Passive Direction-of-Arrival Estimation under High ... - IEEE Xplore

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Abstract—To enhance the passive direction-of-arrival (DOA) estimation performance of a volumetric array in high noise and strong interference background, a ...
Passive Direction-of-Arrival Estimation under High Noise and Strong Interference Condition for Volumetric Array MA Qian, SUN Chao, LIU XiongHou, XIANG LongFeng, LIU ZongWei Institute of Acoustic Engineering, Northwestern Polytechnical University [email protected], [email protected], [email protected], [email protected]

Abstract—To enhance the passive direction-of-arrival (DOA) estimation performance of a volumetric array in high noise and strong interference background, a new DOA estimation method based on Fourier integral method (FIM) and inverse beamforming (IBF) is proposed. This method combines the two methods and thus obtains the interference cancelling capability of IBF and the noise suppression ability of FIM simultaneously. Furthermore, the proposed method is generalized to be feasible for arbitrary volumetric array and also achieves the same robustness as the conventional beamforming (CBF). With theoretical analysis and computer simulations, it is shown that with the proposed method, the volumetric array can obtain better DOA estimation results in low signal-to-ratio (SNR) condition as well as in the presence of interferences, when compared to the conventional CBF method. Index Terms—underwater acoustics, direction of arrival (DOA) estimation, Fourier integral method (FIM), Inverse beamforming (IBF), volumetric array

I.

INTRODUCTION

Volumetric arrays are commonly used in the sonar system for underwater target detection due to its flexibility in structure and non-ambiguity in azimuth. Due to the complex underwater environment, the received signals of sonar arrays could easily be affected by noises and interferences. Specifically, self-noise radiated from the platform often makes the sonar array work in a low SNR environment. Besides, target signals may be sheltered by strong interference, which will cause a wrong DOA estimation result. Therefore, how to obtain good DOA estimation performance for the volumetric array in complicated circumstance has become a critical problem. To deal with the problem of target detection in low SNR circumstance, Bucker proposed a method in 1970s, which can estimate the plane-wave density of the acoustic field directly from the spatial correlations (i.e., covariance matrix) of sensor pairs [1]. Based on the similar idea, Wilson proposed Fourier Series Method (FSM) in 1983, which related the estimated acoustic field to the measured covariance via integral equation [2]. To release the heavy computing burden of FSM, Nuttall et al. proposed an improved algorithm called Fourier integral method (FIM) [3]. The use of Fast Fourier Transformation (FFT) in FIM simplifies the computation and also leads to higher accuracy compared to FSM. In the following research reports, Nuttal and Wilson proved that FIM outperformed

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conventional beamforming (CBF) and adaptive beamforming (ABF) in the passive target detection in low SNR environment when maintaining the same robustness as CBF [4]. To suppress the high sidelobes of FIM, Solomon et al. proposed the Weighted Fourier Integral Method (WFIM) for uninform linear array and sparse linear array by the design of appropriate weighting [5]. However, almost all the works mentioned above have only take uniform or sparse linear arrays into consideration. The use of FIM for volumetric array was only discussed theoretically in [3] with no simulated or experimental DOA estimation results. Besides, FIM is unable to suppress interferences; hence it calls for some interference cancelling processing in the use of FIM for the volumetric flank array. The use of inverse beamforming (IBF) method is and effective way to suppress strong interference. Jeffers proposed the so-called sector inverse beamforming, which is used to recover the signal coming from a certain source in a spatial sector [6]. Afterwards, Zhang and Jiang used IBF in interference shielding, and have successfully suppressed spatial interferences in the underwater passive DOA estimation [7-8]. As a kind of linear processing method, IBF interference cancelling algorithm is robust and convenient in use. Moreover, the outputs of IBF interference cancelling are sensor-domain data and thus, it can be directly used for further processing. Considering shortcomings of existing DOA estimation methods in high noise and strong interference environment, we propose a new passive DOA estimation method for volumetric arrays. This method combines the FIM and IBF methods together, and it contains two steps. Firstly, the IBF method is adopted to process the sensor-domain data in order to eliminate the spatial coherent interferences. Then, the processed sensor-domain data are used as the inputs of FIM. After the FIM beamforming processing, the DOA estimation result with no interference and relatively higher array gain is achieved. And thus, the anti-noise performance and robustness of FIM, together with the interference cancelling ability and simplicity of IBF are obtained simultaneously. In the following sections, the mathematical derivation of FIM and IBF are introduced in Section II and Section III, respectively. In Section IV, the DOA estimation results based on computer simulations are depicted. Finally, conclusions are summarized in Section V.

PM

P3

P2

The integration calculation of θ can be divided into two parts, as:



π

0

θ

dθ = ∫

J (u, v, z ′) = λ 2 ∫

π 2

0

0 ≤θ ≤π 0 ≤ φ ≤ 2π

G f ( x′, y ′, z ′) = ∫ dθ ∫ dφ sin θ N f (θ , φ ) × exp[i (2π λ )( x′ sin θ cos φ + y ′ sin θ sin φ + z ′ cos θ )]

(1)

calculated from the received sensor data, the field directionality could be acquired by solving the integral formula in Eq. 1. Define the parameter J (u , v, z ′) as 2π

(ux′ + vy ′))G f ( x′, y ′, z ′)

(2)

λ where u = sin θ cos φ , v = sin θ sin φ . Substituting Eq. 2 into Eq. 1, yields:

J (u, v, z ′) = ∫ ∫ dxdy exp(−i π



0

0



λ

(ux′ + vy ′))



π



0

0

λ

( x′ sin θ cos φ + y ′ sin θ sin φ + z ′ cos θ )]

= ∫ dθ ∫ dφ sin θ N f (θ , φ )λ 2 × δ (u − sin θ cos φ )δ (v − sin θ sin φ )× exp(i

λ



λ

z ′ cos θ )

(5)

z ′ cos θ ) 2π

λ

z ′ cos θ )]

(3)

u 2 + v2 < 1

(6)

N a (u , v ) = N f [arcsin( u + iv ), arg(u + iv)], N b (u, v) = N f [π − arcsin( u + iv ), arg(u + iv)]

(7)

Substituting Eqs. 6 and 7 into Eq. 5 and calculating the integrals, Eq. 5 can be rewritten as: λ2 ⎧ 2π J (u, v, z ′) = s(u, v) z ′] ⎨ N a (u, v) exp[i s (u, v) ⎩ λ (8) 2π ⎫ s(u, v) z ′]⎬ + N b (u, v) exp[−i λ ⎭ It can be derived from Eq. 8 that: λ2 ⎡ 2π s (u, v) z ′ ⎤ J r (u, v, z ′) = cos ⎢ ⎥ × [ N a (u, v) + Nb (u , v)] (9) s (u , v ) λ ⎣ ⎦ ⎡ 2π s(u , v ) z ′ ⎤ sin ⎢ ⎥ × [ N a (u , v) − N b (u , v ) ] (10) s (u, v ) λ ⎣ ⎦ where J r (u , v, z ′) and J i (u, v, z ′) represent the real part and imaginary part of J (u, v, z ′) , respectively. Specially, when J i (u, v, z ′) =

λ2

z′ = 0 , J (u, v, z ′) turns into:

λ2

× [ N a (u , v) + Nb (u, v) ] (11) s (u , v) Suppose that there are N different spacing values of z ′ for the volumetric array, which can be expressed as zn′ (n = 1, 2," N ) . Thus, according to Eq. 10 and Eq. 11, the directionalities N a (u , v ) and N b (u, v) can be expressed as: J (u, v, 0) =

N a (u , v ) =

N b (u, v) =

× ∫ dθ ∫ dφ sin θ N f (θ , φ ) × exp[i



s (u, v) = (1 − u 2 − v 2 )1 2 ,

The basic idea of FIM is to estimate the acoustic field directionality from the covariance matrix of the array received data. Accordingly, once the covariance G f ( x′, y ′, z ′) can be

J (u, v, z ′) = ∫ ∫ dxdy exp(−i

0

For simplification, define the following variables as:

The directionality of the acoustic field is defined as N f (θ , φ ) , which represents the energy density spectrum of the signal at frequency f and from direction (θ , φ ) . The elevation angle θ ∈ [ 0, π ] is measured down from the z axis and the azimuth angle φ ∈ [ 0, 2π ] is measured clockwise from the x axis, as shown in Fig. 1, where P1 , P2 ," , PM represent the M sensors of the volumetric array, with locations ( xm , ym , z m ), m = 1, 2,..., M . Define x ′, y ′, z ′ as the difference between a pair of sensors and let G f ( x′, y ′, z ′) be the spatial correlation of the sensor pair with the spacing x ′, y ′, z ′ , and an integral equation like Eq. 1 can be used to relate G f ( x′, y ′, z ′) to acoustic field directionality N f (θ , φ ) as [3] 2π

(4)



+ N f (π − θ , φ ) exp(−i

FOURIER INTEGRAL METHOD FOR VOLUMETRIC ARRAY

0



dθ ∫ dφ sin θ

× [ N f (θ , φ ) exp(i

Fig. 1. 3-D coordinate system of a volumetric array

π

π

π 2

× δ (u − sin θ cos φ )δ (v − sin θ sin φ )

φ

0

dθ + ∫

Then, Eq. 3 can be rewritten as:

P1

II.

π 2

0

s (u , v) ⎛ 1 J (u, v, 0) + 2 ⎜ 2λ ⎝ N

∑ sin[2π s(u, v) z′

s (u, v) ⎛ 1 J (u , v, 0) − 2 ⎜ 2λ ⎝ N

∑ sin[2π s(u, v) z ′

J i (u, v, zn′ )

N

n =1

n

J i (u , v, zn′ )

N

n =1

n

⎞ ⎟ (12) λ] ⎠ ⎞ ⎟ (13) λ] ⎠

When θ → π 2 , N a (u , v ) and N b (u, v) will approach to each other, and we have N a (u, v ) ≈ N b (u , v) = N f (π 2, φ ) . Thus, Eq. 9a can be simplified as: N f (π 2, φ )→

s (u , v ) ⎡ 1 4λ 2 ⎢⎣ N

N

∑J n =1

r

⎤ (u, v, zn′ ) ⎥ ⎦

(14)

Recalling Eq. 1, some variables in Eqs. 9, 10 and 11 can be expressed as: 2π J (u, v, 0) = ∫ ∫ dxdy exp(−i (ux ′ + vy ′))G f ( x ′, y ′, 0) (15)

λ

J i (u , v, z n′ ) = Im ∫ ∫ dxdy exp(−i



(ux ′ + vy ′))G f ( x ′, y ′, zn′ ) (16) λ 2π (ux′ + vy ′))G f ( x ′, y ′, z n′ ) (17) J r (u , v, z n′ ) = Re ∫ ∫ dxdy exp( −i λ where G f ( x′, y ′, 0) is obtained from the data of sensor pairs within one plane paralleling with the horizontal plane, and G f ( x ′, y ′, zn′ ) is obtained from the data of sensor pairs in two paralleling planes with the spacing of z n′ . Different from Eq. 72a and Eq. 72b in Ref. [3], for the purpose of improving the DOA estimation performance, some improvements has been made by adding the summation and averaging process in Eqs. 12 and 13. And the process of calculating sound field directionality using FIM can be generalized as: cos θ N f (θ , φ ) = ( J (sin θ cos φ ,sin θ sin φ , 0) 2λ 2 1 N J (sin θ cos φ ,sin θ sin φ , z n′ ) ⎞ + ∑ i ⎟, N n =1 sin(2π cos θ zn′ λ ) ⎠

cos θ ( J (sin θ cos φ ,sin θ sin φ , 0) 2λ 2 1 N J (sin θ cos φ ,sin θ sin φ , zn′ ) ⎞ − ∑ i ⎟, N n =1 sin(2π cos θ zn′ λ ) ⎠

N f (π − θ , φ ) =

(18)

0 ≤ θ < π 2, 0 ≤ φ ≤ 2π where cos θ is the simplification of s (u , v ) . From Eq. 18, we observe that the averaging process of all sensors’ covariance with different spacing z′ is adopted, which is quite different from Ref. [3] where only the sensors’ covariance at a single spacing z′ is used. Furthermore, we can also see that the averaging process of sensors’ covariance with the same spacing z′ is actual a process of suppressing the uncorrelated noise between the sensors. Therefore, the FIM beamforming method can maintain good DOA estimation performance in low SNR circumstance. III. INVERSE BEAMFORMING INTERFERENCE CANCELLING The process of IBF interference cancelling contains two steps: 1) recovering the sensor-domain data of the interference in a certain spatial sector; 2) subtracting the recovered interference data from the original received sensor-domain data. This process of getting the sensor-domain data from beam-domain data is just the inverse process of the conventional beamforming. The signal received at the M-sensor array can be defined as X = S + N, which contains S = [s1 , s 2 ," , s M ] as the signal component and N = [n1 , n 2 ," , n M ] as the noise component, where s m and n m (m = 1, 2," M ) represent the signal component and noise component received at the mth sensor.

And it should be noted that the signal component S consists of both the source signal and the interference signal, and N is assumed to be spatially white. Let B denotes the beamforming matrix where each column of B is the steering vector for each direction. Then the beamforming result of the array can be described as: Y = BX (19) Let C denote the transformation matrix, which make BC a full-rank matrix. By multiplying both sides of Eq. 19 with the matrix inverse (BC) −1 and after manipulate, the original sensor-domain data can be recovered through the IBF transformation, that is: X = C(BC) −1 Y = C(BC)−1 BX (20) It can be seen that: C(BC) −1 B = I (21) Thus, the transformation matrix C has the relationship with B as: C = BH (22) According to Eq. 21 and Eq. 22, the weighting matrix of IBF can be defined as: B H (BB H )−1 (23) Based on the theory of sector inverse beamforming in Ref. [6], if the interference is supposed to be around the direction ϕ , the sector used for recovering the interference signal can be where the values of Δ1 , Δ 2 define as [ϕ − Δ1 , ϕ + Δ 2 ] , determine the range of the sector, which could be adjusted according to different situations. The sector is then divided into several bearings, and the multi-beamforming matrix for the bearings is defined as Bϕ . Then the IBF weighting matrix can be derived from Eq. 23 as BϕH (Bϕ BϕH ) −1 . Thus, the interference signal X′ in this sector can be recovered as: X′ = BϕH (Bϕ BϕH ) −1 Y = BϕH (Bϕ BϕH ) −1 BX (24) Subtracting the interference signal from the received signal, the residual sensor-data Z can be obtained: (25) Z = X − X′ and Z is used as the input of the subsequent processing such as beamforming, with the interference being eliminated. In the IBF processing procedure, the interference is eliminated through subtraction, which makes the process both simple and stable. One of the advantages of using IBF interference cancelling is that the processed data are in sensor level, so the data can be directly used in the subsequent beamforming process like FIM. IV. COMPUTER SIMULATIONS The array considered in simulations is a volumetric array with a half-cylindrical configuration, as depicted in Fig. 2. The array can be divided into seven uniform linear arrays paralleling to x axis or eight arc arrays to yoz plane. Figs. 2(b) and (c) show the expanded views seen from the positive direction of the y-axis and the side view seen from the positive direction of the x-axis, respectively. The dimensions of the array are also indicated in Figs. 2(b) and (c).

In the following computer simulations, the incident signal is considered the far-field plane wave and the noise involved is supposed to be additive white Gaussian noise. SNR and SIR specify the signal-to-noise ratio and signal-to-interference ratio at sensor in dB, respectively. The sound speed in water is set to be 1500 m/s. A. FIM Beamforming Performance The DOA estimation performance of FIM is compared with CBF in different SNRs. Suppose that there are two targets located at the directions θ1 = 85°, φ1 = 80° (target 1) and θ 2 = 85°, φ2 = 110° (target 2), respectively. The frequencies of the signals radiated by the both two targets are 10 kHz. The pulse width is 50ms and the sampling frequency is 48 kHz. SNR is set to be 10dB, 0dB, -10dB and -20dB, separately. From the DOA estimation results depicted in Fig. 3(a), we can see that FIM has almost the same performance at high SNR (i.e., 20dB here). Figs. 3(b)-(d) show that at lower SNRs, FIM shows better DOA estimation performance than CBF. Especially in Figs. 3(c) and (d) (the SNRs are -10dB and -20dB, respectively), FIM far outperforms CBF with lower side-lobe level and outstanding peaks of main-lobes. According to Fig. 3,

it can be concluded that FIM has better DOA estimation performance under low SNR circumstance than CBF. B. IBF Interference Cancelling and FIM DOA Estimation By integrating IBF interference cancelling method with FIM, the DOA estimation performance in presence of interferences is studied. The incident signals come from two target sources at the directions θ1 = 85°, φ1 = 80° (target 1) and θ 2 = 85°, φ2 = 110° (target 2); and the interference signal originates from θ = 85°, φ = 150° . The frequency of both the target signal and the interference signal are 10 kHz, the pulse width is 50ms, and the sampling frequency is 48 kHz. SNR is set to be -8dB and the SIR is -20dB. From Fig. 4(a), it can be seen that by integrating IBF with FIM, IBF-FIM can effectively suppress the strong interference. Fig. 4(b) shows the comparison of the results of DOA estimation using IBF-CBF and IBF-FIM in the relatively low SNR (i.e., -8dB) circumstance. Although IBF-CBF can also suppress the interference, the side-lobes level is much higher than that of IBF-FIM. Thus, the proposed method can suppress the strong interference and maintain good DOA estimation performance in low SNR circumstance. d = 0.075m

(a)

(b)

(c)

Fig.2. The volumetric array with a half-cylindrical configuration. (a) 3-D view of the volumetric array, (b) Expanded view along the y axis, (c) Side view of the volumetric array. 0

0 (a)

(b) -5

-10

Beampattern(dB)

Beampattern(dB)

-5

-15 -20 -25 -30 CBF FIM

-35 -40

0

20 40 60 80 100 120 140 160 180 Azimuth(°)

-10 -15 -20 -25 CBF FIM

-30 -35

0

20 40 60 80 100 120 140 160 180 Azimuth(°)

0

0 (c)

(d) -5 Beampattern(dB)

Beampattern(dB)

-5 -10 -15 -20 -25 -30

-15 -20 -25

CBF FIM 0

-10

-30

20 40 60 80 100 120 140 160 180 Azimuth(°)

CBF FIM 0

20 40 60 80 100 120 140 160 180 Azimuth(°)

(a) SNR=10dB; (b) SNR=0dB; (c) SNR=-10dB; (d) SNR=-20dB Fig.3. DOA estimation results of CBF and FIM. 0

0 (b)

-5

-5

-10

-10

Beampattern(dB)

Beampattern(dB)

(a)

-15 -20 -25 -30 FIM IBF-FIM

-35 -40

0 20 40 60 80 100 120 140 160 180 Azimuth(°)

-15 -20 -25 -30 IBF-CBF IBF-FIM

-35 -40

0

20 40 60 80 100 120 140 160 180 Azimuth(°)

Fig.4. Simulated output for FIM, IBF-FIM and IBF-CBF

V. CONCLUSIONS This paper presents a new DOA estimation method for arbitrary volumetric array which integrates the IBF interference cancelling with the FIM DOA estimation. In this method, the output sensor-domain data of IBF are adopted as the inputs of FIM to get the DOA estimation results. Through computer simulations, it is shown that the proposed method can be used for volumetric arrays to suppress the strong interference and maintain robust DOA estimation performance under low SNR circumstance. Tank experiments will be involved in future works to test the effectiveness of the proposed method. ACKNOWLEDGMENT This work is supported by Foundation for State Key Laboratory of Ocean Acoustics (No. KF200902). REFERENCES

[1] P. B. Homer, “High-resolution cross-sensor beamforming for a uniform line array,” J. Acoust. Soc. Am., Vol. 63, pp. 420-424, 1978.

[2] H. W. James, “Signal detection and location using the Fourier series method (FSM) and cross-sensor data,” J. Acoust. Soc. Am., Vol. 73, pp. 1648-1656, 1983. [3] H. N. Albert, H. W. James, “Estimation of the acoustic field directionality by use of planar and volumetric arrays via the Fourier series method and the Fourier integral method,” J. Acoust. Soc. Am., Vol. 90, pp. 2004-2019, 1991. [4] H. N. Albert, H. W. James, “Adaptive beamforming at very low frequencies in spatially coherent, cluttered noise environments with low signal-to-noise ratio and finite-averaging times,” J. Acoust. Soc. Am., Vol. 108, pp. 2256-2265, 2000. [5] I. S. D. Solomon, A. J. Knight, “Spatial processing of signals received by platform mounted sonar,” IEEE J. Ocean. Eng., Vol. 27, pp. 57-65, 2002. [6] R. Jeffers, B. Breed, Gallemore B., “Passive range estimation and range rate detection,” IEEE Sensor Array and Multichannel Signal Processing Workshop, USA, pp. 112-116, 2000. [7] D. H. Zhang, K. D. Yang, D. H. Wang, “Interference canceling algorithm based on time domain broadband inverse beamforming,” Technical Acoustics, Vol. 26, pp. 130-132, 2007. [8] L. Jiang, L. H. Guo, P. Cai, “Shielding the interference direction based on inverse beamforming algorithm,” in Proc. IEEE ISSCAA. 2nd International Symposium, Shenzhen 2008: 1-5.