Partial field decomposition of multi-source cyclostationary sound field

Applied Acoustics 73 (2012) 524–528

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Technical Note

Partial field decomposition of multi-source cyclostationary sound field Zhimin Chen ⇑, Haichao Zhu, Rongfu Mao Institute of Noise & Vibration, Naval University of Engineering, Wuhan 430033, PR China

a r t i c l e

i n f o

Article history: Received 8 July 2011 Received in revised form 12 October 2011 Accepted 14 October 2011 Available online 26 December 2011 Keywords: Cyclostationary sound field Near-field acoustic holography Cyclic spectral density Singular value decomposition Partial field decomposition

a b s t r a c t The cyclostationary sound field is a kind of special non-stationary field which has obvious modulation phenomena. Conventional planar near-field acoustic holography (PNAH) technique cannot exactly reflect its modulation characteristics. If the cyclic spectral density (CSD) instead of the complex sound pressure is adopted as the reconstruction variable, the modulating wave and carrier wave components of the cyclostationary sound field can be extracted effectively. A new technique called cyclostationary PNAH utilizing the CSD is proposed in this paper. Based on this technique, the problem of partial field decomposition by singular value decomposition (SVD) of multiple incoherent cyclostationary sound sources is researched. The results of numerical simulation and experiments show that the CPNAH technique and SVD decomposition method are effective. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction

2. The characteristics of CSD

Near-field acoustic holography (NAH) is a noise source identification technique which has proved to be effective in practice [1,2]. In NAH the sound field is often assumed to be stationary. This assumption is feasible for many applications but is inappropriate for others, for example, when the noise source is produced by a rotation machinery, because the radiated sound field of rotation machinery is usually periodically time-variant [3–5]. If the field is treated as stationary, the information relating to the change of frequency with time will be lost. The hologram obtained by NAH thus cannot reflect the nature of acoustic field in this case [6,7]. Sound field with periodically time-variant nature is called cyclostationary sound field, to which the analyzing method of second-order cyclic statistics is often applied. Second-order cyclic statistics includes cyclic autocorrelation function and cyclic spectral density (CSD). The CSD can extract the periodical components of cyclostationary sound field, while retain the phase information of signals [8,9], and suppress disturbing noises effectively. In this paper a technique called cyclostationary planar NAH (CPNAH) is presented to identify the noise source of cyclostationary sound field. The CSD instead of complex sound pressure is chosen as the reconstruction variable. A method based on singular value decomposition (SVD) is used to decompose the partial fields of multiple incoherent cyclostationary sound sources. Numerical simulation and experimental results have demonstrated the effectiveness of the method.

2.1. Phase information

⇑ Corresponding author. Tel.: +86 27 83442676; fax: +86 27 83443981. E-mail address: [email protected] (Z. Chen). 0003-682X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2011.10.006

Both CSD and power spectral density (PSD) are all physical quantities indicating the energy of signals. The use of PSD means that the signal is supposed to be stationary, and its phases are uniformly distributed within the range of [p, p]. The stationarity is derived by averaging the signal phases, which eliminates the information about phases. By contrast, the use of CSD means that the signal is periodical in time domain and its time-variant autocorrelation function is derived by time average. A typical cyclostationary signal can be expressed as

xðtÞ ¼ aðtÞ cosð2pf0 t þ hÞ

ð1Þ

where a(t) is a real random stationary signal with the average of zero and h is the initial phase of the carrier wave signal. The cyclic autocorrelation function of the signal is

81 a¼0 > < 2 Ra ðsÞ cosð2pf0 sÞ; 1 j2h Rax ðsÞ ¼ hxðtÞxðt þ sÞej2pat it ¼ R ð s Þe ; a ¼ 2f 0 4 a > : 0; otherwise ð2Þ where hit denotes time average and Ra(s) is the autocorrelation function of a(t). Eq. (2) indicates that when the cyclic frequency a = 0, the cyclic auto-correlation function degrades to the auto -correlation function and the phase information disappears. But at the cyclic frequency ±2f0, the initial phase is retained. The phase information is also retained in the CSD because it is the Fourier transform of cyclic auto-correlation function.

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Z. Chen et al. / Applied Acoustics 73 (2012) 524–528

h

2.2. Time delay

Sarp ðf Þ

i KQ

  ¼ Sarr ðf Þ KK

h

 ai GH f  2 KQ

ð7Þ

The PSD is insensitive to time delay, whereas the CSD is sensitive due to the presence of cyclic frequency domain. Suppose a signal v(t) = x(t  sn), where sn is the time delay. The CSD of v(t) is

h

Sav ðf Þ ¼ Sax ðf Þej2pasn

where the superscript H denotes the Hermite transpose operation, ½Gðf þ a2ÞQ K and ½GH ðf  a2ÞKQ are the spectral functions of the transfer functions which are moved a2 toward the negative and positive direction of the frequency axis respectively. Combining Eqs. (6)–(8) gives

ð3Þ

from which it can be seen that a delay factor ej2pasn exists between the initial signal and the delayed signal. 3. PNAH of cyclostationary sound field

ð5Þ

In the CPNAH technique presented in this paper, time delay between scanning steps is eliminated by using Eq. (5), and the signal phases at the cyclic frequencies are retained. So the spatial phases of data matrices on the two-dimensional hologram surface can be derived. Furthermore, by using the PNAH technique, and adopting the CSD instead of complex sound pressure as the reconstruction variable, the CSD of reconstruction surfurce can be obtained. The CPNAH technique differs from the PNAH technique mainly in that the reference microphone is used to eliminate the time delay and the reconstruction result is the CSD of sound pressure. 4. The CPNAH for multiple sound sources Due to partial coherence between multiple sound sources, the phase relation between the measurement points on the hologram surface usually changes irregularly. In this case, multiple reference microphones allocated near the sources are needed to decompose a composite sound source into mutually incoherent partial fields by using the sound field decomposition technique. Then the sound field reconstruction technique can be applied to each partial source. Suppose that the total number of holography measuring points and the reference microphones is Q and K respectively, and suppose that the cross-spectral matrix of the signals at the measurement points and the reference points are respectively ½Sapr ðf ÞQ K and ½Sarp ðf ÞKQ in which the time delay has been eliminated, and that the cyclic cross-spectral matrix of the signals at these points are respectively ½Sapp ðf ÞQ Q and ½Sarr ðf ÞKK . By using the method of transfer function, the following expressions can be derived,

h

i Sapr ðf Þ

Q K

h  ai  a  ¼ G fþ S ðf Þ KK 2 Q K rr

QQ

h  ai  a  h H  ai ¼ G fþ S ðf Þ KK G f  2 QK rr 2 KQ

ð8Þ

ð9Þ

Applying SVD to the cross-spectral matrix of the reference signals gives

½Sarr ðf ÞKK ¼ USV H

ð10Þ

where S is the diagonal matrix whose elements are singular values, U and V are unitary matrices. So the cyclic cross-spectral matrix of the partial sound field can be obtained, i.e. 1 a ½Saipp ðf ÞQ Q ¼ ½Sapr ðf ÞQ K VD1 i U ½Srp ðf ÞKQ

ð11Þ

where Di is a matrix with the (i, i) th element being the ith singular value and the other elements being zero. The diagonal matrix of Eq. (11) is the cyclic auto-spectral density of the partial sound field of field signals.

ð4Þ

Let the CSD of the pressure measured by the microphone array at the nth scanning step be Sap^n ðf Þ. After eliminating the effect of time delay, the CSD can be written as

Sapn ðf Þ ¼ Spa^n ðf ÞS^ra1 ðf Þ=S^ran ðf Þ

i

a ½Sapp ðf ÞQQ ¼ ½Sapr ðf ÞQ K ½Sarr ðf Þ1 KK ½Srp ðf ÞKQ

In conventional PNAH technique which uses complex sound pressure as the reconstruction variable, the spatial phase of measurement points in the hologram surface is derived by scanning over the plane by a microphone array and using the phase of a fixed point as reference. When the complex pressure is replaced by the CSD as the reconstruction variable, the time delay produced by the time interval between two close scanning steps must be eliminated to ensure spatial phase relationship of the two-dimensional holography data matrix. The delay factor between two close scanning steps can be obtained using a fixed reference microphone in scanning process. Suppose the pressure measured by the reference microphone at the nth scanning step is ^r n ðtÞ ¼ rðt  sn Þ with s1 = 0 and n = 1:N where N is total step number. The CSD of ^r n ðtÞ is S^ran ðf Þ from which the delay factor can be obtained by the following expression

ej2pasn ¼ S^ran ðf Þ=S^ra1 ðf Þ

Sapp ðf Þ

ð6Þ

5. Numerical simulation The sound field used in the numerical simulation is produced by two rigid square pistons embedded in the II and IV quadrant of an infinite baffle respectively. The velocities of the sources are

x1 ðtÞ ¼ a1 ðtÞ cosð2pf0 tÞ

ð12Þ

x2 ðtÞ ¼ a2 ðtÞ cosð2pf0 tÞ

ð13Þ

where a1(t) and a2(t) are random white noises with mean values of zero, f0 = 200 Hz is the carrier wave frequency. Both sources are of typical incoherent cyclostationary signals with the only one cyclic frequency a = 400 Hz. The random amplitude modulation signals can be demodulated by this frequency. The sizes of both pistons are 0.1 m  0.1 m. The surfaces of the pistons are meshed by 10  10 rectangular grids. The sound pressure in time domain at the piston surfaces is calculated by discrete Rayleigh integral. The coordinates of the pistons are (0.5, 0.5, 0) and (0.5, 0.5, 0) respectively in the XOY plane which lies in the baffle plane. The hologram surface of 2 m  2 m is 0.1 m away from the source surface, with 26  26 grids. The reconstruction surface is 0.05 m away from the source surface. The signals are sampled at the frequency of 2048 Hz and with the length of 5 s. Three reference microphones are located in the same plane 0.02 m away from the source surface, their coordinates in the XOY plane being coincident with the centers of pistons and the origin respectively. The snapshot method is used in the simulation. The cyclostationary analysis of the piston sources is implemented with the cyclic frequency a = 400 Hz and spectral frequency f = 40 Hz. The CSD of the sound pressure at the hologram surface is shown in Fig. 1. Applying SVD by Eq. (10) to the incoherent sources at the hologram surface, the two partial fields are decomposed as shown in Figs. 2 and 3. The effectiveness of SVD is shown in Fig. 4, from which it can be seen that multiple incoherent cyclostationary composite sources can be decomposed into mutually incoherent partial fields by SVD.

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Z. Chen et al. / Applied Acoustics 73 (2012) 524–528 0.9

0.3

partial field 1 0.8

0.4

partial field 2

0.25

total of calculation

S/Pa2.Hz-1

0.3

0.2

0.2 0.15 0.1 0.1 0 1 0.5

1 0

0.05

0.5

Y/m

-0.5 -1

X/m

-1

total in theory

0.6 0.5 0.4 0.3 0.2 0.1

0

-0.5

Amplitude/Pa2.Hz.-1

0.7

0 -1

-0.5

0

0.5

1

X/m

Fig. 1. Amplitude distribution of CSD on hologram surface (a = 400 Hz, f = 40 Hz).

Fig. 4. Decomposition effectiveness of SVD.

0.3

0.25

source, the distances and coupling between reference microphones and sound sources, etc. To further understand the SVD decomposition of stationary field, references [11–13] may be referable.

0.2

6. Experiment

0.4

S/Pa2.Hz-1

0.3 0.2 0.15 0.1 0.1

0 1 0.5

1

0.05

0.5

0

Y/m

0

-0.5

-0.5 -1

X/m

-1

Fig. 2. The first partial field.

0.12 0.2 0.1

S/Pa2.Hz-1

0.15 0.08 0.1 0.06

0.05

0.04

0 1 0.5

1 0.5

0

Y/m

0

-0.5

-0.5 -1

0.02

X/m

-1

Fig. 3. The second partial field.

The decomposition of multiple cyclostationary sources by SVD differs that of stationary sources mainly in the preprocess method of hologram surface sound field. Due to the different nature of CSD and PSD, the sound field decomposition methods are inherently different. But the factors causing error in SVD of CPNAH is similar to Hald’s method [10] in which the factors include the coupling between reference microphones, the amplitude of each partial

The experiments to prove the method presented in this paper are carried out in a laboratory of about 12 m  20 m  5 m. The reverberation time of the laboratory is represented in Table 1 and the background noise is 50 dB. From that, the acoustics condition of the laboratory has the small effect on the experiment, but experiment results are acceptable. The two incoherent cyclostationary sources defined by Eqs. (12) and (13) are produced by signal generator and then output to two loudspeakers spaced 14 cm away. The hologram surface is 4 cm away from the loudspeakers and the sound pressure is measured by a scanning microphone array. Three reference microphones are located at the right front of the loudspeakers. The holography measurement points comprise of 13 horizontal points and 11 vertical points which are spaced 5 cm in each direction. The signals are sampled at the frequency of 4096 Hz and with the length of 10 s. The experimental scene is shown in Fig. 5. The delay factors of each scanning step can be calculated by Eq. (4) either through data from one of the reference microphones or through the mean value of the data of all the reference microphones. The decomposition of sound field is implemented by Eqs. (10) and (11) in which the effect of time delay must be eliminated. The analysis frequencies are chosen as a = 400 Hz and f = 90 Hz. The amplitude of the CSD of sound pressure on the hologram surface is shown in Fig. 6. The decomposed partial fields of the loudspeakers are shown in Figs. 7 and 8 respectively. The decomposition effectiveness is shown in Fig. 9. The composite sources of the hologram surface is decomposed into two individual sources which can be reconstructed. The reconstruction plane is at the loudspeakers surface. The results of reconstruction are shown in Figs. 10 and 11 which shows that, despite

Table 1 The reverberation time. Central frequency of 1/3 octave (Hz)

63

125

250

500

1000

Reverberation time (s)

1.4

1.6

1.5

1.5

1.5

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Z. Chen et al. / Applied Acoustics 73 (2012) 524–528 0.25

0.025

0.2 0.15

0.02

0.1

Y/m

0.05

0.015

0 -0.05

0.01

-0.1 -0.15

0.005

-0.2 -0.25 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

X/m Fig. 5. Scene of experiment.

Fig. 8. The second partial field on hologram surface.

0.25

0.035

partial field 1 partial field 2 total of calculation total of measurement

0.04

0.2 0.03

0.035

0.15 0.1

0.02 0

-0.05

0.015

-0.1 0.01

-0.15

Amplitude/Pa2.Hz-1

0.05

Y/m

0.03

0.025

0.025 0.02 0.015 0.01

-0.2

0.005

-0.25 -0.3

-0.2

-0.1

0

0.1

0.2

0.005

0.3

0

X/m

-0.2

-0.1

0

0.1

0.2

0.3

X/m

Fig. 6. Amplitude distribution of CSD on hologram surface (a = 400 Hz, f = 90 Hz). Fig. 9. Decomposition effectiveness of SVD.

0.25

0.03

0.25

0.055

0.2

0.05

0.15

0.045

0.2 0.025

0.15 0.1

0.02

0.015

-0.05 -0.1

0.01

Y/m

Y/m

0.035

0.05

0

0.03

0

0.025 -0.05

0.02

-0.1

-0.15 0.005

-0.2 -0.25 -0.3

0.04

0.1

0.05

0.015

-0.15

0.01

-0.2 -0.2

-0.1

0

0.1

0.2

0.3

X/m Fig. 7. The first partial field on hologram surface.

0.005

-0.25 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

X/m Fig. 10. Result of reconstruction in the first partial field.

interference still exists between two sources, the effect of SVD decomposition is acceptable and the incoherent sources are decomposed successfully. The factors causing interference between the decomposed sources is that the amplitude of two

loudspeakers sources is equal and the coupling exits in two reference microphones.

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0.045

References

0.2

0.04

[1] Maynard JD, Williams EG, Lee Y. Near field acoustic holography I. Theory of generalized holography and the development of NAH. J Acoust Soc Am 1985;78(4):1395–413. [2] Veronesi WA, Maynard JD. Near-field acoustic holography (NAH) II: holographic reconstruction, algorithms and computer implementation. J Acoust Soc Am 1987;81(5):1307–22. [3] Randall RB, Antoni J, Chobsaard S. Comparison of cyclostationary and envelope analysis in the diagnostics of rolling element bearings. ICASSP. In: IEEE international conference on acoustics, speech and signal processing – Proceedings, vol. 6; 2000. p. 3882–5. [4] Randall RB, Antoni J, Chobsaard S. The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals. Mech Syst Signal Process 2001;15(5):945–62. [5] Antoniadis I, Glossiotis G. Cyclostationary analysis of rolling-element bearing vibration signals. J Sound Vib 2001;248(5):829–45. [6] Wan Q, Jiang WKT. He near field acoustic holography technique for cyclostationary sound field and its experimental research. Chinese J Acoust 2005;3:263–70. [7] Chen ZM, Zhu HC, Mao RF. Research on localization of the source of cyclostationary sound field. Acta Physica Sinical 2011;60(10):104305. [8] Prakriya S. Blind identification of nonlinear systems based on higher order cyclic spectra. PhD dissertation, University of Toronto (Canada); 1997. [9] Smith L. Blind channel identification and equalization using second-order cyclostationarity. PhD dissertation, The Pennsylvania State University; 1996. [10] Hald J. STSF – A unique technique for scan-based near-field acoustic holography without restrictions on coherence. B&K Technical Review, vol. 1; 1989. p. l–49. [11] Hallman DL, Bolton JS. A comparison of multi-reference near-field acoustical holography procedures. Proc NOISE-CON 94; 1994. p. 929–34. [12] Kwon HS, Bolton JS. Partial field decomposition in near-field acoustical holography by the use of singular value decomposition and partial coherence procedures. Proc NOISE-CON 98; 1998. p. 649–54. [13] Tomlinson MA. Partial source discrimination in near field acoustic: holography. Appl Acoust 1999;57:243–61.

0.15

0.035

0.1

0.03

Y/m

0.05 0.025 0

0.02

-0.05

0.015

-0.1 -0.15

0.01

-0.2

0.005

-0.25 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

X/m Fig. 11. Result of reconstruction in the second partial field.

7. Conclusions The CSD of the sound pressure instead of complex pressure is used as the reconstruction variable and a new method called CPNAH is presented in this paper to reconstruct the cyclostationary sound field. A decomposition method based on SVD is developed to decompose the sound field comprised of multiple incoherent cyclostationary sources. Numerical simulation and experimental results show that the method is effective in the multiple incoherent cyclostationary sources decomposition problem. The reference microphone allocation principles adopted by the SVD of stationary sound field are also applicable in cyclostationary sound field.