Broadband MVDR Beamformer Applying PSO - Semantic Scholar
Broadband MVDR Beamformer Applying PSO Liang Wang and Zhijie Song Department of Ocean Technology, Ocean University of China, 238 Songling Road, Qingdao, 266100, P.R. China [email protected]
Abstract. In this paper, a broadband MVDR(minimum variance distortionless response) beamforming method based on time-domain (TMVDR) is presented. Using TMVDR beamformer, stable sample matrix estimation could be obtained in short time period. To obtain the stable optimum solution of TMVDR, a numerical searching method optimized by PSO algorithm with constrain condition is introduced. Out-sea experiment show the performance of TMVDR beamformer applying PSO algorithm proposed in this paper. Keywords: DOA estimation, adaptive beamforming, PSO algorithm.
1 Introduction The MVDR beamformer has superior performance on DOA(direction of arrival) estimation in low SNR (Signal to Noise Ratio) condition, and has been widely used in areas like sonar, radar and communication etc. In applying MVDR adaptive beamforming to passive detection of broadband signals two principal concerns arise. The first is snapshot deficient in the present of fast-moving targets. Broadband MVDR beamforming was usually implemented in frequency domain [1],[2], which firstly decomposes the time-domain data into certain sub-bands by Fourier Transform and then processes on each sub band. This method requires multiple snapshots in order to get stable correlation matrix estimation. Snapshot deficient causes distortion of beam pattern, loss of SINR, and related effects [3]. The second concern is unstable inverse when the eigenvalue dispersion of sample matrix is large since MVDR beamforming method currently adapt SMI (Sample Matrix Invert) algorithm [4]. That means tiny disturbance of sample matrix estimation will cause tremendous compute error when existing strong interference. Although diagonal loading [5] method could improve the robustness of the correlation matrix and obtain stable solution with fewer snapshots, but correspondingly the array gain and azimuth resolution ability decreases obviously because extra noise is introduced. In this paper, we discussed a TMVDR (Time-domain MVDR) beamformer applying PSO (Particle Swarm Optimization) algorithm [6]. The array weights of TMVDR beamformer are designed to be a complex vector by treating two-channel orthogonal signals as one complex analytical signal, which minimize the power at its output while providing, at the same time, a fixed response toward the direction of arrival of the signal of interest. Different with frequency domain MVDR, TMVDR doesn’t need block processing since it needn’t FFT transform, thus we can obtain relative stable Y. Tan, Y. Shi, and K.C. Tan (Eds.): ICSI 2010, Part I, LNCS 6145, pp. 152–159, 2010. © Springer-Verlag Berlin Heidelberg 2010
Broadband MVDR Beamformer Applying PSO
153
sample matrix estimation in short time period. To obtain the optimum solution of TMVDR, numerical search method optimized by PSO algorithm is proposed to replace the SMI algorithm. Numerical search algorithm can solve the problem of unstable inverse by searching a group of optimum solution in the weight vector space without diagonal loading. That is, we can always find a set of optimum weight vector to minimize the power of interference and noise by numerical searching method. The paper is organized as follows. In Sec II, a TMVDR beamformer based on time-domain analytical signal is formatted. Sec III proposes a numerical searching method optimized by PSO algorithm to obtain the stable solution of TMVDR. Sec IV investigates the performance of TMVDR beamformer applying PSO algorithm via out-sea experiment. Sec V contains a brief synopsis of this paper and results of our work.
2 TMVDR Beamformer Supposing the broadband passive signal is received by uniform line array composed of M elements, after delayed time of Δi (1 ≤ i ≤ M ) , the signal vector received by array is S = [ s1 , s2 , " sM ]T
(1)
X =S+N
(2)
The array output is:
where N = [ n1 , n2 , " nM ]T is the additive noise vector, T denotes transpose. Eq. (2) is Hilbert transformed to
X =S+N
(3)
where X = H ( X ) , S = H ( S ) , N = H ( N ) . Time domain analytical signal can be constructed as Y = Ys + Yn = X + j ⋅ X
(4)
where Ys denotes the signal part of time-domain analytical signal , Yn denotes the noise part .Thus the adaptive array output is: C = W Y = W Ys + W Yn H
H
H
(5)
where superscript H means transpose and conjugate , W = [ w1 w2 " wM ]T . When the scanning direction is aimed to the incoming signal , there is
s1 + j ⋅ s1 = s2 + j ⋅ s2 " = sM + j ⋅ sM = s + j ⋅ s where si is the ith interested signal received on array .Under the constrain condition
(6)
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L. Wang and Z. Song
M
∑w i =1
i
=1
(7)
Eq. (5) can be written as: C = s + j ⋅ s + W H Yn
(8)
When the interested signal, interference and noise are cross uncorrelated, because the real part and the imaginary part of analytical signal are orthogonal, then the array output power comes to: P = E{CC * } = W H RW = s 2 + ( s ) 2 + W H RnW
(9)
R = E{YY H }
(10)
Rn = E{YnYnH }
(11)
where
Hilbert transformation can’t change the power of the signal of interest, making ( s ) 2 = s 2 a constant. From Eq.(9), a minimum W H RnW means a minimum total output power. Therefore, TMVDR can be expressed as a constrained optimum problem :
min{ N P} W M
s.t.
∑ wi = 1
(12)
i =1
Using Eq. (10), exact sample matrix can be estimated in few snapshots period if the noise and interference is short time stable since TMVDR doesn’t need split the data to blocks for Fourier transform.
3 Numerical Search Method Using TMVDR beamformer, the total output power of array tends to be stable in short time period. The disturbance of sample matrix has little effect on output power. Thus we can obtain the optimum solution of Eq.(12) by numerical search algorithm. However, the computational complexity is huge by Numerical search algorithm than SMI algorithm. In this paper we adapt PSO algorithm to accelerate the convergence speed. 3.1 PSO Algorithm
Supposing a particle swarm composed of N particles, the position vector of the ith particle in the M dimension weights space is Z i = [ z1 z 2 " z M ]T
(13)
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155
and the moving velocity vector is Vi = [v1 v2 " vM ]T
(14)
Then the current position of each particle in the swarm is updated as follows: ⎧Vi l +1 = a ⋅Vi l + c1 ⋅ rand (0,1) ⋅ ( pi − Z il ) ⎪ + c2 ⋅ rand (0,1) ⋅ ( pg − Z il ) ⎨ ⎪ l +1 l l +1 ⎩ Z i = Z i + Vi
(15)
where a is the inertia weight, c1 is the own experience weight and c2 is the global experience weight, rand (0,1) is uniform random number distributing in the range [0,1], pi is the best previous position, pg is the global best position. The iteration terminates when pg remains invariant and then pg is the global optimum solution. PSO algorithm introduces own experience weight and companion’s experience weight, making the particle at the local extreme value easily out of the local optimum position. Hence PSO algorithm is efficient for searching problem and has rapid convergence velocity. 3.2 Range of Search Space
Supposing the optimum solution of (12) is
Wopt = [ w1 , w2 ", wM ]T = C D D
(16)
C = [C1 , C2 , " , CM ]T
(17)
D = [e jϕ1 , e jϕ2 , " , e jϕM ]T
(18)
where
C stands for the mode vector of optimum weight , ϕ1 , ϕ 2 , " , ϕ M the phase of the optimum weight vector. Defining Cmax = max N [Ci ] from Eq. (11), the minimum output power of array i =1, 2,", M
，
is K
M
l =1
i =1
Pmin = ∑ (∑ wi xil )2
(19)
where K is the number of sampling time points. According to constrain condition
∑ w = 1 ，make identity transformation to Eq.(19) M
i
i =1
156
L. Wang and Z. Song
K
M
Pmin = ∑ (∑ l =1
i =1
wi xil
K
M
∑w i =1
i
M
) 2 = ∑ [∑ l =1
i =1
wi xi / Cmax M
∑w /C l =1
i
K
M
]2 = ∑ [∑ l =1
max
i =1
wi xi M
∑w l =1
]2
(20)
i
where wi = wi / Cmax
(21)
the solution of MVDR is unique under ideal conditions, Hence,
W = Wopt
，
(22)
where W = [ w1 , w2 , " wM ]T . Form Eq.(20) and Eq.(21), the maximum mode of Wopt is 1. Therefore we can obtain the range of searching weight vector is Ci ∈ [0,1] Di ∈ [0, 2π ] , i = 1, 2, " , M .
和
3.3 Introduce of Constrain Condition
Supposing the complex weight vector of ith particle is W = [ w1 , w2 , " wM ]T . Constrain condition can be represented as ⎧M ⎪∑ Re( wi ) = 1 ⎪ i =1 ⎨M ⎪ Im( w ) = 0 i ⎪⎩∑ i =1
(23)
1 M ai = M 1 [Re( wi ) + ] ∑ M i =1
(24)
making Re( wi ) +
bi = Im( wi ) − Where
1 M
M
∑ Im(w ) i =1
(25)
i
1 is introduced to ensure ai ≤ 1 . Constructing the vector as follows, M Wc = [ a1 + jb1 , a2 + jb2 , " aM + jbM ]T
(26)
Eq. (26) is just the complex weight vector with constrain condition. Hence the steps of PSO algorithm can be described as follows:
Broadband MVDR Beamformer Applying PSO
1).
Set the initial position vector of particle as Eq.(16),where Ci = rand (0,1) j 2π rand (0,1)
2).
3).
4). 5).
157
，
. Di = e Using Eq.(24)and (25), respectively process the real part and imaginary part of the initial position vector, and then obtain the complex weight vector as Eq. (26). Calculate the objective function P in Eq.(9). Find the minimum objective function Pmin and record the best previous position pi and the best global position pg . Update the position of particle according to Eq. (13). The iteration terminates when Pmin (n) (n is the iteration times) remains invariant( Pmin (n + k ) − Pmin (n) < ε ) .The best global position pg at this time is the optimum solution of TMVDR.
4 Result of Sea Experiment 4.1 Sea Experiment Description
In order to investigate the performance of TMVDR beamformer applying PSO algorithm, the data of out-sea experiment was processed. The array in experiment was uniform line array composed of 32 sensors; the interval between each unit is 1m; the band width is 300-700Hz, the total time period for data processing is 100 seconds, within which the target were near 40º 70º 90º 140º ,150º and a strong interference source was around 0º~20º. Data sampling frequency was 6 KHz.
， ， ，
4.2 Signal Processing
The data was respectively processed by TMVDR method applying PSO algorithm and FMVDR (frequency domain MVDR) using diagonal loading algorithm. FMVDR’s covariance matrix estimation follow the next steps: Split the data into many blocks, each block has 512 sampling time points. There are 256 points overlap with two adjacent block, which is also the data length of each snapshot. After DFT, the cross-spectral density matrix was estimated on each frequency. The number of snapshots for each estimation is 64.That is ,the number of sampling time points for each processing is (64+1)×256=16640. Diagonal loading adapted the optimum factor[5]. TMVDR chooses 300-700Hz band width through band-pass filter, and only uses 1000 sampling time points to estimate the output power for each processing. 4.3 Comparison
Figure 1 displays energy level for 0°~180°azimuth angle v.s. time respectively processed by TMVDR and FMVDR. In Figure 1, as we have expected, TMVDR method has far better signal detection ability than FMVDR method. Tracks of sources are bright in the TMVDR result. Especially for the weak targets near 140°and 150° FMVDR almost
，
158
L. Wang and Z. Song
Fig. 1. Output of TMVDR method and FMVDR method
cannot detect the signal because the diagonal loading cause the degradation of array gain. In addition, the period of TMVDR for each processing is 1000/6000≈0.16s far shorter than FMVDR which period for each processing is 16640/6000≈2.77s.That makes TMVDR has better performance than FMVDR in signal detection for the fast moving targets.
，
，
5 Conclusion A time domain broadband MVDR beamforming method is presented in this paper. On the basis of constructing time domain analytical signal, TMVDR introduces complex weights and could obtain stable sample matrix estimation in short time period. To obtain the stable optimum solution of TMVDR, a numerical searching method optimized by PSO algorithm with constrain conditions is proposed to replace the SMI algorithm. Using PSO algorithm, stable optimum weights vector can be searched with rapid convergence velocity. The performance of TMVDR method applying PSO algorithm is put into test via out-sea experiment. Compared with FMVDR method applying diagonal loading algorithm, TMVDR method has better performance than FMVDR method for the problems of passive detection and azimuth estimation on broadband sound sources.
References 1. Kim, B.-C., Lu, I.-T.: High Resolution Broadband Beamforming Based on the MVDR Method. In: Proc. MTS/IEEE Oceans, vol. 3, pp. 1673–1676 (2000) 2. Yang, Y., Wan, C., Sun, C.: Broadband Beamspace DOA Estimation Algorithms. In: Proceeding of Oceans 2003, vol. 3, pp. 1654–1660 (2003)
Broadband MVDR Beamformer Applying PSO
159
3. Van Trees, H.L.: Optimum Array Processing. John Wiley & Sons, Inc., New York (2002) 4. Manolakis, D.G., Ingle, V.k., Kogon, S.M.: Statistical and Adaptive Signal Processing. Tsinghua University Press (2003) 5. Mestre, X., Lagunas, M.A.: Finite sample size effect on minimum variance beamformers: Optimum diagonal loading factor for large arrays. IEEE Transactions on Signal Processing 54, 69–82 (2006) 6. Li, B., Xiao, Y.: New Evolution Algorithm: Particle Swarm Optimization. Computer Science 30, 19–22 (2003)
Abstract. In this paper, a broadband MVDR(minimum variance distortionless response) beamforming method based on time-domain (TMVDR) is presented. Using TMVDR beamformer, stable sample matrix estimation could be obtained in short time period. To obtain the stable optimum solution of TMVDR, a numerical searching method optimized by PSO algorithm with constrain condition is introduced. Out-sea experiment show the performance of TMVDR beamformer applying PSO algorithm proposed in this paper. Keywords: DOA estimation, adaptive beamforming, PSO algorithm.
1 Introduction The MVDR beamformer has superior performance on DOA(direction of arrival) estimation in low SNR (Signal to Noise Ratio) condition, and has been widely used in areas like sonar, radar and communication etc. In applying MVDR adaptive beamforming to passive detection of broadband signals two principal concerns arise. The first is snapshot deficient in the present of fast-moving targets. Broadband MVDR beamforming was usually implemented in frequency domain [1],[2], which firstly decomposes the time-domain data into certain sub-bands by Fourier Transform and then processes on each sub band. This method requires multiple snapshots in order to get stable correlation matrix estimation. Snapshot deficient causes distortion of beam pattern, loss of SINR, and related effects [3]. The second concern is unstable inverse when the eigenvalue dispersion of sample matrix is large since MVDR beamforming method currently adapt SMI (Sample Matrix Invert) algorithm [4]. That means tiny disturbance of sample matrix estimation will cause tremendous compute error when existing strong interference. Although diagonal loading [5] method could improve the robustness of the correlation matrix and obtain stable solution with fewer snapshots, but correspondingly the array gain and azimuth resolution ability decreases obviously because extra noise is introduced. In this paper, we discussed a TMVDR (Time-domain MVDR) beamformer applying PSO (Particle Swarm Optimization) algorithm [6]. The array weights of TMVDR beamformer are designed to be a complex vector by treating two-channel orthogonal signals as one complex analytical signal, which minimize the power at its output while providing, at the same time, a fixed response toward the direction of arrival of the signal of interest. Different with frequency domain MVDR, TMVDR doesn’t need block processing since it needn’t FFT transform, thus we can obtain relative stable Y. Tan, Y. Shi, and K.C. Tan (Eds.): ICSI 2010, Part I, LNCS 6145, pp. 152–159, 2010. © Springer-Verlag Berlin Heidelberg 2010
Broadband MVDR Beamformer Applying PSO
153
sample matrix estimation in short time period. To obtain the optimum solution of TMVDR, numerical search method optimized by PSO algorithm is proposed to replace the SMI algorithm. Numerical search algorithm can solve the problem of unstable inverse by searching a group of optimum solution in the weight vector space without diagonal loading. That is, we can always find a set of optimum weight vector to minimize the power of interference and noise by numerical searching method. The paper is organized as follows. In Sec II, a TMVDR beamformer based on time-domain analytical signal is formatted. Sec III proposes a numerical searching method optimized by PSO algorithm to obtain the stable solution of TMVDR. Sec IV investigates the performance of TMVDR beamformer applying PSO algorithm via out-sea experiment. Sec V contains a brief synopsis of this paper and results of our work.
2 TMVDR Beamformer Supposing the broadband passive signal is received by uniform line array composed of M elements, after delayed time of Δi (1 ≤ i ≤ M ) , the signal vector received by array is S = [ s1 , s2 , " sM ]T
(1)
X =S+N
(2)
The array output is:
where N = [ n1 , n2 , " nM ]T is the additive noise vector, T denotes transpose. Eq. (2) is Hilbert transformed to
X =S+N
(3)
where X = H ( X ) , S = H ( S ) , N = H ( N ) . Time domain analytical signal can be constructed as Y = Ys + Yn = X + j ⋅ X
(4)
where Ys denotes the signal part of time-domain analytical signal , Yn denotes the noise part .Thus the adaptive array output is: C = W Y = W Ys + W Yn H
H
H
(5)
where superscript H means transpose and conjugate , W = [ w1 w2 " wM ]T . When the scanning direction is aimed to the incoming signal , there is
s1 + j ⋅ s1 = s2 + j ⋅ s2 " = sM + j ⋅ sM = s + j ⋅ s where si is the ith interested signal received on array .Under the constrain condition
(6)
154
L. Wang and Z. Song
M
∑w i =1
i
=1
(7)
Eq. (5) can be written as: C = s + j ⋅ s + W H Yn
(8)
When the interested signal, interference and noise are cross uncorrelated, because the real part and the imaginary part of analytical signal are orthogonal, then the array output power comes to: P = E{CC * } = W H RW = s 2 + ( s ) 2 + W H RnW
(9)
R = E{YY H }
(10)
Rn = E{YnYnH }
(11)
where
Hilbert transformation can’t change the power of the signal of interest, making ( s ) 2 = s 2 a constant. From Eq.(9), a minimum W H RnW means a minimum total output power. Therefore, TMVDR can be expressed as a constrained optimum problem :
min{ N P} W M
s.t.
∑ wi = 1
(12)
i =1
Using Eq. (10), exact sample matrix can be estimated in few snapshots period if the noise and interference is short time stable since TMVDR doesn’t need split the data to blocks for Fourier transform.
3 Numerical Search Method Using TMVDR beamformer, the total output power of array tends to be stable in short time period. The disturbance of sample matrix has little effect on output power. Thus we can obtain the optimum solution of Eq.(12) by numerical search algorithm. However, the computational complexity is huge by Numerical search algorithm than SMI algorithm. In this paper we adapt PSO algorithm to accelerate the convergence speed. 3.1 PSO Algorithm
Supposing a particle swarm composed of N particles, the position vector of the ith particle in the M dimension weights space is Z i = [ z1 z 2 " z M ]T
(13)
Broadband MVDR Beamformer Applying PSO
155
and the moving velocity vector is Vi = [v1 v2 " vM ]T
(14)
Then the current position of each particle in the swarm is updated as follows: ⎧Vi l +1 = a ⋅Vi l + c1 ⋅ rand (0,1) ⋅ ( pi − Z il ) ⎪ + c2 ⋅ rand (0,1) ⋅ ( pg − Z il ) ⎨ ⎪ l +1 l l +1 ⎩ Z i = Z i + Vi
(15)
where a is the inertia weight, c1 is the own experience weight and c2 is the global experience weight, rand (0,1) is uniform random number distributing in the range [0,1], pi is the best previous position, pg is the global best position. The iteration terminates when pg remains invariant and then pg is the global optimum solution. PSO algorithm introduces own experience weight and companion’s experience weight, making the particle at the local extreme value easily out of the local optimum position. Hence PSO algorithm is efficient for searching problem and has rapid convergence velocity. 3.2 Range of Search Space
Supposing the optimum solution of (12) is
Wopt = [ w1 , w2 ", wM ]T = C D D
(16)
C = [C1 , C2 , " , CM ]T
(17)
D = [e jϕ1 , e jϕ2 , " , e jϕM ]T
(18)
where
C stands for the mode vector of optimum weight , ϕ1 , ϕ 2 , " , ϕ M the phase of the optimum weight vector. Defining Cmax = max N [Ci ] from Eq. (11), the minimum output power of array i =1, 2,", M
，
is K
M
l =1
i =1
Pmin = ∑ (∑ wi xil )2
(19)
where K is the number of sampling time points. According to constrain condition
∑ w = 1 ，make identity transformation to Eq.(19) M
i
i =1
156
L. Wang and Z. Song
K
M
Pmin = ∑ (∑ l =1
i =1
wi xil
K
M
∑w i =1
i
M
) 2 = ∑ [∑ l =1
i =1
wi xi / Cmax M
∑w /C l =1
i
K
M
]2 = ∑ [∑ l =1
max
i =1
wi xi M
∑w l =1
]2
(20)
i
where wi = wi / Cmax
(21)
the solution of MVDR is unique under ideal conditions, Hence,
W = Wopt
，
(22)
where W = [ w1 , w2 , " wM ]T . Form Eq.(20) and Eq.(21), the maximum mode of Wopt is 1. Therefore we can obtain the range of searching weight vector is Ci ∈ [0,1] Di ∈ [0, 2π ] , i = 1, 2, " , M .
和
3.3 Introduce of Constrain Condition
Supposing the complex weight vector of ith particle is W = [ w1 , w2 , " wM ]T . Constrain condition can be represented as ⎧M ⎪∑ Re( wi ) = 1 ⎪ i =1 ⎨M ⎪ Im( w ) = 0 i ⎪⎩∑ i =1
(23)
1 M ai = M 1 [Re( wi ) + ] ∑ M i =1
(24)
making Re( wi ) +
bi = Im( wi ) − Where
1 M
M
∑ Im(w ) i =1
(25)
i
1 is introduced to ensure ai ≤ 1 . Constructing the vector as follows, M Wc = [ a1 + jb1 , a2 + jb2 , " aM + jbM ]T
(26)
Eq. (26) is just the complex weight vector with constrain condition. Hence the steps of PSO algorithm can be described as follows:
Broadband MVDR Beamformer Applying PSO
1).
Set the initial position vector of particle as Eq.(16),where Ci = rand (0,1) j 2π rand (0,1)
2).
3).
4). 5).
157
，
. Di = e Using Eq.(24)and (25), respectively process the real part and imaginary part of the initial position vector, and then obtain the complex weight vector as Eq. (26). Calculate the objective function P in Eq.(9). Find the minimum objective function Pmin and record the best previous position pi and the best global position pg . Update the position of particle according to Eq. (13). The iteration terminates when Pmin (n) (n is the iteration times) remains invariant( Pmin (n + k ) − Pmin (n) < ε ) .The best global position pg at this time is the optimum solution of TMVDR.
4 Result of Sea Experiment 4.1 Sea Experiment Description
In order to investigate the performance of TMVDR beamformer applying PSO algorithm, the data of out-sea experiment was processed. The array in experiment was uniform line array composed of 32 sensors; the interval between each unit is 1m; the band width is 300-700Hz, the total time period for data processing is 100 seconds, within which the target were near 40º 70º 90º 140º ,150º and a strong interference source was around 0º~20º. Data sampling frequency was 6 KHz.
， ， ，
4.2 Signal Processing
The data was respectively processed by TMVDR method applying PSO algorithm and FMVDR (frequency domain MVDR) using diagonal loading algorithm. FMVDR’s covariance matrix estimation follow the next steps: Split the data into many blocks, each block has 512 sampling time points. There are 256 points overlap with two adjacent block, which is also the data length of each snapshot. After DFT, the cross-spectral density matrix was estimated on each frequency. The number of snapshots for each estimation is 64.That is ,the number of sampling time points for each processing is (64+1)×256=16640. Diagonal loading adapted the optimum factor[5]. TMVDR chooses 300-700Hz band width through band-pass filter, and only uses 1000 sampling time points to estimate the output power for each processing. 4.3 Comparison
Figure 1 displays energy level for 0°~180°azimuth angle v.s. time respectively processed by TMVDR and FMVDR. In Figure 1, as we have expected, TMVDR method has far better signal detection ability than FMVDR method. Tracks of sources are bright in the TMVDR result. Especially for the weak targets near 140°and 150° FMVDR almost
，
158
L. Wang and Z. Song
Fig. 1. Output of TMVDR method and FMVDR method
cannot detect the signal because the diagonal loading cause the degradation of array gain. In addition, the period of TMVDR for each processing is 1000/6000≈0.16s far shorter than FMVDR which period for each processing is 16640/6000≈2.77s.That makes TMVDR has better performance than FMVDR in signal detection for the fast moving targets.
，
，
5 Conclusion A time domain broadband MVDR beamforming method is presented in this paper. On the basis of constructing time domain analytical signal, TMVDR introduces complex weights and could obtain stable sample matrix estimation in short time period. To obtain the stable optimum solution of TMVDR, a numerical searching method optimized by PSO algorithm with constrain conditions is proposed to replace the SMI algorithm. Using PSO algorithm, stable optimum weights vector can be searched with rapid convergence velocity. The performance of TMVDR method applying PSO algorithm is put into test via out-sea experiment. Compared with FMVDR method applying diagonal loading algorithm, TMVDR method has better performance than FMVDR method for the problems of passive detection and azimuth estimation on broadband sound sources.
References 1. Kim, B.-C., Lu, I.-T.: High Resolution Broadband Beamforming Based on the MVDR Method. In: Proc. MTS/IEEE Oceans, vol. 3, pp. 1673–1676 (2000) 2. Yang, Y., Wan, C., Sun, C.: Broadband Beamspace DOA Estimation Algorithms. In: Proceeding of Oceans 2003, vol. 3, pp. 1654–1660 (2003)
Broadband MVDR Beamformer Applying PSO
159
3. Van Trees, H.L.: Optimum Array Processing. John Wiley & Sons, Inc., New York (2002) 4. Manolakis, D.G., Ingle, V.k., Kogon, S.M.: Statistical and Adaptive Signal Processing. Tsinghua University Press (2003) 5. Mestre, X., Lagunas, M.A.: Finite sample size effect on minimum variance beamformers: Optimum diagonal loading factor for large arrays. IEEE Transactions on Signal Processing 54, 69–82 (2006) 6. Li, B., Xiao, Y.: New Evolution Algorithm: Particle Swarm Optimization. Computer Science 30, 19–22 (2003)
