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Direction-of-Arrival Estimation Based on Combination of Independent-Component-Analysis and Spatial Spectrum [Sensors & Transducers (Canada)](Sensors & Transducers (Canada) Via Acquire Media NewsEdge) Abstract: Conventional space spectrum algorithm of the direction-of-arrival (DOA) estimation such as the multiple-signal-classification (MUSIC), have shown good results. However, the algorithm depends on the prior of signal source number estimation. But the signal source number estimation will be difficult in low signal-tonoise ratio (SNR) and weak signal. In order to improve the soundness DOA estimation in the low SNR and weak signal, we present an algorithm governed by independent-component-analysis (ICA). The algorithm can directly estimate the DOA without the signal source number be determined, in other words it does not depend on the eigenvalues distribution of covariance matrix. Simulation results show that the proposed algorithm is better robustness than conventional space spectrum algorithm to estimate DOA of the low SNR signals and the presence of weak signal. Copyright © 2013IFSA. Keywords: Independent-component-analysis (ICA), Spatial spectrum, Direction-of-arrival (DOA) estimation, Signal source number estimation, Multiple-Signal-Classification (MUSIC). (ProQuest: ... denotes formulae omitted.) 1. Introduction Direction of arrival (DOA) estimation algorithm refers to process the mixed signal received by array and get the azimuth angle that the target is relative to reference point, as shown in Fig. 1. DOA algorithm is widely used in such fields as array signal enhancement, preprocessing of early stage speech recognition, mixed signal separation, communication, radar and sonar. The key technology of signal DOA estimation is spatial spectrum estimation. DOA can be precisely calculated with a variety of spatial spectrum estimation methods in the case of known signal dimension [1]. Specifically, the signal dimension refers to the number of signal source. The signal source number is unknown in real environment, and the conventional spatial spectrum estimation may be less estimated or excess estimated. Less estimation or excess estimation will influence the accuracy of target signal DOA estimation. Even in the case of unknown source number, spatial spectrum estimation method is likely to fail, and so there would be no way to estimate DOA of target signals correctly. Conventional method to solve the spatial spectrum estimation failure is to estimate the number of signal source before DOA estimation. Commonly used methods for source number estimation are hypothesis test [2], accurate testing method based on information theory [3], generalized likelihood ratio method, maximum posteriori probability method [4], signal source number estimation aimed at coherent signal source and signal source number detection based on model [5,6]. Make improvement to them, and the performance is improved. The accuracy of these methods is higher in high SNR, but when hybrid matrix is pathological or weak signal presents, the estimated effect is poorer [7-10]. Therefore, this paper proposed DOA estimation algorithm based on the integration of independent component analysis and spatial spectrum. In the process of algorithm, the DOA estimated by ICA constrains the spatial spectrum method, and can estimate the DOA of target signals directly. The robust of this algorithm is high. It still can estimate the DOA of signal in small SNR and the presence of weak signal [12-15]. 2. Signal Model of Array The array considered in this paper is uniform circular array. As shown in Fig. 1, the radius of the array is 1/2 smaller than the wavelength of source signal, and the array is isotropic. For the N source signals whose center frequency are fc, s(k) = [sl(k),s2(k),---,sN(k)]fi incident to the array. The incident angle of the i01 signal is {0i,(pi), Ot is the included angle between z axes and the signal incidents to the array, (pt is the included angle between x axis and the projection of the signal incidents to the array. Use (1) to describe the observation signal of array element. ...(1) where x(k)^[xl(k),x2(k),...,xM(k)f is data snapshot of Mx 1 dimension; s(k) is the Nxl dimension vector of spatial signal; A = [ax,a2,...,aN] is the direction vector array of MxN dimension; a. = [exp(-y2^/cr".),exp(-y2^/cr2iexp(-y2;r/c z-jf y InCk - 1) is direction vector; .... is c M delay. Here, n is the number of incoming signal; m is the number of array element in array; ai is the direction vector of the ith incoming signal; rki is the delay when the i01 incoming signal gets to the k01 array element and the center of array element; n(&) is the additional noise of array. 3. Signal Separation of Independent Component Analysis Definition 3.1. Mixed signal: the signals sent out by N signal sources are received by M sensors, if ignore the difference between sensors, and think that the signal received by sensor is instantaneous linear superposition, then the mixed signal is ...(2) where x(k) is the sensor output vector of Mxl dimension; s(Ä:) is the signal source vector of Nxl dimension; H is unknown hybrid vector. Particularly, although the writing here is different from formula (1), it is thought that H is an unknown array without any priori knowledge in independent component analysis. n(k) is the noise vector of N x 1 dimension. Hypothesis 3.1: The statistics between Nxl dimension signal source vectors of s(¿) is independent; Hypothesis 3.2: H is full column rank; Hypothesis 3.3: n(&) is Gaussian white noise. Definition 3.2: Independent component analysis (ICA): through searching a disjunct matrix W with full MxN rank, make ...(3) Each component of output vector y(k) = [yx(k),y2(k),...,yM(k)f should independent as far as possible. According to hypothesis 3.1-3.3, when each component of output vector y(k) is independent, y(k) contains s(&) source signal component, namely can separate signal source vector from mixed signal x(k) through ICA. Definition 3.3: Statistics of independent measure function: define R(W) = E{p(W,y)} function, when each component of y(k) is mutually independent, R(W) is minimum, namely minimizing R(fV) can determine disjunct matrix and make the components of output vector y(k) independent. If make /^(yjW) as the probability density function of output vector y(k) = Wi(i) = WHs(&), q{y) as the probability density function of the components of y in statistic independence, then ... (4) Formula (4) is only a reference function. KL divergence can be defined as [16-18] ... (5) When equals the real distribution of source signal, ... (6) Namely, when K[py(y; W) || #(y)] = 0 , y the components of y statistically independent. So can make R(W) = Kpq . In order to minimize R(fV) , Amari introduced natural gradient descent algorithm [19] in Riemannian space, and got iterative form ... (7) f(y) is nonlinear function vector. After promotion, get the real-time form of natural gradient descent algorithm in the sense of lie group [20]. 7... (8) Among them ... (9) and ... (10) q is the delay coefficient. When formula (8) is convergent, get W , then can use formula (3) to separate independent component, namely estimate source signal. 4. DOA Estimation Based on Spatial Spectrum For formula (1), in the situation when signal space dimension is known, consider that n(&) is the Gaussian noise of a variance and 0 mean value, make eigenvalue decomposition signal space and noise space to x(&) covariance matrix R. ... (11) study ... (12) on the other hand, R = ARSAH + a21, then ... (13) Summarize formula (13) and (14), get ... (14) And then UnARsAHUn = 0 . As is known to all, tHQt = 0, if and only if t = 0 , so formula (15) can be equivalent to ... (15) Put A = [a, (0,, (px ), a2 (02,<p2),...,aN(0N, <pN )] in the above formula, then ... (16) Only when (0t<p) = (6yi<pi)A62i<p1)i...i{6Nt<pN) , the above formula can be set up. Defined function: ... (17) Formula (18) described the distribution of spatial parameter (DOA), so it is called spatial spectrum. In view of the actually received data is limited, then x(t) covariance matrix is ... (18) Make eigenvalue decomposition to R ? and get characteristic vector matrix U" of noise space, the spatial spectrum can be defined as ... (19) Formula (19) will generate "spectral peak" in the direction of signal source, and relatively smooth in other directions. Search N maximum spectral peaks in spatial spectrum, and the corresponding angle of the N spectral peaks is the DOA of the N incoming target signals. Use the direction vector got from formula (19) to determine corresponding beam, namely use the minimum variance distortionless MVDR to determine beam: ... (20) Then the incoming signal waveform from direction can be estimated as, ... (21) Formula (21) is the signal separation through the way of spatial spectrum estimation. When making R eigenvalue decomposition, the eigenvalue meet: ... (22) But does not meet: A, >A2 >...>An >AN+i =... = A M=a2, (23) So the signal source number N cannot be determined only from eigenvalue decomposition. So the estimation of signal source number must use AIC criterion, MIBS criterion, and etc. Namely the conventional spatial spectrum estimation algorithm must use signal source number estimation method to estimate the value of N, and then to estimate DOA. Section 5 will present the DOA estimation algorithm of combining ICA with spatial spectrum, which can avoid the situation that must estimate the number of signal source first before DOA estimation. 5. DOA Estimation Algorithm Based on the Combination of ICA and Spatial Spectrum From the analysis in section 3, we can know that make independent component analysis to the mixed signal received by array, when the components of separative signal y,CA(k) are mutually independent, yICA (k) contains the source signal component s(&) to be estimated. Again, from the analysis in section 4, we can know that when the DOA of correct signal source is found, use the beam formed by DOA, the signal source yP(i) (t) in this direction can be got from mixed signal x(t) (i = 1,2,...,N) . According to hypothesis 3.2 in section 3, yP(i) (t) are mutually independent. Definition 5.1: Use the signal yP(i)(k) (i = 1,2,...,«) separated by spatial spectrum and the components in yICA(k) analyzed by independent component to make correlation calculation. Mark the maximum correlation coefficient got from it as rmax n, and the minimum correlation coefficient as rminJ. The smallest gap between them is ... (24) Definition 5.2: The correlation coefficient estimation of the 11,1 component of yP{i)(t) and y,CA(k) : ... (25) Theorem 5.1: When n = N , formula (24) is maximum. Demonstration: know from definition 3.2, the components of yICA (k) independent mutually, and is the estimation of s(£) ; Again, because when n = N , spatial spectrum estimation can correctly estimate DOA, so the separated signal yP(i) (t) is source signal; So the components of yP(i) (t) only have maximum correlation with one component of y iCA (k), and the correlation with other components is zero; So when n = N, formula (24) is maximum, (has been proved). 5.1. Basic Steps of the Algorithm According to the previous analysis, the basic steps of DOA estimation algorithm based on ICA and space spectrum are as follows: 1) Firstly, use independent component analysis, including formula (3)-(10), to separate received mixed signal x(k) into independent component yICA (k) ; 2) Then, suppose the scope of the unknown signal source number n is 1 - M -1 , so there areM-1 space spectrums. Use formula (18)-(21) to calculate the signal that each space spectrum separated, and mark corresponding DOA. Gx = {yPm Corresponding DOA is {(0X, (px )} ; G2={ypX{k\yP2{k)} Corresponding DOA is {(6X, (px ), (02, (p2 )} ; ... Gfj = {yp\{}d),yp2(Jt),...,ypN(Jc)} Corresponding DOA is ... Gm-\ = {yP\(k),yP2(k),...,ynM_X)(k)} Corresponding DOA is {(^p^l )» (^2 ' ^2 )»***» (^Ai-l) » ^(A/-l) )} >K 3) Finally, use formula (24) and (25) to make correlation calculation to the components of G and yICA(k). When mn(i) reaches its first peak, the DOA corresponding to is the DOA to Gi be estimated. 5.2. Algorithm Correction When there exists signal with larger comparison in strong or weak, if there is one signal is strong, the other are weak. When using the basic steps to conduct DOA estimation, G, gets to its first peak, then there is only DOA with stronger signal. Therefore, when G, gets to its first peak, it must add some measures for amendment. After step 1) of the algorithm, calculate the separated performance index (PI) value, ... (26) Among them ... (21} In step 2), the side lobes of the corresponding spatial spectrum of G, have "the highest main lobe * (l-PI)/2" are the "spectrum peaks" formed from the incoming signal. If there is still no any side lobe is greater than the "highest main lobe * ( 1 -PI)/2" after modification, the DOA corresponding to is thought to be estimated DOA. Or when the acquired mn(i) gets to its second peak, the DOA corresponding to Gi is thought to be estimated DOA. 6. Comparisons and Analysis of Algorithm Performance In order o test the performance of the DOA estimation algorithm (ICA-P) based on the combination of ICA and spatial spectrum, the computer simulation experiment was carried out, and compared with conventional DOA spatial spectrum estimation algorithm, such as AIC-MUSIC, MIBS-MUSIC. The signal source used in simulation experiments is three narrowband signals. They are: sx = ün(2xf(0.\sawtooth(\0xt) + 0.2)) s2 = sin(2;r / (0. \sawtooth{9xt) + 0.2)) s3 = sin(2;r / (0. \sawtooth{7 xt) + 0.2)) Among them, sawtooth(2xfat) is the saw tooth wave whose frequency is fa.f = 1000Hz , sample frequency is 2000 Hz. Sensor array model, as shown in Fig. 1 and formula (1), the number of array source is 12, isotropic uniform circular array, radius is half of signal wavelength. Experiment 1: the DOA estimation under different signal-to-noise ratios. Incoming signal is H s1.s2.s3, and angle is (25° ,15°), (30°, 45°), (40°, 80°) respectively. Under each different SNR, do experiments for 100 times on AIC-MUSIC, MIBSMUSIC and ICA-P that the algorithm proposed in this paper respectively. The eigenvalues distribution of covariance matrix of different SNR shown in Fig. 2, Fig. 3, Fig. 4, Fig. 5. The accuracy of different algorithms to DOA estimation shown in Table 1, Table 2, Table 3. Conventional space spectrum method, such as AIC-MUSIC and MIBS-MUSIC, is to first use AIC and MIBS to estimate the number of signal source separately, and then use MUSIC algorithm to estimate DOA. It can be seen from Table 1, 2, 3 that AIC-MUSIC and MIBS-MUSIC are failure when the SNR is less than -5dB. It is because AIC and MIBS are depended on the eigenvalues distribution of covariance matrix, and no significant difference between the eigenvalues of signals and noise (shown in Fig. 2, Fig. 3), cause the failure of AIC and MIBS. And then cause the failure of MUSIC. While the ICA-P proposed in this paper, because does not need to estimate signal source number in advance, algorithm is still valid in low SNR, shown in Table 3. Experiment 2: There are signals with large comparison in signal strength in incoming signal. Si incidents from (25°, 15°), and S2 incidents from (40°, 80°). Si is strong signal, S2 is weak signal. In the comparison of different signal strength, do experiments for 100 times on AIC-MUSIC, MIBSMUSIC and ICA-P. The experimental results are shown in Table 4, Table 5 and Table 6. In the presence of weak signal, AIC and MIBS consider the eigenvalues of weak signal as the eigenvalues of noise. So the estimation of source number is wrong, and the DOA of weak signal cannot be estimated. And the DOA estimation algorithm of ICA-P proposed in this paper, in the three cases of the experiment, has higher accuracy of DOA estimation. Therefore, when using the algorithm proposed in this paper to estimate the DOA of signal, the phenomenon that the DOA of strong signal floods the DOA of weak signal will not exist, as shown in Table 4, Table 5, Table 6. 7. Conclusions Based on systemic analysis of ICA and spatial spectrum, this paper got the relationship between signal yP(i) (t) separated by spatial spectrum and the components of y,CA(k) analyzed by independent component. Proposed the DOA estimation algorithm constrained by ICA. When using this algorithm to estimate DOA, it is no need to estimate the number of signal source first, the number of signal source is obtained directly in the process of iterative calculation, at the same time, obtained the corresponding DOA of each source signal. Because the algorithm does not depend on the number of signal source, it has the advantage of robust DOA estimation to low SNR and weak signals. In view of the possibility of existing signal with bigger strength difference, also correct of the algorithm. Through the simulation experiment, we can see that the algorithm in this paper is more robust than conventional spatial spectrum estimation algorithm in low SNR and the presence of weak signal. Acknowledgements This work is supported by the Education Department of Guangxi, China (project approval number: 2012JGA267), and Guangxi Office for Education Sciences Planning, China (project approval number: 2013008), and Guangxi Provincial Natural Science Research Project for Universities (project approval number: 2013YB378), and Characteristic professional project fund of the Education Department of Guangxi, China (project approval number: GXTSZY277) and Education Science fund of the scientific research projects of Guangxi College of Education, (project approval number: A2012002). References [1] , R. O. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Transactions on Antennas and Propagation, Vol. 34, Issue 3, 1986, pp. 276-280. [2] , S. Kritchman, Non-parametric detection of the number of signals: hypothesis testing and random matrix theory, IEEE Transactions on Signal Processing, Vol. 57, Issue 10,2009, pp. 3930-3941. [3] . 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Jinde HUANG, Ping WANG Mathematics and Computer Science Department, Guangxi College of Education, Nanning, Guangxi, 530023, China Tel: 0086 15907719073 E-mail: [email protected] Received: 4 October 2013 /Accepted: 22 November 2013 /Published: 30 December 2013 (c) 2013 International Frequency Sensor Association |