A DOA Estimation Algorithm of Virtual Array Based on Beam forming

Qingwei Li College of Information and

Communication Engineering Harbin Engineering University

Harbin, China xinxishe@126.com

Yi Jiang College of Information and

Communication Engineering Harbin Engineering University

Harbin, China jiangyi@hrbeu.edu.cn

Xinying Diao College of Information and

Communication Engineering Harbin Engineering University

Harbin, China diaoxinying@hrbeu.edu.cn

Abstract—Virtual Array Transformation of DOA Algorithm is widely used in actual engineering DOA estimation of the uniform linear array (ULA) of spacing of array elements greater than half a wavelength or any non-uniform linear array (NULA).However, when virtual array is transformed, if there is a signal outside the current transformation sensor, a serious impact will be had on virtual array DOA estimation performance of the signal in the current transformation interval. In this paper, a virtual array transformation algorithm is proposed based on beam forming methods, which can effectively solve the existing problems of traditional virtual array transformation algorithms, and the mathematical model and test simulation results of the algorithm are given.

Keywords-DOA estimation; virtual array transformation; transformation sectors; beam forming

I. INTRODUCTION Conventional MUSIC algorithm and spatial smoothing

algorithm are always considered for ULA of Array element spacing 0.5d λ≤ . But it is very hard to ensure 0.5d λ≤ ULA in actual engineering, which makes the classic MUSIC algorithm and spatial smoothing DOA estimation algorithm restricted. However, the virtual array transformation method can transform any NULA or ULA with spacing of array elements 0.5d λ> to ULA of spacing of array elements

0.5d λ≤ , what not only makes the study of previously ULA DOA estimation algorithm can be used, but also increase the aperture of array for using the virtual array elements.

When virtual transformation is taken, we should calculate the transformation matrix. If the transformation interval is too large, the transformation matrix obtained makes the error of DOA estimated angle too large. Therefore, the intervals divided of the scanned area should be small enough to ensure the accuracy of DOA estimation of virtual transform in the divided interval. When a scan range is selected, first we divide the range into several intervals. When we take virtual transform in a certain interval, if a signal exists outside this interval, the DOA estimation of virtual transformation of signals in this interval will be inaccurate.

As a result, this paper presents a virtual array DOA method based on beam-forming. Before we have the DOA estimation of virtual transform in a certain interval, we can first use the

weighting factors for beam-forming to weight every sub-array. Then we can filter the signals outside the current transform interval and can have accurate DOA estimation of virtual transformation.

II. THE PRINCIPLE OF VIRTUAL ELEMENTS TRANSFORMATION

In this section we describe the principle of virtual array transformation. Let us divide the field of view of the array into u sectors, which is 1 2 u, , ........., .Now we discuss the principle of virtual transformation in the j th sector j . And then the sector is

[ , , 2 ,...... ]l l l rθ θ θ θ θ θ= + Δ + Δj (1)

where θΔ is the step of angle. Then the steering vector of actual array can be obtained

[ ( ) ( ) ......... ( )]L L rθ θ θ θ= + ΔA (2)

And the steering vector of virtual array is

[ ( ) ( ) ......... ( )]L L rθ θ θ θ= + Δ (3)

In other words, A is the response of the real array to signals in the sector j ,and is the response of interpolated array to the same signals. The spacing of the virtual array elements is a half wavelength. Then, there exists a constant matrix B between the actual array and the virtual array, which can be written as

= ⇔ =H HBA A BAA AA (4)

From above formula, we can obtain that

H H -1B = AA (AA ) (5)

Assume that the actual data covariance matrix is R and the noise of system is additive Gaussian white noise, the average power of which is 2δ . Then

2δ= +SR AR A I (6)

Hence the data covariance matrix of virtual array is

945978-1-4244-8165-1/11/$26.00 ©2011 IEEE

2δ+H H HSR = BRB = BAR A B BB (7)

Because ≠HBB I , and from (6) and (7), we know that the white noise has become color noise. Therefore the noise received is independent. So we should revise the matrix of transformation B as that

H -1/ 2T = (BB ) B (8)

We can prove that

HTT = I (9)

So, the covariance matrix of rectification is

2δ+H H HSR = TRT = TAR A T I (10)

III. DOA ESTIMATION OF VIRTUAL ARRAY TRANSFORMATION BASED ON BEAM FORMING

2x 3x1x Mx

1v 2v 3v 4v pv

m u×

Figure 1. The structure chart of DOA estimation of virtual array

transformation based on beam forming

Fig. 1 shows a Uniform Linear Array of M elements, the spacing of which is d .Let us divide the array into numbers of sub-arrays as follow:

We give a group of serial number to the elements of the array from the left one to the right. So we can get the serial number is 1 2 …….M. The length of every sub-array ism . Hence the serial number of the k th sub-array can describe as , 1,......, 1k k k m+ + − . Assuming we can divide the array into p sub-arrays, therefore we can obtain that

1p M m= + − (11)

According the description of above, we have divided the field of view into u sectors, which is 1 2 u, , ........., .

Assume there are N signals received by the elements of the array. The angles of these signals are 1 2( , ,....... )Nθ θ θ .And

the j th signal ( )js t is located in the j th sector j . The other incoming signals are located in others sectors.

The general formula of incoming signals can be written as

( ( ))( ) ( ) , 1, 2.........ic ij w t ti is t u t e i Nφ+= = (12)

According to the Snapshot representation of Array signal processing we can obtain that

;= +X AS N (13)

where [ ]1 2 ......... TNs s s=S and

1 2[ ( ) ( ) ....... ( )]Nθ θ θ=A , which can be called Array Flow Pattern Matrix. And ( )iθ can be called Steering Vector. Where 1,2.........i N= .

In fig.1, we use kx to express the data received by the k th element. And kX stands for the data vector received by the k th sub-array. Then

( 1) ;k− +kX = AD S N (14)

where 1,2....k p= and [ ]1 1...... Tk k k k mx x x+ + −=X ,

1

2

0 00 0

0 0 N

j

j

jN N

e

e

e

β

β

β

−

−

−×

=D ,

and2 sin

, 1,2......ii

di N

π θβλ

= = .

For the u sectors, there are u groups of weighting factors corresponding to them. The weighting factors corresponding to every sector is calculated by beam forming algorithm. And the beam pointing of every group weighting factor is to corresponding sector.

In the net of beam weighting,

[ ]= 1 2 uT W W ....... W (15)

where iW is the weighting factor corresponding to the i th

sector. And ,1 ,2 ,...... , 1, 2......,T

i i i mw w w i u= =iW .So the combining data of the k th sub-array is

kv = Ti KW X (16)

where kv is the bombing data of the k th sub-array, and

[ ]1 1...... Tk k k k mx x x+ + −=X

We can work out

,1 ,2 1 , 1.........k j k j k j m k mv w x w x w x+ + −= + + + (17)

where 1,2.....k p= .

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If we write the data vector of each sub array as 1 2 .......

T

pv v v=V , we can obtain the actual data covariance matrix of all the sub arrays R

[ ]HE=R V * V (18)

And then, according to (5) and (8) we can obtain the matrix of transformation. After that we can work out the virtual data covariance matrix R by (10). In this procedure we expand p branches of date to 2 1p − . Next, we can get the eigenvalues

of R and then obtain the angle of the incoming signal ( )js t of current transformation sector j by MUSIC algorithm.

By the same method and step, we can get the angles of different incoming signals in different sectors, avoiding the impact to the signal in current transformation sector by signals in other sectors.

IV. SIMULATION

Assuming that the scanning range is [ 90 , 90 ]° °− − in the test, the entire scanning range is evenly divided into three sectors that are [ 90 , 30 ]° °= − −1 , [ 30 ,30 ]° °= −2

[30 ,90 ]° °=3 .

The same simulation conditions of all the experiments are given as follows

The spacing of the array elements is ULA of spacing of elements 0.7d λ= . The number of array elements is

12M = .And the number of every sub array elements is 6m = . The number of sub arrays is 1 7p M m= + − = .The SNR is 0dB, step of angle is 0.01°and sampling snapshot is 500. The incoming signals are two and their angles are 0°and 80°.

4.1Take the transformation sector as [ 30 30 ]° °= −2 . We directly have virtual transformation in the sector without any preprocessing. The result is given as follows.

-30 -20 -10 0 10 20 30-15

-10

-5

0

5

10

15

20DOA estimation of Virtual transformation without any preprocessing

Angle(°)

Spa

tial S

pect

rum

(dB

)

Figure 2. The figure of DOA estimation of virtual transformation without

any preprocessing

From the result of simulation we can have a conclusion. For there is a signal outside of the current transformation sector, as a result, there is a pseudo peak of wave. If we take virtual transformation without any processing to the signal outside the current sector, the algorithm of virtual transformation of DOA estimation will lose its efficacy.

.2 Validation of interference suppression in DOA estimate by beam forming

All the sub arrays are Hamming Weighted, beam pointing to 0 . Results show as follow:

-100 -80 -60 -40 -20 0 20 40 60 80 100-10

-5

0

5

10

15

20

25DOA estimation after beam forming

Angle(°)

Spa

tial S

pect

rum

(dB

)

Figure 3. DOA estimatation after beam forming

Fig.3 shows the signal outside the current selected sector 2 can be filtered efficiently by weighting factors to every sub

array for beam forming.

4.3 Before taking virtual transformation in the simulation condition of 4.1, first we give Hamming Weighting factors to every sub array for beam pointing to 0 and take [ 30 30 ]° °= −2 as the transforming sector. The result shows as follows.

-30 -20 -10 0 10 20 30-10

-5

0

5

10

15

20

25

30DOA estimation based on virtual array of beam forming

Angle(°)

Spa

tial S

pect

rum

(dB

)

Figure 4. DOA estimatation based on virtual array based on beam forming

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Compared with Fig. 2, Fig. 4 shows interference of signals outside the current transformation sector can be effectively restrained by filtering in spatial domain with beam forming for each sub array before transformation.

Fig. 4 also shows that when SNR=0 dB, high precision of DOA estimation can be achieved by beam forming of sub array which has strong filtering capability for interference. As to the signal in the sector [30 ,90 ]° °=3 , the same method can be applied into use. Select the transform sector [30 ,90 ]° °=3 , sub arrays are weighted by the weighting factors of beam pointing to 60 . And the result shows as follow:

30 40 50 60 70 80 90-10

-5

0

5

10

15

20

25

30DOA estimatation based on virtual array of beam forming for signal of 80°

Angle(°)

Spa

tial S

pect

rum

(dB

)

Figure 5. DOA estimatation based on virtual array of beam forming for

signal of 80

V. CONCLUSIONS A new method that applies beam forming to DOA

estimation of virtual array is proposed in this paper. At the time, this paper gives out the mathematical model of the virtual array of DOA estimation based on beam forming and the results of simulation .By weighting every sub array with corresponding weighting factors, it can effectively filter the signal outside the current transforming sector. Not only can it effectively overcome the shortage of directly taking virtual transformation, but also can it filter all the noise outside the beam pointing. For the whole array is divided into different sub-arrays, which makes array aperture reduced. But this can be compensated through next virtual array transformation.

ACKNOWLEDGMENT This paper is funded by the International Exchange

Program of Harbin Engineering University for Innovation-oriented Talents Cultivation.

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