Cluster Computing https://doi.org/10.1007/s10586-017-1493-0
Energy DOA estimation of MUSIC symmetrical compressed spectrum on vector sensors array Biao Wang1 · Feng Chen1 · Yingchun Chen2 · Shiqi Mo2 Received: 30 October 2017 / Revised: 20 November 2017 / Accepted: 22 November 2017 © Springer Science+Business Media, LLC, part of Springer Nature 2017
Abstract Aiming at the shortcomings that MUSIC symmetrical compressed spectrum (MSCS) was only applicable to the sound pressure array, and the half spectrum search couldn’t distinguish the space radiation source and the mirror radiation source, a vector sensors array MSCS energy azimuth estimation algorithm was proposed. The new study used the idea of dimensionality reduction to combine the sound pressure and the vibration velocity to reduce the 3D guidance vector to one dimension, so that MSCS algorithm could be applied to vector sensors array, and then introduced spatial energy spectral function to optimize MSCS to solve the problem that MSCS half spectrum search could not distinguish between the space radiation source and the mirror radiation source. As the energy spectrum function has good anti-noise performance, so the new algorithm’s antinoise performance can be greatly improved. Finally, the correctness and validity of the algorithm are verified by simulation experiments. Keywords Multiple signal classification (MUSIC) · MUSIC symmetrical Compressed spectrum (MSCS) · Dimensionality reduction · Energy spectrum
Introduction As a hot issue of array signal processing, direction of arrival (DOA) [1–4] has been favored by many experts and scholars. The traditional beamforming is subject to Rayleigh limit, which seriously affects the performance of position estimation. In order to solve this problem, Schmidt introduced the classic multiple signal classification (MUSIC) [5] algorithm in 1986. In this method, the estimation function utilized the orthogonality of the signal subspace and the noise subspace to be constructed, so that the target orientation information can be obtained. Although the performance of this method is super-resolution, the complexity of the algorithm was greatly improved due to the processing of data covariance and the amount of spectral search, which is difficult to be applied to practical engineering [6,7]. In order to solve this problem, ESPRIT [8,9] and root-MUSIC [10,11] came into being
B
Feng Chen
[email protected]
1
School of Electronic and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China
2
Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China
which do not need spectrum search. However, these two methods have their own limitations. ESPRIT method requires the array with rotation invariance, so ESPRIT can’t be applied to arbitrary formation. Root-MUSIC requires the data array structure to meet the Vandermond structure, so rootMUSIC is mostly used in uniform linear array. In literature [12], the Nystrom method was introduced into the MUSIC algorithm, which uses the idea of low rank approximation to construct the signal subspace. Although the complexity of the operation is reduced to some extent, the estimated performance is greatly reduced through this algorithm. In literature [13], an online MUSIC method is proposed by Cao et al., which combines the candid covariance-free incremental principal component analysis (CCIPCA) algorithm with the MUSIC algorithm in the field of pattern recognition. This method does not need to construct and decompose the data covariance, and thereby reducing the complexity of MUSIC algorithm. But in the conditions of small snapshots, the estimated performance deteriorates dramatically. The above data processing is directed against data covariance. Compared with the complexity of spectral search, the processing complexity of covariance is negligible. As a result, Yan et al. proposed an MSCS algorithm in literature [14,15]. This method constructed mirror radiation source mapped in
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the symmetric interval, and generated a peak amplitude. So, conducting half spectrum search of the data and then doing discrimination are enough. By this means, the azimuth information will be obtained. Although this method reduced the computation of spectral search by half, it is necessary to use other follow-up methods to obtain the azimuth information and its noise immunity is poor. As all know, the vector hydrophone is a sensor capable of obtaining vibration velocity and pressure information in the sound field simultaneously [16,17]. Compared with the sound pressure sensor, it provides more abundant information of the vector data. However, most vector array algorithms treat sound pressure and vibration velocity as independent units without considering the correlation between sound pressure and vibration velocity [18,19]. Therefore, Yang et al [20] proposed a method based on the combination of sound pressure and vibration velocity. This method considered the sound pressure and the vibration velocity are related to the far-field sound field, However, in the isotropic noise field, the sound pressure and the vibration velocity are not related. Based on the concept of sound intensity, a joint method of sound pressure and vibration velocity was proposed to suppress the isotropic noise. In recent years, Shi et al. [21] Combined the method of joint processing of sound pressure and vibration velocity to the source number detection, which greatly enhanced the detection and detection performance. Based on the above, this paper makes full use of the correlation of sound pressure and particle velocity. Special combination of the information vector hydrophone is done to overcome the disadvantages of the MSCS not applying to space vector sensors array. Based on the idea of space energy spectrum, an MSCS method of energy DOA estimation on vector sensors array was proposed to solve the problem that the MSCS can’t distinguish the spatial radiation source and the mirror radiation source when doing half spectrum search. In this paper, the shortcoming that traditional MSCS is not suitable for the vector sensors array was overcame based on the idea of dimensionality reduction. At the same time, an energy spectrum function is introduced to distinguish the radiation source and the mirror radiation source, which improved performances of the algorithm in anti-noise and resolution greatly.
X(t) = A(θ )S(t) + N(t)
(1)
Among them, A(θ ) = [a(θ 1), . . . , a(θ K )]
(2)
= [1, e−j2π d sin(θk )/λ , . . . , Among them, a(θk ) −j2π(M−1)d sin(θ )/λ T k ] is a steering vector of the signal. e S(t) = [sT1 (t), . . . , sTK (t)]T is a signal source vector. N(t) = [n1 (t), . . . n M (t)]T is a M∗ 1 dimensional noise vector. The symbol T stands for transpose operations. Assuming that when the noise of each element is zero mean Gauss white noise and they are independent of each other, the covariance matrix of the received data is represented by R. Then R can be obtained by the following expression: H (3) R = E X(t)X (t) = A(θ )R S AH (θ ) + σ 2 I Among them, R S = E[S(t)S H (t)]. As a result, R can be considered as composed of signal subspace and noise subspace. H R = U S ΛS U H S + U N ΛN U N
(4)
1.2 MSCS algorithm Although the MUSIC algorithm is a high-resolution algorithm, but it requires a smaller spectral search step. So, MUSIC algorithm spectral search calculation is huge. MSCS is a fast algorithm which is evolved on MUSIC algorithm. The main principle of this method is to compress the MUSIC spatial spectrum symmetrically so that the spatial radiation source has an equal amplitude mirror radiation source in the y-axis symmetrical position. After the half spectrum search, do follow-up processing to get the source azimuth information. Thus, this algorithm can reduce the computational complexity of spectral search. Assuming that there is a radiation source S1 in the space under far-field conditions and the incident angle is θ1 . According to the orthogonality principle proposed by Schmidt, it can be known: a H (θ1 )U N = 0
(5)
1 Traditional DOA method
Among them, a(θ1 ) is a steering vector. From the Euler formula it can be known:
1.1 Receiving data model of acoustic pressure array
a(−θ1 ) = a∗ (θ1 )
Assuming that under the far field waveguide condition, K narrowband signal sources is received by the uniform line array composed of M hydrophones. The incident angle is θ and θ = {θ1 , θ2 , . . . , θ K }. The distance between each element is d, so the element receiving data model is as follows:
In Eq. (6), the symbol ∗ represents complex conjugate operation. According to Eqs. (5) and (6), it can be obtained: ∗ aH (θ1 )U N = a H (−θ1 )U ∗N = 0 H ∗ (7) a (θ1 ) U ∗N = aT (θ1 )U ∗N = 0
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(6)
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Do conjugate operations of the left and right sides of the Eq. (7): ∗ ∗ a(−θ1 ) H U ∗N = a(−θ1 )T U N = 0
(8)
It can be known from the formula above that there is a mirror radiation source S1’ in the symmetry interval of the radiation source S1, and the incident angles of both are opposite numbers. Therefore, the steering vectors of the two are complex conjugate relations. If the steering vectors of the former are orthogonal to U N , the steering vectors of the latter are orthogonal to U ∗N ]. According to this property, the spatial spectrum of MUSIC algorithm can be compressed symmetrically so as to construct a new space spectrum function. P M SC S (θ ) =
1 AH (θ )U
H ∗ T N U N U N U N A(θ )
(9)
Do spectrum search on Eq. (9) in the [− 90◦ , 0◦ ] interval or its symmetry interval. Get the peak position then and do follow-up processing to remove the mirror value. Finally, it can get the target orientation information.
2 Energy DOA estimation of MSCS on vector sensors array 2.1 The dimensionality reduction of vector sensors array The vector hydrophone is a sensor that can obtain the sound pressure and velocity information at the same time. Under the condition of far field, assuming the wave impedance is ρc = 1. Considering the two-dimensional case, the pitch angle is assumed to be 90◦ , so the receiving data model of the vector hydrophone can be expressed as: ⎧ ⎨ Vx = p cos θ V = p sin θ ⎩ y p = p(r , t)
(10)
A(θ ) = [a(θ1 ), a(θ2 ), . . . , a(θk )] a(θk ) = [1, exp(−jωτk ), . . . , exp(−jω(M − 1)τk )]T τk = d sin(θk )/c S(t) = [s1 (t), s2 (t), . . . , sk (t)]T
ψ 1 = diag{ejθ1 , ejθ2 , . . . , ejθ K } ψ 2 = diag{e−jθ1 , e−jθ2 , . . . , e−jθ K } ⎧ ⎨ P(t) = [p1 (t), p2 (t), . . . , pM (t)]T V 1 (t) = [v11 (t), v12 (t), . . . , v1M (t)]T ⎩ V2 (t) = [v21 (t), v22 (t), . . . , v2M (t)]T ⎧ ⎨ N p (t) = [np1 (t), np2 (t), . . . , npM (t)]T N 1 (t) = [n11 (t), n12 (t), . . . , n1M (t)]T ⎩ N 2 (t) = [n21 (t), n22 (t), . . . , n2M (t)]T So the steering vectors can be expressed as follows B(θ ) = [b(θ1 ), . . . , b(θ K )]
(13)
where b(θk ) = a(θk )⊗h(θk ) is the k-th column, the symbol ⊗ represents Kronecker operation and h(θk ) can be expressed as h(θk ) = [1 cos θk sin θk ]T . Do conjugate operations to b(θk ) b∗ (θk ) = a∗ (θk ) ⊗ h∗ (θk ) = a(−θk ) ⊗ [1 cos θk sin θk ]T (14) so b(−θk ) can be expressed as b(−θk ) = a(−θk ) ⊗ h(−θk ) = a(−θk ) ⊗ [1 cos θk − sin θk ]T (15)
In order to make full use of the vibration velocity information, combine the vibration velocity part of formula (10): ⎧ ⎨ V1 = Vx + j Vy V2 = Vx − j Vy ⎩ p = p(r , t)
Among them, P(t) represents a sound pressure signal. V1 (t) and V2 (t) represent two kinds of combined vibration velocity signals. A(θ ) is a steering vector of sound pressure array and S(t) is the incident signal. N p t) is the received noise on the sound pressure unit. N 1 (t) and N 2 (t) are the received noise on the combined vibration velocity signals. ψ 1 and ψ 2 represent the combination coefficients of the two combined vibration velocity signals which can be expressed as follows:
(11)
Extend the formula above to the array form, then the next formula is available: ⎧ ⎨ P(t) = A(θ )S(t) + N p (t) (12) V (t) = A(θ )ψ 1 S(t) + N 1 (t) ⎩ 1 V 2 (t) = A(θ )ψ 2 S(t) + N 2 (t)
so, the vector matrix of this receiving method does not satisfy formula (6) and (7) which are the preconditions of MSCS. In order to apply MSCS to the vector sensors array, there are two methods generally. The first method is to construct a steering vector to satisfy formula (6) and formula (7). The second method is to reduce the dimensionality of the array flow pattern of the vector hydrophone. In this paper, the second method is choosed. A new velocity model is needed to be constructed. Through the electronic rotation, the combined vibration velocity in the reference direction is available and its expression is as follows [20,21]:
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V c (t) = V 1 (t) cos(θr ) + V 2 (t) sin(θr ) = A(θ )ψ 1 cos(θr )S(t) + A(θ )ψ 2 sin(θr )S(t) + N c (t) = A(θ )ψ c S(t) + N c (t)
(16)
Among them, ψ c = diag{ejθ1 cos θr + e−jθ1 sin θr , . . . , cos θr + e−jθk sin θr }. N c (t) = N 1 (t) cos(θr ) + N 2 (t) sin (θr ) is the combined vector noise. The covariance function of sound pressure and combined velocity can be expressed as: (t) Rpv = E P(t)V H c (t) = E A(θ )S(t)SH (t)ψ c AH (θ ) + E N p (t)N H c ejθk
(17) Since in the isotropic noise field, the noise signals obtained by different types of received elements are not relevant. Therefore, in the uniform vector line array, the noise signals obtained by the three different types of receiving units are also uncorrelated. As a result, the noise signal received by the sound pressure unit is not related to the combined noise signal. H (18) E N p (t)N c (t) = 0 Assume that P = E[S(t)S H (t)]. Substitute formula (18) into formula (17), and then follow formula can be gained: Rpv = A(θ )Pψ c AH (θ )
(19)
At this point the steering vector has been degraded into steering vector and it satisfies formula (6) and formula (7). Therefore, MSCS can be used in vector sensors array processing to solve the problem that MSCS can’t be applied to vector sensors arrays. From the formula (18), it can be known that the joint processing method based on sound pressure and vibration velocity can play the role of anti-simultaneous noise, and reduce the signal-to-noise ratio threshold of the DOA estimation.
Assuming that the K signals are uncorrelated and they are independent of each other. Do eigenvalue decomposition on the covariance matrix obtained from formula (19): M
H H λi ei eH i = U S ΛS U S + U N ΛN U N
(20)
i=1
Among them, λi is the eigenvalue obtained by eigen decomposition. ei is the corresponding eigenvector of
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λ1 ≥ λ2 ≥ · · · ≥ λ K > λ K +1 = λ K +2 = . . . λ M
(21)
But in reality, due to the limited conditions like the limited number of snapshots et al., the noise characteristic value may not equal, i.e.: λ1 ≥ λ2 ≥ · · · ≥ λ K > λ K +1 > λ K +2 > . . . λ M
(22)
The top K eigenvalues obtained from formula (19) have the following characteristics: λi = σsi2 + σn2 (i = 1, 2, . . . , K ) σn2 = tr (ΛN ) (M − K)
(23) (24)
Among them, σn2 represents the average power of noise, σsi2 corresponds to the power of the previous K signals, and tr stands for the trace operation. From formula (23), it can be known that the K eigenvalues corresponding to the signal subspace are actually composed of the sum of the signal power and the average power of noise. Construct a diagonal matrix called ΛS which only have signal power by using the first k eigenvalues obtained by formula (20). The expression is as follows: ΛS
2 2 2 = diag λ1 − σn , λ2 − σn , . . . , λ K − σn
(25)
According to the calculation of the power of the signal source, the matrix RA is defined as follows: RA = A(θ )RS AH (θ ) = A(θ )PAH (θ ) = U S ΛS U HS
(26)
Only take the k th steering vector of the signal and proceed to the next step:
2.2 The construction of spatial energy spectrum function
Rpv =
λi . ΛS = diag(λ1 , λ2 , . . . , λ K )is a signal eigenvalue matrix consisting of larger eigenvalue while ΛN = diag(λ K +1 , λ K +2 , . . . , λ M ) is a noise eigenvalue matrix consisting of the smaller one. U S = [e1 , e2 , . . . , e K ] and span(U S ) = span(A) is the signal subspace. U N = [eK+1 , eK+2 , . . . , eM ] and span(U N ) is the noise subspace. In theoretical research, eigenvalue has the following characteristic:
H H + aH (θk )R+ A a(θk ) = a (θk )(A(θ )PA (θ )) a(θk )
= [A+ (θ )a(θk )]H P + A+ (θ )a(θk ) = δ Tk P + δ k
= [P + ]kk = 1/ pk
(27)
In the formula (27), (·)+ represents the pseudo inverse operation. δ k = [0, . . . , 1, 0, . . . , 0]T is a M-dimensional vector, in which the first k element is 1. pk is the power of the k th signal. From the above equation it can be known that the extreme value of the signal energy spectrum will only appear
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in the corresponding position of the DOA. With this feature, MSCS can be combined with the energy spectrum:
-10 ∗ T A H (θ )U N U H N U N U N A(θ )
(28)
When carrying out the half spectrum search in the [0◦ , 90◦ ] range on formula (28), the source localization is whereθ = ∗ T θi (i = 1, 2, . . . , k), A H (θ )U N U H N U N U N A(θ ) = 0 and + H A (θ )RA A(θ ) = 1/ pi . The peak of the MSCS spatial radiation source is amplified by the corresponding maximum value of the energy spectrum to form the main peak, but there is not corresponding energy spectrum peak at the mirror radiation source where forming the spurious peak. Therefore, when the spatial spectrum function of P N−MSCS carries on the half spectrum search, the main peak value appears at θ = θi (i = 1, 2, . . . , K ) which is the real radiation source. However, spurious peak corresponds to the mirror radiation source θ = −θi (i = 1, 2, . . . , K ).
-20 Power/dB
P N−MSCS (θ ) =
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Assuming that two equal-intensity sources are incident on a 12 elements uniform vector line array and a uniform line array and the incident angles of the two sources are 40◦ and −20◦ , respectively. When SNR is 5 dB, the search step is 1◦ , the snapshots called L is 100, the array element range is halfwavelength, and the searching range is [0◦ , 90◦ ], do space spectrum contrast test on algorithm in literature [14] and the algorithm in this paper and the result is shown in Fig. 1. In literature [14], two equal amplitude spatial spectra are formed on 20◦ and 40◦ which can’t distinguish the radiation source and the mirror radiation source from the figure. The algorithm in this paper forms a main peak at 40◦ and a pseudo peak at 20◦ which could distinguish between the radiation source and the mirror radiation source precisely. The method proposed in this paper can form a more sharp peak and lower side lobe suppression than MSCS method. As a result, the proposed algorithm has better anti-noise performance than MSCS algorithm. The spatial spectrum contrast of the MSCS method and the proposed method in this paper is conducted when the radiation source is symmetrical. The real angles are 20◦ and −20◦ and the results are shown in Fig. 2. At this point, the mirror peak values of the two methods coincide with the true peak values at 20◦ and −20◦ . However, the spectrum of the proposed algorithm is sharper, the band is narrower, and the side lobe suppression is lower, which explain the superiorities of the algorithm this paper proposed.
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In order to verify the estimated performance of the two methods at low signal-to-noise ratio, small snapshots and close angle, follow simulation experiments have been done. Keep the above experimental conditions constant, set the incident angle as −20◦ and 22◦ , set snapshots as 10 and set SNR as 0 dB then conduct the comparison test. The results are shown in Fig. 3. It can be seen from Fig. 3a that the MSCS method can only form one peak in the half-spectrum range, which can no longer estimate the radiation source and the mirror radiation source with close incident angle. From Fig. 3b, the method proposed in this paper can still distinguish
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two sources whose incident angles are −20◦ and 22◦ under this condition, which shows that the algorithm proposed in this paper has better resolution. Considering the estimated performance of the algorithm in the situation of multi-objective, set the incident angle as −20◦ , 40◦ and 60◦ respectively. Set the number of snapshots as 100 and keep the other experimental conditions unchanged. Then, carry out the multi-objective performance estimation experiment. The result is shown in Fig. 4. It can be seen from the figure that the algorithm this paper proposed can accurately estimate the azimuth information of the target, while the MSCS method has a certain degree of deviation at −60◦ , 60◦ , and can’t obtain the azimuth information accurately. In the case of multi-target, compared with the traditional MSCS algorithm, the proposed algorithm’s main peak and pseudo-peak are sharper. So this algorithm has a higher resolution than the traditional MSCS algorithm. In conclude, the algorithm this paper proposed is more suitable for multi-objective than the traditional MSCS algorithm.
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3.2 Comparison of the success rates between two methods Set the incident angles to 20◦ and 40◦ respectively, increase the signal-to-noise ratio from – 10 to 10 dB at intervals of 2 dB make the snapshot number L equal to 10, 20 and 50, and then conduct 300 times independent computer simulation experiments respectively. When the deviation is found within 1◦ , the algorithm is successful and the results are shown in Fig. 5. It can be seen from the figure,when the signal-tonoise ratio is constant, the two algorithms’ success rates have increased with increase of the snapshots. On the whole, the proposed algorithm has better performance than MSCS in the same SNR and snapshots. The main reason is that this method utilizes the anti-noise performance of the power spectrum function, which makes the performance more stable under the conditions of small snapshots and low SNR.
Cluster Computing 12 MSCS(L=10) MSCS(L=20) MSCS(L=50) Proposed method(L=10) Proposed method(L=20) Proposed method(L=50)
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MSCS sacrifices a small amount of computation to construct the energy spectrum function. Using the signal energy spectrum function to distinguish the real sound source and the mirror radiation source not only can achieve the desired effect but also suppress the side lobe and greatly enhance the spectral peak so that the peak is easier to be searched. These advantages are mainly due to the full use of the anti-noise performance of the signal subspace. Although the proposed algorithm has successfully solved the drawbacks of traditional MSCS, it still needs to partition the space which means the number of the source must be taken as prior knowledge.
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3.3 Comparison of the standard deviation between two methods In order to verify the performance of the algorithm further, standard deviation analysis experiment on the premise was done whose experimental conditions were constant. Figure 6 shows that as the number of the snapshots increase, the performances of the two algorithms become better. The performances of the two algorithms are greatly improved when the snapshots are 50. But on the whole, the performance of the algorithm proposed in this paper is better than that of MSCS, especially under the condition of low signal-to-noise ratio.
4 Conclusion Aiming at the shortcomings that the traditional acoustic pressure sensors array MSCS algorithm can’t be directly applied to three dimensional vector sensors array and can’t distinguish the real sound source and the mirror radiation source during the half spectral search (DOA ambiguity), this paper proposed a new algorithm called energy DOA estimation of MSCS based on vector sensors array after studying on the characteristics of vector array steering vector and signal energy spectrum function. Through the studies of energy DOA estimation of MSCS based on vector sensors array, following conclusions can be obtained. Compared with the method of acoustic pressure sensors array MSCS algorithm, energy DOA estimation of MSCS based on vector sensors array this paper proposed resists isotropic noise to some extent after doing combined processing of the acoustic pressure and the acoustic particle velocity. The energy estimation algorithm of vector array
Acknowledgements This work was supported by National Natural Science Foundation (11574120, U1636117), the Natural Science Foundation of Jiangsu province of china (BK20161359), Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX17_0604). the Open Project Program of the Key Laboratory of Underwater Acoustic Signal Processing, Ministry of Education, China (UASP1503) and Six Talent Peaks project of Jiangsu Province.
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Biao Wang received his M.S. and Ph.D. degree in information and signal processing from Institute of Acoustics, Chinese Academy of Sciences (IACAS), respectively at 2005 and 2009. Since 2009, He is currently working at School of Information and Electronic Engineering, Jiangsu University of Science and Technology. His research interests are array signal processing, target localization and tracking, underwater acoustic sensor networks.
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Feng Chen was born in Hubei, China. He is a postgraduate at Jiangsu University of Science and Technology. His current research interests include vector-sensor array processing and signal DOA estimation.
Yingchun Chen was born in Jiangsu, China. She is studying in Harbin Engineering University for master degree. Her current research interests include Information and Communication Engineering, Underwater Acoustic Engineering.
Shiqi Mo was born in Heilongjiang, China. He received the Ph.D. degree in signal and acoustic processing from the Harbin Engineering University .He is currently a Full Professor at the Harbin Engineering University. His current research interests include vector-sensor array processing, Underwater acoustic channel and sonar system environment.