Four-element planar arrays focus a point-like source based on the artificial iterative phase conjugated processing

Ting Li 1 Department of underwater weaponry and chemical defense, Dalian Naval Academy

Dalian China Yi Zhang Department of underwater weaponry and chemical defense, Dalian Naval Academy

Dalian China Liufang Fu Department of underwater weaponry and chemical defense, Dalian Naval Academy

Dalian China

ABSTRACT The artificial iterative phase conjugated processing is an improved algorithm of phase con- jugation and has been proven to focus a sharp focal spot using linear array. Because cross- shaped four-element planar array and triangular four-element planar array are widely used in the situation of little acoustic measuring points, their focal patterns by the artificial itera- tive phase conjugated processing are discussed in this paper. Numerical simulations gives conclusions. Based on the artificial iterative phase conjugated processing, two arrays focus a smaller focal spot than that by phase conjugation. As their iteration number increases, focal spot size decreases but the side-lobe amplitude becomes big. Considering the focal spot size and side-lobe interference, the triangular four-element array has a clearer pattern than the cross-shaped one.

1. INTRODUCTION

Because a sharp focal spot can provide an exact location of the source, it is always a popular top- ic to reduce the focus for noise imaging and location domain [1-9], especially for low-frequency case[10].Phase conjugation is also referred to as time reversal in time domain. It can refocus the incident wave back and further achieving self-adaptive focusing. However, because of the diffrac- tion limitation, focal spot size focused by phase conjugation is larger than half a wavelength. Artifi- cial iterative phase conjugated processing (AIPCP) has been proven to enhance the focus and the diffraction limitation is overcome using uniform linear array[11]. Cross-shaped four-element planar array and triangular four-element planar array are widely used in the situation of little acoustic measuring points[12]. It may be an interesting study on their focal patterns by phase conjugation method (PCM) and artificial iterative phase conjugated processing (AIPCP). In this paper, we em- ploy two kinds of four-element planar arrays to focus a point-like source in the free space using

1 litingyouxiang@sina.com

worm 2022

PCM and AIPCP. Research is done by numerical simulation. Conclusions come out that focal spot calculated by artificial iterative phase conjugated processing is smaller than that calculated by phase conjugation. Increasing the iteration number, focal spot becomes sharp but the side-lobe disturbance increasing. The dipole form of AIPCP using triangular four-element array has a sharp focus and an acceptable side-lobe disturbance with a small iteration number. 2. ARTIFICIAL ITERATIVE PHASE CONJUGATED PROCESSING

s r In an infinite, homogeneous and isotropic medium, the point-like source is located at in Car- tesian coordinates in figure 1. Figure 1 takes the cross-shaped four-element planar array for exam-

j r ple. The array is placed on XOY plane in Cartesian coordinates. Element j is located at and re- ceives the sound from the source.

worm 2022

Figure 1: Sketch map of the four-element planar array, source, and field points There are two forms of PCM, the monopole one and the dipole one. The pressure received by transducer j are conjugated and then sent by monopole transducers. This monopole form is called PC/M.

        , , , , , , * / r r r r r r j j s j G G I M jPC  (1)

The asterisk denotes a complex conjugation. The normal derivate of pressure is conjugated and emitted by dipole sources. This array form of PCM is called PC/D.

     



     r r r r r r

*

, ,

, , , ,

G G I jPC (2)

j j s j

D / 



n

n

In AIPCP, each receiver does a PCM in one iteration loop. Outputs of all transducers are summed together after each transducer has finished their iteration loops [11]. According to the two forms of PCM, there are two forms of AIPCP, the monopole one and the di- pole one described as follows after N iteration loops[11]:

      

     r r r r r r j j s j G G I PC (3)

1 N * M / j N , , , , , ,

     



1 *

N







 

G G I    r r r r r r

, ,

, , , ,

j j s j (4)

N jPC/D

  

  





n

n

 where denotes the normal derivative operator of the array.

n 

3. FOCUSING ANALYSIS BY NUMERICAL SIMULATION

worm 2022

In this section, a point-like source locates at the origin of the Cartesian coordinate system in the free space. Two kinds of four-element planar arrays are employed to focus the source, their sketch map are shown in figures 2 and 3. Focal spot size by PCM and AIPCP using planar arrays is dis- cussed in quite near distance, near field and far field of the source. Effects of iteration number are also discussed for AIPCP.

Figure 2: Sketch map of cross-shaped four-element planar array

Figure 3: Sketch map of triangular four-element array

3.1. Criterion of near field and far field

The twin-line planar array has two linear arrays that parallel to the X-axis and has Z-axis sym- metry as shown in figure 2. The scale of planar array is 2 meters. A point-like source located at the origin of the Cartesian coordinates sends a single frequency wave at 344Hz and the wavelength is 1 meter. According to the criterion of near field and far field shown as follows:

1.356 Z L  Quite near field:

2 1.356 L L Z 

   Near field:

1.356 L Z L 

2

   Far field:

3.2. Comparisons between PCM and AIPCP

Coordinates of the cross-shaped four-element planar array are (1,0), (0,1), (-1,0), (0,-1). Coordi-

2 2

2 2 nates of the triangular four-element array are ( ,-0.5), (0,1), (- ,-0.5), (0,0).The array locates

at 2m, 5m and 30m away from the source. They are corresponding to a quite near distance, a near field and a far field away from source according to the criterion in section 3.1. In the legend, “PC/Dcros” means the focus is calculated by PC/D using the cross-shaped four-element planar ar- ray. “AIPC/M2tria” means the focus is calculated by AIPC/M using the triangular four-element ar- ray iterating 2 times.

worm 2022

Figure 4: Comparisons between PCM and AIPCP recorded at 2m away from the source

“dwiy (Won

Figure 5: Comparisons between PCM and AIPCP recorded at 5m away from the source

Norm. Amp.

worm 2022

Figure 6: Comparisons between PCM and AIPCP recorded at 30m away from the source For PCM, the focal spot size by cross-shaped array is smaller than that by triangular array, but their amplitudes of side lobe are large. For AIPCP, the focal spot size by cross-shaped array is smaller than that by triangular array, but their amplitudes of side lobe are large with the same itera- tion number. Considering the focal spot size and side lobe disturbance, AIPC/D using triangular array has the best performance.

3.3. Focal patterns by triangular four-element planar array with different iteration number

In this part, focal spots are compared calculated by AIPC/D using the triangular four-element ar- ray with different iteration number in quite near field, near field and far field. In the legend, “AIPC/D2tria” means the focus is calculated by AIPC/D using the triangular four-element array iterating 2 times.

Norm. Amp.

Figure 7: Comparisons with different iteration number for AIPC/D recorded at 2m away from the

source.

08 o7 6 Eos! os Fm. A Soa 03 02 os 02 os 06 08

worm 2022

Figure 8: Comparisons with different iteration number for AIPC/D recorded at 5m away from the

source.

Figure 9: Comparisons with different iteration number for AIPC/D recorded at 30m away from the

source. As their iterate number increases, focal spot size decreases but the side-lobe amplitude becomes big as shown in figure 7 to figure 9. Increasing iteration number can deduce the focal spot size but increasing the side lobe amplitude. 4. CONCLUSIONS

08 o7 6 Eos! os Fm. A Soa 03 02 os 02 os 06 08

To check the focusing ability of artificial iterative phase conjugated the four-element planar ar- ray, two kinds of the four-element planar array, the cross-shaped four-element and triangular four- element one, are employed to focus a point-like source by numerical simulation. Conclusions come out that contrasting with phase conjugation, artificial iterative phase conjugated planar array focuses a small focal spot but a big side lobe. Although increasing the iterate number can sharp the focal spot, the side-lobe amplitude becomes big. Considering the focal spot size and side-lobe interfer- ence, AIPC/D using the triangular four-element array with a small iteration number is the best choice. 5. ACKNOWLEDGEMENTS

We would like to gratefully acknowledge the support of the College Development Found of Da- lian Naval Academy (Grant No.DJYKYKT2021-024) and the Key discipline construction project.

Norm. Amp. 2

6. REFERENCES

1. J. de Rosny , M. Fink. Focusing properties of near-field time reversal. Phys. Rev. A, 76(6) , 637-

640(2007). 2. Fink M. Time-Reversal Waves and Super Resolution. 4th International Conference and the 1st

Congress of the IPIA2008 ,pp010024.Canada. 3. G. Lerosey, J. De Rosny, A. Tourin, and M. Fink. Focusing beyond the diffraction limit with

far-field time reversal. Science, 315(5815) , 1120-1122(2007). 4. P. Blomgren, G. Papanicolaou, and H. K. Zhao. Super-resolution in time-reversal acoustics. J.

Acoust. Soc. Am, 111(1), 230-248(2002). 5. E. Bavu and A. Berry.High-Resolution Imaging of Sound Sources in Free Field Using a Numer-

ical Time-Reversal Sink. ACTA ACUST UNITED AC, 95(4) ,595-606(2009) 6. C. Heedong, Y. Ogawa, T. Nishimura, and T. Ohgane, Iterative angle-and-time-domain gating

technique for time-reversal MUSIC imaging. Signal Processing , 111(C) , 39-49(2015). 7. J. Garnier and M. Fink. Super-resolution in time-reversal focusing on a moving source. Wave

Motion, 53 , 80-93(2015). 8. H. Ma, X. Zeng, and H. Wang. A novel dual-channel matching method based on time reversal

and its performance for sound source localization in enclosed space. Acoust Aust, 44 , 417- 428(2016). 9. J. de Rosny and M. Fink.Overcoming the diffraction limit in wave physics using a time-reversal

mirror and a novel acoustic sink. Phys. Rev. Lett., 89 , 16(2002) 10. C.-X. Bi, Y.-C. Li, Y.-B. Zhang, and L. xu.Super-resolution imaging of low-frequency sound

sources using a corrected monopole time reversal method. J Sound Vib, 410 , 303-317(2017). 11. T. Li, S. Li, and S. Liu. An Enhanced Sharpness on Acoustic Focus Via Artificial Iterative

Phase Conjugation Processing. Acoustics Australia, 46(2) , 227-240(2018). 12. Xu zhiyong, Zhao zhao. Pertubation sensitivity analysis of azimuth estimation of the acoustic

source for the planar microphone array. Journal of Xidian University, 44(4) , 95-100(2017).

worm 2022