Sparse recovery method for far-field and near-field sources localization using oblique projection

June 2013, 20(3): 90–96 www.sciencedirect.com/science/journal/10058885 http://jcupt.xsw.bupt.cn

The Journal of China Universities of Posts and Telecommunications

Sparse recovery method for far-field and near-field sources localization using oblique projection

WANG Bo (*), ZHAO Yan-ping, LIU Juan-juan

College of Communication Engineering, Jilin University, Changchun 130025, China

Abstract

In this paper, we propose a novel source localization method to estimate parameters of arbitrary field sources, which may lie in near-field region or far-field region of array aperture. The proposed method primarily constructs two special spatial-temporal covariance matrixes which can avoid the array aperture loss, and then estimates the frequencies of signals to obtain the oblique projection matrixes. By using the oblique projection technique, the covariance matrixes can be transformed into several data matrixes which only contain single source information, respectively. At last, based on the sparse signal recovery method, these data matrixes are utilized to solve the source localization problem. Compared with the existing typical source localization algorithms, the proposed method improves the estimation accuracy, and provides higher angle resolution for closely spaced sources scenario. Simulation results are given to demonstrate the performance of the proposed algorithm.

Keywords array signal processing, near-field, far-field, sparse signal recovery, oblique projection

1 Introduction

In the past three decades, source localization as an important research area in array signal processing has received tremendous attention and can be applied to a variety of application fields, such as radar, sonar, communication, etc. According to the distance between sources and array, source localization problem can be divided into direction-of-arrival (DOA) estimation and near-field source localization. When the sources lie in the far-field region of array, the signals received by the array sensors are planar waves, and the DOAs of the sources need to be estimated. Numerous DOA estimation algorithms based on the planar wave assumption have been developed, for example, the classical multiple signal classification (MUSIC) [1], estimation of signal parameters via rotational invariance techniques (ESPRIT) [2] and their variants. However, when the sources are located at the fresnel region, i.e., the near-field region, the planar wave assumption can not hold at the situation, and the signal Received date: 24-09-2012 Corresponding author: WANG Bo, E-mail: [email protected] DOI: 10.1016/S1005-8885(13)60055-2

wavefront can be characterized by the DOA and range. Under the near-field assumption, various algorithms, such as 2-D MUSIC [3], path-following [4], and high-order statistics (HOS) ESPRIT [5] and so on, have been proposed to solve the near-field localization problem.

In practice, all the sources may not just exist in the one area of far-field and near-field regions, in other words, there may be far-field sources and near-field sources (mixed sources scenario) at the same time. Therefore, we need a source localization method which can be applied to any kind of scenarios. But the aforementioned algorithms only consider the far-field or near-field situation, and may not perform the arbitrary field sources. Recently, based on conventional subspace approach, the HOS-based MUSIC method [6] and the second order statistics (SOS)-based MUSIC method [7] are presented, which are suitable for the situation. On the other hand, by combining HOS with the sparse recovery method, a mixed sources localization method [8] is developed in our recent work.

In this paper, a novel localization method is presented to estimate the parameters of far-field and near-field sources.

Issue 3 WANG Bo, et al. / Sparse recovery method for far-field and near-field sources localization using oblique projection 91

To avoid using HOS, spatial-temporal SOS matrixes which remain array aperture are introduced in the proposed method. Next, we take the signal characteristic into account and utilize the oblique projection technique [9] to achieve the signal separation. Consequently, by using sparse signal representation framework [10], which is an emerging research topic in signal processing society and has been applied to the DOA estimation area [11–13], the parameters of each source are estimated, respectively. Compared with the typical algorithms, the proposed method has higher estimation accuracy and improved resolution.

2 Data model

Assume that M narrowband sources, which include 1M near-field sources and 1M M− far-field sources, impinge a uniform linear array (ULA) composed of 2 +1L sensors with inter-sensor spacing d. Let the center sensor of the array be phase reference point, the output of the thl sensor can be represented as

( ) ( ) ( ) ( ) ( )1 2

1

j j

1 1e e ; m m m

M Ml l ll m m l

m m Mx t s t s t n tγ φ γ+

= = +

= + +∑ ∑

1 0, ,t N= … − (1) where [ ],l L L∈ − , ( )ln t denotes the additive complex

Gaussian white noise of the thl sensor with power 2nσ ,

( )ms t is the thm source signal. The mγ and mφ have the following forms:

22

2π sin

π cos

m m

m mm

dr

dγ θλ

φ θλ

= − =

(2)

where λ is the carrier wavelength, mθ and mr denote the DOA and range of the thm source, respectively. Similar to Ref. [14], we assume that the ( )ms t can be represented

as ( ) je m

m

w tm ss t σ= , in which

msσ and mw denote the

amplitude and radian frequency of the thm signal, respectively. And we further assume that all the signals have different radian frequencies.

Let T denote the transposition operator, Define

( ) ( ) ( ) T, , ,L Lt x t x t−= … x ( ) ( ) ( ) T

1 ,, , Mt s t s t= … s

and ( ) ( ) ( ) T, ,L Lt n t n t−= … n , we rewrite Eq. (1) in

vector notation as ( ) ( ) ( ) ( ),t r t tθ= +x A s n (3)

where ( ), rθA is the array manifold matrix, and

( ) ( ) ( ) ( ) ( )2j1 1, , , , , , , e ,m mL L

M M m mr r r r γ φθ θ θ θ

− += … = A a a a

( )2 Tj, e .m mL Lγ φ+ …

Note that the range mr should be ∞ (i.e., 0mφ = ) in the steering vector ( ),m mrθa when the thm source locates at

the far-field region. In order to avoid the phase ambiguity, we let / 4d λ≤ in this paper.

3 The proposed method

3.1 Construction of spatial-temporal matrixes and frequencies estimation

Based on the model assumption, we obtain

( ) ( ) ( ) ( )j j2* 2 21, ,

1e δm m

m

Mw l

l l l s n l lm

r E x t x t eτ γτ τ σ σ τ− −=

= − = + ∑

(4) where * denotes the complex conjugate, ( ),δl l τ− representing the Dirac delta function is equal to 0 if 0l ≠ or 0τ ≠ . By collecting the ( )1,lr τ for 1, , 2 1Lτ = … + ,

the ( )2 1 1L + × covariance vector 1,lr can be expressed

as

( ) ( ) T

1, 1, 1,1 , , 2 1l l lr r L = … + r (5)

Note that the noise term in covariance vector 1,lr is eliminated for the time lag τ is not equal to 0. Because l belongs to the interval [ ], ,L L− the ( ) ( )2 1 2 1L L+ × + spatial-temporal covariance matrix 1R is obtained as

( ) ( )T1 1, 1,, ,L L sw Λ γ− = … = R r r A A (6)

where ( ) ( ) ( ) ( )1 2, , , ,Mw w w w= … A a a a ( ) je ,mwmw = a

( ) ( ) ( ) ( ) ( ) ( )Tj 2 1

1 2, e , , , , ,mw LM mγ γ γ γ γ+ … = … = A a a a a

Tj2 j2e , ,e ,m mL Lγ γ− … and sΛ is the diagonal matrix

composed of the elements { }1 2

2 2 2, , ,Ms s sσ σ σ… . In a similar

manner as Eq. (4), we obtain

( ) ( ) ( ) ( ) ( )2jj* 2 2

2, 0 n ,01

e e δm mm

m

M l lwl l s l

mr E x t x t γ φττ τ σ σ τ

+

=

= − = + ∑ (7)

the ( )2,lr τ for 1, , 2 1Lτ = … + consists of the

covariance vector ( ) ( ) T

2, 2, 2,1 , , 2 1l l lr r L = … + r , and the

( ) ( )2 1 2 1L L+ × + spatial-temporal covariance matrix 2R

is given by

92 The Journal of China Universities of Posts and Telecommunications 2013

( ) ( )T2 2, 2,, , ,L L sw rΛ θ− = … = R r r A A (8)

Observe the two spatial-temporal covariance matrixes 1R and 2R , it is easily shown that the overcomplete

representations of the spatial-temporal matrixes can be obtained by dividing the frequency area into discrete set. Therefore, based on the spatial-temporal matrix 1R or

2R , we can estimate the frequencies of signals by casting the problem as a sparse recovery problem. Here, we use the matrix 1R to estimate the signals frequencies. Let

( )TsΛ γ=B A , then ( )1 .w=R A B

We uniformly divide the whole frequency area into 1N

grids and form the frequency set 11 2, , , Nw w w = … w ,

then the sparse representation of 1R is

( )1 s=R A w B ( 9 )

where ( ) ( ) ( ) ( )11 2, , , Nw w w = … A w a a a is the overco-

mplete basis, and sB is the sparse representation of B in terms of the overcomplete basis ( )A w . In fact, because

we can only obtain the sample covariance matrix 1R̂ of the 1R based on the limited snapshots, the Eq. (9) should

be substituted for ( )1ˆ

s≈R A w B . Define that 2

m

lfb

denotes the 2l norm of the thm row of the matrix sB , and the 1 1N × vector 2l

fb consists of the 2

m

lfb for

11 m N≤ ≤ . Consequently, the frequencies of signals can be obtained by solving the following objective function

( ) ( ) 22

1 1Fˆmin 1 l

s fβ β− − +R A w B b (10)

which is a convex optimization problem and can be handled by second order cone programming (SOCP) solved by the software package SeDuMi [15]. Where

1·

and F

· denote the 1l norm and the Frobenius norm, respectively. β is the regularized parameter and limited in the interval ( )0,1 .

To decrease computation burden and enhance the performance for solving the convex optimization problem, the singular value decomposition (SVD) of 1R̂ and the weights of 1l norm term are introduced in Eq. (10). Let

SV1R̂ and SV

sB represent the SVD forms of the 1R̂ and

sB , respectively (the more details about SVD may be referred to Ref. [11]), the ( ) ( )2 1 2 1L L M+ × + − matrix

1nU contains 2 1L M+ − singular vectors in the noise

subspace of the 1R̂ . The 1 1N × weights vector fα

(like Ref. [16]) associated with the 1l norm term in Eq. (10) is given by

( ) ( ) ( )HH 1 1diagf n n =

A w U U A wα (11)

where diag( )⋅ represents the diagonal elements, and H denotes the conjugate transposition. Therefore, we have the following objective function (so called weighted 1l minimization problem) corresponding to the Eq. (10)

( ) ( ) 22SV SV T

1 Fˆˆmin 1 l

s f fβ β− − +R A w B bα (12)

where the 1 1N × vector 2ˆlfb corresponding to SV

sB has

the similar definition as 2 .lfb Consequently, the

frequencies { }1 2, , , Mw w w… of signals can be obtained

by solving Eq. (12). Remark A The computation complexity for solving

the cost Eq. (10) by using an interior point method is 1

3{[(2 1 ]) }O L N+ × [11]. However, the objection Eq. (12) requires 3

1[( ) ]O M N× . Because the source number M is smaller than the sensor number 2 1L + in general, the SVD of 1R̂ reduces the computation burden of the algorithm.

Remark B The key idea of the weighted 1l minimization [16–17] is that applying a smaller weight to the elements of the vector like 2ˆl

fb that should largely

improve the recovery performance of algorithm. Because the approximate orthogonality between 1

nU and ( )wA , the thm entry of the weight vector fα corresponding to

the frequency nw may be close to zero if ( )mw =a

( )nwa for 1 n M≤ ≤ . As a result, the weights vector

introduced in Eq. (12) enhances the performance of frequency estimation.

3.2 Signal separation

Once the frequencies of signals are obtained, the oblique projection matrixes can be constructed to realize the signal separation of the spatial-temporal matrixes 1R and 2R . For simplicity, we use ma to denote ( )mwa , and define

that the ( ) ( )2 1 1L M+ × − matrix mA is

[ ]1 2 1 1, , , , , ,m m m M− += … …A a a a a a (13) Then, we have the oblique projection matrix

m ma AE

given by Ref. [9]

( ) 1H Hm m m mm m m m

−⊥ ⊥=a A A AE a a P a a P (14)


where m

⊥AP represents the orthogonal projection matrix on

the null space of mA . The oblique projection matrix

m ma AE has the following properties

0m m

m m

m m

m

= =

a A

a A

E a aE A

(15)

Therefore, we can get the ( ) ( )2 1 2 1L L+ × + single component spatial-temporal matrix 1mR which only contains the thm signal information and is expressed as

( )2 T1 1m m mm m sσ γ= =a AR E R a a (16)

Similarly, the ( ) ( )2 1 2 1L L+ × + single component matrix 2mR corresponding to the 2R is described as

( )2 T2 2 ,

m m mm m s m mrσ θ= =a AR E R a a (17)

Because 1mR and 2mR for 1, 2, ,m M= … only retain one signal information, respectively, we can independently estimate the parameters of each source. The details are shown in the next subsection.

Note that in order to realize the signal separation of the spatial-temporal matrixes 1R and 2R successfully, it requires that the oblique projection matrix

m ma AE , which

depends on the frequency mw of the signal, should be obtained correctly. Therefore, the estimation accuracy of the frequencies { }1 2, , , Mw w w… of the proposed

algorithm is the key point to ensure the signal separation successfully. As mentioned above, the weighted 1l minimization is applied to estimate the frequencies { }1 2, , , Mw w w… , which can provide the high accuracy of

frequency estimation. Consequently, the proposed algorithm can deal with the signal separation effectively.

3.3 DOA and range estimation

By transposing the matrixes 1mR and 2mR , we obtain the ( ) ( )2 1 2 1L L+ × + single component data matrixes

3mR and 4mR for 1, 2, ,m M= … , which can be formulated as follow.

( )T 2 T3 1 mm m s mγ σ= =R R a a (18)

( )T 2 T4 2 ,

mm m m m s mrθ σ= =R R a a (19)

Let ( )1 2 1L× + vector 2 Tmm s mσ=b a , the 3mR and

4mR can be rewritten as

( )3m mγ=R a b (20)

( )4 ,m m m mrθ=R a b (21)

Observe the Eqs. (20) and (21), it is obvious that 3mR and 4mR associated with the DOA or range of the thm source have the similar structure as ( )1 w=R A B which

is used for estimating the frequencies of signals via sparse recovery method in the Sect. 3.1. Therefore, the DOA and range estimation problem can also be cast as the SOCP problem by utilizing 3mR and 4mR . To estimate DOA of source, we divide the DOA domain with the same interval into a grid set { }21= , , Nθ θ…θ . Then, the overcomplete

representation of ( )γa can be expressed as

( ) ( ) ( )21 , , Nθ θ θ = … A a a (22)

( ) ( ) ( )T4π 4πj sin j sin

e , ,em md dL L

m

θ θλ λθ

− = …

a (23)

we have a sparse model ( )3 ,m smθ=R A B where the

( )2 2 1N L× + matrix smB is the sparse representation of

the ( )1 2 1L× + vector mb according to the

overcomplete matrix ( )θA . Due to the fact that 3mR is

transformed from 1mR , we define 3ˆ

mR corresponding to the

available 1ˆ

mR of 1mR for the limited snapshots. In a similar manner as Eq. (12), the weighted 1l minimization for DOA estimation can be given by

( ) ( ) 22SV SV T

3 3ˆmin 1 l

m sm m mFβ θ βα− − +R A B b (24)

where SV1ˆ

mR and SVsmB corresponds to the SVD forms of

3ˆ

mR and smB , separately. 2lmb is obtained through the 2l

norm operating on the rows of the matrix smB . The

2 1N × weights vector 3mα is

( ) ( ) ( )HH 3 33 diag m m

m n nα θ θ = A U U A (25)

where 3mnU represents the noise subspace matrix

corresponding to the 2 1L M+ − smallest singular values of 3

ˆmR . Consequently, the DOAs { }1 2, , , Mθ θ θ… are

achieved by solving Eq. (24) for 1, 2, ,m M= … . After the DOAs { }1 2, , , Mθ θ θ… are given, we can acquire the

overcomplete representation ( ),mθA r of ( ),m mrθa by

dividing the range domain into grids set r to estimate range of source. For near-field scenario, the range domain

is so called fresnel region ( )1/ 23 20.62 / , 2 /D Dλ λ

[18]

with D denoting the array aperture. However, for far-field source, the range mr is ∞ . Therefore, the range grids


set r should include the fresnel region grids to represent the potential range parameters of near-field sources, and few far-field grids which are bigger than 22 /D λ just for distinguishing far-field and near-field sources. Assume that the range grids set is { }31= , , ,Nr r…r then

( ) ( ) ( )31, , , , , ,m m m Nr rθ θ θ = … A r a a and the sparse

representation of 4mR is given by

( )4 ,m m smθ=R A r B (26)

where smB is the sparse representation of mb according to the overcomplete matrix ( ),mθA r . We note that the

Eq. (26) may be approximately equal for far-field source scenario, because the range grids set can only approximately represent the range of far-field source which is ∞ . Obviously, the matrixes 3mR and 4mR have the similar sparse model. Therefore, using the similar processing procedure as DOA estimation, the range estimation problem can be transformed into the weighted

1l minimization like Eq. (24) to obtain range parameter. The details about converting the Eq. (26) into the convex optimization cost function are omitted here because the range estimation and DOA estimation almost share the same transforming process. Note that when the estimated range parameter is bigger than 22 /D λ , the corresponding source belongs to the far-field source and the estimated range should be replaced by ∞ .

Remark C To estimate near-field source parameters, the SOS-based MUSIC algorithm constructs a special covariance matrix suffering aperture loss, which may lead to performance attenuation. However, the proposed method exploits spatial and temporal information of array output to build two spatial-temporal covariance matrixes which can avoid the aperture loss. Furthermore, our method uses oblique projection technique to realize signal separation which makes that the parameters of each source are individually estimated, namely, the DOA and range estimation of each source has no effect on others even if the closely spaced sources scenario is met. Therefore, the proposed method has higher ability for resolving closely spaced sources and provides better estimation accuracy than the SOS-based MUSIC algorithm and the HOS-based MUSIC algorithm.

4 Simulations

This section provides several simulations to demonstrate

the performance of the proposed method, compared with the SOS-based MUSIC method and the HOS-based MUSIC method.

In the following experiments, a ULA with / 4d λ= is considered, the snapshots number is 400, and the regularized parameter β is 0.7. For simplicity, let the signal amplitude 1

msσ = for 1, 2, ,m M= … . The whole

frequency domain from π− to π is divided into the frequency grids set w with 0.01π resolution, and the DOA domain from 90− ° to 90° is sampled to form the coarse DOA grids set θ with 1° interval. Note that we use the coarse DOA grids set to obtain the initial DOA estimation and construct the local refined grids set around the estimated angles with 0.1° resolution to achieve the final results (i.e., the multiresolution grid in Ref. [11]). For range grids set ,r the fresnel region

( )1/ 23 20.62 / , 2 /D Dλ λ

are divided with 0.05λ

resolution, and the range grids section for far-field scenario is sampled from 24 /D λ to 210 /D λ with

22 /D λ interval to approximately represent far-field sources. The following experiment results are obtained through 500 times Monte-Carlo simulations, and SOS-based method and HOS-based method in the figures of the following experiments represent SOS-based MUSIC method and HOS-based MUSIC method, respectively.

In the first experiment, we use a ULA consisting of 7 sensors, and two signals with radian frequency 0.1π and 0.3π are emitted by two sources, respectively. we evaluate the performance of the proposed method for the mixed sources scenario, which includes one far-field source and one near-field source. The parameters of the two sources are set as { }1 110.5 , rθ = ° = ∞ and

{ }2 225 , 2rθ λ= ° = , respectively. The RMSEs for the two

sources are illustrated in Figs. 1 and 2.

Fig. 1 RMSE of DOA vs. SNR


Fig. 2 RMSE of range vs. SNR

The simulation results show that the proposed method has better estimation accuracy of DOA and range than the conventional algorithms for mixed sources scenario when SNR is bigger than 0 dB.

In the second experiment, to test the robustness of our method, we examine a special example that one near-field source and one far-field source share the same DOA, and two near-field sources belong to the closely spaced sources. Here, four sources with radian frequencies 0.3π− , 0.1π , 0.3π and 0.4π which are located at { }1 120 , rθ = − ° = ∞ ,

{ }2 210 , ,rθ = ° = ∞ { }3 310 , 3 ,rθ λ= ° = and { 4 13 ,θ = °

}4 2r λ= respectively, are considered. The SNR and

number of sensors are set as 30 dB and 9, separately. Fig. 3 shows that the SOS-based MUSIC method can estimate the DOAs of the far-field sources correctly, however, the two near-field sources form a peak in the spatial spectrum for the two sources are too close to resolve. Fig. 4 illustrates that the proposed method can resolve the DOAs of all sources successfully even if the second source (far-field source) and the third source (near-field source) have the same DOA, and the last two sources are closely spaced sources.

Fig. 3 Spatial spectrum of DOA for the SOS-based MUSIC method

Fig. 4 Spatial spectrum of DOA for the proposed method

In the last experiment, to test the performance of the proposed algorithm in the broad angle domain, we consider the mixed sources scenario including one far-field source and one near-field source that the DOA 1θ of the far-field sources with radian frequency 0.1π varies from

40− ° to 50° with 15° gap, and the DOA 2θ of the near-field sources with radian frequency 0.3π is

12 10θ θ= + ° , whereas the range 2 2r λ= . The SNR and number of sensors are set as 5 dB and 7, respectively. The performance of the proposed algorithm is also compared with the conventional subspace approaches and cramer-rao bound (CRB) in Ref. [7]. Figs. 5 and 6 provide the RMSEs of DOA and range estimation versus the DOA of the far-field source. We see that the performance of the proposed algorithm outperforms the conventional subspace methods. Moreover, it is worth noting that the RMSEs of both DOA and range estimation are lower than the CRB, because the signal separation technique of the proposed method transforms the multiple sources localization into the parameters estimation of the single source.

Fig. 5 RMSE of DOA vs. DOA of source 1


Fig. 6 RMSE of range vs. DOA of source 1

5 Conclusions

In this paper, a source localization method based on sparse signal representation which can be suitable for arbitrary field source is presented. By constructing spatial-temporal covariance matrixes, the aperture loss is avoided. Because the oblique projection technique is utilized to realize signal separation, the proposed method can estimate the parameters of each source individually. Therefore, the sparse localization method can provide higher estimation accuracy and better resolution than the typical algorithms. In addition, the proposed method can be applied to the special scenario that two sources located at the different fields of array aperture share the same DOA.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (60901060).

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(Editor: ZHANG Ying)

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Sparse recovery method for far-field and near-field sources localization using oblique projection