significant. This problem motivates us to find the way to reduce the effect of noise, which is normally not likely possible in most case unless we have enough set of data to be averaged. 3. Moving periodic impulsive source localization The first step to apply the moving frame acoustic holography method to localizing the periodic impulsive source is, therefore to design the band pass filters. These enable us to separate each envelope part of the signal (Fig. 2(b)). Because we can regard the moving periodic impulsive signal as the combination of several moving pure tones, we can readily separate each band signal unless the two nearby bands overlap each other: the side band overlapping problem [1]. Let fhi be the ith pure tone component of periodic impulsive source and let us call this ith center frequency: fhi ¼ i/DT. The Dopplerized signal of ith center frequency have the frequency band between 1/(1+2M)fhi and 1/(12M)fhi [1], where M is Mach number. Then the filter passing the ith band can be written as ( i 1 i 1 1; 1þ2M DT of o 12M DT ; Bi ðf Þ ¼ (7) 0 otherwise:
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Within the frequency range that does not have the side band overlapping, that is, for the center frequency satisfying the following equation: 1 2M 1 . 4M DT Eq. (7) gives us the Dopplerized pure tone embedded by the band pass filtered noise: f hi o
(8)
Bi ðf ÞPðxm ; ym ; zm ; f Þ ¼ Pf hi ðxm ; ym ; zm ; f Þ þ Bi ðf ÞNðxm ; ym ; zm ; f Þ.
(9)
Fig. 3 illustrates the band pass filtering process especially for the third band. By using Eq. (1), Eq. (9) can be rewritten as Df ^ Ph ðkx ; ym ; zm ; f hi Þ þ Bi ðf ÞNðxm ; ym ; zm ; f Þ, (10) u where kx ¼ 2p(ffhi)/u in this case. Inserting Eq. (10) into Eq. (3) brings us the hologram with spatially distributed noise Nh(xm, ym, zm; fhi): Z u 1 1 Bi ðf ÞPðxm ; ym ; zm ; f Þejkx xh dkx ¼ Ph ðxh ; ym ; zm ; f hi Þ þ N h ðxh ; ym ; zm ; f hi Þ, (11) Df 2p 1 Bi ðf ÞPðxm ; ym ; zm ; f Þ ¼
where Nh(xm, ym, zm; fhi) can be written as Z u 1 1 N h ðxh ; ym ; zm ; f hi Þ ¼ Bi ðf ÞN h ðxm ; ym ; zm ; f Þejkx xh dkx . Df 2p 1
(12)
By using the conventional acoustic holography process [4,5], we can obtain the sound field on the source plane embedded by noise. To reduce the effect of noise, we can use distinctive characteristics of periodic impulsive signals and the noise. First, we can obtain several holograms for center frequencies fh1, fh2, y . Furthermore, whereas the sound field due to periodic impulsive sources has its maximum value at the source position, noise uncorrelated with the periodic impulsive signals is randomly distributed. These characteristics motivate us to average holograms: PN a ðxh ; ym ; 0Þ ¼
Na 1 X Ph ðxh ; ym ; 0; f hi Þ þ N h ðxh ; ym ; 0; f hi Þ , N a i¼1
(13)
where PN a ðxh ; ym ; 0Þ is the averaged hologram and Na is the number of hologram we can obtain. The effect of noise can be reduced through the average process, so that we can obtain the averaged hologram on the source
Fig. 3. The band pass filtering process for the third band.
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plane for high Na: PN a ðxh ; ym ; 0Þ
Na 1 X Ph ðxh ; ym ; 0; f hi Þ. N a i¼1
(14)
It is noteworthy that the sound pressure at a single frequency has spherical wave front. Thus Eq. (14) can be rewritten as PN a ðxh ; ym ; 0Þ
Na 1 P0 X ej2pf hi ðt0 þR=cÞ , N a R i¼1
(15)
where R is the distance between the sound source and (xh, ym, 0) on the hologram plane. Through the above equation, we can find that the averaged hologram has its maximum magnitude at the source position (RE0), which enables us to localize the moving periodic impulsive source. Therefore, we can summarize as follows. The periodic impulsive signal that is moving is found to be extracted by designing appropriate filter in the frequency domain. This is the solution of the first problem. The second problem that is related with noise, is found to be easily solved by averaging the hologram data that are normally available. Next attempt is to see whether this proposed means can practically work. 4. Example and conclusion A simple example: the single source case shows the effect of hologram average. The position of source is (Dxh, Dyh, zm) ¼ (5, 3, 0)m with respect to the hologram coordinate in Fig. 1, and the period of impulse is 0.1 s. The sound source is moving along the x-axis with the speed 3.4 m/s. The impulsive signal is imbedded by the white noise whose signal-to-noise ratio is 0 dB. Fig. 4(a and b) shows the moving frame acoustic holography results at f ¼ 120 Hz with and without noise, respectively. When there is no noise, the position of source can be easily found (Fig. 4(a)). However the effect of noise degrades the capability to detect the position of source (Fig. 4(b)). Fig. 5(a and b) shows the hologram average result within the frequency 120–240 Hz with and without noise. As shown in the figures, the effect of noise is considerably decreased compared with those without averaging process, so that we can find the position of source well. Fig. 5(c) shows the spatially averaged error [7] which shows the difference between the average result with and without noise. It can be seen that the error decreases as number of average increases. We can, therefore, conclude the following. We have proposed a method to localize the moving periodic impulsive source in a noisy environment. It was verified that the sound field from that moving periodic impulsive noise could be visualized through the moving frame acoustic holography method by using the band
Fig. 4. Hologram at f ¼ 120 Hz (a) when there is no noise and (b) when there exists noise.
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Fig. 5. Hologram average results between f ¼ 120 and 240 Hz (a) without noise, and (b) with noise of S/N ratio 0 dB, and (c) spatially averaged error with respect to number of averages.
pass filter. Also we have proposed a method to average holograms from several pure tone components to remove the effect of background noise. The method has been verified mathematically, and the feasibility of the proposed method was studied by the simulation. This method can be used to monitor whether or not there are lose parts on moving vehicles, including high-speed train. It can be also applied to monitor the assembly quality of products. Acknowledgment This study was supported by Korea Railroad Research Institute and the Ministry of Education and Human Resources Development of Korea _Brain Korea 21 project.
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