A Simple and Efficient Algorithm of 3-D Single-Source Localization with Uniform Cross Array Bing Xue 1 2 a) * Guangyou Fang 1 2 b and Yicai Ji 1 2 c)

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1 A Simpe Efficient Agorithm of 3-D Singe-Source Locaization with Uniform Cross Array Bing Xue a * Guangyou Fang b Yicai Ji c Key Laboratory of Eectromagnetic Radiation Sensing Technoogy, Institute of Eectronics, Chinese Academy of Sciences, Beijing 9, China University of Chinese Academy of Sciences, Beijing 9, China Corresponding Author* a: xuebing4@mais.ucas.ac.cn b gyfang@mai.ie.ac.cn c ycji@mai.ie.ac.cn Abstract A simpe efficient agorithm is presented for estimating the three-dimensiona (3-D ocation (two dimensiona direction-of-arriva (DOA range of a singe source with uniform cross array (UCSA. Based on a correation function, the DOA of source can be estimated with the uniform inear arrays (ULAs existing in UCSA. Then, another correation function is buit to estimate the range. The proposed agorithm provides the ow computationa compexity the high accuracy for the estimated vaues. Compared with two-dimensiona (-D mutipe signa cassification (MUSIC agorithm, numerica exampes are presented to demonstrate the performance of the proposed agorithm. Key words: Array signa processing, uniform cross array, three-dimensiona ocaization, passive sources ocaization. Introduction There is important topic in the passive sources ocaization with an array of spatiay dispersed sensors, which has many considerabe appication areas such as radar, artificia source ocaization sonar []. Various high-performance subspace identification agorithms ike MUSIC agorithm [, 3] ESPRIT agorithm [4] have been used to dea with the DOA estimation of the Far-fied sources. However, both DOA range of the source that is ocated at the Fresne region need be concerned in some practica appications ike sonar speaker guidance systems [5]. Therefore, in such scenarios, severa soutions are avaiabe to process the range of the source ocaization. Using ESPRIT agorithm, Zhi et a [] gave a method transforming -D search, incuded in the ocation estimations, to -D search in ULA, which estimates the eevation ange range of the source. Jiang et a [7] presented an agorithm that estimates DOAs of mixed sources by ESPRIT estimates the range of near-fied source by MUSIC, which aso estimates the eevation ange range of the source. However, the aforementioned high-resoution agorithms are a aimed at -D ocaization. For some 3-D probems (azimuth ange, eevation ange range, there are many researches that need to be done. Recenty, there are a significant amount of attentions paid to the 3-D ocaization. Wu et a [8] used the two-stage MUSIC to get the 3-D ocaization of the near-fied sources with uniform circuar array (UCA, which uses the information substantiay. Jung Lee [9] presented a singe source ocaization method using phases of the correation function with UCA, which has the ower computationa compexity than MUSIC agorithm. Jiang et a [] introduced the aperture extension method used in 3-D near-fied source ocaization with nonuniform cross array, which extends the range of appication for the MUSIC agorithm. The parameters of inear arrays are simper than the circuar arrays the MUSIC ESPRIT agorithm shoud compete the space search. In order to decine the cacuation compexity in 3-D ocation estimation with inear arrays, in this paper, a nove soution for 3-D ocation estimation of a singe narrowb source with a UCSA is deveoped, which deveops two nove correation functions to estimates the two eevation anges using two ULAs respectivey synthesizes them to compete the DOA range estimations. As a resut, the proposed agorithm is computationay simpe than the -D MUSIC requiring -D search eigen vaue decomposition. The accurate performance of the proposed agorithm is cose to MUSIC agorithm, it sti shows the great performance.

2 Probem formuation: Fig.. The array configuration structure. A UCSA composed of two ULAs with M isotropic sensors impinged by a narrowb signa that is radiated by a singe source. The inter-eement spacing is d, the phase reference ocation is the array center. UCSA geometry is shown in Fig.. Besides, the source is ocated at, is the eevation ange measured from the ULA in x axis, is eevation ange measured from the ULA in y axis r is the range measured from the UCSA centre. Assume that is the eevation ange is the azimuth ange in the 3-D coordinate system. the equations ( cos cos cos cos sin cos can be obtained to get the DOA. For each ULA, the sampe signa received by the -th sensor can be expressed as s(t (,, r is the source signa, n(t [, time of the signa from UCSA center to -th sensor. v ( M /. x, x j x ( t s( t e n ( t [, presents the additive sensor white noise, represents the deay reated with the propagation, y y can be written as v v ( can be respectivey expressed as y d d sin y cos, are caed eectric anges of ULA in x axis, r of ULA in y axis, denotes the source signa waveength. Using matrix form, ( can be written as x (t x x y (t x x x( [ xx( t, xx( t xm T ( x d d sin, x cos, r y y are eectric anges x t x ( t] (3 y( [ xy( t, xy( t ym T x t x ( t] (4 is represented by the eevation ange range of the source ocation using ULA in x axis ULA in y axis, respectivey. Therefore, the proposed agorithm estimates the ocation of the source by the phases of the correation function using two certain sensors. For this paper, the foowing hypotheses are need to hod: The source signa is a zero-mean narrowb stationary process. The sensor noise is spatiay uniform white independent from the signa. Assume that M is even. For the ULA in x axis, we shoud buid a correation function defined as j M x P E x n x n e (5 ( ( x x M s n,,, M, ( presents the compex conjugate operator, denotes the power of a compex narrowb signa s (t n represents the power of a white compex noise. Then, considering that the received signa data is noiseess, we can get the phase function expressed as U arg( P d P sin (6,,, M. By (6, can be obtained as In the simiar way, for the ULA in y axis, can be written as s arcsin ( d (7 U

3 3 V is the phase function cacuated by the correation function arcsin ( d (8 V Q,which is obtained by the In order to get the range of the source, we buid the second correation function. For the ULA in x axis, this correation function can be defined as j x x R E x n x n e (9 [ ( ] ( x x s n (n x y,,, M. Then, assuming that the received data is noiseess, too, we can get the phase function expressed as S arg( R d R cos r (,,, M. From (, r can be obtained as In the simiar way, using the ULA in y axis, r can be aso obtained as C is the phase function cacuated by the correation function N N n We can use ( / to estimate Equay, ˆ can be represented as E d r cos ( S d r cos ( C B, which is obtained by the (n x y. So, in order to use the array adequatey, ˆ can be represented as ˆ d M arcsin(/( M Uˆ (3 ˆ M arcsin(/( M ˆ V d (4 For getting the range estimation, using a the information of UCSA, we can obtain ˆ ( M d cos cos ˆ rˆ [ ] (5 M ˆ M ˆ S C It is worth noting that, if d is equa to or ess than is reated to d so that we can choose the vaue of d according to the approximate range for the actua scenarios. When we consider the computationa compexity, our agorithm requires MN mutipications the first correation function (5 phase function (6 respectivey, O(M in computing the DOA. Simiary, it needs /3, the proposed agorithm admits various array aperture. The range of source.. M mutipications to assess ( M N mutipications M mutipications to assess the second correation function (9 phase function ( respectivey, O(M in computing the DOA. Meanwhie, the -D MUSIC estimator needs M N mutipications for computing 3 the covariance matrix, O(3M / 4 for computing the eigen vaue decomposition of the covariance matrix, the additiona computing costs in -D search. 3 Simuation resuts: Some simuation resuts are used to assess the proposed agorithm performance. A symmetric UCSA with M 8 d 4 is given. A narrowb signa source ocated at (46. o, 6.4 o,.8λ is radiating on this UCSA. We use the spatia white compex Gaussian rom process as the additive noise. As the performance comparison, -D MUSIC agorithm that estimates DOA using ULA in x axis ULA in y axis respectivey estimates the range by the average of the ranges cacuated in ULA in x axis ULA in y axis are executed. And we aso execute the reated Cramer-Rao Bound (CRB. The signa-to-noise ratio (SNR is given reative to each source signa. The resuts shown next are assessed by the estimated root man square error (RMSE from the average resuts of independent Monte-Caro simuations. We determine the SNR from to db the snapshot number is 6. The RMSEs of the, r estimations by using the proposed agorithm are denoted in Fig., Fig.3 Fig.4, respectivey. The -D MUSIC agorithm is executed by using a search step size of (,, r = (. o,. o,.λ.

4 4.5 proposed agorithm ange D-MUSIC agorithm ange CRB ange RMSE (degree in db SNR (db Fig.. RMSEs of ange estimation for singe source versus SNRs. (46. o, 6.4 o,.8λ, the snapshot number is 6. independent trais. As shown in Fig., Fig.3 Fig.4. The performance of the proposed agorithm is very cose to that of -D MUSIC agorithm CRB over a wide range of SNR. When the computationa compexity is concerned, we compare the CPU times between them. For the -D MUSIC agorithm, we assume search grids of the r assume search grids of the r. The computed resuts are obtained by using the Inte i7 processors to run at.4ghz. The running time of the proposed agorithm is.55735s. The running time of the two -D MUSIC estimators is s. As our expecting, the CPU running times indicated that our agorithm is simper in computation than the -D MUSIC agorithm..5 proposed agorithm ange D-MUSIC agorithm ange CRB ange RMSE (degree in db SNR (db Fig.3. RMSEs of ange estimation for singe source versus SNRs. (46. o, 6.4 o,.8λ, the snapshot number is 6. independent trais..5 proposed agorithm range D-MUSIC agorithm range CRB range RMSE (range in db Concusion SNR (db Fig.4. RMSEs of range estimation for singe source versus SNRs. (46. o, 6.4 o,.8λ, the snapshot number is 6. independent trais.

5 5 This paper has given a three dimensiona scenario for singe source ocaization using UCSA. Our investigation has shown that unike the MUSIC agorithm, the nove agorithm is very simpe has the ow cacuation compexity. In addition, the estimation performance of the proposed agorithm is cose to that of MUSIC agorithm. References [] H. Krim M. Viberg, Two decades of array signa processing research: The parametric approach, IEEE Signa Process. Mag., vo. 3, no. 4, pp , 996. [] T.Xia, Y.Zheng, Q.Wan, X.Wang, Decouped estimation of -D anges of arriva using two parae uniform inear arrays, IEEE Trans. Antennas Propag., vo. 55, no. 9, pp , Sep. 7. [3] H.Tao, J. Xin, J. Wang, N. Zheng, A. Sano, Two-Dimensiona Direction Estimation for a Mixture of Noncoherent Coherent Signas, IEEE Trans. Signa Process., vo. 63, no., pp , Jan. 5. [4] R. Rot T. Kaiath, Esprit-estimation of signa parameters via rotationa invariance techniques, IEEE Trans. Acoust., Speech. Signa Proces., vo. 37, no. 7, pp , Ju [5] J. Liang D. Liu, Passive ocaization of mixed near-fied farfied sources using two-stage music agorithm, IEEE Trans. Signa Process., vo. 58, no., pp. 8, Jan.. [6] W. Zhi M. Y. Chia, Near-Fied Source Locaization via Symmetric Subarrays, Internationa Conference on Acoustics, Speech, Signa Processing, pp. 4, 7. [7] J. Jiang, F. Duan, J. Chen, Y. Li, X. Hua, Mixed Near-Fied Far-Fied Sources Locaization Using the Uniform Linear Sensor Array, IEEE Sensors Journa, vo. 3, no. 8, pp , Aug. 3. [8] Y. Wu, H. Wang, Y. Zhang, Y. Wang, Mutipe near-fied source ocaisation with uniform circuar array, Eectronics Letters, vo. 49, no. 4, pp. 59 5, Nov. 3. [9] T. Jung K. Lee, Cosed-Form Agorithm for 3-D Singe-Source Locaization With Uniform Circuar Array, IEEE Antennas Wireess Propag. Lett., vo. 3, no. 6, pp , Jun. 4. [] J. Jiang, F. Duan, J. Chen, Y. Li, X. Hua, Locaization of 3D Near-fied Source using the Aperture Extension Method Nonuniform Cross Array, Progress In Eectromagnetics Research B, vo. 55, pp , 3.