Performance of underwater adaptive array including mutual coupling effects

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Applied Acoustics 70 (2009) 190–193 www.elsevier.com/locate/apacoust

Technical Note

Performance of underwater adaptive array including mutual coupling effects Kun-Chou Lee *, Jhen-Yan Jhang, Min-Chih Huang Department of Systems and Naval Mechatronic Engineering, National Cheng-Kung University, Tainan 701, Taiwan Received 26 October 2007; received in revised form 10 January 2008; accepted 15 January 2008 Available online 4 March 2008

Abstract In this paper, the mutual coupling effects on the performance of underwater adaptive arrays are analyzed. This study manifests the discrepancies between cases of considering and ignoring the mutual coupling effects. The least mean square (LMS) based adaptive array is utilized to illustrate the mutual coupling effects. Numerical examples show that the performance of an underwater adaptive array can be improved as mutual coupling effects are considered. The consideration of mutual coupling effects is necessary in analyzing underwater adaptive arrays. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Underwater array; Adaptive array; Mutual coupling

1. Introduction An adaptive array can adjust its pattern automatically. It has been applied to radar and sonar systems [1–3] for many years. Adjusting weights of elements is usually required in an adaptive sonar array. The received signals of the array elements together with their weights are linearly combined to produce the array output. There are many algorithms to calculate the weights for elements of adaptive arrays, e.g., the least mean square (LMS) algorithm [4], the recursive least squares (RLS) algorithm [5], the generalized eigenvalue (GE) algorithm [6], etc. In [7–9], mutual coupling effects between elements of antenna arrays are studied. For underwater arrays, the sonar elements are usually assumed to be isolated, i.e., the interactions between array elements are ignored [10– 12]. However, in practice, there exist mutual coupling effects between sonar elements of underwater arrays. The

mutual coupling effects will affect the pattern, gain, beamwidth, etc., of an underwater array. In this study, the mutual coupling effects between sonar elements of an underwater adaptive array are analyzed. This treatment of mutual coupling can be applied to all kinds of underwater arrays [10–12]. To illustrate the mutual coupling effects, the least mean square (LMS) algorithm is utilized to calculate the steady state weights of adaptive arrays. This is because the LMS algorithm is easy to implement [13]. The mutual coupling effects of underwater adaptive arrays can be treated by the impedance matrix [14,15]. Numerical examples show that the performance of an underwater adaptive array is improved as mutual coupling effects are considered. Neglecting the mutual coupling effects will cause some errors in predicting the desired signal direction. The proposed mutual coupling analysis can reduce these errors. 2. Formulation

* Corresponding author. Tel.: +886 6 2757575x63536; fax: +886 6 2437102. E-mail addresses: [email protected], [email protected] (K.-C. Lee).

0003-682X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2008.01.004

Consider an adaptive array in the underwater receiving system, as shown in Fig. 1. The array output contains the desired signal, the interference signal and the thermal noise.

K.-C. Lee et al. / Applied Acoustics 70 (2009) 190–193

191

Nomenclature Ad Ai Ar R(t) S W Ud Ui X

desired signal amplitude interference signal amplitude reference signal amplitude reference signal reference correlation vector steady state weight vector normalized phasor vector of the desired signal normalized phasor vector of the interference signal measured signal

As the element spacing is one-half wavelength, the /d and /i in Eq. (3) become

θd element #1 w1

desired signal vector interference signal vector thermal noise vector impedance matrix covariance matrix incidence angle of the desired signal incidence angle of the interference signal phase of the desired signal phase of the interference signal frequency of signals

Xd Xi Xn Z U hd hi wd wi x0

/d ¼ p sin hd ; /i ¼ p sin hi ;

#2

#N w N ····

w2

where hd and hi are incidence angles of the desired and interference signals, respectively. For m jammers, the interference signal vector X i in Eq. (1) becomes m m X X Xi ¼ X ik ¼ Aik ejðx0 tþwik Þ U ik ; ð5Þ

∑

k¼1

Fig. 1. The LMS underwater adaptive array.

In practical situation, there exist mutual coupling effects between pistons of arrays. However, these mutual coupling effects are usually ignored in many existing studies, i.e., the pistons are assumed to be isolated. The mutual coupling effects can be described by an impedance matrix Z [14,15]. Therefore, the measured signal including mutual coupling effects can be given as ð1Þ

where X d is the desired signal vector, X i the interference signal vector, and X n the thermal noise vector. They can be given by

W ¼ U1 S:

ð6Þ

The covariance matrix U in Eq. (6) is defined as U ¼ EðX  X T Þ;

ð7Þ

where the asterisk denotes complex conjugate and E() denotes expectation. The reference correlation vector is given as S ¼ EðX  RðtÞÞ:

ð8Þ

In Eq. (8), the R(t) is the reference signal and can be given as RðtÞ ¼ Ar ejðx0 tþwd Þ ;

X d ¼ Ad ejðx0 tþwd Þ U d ; X i ¼ Ai ejðx0 tþwi Þ U i ; X n ¼ ½ n1 ðtÞ n2 ðtÞ   

k¼1

where Aik is the interference signal amplitude of the kth jammer. From the LMS algorithm [13], the steady state weight vector W of the underwater adaptive array can be given as

array output

X ¼ ZðX d þ X i Þ þ X n ;

ð4Þ

ð2Þ nN ðtÞ T ;

where T denotes transpose. In Eq. (2), x0 is the frequency, wd is the phase of the desired signal, wi is the phase of the interference signal, Ad is the desired signal amplitude and Ai is the interference signal amplitude. The U d and U i in Eq. (2) are phasor vectors and are normalized so that the first component becomes 1, i.e.,  T U d ¼ 1 ej/d    ej/d ðN 1Þ ; ð3Þ  T U i ¼ 1 ej/i    ej/i ðN 1Þ Þ :

ð9Þ

where Ar is the reference signal amplitude. Since R(t) and X n are statistically independent and only X d correlates with R(t), Eq. (8) becomes S ¼ Ar Ad Z  U d :

ð10Þ

From Eqs. (1) and (7), we have 2

U¼r I þZ



A2d U d U Td

þ

m X

A2ik U ik U Tik

! ZT;

ð11Þ

k¼1

where r is the standard deviation of noise power and I is the unit matrix. From (6), (10), (11), the weights W of the array elements can be obtained.

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K.-C. Lee et al. / Applied Acoustics 70 (2009) 190–193

Consider next the pattern of the array with this weight vector. To compute the pattern, we suppose a unit amplitude signal propagates into the array from angle h. This will produce a signal vector in the array  T ð12Þ X ¼ ejx0 t 1 ej/ðhÞ    ejðN 1Þ/ðhÞ ;

90

1

120

60 0.8 0.6 30

150

where

0.4

/ðhÞ ¼ p sin h:

ð13Þ 0.2

The array pattern including mutual coupling effects will be given as jW T ðZ 1 X Þj:

ð14Þ

As the array mutual coupling effects are ignored, we have Z ¼ I. Therefore, Eqs. (1), (8) and (11) are simplified as X ¼ X d þ X i þ X n; S¼

ð15Þ

Ar Ad U d

ð16Þ

210

330

and U ¼ r2 I þ A2d U d U Td þ

m X

A2ik U ik U Tik ;

ð17Þ

240

k¼1

respectively. The array pattern ignoring mutual coupling effects can be obtained from (14) by replacing Z with I. Under such simplification, it will cause some errors in the array performance. 3. Numerical examples In the simulation, the element spacing is assumed to be one-half wavelength, the desired signal amplitude is one and the interference signal amplitude is ten. The number of array elements is chosen as N = 6. Each sonar element is piston with dimension of one-half wavelength square. The standard deviation of the noise component is assumed to be r = 1. In order to illustrate the array mutual coupling effects, we consider two cases, i.e., the non-interference and the mono-interference cases. In both cases, the desired signal direction is from hd = 50°.

300 270

Fig. 2. The array pattern of six sonar elements by assuming there is no interference.

3.2. Case 2: mono-interference Fig. 3 shows the array pattern of six sonar elements by assuming there is a mono-interference from hi = 90°. The

90

60 0.8 0.6 30

150 0.4 0.2

3.1. Case 1: non-interference Fig. 2 shows the array pattern of six sonar elements by assuming there is no interference. The solid line is the array pattern by our mutual coupling analysis and the main peak occurs at h = 49.9°. The dotted line is the array pattern by ignoring the mutual coupling effects and the main peak occurs at h = 51.6°. In this case, the main peak is expected to occur at h = 50°. The errors are 1.6° by ignoring the mutual coupling effects and 0.1° by considering the mutual coupling effects, respectively. The spatial error Ds will be proportional to the distance r of signal sources, i.e., Ds = r  Dh. For example, as the distance of signal sources is r = 1000 m, the spatial errors are 27.93 m by ignoring the mutual coupling effects and 1.75 m by considering the mutual coupling effects, respectively.

1

120

210

330

240

300 270

Fig. 3. The array pattern of six sonar elements by assuming there is a mono-interference from hi = 90°.

K.-C. Lee et al. / Applied Acoustics 70 (2009) 190–193 Table 1 Comparisons between the simulation results of case 1 (non-interference) and case 2 (mono-interference)

Considering mutual coupling Neglecting mutual coupling

Desired signal

Noninterference

Monointerference

50°

49.9°

49.0°

50°

51.6°

46.4°

solid line is the array pattern by our mutual coupling analysis and the main peak occurs at h = 49.0°. The dotted line is the array pattern by ignoring the mutual coupling effects and the main peak occurs at h = 46.4°. In this case, the main peak is expected to occur at h = 50°. The errors are 3.6° by ignoring the mutual coupling effects and 1° by considering the mutual coupling effects, respectively. As the distance of signal sources is r = 1000 m, the spatial errors will be 62.83 m by ignoring the mutual coupling effects and 17.45 m by considering the mutual coupling effects, respectively. Comparisons between the simulation results of the above two examples are shown in Table 1. 4. Conclusion In general, neglecting the array mutual coupling effects can simplify the computation complexity, but is not practical. In this study, we utilize a matrix model to analyze the mutual coupling effects in an underwater adaptive array. Numerical examples show the performance of an underwater adaptive array is improved by considering the mutual coupling effects. The mutual coupling analysis in this study is very straightforward and easy to implement. Some related mutual coupling models to treat the underwater arrays were shown in [16–18]. In our model, the directivity of transducers is assumed to be omnidirectional for simplicity. In underwater sonar, it is difficult to get omnidirectional elements in a realistic condition. As the transducers are not omnidirectional, the received signals of array elements, i.e., components of U d and U i in (3), should be multiplied by a factor to represent the directivity effect. This will be the future work of our study.

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Acknowledgements The work in this paper was supported by the National Science Council, Taiwan, under Grant No. NSC 96-2628E-006-250-MY3. References [1] Park YC, Sommerfeldt SD. A fast adaptive noise control algorithm based on the lattice structure. Appl Acoust 1996;47(1):1–25. [2] Chen KT. An adaptive active control of sound transmission through an aperture at low frequencies. Appl Acoust 1995;46(2):153–74. [3] Chen KT, Chen YN, Chen W, Liu YH. Adaptive active control on the acoustic transmission of the acoustic sources in aperture at low frequencies. Appl Acoust 1998;54(2):141–64. [4] Wright AB, Xie B, Karthikeyan A. A comparison of white test signals used in active sound cancellation. Appl Acoust 2000;59(4):337–52. [5] Jones RW, Olsen BL, Mace BR. Comparison of convergence characteristics of adaptive IIR and FIR filters for active noise control in a duct. Appl Acoust 2007;68(7):729–38. [6] Haykin S. Adaptive filter theory. Singapore: Pearson Education; 2003. [7] Lee KC, Jhang JY. Application of particle swarm algorithm to the optimization of unequally spaced antenna arrays. J Electromagnet Wave 2006;20(14):2001–12. [8] Lee KC. A genetic algorithm based direction finding technique with compensation of mutual coupling effects. J Electromagnet Wave 2003;17(11):1611–22. [9] Lee KC. Genetic algorithms based analyses of nonlinearly loaded antenna arrays including mutual coupling effects. IEEE Trans Antennas Propag 2003;51:776–81. [10] Rigelsford JM, Tennant A. A 64 element acoustic volumetric array. Appl Acoust 2000;61(4):469–75. [11] Mao Q, Xu B, Jiang Z, Gu J. A piezoelectric array for sensing radiation modes. Appl Acoust 2003;64(7):669–80. [12] Xia R, Shou W, Chen G, Zhang M. A new-style phased array transducer for HIFU. Appl Acoust 2002;63(9):957–64. [13] Compton RT. Adaptive antenna: concepts and performance. Englewood Cliffs (NJ): Prentice-Hall; 1988 [chapter 2]. [14] Burnett DS, Soroka WW. Tables of rectangular piston radiation impedance function with application to sound transmission loss through deep apertures. J Acoust Soc Am 1972;51(5):1618–23. [15] Arase EM. Mutual radiation impedance of square and rectangular pistons in a rigid infinite baffle. J Acoust Soc Am 1964;36(8):1521–5. [16] Lee H, Tak J, Moona W, Lim G. Effects of mutual impedance on the radiation characteristics of transducer arrays. J Acoust Soc Am 2004;115(2):666–779. [17] Mikami D, Yokoyama T, Hasegawa A, Kikuchi T. Sound power reduction of underwater array projector caused by mutual radiation impedance. Jpn J Appl Phys 1997;36:3340–4. [18] Yokoyama T, Henmi M, Hasegawa A, Kikuchi T. Effects of mutual interactions on a phased transducer array. Jpn J Appl Phys 1998;37:3166–71.