# Performance Analysis of MUSIC and MVDR DOA Estimation Algorithm

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2 In first section of paper, the performance of two algorithms for direction of arrival i.e. MUSIC and MDR respectively are investigated. The performance of these algorithms is analyzed by considering parameters like number of array elements, user space distribution, number of snapshots, signal to noise ratio, Mean Square Error MSE), which results in optimum array design in SAS. In second section the simulation result, conclusion and future work is presented. The algorithms have been simulated in MATLAB 7.4 version. II. DOA ESTIMATION ALGORITHMS The algorithms based on DOA are classified as nonsubspace or quadratic type and subspace type [6]. The Barltett and Capon Minimum Variance Distortionless Response) [6] are quadratic type algorithms. Both the methods are highly dependent on physical size of array aperture, which results in poor resolution and accuracy, [5], [7], [9], [11], [12], [13]. Subspace based DOA estimation method is based on the Eigen decomposition [8]. The subspace based DOA estimation algorithm MUSIC provides high resolution, and is more accurate and not limited to physical size of array aperture [2] [7]. The various DOA algorithm performance is analyzed based on number of snapshots, number of users, user space distribution, number of array elements, SNR and MSE. III. MUSIC MUSIC means Multiple Signal Classification. It is one of the high resolution subspace DOA algorithms, which gives the estimation of number of signals arrived, hence their direction of arrival [6]. MUSIC deals with the decomposition of covariance matrix into two orthogonal matrices, i.e., signal-subspace and noise-subspace. Estimation of DOA is performed from one of these subspaces, assuming that noise in each channel is highly uncorrelated. This makes the covariance matrix diagonal. Consider the model equation when M signals incident on array and corrupted by noise: 1) Using equation 1) x = S + n 2) ] 3) = [ 4) The matrix S is an N M order matrix of the M steering vectors. Assuming that the different signals are uncorrelated, the correlation matrix of x can be given by: R = E [ ] 5) = E [ ]+ E [ ], = +, 6) =, 7) = 8) [ ]9) The signal covariance matrix,, is clearly a N N order matrix with rank M. Therefore it has N M eigenvectors corresponding to the zero Eigen value. Let be such an eigenvector. So, 10), 11) 12) The validity of final equation holds as the matrix A is clearly positive definite. Equation 12) implies that all N M eigenvectors ) of corresponding to the zero eigenvalue are orthogonal to all M signal steering vectors. This is the very basis for MUSIC. Let be the N N M) matrix of these eigenvectors. MUSIC plots the pseudospectrum can be given by; = = = 13) Since the eigenvectors making up are orthogonal to the signal steering vectors, the denominator becomes zero when is a signal direction. Therefore, the estimated signal directions are the M largest peaks in the pseudo-spectrum. However, in any practical situation, the signal covariance matrix would not be available. The most we can expect is to be able to estimate R the signal covariance matrix. The key is that the eigenvectors in can be estimated from the eigenvectors of R. For any eigenvector ), 14) i.e. any eigenvector of is also an eigenvector of R with corresponding Eigen value +. Let = QΛ. Therefore, R=Q [Λ + ] =Q Based on this Eigen decomposition, the eigenvector matrix Q can be portioned into a signal matrix with M columns, corresponding to the M signal Eigen values, and a matrix, with N M) columns, corresponding the noise Eigen values ). Note that, the N N M) matrix of eigenvectors corresponding to the noise Eigen value ), is exactly the same as the matrix of eigenvectors of Rs 51 Copyright IJEMR. All Rights Reserved.

3 corresponding to the zero-eigen value. This is the matrix used in Eqn. 13). Defines the signal subspace, while, defines the noise subspace.there are few important observations to be made: The signal Eigen value is given by = N + The smallest Eigen values of R are the noise Eigen values and are all equal to, i.e., one way of distinguishing between the signal and noise Eigen values equivalently the signal and noise subspaces) is to determine the number of small Eigen values that are equal. By orthogonality of Q, Using the final two observations, we see that all noise eigenvectors are orthogonal to the signal steering vectors. This is the basis for MUSIC. Consider the following function of: = = 15) Where is one of the N M) noise eigenvectors. If is equal to DOA one of the signals, s ) and the denominator is identically zero. MUSIC, therefore, identifies as the directions of arrival, the peaks of the function. IV. MINIMUM VARIANCE DISTORTION LESS RESPONSE MVDR means Minimum Variance Distortionless Response. In Figure-1 a uniform linear array ULA) of N equally spaced sensors is shown. A number of plane waves from M narrowband sources impinging from different angles, i = 1, 2 M. At a particular instant of time t, t=1, 2 K, where K is the total number of snapshots taken, the array output will consist of the signal, and in addition to that noise components The signal vector x t) can be defined as different angles, i = 1, 2 M. Where s t) is an M 1 vector of source waveforms and for a particular source at direction θ from the array bore sight; aθ) is an N 1 vector referred to as the array response to that source or array steering vector for that direction. It is given by: [ ] Fig.-1. Uniform Linear Array with M- Element Where T is the transposition operator and Ф represents the electrical phase shift from element to element along the array. This can be defined by: Where d is the element spacing and λ is the wavelength of the received signal. The signal vector x t) of size N 1 can be written as: Where, is an N M matrix of steering vectors. The array output consists of the signal plus noise components, and it can be defined as: Where are signal and noise components. If there are D signals incident on the array, the received input data vector at an M-element array can be expressed as a linear combination of the D incident waveforms and noise. Where A is the matrix of steering vectors, Where is the signal vector, and is a noise vector with components of variance. Now equation -18 can be written in the form of matrix of size N K as: U = A.S + N Where S=[s 1) s K)] is an M K matrix of source waveforms and N= [n 1). N K)] is an N K matrix 52 Copyright IJEMR. All Rights Reserved.

4 of sensor noise. The spatial correlation matrix R of the observed signal vector u t) can be defined as: Where E and H are the expectation and conjugate transpose operators, respectively. The spatial correlation matrix R can now be express can now be expressed as: The peaks in the MVDR angular spectrum occur whenever the steering vector is orthogonal to the noise subspace. This technique minimizes the contribution of the undesired interferences by minimizing the output power while maintaining the gain along the look direction to be constant, usually unity. That is Using Lagrange multiplier, the weight vector can be given by, The output power of the array as a function of the DOA estimation, using MVDR beamforming method, is given by MVDR spatial spectrum as, Fig.2 -DOA estimation at angle-15 The angles of arrival are estimated by detecting the peaks in the angular spectrum. V. SIMULATION RESULT The MUSIC and MVDR techniques for DOA estimations are simulated using MATLAB. Performance of the algorithm has been analyzed by considering function of array elements, of SNR and as a function of snapshots. The simulation has been run for signals coming from different angles at 15, 20, 25, 30 and 35 for 200 snapshots, power of incoming signal 15 db, SNR of 50dB, and 8 array elements. MUSIC algorithm estimates the number of incident signals on the array and their directions of arrival. The first step is the calculation of the covariance matrix. The next step involves calculation of the Eigen values and Eigen vectors. On knowing the minimum Eigen value the multiplicity of this Eigen value is obtained. The next step is to obtain the noise eigenvector matrix of these minimum Eigen values and form the Noise Eigen Vector. Using this Noise Eigen vector the direction of arrival can be plotted in a single direction or multiple dimensions. Fig.3 -DOA estimation at angle-20 Fig.4 -DOA estimation at angle Copyright IJEMR. All Rights Reserved.

5 Fig.5 -DOA estimation at angle-30 Fig.8 -DOA estimation at angle-20 Fig.6 -DOA estimation at angle-35 Fig.9 -DOA estimation at angle-25 It has been observed that in MVDR spectrum there is a sharp peak in an angular spectrum and a lower noise floor compared to the MUSIC algorithm. In MUSIC spectrum sharper peak indicates the location of desired user where as in MVDR power plot maximum power at an angle indicates the location of desired user. Fig-2-11 shows the comparison of both the algorithms. Fig.10 -DOA estimation at angle-30 Fig.7 -DOA estimation at angle Copyright IJEMR. All Rights Reserved.

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