Diagonal Loading of Robust General-Rank Beamformer for Direction of Arrival Mismatch

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1 Reearch Journal of Applied Science, Engineering and Technology 5(17): , 013 ISSN: ; e-issn: Maxwell Scientific Organization, 013 Submitted: June 8, 01 Accepted: Augut 08, 01 Publihed: May 01, 013 Diagonal Loading of Robut General-Rank Beamformer for Direction of Arrival Mimatch 1 Z.U. Khan, A. Naveed, 1 A. Safeer and 1 F. Zaman 1 Department of Electronic Engineering, IIU, H-10, Ilamabad, Pakitan School of Engineering and Applied Science, ISRA, Univerity, Ilamabad, Pakitan Abtract: Thi tudy preent a technique which utilize the movement of the peak of the main beam toward the preumed ignal direction with negative diagonal loading for robut general-rank beamformer. The main beam ymmetry along preumed ignal direction i improved by thi movement. When deired ignal i contained in the data naphot, the conventional beamformer face the problem of performance degradation even if there i a mall mimatch between the preumed and the actual ignal direction. Diagonal loading i a popular technique to mitigate thi problem. There i no definite criterion to find diagonal loading level. A new diagonal loading method ha been propoed in the literature which utilize the movement of the peak of main beam toward the preumed ignal direction with poitive diagonal loading. The propoed technique work iteratively for the election of negative diagonal loading level to move the main beam at a poition to get the beam ymmetry at deired level and hence the deired robutne. The mimatched ignal will not be cancelled a long a it i within the half of the width of the main beam. But there i the tradeoff between thi robutne and interference cancelling capability. Keyword: Adaptive beamforg, Minimum Variance Ditortionle Repone (MVDR) beamformer, robut adaptive beamforg INTRODUCTION Adaptive beamforg i a popular patial filtering technique which utilize antenna array for ignal etimation from deired direction (Zaman et al., 01a, b and c) and placing null in the direction of undeired ignal (Khan et al., 011). The weight of the beamformer are optimized according to ome pecific criteria, uch a imum variance, maximum entropy and maximum Signal to Interference-plu-Noie Ratio (SINR). Beamforg find it application in the field of radar, onar, medical imaging and wirele communication (Synnevag et al., 009; Sharma et al., 008; Roy et al., 009). Minimum Variance Ditortionle Repone (MVDR), Sample Matrix Inverion (SMI), Linearly Contrained Minimum Variance (LCMV) and Generalized Sidelobe Canceller (GSC) are the popular adaptive beamformer. When deired ignal i preent in data naphot and there i a mimatch between the preumed and actual ignal direction, the deired ignal i taken a interference. In uch ituation, the deired ignal i cancelled and the performance of the beamformer degrade everely. Effort are in progre to develop robut algorithm for uch mimatche. Diagonal loading (Carlon, 1988) i a popular technique robut againt direction of arrival mimatch but ha no uitable way to find the diagonal loading factor. Another attractive approach i robut adaptive beamforg uing wort cae performance optimization (Vorobyov et al., 003). But it performance i quite cloe to the imple algorithm known a diagonal Loading of the Sample Matrix Inverion (LSMI) algorithm. Thee beamformer utilize poitive diagonal loading and their performance depend upon the proper election of diagonal loading factor. A new diagonal loading technique appear in (Wang and Wu, 011) which find diagonal loading level by utilizing the fact that the peak of the main beam move toward the preumed direction by increaing the diagonal loading level. Thi technique i effective if the mimatched ignal appear within the half of the width of the main beam. There i a tradeoff between robutne againt ignal look direction error (controlled by the movement of the peak of the main beam) and the interference uppreion capability. Robut general-rank beamformer (Shahbazpanahi et al., 003) utilize negative and poitive diagonal loading for preumed and received ignal covariance matrice repectively. Negative diagonal loading i meant for the robutne againt direction of arrival mimatch. If thi loading level exceed beyond a certain limit, the preumed ignal covariance matrix no longer remain poitive emi-definite and become uele. Thi tudy utilize the movement of the peak of the main beam toward the preumed ignal direction by negative diagonal loading and the expreion for maximum value of the diagonal loading level, that Correponding Author: Z.U. Khan, Department of Electronic Engineering, IIU, H-10, Ilamabad, Pakitan 457

2 Re. J. Appl. Sci. Eng. Technol., 5(17): , 013 maintain the poitive definite property of the negative diagonal loaded matrix, i given. Diagonal loading level i increaed from imum toward maximum iteratively to achieve the beam ymmetry up to the deired level for robut general-rank beamformer. The technique given in (Chen and Vaidyanathan, 007) utilize angular region of mimatch for the robutne of the beamformer. Same diagonal loading level i applied to the ignal regardle of mall or large mimatch in that region i.e., ame compromie for null depth regardle of mall or large DOA mimatch. On the other hand thi technique utilize beam ymmetry to elect diagonal loading level. If mimatch i mall, mall value of diagonal loading level will give deired beam ymmetry and large diagonal loading level will be required to achieve deired beam ymmetry for large mimatch i.e., compromie on null depth i different for different ituation. LITERATURE REVIEW Mathematical model: Conider a uniform linear array of M antenna element with inter-element pacing λ/, where λ i the wavelength of incog narrow band ignal of interet. Let the array receive ignal from K far-field ource. The output of the i th antenna element ii=mm i.e.,{yy ii (nn)} ii=1 i given by: KK yy ii (nn) = ee jj (ii 1)ππππππππ θθ ll ll (nn) + vv ii (nn) ll=1 In the above expreion, ll (n) repreent the ignal amplitude received from l th ource and vv ii (nn) i the additive white noie added at the output of i th enor. The output vector y(n) contain the individual output of all the element and i given a: yy(nn) = [yy 1 (nn)yy (nn) yy MM (nn)] TT (1) aa(θθ ll ) = [1ee jjjjjj ee jjφφφφ ee jj (MM 1)φφφφ ] TT where, ll = ππππππππθθ ll Thee vector can be placed in a ingle matrix A given a: AA = [aa(θθ 1 )aa(θθ ) aa(θθ kk )] y(n) Can be expreed a: y (n) = A (n) + v (n) where, v(n) i the noie vector having uncorrelated component and hence it correlation matrix i given a: RR vv = EE[vv(nn)vv HH (nn)] = σσ vv II MM The correlation matrix R y by: of received ignal i given RR yy = EE[yy(nn)yy HH (nn)] = AAAA AA HH + σσ VV II MM () MVDR Beamformer: Thi beamformer utilize econd order tatitic of the array output. It imize the variance (average output power) of the beamformer and maintain the ditortionle repone in the deired ignal direction. Let aa(θθ ) be the teering vector in the deired ignal direction and R i the received ignal covariance matrix, the optimization problem and it olution in term of beamformer weight vector w Mv are given in (3) and (4) repectively (Liu et al., 003): ww ww HH RRRR ubject to ww HH aa(θθ ) = 1 (3) ww MMMM = RR 11 aa(θθ ) aa HH (θθ )RR 11 aa(θθ ) (4) SMI Beamformer: Thi beamformer imize Signal to Interference-plu-Noie Ratio (SINR) of the array output. Let R be the ignal covariance matrix, the optimization problem for thi beamformer i tated a: The ource ignal vector (n) repreenting ignal amplitude from K ource and i given a: (nn) = [ 1 (nn) (nn). KK (nn)] TT Thee ource are conidered to be uncorrelated to each other and their correlation matrix R i given by: σσ RR = 0 σσ In the above expreion, {σσ } ll=kk ll=1 repreent the power of the ignal received from l th ource. A et of teering vector aa(θθ ll ): ll = 1, KKcan be defined a: ww ww H RRRR ubject to ww H RR ww = 1 (5) The olution to thi optimization problem a given in (Shahbazpanahi et al., 003), i w opt = P{R -1 R }, where P{.} i the Eigen vector correponding to the maximal Eigen value and w opt i the optimized weight vector. ROBUST ADAPTIVE BEAMFORMERS The performance of traditional beamformer degrade everely due to error in the ignal look direction. Robut algorithm have been developed to mitigate thi problem. In thi ection, two robut beamforg algorithm i.e., Loaded SMI and General- Rank beamformer are being dicued. 458

3 Re. J. Appl. Sci. Eng. Technol., 5(17): , 013 Loaded SMI beamformer: The key idea for thi beamformer i to add ome quadratic penalty to regularize the olution for optimum weight vector. The optimization problem for thi beamformer i defined a: (RR + γγii)ww tttt ww HH RR ww = 1 w ww HH The robut weight vector for thi beamformer come out to be a: For thi problem, a given in (Shahbazpanahi et al., 003) come out to be: = εε wwwwhh ww (7) By putting the value of, the optimization problem (5) become a: RRRR ubject to ww HH (R εεii)ww = 1 ww ww HH ww LLLLLLLL = PP{(RR + γγii) 1 RR In the above expreion, RR + γγii and I are the diagonally loaded ample covariance matrix and identity matrix repectively. The variance of artificial noie i increaed by an amount γγ in thi method. Thi approach put more effort to uppre white noie rather than interference. Due to above mentioned modification; LSMI improve the performance of Sample Matrix Inverion (SMI) method in the preence of an arbitrary teering vector mimatch. But thi improvement i not o ignificant in cae of look direction error vector with large norm (Song et al., 006). Moreover, another eriou hortcog of thi approach i that there i no reliable way to chooe proper value for the loading factor γγ, a the optimal choice of γγ i dependent on unknown parameter of ignal and interference (Shahbazpanahi et al., 003). However recommended loading factor i σσ nn σσ LL < 10σσ nn where σσ LL i the diagonal loading level and σσ nn i the noie power (Jeyali and Sukaneh, 011) o, the imum loading level mut be equal to noie power σσ nn i.e., imal Eigen value of R. Robut general-rank beamformer: Thi beamformer aume that the mimatch in deired ignal direction of arrival caue an error matrix in R. Let be bounded by ome known poitive contant ε i.e., ε. Where. denote Frobeniu norm of a matrix. In (Shahbazpanahi et al., 003), the SINR maximization problem ha been modified for the robutne of the beamformer againt DOA mimatch a given below: w ww HHRRRR ubject to wwhh (RR + )ww 1 for all εε (6) For the wort cae performance, can be found by olving the following optimization problem. ww HH (RR + )ww ubject to εε 459 The optimum weight vector for the robut beamformer come out to be: ww rrrrrr = PP{RR 1 (RR εεii)} (8) To overcome other array imperfection, another imilar mimatch matrix 1 i conidered in R, with the condition 1 γγ. The robut weight vector, a given in (Shahbazpanahi et al., 003) come out to be: ww rrrrrr. = PP{(RR + γγii) 1 (RR εεii)} (9) If we put εε = 0, ww rrrrrr. = ww LLLLLLLL. Thi how that LSMI beamformer i the pecial cae of General-Rank beamformer. PROPOSED DIAGONAL LOADING ALGORITHM In thi ection, we will dicu the flow chart with parameter and tep of the propoed algorithm for optimum diagonal loading level. The propoed algorithm preent iterative approach to find diagonal loading level from the range of ε and γγ to get deired beam ymmetry. Since LSMI beamformer i the pecial cae of General-Rank beamformer, therefore the propoed algorithm i developed for General-Rank beamformer. The parameter ued in the propoed algorithm are dicued below. In cae of direction of arrival mimatch if the diagonal loading level i zero, the General-Rank robut beamformer become SMI beamformer and the deired ignal i cancelled. A the diagonal loading level i increaed, the peak of the main beam move gradually toward the preumed ignal direction and the deired ignal cancellation along with interference uppreion capability will be reduced (Wang and Wu, 011). If the peak of the main beam i moved at the preumed ignal direction and the direction of arrival mimatch i within the half of the width of the main beam, the deired

4 Re. J. Appl. Sci. Eng. Technol., 5(17): , 013 ignal will not be uppreed heavily. A ufficient condition to guarantee the peak of the main beam at the preumed ignal direction i the exact left-right ymmetry at the preumed ignal direction. For a uniform linear array of M antenna element with interelement pacing d and wavelength of incog narrow band ignal equal to λ, the approximate width of the main beam i given a θθ mmmm = 50.7λλ (Wang and Wu, MMMM 011). When the ideal diagonal loading hift the peak of the main beam at the preumed ignal direction, the following condition will be atified: HH 10llllll 10 ww rrrrrr, aa(θθ + δδδδ) HH 10llllll 10 ww rrrrrr, aa(θθ δδδδ) = 0 (10) where, δδδδ 0.5θθ mmmm. Under thi condition beam i ideally ymmetric along the preumed ignal direction. But thi will require diagonal loading level very high (ideally infinity) with very mall interference uppreion capability. The above expreion can be modified a given below (Wang and Wu, 011): HH 10llllll 10 ww rrrrrr, aa(θθ + δδδδ) HH 10llllll 10 ww rrrrrr, aa(θθ δδθθ= μμ (11) where, μμ i the trade off parameter which we call ymmetry of the main beam at a ditance δδδδ from preumed ignal direction. Thi parameter control robutne and interference uppreion capability. An expreion for negative diagonal loading factor εε for robut general-rank beamformer i given below (Chen and Vaidyanathan, 007): εε = max θθ δδδδ θθ θθ +δδδδ aa(θθ)aahh (θθ) aa(θθ )aa HH (θθ ) (1) where. repreent Frobeniu norm of a matrix and the deired ignal mimatch region i repreented by θθ δδδδ θθ θθ + δδδδ. Thi expreion limit the value of εε up to the level to guarantee the negative diagonally loaded ignal covariance matrix (RR εεii) to remain poitive definite. The other parameter ued in the flowchart are given a: εε mmmmmm = max θθ δδδδ θθ θθ +δδδδ aa(θθ)aahh (θθ) aa(θθ )aa HH (θθ ) εε mmmmmm = max θθ δδδδ nn θθ θθ θθ +δδδδ +nn θθ aa(θθ)aahh (θθ) aa(θθ )aa HH (θθ ) It mean εε mmmmmm i ought by expanding the region around θθ. Thi expand and eek proce continue until εε mmmmmm i achieved. It i the upper limit of negative diagonal loading factor: δδδδ = The tep ize for the poitive diagonal loading factor γγ δδεε = The tep ize for the negative diagonal loading factor ε m = log w H a( θ + δθ) = output power of beamformer at θθ + δδδδ for certain value of ε and γγ which give weight vector w during an iteration m = 10llllll 10 ww HH aa(θθ δδδδ) = output power of beamformer at θθ + δδδδ for certain value of εε and γγ which give weight vector w during an iteration ww = PP{(RR + γγii) 1 (RR εεii) = weight vector for certain value of εε and γγ during an iteration Step 1: Initialization: In thi tep imum and maximum value of diagonal loading level are initialized a hown in flow chart. Step : Evaluation of w: Weight vector w with initial value of γγ and ε i evaluated. Step 3: Comparion: Contraint given in (6) i checked uing weight vector. Step 4: Evaluation of m 1 and m : If contraint in (6) i atified, m 1 and m are evaluated otherwie poitive diagonal loading level i increaed gradually with tep ize δδδδ and go to tep until contraint in (6) i atified. It mut be noted that for a certain value of ε there i a particular imum value of γγ which atifie contraint (6). The algorithm doe not converge if γγ i below that particular value and converge for all higher value of γγ. Step 5: If deired beam ymmetry i achieved, Stop, otherwie increae negative diagonal loading level gradually with tep ize δδδδ and go to tep. Ele top with bet available weight vector. Weight vector for only thoe imum value of γγ and ε i elected a w rob, which give bet ymmetry in the half power beam width in the ignal mimatch region. Since thi weight vector correpond to imum value of γγ and ε, o give better null depth in addition to better performance in the ignal mimatch region. 460

5 Re. J. Appl. Sci. Eng. Technol., 5(17): , 013 γ=γ, γ, max Initialize ε=ε ε, µ, max Evaluate w γ=γ+δ γ No If w H a( θ ) 1, For θ δθ θ θ + δθ Ye γ=γ Evaluate m m 1, If ab( m m ) µ 1 No ε=ε+δ ε Ye wrob, = w No If ε ε max Ye Flow chart for the propoed algorithm SIMULATION RESULTS FOR ROBUST ADAPTIVE BEAMFORMERS A uniform linear array of 15 antenna element ha been ued with inter element pacing λλ/. One deired ignal with preumed direction along 0º and two interference at 35º and 70º are ued. The SNR and INR are 10 db and 30 db repectively. Simulation are carried out in MATLAB and all the reult are averaged over 500 naphot. Performance of MVDR beamformer: In thi cae, the Performance of MVDR beamformer i dicued 461 for both with and without DOA mimatch. Without DOA mimatch, the output power of the beamformer at 3 i db. While for the DOA mimatch cae, the performance of beamformer degraded i.e., the output power of the beamformer at 3 i db. The comparion of both ituation can be oberved from Fig. 1a and b. Performance of robut general rank beamformer: In thi cae, we evaluate the performance of robut general rank beamformer. The actual DOA of the ignal i taken along 3. By uing expreion (1),

6 Re. J. Appl. Sci. Eng. Technol., 5(17): , 013 Output power (db) Output power (db) (a) (b) Fig. 1: Performance of MVDR beamformer: (a) without mimatch, (b) with mimatch Output power (db) Angle of arrival (degree) Angle of arrival (degree) Angle of arrival (degree) Fig. : Performance of robut general-rank beamformer for εε =4.54, γγ = 140 and μμ =3.4 db at ±3 we get εε = 4.54 and for εε = 4.54, μμ = 7.33 db at ±3, and the propoed algorithm converge for γ = 39. So clearly, one can oberve the improvement by comparing Fig. 1b and. Beam ymmetry with ε and γ: In thi ub-ection, we dicu the beam ymmetry which i defined a: 10llllll 10 (mm 1 ) 10llllll 10 (mm ) We ue a uniform linear array of 15 element to oberve beam ymmetry by varying εε and γ. In Table 1 beam ymmetry i elaborated for increaing value of ε. It i clear from Table 1, that a εε increae, beam ymmetry improve and the deired ignal cancellation in the mimatch region decreae. In ideal ituation the beam ymmetry i zero. In Table, we dicued the beam ymmetry for increaing value of γ. The beam ymmetry improve 46 Table 1: Beam ymmetry with increaing value of εε εε γγ Symmetry at ±3 Symmetry at ± Symmetry at = dB db 4.66 db db db db db db db db db db Table : Beam ymmetry with increaing value of γγ εε γγ Symmetry at±3 Symmetry at ± Symmetry at ± db db.3795 db db db db db.834 db db db.3349 db db db db db db db db db db db db db db db db db Table 3: Comparion of Null depth with increaing value of εε and γγ γγ εε Signal (db) Null_1 (db) Null_ (db) for the increaing value of γ. From both table, it i quite clear that the beam ymmetry i much improved for mall increaing value of εε a compare to γ. Null depth and diagonal loading level: One can oberve from Table 3 the tradeoff between ignal trength and null depth. A the value of ε or γ increae the null depth decreae while the trength of ignal improve. CONCLUSION AND FUTURE WORK In thi tudy, we dicued diagonal loading for robut general rank beamformer. Diagonal loading i elected iteratively on the bai of deired beam ymmetry at an angle in the ignal mimatch region. In future, we will dicu it for three dimenional array. REFERENCES Carlon, B.D., Covariance matrix etimation error and diagonal loading in adaptive array. IEEE T. Aero. Elec. Sy., 4(4): Chen, C.Y. and P.P. Vaidyanathan, 007. Quadratically contrained beamforg robut againt direction-of-arrival mimatch. IEEE T. Signal Proce., 55(8): Jeyali, T.S. and R. Sukaneh, 011. Robut adaptive beamformer uing diagonal loading. Cyber Journal: Multidicip. J. Sci. Tech., J. Selected Area Telecommun. (JSAT), March Edn., pp:

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