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Diagonally loaded SMI algorithm based on inverse matrix recursion
Journal of Systems Engineering and Electronics, Vol. 18, No. 1, 2007, pp. 160-163
Diagonally loaded SMI algorithm based on inverse matrix recursion Cao Jianshu & Wang Xuegang Coll. of Electronic Engineering , Univ. of Electronic Science and Technology, Chengdu 610054, P. R. China (Received July 28,2005)
Abstract: The derivation of a diagonally loaded sample-matrix inversion (LSMI) algorithm on the busis of inverse matrix recursion (i.e.LSMI-IMR algorithm) is conducted by reconstructing the recursive formulation of covariance matrix. For the new algorithm, diagonal loading is by setting initial inverse matrix without any addition of computation. In addition, a corresponding improved recursive algorithm is presented, which is low computational complexity. This eliminates the complex multiplications of the scalar coefficient and updating matrix, resulting in significant computational savings. Simulations show that the LSMI-MR algorithm is valid. Keywords: Robust adaptive beamforming, Space-time adaptive processing (STAP), Diagonal loading, Inverse matrix recursion.
1. Introduction The purpose of this study is to develop a diagonally loaded sample-matrix inversion (LSMI) algorithm on the basis of inverse matrix recursion (i.e. LSMI-IMR algorithm). Diagonal loading is a well-known technique for adaptive beamforming, because it provides robustness to a variety of types of mismatch, including steering vector direction of arrival (DOA) mismatch, element position, gain and/or phase perturbations, and "statistical" mismatch due to finite sample support"41. Traditional sample-matrix inversion (SMI) algorithm on the basis of inverse matrix recursion (SMI-IMR algorithm)[5*61 does not work for diagonal loading. However, as a principal approach of generating the adaptive weights in space-time adaptive processing (STAP), the attractiveness of the SMI-IMR approach is that it can update the inverse of the covariance matrix rapidly and is easily mapped onto systolic array structures for a parallel implementation[61,which is not possible with the general diagonally loaded sample-matrix inversion (LSMI) algorithm. Therefore, there arises the need to develop a novel and efficient LSMI-IMRalgorithm, which retains the above-mentioned appealing properties of the SMI-IMR algorithm and requires only setting initial inverse matrix for diagonal loading implementation.
2. LSMI-IMR algorithm derivation Consider airborne array radar. Xl denotes the N K x l inputdata vector on the Ith range cell, where N and K are the number of array elements and coherent
pulses, respectively. L indicates the number of the selected range cells. The entire 2-D spatial-temporal inputdata is N K x L matrix X = [ X I ; . . , X l , . . . 3 ~ 1 . The expression of linearly constrained minimum variance (LCMV) optimization problem is as follows min WHRW s.t. W H S=1 where, R denotes the covariance matrix, S indicates the normalized spatial-temporal steering vector, (.)" represents the complex conjugate transpose. The solution of the optimal weight vector W is
w = R-'sI(s~R-'s) (2) For sample-matrix inversion (SMI) algorithm, R is obtained from its maximum likelihood (ML) estimation l L RL x,x; (3)
=TC 'L 1=1
The expression of the diagonally loaded covariance matrix kL is
iiL=RL++21
(4) where I is the K N X KN identity matrix and CJ indicates a real constant representing the diagonal loading value. This is a simple and efficient method for robust adaptive beamforming '1-41. Replacing R-' with i,' in Eq. (2), the weight vector W with diagonal loading is
w = R;'S/(SHR,-'S)
(5) Equation ( 5 ) represents a general diagonally loaded SMI (LSMI) algorithm, where EL1 is obtained by directly inverting matrix i L .The derivation of the LSMI-IMR algorithm is as follows.
Diagonally loaded SMI algorithm based on inverse matrix recursion For obtaining the new inverse matrix recursion formulations with respect to diagonal loading, it is necessary to construct the recursive formulations of covariance matrix. This is a key step. On the basis of Eq. (3), the definition of recursive formulations are as follows [Rl = R,, + X I X $
where, ROdenotes the initial matrix. Then, according to the matrix inversion lemma '71
M i =Mi-1+ X , X , H , Z = l , . . . , L
Where, M, = R, = (La2)Z. Then, according to the matrix inversion lemma
Exploiting Eqs. (10) and (6), the results are M ,=1R, Thus M,-'= R;' / 1 This indicates that Eq. (11) is the recursive equivalence of Eq. (7). Eq. (11) is further simplified. Since matrix MI is a positive Hermite matrix, hence, matrix M;' is also a Hermite matrix MY1=
Let
where, R,,-' indicates the initial inverse matrix. From Eqs. (3) and (6) R , =iL+ R, L
(8) Observing Eqs. (8) and (4), clearly, RL is the equal of ii, if R, / L = ~ ~ his I . means the insertion of diagonal loading into the recursion Eq. (7) is by the initialization R,' = I /(La2)
(9)
Equation (9) establishes an appearance of equivalence between the LSMI-IMR algorithm initialization and diagonal loading. The accomplishment of the diagonal loading is by setting the matrix R,' without the need for any additional computation. Meanwhile, the optimal initial matrix R,' can be determined on the basis of diagonal loading analysis. There are certain methods 18, 91 which offer an estimation of diagonal loading.
3. Improving the iterative process If the iteration operations are directly completed via Eq. (7) [5,61, the (NK)(NK+1)/2 complex multiplications of the scalar coefficient U(1-1) and the updating matrix (i.e. the result of ( R Z , - (RL!lxl)(X:Ri21) )) (1 - 1)+ X ; R z l X l will be generated for every inputdata vector X,. Nevertheless, the above-mentioned multiplications undergo elimination. The details are as follows. Eq. (10) defined as
A,-' =Mi:X ,
(14)
iql= X;M;l
(15)
Consequently Substituting Eqs. (14) and (15) into Eq. (11) as
In Eq.(16), the complete iterative process comprises of three steps =M:lXl
M;' = ML: - (4-,/ b ) ( & / b)H
(17)
(19)
Where, l=l;*.,L, and initial matrix Mi' = I / ( L a 2 ) . Updating computation in Eqs. (16) and. (7) for per inputdata vector Xl requires approximate 3(Na2/2+5(NK)/2 and 2(NQ2+3(NK) complex multiplications, respectively. It is obvious that this improved algorithm given in Eq. (16) reduces the computational complexity by nearly 25%. Furthermore, the dominant multiplications and the matrix-vector product are fit for parallel pipeline implementation with systolic array, which is attractive for the development of the hardware system in practical application. Finally, replacing R-' with Ril = LM;' in Eq. (2), the expression of optimal weight vector W with diagonal loading is w = M i ' s /(sHM;ls) (20)
162
Cao Jianshu & Wang Xuegang
For convenience, the cummarization of the computing steps about the algorithm is as follows. step 1 Initialization: M,' = I /(LO')
Step 2 Iterative operations: for 1= 1, ..- ,L =M;J1X,
b = Jl+ xpt4-l
mH
M,' = MZ1- (A&l /b)(A&,
same impact as the LSMI algorithm for diagonal loading. 0 -5 -10 9 -15 -20
p72
-25
-30
Step 3 Optimal adaptive weight vector w = M,'S /(SHM,'S)
4. Simulations In this section, the examination of the LSMI-IMR algorithm is carried out by conducting simulations. A sidelooking airborne phased array is considered herein, which is an equivalent linear array synthesized by 4 rowxl2 column rectangular array. The principal parameters in the simulation are listed as: the number of array elements N=12, the number of coherent pulses K=24, the radar wavelength 1=0.2 m, the space of array elements d=O.l m, the platform altitude h=6 OOO m, the pulse repetition frequency (PRF) f;=2 500 Hz, the platform velocity V=250 d s and the array amplitude and phase errors 3%. The transmitting antenna boresight points toward array normal, the input clutter-to-noise-ratio (CNRJ at each element 60 dB, the Chebyshev weights for transmiting antenna, in azimuth direction -40 dB, and in elevation direction -20 dB. The explanation of the clutter model is in Ref. [lo]. Without loss of generality, we take (cos(ty0)=0.3, 2 f d / f , = -0.5) as spatial and temporal domain normalized frequency of the steering vector S, in whichfd is Doppler frequency, let diagonal loading value o2= 0 dB and o2= 20 dB. For each loading value,the LSMI-IMR and LSMI algorithms are utilized to calculate the weight vector W. To compare the calculation results, it is convenient to plot adaptive antenna patterns of spatial domain (where the Doppler channel normalized frequency 2f d / f, = -0.5 ). Figure 1 and Figure 2 demonstrate that the adaptive antenna patterns form deep nulls in the direction of the ground clutter and completely coincide with each other when o2=OdB and o2=20dB. The reason for this is that both the algorithms produce identical weight vectors. Hence, the LSMI-IMR algorithm has the
05
-05
-.urn. .
4 V ) I
1.0
LSMI-IMR
Fig. 1 The adaptive antenna patterns of spatial domain for
LSMI-IMRand LSMI algorithms with 0 dB loading.
-5 -10
. 5
-15
-20
8 -25 -30 -35
-1.0
-05
0 *W)
-:LSW ;
0.5
1.0
:LSW-M
Fig2 The adaptive antenna patterns of spatial domain for
LSMI-IMR and B M algorithms with 20 dB loading.
5. Conclusion In this study, the introduction of a new LSMI-IMR algorithm led to a fast implementation of diagonal loading. The LSMI-IMR algorithm initialization is equivalent to diagonal loading. The accomplishment of diagonal loading is by setting initial inverse matrix without any addition of computation; its analysis provides insights into the choice of the initial inverse matrix. Furthermore, a corresponding improved recursive algorithm is presented, whose computational complexity is low. Simulations show that the LSMIIMR algorithm is valid.
References [ l ] Carlson B D. Covariance matrix estimation errors and
diagonal loading in adaptive arrays. IEEE Trans. on Aerospace and Electronic Systems, 1988,24(4): 397-401. [2] Harmanci K, Tabrikian J, Krolik J L. Relationships between
adaptive minimum variance beamforming and optimal source localization. IEEE Trans. on Signal Processing, 2000,48(1):1-12.
[3] Kogon S M. Eigenvectors, diagonal loading and white noise gain constraints for robust adaptive beamfonning. Signals,
Diagonally loaded SMI algorithm based on inverse matrix recursion Systems and Computers, 2003, 1853-1857.
[4] Nadakuditi R R, Edelman A. The bias of the MVDR beamformer outputs under diagonal loading. Acoustics, Speech, and Signal Processing, 2005,4: 793-796.
163
loading. Radal; Sonar and Navigation, IEE Proceedings, 2004,151(6): 337-343. [lo] Wang Y L, Peng Y N. Space-Time Adaptive Processing. Beijing: Press of tsinghua, 2O00:30-32.
[5] Brennan L, Mallett J, Reed I. Adaptive arrays in iiirborne MTI radar. IEEE Trans. on Antennas and Ptvpagation,
1976,24(5):607-615. [6] Xiong Jun, Liao Guisheng. Recursive algorithm of adaptive weight extraction of space-time signal processing for airborne radars. Radar 1996Proceedings, 1996: 86-90.
Cao Jianshu was born in 1970. Now he is pursuiting the Ph.D. degree in University of Electronic Science and Technology of China. His research interests are signal processing for airborne radar and real-time signal processing. E-mail: js-cao@ 163.com
171 B a n g Xianda. Matrix analysis and applications. Beijing:
Press of Tsinghua, 2004: 68.
[8] Ning Ma, Joo Thiam Goh. Efficient method to determine
diagonal loading value
. Acoustics,
Speech, and Signal
Processing, 2003, (5):341-344.
[9] Vincent F, Besson 0. Steering vector errors and diagonal
WangXuegang was born in 1962. Now he is a professor of University of Electronic Science and Technology of China. His scientific interests are radar signal processing and real-time signal processing.
Diagonally loaded SMI algorithm based on inverse matrix recursion Cao Jianshu & Wang Xuegang Coll. of Electronic Engineering , Univ. of Electronic Science and Technology, Chengdu 610054, P. R. China (Received July 28,2005)
Abstract: The derivation of a diagonally loaded sample-matrix inversion (LSMI) algorithm on the busis of inverse matrix recursion (i.e.LSMI-IMR algorithm) is conducted by reconstructing the recursive formulation of covariance matrix. For the new algorithm, diagonal loading is by setting initial inverse matrix without any addition of computation. In addition, a corresponding improved recursive algorithm is presented, which is low computational complexity. This eliminates the complex multiplications of the scalar coefficient and updating matrix, resulting in significant computational savings. Simulations show that the LSMI-MR algorithm is valid. Keywords: Robust adaptive beamforming, Space-time adaptive processing (STAP), Diagonal loading, Inverse matrix recursion.
1. Introduction The purpose of this study is to develop a diagonally loaded sample-matrix inversion (LSMI) algorithm on the basis of inverse matrix recursion (i.e. LSMI-IMR algorithm). Diagonal loading is a well-known technique for adaptive beamforming, because it provides robustness to a variety of types of mismatch, including steering vector direction of arrival (DOA) mismatch, element position, gain and/or phase perturbations, and "statistical" mismatch due to finite sample support"41. Traditional sample-matrix inversion (SMI) algorithm on the basis of inverse matrix recursion (SMI-IMR algorithm)[5*61 does not work for diagonal loading. However, as a principal approach of generating the adaptive weights in space-time adaptive processing (STAP), the attractiveness of the SMI-IMR approach is that it can update the inverse of the covariance matrix rapidly and is easily mapped onto systolic array structures for a parallel implementation[61,which is not possible with the general diagonally loaded sample-matrix inversion (LSMI) algorithm. Therefore, there arises the need to develop a novel and efficient LSMI-IMRalgorithm, which retains the above-mentioned appealing properties of the SMI-IMR algorithm and requires only setting initial inverse matrix for diagonal loading implementation.
2. LSMI-IMR algorithm derivation Consider airborne array radar. Xl denotes the N K x l inputdata vector on the Ith range cell, where N and K are the number of array elements and coherent
pulses, respectively. L indicates the number of the selected range cells. The entire 2-D spatial-temporal inputdata is N K x L matrix X = [ X I ; . . , X l , . . . 3 ~ 1 . The expression of linearly constrained minimum variance (LCMV) optimization problem is as follows min WHRW s.t. W H S=1 where, R denotes the covariance matrix, S indicates the normalized spatial-temporal steering vector, (.)" represents the complex conjugate transpose. The solution of the optimal weight vector W is
w = R-'sI(s~R-'s) (2) For sample-matrix inversion (SMI) algorithm, R is obtained from its maximum likelihood (ML) estimation l L RL x,x; (3)
=TC 'L 1=1
The expression of the diagonally loaded covariance matrix kL is
iiL=RL++21
(4) where I is the K N X KN identity matrix and CJ indicates a real constant representing the diagonal loading value. This is a simple and efficient method for robust adaptive beamforming '1-41. Replacing R-' with i,' in Eq. (2), the weight vector W with diagonal loading is
w = R;'S/(SHR,-'S)
(5) Equation ( 5 ) represents a general diagonally loaded SMI (LSMI) algorithm, where EL1 is obtained by directly inverting matrix i L .The derivation of the LSMI-IMR algorithm is as follows.
Diagonally loaded SMI algorithm based on inverse matrix recursion For obtaining the new inverse matrix recursion formulations with respect to diagonal loading, it is necessary to construct the recursive formulations of covariance matrix. This is a key step. On the basis of Eq. (3), the definition of recursive formulations are as follows [Rl = R,, + X I X $
where, ROdenotes the initial matrix. Then, according to the matrix inversion lemma '71
M i =Mi-1+ X , X , H , Z = l , . . . , L
Where, M, = R, = (La2)Z. Then, according to the matrix inversion lemma
Exploiting Eqs. (10) and (6), the results are M ,=1R, Thus M,-'= R;' / 1 This indicates that Eq. (11) is the recursive equivalence of Eq. (7). Eq. (11) is further simplified. Since matrix MI is a positive Hermite matrix, hence, matrix M;' is also a Hermite matrix MY1=
Let
where, R,,-' indicates the initial inverse matrix. From Eqs. (3) and (6) R , =iL+ R, L
(8) Observing Eqs. (8) and (4), clearly, RL is the equal of ii, if R, / L = ~ ~ his I . means the insertion of diagonal loading into the recursion Eq. (7) is by the initialization R,' = I /(La2)
(9)
Equation (9) establishes an appearance of equivalence between the LSMI-IMR algorithm initialization and diagonal loading. The accomplishment of the diagonal loading is by setting the matrix R,' without the need for any additional computation. Meanwhile, the optimal initial matrix R,' can be determined on the basis of diagonal loading analysis. There are certain methods 18, 91 which offer an estimation of diagonal loading.
3. Improving the iterative process If the iteration operations are directly completed via Eq. (7) [5,61, the (NK)(NK+1)/2 complex multiplications of the scalar coefficient U(1-1) and the updating matrix (i.e. the result of ( R Z , - (RL!lxl)(X:Ri21) )) (1 - 1)+ X ; R z l X l will be generated for every inputdata vector X,. Nevertheless, the above-mentioned multiplications undergo elimination. The details are as follows. Eq. (10) defined as
A,-' =Mi:X ,
(14)
iql= X;M;l
(15)
Consequently Substituting Eqs. (14) and (15) into Eq. (11) as
In Eq.(16), the complete iterative process comprises of three steps =M:lXl
M;' = ML: - (4-,/ b ) ( & / b)H
(17)
(19)
Where, l=l;*.,L, and initial matrix Mi' = I / ( L a 2 ) . Updating computation in Eqs. (16) and. (7) for per inputdata vector Xl requires approximate 3(Na2/2+5(NK)/2 and 2(NQ2+3(NK) complex multiplications, respectively. It is obvious that this improved algorithm given in Eq. (16) reduces the computational complexity by nearly 25%. Furthermore, the dominant multiplications and the matrix-vector product are fit for parallel pipeline implementation with systolic array, which is attractive for the development of the hardware system in practical application. Finally, replacing R-' with Ril = LM;' in Eq. (2), the expression of optimal weight vector W with diagonal loading is w = M i ' s /(sHM;ls) (20)
162
Cao Jianshu & Wang Xuegang
For convenience, the cummarization of the computing steps about the algorithm is as follows. step 1 Initialization: M,' = I /(LO')
Step 2 Iterative operations: for 1= 1, ..- ,L =M;J1X,
b = Jl+ xpt4-l
mH
M,' = MZ1- (A&l /b)(A&,
same impact as the LSMI algorithm for diagonal loading. 0 -5 -10 9 -15 -20
p72
-25
-30
Step 3 Optimal adaptive weight vector w = M,'S /(SHM,'S)
4. Simulations In this section, the examination of the LSMI-IMR algorithm is carried out by conducting simulations. A sidelooking airborne phased array is considered herein, which is an equivalent linear array synthesized by 4 rowxl2 column rectangular array. The principal parameters in the simulation are listed as: the number of array elements N=12, the number of coherent pulses K=24, the radar wavelength 1=0.2 m, the space of array elements d=O.l m, the platform altitude h=6 OOO m, the pulse repetition frequency (PRF) f;=2 500 Hz, the platform velocity V=250 d s and the array amplitude and phase errors 3%. The transmitting antenna boresight points toward array normal, the input clutter-to-noise-ratio (CNRJ at each element 60 dB, the Chebyshev weights for transmiting antenna, in azimuth direction -40 dB, and in elevation direction -20 dB. The explanation of the clutter model is in Ref. [lo]. Without loss of generality, we take (cos(ty0)=0.3, 2 f d / f , = -0.5) as spatial and temporal domain normalized frequency of the steering vector S, in whichfd is Doppler frequency, let diagonal loading value o2= 0 dB and o2= 20 dB. For each loading value,the LSMI-IMR and LSMI algorithms are utilized to calculate the weight vector W. To compare the calculation results, it is convenient to plot adaptive antenna patterns of spatial domain (where the Doppler channel normalized frequency 2f d / f, = -0.5 ). Figure 1 and Figure 2 demonstrate that the adaptive antenna patterns form deep nulls in the direction of the ground clutter and completely coincide with each other when o2=OdB and o2=20dB. The reason for this is that both the algorithms produce identical weight vectors. Hence, the LSMI-IMR algorithm has the
05
-05
-.urn. .
4 V ) I
1.0
LSMI-IMR
Fig. 1 The adaptive antenna patterns of spatial domain for
LSMI-IMRand LSMI algorithms with 0 dB loading.
-5 -10
. 5
-15
-20
8 -25 -30 -35
-1.0
-05
0 *W)
-:LSW ;
0.5
1.0
:LSW-M
Fig2 The adaptive antenna patterns of spatial domain for
LSMI-IMR and B M algorithms with 20 dB loading.
5. Conclusion In this study, the introduction of a new LSMI-IMR algorithm led to a fast implementation of diagonal loading. The LSMI-IMR algorithm initialization is equivalent to diagonal loading. The accomplishment of diagonal loading is by setting initial inverse matrix without any addition of computation; its analysis provides insights into the choice of the initial inverse matrix. Furthermore, a corresponding improved recursive algorithm is presented, whose computational complexity is low. Simulations show that the LSMIIMR algorithm is valid.
References [ l ] Carlson B D. Covariance matrix estimation errors and
diagonal loading in adaptive arrays. IEEE Trans. on Aerospace and Electronic Systems, 1988,24(4): 397-401. [2] Harmanci K, Tabrikian J, Krolik J L. Relationships between
adaptive minimum variance beamforming and optimal source localization. IEEE Trans. on Signal Processing, 2000,48(1):1-12.
[3] Kogon S M. Eigenvectors, diagonal loading and white noise gain constraints for robust adaptive beamfonning. Signals,
Diagonally loaded SMI algorithm based on inverse matrix recursion Systems and Computers, 2003, 1853-1857.
[4] Nadakuditi R R, Edelman A. The bias of the MVDR beamformer outputs under diagonal loading. Acoustics, Speech, and Signal Processing, 2005,4: 793-796.
163
loading. Radal; Sonar and Navigation, IEE Proceedings, 2004,151(6): 337-343. [lo] Wang Y L, Peng Y N. Space-Time Adaptive Processing. Beijing: Press of tsinghua, 2O00:30-32.
[5] Brennan L, Mallett J, Reed I. Adaptive arrays in iiirborne MTI radar. IEEE Trans. on Antennas and Ptvpagation,
1976,24(5):607-615. [6] Xiong Jun, Liao Guisheng. Recursive algorithm of adaptive weight extraction of space-time signal processing for airborne radars. Radar 1996Proceedings, 1996: 86-90.
Cao Jianshu was born in 1970. Now he is pursuiting the Ph.D. degree in University of Electronic Science and Technology of China. His research interests are signal processing for airborne radar and real-time signal processing. E-mail: js-cao@ 163.com
171 B a n g Xianda. Matrix analysis and applications. Beijing:
Press of Tsinghua, 2004: 68.
[8] Ning Ma, Joo Thiam Goh. Efficient method to determine
diagonal loading value
. Acoustics,
Speech, and Signal
Processing, 2003, (5):341-344.
[9] Vincent F, Besson 0. Steering vector errors and diagonal
WangXuegang was born in 1962. Now he is a professor of University of Electronic Science and Technology of China. His scientific interests are radar signal processing and real-time signal processing.