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NEAR FIELD CYCLOSTATIONARY SOURCE LOCALIZATION BY SECOND ORDER CYCLIC CUMMULANT

Xuezhi Yan, Shuxun Wang, Zhongsheng Ma and Yukuan Ma College of Communication Engineering Jilin University Changchun, China

Abstract: Most exiting array signal processing techniques for direction finding rely heavily on the far-field assumption. When the sources are located close to the array, these techniques may no longer perform satisfactorily. Most modulated communication signals exhibit cyclostationarity. Many cyclostationarity-based methods have been proposed to improve signal detection capability. In this paper a new algorithm for near-field sources localization is presented based on second order cyclic cummulant. Under the cases of existing interference and different distributed noises, the simulation results indicate that the proposed method can remove cyclostationary interference with different cyclic frequency from the desired signals. Furthermore, it can effectively suppress additive stationary noises with any distribution. Keywords: near-field source, cyclostationarity, noise

Introduction: An important problem in a wide variety of application such as radar, sonar, speech, communication, etc. is locating in space the sources of signals received by an array of sensors. Most attention has been restricted to the model of far-field. In 2-D far-field source localization, a source's location is characterized only by its bearing angle because the wavefront from a far-field source can be assumed to be planar for a uniform linear array of sensor (ULA). However, when sources are located close to the array the assumption is no longer valid. The wavefront in such cases is spherical. In 2-D near-field source localization, a source's location is characterized by its bearing angle and range. The increase in number of parameter caused the algorithm based on the far-field assumptation can no longer be directly applied so substitution must be found. Some work addressing the near-field problem in 2-D has been performed. Spatial Wigner-Ville[1] algorithm proposed by A.L. Swindlehurst et al. can locate the near-field sources but it requires large array if high resolution is needed. Huang et al[2] proposed the 2-D MUSIC algorithm which requires 2-D grid search on the bearing angle and range to identify the peak of MUSIC spectrum. This algorithm is easily to simulate but it is computed inefficiently. Weiss et al. [3] estimated source locations using a polynomial rooting method thereby reducing the computational burden to some extent, however, the algorithm still requires bulk of computations. Starer et al. [4] and Lee et al. [5] proposed pass-following algorithms to reduce the computational burden with two 1-D searches. In wireless communications, since most signals are non-Gaussian and cyclostationary due to modulation [6], Lee et al. [7] proposed a GIBE method to estimate the near-field cyclostationary signals, but it can only estimate the bearing and can not estimate the range parameter. In this paper we present a MUSIC algorithm based on second order cyclic cummulant which can not only remove cyclostationary interference with different cyclic frequency from the desired signals but also can suppress additive stationary noises with any distribution. Furthermore, it can effectively estimate the bearing and range of near-field multiple sources.

Results: 1. 2-D localization signal model and assumption 2-D near field source localization geometry model is shown in figure 1

ith source

i

θ

i

r

r

mi

-1 1

Fig 1 2-D near-field source localization geometry model Consider I near-field, narrowband, independent sources observed by a ULA of

sensors with interelement spacing . Let the array center be the phase reference

point. The signal received by the mth sensor is expressed as

0 m -p p

1 + 2 = p M d

p m p t n e t s t x m j

) (

I

⁼
∑
),

) (

m

-

τ ω (1) where denotes the ith source signal, is the additive noise,

mi i

+

(

≤ ≤ -

i

=

1

i

) (t s_i

) (t n_m

ω is the carrier angle

i

τ is delay association with the ith source signal

propagation time between sensor '0' and sensor . By simple geometry, m mi

τ is

frequency of the ith source signal, and mi

τ )

mi i mi mi r r

( 2 - = - = - = (2)

π λ

r

r

r

r

c

f

λω

i

mi

i

i

i

r i

where is the distance between the ith source and the mth sensor, r is the distance

between the ith source and the 0th sensor, c is the signal propagation speed and is the

carrier frequency of the ith source signal

mi

f

i

r

mi r md r

= -

d m r r

+

-

2 2

i

1 ( 2

-

) 1 sin 2

θ (3)so

i

i

i

i

π τ (4)

mi r md r

=

2 2

d m r θ λω

1 ( 2

i

+

-

) 1 sin 2

i

-

i

i

i

2

r λ is its wavelength. Using binomial

expansion and the so-called Freshnel approximation, we can write:

2

θ , are the bearing and range of the ith source and

i

i

π τ ω +

=

-

-

mi i ϕ

d m r

2 2

≈ (5)

2 m m

i

2 (

sin

θ

d m

md r

i

φ θ

i

)

2 2

sin

2

i

i

i

λ

2

r

2

r

2

i

i

where the parameter i

φ and

ϕ are nonlinear function of azimuth i

i

i

θ and range of the ith

r

source:

) sin( 2 i i d θ

λ π φ - = , )

(

²^cos2 i i i r d θ λ π ϕ =

(6)

thus, the signal model can be approximately expressed as:

t x i i

m t n

) (

^≈
∑
+

I

i

e t s

) (

m m j i

( 2 ϕ φ (7)

It can be expressed using vector notion as following:

+

)

) (

m

A x +

) , ( ) ( t t r

t n s

= θ

) (

) (

(8) where

) ( 1

t )]

[

), (

=ΛΛ x

T

-

^xp t x t

x t

), (

t

x t x

) (

), (

(

T

1 - + -

p

0

p

p

= s

) ( 2

1

t )]

), ( [

t s t s

),...... (

I t s

(

=ΛΛ n

) ( 1

-

n t )]

(

[

p t n t

1 - + -

p

0

p

p

), (

n t

), (

n t n t

) (

), (

T

A =

) , ( 2

θ θ θ a a a

r θ

), , ( [

r

),...... , (

r

)] , (

1 1 I

2

I r

a
is array manifold.

The following assumption are considered to hold through the work: H1 The desired source has zero-mean with the cycle frequency α

) , (

θ

r

= Λ

[

e

j

( 2

2 p p j p p

+ -

ϕ φ

i

i

)

,

Λ

, 1 ,

e

(

ϕ φ

i

+

i

)

]

i

i

) 0 ( ≠

α and is cyclically

independent with other signals for this cycle frequency.

H2 The noises are stationary and spatially uncorrelated with the covariance .

σ

2

H3

4 / λ = d and I

M 2

> .

The task of 2-D near field source localization is to estimate the parameter pair I i i

} , {θ

i r ≤ ≤ 1

.

2. Second order Cyclic MUSIC algorithm based 2-D localization

From the definition of the cyclic correlation function[6], and under the assumptions for the source signals and noise, we have the cyclic array covariance matrixes(CACM)

t

j

H

R

α τ

x e t

) (

τ

+ = x x

(

t

τ

) 2 /

(

-

) 2 /

⋅

-

2

πα

=

A

) , (

τ θ

r

s

(

t

+

) 2 /

s

H θ

(

t

-

) 2 /

^τ⋅2 r e

j

^-πα A

H t

) , (

α A R A = (9)

where
()

∑
- =

) ( ) , ( r r

τ θ

s θ

H

) , (

N N , H denote Hermitian transpose, and is the cyclic source covariance matrix (CSCM)

∞

^⋅=→ ⋅

) / 1 ( lim

N t

1 0

R

α s

) (τ

when the source signals are mutually not cyclically correlated will be diagonal.

R

α s

) (τ

R

⁼diag R (10)

s R

α

) ( 1 Λ

s

sI

τ

), ( (

^αΛ τ τ

α

), (

0

) 0 ,

where is the cyclic autocorrelation function (CACF) of the signal expressed

by

R )

α si

) (τ

(t s_i

R

α τ

si e t s

i

) (

τ

∗ - + = (11)

t s

(

τ

) 2 /

(

) 2 /

-

j

2

πα

t

i

∗

˄ R

denotes conjugate transpose. can be of asymmetry form and can be get based on the

R

) ( ˄ τ

α x

^α, τ xk xj

) (

τ

R

˄ -

α τ

xk xj e t x t x

, )

) (

^τ=
∑
+ for 0

1

N t

- - =

1 0

N

) (

(

j

2

πα

t

≥ τ (12)

R

˄ - -

- =

α τ

1

, )

) (

^τ=
∑
+ for 0

N t xk xj e t x t x

) (

(

1

N

j

2

πα

t

≤ τ (13)

τ

Rank order the eigen values of in the descending order to obtain

R

˄

α x

λ λ λ Λ

≥ ≥ 2 1

M

Let eigen vector u ,..., corresponding to the (M-I) smaller eigen values,

1 + I 2

+ I u M

u

λ , 2 + I

1 + I

λ ,..., M

λ ,we construct the noise eigen matrix from the eigen vetors as

G

I u u u G Λ

=

[ 2

1 M

+

I

+

,

]

) , ( r a θ

,corresponding to the coordinate pair the {

} , r θ is the continuum of all possible

near-field steering vector.

The spatial spectrum of second order cyclic MUSIC algorithm is

1

) , ( ) , (

P

H

r CMUSIC θ

θ a

= r r

GG a (14)

The peak of the spectrum, are the location of the source,

) , ( r CMUSIC

) , (

θ

H

-

P θ }

, { i i r θ I i Λ

, 2 , 1 = .

Discussion: Simulation of remove cyclostationary interference signal and suppress additive stationary noises any distribution

According to [8], the range of the near field is approximately r where

L ≤

2

2

λ /

) 1 ( = - = d

M L is the array's aperture. So in this paper λ

λ 4

32 ≤ r

is the near-field region.

In order to show the effectiveness of the proposed algorithm, computer simulations for testing signal selectivity, noise suppression are conducted. We consider seventeen-element array spaced as in Figure1. The sensor outputs are collected at the rate =4MHz, the speed of

propagation is c m/s. The length of samples is

s

f

10 3× = =

8

N and the signals of

128

interesting (SOI) from

λ 0

. 15 , 0 . 20 °

and λ

. ° 40 are all BPSK signals with 2MHz baud

0 . 20 , 0

0 . 25 , 0 . 60 °

5 . 0 =

rate and interference from λ

is also BPSK signals with 1.6MHz baud rate. So the SOI's cyclic frequency is α

, and the interference's cyclic frequency is

. 0 = α . The

4

SNR is 20 db. Based on the proposed method the spatial spectrum contour figures are shown in figure 2 and figure 3 with 20 times monte carlo simulation each.

In Figure2 the noise is allow exponential distribution, In Figure3 the noise is allow logarithmic normal distribution, From the experiment results we can see that the cyclostationary interference with different cyclic frequency is removed and the noise is suppressed with the proposed algorithm.

Conclusion: In this paper we present an algorithm based on second order cyclic cummulant which can remove cyclostationary interference with different cyclic frequency from the desired signals and can suppress additive stationary noises with any distribution. Furthermore, it can effectively estimate the bearing and range of near-field multiple sources.

Fig 2 Remove cyc and suppress add o with near-field cy

lostationary itive exponentia clic MUSIC a

inter l n lgori

ference ise thm

Fig 3 Remove cy nary interference and suppress addi nential noise with near-field cy SIC algorithm

clostatio tive expo clic MU

Reference:

[1] A.L. Swindlehurst, T. Kailath, Near-Field Source Parameter Estimation Using a Spatial Wigner Distribution Approach. Proc. SPIE Conf. 975, San Diego, CA, August, 1988, pp.86-92 [2] Y.D. Huang, M.Barkat, Near-field Multiple Source Localization by Passive Sensor Array. IEEE Trans. Antennas Propag. Vol.39, No.7, 1991, pp.968-974 [3] A.J. Weiss, B. Friedlander, Range and Bearing Estimation via Polynomial Rooting, IEEE Trans. Ocean Engineering, Vol.18, No.2, April 1993, pp.130-137 [4] D. Starer, A. Nehorai, Passive Localization of Near-Field Sources by Path Following, IEEE Trans. Signal Processing, vol. 42, Mar. 1994, pp. 677-680 [5] J.H. Lee, C.M. Lee, and K. K. Lee, A Modified Path-Following Algorithm Using a Known Algebraic Path. IEEE Trans. signal processing, Vol. 47, No. 5, May. 1999, pp. 1407-1409 [6] W.A.Gardner, Exploitation of spectral redundancy in cyclostationary signals, IEEE Signal Processing Mag ,vol.8, April 1991 pp.14-37 [7] J.H. Lee, C.H. Tung, Estimating the bearings of near-field cyclostationary signals, IEEE

Trans. on Signal Processing, Vol.50, No.1, January 2002, pp.110-118 [8] R.Kennedy, T.Abhayapala, and D.ward, Broadband Nearfield Beamforming using a Radial Beampattern Transformation, IEEE Trans. Signal Process, vol.46, no.8, Aug. 1998, pp.2147-2156.