Low Complexity Method for Wideband DOA Estimation Based on Sparse Representation Using

Rotational Signal Subspace

Yonghong Zhao, Linrang Zhang, Juan Zhang, Yumei Guo, Yabin Gu National Laboratory of Radar Signal Processing

Department of Electronic Engineering, Xidian University Xi'an, 710071, China

Email: {Zhaoyh_2014, yabingu} @163.com, {lrzhang, jzhang}@xidian.edu.cn, {875042618}@qq.com

Abstract—A low complexity method for wideband direction finding based on sparse representation is proposed, which combines the rotational signal subspace (RSS) with weighted subspace fitting (WSF) to improve the performance of DOA estimation. Exploiting the result of the focusing operation, the covariance matrix at the focusing frequency can be obtained and used as the data for sparse recovery to get wideband DOA estimates. The WSF is employed to reduce the sensitivity to the noise and the regularization parameter is given by the asymptotic distribution of the WSF criterion. Simulations are provided to prove the efficiency and performance of the proposed method.

Keywords—direction-of-arrival (DOA) estimation; sparse representation; sensor array processing.

I. INTRODUCTION Recent years, sparse signal representation (SSR) has drawn

much attention from researchers, which exchange the problem of parameter estimation of source localization to the problem of sparse spectrum estimation [1]-[5]. In [6], the authors introduce WSF into narrow sparse representation and obtain asymptotically efficient DOA estimates. The algorithm named

1-SVD [7] successfully applies SSR model in DOA estimation and gives the process in wideband cases like incoherent signal subspace methods (ISSM), which is followed by frequency decomposition and narrowband process. There has also been some emerging investigation of coherently exploiting the frequency diversity in the context of wideband DOA estimation. The idea in [8], [9] is to combine the whole frequency information of the received signal instead of processing each sub-band individually. The algorithm named W-SpSF [10], applies the vectorisation operator on the covariance matrix of each band to build a single measurement vector, in which the overcomplete basis is a diagonal matrix with the corresponding overcomplete basis of each band as diagonal elements.

Joint sparsity endows the algorithms with the superiority of higher resolution, even under the condition when the signals are coherent. However, the SSR-based methods for wideband DOA estimation have a higher computational complexity. In this paper, we introduce RSS [11] into the SSR model and propose an efficient dimension reduction methodology to greatly reduce the complexity of solving the problem of SSR-based wideband DOA estimation.

II. PROBLEM FORMULATION

A. Wideband signal model Consider K wideband far-field signals impinging on a

sensor array constituted by M omni-directional elements from directions of 1= , K θ ， , which keep constant during the observation period. The array output is filtered by a filter bank or the DFT into J narrowband ones. The wideband array output can be modeled as [12]:

= , + , =1, , , 1, ,l j j l j l jf f f f l L j Jθ y A s n (1)

where Ml jf y is the DFT coefficients of sub-band jf ,

coming from the l -th segment received data. The source signal vector K

l jf s is assumed to follow Gaussian distribution, satisfying

1 2

H 2 2 21, 2, , 1 2, , , -l j l j j j K jE f f diag l l

s s

and independent of each frequency, where E , H , diag ,

, and 2,k j denote the expectation operator, conjugation

transpose, the diagonal operator, the Dirac function, and the k -th signal power of sub-band jf ,respectively. l jfn is the complex Gaussian white noise vector and uncorrelated with sources.

1 2, = , , , , , , M Kj j j j Kf f f f θ A a a a

is the steering matrix at the frequency bin jf with the 1M steering vector

11 1 12 sin 2 sin 2 sin, = , , ,j k j m k j M kj f d c j f d c j f d cj kf e e e

a

where 1md is the distance between the m -th sensor and the first sensor for =1,2,m M , c is the velocity of signals.

Under the above hypotheses, the j -th narrowband output covariance matrix is

978-1-4673-8823-8/17/$31.00 ©2017 IEEE 0460

H 2,, ,j j Sj j n j Mf f θ θR A R A I (2)

where H

Sj l j l jE f f R s s denotes the signal covariance

matrix. 2,n j is the noise power at the frequency jf with an

assumption that 2 2 2 2,1 ,2 ,, ,n n n J n . MI is M M

identity matrix.

B. Rotational Signal Subspace For wideband signals, the conventional way combines the

direction of arrival information embedded in each band using ISSM or coherent signal subspace methods (CSSM). The CSSM can cope with coherent wavefronts and attain a better performance than ISSM. RSS is a class of CSSM, which focusing matrix jf M M T satisfying the following constrained minimization problem:

0

H

ˆ ˆmin , - , , =

subj

1,

ect

,

to:

jj j

f F

j j M

f f f j J

f f

T

A T A

T T I (3)

where F

is the Frobenius matrix norm, 0f is the focusing

frequency, and ̂ is the preselected angles or preliminary estimation angles.

One solution to (3) is given by

H=j j jf f fT V U (4)

where jfU and jfV are the left and right singular

vectors of H

0ˆ ˆ, ,jf f A A , respectively. Using focusing

operation to the steering matrix ,jf θA , we can get

0 , ,j jf f fθ θA T A (5)

Then the covariance matrix at the focusing frequency 0f can be casted as following:

00

1

H

1

H H 2

1

1

1

1 , ,

J

jj

J

l j l jj

J

j j Sj j j n Mj

J

E f fJ

f f f fJ

θ θ

R R

x x

T A R A T I

(6)

where 0jR and l j j l jf f fx T y denote the covariance

matrix and receive data after focusing operation, respectively. In practice, 0R is estimated from the L available snapshots, that is

H

01 1

1ˆJ L

l j l jj l

f fJ

R x x . (7)

III. WIDEBAND SPARSE REPRESENTATION WSF ALGORITHM

In this section, we will propose a sparse representation model based on the WSF criterion for wideband direction finding, which has an efficient dimension reduction to greatly reduce the complexity of solving the problem of SSR-based wideband DOA estimation, owing the RSS method.

First, performing the eigendecomposition on the covariance matrix 0R , we have

H H 2 H0 0 0 0 0 0

1

M

i i i s s s n n ni

e e

R E E E E (8)

where ie and i denote the eigenvector and eigenvalue, respectively. The diagonal matrix 0s contains the 'K largest eigenvalues. The corresponding eigenvectors construct the signal subspace 0sE , and the others construct its orthogonal complement, noise subspace, 0nE , where 'K is the number of uncorrelated signals.

According to [13], the weighted subspace fitting can be expressed as

21/2

0 0, arg min s F θθ

,BB E W A B (9)

where B is a 'K K matrix of full rank, 0A is the steering matrix at the focusing frequency 0f , and arg min refers to the minimizing argument of the preceding expression. W is a positive define weighting matrix and the optimal value is given by [13], i.e. 2 1

0ˆ

s W , where 0

ˆs is the estimation of 0s ,

20

ˆs n M I .

Combining the WSF with sparse representation, we can get a new sparse representation model for wideband DOA estimation:

2,1

1/20 0

min

. . s Fs t

B

E W A B (10)

where 0A is the overcomplete basis, namely,

0 0 0 1 0, , , , , Nf f f Θ A A a a

where 1= , , N Θ is the set of directions getting from sampling the potential space of the incident signals uniformly or non-uniformly. B is a 'N K matrix having the common

sparsity among different columns and 2

2,1 21

Nn

nB b , where

n is the n -th row of the matrix. Then we analyze how to choose a proper regularization

parameter via the distribution property. Let

21/ 2

0 0s FV E W A B (11)

according to [13] and with the assumption of sparse recovery exactly, we can obtain

2

2ˆ~ 2 '

2n V K M KL

(12)

978-1-4673-8823-8/17/$31.00 ©2017 IEEE 0461

where 2 2 'K M K denotes a 2 distribution with

2 'K M K degrees of freedom. Based on the analysis above, the regularization parameter can be determined by

2

2ˆ, 2 '

2n p P K M KL

(13)

where P , 2 , and p represent the probability of event,

cumulative distribution function of 2 , and probability, respectively.

To solve the optimization problem in (10), we manipulate this optimization into a second order cone programming (SOCP) form:

1

1/20 0

2

2

min

. .

where

N

nn

s F

nn

b

s t q b

q

E W A B

b

(14)

then a package for optimization, called CVX is used to solve our SOCP problem. The vector 1 2, , , Nq q q q constitutes the angle spectrum, and we can get the DOA of wideband signals from its peaks location.

An important part of our proposed algorithm is that the choice of the preliminary DOA estimation, which determines the focusing matrix. Unlike [11], we address the selection of the preliminary DOA directly via the static angles which isn’t related to the target angles. Then the focusing matrix can be obtained offline. In order to achieve better precision, we explore the idea of adaptively refining the grid like [7]. A rough grid of potential source location can be created firstly, such as a uniform grid with1 . Considering the computational complexity and precision, we exploit a non-uniform grid refinement which has dense grids around the peaks obtained by using the uniform grid and coarse grids in the other regions, i.e.,

1 : : : : : :c k k d d k d k c N

where c , d , and denote the coarse grid spacing, dense grid spacing, and the interval around the k -th peak of the spectrum.

In addition, we provide the computational complexity analysis in terms of sparse recovery. Using the focusing operation, we build a new overcomplete basis for wideband direction finding based on sparse representation. Referring to the conclusion in [7], we can derive that the cost for the

proposed wideband sparse frame is 3'K N . In contrast to

[10], [8], the computational complexity are significantly large,

which are 3JN and 3'K JN , respectively.

IV. EXPERIMENT In this section, the performance of our proposed method is

investigated. We consider a uniform array with 16M sensors, which the interspacing is half of the wavelength

corresponding to the lowest frequency. Suppose the far-field wideband sources have the same center frequency, 100Hzcf , and the same bandwidth, 40HzB .

-90 -60 -30 0 30 60 90-200

-150

-100

-50

0

DOA(Deg)

Nom

aliz

ed S

patia

l Spe

ctru

m (d

B)

W-SpSFRSSProposed

(a)

-10 -5 0 5 10 15 20-30

-25

-20

-15

-10

-5

0

DOA(Deg)

Nom

aliz

ed S

patia

l Spe

ctru

m (d

B)

W-SpSFRSSProposed

(b)

Fig.1. Spatial spectra of RSS, W-SpSF and proposed methods for three uncorrelated sources: (a) global spatial spectra; (b) local spatial spectra.

-90 -60 -30 0 30 60 90-200

-150

-100

-50

0

DOA(Deg)

Nom

aliz

ed S

patia

l Spe

ctru

m (d

B)

W-SpSFRSSProposed

(a)

-10 -5 0 5 10 15 20-30

-25

-20

-15

-10

-5

0

DOA(Deg)

Nom

aliz

ed S

patia

l Spe

ctru

m (d

B)

W-SpSFRSSProposed

(b)

Fig.2. Spatial spectra of RSS, W-SpSF and proposed methods for three sources which the first two sources are correlated and the third one is uncorrelated to them: (a) global spatial spectra; (b) local spatial spectra.

978-1-4673-8823-8/17/$31.00 ©2017 IEEE 0462

In the experiment, we show the resolution performance under the scenario where SNR 0 dB , 200L , 33J , and

3K signals from directions 0 ,3 ,14 . In Fig. 1, we

compare the spectra obtained using the proposed method with those of RSS and W-SpSF method where the three sources are uncorrelated. Fig. 2 gives the corresponding results under the scenario where the sources at 0 and 3 are correlated with a correlation coefficient of 0.98 and the source at 14 is uncorrelated to the first two sources. The sign “o” indicates actual location of the DOAs in the figures. As can be seen from Figs.1 and 2, the proposed method and W-SpSF can succeed in resolving the first two close-up target sources whether they are correlated or not, whereas RSS method merges the two peaks, and the proposed method achieves more accurate results and sharper peak than W-SpSF. A spurious peak shown in Fig. 2 due to noise appears in the plot and can be removed by increasing the regularization parameter given by (13) appropriately.

V. CONCLUSION In this paper, we introduce the rotational signal subspace

into the SSR model and propose an efficient dimension reduction methodology to greatly reduce the complexity of solving the problem of SSR-based wideband DOA estimation. The proposed method can be cast as the sparse recovery problem of only the covariance matrix at the focusing frequency. Moreover, in order to reduce the sensitivity to noise, we incorporate WSF into the proposed method and build the corresponding sparse model. The computational complexity analysis gives that the proposed method has lower computational burden and simulations show that the proposed method has better performance than the compared methods.

REFERENCES [1] Z. Tang, G. Blacquiere and G. Leus, “Aliasing-free wideband

beamforming using sparse signal representation,” IEEE Transactions on Signal Processing, vol. 59, no. 7, pp. 3464-3469, July 2011.

[2] J. Zhang, N. Hu, M. Bao, X. Li and W. He, “Wideband DOA estimation based on block FOCUSS with limited samples,” Global Conference on Signal and Information Processing (GlobalSIP), 2013 IEEE, Austin, TX, 2013, pp. 634-637.

[3] C. Liu, Y. V. Zakharov and T. Chen, “Broadband underwater localization of multiple sources using basis pursuit de-noising,” IEEE Transactions on Signal Processing, vol. 60, no. 4, pp. 1708-1717, Aprial 2012.

[4] M. M. Hyder and K. Mahata, “Direction-of-arrival estimation using a mixed 2,0 -norm approximation,” IEEE Transactions on Signal Processing, vol. 58, no. 9, pp. 4646-4655, September,2010.

[5] Z. Liu, Z. Huang and Y. Zhou, “An efficient maximum likelihood method for direction-of-arrival estimation via sparsebayesian learning,” IEEE Transactions on Wireless Communication, vol. 11, no. 10, pp. 1-11, October,2012.

[6] N. Hu, Z. Ye, D. Xu and S. Cao, “A sparse recovery algorithm for DOA estimation using weighted subspace fitting,” SignalProcessing, vol. 92, no. 10, pp. 2566-2570, April 2012.

[7] D. Malioutov, M. Cetin and A. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 3010-3022, August 2005.

[8] G. Zhao, Z. Liu, J. Lin, G. Shi and F. Shen, “Wideband DOA estimation based on sparse representation in 2-D frequency domain,” IEEE Sensors Journal, vol. 15, no. 1, pp. 227-233, Jan. 2015.

[9] Z. Liu, X. Wang, G. Zhao, and Z. Gao, “Wideband DOA estimation based on sparse representation-an extension of 1-SVD in widebang cases,” in IEEE International Conference on Signal Processing, Communication and Computing (ICSPCC), Kunming, 2013, pp. 1-4.

[10] Z. Q. He, Z. P. Shi, L. Huang and H. C. So, “Underdetermined DOA estimation for wideband signals using robust sparse covariance fitting,” IEEE Signal Processing Letters, vol. 22, no. 4, pp. 435-439, April 2015.

[11] H. Hung and M. Kaveh, “Focussing matrices for coherent signal-subspace processing,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 36, no. 8, pp. 1272-1281, Aug 1988.

[12] M. A. Doron and A. J. Weiss, “on focusing matrices for wideband array processing,” IEEE Transactions on Signal Processing, vol. 40, no. 6, pp. 1295-1302, June 1992.

[13] M. Viberg, B. Ottersten and T. Kailath, “Detection and estimation in sensor arrays using weighted subspace fitting,” IEEE Transactions on Signal Processing, vol. 39, no. 11, pp. 2436-2449, Nov 1991.

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