NAH method in non-free sound field reconstruction of the sound source will lead to considerable error. This paper proposes a method to reconstruct the sound source in a closed space. In the closed space, the total sound pressure is the sum of the source radiation pressure and the reflected sound pressure from medium interface. Reflections from medium interface is a convolution noise, rather than additive noise. In order to reconstruct a sound source in the closed space, in this paper we first adopt the complex cepstrum method to separate and radiation pressure and the reflected sound pressure, then filter them to reduction of reflected sound pressure, and then reconstruct the sound source. The simulation results show the correctness and effectiveness of the method

Duo Qu

Jingjun Lou

Xianglin Tian

Yangyang Zhang

English

Vibroengineering PROCEDIA

 50 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 9. ISSN 2345-0533  
The NAH based on complex cepstrum method in a 
closed space 
Xianglin Tian1, Jingjun Lou2, Yangyang Zhang3, Duo Qu4 
1, 3, 4College of Power Engineering, Naval University of Engineering, Wuhan 430033, China 
2Scientific Research Ministry of Naval University of Engineering, Wuhan 430033, China 
1Corresponding author 
E-mail: 1txl19830107@126.com, 2jingjun_lou@hotmail.com, 315871778743@yeah.net, 
4312237642@qq.com 
(Received 19 September 2016; accepted 20 September 2016) 
Abstract. NAH method in non-free sound field reconstruction of the sound source will lead to 
considerable error. This paper proposes a method to reconstruct the sound source in a closed space. 
In the closed space, the total sound pressure is the sum of the source radiation pressure and the 
reflected sound pressure from medium interface. Reflections from medium interface is a 
convolution noise, rather than additive noise. In order to reconstruct a sound source in the closed 
space, in this paper we first adopt the complex cepstrum method to separate and radiation pressure 
and the reflected sound pressure, then filter them to reduction of reflected sound pressure, and 
then reconstruct the sound source. The simulation results show the correctness and effectiveness 
of the method. 
Keywords: enclosed space, the complex cepstrum method, the sound source reconstruction, NAH. 
1. Introduction 
In the early 1980s, american scholar E. G Williams put forward the concept [1, 2]. NAH 
considering the effect of evanescent waves, breaking the Rayleigh criterion limiting the resolution 
of reconstruction in recent years has been widely used. NAH is different from of the far field 
acoustic holography. In the holographic surface near the sound source surface, the measurement 
data is recorded. The sound pressure is then reconstruct three-dimensional sound field, sound 
intensity, and particle velocity by the Fourier technology. Since the limit of the two 
-dimensional Fourier transform itself, planar NAH required the measured surface should be 
conformal of the sound source [3], in order to make NAH can be applied to any surface shape of 
the sound source, the boundary element method [4] have been proposed. Xinzhao Chen propose 
the distributed source boundary point method [5], further simplifying the tedious calculations 
boundary element method for the coefficient matrix. In order to improve the accuracy of the 
reconstruction effectively, many scholars in-depth study of the data interpolation and 
extrapolation methods [6-10]. 
Planar NAH basing on spatial sound field transformation requires sound field to be  
free-field measurements, which severely limits the planar NAH applications. Dual holographic 
[11] located on different sides of the holographic interference source separation plane NAH further 
expand its application, but the application plane NAH reconstruction of the sound source in an 
enclosed space will still produce large errors. In this paper, the complex cepstrum preprocess the 
measured data before reconstruction, to eliminate or weaken the effects of reflected sound. 
2. The theory of the planar acoustic holography 
The acoustic wave propagation satisfies the equation [12] in the three-dimensional ideal fluid 
medium: 
ߩ଴
∂ݒԦ(ݎ, ݐ)
∂ݐ = −∇݌(ݎ, ݐ) , − ߩ଴݀݅ݒݒԦ(ݎ, ݐ) =
∂ߩᇱ(ݎ, ݐ)
∂ݐ , ݌(ݎ, ݐ) = ܿ
ଶߩᇱ(ݎ, ݐ). (1)
The solutions of the above three equations is: 
THE NAH BASED ON COMPLEX CEPSTRUM METHOD IN A CLOSED SPACE.  
XIANGLIN TIAN, JINGJUN LOU, YANGYANG ZHANG, DUO QU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 9. ISSN 2345-0533 51 
∇ଶ݌(ݔ, ݕ, ݖ, ݐ) = 1ܿ଴ଶ
∂ଶ݌(ݔ, ݕ, ݖ, ݐ)
∂ଶݐ . (2)
Fourier transform on both sides, we can obtain: 
∇ଶ݌(ݔ, ݕ, ݖ) + ݇ଶ݌(ݔ, ݕ, ݖ) = 0, (3)
where ݌(ݔ, ݕ, ݖ) is the frequency domain pressure; ݇ = ߱/ܿ is the wave number; ߱ is the circular 
frequency; ܿ is the speed of sound. 
The formula is Helmhotz equation, the equation always hold to steady sound. The solution of 
Eq. (3) can be expressed as the Kirchhoff integral: 
݌(ݔ, ݕ, ݖ) = ඵ ቈ݃(ݎ, ݎௌ)
∂݌(ܵ)
∂݊ − ݌(ܵ)
∂݃(ݎ, ݎௌ)
∂݊ ቉
ௌ
݀ݏ, (4)
where ݏ is a closed surface, ݎ is the distance from the point to the origin from the field, ݎௌ is the 
distance from the sound source surface to the point in the sound field, ݌(ܵ) is the complex sound 
pressure, ݃(ݎ, ݎௌ) is Green’s function. 
In an infinite radius hemispherical surface of the sound field, Eq. (4) can be expressed as: 
݌(ݔ, ݕ, ݖ) = ඵ ቈ݃(ݎ, ݎௌ)
∂݌(ܵ)
∂݊ − ݌(ܵ)
∂݃(ݎ, ݎௌ)
∂݊ ቉
ௌ೥ೞ
݀ݏ
      + ඵ ቈ݃(ݎ, ݎௌ)
∂݌(ܵ)
∂݊ − ݌(ܵ)
∂݃(ݎ, ݎௌ)
∂݊ ቉
ௌಮ
݀ݏ.
(5)
Semi-free sound field infinity meet sommerfeld radiation conditions, Eq. (5) in the second 
integral is zero, so Eq. (5) simplifies to: 
݌(ݔ, ݕ, ݖ) = ඵ ቈ݃(ݎ, ݎௌ)
∂݌(ܵ)
∂݊ − ݌(ܵ)
∂݃(ݎ, ݎௌ)
∂݊ ቉
ௌ೥ೞ
݀ݏ. (6)
In actual sound field, the sound pressure distribution satisfies Dirichlet conditions, the Green’s 
function is: 
ቐ
݃(ݎ, ݎௌ)|௥ೄ = 0,
− ∂݃(ݎ, ݎௌ)∂݊ = ݃஽(ݎ, ݎௌ).
(7)
Meet the conditions for the Green’s function expression: 
݃஽(ݎ, ݎௌ) =
ݖ(1 − ݅݇ݎᇱ)݁௜௞௥
2ߨݎᇱଷ , (8)
where ݎᇱ is the distance from the sound source surface, Eq. (6) simplifies to: 
݌(ݎ) = ඵ ݌(ܵ)
ௌ೥ೞ
݃஽(ݎ, ݎௌ)݀ݏ. (9)
In the Cartesian coordinate system, Eq. (9) can be express as: 
THE NAH BASED ON COMPLEX CEPSTRUM METHOD IN A CLOSED SPACE.  
XIANGLIN TIAN, JINGJUN LOU, YANGYANG ZHANG, DUO QU 
52 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 9. ISSN 2345-0533  
݌(ݔ, ݕ, ݖ) = ඵ ݌(ݔ௭௦, ݕ௭௦, ݖ௭௦)
ௌ೥ೞ
݃஽(ݔ − ݔ௭௦, ݕ − ݕ௭௦, ݖ − ݖ௭௦)݀ݏ. (10)
According to the convolution theorem there is: 
෠ܲ(݇௫, ݇௬, ݖ) = ෠ܲ(݇௫, ݇௬, ݖ௦)ܩ෠஽(݇௫, ݇௬, ݖ − ݖ௦) (11)
where: 
ܩ෠஽(݇௫, ݇௬, ݖ − ݖ௦) =
ۖە
۔
ۖۓexp ቆ݅(ݖ − ݖ௦)ට݇ଶ − ൫݇௫ଶ + ݇௬ଶ൯ቇ, ݇ଶ > ݇௫ଶ + ݇௬ଶ,
exp ቆ−(ݖ − ݖ௦)ට൫݇௫ଶ + ݇௬ଶ൯ − ݇ଶቇ, ݇ଶ < ݇௫ଶ + ݇௬ଶ.
(12)
From Eq. (12) can be introduced: 
෠ܲ൫݇௫, ݇௬, ݖ௦൯ = ෠ܲ൫݇௫, ݇௬, ݖு൯ܩ෠஽ି ଵ൫݇௫, ݇௬, ݖு − ݖ௦൯. (13)
Calculated in accordance with Eq. (13) can reconstruct surface sound pressure angular 
spectrum, then the two-dimensional inverse Fourier transform to seek sound pressure. 
Sound field model in the enclosed space and complex cepstrum technology  
Due to the presence of free-field conditions strictly, sound field approximation we encounter 
are generally enclosed spaces non-free field. In an enclosed space, sound waves are medium 
interface reflection, radiation, the presence of reflected waves in the sound field. In the closed 
space of the sound in a schematic view [13] as shown in Fig. 1. 
 
Fig. 1. Enclosed space sound field schematic 
As can be seen from Fig. 1: 
ݔ(ܯ) = ݑ(ܯ) + ෍ ߩ௜
ஶ
௜ୀ଴
ݏ(ܯ − ݅ܯ௜), (14)
where ݑ(ܯ) is the noise, ߩ௜ is the Reflection coefficient, ݏ(ܯ − ݅ܯ௜) is the signal of reflection, 
ݏ(ܯ) is the signal of sound source, ݔ(ܯ) is the total signal. Ignore the noise, then Eq. (14) can 
express: 
ݔ(ܯ) = ෍ ߩ௜
ஶ
௜ୀ଴
ݏ(ܯ − ݅ܯ௜). (15)
THE NAH BASED ON COMPLEX CEPSTRUM METHOD IN A CLOSED SPACE.  
XIANGLIN TIAN, JINGJUN LOU, YANGYANG ZHANG, DUO QU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 9. ISSN 2345-0533 53 
Convolution represented by the formula: 
ݔ(ܯ) = ݏ(ܯ) ∗ ෍ ߩ௜
ஶ
௜ୀ଴
ߜ(ܯ − ݅ܯ௜), (16)
ݔ(ܯ) = ݏ(ܯ) ∗ ℎ(ܯ), (17)
where ℎ(ܯ) is impulse response of the room. From the Eq. (17), acoustic reflection can be seen 
as a form of noise Convolution noise. While taking the Eq. (17) on both sides do Fourier transform 
and logarithmic: 
෠ܺ(߱) = መܵ(߱) + ܪ෡(߱). (18)
Both sides of the equation called cepstral, design a passband filter make the ܪ෡(߱) disappear, 
and make Inverse Fourier transform of መܵ(߱), then we can obtain ݏ(݊): 
Error is defined as: 
݁ݎݎ݋ݎ = ‖݌௟(ݏ) − ݌௛(ݏ)‖ଶ‖݌௟(ݏ)‖ଶ × 100 %. (19)
3. Numerical simulation 
In order to verify that difference of the sound signal before and after data processing, 
respectively, two and four-point source source reconstruction simulation have been done. Of the 
parameters of the simulation is the same, the enclosed space length, width and height, respectively, 
1 m, 1 m, 1 m. Analysis of the frequency are 800 Hz, surface vibration velocity of the ball is 
0.25 m/s, holographic measurement distance is 0.1 m, reconstructed surface distance is 0.05 m, 
holographic surface is 64×64 square planar array, planar array of edge length is 0.8 m. The close 
space impulse response is shown in Fig. 2. The characteristics of passband filter is shown in Fig. 3. 
Two-point and four-point source source simulation results shown in Fig. 4 to Fig. 9. 
 
Fig. 2. The close space impulse response 
 
Fig. 3. Bandpass filter amplitude-frequency characteristics 
THE NAH BASED ON COMPLEX CEPSTRUM METHOD IN A CLOSED SPACE.  
XIANGLIN TIAN, JINGJUN LOU, YANGYANG ZHANG, DUO QU 
54 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 9. ISSN 2345-0533  
Fig. 4. Theoretical sound  
pressure contours 
of two sound sources 
Fig. 5. The reconstruction of two 
point sources acoustic pressure 
contours after data processing 
 
Fig. 6. The reconstruction of two 
point sources acoustic pressure 
contours before data processing 
Fig. 7. Theoretical sound pressure 
contours of four sound sources 
 
Fig. 8. The reconstruction of four 
point sources acoustic pressure 
contours after data processing 
 
Fig. 9. The reconstruction of four 
point sources acoustic pressure 
contours before data processing 
As can be seen from the simulation graph, the data of the two-point sources and four-point 
sources after the data processing could reconstruct the sound source precisely, there will be greater 
reconstruction error before data processing. For two sources, the sound source overlap portion is 
more, and cannot distinguish between two sources. For a four-point source, the sound pressure 
cloud also has a relatively large distortion. Error is shown in Fig. 10. 
 
Fig. 10. Error analysis figure 
From Fig. 10, in the low frequency the error can be controlled within 20 %, the error of two 
sources is less than the error of four-point sources. In 1300 Hz to 1900 Hz band, two sources of 
error control in about 8 %, but with increased frequency, the error increases. This is caused by the 
complex cepstrum algorithm itself restricted, optimizing the parameters or combining with other 
reverberation algorithms the error can be controlled within a smaller range. 
4. Conclusions 
This paper proposes a method about reconstruction of the sound source in closed space with 
NAH. After receiving the flat plane array sound pressure data by the method of complex cepstrum 
data processing, data processed planar NAH sound source reconstruction. Simulation results show 
THE NAH BASED ON COMPLEX CEPSTRUM METHOD IN A CLOSED SPACE.  
XIANGLIN TIAN, JINGJUN LOU, YANGYANG ZHANG, DUO QU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 9. ISSN 2345-0533 55 
that the sound pressure level is very similar to the theoretical value of the post processed 
reconstruction of the sound source. In low frequencies, error analysis shows that the method of 
processing data for complex cepstrum method based on 1300 Hz to 1900 Hz frequency error 
control is good. The reason of the increasing of the error is that the parameters of the complex 
cepstrum algorithm cannot adjust adaptive the frequency increases, optimizing the parameters or 
combining with other reverberation algorithms could control the error within a smaller range. 
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The NAH based on complex cepstrum method in a closed space

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