Self-adapting root-MUSIC algorithm and its real-valued formulation for acoustic vector sensor array: [Doc 387]

Abstract

In this paper, based on the root-MUSIC algorithm for acoustic pressure sensor array, a new self-adapting root-MUSIC algorithm for acoustic vector sensor array is proposed by self-adaptive selecting the lead orientation vector, and its real-valued formulation by Forward-Backward(FB) smoothing and real-valued inverse covariance matrix is also proposed, which can reduce the computational complexity and distinguish the coherent signals. The simulation experiment results show the better performance of two new algorithm with low Signal-to-Noise (SNR) in direction of arrival (DOA) estimation than traditional MUSIC algorithm, and the experiment results using MEMS vector hydrophone array in lake trails show the engineering practicability of two new algorithms.[PUBLICATION ABSTRACT]

Wang et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:228 http://asp.eurasipjournals.com/content/2012/1/228

RESEARCH Open Access

Self-adapting root-MUSIC algorithm and its real-valued formulation for acoustic vector sensor array

Peng Wang1,3, Guo-jun Zhang1,2, Chen-yang Xue1,2, Wen-dong Zhang1,2 and Ji-jun Xiong1,2*

Abstract

In this paper, based on the root-MUSIC algorithm for acoustic pressure sensor array, a new self-adapting root-MUSIC algorithm for acoustic vector sensor array is proposed by self-adaptive selecting the lead orientation vector, and its real-valued formulation by Forward-Backward(FB) smoothing and real-valued inverse covariance matrix is also proposed, which can reduce the computational complexity and distinguish the coherent signals. The simulation experiment results show the better performance of two new algorithm with low Signal-to-Noise (SNR) in direction of arrival (DOA) estimation than traditional MUSIC algorithm, and the experiment results using MEMS vector hydrophone array in lake trails show the engineering practicability of two new algorithms.

Keywords: Acoustic vector sensor, DOA, MUSIC

1.BackgroundCompared to traditional acoustic pressure sensor, the acoustic vector sensor can measure both the scalar acoustic pressure and the acoustic particle velocity vector at a certain point of the acoustic field. So it possesses higher direction sensitivity and can acquire more measurement information [1-3]. By taking advantage of the extra information, vector sensors arrays are able to improve the direction-of-arrival (DOA) estimation performance without increasing array aperture size.Nehorai and Paldi have developed the measurement model of the acoustic vector sensor array for dealing with narrowband sources [4], many methods such as MUSIC algorithms have been proposed for applying acoustic vector sensor array to DOA estimation problems [5-8].

Root-MUSIC algorithm is a polynomial form of MUSIC algorithm [7,8]. This algorithm adopts the roots of a polynomial to replace the search for spatial spectrum in MUSIC algorithm, reducing the calculation amount

and improving estimation performance. Nevertheless, it is mainly applied to acoustic pressure sensor array.

Combining the Micro Electronic Mechanical Systems (MEMS) technology with design of vector hydrophone, it can break the performance limitation of existing hydrophone. A novel biomimetic MEMS vector hydro-phone has been developed by Xue and co-authous (Figure 1), and has been measured for index [9-12].

In this paper, a self-adapting root-MUSIC algorithm and its real-valued formulation for acoustic vector sensor array are proposed. Furthermore, the comparison of performance between this algorithm and MUSIC algorithm has been made by simulation method. Finally, the engineering practicability has been tested according to the experimental data of MEMS vector hydrophone array in lake trials.

2. Signal model of acoustic vector sensor array Consider N far-field narrowband signals incident on an uniform line array of M acoustic vector sensors along the x-axis in space, from directions = [1, 2, , N]T,

the received signal vector of the array can be expressed as

Z t

A

S t

Nv t

* Correspondence: mailto:[email protected]

Web End [email protected]

1Key Laboratory of Instrumentation Science & Dynamic Measurement, North University of China, Taiyuan 030051, China

2Science and Technology on Electronic Test & Measurement Laboratory, North University of China, Taiyuan 030051, ChinaFull list of author information is available at the end of the article

2012 Wang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

; 1 where Z(t) is the 3M 1 snapshot data vector of the array, S(t) is the N 1 vector of the signal, Nv (t) is the

Wang et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:228 Page 2 of 8 http://asp.eurasipjournals.com/content/2012/1/228

In practical calculation, the received data are finite, so the covariance matrix R can be estimated as

^

R

1 L

3M 1 vector of the Gaussian noise data vector, and the noise and the signal are independent, A() is the steering vector matrix of the acoustic vector sensor array.

A

a 1

; a 2

; ; a N

a1 1

u1; a2 2

u2; ; aN N

uN

;

2

XLi1Z t ZH t ; 5

where L is the number of snapshots.

3. Self-adapting root-MUSIC algorithm for vector sensor arrayThe basic idea of self-adapting root-MUSIC algorithm is: firstly weight summation for three-way signal of vector sensor, select the self-adaptive lead orientation, then construct polynomial by noise subspace, and finally estimate DOA of signals by finding the roots of polynomial.

Selection of lead orientation vector

Weight 1, cos , sin to the output signal pi(t),vix(t), viy(t) of ith vector sensor respectively, and make sum

yi t pi t

vix t

cos viy t

sin; 6

then the average power is P i() = E[|yi(t)|2].

The function of weight corresponds to make electronic rotary for the output of the vector sensor, the direction which reflects the maximum energy is the signal direction [13].

P i() is the output of spatial spectrum of ith vector sensor with relevant, reflects the energy distribution in space. It is the equivalent of a spatial filter, and can implement the signal and noise separation based on the orientation difference of the signal and interference.

The vector form of (6) for vector sensor array can be written as

Y WZ;

where W = diag[1, cos , sin , , 1, cos , sin ].

Take

P

1 M

Figure 1 MEMS vector hydrophone.

where ak k 1; e j

k ; e j2k ; ; e j M 1k

T

is the

acoustic pressure corresponding of the kth signal, k

2

d sink , in which d is the inter-element spacing, and is the wavelength corresponding to the maximum frequency of signals. uk = [1, cos k, sin k]T is the direction vector of the kth signal, and the notation denotes the Kronecker product.

So the covariance matrix for the array received signal is given by

R E Z t

ZH t

AE S t

SH t

AH

E N t

NH t

ARSAH 2I; 3

where RS is the signal covariance matrix, 2 is the energy of Gaussian white noise, I is the normalized noise covariance matrix, and ( )H stands for complex conjugate transpose.

From the theory of subspace decomposition, the eigen-decomposition is

R USSUHS UNNUHN; 4

where US is the signal subspace spanned by eigenvectors corresponding to major eigenvalues of matrix R, UN is

the noise subspace spanned by eigenvectors corresponding to small eigenvalues of matrix R.

XMi1Pi ; 7

where P() is spatial spectrum of array. The lead orientation 0 can be obtained from the maximum of P() for [0, 2].

The lead orientation vector can be received as

u 1; cos0; sin0

T; 8

where u is also known as self-adaptive lead vector.

Wang et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:228 Page 3 of 8 http://asp.eurasipjournals.com/content/2012/1/228

4. RV-Root-MUSIC algorithmIn the above method, the computational complexity will be reduced greatly if making eigendecomposition for a real-valued matrix instead of complex covariance matrix R[14]. The specific process is as follows:

Define

J3M JMI3; 13

where JM is the M M exchange matrix with ones on its antidiagonal and zeros elsewhere, and I3 is a 3 3 identity matrix.

Import the Forward-Backward(FB) smoothing matrix RFB as[15],

RFB

1

2 R J3MR J3M

; 14

where ( )* stands for complex conjugate.

The real-valued covariance matrix C can be obtained by

C PHR

FBP; 15 where P = Q I3 is a sparse matrix with real-valued conversion [14], and matrix Q can be chosen for arrays with an even and odd number of sensors respectively by(16) and (17).

Q2n

1

Construction of the polynomial

Define the polynomial

f z

zM 1FT 1=z

UNUHNF z

; 9

where F(z) = [1, z, , zM 1]T u, z = exp(j), = (2/)d sin , and is the azimuth angle of the signals to be estimated.

Let

B

b11 b12 b1M

b21 b22 b2M

bM1 bM2 bMM

0

B

B

@

1

C

C

A

UNUHN; 10

where bij (i, j = 1, 2, , M) are 3 3 symmetry sub-matrix.

Then

f z

zM 1FT 1=z

BF z

ubM1uH

zu

X2i1biM 2;iuT zM 1u

XMi1 bi;iuT

zMu

XM 1i1bi;i1uT z2M 3u

X2i1 bi;iM 2uT

z2M 2ub1MuT

XMk1u

!

zk 1

Xki1 biM k;iuT

XM 1k1u

!

XM ki1 bi;ikuT

p

; 16

In jIn

2 Jn jJn

zMk 1; 11

So the order of the polynomial f(z)is 2(M 1), it has (M 1) pair roots which every two conjugate with each another. and there are N roots which lie on the unit circle,

zi exp ji ;

Q2n1

1

p

2

0

@

In 0 jIn

0T

2

p 0T Jn 0 jJn

1

A; 17

where 0 is the n 1zero vector.

It is proved that C is a real-valued covariance matrix as follows.

Because of Q* = JQ,JQ* = QandJH = J, then

PHJ3MR J3MP QI3

H JMI3

R JMI3

QI3

QHI3

JMI3 R JMI3

QI3

i 1; 2; ; N:In practical calculation, considering the error of covariance matrix, the N roots ^zi nearest to the unit circle can be estimated as the DOAs of the signals.

^

i arcsin

2d arg ^zi

f g

; i 1; 2; ; N: 12

To sum up, the self-adapting root-MUSIC algorithm can be formulated as the following six-step procedure:

Step 1: Compute R by (3), and the estimate is given by(5).

Step 2: Obtain UN from the eigendecomposition of R

by (4).Step 3: Compute the lead vector u by (8).

Step 4: Construct the polynomial f(z) by (11).

Step 5: Find the root of the polynomial f(z), and select the roots ^zi that are nearest to the unit circle as being the roots corresponding to the DOA estimates.

Step 6: Receive ^

i to the DOA estimates by (12).

QHJM

R

JMQ I3

3

Q

HI3 h i

R Q I3

QI3

I

HR QI3

P

HR P PHRP

;

18

So,

C PHR

FBP

12 PHRP PHJ3MR J3MP

Re PHRP

;

19

where Re() is the real part operator.

Wang et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:228 Page 4 of 8 http://asp.eurasipjournals.com/content/2012/1/228

Let the eigendecompositions of the matrix C be defined in a standard way

C ESSEHS ENNEHN; 20

Similarly to (9), the real-valued root-MUSIC polynomial can be used

f z

zM 1FT 1=z

ENEHNF z

; 21

The computational complexity of self-adapting root-MUSIC algorithm and its real-valued formulation is discussed as follows.

The mainly difference between two methods is that the processing of the covariance matrix. Firstly, the reconstruction of covariance matrix R by (14) and (15) is necessary for RV-Root-MUSIC algorithm, Since the array covariance matrixR is a 3M 3M complex matrix, the matrix Ccan be constructed using 2 (3M)3 real multiplications and (3M)2(3M 1) real additions by (19).

Secondly, the velocity of convergence for eigendecom-position of the complex matrix C and the real matrix R is O(n3). simultaneously, the noise subspace of the complex matrix R is also complex, and the noise subspace of the real matrix C is also real.

Finally, the polynomial f(z) can be constructed via complex matrix R using 4[(3M)2(3M N) + (3M)2 + 3M] real multiplications and 3[(3M)2(3M N 1) + (3M + 1) (3M 1)] real additions by (9), but the polynomial f(z) can be constructed via real matrix C using [(3M)2 (3M N) + (3M)2 + 3M] real multiplications and [(3M)2 (3M N 1) + (3M + 1)(3M 1)] real additions by (21), so it is possible that the computational complexity for real matrix Ccan be reduced up to 75% real

multiplications and 66.7% real additions compared to the complex matrix R.

From the above analysis, the computational complexity of the RV-Root-MUSIC algorithm is significantly lower than the self-adapting root-MUSIC algorithm thanks to the eigendecomposition of the real-valued matrix C instead of that of the complex matrix R. On the other hand, due to the inherent forward-backward averaging effect by (14), RV-Root-MUSIC algorithm can separate two completely coherent sources and provides improved estimates for correlated signals. This will be validated in the last experiment of lake trials.

5. Simulation experimentTo verify the performance of the proposed self-adapting root-MUSIC algorithm and RV-Root-MUSIC algorithm, simulation experiments are carried out in the following.

The experiment employs the uniform linear array composed of four vector sensors, receives a signal with the frequency being 1kHz and the angle of incidence being 30, in which inter-element spacing is half wavelength and the adding noise is Gaussian white noise, and assumes the Signal-to-Noise (SNR) being 0dB and the number of snapshots being 200. The DOA estimation using self-adapting root-MUSIC algorithm is shown in Figure 2,

Figure 3 The curve between RMSE and SNR of three methods.

Figure 2 DOA estimation of self-adapting root-MUSIC algorithm.

Figure 4 Field experiment.

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Table 1 The DOA estimation result of different frequency signal using two methods

Signal frequency

Average of DOA estimation of MUSIC algorithm()

Average of DOA estimation of self-adapting root-MUSIC algorithm()

331Hz 90.2768 90.0377

800Hz 89.7857 89.8085

1kHz 89.8571 89.8598

1.5kHz 89.8636 89.4449

3kHz 89.6324 89.7719

where the notation * stands for all roots of polynomial, o for the DOA estimation, and - for the unit circle. From Figure 2, it can be seen that the DOA of signal can be correctly estimated using self-adapting root-MUSIC algorithm.

In addition, the performance between the proposed self-adapting root-MUSIC algorithm, the RV-Root-MUSIC algorithm and the traditional MUSIC algorithm is compared. In Figure 3, the root mean square error (RMSE) using 500 independent Monte Carlo trials for each SNR is shown when SNR changes from 20dB to 20dB. The proposed self-adapting root-MUSIC algorithm and the RV-Root-MUSIC algorithm have identical performance, and they have better performance for low SNRs and almost the same estimation performance for high SNRs with MUSIC algorithm.

Finally, In the above simulation conditions, the statistics for computing time of two algorithms has been made, and it is shown that the integrated computing time of the RV-Root-MUSIC algorithm is average less

about 23% than the self-adapting root-MUSIC algorithm by comparing two methods. Certainly, the computing time of the RV-Root-MUSIC algorithm can be reduced more with the increase of the number of array elements.

6. Lake trialsThe test experiment has been made in the Fenhe lake (Figure 4). The line array has been composed of two MEMS vector hydrophone with inter-element spacing being 0.5 m, and it has been fixed underwater 10 m at the side of the ship. The arrays compass could take real-time measurement for its pose to keep the arrays horizontality. Three experiments have been made respectively.

Experiment 1

The acoustic emission transducer has been placed in the direction with 90 of the array, launched 331Hz, 800Hz, 1kHz, 1.5kHz,3kHz continuous single-frequency signal respectively, the DOA has been estimated for receiving data, once per second. Table 1 is the average result of DOA estimation in different frequency signal using MUSIC algorithm and self-adapting root-MUSIC algorithm, Figure 5 is the time-bearing display of a single-frequency signal with 1.5kHz using two methods. The result shows the better performance of two methods.

Experiment 2

The experiment used a motor boat for moving target, which run from about 10 to about 160 position, tested track time is about 160s. Broadband noise which motor boat radiate has been narrowband filtered as 800Hz

Figure 5 Time-bearing display of signal with 1.5kHz.

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for the center frequency, once per second. Figure 6 shows the time-bearing display of target ship using MUSIC algorithm, conventional beam-forming method (which is also known as Bartlett beam-forming for acoustic vector sensor array) and self-adapting root-MUSIC algorithm respectively, the results are basically consistent with the actual trajectory of motor boat.

Experiment 3

The experiment used motor boat and emission transducer for two acoustic sources. The acoustic emission transducer has been placed in the direction with 180 of the array, launched 800Hz continuous single-frequency signal, simultaneously, the motor boat run from about 10 to about 180 position, tested track time is about 108s. Broadband noise which motor boat radiate has

been narrowband filtered as 800Hz for the center frequency, once per second.

Here, these two sources can be seen the coherent signals. First the real-valued covariance matrix C is homologous used to replace the complex matrix R in MUSIC algorithm and conventional beam-forming method, and then the time-bearing display of two sources using three methods respectively can be seen in Figure 7. The MUSIC algorithm can be more clearly distinguish between these two sources at the outset, but there will be some ambiguity when two sources approached (Figure 7(a)), and conventional beam-forming method is completely unable to distinguish (Figure 7(b)), but the RV-Root-MUSIC algorithm can clearly distinguish (Figure 7(c)), the results are basically consistent with the actual trajectory of motor boat and emission transducer.

Figure 6 Time-bearing display of motor boat.

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Figure 7 Time-bearing display of motor boat and emission transducer.

7. ConclusionsThe results of simulation experiment show the higher DOA estimation accuracy and lower RMSE of the new self-adapting root-MUSIC algorithm and the RV-Root-MUSIC algorithm than the traditional MUSIC algorithm, and the results in lake trails show the engineering practicability of two new algorithms, it can be verified that the performance of RV-Root-MUSIC algorithm distinguishing the coherent signals.

Competing interestsThe authors declare that they have no competing interests.

AcknowledgementsThis work is supported by the National Nature Science Foundation of China (Grant No. 61127008) and International Science & Technology Cooperation Program of China (Grant No.2010DFB10480).

Author details

1Key Laboratory of Instrumentation Science & Dynamic Measurement, North University of China, Taiyuan 030051, China. 2Science and Technology on Electronic Test & Measurement Laboratory, North University of China, Taiyuan 030051, China. 3School of Science, North University of China, Taiyuan 030051, China.

Received: 17 February 2012 Accepted: 10 October 2012 Published: 25 October 2012

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doi:10.1186/1687-6180-2012-228Cite this article as: Wang et al.: Self-adapting root-MUSIC algorithm and its real-valued formulation for acoustic vector sensor array. EURASIP Journal on Advances in Signal Processing 2012 2012:228.

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