Near-field time-frequency localization method using sparse representation

The Journal of China Universities of Posts and Telecommunications December 2012, 19(6): 29–34 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Near-field time-frequency localization method using sparse representation WANG Bo, LIU Juan-juan ( ), SUN Xiao-ying , ZHANG Yan-jun College of Communication Engineering, Jilin University, Changchun 130025, China

Abstract This paper presents a novel near-field source localization method based on the time-frequency sparse model. Firstly, the method converts the time domain data of array output into time-frequency domain by time-frequency transform; then constructs sparse localization model by utilizing the specially selected time-frequency points, and finally the greedy algorithms are chosen to solve the sparse problem to localize the source. When the coherent sources exist, we propose an additional iterative selection procedure to improve the estimation performance. The proposed method is suitable for uncorrelated and coherent sources, moreover, the improved estimation accuracy and the robustness to low signal to noise ratio (SNR) are achieved. Simulations results verify the efficiency of the proposed algorithm. Keywords near-field source, time-frequency distribution, sparse representation, DOA estimation, range estimation, greedy algorithm

1

Introduction 

Near-field source localization has been applied in many areas [1] such as radar, sonar, modern wireless communication and so on, and also has received a tremendous attention over the years. Many algorithms have been proposed to solve the localization problem, such as two-dimensional multiple signal classification (2-D MUSIC) [2], estimation of signal parameters via rotational invariance techniques (ESPRIT) based on higher-order cumulant [3] and other subspace algorithms [4–6]. However, these high resolution methods will fail to work when there are coherent signals caused by multipath propagation. To decorrelate the coherent signals, some preprocessing methods are proposed, such as the spatial smoothing technique [7] and difference technique [8], which have been applied to localize the far-field coherent sources. But it is hard to apply the techniques to the near-field coherent sources localization due to the nonlinear of the localization model. Recently, the sparse representation has been introduced in direction of

Received date: 28-05-2012 Corresponding author: LIU Juan-juan, E-mail: [email protected] DOI: 10.1016/S1005-8885(11)60315-4

arrival (DOA) estimation [9–13], which can efficiently deal with the coherent source problem without any decorrelated operation and will not be affected by the model nonlinear. These localization methods mentioned above do not take the characteristics of source signals into account. When source signals are unstationary and their time-frequency (T-F) characteristics are known, the T-F information can be incorporate into the localization algorithms to improve the localization performance. The technique has been applied to DOA estimation [14] and the performance of the class of algorithms are analyzed theoretically in Ref. [15], which verifies that T-F characteristics bring many advantages including high estimation accuracy, supper resolution and no requirement about the less source number than sensors. In this paper, combining the T-F distributions with sparse representation, we propose a new localization algorithm of near-field sources, which can estimate the localization parameters of all sources, no matter whether the sources are uncorrelated or coherent. Firstly, the method converts the time domain data of array output into T-F domain by T-F transform; then selects the special T-F points to construct the localization sparse model; finally

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The Journal of China Universities of Posts and Telecommunications

the greedy algorithms [16–17] are chosen for solving the sparse problem to estimate the localization parameters of sources. For coherent sources, the iterative selection procedure is added on the basis of the above algorithm to improve the estimation performance. In the simulations, we compare the performance of proposed algorithm with 2-D MUSIC method and Cramer-Rao bound (CRB) [1] and the results show that the method has the advantages of high estimation accuracy and robustness to low SNR.

2

Data model

2.1 Received signal model Consider the scenario that K nonstationary sources

^sk (t )`k 1 K

impinge on a uniform linear array (ULA) of M

omnidirectional sensors with inter-spacing d. Let the first sensor as reference sensor, the output of sensors y  t at time index t can be expressed as y (t ) [ y1 (t ), y2 (t ),..., yM (t )]T

K

¦ a T

k

where H is the conjugate transpose. By substituting Eqs. (1) and (2) into Eq. (3), we can denote the ith column elements of matrix D yy (t , f ) in following form D yyi  t , f

f

k f

H i

f

W e  j4f W

B T , r Dss (t , f )biH T , r  B T , r ˜

Dsni (t , f )  Dni s  t , f biH T , r  Dnni (t , f ) ( 4 )

Based on the uncorrelated assumption between source and noise, we have Dyyi  t , f B T , r Dss (t , f )biH T , r  Dnni (t , f )ei (5) where ei is the M u1 vector given i P ei [0, },1, }, 0]T , in which only ith element is one.

3

by

Proposed method

3.1 The T-F sparse model of near-field localization

[0.62  D 3 O

1/ 2

(1)

1, 2,..., N

f

¦ W¦ I  k ,W y  t  k  W y  t  k 

Consider that the near-field region of array is uniformly divided into N r NT grids, namely, the range domain

, rk sk (t )+n(t )

k 1

B T , r s (t )  n(t ); t

2012

, 2 D 2 O ] , which belongs to the Fresnel

where N represents the total sample number and a (T k , rk )

zone of array [18], is uniformly divided into N r intervals,

is the steering vector of kth source, which can be defined as

domain [90q,90q] are uniformly divided into

ª «¬1, 2,..., e

T

º »¼ , where parameters (T k , rk ) represent the direction of a T k , rk

ª 2( M 1) d sin T k ( M 1)2 d 2 cos 2 T k º j«  » O O rk »¼ ¬«

arrival (DOA) and range of kth source, respectively, O denotes the signal wavelength and T is the transpose symbol. The array manifold matrix B (T , r ) is given by B (T , r ) [a (T1 , r1 ),..., a (T K , rK )] . The output yi (t ) of ith sensor can be represented as yi (t ) bi (T , r ) s (t )  ni (t ); t 1, 2,..., N (2) where the bi (T , r ) is ith row vector of B (T , r ) , and ni (t ) denotes the additive Gaussian white noise of ith sensor, which is independent from the source signals. 2.2 Spatial T-F distribution The spatial Cohen’s class T-F distribution matrix of array output y  t can be obtained as follow [14] D yy  t , f

f

¦

f

¦ I  k ,W y  t  k  W y H  t  k  W e j4f W

k f W f

(3)

here D =  M  1 d is the array aperture. And the DOA NT

intervals as shown by G {(T1 , r1 ),! , (T1 , rNr ),! , (Tk , r1 ),! , (Tk , rNr ),! , (TNT , r1 )! , (TNT , rNr )}

where we assume the source locations exactly lie at the grids, and the overcomplete basis A T , r can be given by A T , r

^a T , r ,!, a T , r ,!, a T , r ,!, a T , r ,! , a T , r ,! , a T , r ` 1

k

1

Nr

1

NT

Nr

k

1

1

NT

Nr

(6)

Assume the total number of atoms N r NT !! K , that the Eq. (5) can be translated under-determined system of equations. D yyi (t , f ) A(T , r ) xi (t , f )  i (t , f ) where i (t , f )

into

following (7)

Dni ni (t , f )ei , and xi (t , f ) is the sparse

representations of Dss (t , f )biH (T , r ) corresponding to the overcomplete basis A T , r . Note that the non-zero

elements of the sparse vector xi  t , f are associated with the DOA and range parameters of near-field sources. And Eq. (7) can be denoted in multiple measurement vectors

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WANG Bo, et al. / Near-field time-frequency localization method using sparse representation

form as D yy  t , f [ D yy1  t , f ,! , D yyM  t , f ] A T , r X  t , f  E  t , f

(8)

> x1 (t , f ),! , xM (t , f )@ , > 1 (t , f ),! , M (t , f )@ .

where X  t , f

and E  t , f

It is obvious that the Eq. (8) holds true for every T-F point in T-F domain, and the T-F points along the instantaneous frequency (IF) of kth source are chosen to construct the sparse localization model. As in Refs. [14– 15], we take pseudo Wigner-Ville distribution (PWVD) to describe the localization algorithm, in which the rectangular window length L of PWVD is odd. Leaving out the rising or falling power distributions of kth source and utilizing N  L  1 T-F points along IF of kth source, the sparse representation of Dk associated with kth source is given by Dk [ D yy (t L 2 , f k , L 2 ),..., D yy (tL 2+1 , f k , L 2+1 ),! , A T , r [ X (tL 2 , f k , L 2 ),! ,

D yy (t N  L 2 , f k , N  L 2 )]

X (t N  L 2 , f k , N  L 2 )]  [ E (t L 2 , f k , L 2 ),! , E (t N  L 2 , f k , N  L 2 )]

where Ek

Xk

A T , r X k  Ek

(9)

[ X (t L 2 , f k , L 2 ),! , X (t N  L 2 , f k , N  L 2 )] , and

[ E (tL 2 , f k , L 2 ),! , E (t N  L 2 , f k , N  L 2 )] .

Note that when the kth source is uncorrelated with others, just one row corresponding to the kth source in sparse matrix X k is non-zero. However, if it is coherent with other sources, the rows corresponding to other sources also are non-zeros. Through solving the sparse linear inverse problem with multiple measurement vectors, we can localize the kth source and the sources which are coherent with the kth source. To recover the sparse signal matrix from multiple observations contaminated with additive noise as Eq. (9), suppose the bound on the total energy of Ek is

N L 2 K

¦ ¦  t , f i

j

k, j

2 2

İG 2 , we can solve

j L2 i 1

the follow convex relaxation problem [19] as ½ min R ( X k ) °° s.t. ¾ ° Dk  A T , r X k F İG °¿

(10)

where R (˜) denotes the number of nonzero rows, and ˜ F is the Frobenius norm. The optimization problem can

be solved by convex relaxation approach [19] which has considerable computation burden. On the contrary, the greedy algorithm [16–17] owns lower computation burden

31

compared with the convex relaxation method, thereforeˈit is utilized to solve the sparse representation problem in this paper. At the same time, the priori information about the source number, T-F information of sources and the correlation information can be incorporated into the greedy algorithms to improve the algorithm performance. 3.2 Localization of uncorrelated sources Uncorrelated sources have distinctive T-F characteristics, therefore, only the localization information of kth source is contained in the sparse representation Eq. (9). Consequently, we only select the most correlative atom with Dk by using the greedy algorithm (orthogonal matching pursuit) [16–17] to obtain the localization parameters of kth source. The proposed algorithm is summarized as follow steps. 1) Construct the T-F sparse localization model by using Eq. (9). 2) Initialization: set residual rres Dk , and the selected atom set aatom ‡ , whose index set iindex ‡ , iter 0 , and set iteration number z 1 . 3) Select atoms: Find

 p, q

max

p1,!, NT q1,!, N r

 p, q , aatom ˖, iter

and, iindex  iter







a T Tp , rq rres

1



a Tp , rq , aatom ˖ , iter

denotes ith column of matrix a atom . 4) Residual updating: rres the

 a atom

†

a

H atom

Dk  a atom a atom† Dk , where

H a atom aatom .

1

5) Return to step 2) if iter  z . 6) The localization estimations [(Tˆ1 , rˆ1 ),! , (Tˆn , rˆn )] G (i index ) are obtained from the selected atoms index set i index . Note that because Dk only contains the localization information of kth source for uncorrelated source scenario, the iteration number z 1 . Therefore, based on the Dk for k=1, 2, …, K, we need execute the above procedure repeatedly for K times to obtain the parameters of all sources. In the proposed algorithm, the spatial T-F distribution is introduced to enhance the SNR, moreover, all sources can be localized independently because each source has the discriminative T-F characteristics. Therefore, our method can improve the estimation accuracy. 3.3 Localization of coherent sources In fact, if the coherent sources exist, the

Dk

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The Journal of China Universities of Posts and Telecommunications

corresponds to kc (kc ! 1) coherent sources (containing

(RMSE) RMSE [1], which is defined as

kth source) which share the same T-F signatures with kth source. Let iteration number z kc , we can select kc atoms based on Dk to estimate the parameters of kc

RMSE

coherent sources by using the algorithm as shown in Sect. 3.2. Because of the correlation of atoms corresponding to kc coherent sources, the algorithm may suffer from poor performance. To improve the estimate accuracy, the key point is to reduce the impact of coherent sources on algorithm performance. Aiming at this problem, we propose additional Iterative Selection steps to improve the estimation performance for coherent sources. The algorithm with additional iterative selection steps for coherent sources is summarized as follow. 1) Obtain the initial atom set a atom which contains kc

1 Mr

Mr

¦

xˆr  x

2

2012

,

r 1

where xˆr denotes the estimates of x at the rth Monte Carlo trial. In Figs. 1~4, the RMSEs versus SNR are shown, which are obtained by averaging over M r =100 Monte Carlo trials. From Figs. 1 to 4, we can see that the proposed method outperforms conventional MUSIC.

atoms, by utilizing the greedy algorithm as in Sect. 3.2 based on Dk . 2) Iterative updating: go through the following procedure to update the selected ith atom in turn, i 1, 2! , n . a) calculate the residual rres about ith atom rres

Fig. 1 The RMSE of DOA estimation versus SNR for source 1

i i i Dk  aatom denotes the rest  aatom Dk , where aatom †

of a atom except ith column. b) select and update ith atom  p, q max a T Tp , rq rres , aatom ˖, i p1,!, NT q1,!, N r





1





a Tp , rq .

3) Execute step 2) repeatly until the all selected atoms remained unchanged.

4

Performance analysis

Fig. 2 The RMSE of DOA estimation versus SNR for source 2

In this section, several experiments are done to validate effectiveness of proposed algorithms. In all simulations we consider that the sensor number M 11 of ULA with inter-spacing d O 4 . The snapshots is 256, and the rectangular window of PWVD length L=129. In the first simulation, we consider two uncorrelated linear modulation frequency (LFM) sources with DOA and range parameters  0q, 5O and 10q, 10O , and the initial and final normalized frequency of two sources are  0.1, 0.5 and  0.0, 0.4 , respectively. The range domain is uniformly sampled with grid 0.1O , and DOA domain is sampled with grid 0.1q . We compare the performance of proposed method with 2-D MUSIC method [2] and CRB [1]. The performance is measured by the root mean square error

Fig. 3 The RMSE of range estimation versus SNR for source 1

In the second simulation, two coherent sources, which share the same initial and final normalized frequency  0.1, 0.5 , are located at  40q, 3O and  40q, 10O , respectively. The Fresnel region of array and DOA domain are uniformly sampled with 0.1O in range space and 1°

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WANG Bo, et al. / Near-field time-frequency localization method using sparse representation

in DOA spaceˈrespectivelyˈand let SNR be 10 dB. We compare the proposed method with 2-D MUSIC method. The simulation results of 10 trials are shown in Figs. 5 and 6. Note that Fig. 6 shows the contour map of the 2-D MUSIC spatial spectrum for 10 trials, and similar to Fig. 5, the circles in Fig. 6 indicate the true position of sources. As observed in Fig. 5, the proposed method can resolve the coherent sources successfully and provides good results. However, we can see that the 2-D MUSIC method fails to estimate the coherent sources from Fig. 6.

33

0 dB to 25 dB, the RMSEs of proposed method are shown in Figs. 7 and 8, in which M r =100 Monte-Carlo trials is done for the results. We can see that the proposed method provides good performance even in low SNR.

Fig. 7 The RMSE of DOA estimation versus SNR for two coherent sources

Fig. 4 The RMSE of range estimation versus SNR for source 2

Fig. 8 The RMSE of range estimation versus SNR for two coherent sources

5

Fig. 5 The parameters estimation of two coherent sources using proposed method

Conclusions

Based on the T-F sparse representation, we propose a new near-field localization method, which constructs sparse localization model through the time-frequency transform, and chooses the greedy algorithm to solve the sparse problem. The proposed method can efficiently localize the uncorrelated sources and coherent sources. From the simulations results, it has been shown that improvements in accuracy and robustness to low SNR can be achieved. Acknowledgements This work was supported by the National Natural Science

Fig. 6 The contour map of 2-D MUSIC spatial spectrum for two coherent sources

In the last simulation, we consider the same coherent sources as the second simulation. Varying the SNR from

Foundation of China (60901060).

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The Journal of China Universities of Posts and Telecommunications

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(Editor: WANG Xu-ying)