Default Normal Template N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 70 Al-khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol.2, No.1,pp 70-77, (2006) Direction Finding Using GHA Neural Networks N. H. Abbas Electrical Eng. Dept., College of Eng./ University of Baghdad (Received 4 September 2005; accepted 4 April 2006) Abstract:- This paper adapted the neural network for the estimating of the direction of arrival (DOA). It uses an unsupervised adaptive neural network with GHA algorithm to extract the principal components that in turn, are used by Capon method to estimate the DOA, where by the PCA neural network we take signal subspace only and use it in Capon (i.e. we will ignore the noise subspace, and take the signal subspace only). Keywords: Direction of arrival (DOA), Generalized Hebbian Algorithm (GHA), Principal component analysis (PCA), Capon. 1.Introduction Estimating the DOA of the source is a central problem in the array signal processing. Many methods for estimation the DOA have been proposed, including the Maximum Likelihood (ML) technique [1], the minimum variance method of Capon [2], and the MUSIC method of Schmidt [9]. The ML method has the best performance. Because of high computational load of the multivariate nonlinear maximization problem involved, the ML technique did not becomes popular. Suboptimal methods are more prevalent than the ML technique when the signal_to_noise ratio and number of samples are both not too small, because the suboptimal methods involve solving only a one-dimensional maximization problem and subspace (signal subspace or noise subspace). This paper uses adaptive algorithm for extracting the subspace information based on the PCA neural network. Where we extract the principal component (i.e. the signal subspace only) by using of GHA algorithm with adaptive learning rate, which then used by a Capon to find the DOA. The mapping of this paper is as follows: Section two provides some background information. Where the data model and the subspaces are present. In section three an expression for the Capon method is presented. In section four an expression for neural estimator is presented. In section five a computer simulation using matlab 6.0 is provided to support the theoretical observations. 2.General Consideration Assume that plane waves emitted by D1 narrow band sources impinged on a uniform circular array (UCA) consisting of N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 71 M sensors, and the DOAs of these sources (the azimuth angle is measured with respect to reference sensor, and elevation is measured with respect to z-axis as shown in figure (1) are [(θ1, Φ1), (θ2, Φ2),…, (θD, ΦD)].The array output vector at the k_th snapshot can then be expressed as x(k) =As(k)+n(k) (1) Where s(k) is the D×1 vector of incident signals which are assumed to be zero mean stationary, complex and Gaussian random processes n(k) is M×1 vector of additive noises which assumed to be zero mean random processes that are uncorrelated with each other and with the signals, and A is the steering (or direction) matrix given by A= [a(θ1,Φ1), a(θ2,Φ2),. . . a(θD,ΦD)].The steering vector corresponding to the i_th DOA(θ,Φ) is given by a(θi,Φi)=[ a1(θ1,Φ1)e -j*w 0 *τ1(θi,Φi),. . ., aM(θD,ΦD) e -j*w 0 *τM(θi,Φi)] (2) Where am (θi,Φi) denotes the complex gain response of m_th sensor to a wave front arriving from direction (θi,Φi),w0 denotes the center frequency of the signals ,and τm(θi,Φi) denote the propagation delay between the sensors for a wave front impinging from direction (θi,Φi) which is given by τi=j2ΠR/λsin (Φi)cos (2Πm/M- θi) (3) Where R is the radius of the circular array, and λ is the wavelength. The covariance matrix of the array signal vector is given by R =E[x(k)x(k) H] =ASAH +ỏn 2 I =Rs +Rn (4) Where the superscript "H" denote the conjugate transpose .S=E[s (k) ×s (k) H] and ỏn 2 is the variance of the additive noise, let λ1≥ λ2≥ . . . λD≥ λD+1= . . . λM= ỏn 2, denote the eigenvalues of R and e1, e2. . .eM denotes the corresponding eigenvectors. If the matrices Es and En are formed as Es= [e1, e2,. . ., eD] (5) And En=[eD+1, eD+2 . . .eM] (6) Then the linear span of Es, known as the signal space, is same as spanned by the columns of A. The linear span of En, known as the noise subspace, is the orthogonal component of the signal subspace, then En H×a(θk,Φk)=0 ,k=1,2,. . .,D. 3.The Capon Method The Capon method tries to optimize the beamforming process according to the time varying covariance matrix .The spectrum is given by PMV (θ, Φ) =1/(a H (θ, Φ)R-1 a (θ, Φ)) (7) The method minimizes the power contributed by the noise and signals originating from other direction the current steering direction. Because R is consisting of a signal subspace and noise subspace, then we will take only the signal subspace RS, which is equal to ES ΛS ES (8) Where ΛS is a diagonal matrix of the signal eigenvealues, and ES is the corresponding eigenvectors as stated in (5).This feature of selection the signal subspace can be obtained by applying the Principal Component Analysis (PCA) Neural Network which extract the principal component i.e. λ1, λ2,. . . , λD and its corresponding eigenvectors e1,e2,. . . ,eD as illustrated in the next section. The number of sources is assumed known or we can find it by AIC, or MDL [10]. 4.The Neural Estimator In the last years several papers dealing with PCA neural networks [3],[4],[5],[6],[7],[8],[11]and [12] have discussed the advantages ,problems, and difficulties of such neural network (which is shown in figure 2) .In what follow we make use of an GHA algorithm with adaptive learning rate . Our neural estimator can be summarized in the following steps: N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 72 1) The GHA Algorithms FOR signal samples x(n): ferom n=1 To N (i) set error signal e0(n)=x(n) FOR every neuron: FROM j=1 To D (2) set wj(0) randomly (3) set ηj in accordance with input variance FOR input samples: FROM n=1 To N (4) yj(n)=x(n) H×wj(n-1) (5) wj(n)= wj(n-1)+ηj yj(n)[ej-1(n)-wj(n-1) × yj(n)] IF │ wj(k)- wj(k-1) │ < Є (where Є is very small value) THEN (6) wj=wj(n); GO TO STABLE (7) decrease ηj exponentially STABLE: FOR input samples: FROM n=1 TO N (8) set yj(n)=x(n) H×wj (9) set error ej(n)=ej-1(n)-yj(n)wj 2) wj qj ,and yj jv Where qj are the eigenvectors corresponding to eigenvalues vj. 3) DOA estimator, we use the modified Capon method where we take the synaptic weight as the eigenvector of the signal- subspace and the output of the PCA neural network as the squire roots of the output [7]. Then DOA are obtained as the peak location of the function according to the equation PMCapon (θ, Φ) =1/(a H (θ, Φ)ŘS -1 a (θ, Φ)). (9) Where ŘS -1=inverse of ŘS, and ŘS=∑ D i=1 (yj) 2 wi wi H . 5.Simulation In our simulation we will use a uniform circular array consisting of eight sensors of radius 5λ/Π as shown in figure 1. Assuming two noncoherent signals with the signal to noise ratio equal to one are impinging on the array and the first source have elevation angle θ1=50,and azimuth angle Φ1=30, while the second source have elevation angle θ2=60, and azimuth angle Φ2=40.Then by the use of the GHA algorithm we will find the signal subspace, that in turn are used by Capon method to find the DOA of impending sources as shown in figure 3,4, and 5,where figure 3 show the DOA(θi,Φi),figure 4 , show the azimuth angle only, while figure 5 show the elevation angle only, where the angle will be corresponding to the peak locations in the N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 73 spectrum of the Capon as shown in previous figures. 6. Discussion and Conclusions This paper described a simple ,but efficient methods based on PCA Neural Network to find the DOA ,where by use of the PCA neural network we don’t need to compute the correlation matrix R rather the first D eigenvectors of R are computed by the algorithm directly from the input data .the resulting computational saving can been enormous especially if the number of element M in the input vector is very large ,and the required number of the eigenvectors associated with the D largest eigenvalues of the correlation matrix R is small fraction of M..Then the DOA is achieved by incorporating the neural network with Capon method, with use of signal subspace only. Thus by the use of PCA neural network we neglect part of the noise, due to the neglecting of the noise subspace. 7.Refereneces [1] Blester, Y and Macorski A., 1986 "Exact maximum likelihood parameter Estimation of superimposed exponential signals in noise", IEEE Trans. On Acoustic, Speech, and Signal processing, vol. 34, pp. 1081-1089. [2] Capon, J., 1969,"High- resolution frequency – wavenumber spectrum analysis", Proc. IEEE, vol. 57, No. 8, August, pp. 1408 – 18. [3] Chatterjee, C., kang, Z., and Roychowdhury, V., 2000, "Algorithm for accelerated convergence of adaptive PCA", IEEE Trans. On neural network, Vol.11, No.2. [4] Chchocki, A., Kasprzak, W., and Skarbek, W., 1996,"Adaptive learning algorithm for principal component analysis with partial data", Austrian society for syberenetic studies, Vienna, Austria. [5] Chen ,T. ,and Yan ,W. , 1998, "Global convergence of Oja׳s subspace algorithm for principal component extractions",Vol. 9 ,No. 1. [6]Chichoki, A., Swiniarski, R., W., and Bogner, R., E.,1996 ,"Hierchical neural networks for robust PCA computation of complex valued signals". [7] Haykin, S., 1994, "Neural network, a comprehensive foundation", © by Macmillan college publishing company. Inc. [8] Kung, S., Y., 1993, "Digital Neural Network", © by PTR prentice Hall, Inc. [9] Schmidt, R.O., 1986, "Multiple emitter location and signal parameter estimation ", IEEE Trans.on antenna and propagation, vol. 34, No. 3, pp. 276 – 80. [10] Wax, M., .and Kailath, T., 1985, " Detection of signals by information theoretic criteria", IEEE Trans. On Acoustic, speech, and signal processing, vol. ASSP-33, No.2. [11] Weinglessel, A., and Hornik, K., 2000, "Local PCA algorithms", IEEE Trans. On neural network, Vol. 11, No.6. [12] Weinglessel, A., and Hornick, K., 1996, "Neural network algorithms for online principal component analysis and singular value decomposition ", Tuien-central for computational Intelligence. N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 74 X Z θi Φi Ith plane wave Figure 1: Uniform Circular Array The reference sensor xn . x2 y1 y2 ym    . . . . . Figure 2: The PCA neural network. x1 N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 75 Figure 4:The azimuth angle in degree The azimuth angle in degree T h e n o rm a li z e d s p e c tr u m Figure 3: The DOA( , ). N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 76 T h e n o rm a li z e d s p e c tr u m Figure 5:The elevation angle in degree N. H. Abbas /Al-khwarizmi Engineering Journal, Vol.2, No. 1,PP 70-77 (2006) 77 مع الشبكات العصبية GHAايجاد االتجاه باستخدام خوارزمية نزار هادي عباس ة /جامعة بغدادكلية الهندس /قسم الهندسة الكهربائية :الخالصة خوارزمياة ماع الاذاتي التعنايم ذات العصابية الشابكة اساتعمننا حياث. الوصاو زاوياة إليجااد عصبية شبكة تصميم تم البحث هذا في GHA كاابون بطريقاة تستخدم بدورها التيو الدائري الترتيب ذات الهوائيات قب من ةالمستنم لإلشارة األساسية المركبات النتزاع (كاابون بطريقاة يساتخدم بادوره الاذيتاخاذ جازم مان ف اام االشاارة PCA أ العصابية الشبكةان حيث. الوصو زاوية يجادإل .ال و ام ف ام هم وت