# Error Self-Calibration Algorithm for Acoustic Vector Sensor Array

Authors: Peng Wang; Yujun Kong; Mingxing Zhang
Journal: Journal of Sensors
 (2019)

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## Abstract

In this paper, the errors of acoustic vector sensor array are classified, the impact factor of each error for the array signal model is derived, and the influence of each type of error on the direction-of-arrival (DOA) estimation performance of the array is compared by Monte Carlo experiments. Converting the directional error and location error to amplitude and phase errors, the optimization model and error self-calibration algorithm for acoustic vector sensor array are proposed. The simulation experiments and field experiment data processing of MEMS vector sensor array show that the proposed self-calibration algorithm has good parameter estimation performance and certain engineering practicability.


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## Body

## 1. Introduction

Compared 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, 2].A novel biomimetic MEMS vector hydrophone in Figure1 has been developed for acoustic detection application [3–6], which is based on the meso-piezoresistive effect and the acoustics theory of cylinder. It exploits the thin film of quantum as well as the sensitive unit, which adopts cilia bionic structure. The bionic structure is adapted by the combination of micro and nano cilia, and the underwater acoustic signal is detected by detecting the change of resistance of the four-beam microstructures. Through the experimental tests, the sensitivity of MEMS vector hydrophone can reach up to −191.9 dB and −188.1 dB (0dB=1V/μPa), which conforms to the sensitivity increments of 6 dB per octave in the working bandwidth ranging from 20 Hz to 1 KHz, and the directivity curve follows “8” cosine orientability, which conforms to avoid the left-right ambiguity. Because the micro-nano-sensitive structure can realize batch manufacturing and one-time integration, the MEMS vector hydrophone has the characteristics of small size, good consistency, and better suitability for the construction of arrays.Figure 1
MEMS vector sensor and microstructure.
(a)
Vector sensor
(b)
MicrostructureBy taking advantage of the extra information, acoustic vector sensor array is 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 [1]; many methods such as MUSIC algorithm and root-MUSIC algorithm have been proposed for applying acoustic vector sensor array to DOA estimation problems [6–9].In most traditional DOA estimation algorithms, the array steering matrix is all known, so these algorithms have excellent performance. However, in practice, due to various factors, all types of error are inevitable, so the array steering matrix will always be a certain degree of deviation or disturbance. Therefore, the performance of these high resolution DOA estimation algorithms will be seriously deteriorated and will even fail. In [6], a self-adapting root-MUSIC algorithm is proposed and successfully applied in lake trials of MEMS vector hydrophone array. In order to ensure the estimation accuracy of the proposed algorithm, we corrected the array which is composed of two MEMS hydrophones in advance, so we got better results. But if the array error is not corrected, the performance of the algorithm will be greatly reduced.So, array error is a technical bottleneck when these methods are be applied in practical. In the early days, array error correction was realized by discrete measurement, interpolation, and storage of array steering matrix, but these methods are costly and ineffective. Since 1990s, some scholars convert array error correction to a parameter estimation problem through the modeling of array disturbances. These array error calibration methods can be divided into two categories: auxiliary source correction [10] and adaptive correction [11]. Auxiliary source correction (ASC) estimates the array disturbance parameters offline by setting an auxiliary source in space. However, ASC method requires very high accurate azimuth information of the auxiliary source. When the azimuth information of the auxiliary source is biased, it will bring about large deviation. The adaptive correction method to carry out the joint estimation to the azimuth of the signal source and the array perturbation parameters is realized according to some optimization functions, which do not need the auxiliary source with known azimuth information, and can be corrected online in real time to complete the azimuth estimation, so its correction accuracy is higher, but it also brings a large amount of computation.Many scholars have done many researches about the array error correction. Liu et.al describe the attitude error of the array elements on the basis of using the Generalized Euler Angle from kinematics and built Gauss Perturbation Models of array errors [12]. Liu and Jin proposed an error correction for phase error in vector sensor array by using the projection of array steering matrix in signal subspace [13]. Liang et.al proposed an array correction method based on iterative optimization and subspace fitting and combined processing by using pressure and particle velocity [14]. Lim et.al presented two algorithms that are capable of estimating and compensating rotational perturbation angles based on steering vector error subspace criteria [15]. Liang et al. presented a self-calibration algorithm of altitude error, which can estimate the attitude error parameter and DOA of sources simultaneously by utilizing nonlinear iteration to minimize the objective function [16]. Zhang et al. proposed a calibration algorithm based on eigendecomposition for location, gain, and phase errors of acoustic vector sensor array, which require at least three DOA information from the cooperation source [17].But these methods are according to the acoustic pressure sensor array that the sensor does not have in directivity. But for the vector sensor array, each sensor has directional errors leading to the array which also has directional errors. The main types can be classified into three categories. The first type is amplitude and phase distortion, which needs to consider the error between the different sensors and the error between the acoustic pressure and the vibration velocity of one sensor. The second type is directional distortion including the directional deviation of the different sensors and directional deviation between vibration velocitiesvx and vy of one sensor. The third type is the position distortion. This is the position deviation of sensor placement in practical application.In this paper, the amplitude error, phase error, directional error, and position error are discussed, and a new error self-calibration algorithm for acoustic vector sensor array is proposed. The organization of this paper is as follows: the signal model, MUSIC algorithm, error model of AVS array, and the influences for the performance of DOA estimation are presented and discussed in Section2; the error self-calibration algorithm is proposed in Section 3; the simulation experiments and lake test of MEMS vector sensor array are made in Section 4; and the paper is summarized in Section 5.


## 2. Models

### 2.1. Signal Model of AVS Array

ConsiderN far-field narrowband signals’ incident on an uniform line array of M2-D acoustic vector sensor along the x-axis in geometric plane surface (Figure 2), from directions θ=θ1,θ2,⋯,θNT, the received signal vector of the array can be expressed as follows:
(1)Zt=AθSt+nt,where Zt is the 3M×1 snapshot data vector of the array, St is the N×1 vector of the signal, nt is the 3M×1 vector of the Gaussian noise data vector, the noise and the signal are independent, and Aθ is the 3M×N steering vector matrix of the acoustic vector sensor array
(2)Aθ=aθ1,aθ2,⋯,aθN,where aθk=1,ejβk,e2jβk,⋯,eM−1jβkT⊗uk,k=1,2,⋯,N, is the acoustic pressure corresponding of kth signal, βk=2π/λdsinθk, in which d is the interelement spacing, and λ is the wavelength corresponding to the maximum frequency of signals. uk=1,cosθk,sinθkT is the direction vector of the kth signal, and the notation ⊗ denotes the Kronecker product.Figure 2
MEMS vector sensor and microstructure.The covariance matrix ofZt is given by
(3)R=EZtZHt=AEStSHtAH+σn2I=ARsAH+σn2I,where RS is signal covariance matrix, σ2 is the power of Gaussian white noise, and
(4)Iv=1,12,12,⋯,1,12,12is the normalized noise covariance matrix.By the definition of the array covariance,R is a Hermitian matrix, so the eigenvalues λii=1,2,⋯,3M are all positive. Suppose λ1≥λ2≥⋯≥λ3M>0, and the corresponding eigenvector is e1,e2,⋯,e3M, then the eigendecomposition of R is
(5)R=U∑UH=∑m=13MλmememH,where Σ=diagλ1,λ2,⋯,λ3M is a diagonal matrix composed of all eigenvalues of matrix R and U=e1,e2,⋯,e3M is a matrix composed of eigenvectors of matrix R.The eigenvalues of the covariance matrixR can be used to distinguish the signal and noise, the 3M eigenvalues are divided into the front N large eigenvalues λ1≥λ2≥⋯≥λN>σn2 and 3M−N smaller eigenvalues, in which one part is σn2 and the other part is σn2/2, and the corresponding eigenvectors are constituted in the matrix Us=e1,e2,⋯,eN and Un=eN+1,eN+2,⋯,e3M, respectively, which are called the signal subspace and the noise subspace. In theory, the signal subspace and the noise subspace are orthogonal.


### 2.2. MUSIC Algorithm for AVS Array

From the theory of subspace decomposition, formula (5) can be written as
(6)R=Us∑sUsH+Un∑nUnH=∑m=1NλmememH+∑m=N+13MλmememH.It is obvious that each column vector of array steering matrix is located in the signal subspace, so the signal subspaceUs is the same as the space spanned by the column vector of signal subspace array steering matrix Aθ, then Aθ and noise subspace Un are orthogonal,
(7)aHθUn=0.In practice, considering the finite sample data, the vector sensor covariance matrix can be estimated as(8)R~=1L∑i=1LZtZHt,where L is the number of snapshot. And due to the existence of noise, aθ and Un cannot be completely orthogonal; DOA estimation of the MUSIC algorithm is actually achieved by minimum optimization
(9)θ~MUSIC=argminaHθU~nU~nHaθ,and the spectral estimation formula of MUSIC algorithm is
(10)θ~MUSIC=argmax1aHθU~nU~nHaθ.


### 2.3. Error Model and Simulation for AVS Array

#### 2.3.1. Amplitude Distortion

The array amplitude distortion includes the amplitude distortion between the sensors and the amplitude distortion between the components of the sensors. Ideally, it is consistent between the sensitivity of each sensor and between different components of the sensors. However, due to the processing technology and other issues, it is not be completely consistent. The sensitivity of themth vector sensor is
(11)gm=gmp,gmvx,gmvyT,m=1,2,⋯,M,where gmp, gmvx, and gmvy are the sensitivities of the acoustic pressure component p, the vibration velocity component vx, and the vibration velocity vy, respectively, in the mth sensor
(12)G=diagg1p,g1vx,g1vy,g2p,g2vx,g2vy,⋯,gMp,gMvx,gMvy.Suppose a single signal from a 40-degree incident on an uniform line array composed of 6 MEMS vector hydrophones, the interelement spacing is half wavelength, the adding noise is Gaussian white noise, SNR being −10 dB, 0 dB, and 10 dB, respectively, and the amplitude distortion satisfies the normal distribution in which the mean value is 0 and the variance is constant. When the variance of amplitude distortion changes from 0 to 2, using 500 independent Monte Carlo trials for each value, the curves between the RMSE of DOA estimation and the variance of amplitude distortion are shown in Figure3; it can be seen that the RMSE of DOA estimation becomes large with the increase of variance of array amplitude distortion, and the performance is close for different SNRs. This shows that the influence for the performance of DOA estimation from array amplitude distortion is more serious than from the noise.Figure 3
The curves between RMSE of DOA estimation and the variance of amplitude distortion with different SNRs.


#### 2.3.2. Phase Distortion

The phase error includes the phase error of the elements and the phase error of the component of elements. Due technological reasons, the phase of the sensor and the different components of the sensor are not the same.Suppose the phase response of themth vector sensor is
(13)ejϕm=ejφmp,ejφmvx,ejφmvyT,in which ejφmp,ejφmvx,andejφmvy are the phase responses of the acoustic pressure p, the vibration velocity component vx, and the vibration velocity vy, respectively, in the mth sensor, so the phase response of the vector sensor array is
(14)Φ=diagejϕ1p,ejϕ1vx,ejϕ1vy,ejϕ2p,ejϕ2vx,ejϕ2vy,⋯,ejϕmp,ejϕmvx,ejϕmvy.Under the above simulation conditions, suppose the phase distortion satisfies the norm distribution with the mean value is 0 and the variance is constant. When the variance of phase distortion changes from 0 to 2, the curves between the RMSE of DOA estimation and the variance of phase distortion are shown in Figure4 under 500 independent Monte Carlo trials for each value. It can be seen that the performance of DOA estimation under different SNRs is almost the same, which shows that the influence of phase distortion on array performance is more serious than that of noise. Even when the variance of phase error is small, the RMSE of DOA estimation is larger.Figure 4
The curves between RMSE of DOA estimation and the variance of phase distortion with different SNRs.


#### 2.3.3. Directional Distortion

An acoustic vector sensor has to be of an intrinsic character in the directional sensitivity. Ideally, the output of the single vector sensor is(15)q=1,cosθ,sinθT.When the array (such as linear array) is composed, the vibration velocity component of each sensor is pointed to the same direction, and the components of the vibration velocityvx and the vibration velocity vy are vertical. But in practice, it cannot completely guarantee the full alignment of the vibration velocity components, so it will bring the distortion of the direction of the sensor. At the same time, for the combined vector sensor, the orthogonality between the components of the vibration velocity and the vibration velocity cannot be guaranteed, so it can cause directional distortion.Under the influence of the directional distortion, the output of themth vector sensor is
(16)qm=1qmvxqmvy=1cosθ+Δθmxsinθ+Δθmx+Δθmy,m=1,2,⋯,M,in which Δθmx indicates the direction deviation of the vibration velocity vx and Δθmy indicates the vertical deviation of the components of the vibration velocity vx and vibration velocity vy in Figure 5.Figure 5
The diagram of directional distortion.The distortion matrix of the vector sensor array influenced by the directional error of elements is(17)Q=diag1,q1vx,q1vy,1,q2vx,q2vy,⋯,1,qMvx,qMvy.Similar to the above simulation conditions, the curves between the RMSE of DOA estimation and the variance of directional distortion are shown in Figure6 under 500 independent Monte Carlo trials for each value.Figure 6
The curves between RMSE of DOA estimation and the variance of directional distortion with different SNRs.


#### 2.3.4. Position Distortion

When the vector sensor array is formed in a certain way, the position of its installation location will inevitably produce a deviation, that is, the position error of the array element. At this time, the position error of array element affects the wave path difference and phase difference, so it influences orientation estimation of array.For uniform linear array, the position error of themth sensor is Δxm,Δym. The wave path difference between the mth and the referenced sensor is m−1d+Δxmcosθ+Δymsinθ, where d represents the interelement spacing. At this point, the corresponding phase difference is −2πm−1d+Δxmcosθ+Δymsinθ/λ. Compared with no error, the phase error of the mth sensor is −2πΔxmcosθ+Δymsinθ/λ.Therefore, the error of the array element will eventually lead to the phase error of the array element. For the error term, the vector sensor is a whole. So the phase error of the acoustic pressure and velocity of the same sensor are consistent. At this point, the effect of the whole array is(18)Ψ=diagφ1,φ1,φ1,φ2,φ2,φ2,⋯,φM,φM,φM,where φm=e−j2πΔxmcosθ+Δymsinθ/λ,m=1,2,⋯,M.Similar to the above simulation conditions, the curves between the RMSE of DOA estimation and the variance of directional distortion are shown in Figure7 under 500 independent Monte Carlo trials for each value. It can be seen that the position distortion has the greatest influence on the performance of DOA estimation.Figure 7
The curves between RMSE of DOA estimation and the variance of position distortion with different SNRs.


#### 2.3.5. Comprehensive Distortion

Considering the effects of the above four kinds of errors, the output of themth sensor is
(19)qm=qmpqmvxqmvy=gmpejϕnpφmgmvxejϕnvxφmcosθ+Δθmxgmvyejϕnpφmsinθ+Δθmx+Δθmy,m=1,2,⋯,M.Then, the output of the whole array is(20)Zt=ΓAθSt+nt,where Γ=G⋅Φ⋅Q⋅Ψ represents the comprehensive distortional matrix of array.Suppose an uniform line array is composed of 6 vector sensors, interelement spacing is half wavelength, and the adding noise is Gaussian white noise and assumes SNR being 0 dB and the number of snapshots being 500.Suppose two signals from 30° and 100° incident on the array, the SNR is 10 dB, the spectrum of MUSIC algorithm with no errors, and the comprehensive error shown in Figure8, it can be seen that the MUSIC algorithm almost failed to determine the orientation of the signal under the comprehensive effects of various errors.Figure 8
The MUSIC spectrum in the influence of comprehensive error.


## 2.1. Signal Model of AVS Array

ConsiderN far-field narrowband signals’ incident on an uniform line array of M2-D acoustic vector sensor along the x-axis in geometric plane surface (Figure 2), from directions θ=θ1,θ2,⋯,θNT, the received signal vector of the array can be expressed as follows:
(1)Zt=AθSt+nt,where Zt is the 3M×1 snapshot data vector of the array, St is the N×1 vector of the signal, nt is the 3M×1 vector of the Gaussian noise data vector, the noise and the signal are independent, and Aθ is the 3M×N steering vector matrix of the acoustic vector sensor array
(2)Aθ=aθ1,aθ2,⋯,aθN,where aθk=1,ejβk,e2jβk,⋯,eM−1jβkT⊗uk,k=1,2,⋯,N, is the acoustic pressure corresponding of kth signal, βk=2π/λdsinθk, in which d is the interelement spacing, and λ is the wavelength corresponding to the maximum frequency of signals. uk=1,cosθk,sinθkT is the direction vector of the kth signal, and the notation ⊗ denotes the Kronecker product.Figure 2
MEMS vector sensor and microstructure.The covariance matrix ofZt is given by
(3)R=EZtZHt=AEStSHtAH+σn2I=ARsAH+σn2I,where RS is signal covariance matrix, σ2 is the power of Gaussian white noise, and
(4)Iv=1,12,12,⋯,1,12,12is the normalized noise covariance matrix.By the definition of the array covariance,R is a Hermitian matrix, so the eigenvalues λii=1,2,⋯,3M are all positive. Suppose λ1≥λ2≥⋯≥λ3M>0, and the corresponding eigenvector is e1,e2,⋯,e3M, then the eigendecomposition of R is
(5)R=U∑UH=∑m=13MλmememH,where Σ=diagλ1,λ2,⋯,λ3M is a diagonal matrix composed of all eigenvalues of matrix R and U=e1,e2,⋯,e3M is a matrix composed of eigenvectors of matrix R.The eigenvalues of the covariance matrixR can be used to distinguish the signal and noise, the 3M eigenvalues are divided into the front N large eigenvalues λ1≥λ2≥⋯≥λN>σn2 and 3M−N smaller eigenvalues, in which one part is σn2 and the other part is σn2/2, and the corresponding eigenvectors are constituted in the matrix Us=e1,e2,⋯,eN and Un=eN+1,eN+2,⋯,e3M, respectively, which are called the signal subspace and the noise subspace. In theory, the signal subspace and the noise subspace are orthogonal.


## 2.2. MUSIC Algorithm for AVS Array

From the theory of subspace decomposition, formula (5) can be written as
(6)R=Us∑sUsH+Un∑nUnH=∑m=1NλmememH+∑m=N+13MλmememH.It is obvious that each column vector of array steering matrix is located in the signal subspace, so the signal subspaceUs is the same as the space spanned by the column vector of signal subspace array steering matrix Aθ, then Aθ and noise subspace Un are orthogonal,
(7)aHθUn=0.In practice, considering the finite sample data, the vector sensor covariance matrix can be estimated as(8)R~=1L∑i=1LZtZHt,where L is the number of snapshot. And due to the existence of noise, aθ and Un cannot be completely orthogonal; DOA estimation of the MUSIC algorithm is actually achieved by minimum optimization
(9)θ~MUSIC=argminaHθU~nU~nHaθ,and the spectral estimation formula of MUSIC algorithm is
(10)θ~MUSIC=argmax1aHθU~nU~nHaθ.


## 2.3. Error Model and Simulation for AVS Array

### 2.3.1. Amplitude Distortion

The array amplitude distortion includes the amplitude distortion between the sensors and the amplitude distortion between the components of the sensors. Ideally, it is consistent between the sensitivity of each sensor and between different components of the sensors. However, due to the processing technology and other issues, it is not be completely consistent. The sensitivity of themth vector sensor is
(11)gm=gmp,gmvx,gmvyT,m=1,2,⋯,M,where gmp, gmvx, and gmvy are the sensitivities of the acoustic pressure component p, the vibration velocity component vx, and the vibration velocity vy, respectively, in the mth sensor
(12)G=diagg1p,g1vx,g1vy,g2p,g2vx,g2vy,⋯,gMp,gMvx,gMvy.Suppose a single signal from a 40-degree incident on an uniform line array composed of 6 MEMS vector hydrophones, the interelement spacing is half wavelength, the adding noise is Gaussian white noise, SNR being −10 dB, 0 dB, and 10 dB, respectively, and the amplitude distortion satisfies the normal distribution in which the mean value is 0 and the variance is constant. When the variance of amplitude distortion changes from 0 to 2, using 500 independent Monte Carlo trials for each value, the curves between the RMSE of DOA estimation and the variance of amplitude distortion are shown in Figure3; it can be seen that the RMSE of DOA estimation becomes large with the increase of variance of array amplitude distortion, and the performance is close for different SNRs. This shows that the influence for the performance of DOA estimation from array amplitude distortion is more serious than from the noise.Figure 3
The curves between RMSE of DOA estimation and the variance of amplitude distortion with different SNRs.


### 2.3.2. Phase Distortion

The phase error includes the phase error of the elements and the phase error of the component of elements. Due technological reasons, the phase of the sensor and the different components of the sensor are not the same.Suppose the phase response of themth vector sensor is
(13)ejϕm=ejφmp,ejφmvx,ejφmvyT,in which ejφmp,ejφmvx,andejφmvy are the phase responses of the acoustic pressure p, the vibration velocity component vx, and the vibration velocity vy, respectively, in the mth sensor, so the phase response of the vector sensor array is
(14)Φ=diagejϕ1p,ejϕ1vx,ejϕ1vy,ejϕ2p,ejϕ2vx,ejϕ2vy,⋯,ejϕmp,ejϕmvx,ejϕmvy.Under the above simulation conditions, suppose the phase distortion satisfies the norm distribution with the mean value is 0 and the variance is constant. When the variance of phase distortion changes from 0 to 2, the curves between the RMSE of DOA estimation and the variance of phase distortion are shown in Figure4 under 500 independent Monte Carlo trials for each value. It can be seen that the performance of DOA estimation under different SNRs is almost the same, which shows that the influence of phase distortion on array performance is more serious than that of noise. Even when the variance of phase error is small, the RMSE of DOA estimation is larger.Figure 4
The curves between RMSE of DOA estimation and the variance of phase distortion with different SNRs.


### 2.3.3. Directional Distortion

An acoustic vector sensor has to be of an intrinsic character in the directional sensitivity. Ideally, the output of the single vector sensor is(15)q=1,cosθ,sinθT.When the array (such as linear array) is composed, the vibration velocity component of each sensor is pointed to the same direction, and the components of the vibration velocityvx and the vibration velocity vy are vertical. But in practice, it cannot completely guarantee the full alignment of the vibration velocity components, so it will bring the distortion of the direction of the sensor. At the same time, for the combined vector sensor, the orthogonality between the components of the vibration velocity and the vibration velocity cannot be guaranteed, so it can cause directional distortion.Under the influence of the directional distortion, the output of themth vector sensor is
(16)qm=1qmvxqmvy=1cosθ+Δθmxsinθ+Δθmx+Δθmy,m=1,2,⋯,M,in which Δθmx indicates the direction deviation of the vibration velocity vx and Δθmy indicates the vertical deviation of the components of the vibration velocity vx and vibration velocity vy in Figure 5.Figure 5
The diagram of directional distortion.The distortion matrix of the vector sensor array influenced by the directional error of elements is(17)Q=diag1,q1vx,q1vy,1,q2vx,q2vy,⋯,1,qMvx,qMvy.Similar to the above simulation conditions, the curves between the RMSE of DOA estimation and the variance of directional distortion are shown in Figure6 under 500 independent Monte Carlo trials for each value.Figure 6
The curves between RMSE of DOA estimation and the variance of directional distortion with different SNRs.


### 2.3.4. Position Distortion

When the vector sensor array is formed in a certain way, the position of its installation location will inevitably produce a deviation, that is, the position error of the array element. At this time, the position error of array element affects the wave path difference and phase difference, so it influences orientation estimation of array.For uniform linear array, the position error of themth sensor is Δxm,Δym. The wave path difference between the mth and the referenced sensor is m−1d+Δxmcosθ+Δymsinθ, where d represents the interelement spacing. At this point, the corresponding phase difference is −2πm−1d+Δxmcosθ+Δymsinθ/λ. Compared with no error, the phase error of the mth sensor is −2πΔxmcosθ+Δymsinθ/λ.Therefore, the error of the array element will eventually lead to the phase error of the array element. For the error term, the vector sensor is a whole. So the phase error of the acoustic pressure and velocity of the same sensor are consistent. At this point, the effect of the whole array is(18)Ψ=diagφ1,φ1,φ1,φ2,φ2,φ2,⋯,φM,φM,φM,where φm=e−j2πΔxmcosθ+Δymsinθ/λ,m=1,2,⋯,M.Similar to the above simulation conditions, the curves between the RMSE of DOA estimation and the variance of directional distortion are shown in Figure7 under 500 independent Monte Carlo trials for each value. It can be seen that the position distortion has the greatest influence on the performance of DOA estimation.Figure 7
The curves between RMSE of DOA estimation and the variance of position distortion with different SNRs.


### 2.3.5. Comprehensive Distortion

Considering the effects of the above four kinds of errors, the output of themth sensor is
(19)qm=qmpqmvxqmvy=gmpejϕnpφmgmvxejϕnvxφmcosθ+Δθmxgmvyejϕnpφmsinθ+Δθmx+Δθmy,m=1,2,⋯,M.Then, the output of the whole array is(20)Zt=ΓAθSt+nt,where Γ=G⋅Φ⋅Q⋅Ψ represents the comprehensive distortional matrix of array.Suppose an uniform line array is composed of 6 vector sensors, interelement spacing is half wavelength, and the adding noise is Gaussian white noise and assumes SNR being 0 dB and the number of snapshots being 500.Suppose two signals from 30° and 100° incident on the array, the SNR is 10 dB, the spectrum of MUSIC algorithm with no errors, and the comprehensive error shown in Figure8, it can be seen that the MUSIC algorithm almost failed to determine the orientation of the signal under the comprehensive effects of various errors.Figure 8
The MUSIC spectrum in the influence of comprehensive error.


## 2.3.1. Amplitude Distortion

The array amplitude distortion includes the amplitude distortion between the sensors and the amplitude distortion between the components of the sensors. Ideally, it is consistent between the sensitivity of each sensor and between different components of the sensors. However, due to the processing technology and other issues, it is not be completely consistent. The sensitivity of themth vector sensor is
(11)gm=gmp,gmvx,gmvyT,m=1,2,⋯,M,where gmp, gmvx, and gmvy are the sensitivities of the acoustic pressure component p, the vibration velocity component vx, and the vibration velocity vy, respectively, in the mth sensor
(12)G=diagg1p,g1vx,g1vy,g2p,g2vx,g2vy,⋯,gMp,gMvx,gMvy.Suppose a single signal from a 40-degree incident on an uniform line array composed of 6 MEMS vector hydrophones, the interelement spacing is half wavelength, the adding noise is Gaussian white noise, SNR being −10 dB, 0 dB, and 10 dB, respectively, and the amplitude distortion satisfies the normal distribution in which the mean value is 0 and the variance is constant. When the variance of amplitude distortion changes from 0 to 2, using 500 independent Monte Carlo trials for each value, the curves between the RMSE of DOA estimation and the variance of amplitude distortion are shown in Figure3; it can be seen that the RMSE of DOA estimation becomes large with the increase of variance of array amplitude distortion, and the performance is close for different SNRs. This shows that the influence for the performance of DOA estimation from array amplitude distortion is more serious than from the noise.Figure 3
The curves between RMSE of DOA estimation and the variance of amplitude distortion with different SNRs.


## 2.3.2. Phase Distortion

The phase error includes the phase error of the elements and the phase error of the component of elements. Due technological reasons, the phase of the sensor and the different components of the sensor are not the same.Suppose the phase response of themth vector sensor is
(13)ejϕm=ejφmp,ejφmvx,ejφmvyT,in which ejφmp,ejφmvx,andejφmvy are the phase responses of the acoustic pressure p, the vibration velocity component vx, and the vibration velocity vy, respectively, in the mth sensor, so the phase response of the vector sensor array is
(14)Φ=diagejϕ1p,ejϕ1vx,ejϕ1vy,ejϕ2p,ejϕ2vx,ejϕ2vy,⋯,ejϕmp,ejϕmvx,ejϕmvy.Under the above simulation conditions, suppose the phase distortion satisfies the norm distribution with the mean value is 0 and the variance is constant. When the variance of phase distortion changes from 0 to 2, the curves between the RMSE of DOA estimation and the variance of phase distortion are shown in Figure4 under 500 independent Monte Carlo trials for each value. It can be seen that the performance of DOA estimation under different SNRs is almost the same, which shows that the influence of phase distortion on array performance is more serious than that of noise. Even when the variance of phase error is small, the RMSE of DOA estimation is larger.Figure 4
The curves between RMSE of DOA estimation and the variance of phase distortion with different SNRs.


## 2.3.3. Directional Distortion

An acoustic vector sensor has to be of an intrinsic character in the directional sensitivity. Ideally, the output of the single vector sensor is(15)q=1,cosθ,sinθT.When the array (such as linear array) is composed, the vibration velocity component of each sensor is pointed to the same direction, and the components of the vibration velocityvx and the vibration velocity vy are vertical. But in practice, it cannot completely guarantee the full alignment of the vibration velocity components, so it will bring the distortion of the direction of the sensor. At the same time, for the combined vector sensor, the orthogonality between the components of the vibration velocity and the vibration velocity cannot be guaranteed, so it can cause directional distortion.Under the influence of the directional distortion, the output of themth vector sensor is
(16)qm=1qmvxqmvy=1cosθ+Δθmxsinθ+Δθmx+Δθmy,m=1,2,⋯,M,in which Δθmx indicates the direction deviation of the vibration velocity vx and Δθmy indicates the vertical deviation of the components of the vibration velocity vx and vibration velocity vy in Figure 5.Figure 5
The diagram of directional distortion.The distortion matrix of the vector sensor array influenced by the directional error of elements is(17)Q=diag1,q1vx,q1vy,1,q2vx,q2vy,⋯,1,qMvx,qMvy.Similar to the above simulation conditions, the curves between the RMSE of DOA estimation and the variance of directional distortion are shown in Figure6 under 500 independent Monte Carlo trials for each value.Figure 6
The curves between RMSE of DOA estimation and the variance of directional distortion with different SNRs.


## 2.3.4. Position Distortion

When the vector sensor array is formed in a certain way, the position of its installation location will inevitably produce a deviation, that is, the position error of the array element. At this time, the position error of array element affects the wave path difference and phase difference, so it influences orientation estimation of array.For uniform linear array, the position error of themth sensor is Δxm,Δym. The wave path difference between the mth and the referenced sensor is m−1d+Δxmcosθ+Δymsinθ, where d represents the interelement spacing. At this point, the corresponding phase difference is −2πm−1d+Δxmcosθ+Δymsinθ/λ. Compared with no error, the phase error of the mth sensor is −2πΔxmcosθ+Δymsinθ/λ.Therefore, the error of the array element will eventually lead to the phase error of the array element. For the error term, the vector sensor is a whole. So the phase error of the acoustic pressure and velocity of the same sensor are consistent. At this point, the effect of the whole array is(18)Ψ=diagφ1,φ1,φ1,φ2,φ2,φ2,⋯,φM,φM,φM,where φm=e−j2πΔxmcosθ+Δymsinθ/λ,m=1,2,⋯,M.Similar to the above simulation conditions, the curves between the RMSE of DOA estimation and the variance of directional distortion are shown in Figure7 under 500 independent Monte Carlo trials for each value. It can be seen that the position distortion has the greatest influence on the performance of DOA estimation.Figure 7
The curves between RMSE of DOA estimation and the variance of position distortion with different SNRs.


## 2.3.5. Comprehensive Distortion

Considering the effects of the above four kinds of errors, the output of themth sensor is
(19)qm=qmpqmvxqmvy=gmpejϕnpφmgmvxejϕnvxφmcosθ+Δθmxgmvyejϕnpφmsinθ+Δθmx+Δθmy,m=1,2,⋯,M.Then, the output of the whole array is(20)Zt=ΓAθSt+nt,where Γ=G⋅Φ⋅Q⋅Ψ represents the comprehensive distortional matrix of array.Suppose an uniform line array is composed of 6 vector sensors, interelement spacing is half wavelength, and the adding noise is Gaussian white noise and assumes SNR being 0 dB and the number of snapshots being 500.Suppose two signals from 30° and 100° incident on the array, the SNR is 10 dB, the spectrum of MUSIC algorithm with no errors, and the comprehensive error shown in Figure8, it can be seen that the MUSIC algorithm almost failed to determine the orientation of the signal under the comprehensive effects of various errors.Figure 8
The MUSIC spectrum in the influence of comprehensive error.


## 3. Error Self-Calibration Algorithm for Vector Sensor Array

The statistical analysis and simulation results show that all errors can be attributed to the amplitude and phase distortions of the vector array. The presence of these model errors will affect the DOA estimation of the vector array. The amplitude distortions will change the eigenvalue of the covariance matrix received by the array and then change the peak height and increase the width of the main peak and the height of the side lobe in the spectrum, while the phase error changes the eigenvector of the covariance matrix, which results in the offset of the maximum of the spectrum and the estimation error. On the basis of the classical self-calibration algorithm for array errors of acoustic pressure [18, 19], a self-calibration algorithm for vector sensor array error is proposed.The basic principle of the self-calibration algorithm is the use of noise subspace and signal subspace orthogonal, even space of noise subspace and array steering vector are orthogonal. Then, the cost function can be obtained by(21)J=∑i=1NUnHΓaθi2=∑i=1NaHθiΓHUnUnHΓaθi.The self-calibration of array error can be achieved by the iterative solution for optimizing the cost functionJ, i.e.,
(22)Γ^,θ^=argminJΓ,θ.Let(23)αθ=diagaθ1diagaθ2⋱diagaθN3M2×3M2,δ=vecΓ3M2×1,which represent the straightening operation of matrix Γ, then
(24)Γaθ=αθδ,so the cost function of formula (21) can be deformed to
(25)J=∑i=1NaHθiΓHUnUnHΓaθi=δHΩθδ,(26)Ωθ=∑i=1NαHθiUnUnHαθi.Hence, we want to minimize formula (22) with respect to δ under the constraint δHw=1, in which
(27)w=1,0,0,⋯,0⏟3MT.The result of this quadratic minimization problem under linear constraints is well known by Lagrange multiplier method and given by(28)δ=Ω−1θwwTΩ−1θw.The solution of the above optimization problem can be solved by iteration of alternating calculation of parametersθ and Γ (Algorithm 1).Algorithm 1
Error self-calibration algorithm for AVS array.
(a) Initialization:k=0,Γ^k=Γ0
(b) Use the matrix of vector sensor array of amplitude and phase distortionΓ^k and calculate spectrum of MUSIC algorithm based on formula (10)
PθΓ^k=1aHθiΓHUnUnHΓaθi,
and the azimuth corresponding to theN spectral peaks of the spatial spectrum is used as the DOA estimation θ^k of the signal source
(c) Fix azimuth estimationθ^k and calculate Γ^k+1 from the vector δ^ by formula (28)
Γ^k+1=diagδ^k+1
(d) Compute the cost function value by formula (25) to determine whether the convergence will stop with threshold ε, and the convergence condition is
Jk+1−Jk≤ε,
if it is convergent, then the iteration stops. Otherwise,k=k+1, go back to (b).In each step of the iterative optimization process and in every direction and every step of frequency iteration estimation, the cost functionJ will be reduced so that
(29)J0≥J1≥⋯≥Jk≥0,so Jk is a convergent series and can converge to an optimal solution. In a large number of simulation experiments, we find that the initial value of Γ0 is taken as the 3M-order unit matrix I3M and the effect is better; this is equivalent to the choice of initial values which are assumed to be error-free.


## 4. Results and Discussion

### 4.1. Simulation Experiments

Suppose two signal sources from 30° and 100° incident on the uniform linear array are composed of 6 AVSs, where the interelement spacing is half wavelength, adding noise is Gaussian white noise, SNR is 10 dB, and the number of snapshots is 500. When the amplitude response coefficient satisfies the normal distribution in which the mean value is 1 and variance is 0.5, the phase response satisfies the normal distribution in which the mean value is 0 and variance is 1, the error calibration effect of the vector sensor array using Algorithm1 is shown in Figure 9.Figure 9
The MUSIC spectrum of self-calibration iteration.In the iterative process, the cost function value and the position of the two signal sources are estimated as shown in Table1, and the curve of iteration process is shown in Figure 10. With the decrease of the cost function value, the orientation of the two signal sources is more and more closer to the real orientation. For the threshold ε=1E−4, the cost function value is 0.002 after the fifth calibration, and its value remains unchanged in the next iteration; so the iterative process converges, it also shows that the convergence speed of the proposed algorithm is very fast. The calibration result of the amplitude and phase distortion and relative errors of the sensor array of 6 elements (18 channels) are shown in Table 2. It is seen that the maximum relative error of amplitude error estimate is 1.35% and the maximum relative error of phase error estimation is 10.87%.Table 1
Self-calibration iterative process of array error of vector sensor.
Iteration number
Signal 1
Signal 2
Cost function value ofJ
1
31.7
99
10
2
30.4
99.7
0.1703
3
30.0
99.9
0.013
4
29.9
99.9
0.0005
5
29.9
99.9
0.0002
6
29.9
99.9
0.0002Figure 10
The iteration curve of cost function.Table 2
Error self-calibration result of vector sensor array.
Vector
Actual amplitude error
Calibration amplitude error
Relative error (%)
Actual phase error (rad)
Calibration phase error (rad)
Relative error (%)
1
Pressure
1
1
0.00
0
0
0.00
Velocityx
0.6603
0.6514
1.35
-1.3566
-1.3586
0.15
Velocityy
1.4775
1.487
0.64
0.7286
0.7283
0.04
2
Pressure
0.9364
0.9324
0.24
0.6072
0.6315
4.00
Velocityx
1.3247
1.3076
1.29
-0.8283
-0.8023
3.14
Velocityy
1.2993
1.3060
0.52
2.8876
2.9122
0.85
3
Pressure
0.9073
0.9031
0.07
0.9191
0.9728
5.84
Velocityx
0.0096
0.0095
1.04
0.4058
0.4379
7.91
Velocityy
1.1815
1.1892
0.65
1.9162
1.9696
2.79
4
Pressure
1.352
1.3486
0.25
-2.4557
-2.3770
3.20
Velocityx
0.6237
0.6157
1.28
-0.0911
-0.0812
10.87
Velocityy
0.9915
0.9966
0.51
-0.0247
-0.0223
9.72
5
Pressure
0.7792
0.7786
0.0800
0.1999
0.2072
3.65
Velocityx
1.8507
1.8257
1.35
-1.0383
-0.9320
10.24
Velocityy
0.9847
0.9912
0.66
0.5294
0.5365
1.34
6
Pressure
1.102
1.0992
0.25
-2.2171
-2.0839
6.01
Velocityx
0.4761
0.47
1.28
0.1074
0.1011
5.87
Velocityy
0.6323
0.6355
0.51
0.4389
0.4722
7.59So the results of the algorithm for the calibration of amplitude and phase errors are satisfactory.


### 4.2. Lake Experiment

The test experiment has been made in Fenhe lake. The line array has been composed of four MEMS vector sensor with interelement spacing being 0.5 m, and it has been fixed underwater 10 m at the side of the ship. The array’s compass could take real-time measurement for its pose to keep the array’s horizontality.The experiment used a motor boat as the moving target, which run from about 130° to about 70°, tested track time is about 80 s. Broadband noise in which motor boat radiates has been narrowband-filtered as 800 Hz for the center frequency, once per second. The tracking results of the conventional beamforming, MUSIC algorithm, and MUSIC algorithm by error self-calibration are shown in Figure11. The results shown that the estimation results of conventional beamforming are very fuzzy; the results of the MUSIC algorithm are clear than the conventional beamforming but not clear in some directions. After using the MUSIC algorithm of the array error self-calibration, it can be very clear to show the operating trajectory of the motor ship.Figure 11
Displacement process diagrams of three methods.
(a)
Conventional beamforming
(b)
MUSIC algorithm
(c)
MUSIC algorithm by error self-calibration


## 4.1. Simulation Experiments

Suppose two signal sources from 30° and 100° incident on the uniform linear array are composed of 6 AVSs, where the interelement spacing is half wavelength, adding noise is Gaussian white noise, SNR is 10 dB, and the number of snapshots is 500. When the amplitude response coefficient satisfies the normal distribution in which the mean value is 1 and variance is 0.5, the phase response satisfies the normal distribution in which the mean value is 0 and variance is 1, the error calibration effect of the vector sensor array using Algorithm1 is shown in Figure 9.Figure 9
The MUSIC spectrum of self-calibration iteration.In the iterative process, the cost function value and the position of the two signal sources are estimated as shown in Table1, and the curve of iteration process is shown in Figure 10. With the decrease of the cost function value, the orientation of the two signal sources is more and more closer to the real orientation. For the threshold ε=1E−4, the cost function value is 0.002 after the fifth calibration, and its value remains unchanged in the next iteration; so the iterative process converges, it also shows that the convergence speed of the proposed algorithm is very fast. The calibration result of the amplitude and phase distortion and relative errors of the sensor array of 6 elements (18 channels) are shown in Table 2. It is seen that the maximum relative error of amplitude error estimate is 1.35% and the maximum relative error of phase error estimation is 10.87%.Table 1
Self-calibration iterative process of array error of vector sensor.
Iteration number
Signal 1
Signal 2
Cost function value ofJ
1
31.7
99
10
2
30.4
99.7
0.1703
3
30.0
99.9
0.013
4
29.9
99.9
0.0005
5
29.9
99.9
0.0002
6
29.9
99.9
0.0002Figure 10
The iteration curve of cost function.Table 2
Error self-calibration result of vector sensor array.
Vector
Actual amplitude error
Calibration amplitude error
Relative error (%)
Actual phase error (rad)
Calibration phase error (rad)
Relative error (%)
1
Pressure
1
1
0.00
0
0
0.00
Velocityx
0.6603
0.6514
1.35
-1.3566
-1.3586
0.15
Velocityy
1.4775
1.487
0.64
0.7286
0.7283
0.04
2
Pressure
0.9364
0.9324
0.24
0.6072
0.6315
4.00
Velocityx
1.3247
1.3076
1.29
-0.8283
-0.8023
3.14
Velocityy
1.2993
1.3060
0.52
2.8876
2.9122
0.85
3
Pressure
0.9073
0.9031
0.07
0.9191
0.9728
5.84
Velocityx
0.0096
0.0095
1.04
0.4058
0.4379
7.91
Velocityy
1.1815
1.1892
0.65
1.9162
1.9696
2.79
4
Pressure
1.352
1.3486
0.25
-2.4557
-2.3770
3.20
Velocityx
0.6237
0.6157
1.28
-0.0911
-0.0812
10.87
Velocityy
0.9915
0.9966
0.51
-0.0247
-0.0223
9.72
5
Pressure
0.7792
0.7786
0.0800
0.1999
0.2072
3.65
Velocityx
1.8507
1.8257
1.35
-1.0383
-0.9320
10.24
Velocityy
0.9847
0.9912
0.66
0.5294
0.5365
1.34
6
Pressure
1.102
1.0992
0.25
-2.2171
-2.0839
6.01
Velocityx
0.4761
0.47
1.28
0.1074
0.1011
5.87
Velocityy
0.6323
0.6355
0.51
0.4389
0.4722
7.59So the results of the algorithm for the calibration of amplitude and phase errors are satisfactory.


## 4.2. Lake Experiment

The test experiment has been made in Fenhe lake. The line array has been composed of four MEMS vector sensor with interelement spacing being 0.5 m, and it has been fixed underwater 10 m at the side of the ship. The array’s compass could take real-time measurement for its pose to keep the array’s horizontality.The experiment used a motor boat as the moving target, which run from about 130° to about 70°, tested track time is about 80 s. Broadband noise in which motor boat radiates has been narrowband-filtered as 800 Hz for the center frequency, once per second. The tracking results of the conventional beamforming, MUSIC algorithm, and MUSIC algorithm by error self-calibration are shown in Figure11. The results shown that the estimation results of conventional beamforming are very fuzzy; the results of the MUSIC algorithm are clear than the conventional beamforming but not clear in some directions. After using the MUSIC algorithm of the array error self-calibration, it can be very clear to show the operating trajectory of the motor ship.Figure 11
Displacement process diagrams of three methods.
(a)
Conventional beamforming
(b)
MUSIC algorithm
(c)
MUSIC algorithm by error self-calibration


## 5. Conclusion

In this paper, according to the characteristics of the vector sensor array, the influence of DOA estimation performance by the array’s amplitude distortion, phase distortion, array element directional distortion, and position distortion is analyzed, and the vector array signal model under comprehensive error is built. Based on this, an error self-calibration algorithm based on vector array is proposed. Finally, the performance of the algorithm is verified by simulation experiments and lake experiments. These shown that the proposed algorithm has important value for the engineering application of MEMS vector sensor array.



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*Source: 9052547-2019-09-04.xml*