SETIT 2007
4th International Conference: Sciences of Electronic, Technologies of Information and Telecommunications March 25-29, 2007 – TUNISIA
Direction of Arrival Estimation Using the Extended Kalman Filter Ferid Harabi*, Hatem Changuel*, Ali Gharsallah* *Equipe d’Electronique, Département de Physique, Faculté des Sciences de Tunis 2092, El-Manar Tunis.
[email protected] [email protected] [email protected]
Abstract: The technique for estimating the parameters of multiple waves provides a convenient tool for the analysis of multiple-waves fields and eventually for actual applications to mobile communications. Based on the extended Kalman filter (EKF), we analyze a recursive procedure for 2-dimensional directions of arrival (DOA) estimation and we will employ the two-L-shape arrays. A new space-variable model which we call a spatial state equation is presented using array element locations and incident angles. In this paper we briefly recapitulate the most important features of the extended Kalman filter (EKF). The performance of the proposed approach is examined by a simulation study with three signals model. The simulation results show a good estimate performance. Key words: elevation angles, azimuth angles, spatial state equation, extended Kalman filter.
discussed in several papers [Shu 03]. On the contrary, we derive a suitable space-variable model in the present paper which we call a
INTRODUCTION The authors have proposed to use the Kalman filter technique for adaptive antennas [Greg 04], and showed the advantage of fast convergence through simulation and the principle of initial parameter selection.
spatial state equation using array element locations and incident angles. Simulation results show the performance of our method even at low SNRs. The rest of the paper is organized as follows: The data model is presented in section 1, the new spatial state equation for incident waves is developed in section 2, in section 3 the extended kalman filter is recapitulated, section 4 presents the 2-D direction of arrival estimation algorithm, section 5 shows simulation results and section 6 makes conclusions.
The problem of estimating the two-dimensional directions of arrival (DOAs), namely, the azimuth and elevation angles, of multiple sources was the topic of several researches [Krekel 89- Wu 03- Li 91- Liu 05]. We will employ the two-L-shape arrays that showed [Tayem 05] better performances than the one L-shape [Hua 91] and the parallel shape arrays [Marcos 93].
1. Data Model Consider the two-L-shape uniform linear array (ULAs) in the x-z and the y-z planes shown in fig.1 with inter-element equals d, using three array elements placed on the x, y and z axes. Each linear array consists of N elements. The element placed at the origin is common for referencing purposes.
In this paper the detection and estimation of the parameters of multiple waves are discussed. Being important in radar and communications systems [Kikuma 90], the technique for estimating the parameters of multiple waves is currently the subject of extensive investigation. We will present a method for estimating the elevation and azimuth arriving waves based on the extended Kalman filter [Jinkuan 95].
Suppose that there are K narrow band sources, S(t), with the same wavelength λ impinging on the array, such that kth source has an elevation angle θ k and an azimuth angle φk , k=1, …., K.
The general characteristics of a Kalman filter such as most-likelihood and convergence have been
We put the complex base-band representation of
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SETIT2007 the signal received by the nth element of one subarray as x n (t ) (n = 1, 2. . . N), the signal sources are far
z
apart from the subarray.
N
The subarray output vector at the snapshot t is then given by:
N
θk
x ( t ) = ( x 1 ( t ), x 2 ( t ),.... x
=
K
∑
k =1
N
( t ))
2
T
2
a (θ k , φ k ) s k ( t )
(1), t=1,…..,P N
N
1
2
N
y
Φk
x
“T ” is the transpose, and a( θ k , φk ) is the steering vector defined by : a( θ k , φk ) = [(1,
with
e − jϕk , . . . , e − jϕ k ( N −1) ) ]T
ϕ k depends
Figure 1: The 2-L-shape array configuration used for the joint azimuth and elevation ( φ , θ ) DOA estimation
(2)
on the position of the subarray.
The received signal is given by:
2. A new spatial state equation for incident waves
y ( t ) = Hx ( t ) + η ( t ) Where
η
Equation (5) could be called a spatial state equation. Correspondingly, (3) is called a measurement equation. The element of X, A and y are all complex in general.
(3)
is an N-dimensional complex white
We reformulate the problem involving complex quantities in terms of real quantities, in order to carry out the parameter estimation of incident waves. Let the real and the imaginary parts of x1,n be denoted by z1 and z2 respectively, and the real and imaginary
noise vector with mean zero and covariance σ2I , I is the identity matrix of size N. H= [11…1]
(4)
with K-components.
parts of exp( − jφ1 ) be denoted by α1 and α 2 , respectively. Continuing in this manner we have
At the (n+1)th element position, the incident wave vector is derived from that at the nth element position by :
x ( n + 1) = A (θ , φ ) x ( n )
(5)
Where
(7)
exp( − j φ k ) = α 2 k −1 + j α 2 k
(8)
It follows that the L-component complex vector X(n) can be completely represented by the following real vector Xr(n) with 2L-components
θ = [θ 1 , θ 2 , ...θ K ] , φ = [φ 1 , φ 2 , ... φ K ]
−jϕ1) 0 ⎤ ⎡exp( ⎢ ⎥ ... A(θ,φ) =⎢ ⎥ ⎢⎣ 0 exp( −jϕK)⎥⎦
x k , n = z 2 k −1, n + jz 2 k , n
X r ( n ) = [ z 1, n z 2 , n ... z 2 k −1, n z 2 k , n ]
(6) By this way, we can rewrite (5) in terms of real vector as:
X r ( n + 1) = Ar (α ) X r ( n ) -2-
(9)
SETIT2007
h(.) are nonlinear matrix functions. We will assume f(.) et h(.) are differentiable . W(t) and η(t) are vectors with white Gaussian random elements of zero mean, and their covariance matrices are as follows:
Where
0⎤ ⎡α1 −α2 ⎥ ⎢α −α 1 ⎥ ⎢ 2 ⎥ Ar (α) = ⎢ ... ⎥ ⎢ α2k−1 −α2k ⎥ ⎢ ⎢⎣ 0 α2k α2k−1 ...⎥⎦
Cov ( W ( t ) W ( t ) T ) = Q (10)
The extended Kalman filter is described by the following equations:
α = [α 1α 2 ....α 2 K ] , k=1,2,…K The
Ar (α )
is 2L × 2L square
ξˆ ( t + 1 / t ) = f ( t , ξˆ ( t )) matrix,
(18)
ξˆ ( t / t ) = ξˆ ( t / t − 1 ) + K ( t )[ V ( t ) − h ( t , ξˆ ( t ))]
correspondingly, (3) can be rewritten as:
Yr ( n ) = H r ( n ) X r ( n ) + N r ( n )
(17)
Cov (η ( t )η ( t ) T ) = R
T K (t ) = P( K / K − 1)∏(t,ξˆ(t / t − 1))
(11)
[∏(t ,ξˆ(t / t − 1))P(t / t − 1)∏(t ,ξˆ(t / t − 1))T + R]−1 (20)
Where Y r ( n ) = [Re( y ( n )) Im( y ( n ))] T
(12)
⎡1 0 1 0 ... 1 0 ⎤ Hr = ⎢ ⎥ ⎣ 0 1 0 1 ... 0 1 ⎦
(13)
N r ( n ) = [Re( η ( n )) Im(η ( n ))] T
(14)
P(t / K ) = P(t / t −1) − K (t)
∏ (t , ξˆ (t / t − 1) P (t / t − 1))
(21)
P ( t / t + 1) = (22) F ( t , ξˆ ( t / t )) P ( t / t ) F ( t , ξˆ ( t / t )) T + Q Where ξˆ(t / t ) and ξˆ(t / t − 1) denote the conditional mean estimates of ξˆ(t ) , based on the measurements
The size of H r is 2 × 2L, and Re(a) and Im(a) in
V(t) and V(t-1), respectively. F (t , ξˆ(t / t )) and
(14) represent the real and imaginary parts of a complex variable a.
∏ (t , ξˆ(t / t − 1)
denote the following Jacobean
matrices evaluated at the values of
3. The Extended Kalman Filter
ξˆ(t / t − 1) , respectively.
The Kalman filtering approach for a non linear system is based on the first order linearization of a nonlinear state equation using a previous estimate as the centre of the updated linear Taylor approximation. The nonlinear system is given by
ξ ( t + 1) = f ( t , ξ ( t )) + W ( t )
(15)
V ( t ) = h ( t , ξ ( t )) + η ( t )
(16)
ξˆ(t / t ) and
∂ F (k , ξˆ(k / k )) = f (k , ξ (k )) ∂ξ ξ =ξ
∂
∏ ( k , ξˆ ( k / k − 1)) = ∂ ξ
h ( k , ξ ( k ))
ξ =ξ
(23)
(24)
ξ (t ) : a state vector, V(t): measurement vector, W(t): process noise, η(t): measurement noise, and f(.) and
K(t) denotes the Kalman gain matrix. P(t/t) and -3-
(19)
SETIT2007 P(t/t-1) denote the estimation error covariance matrices.
4. The 2-D direction of arrival estimation algorithm
The above extended Kalman filter can also be used to estimate unknown parameters in a linear system. In the following, we apply it to the problem where a vector α is given by:
X k (t + 1) = Λ (α ) X k (t ) + W (t )
4.1. Estimation of the elevation angle
Let v(t) be the Nx1 signal received at the linear subarray in the z axes at the snapshot t.
v ( t ) = ( v 1 ( t ), v 2 ( t ),.... v N ( t ))
(25)
= Yk (t + 1) = C (α ) X k (t ) + η (t )
a( θ k ) = [(1,
The extended Kalman filter to determine the unknown parameter vector α now can be obtained by extending the state vector X h to the vector ξ ( k ) by
and
⎡ X k (t )⎤ ⎥ ⎣ α (t ) ⎦
ϕ k ,n =
jϕ k
,...,
e − jϕ k ( N −1) ) ]T
2 π ( n − 1) d cos θ k
(33)
(34)
λ
which
source signal at the snapshot t.
(27)
4.2. Estimation of the azimuth angle
Let x(t) be the Nx1 signal received at the linear subarray in the x axes at the snapshot t.
Equations (25) and (26) are rewritten as follows:
⎡W ( t ) ⎤
ξ ( t + 1) = f ( t , ξ ( t )) + ⎢ ⎥ ⎣ 0 ⎦
x ( t ) = ( x 1 ( t ), x 2 ( t ),.... x
(28)
=
K
∑ a (θ k =1
Y k ( t ) = h ( t , ξ ( t )) + η ( t )
(29)
k
N
( t ))
T
, φ kx ) s k ( t ) + η kx ( t )
(35)
t=1,…..,P Where
Where
h ( t , ξ ( t )) = C (α ) X k ( t )
e−
θ k is the elevation angle of the kth source signal. η kz (t ) is the additive White Gaussian noise of the kth
ξ (t ) = ⎢
⎡ Λ (α ) X f ( t , ξ ( t )) = ⎢ α ⎣
(32)
Where
The matrices Λ (α ) et C( α ) are of finite dimensions. It is assumed that the matrix elements are differentiable with respect to α . W(t) and η(t) are independent of α .
α = α (k)
)s k ( t ) + η kz ( t ) , t=1,…..,P
k
k =1
(26)
including the parameter vector depends on the parameter k,
K
∑ a (θ
T
k
(t )⎤ ⎥ ⎦
a( θ k , φ k ) = [(1,
(30)
and
ϕ k ,n =
(31)
φk
Because (28) and (29) have the same forms as (15) and (16), the extended Kalman filter in (18)-(22) can be applied to (28) and (29). It shows that the parameter vector α can be estimated by the extended Kalman filter.
e − jϕ k , . . . , e − jϕ k ( N −1) ) ]T
2π (n − 1) d sin θ k cos φ k
λ
(36)
(37)
is the azimuth angle of the kth source signal.
η kx (t ) is the additive White Gaussian noise of the kth source signal at the snapshot t in the x subarray with mean zero and covariance σ 2 I . In the same way we estimate the azimuth angle -4-
SETIT2007
φˆky using the y axis subarray. The Nx1 signal received
by H r and Yr ( m) in (11), respectively.
y(t) at the linear subarray in the y axes at the snapshot t can be rewritten as:
2) Extending the state vector by including the unknown incident angles in the state vector. It is obtained from (2), (7) and (27) as follows:
y (t ) = ( y1 (t ), y 2 (t ),.... y N (t )) T K
∑
=
a (θ
k =1
k
,φ
ky
)s k ( t ) + η
ξ ( n ) = [ z 1 , n z 2 , n .. z 2 K , n α 1 α 2 ..α ky
(t ) ,
ϕk =
(42)
4) Using the nonlinear state equation, the extended Kalman filter (18)... (22) is carried out to obtain the estimation of the elevation or the azimuth arrival waves.
Where
and
]T
3) Adapting the extended state vector (42) to (28) and (29), to obtain the nonlinear state equation.
(38)
t=1,…..,P
a( θ k , φ ky ) = [(1,
2K
e − jϕ k , . . . , e − jϕ k ( N −1) ) ]T
2π (n − 1)d sin θ k sin φ k
λ
(39)
5. Simulation Results Computer simulations have been conducted to evaluate the 2-D DOA estimation performance of the proposed method. The parameters used in the simulation are as follows:
(40)
The sensors displacement d is taken to be half the wave length of the signal waves. The incident waves are three waves K=3, with directions of arrival DOA ( θ k , φk ), k=1...K. The additive noise is white
Finally, the azimuth angle estimation φˆk can be written [Tayem 05] as:
Gaussian processes. The L=250 is the number of snapshots per trial.
⎧1 ˆ ˆ ˆ ˆ ⎪ 2 (φ kx + φ ky ) if both φ kx andφ ky are real ⎪ ⎪ ⎪ ⎪φˆkx if φˆky is complex ⎪ ⎪ (41) φˆk = ⎨ ⎪ ⎪ ⎪φˆ if φˆkx is complex ⎪ ky ⎪ ⎪ ⎪ Failure if both are complex ⎩
All the waves are assumed to be direct waves and the power levels are a constant 2, 3.2 and 5, respectively. The measurement noise η(t) in (16) is assumed to be 0.1. The process noise W(t) is assumed to be zero as the state (15) represents the elements state at the same time. The convergence of estimated values of the elevation and azimuth angles are plotted in fig.2 and fig.3, respectively. The abscissa is the number of iterations, and means the numbers of necessary elements. In figures.2 (a) and (b) the waves are located at (60°,20º), (70°,30º) and (80°,40º), respectively and signal to noise ratio (SNR)=10 dB. This is an example of the most common case because the incident angles differences are medium and, accordingly, the convergence is not bad. It is shown that the estimated values steadily converge to the actual values. About 110 iterations are required to achieve a steady state for the elevation angles. In other words, 110 out of 250 elements for the z axes array are used for convergence. For the azimuth angles, only 85 iterations are required to obtain the convergence to the real azimuth angles.
The problem to estimate the elevation angle θ k and the azimuth angle φ k in (1) or α 2 k −1 and α 2 k in (8), can be solved with the same procedure as parameters estimation in linear systems. We should apply an extended Kalman filter approach by means of spatial state equations of the incident waves as mentioned above. The procedure is summarized as follows:
In figures.3 (a) and (b), three waves are located at (70°,20º), (74°,24º) and (78°,28º), respectively. Despite the proximity of the elevation and azimuth angles, the convergence characteristics of angles estimations show no degradation from fig.2. Although the variations of the elevation angles are larger and the convergence speeds are slower than in fig.2, the estimates errors are quite small after achieving the convergence state after 180 iterations.
1) Defining a space-variable vector expressing the incident waves, to obtain a spatial state equation of the incident waves. The matrix Λ (α ) and X h (k) in
(25) is replaced by Ar (α ) and X r (m) in (9), respectively. And the matrix C( α ) and Yh ( k ) in (26) -5-
SETIT2007 Figures.4 (a) and (b) show the histogram plots for the joint elevation and azimuth angles, respectively, for a single source with DOA located at (60°,20º) and signal to noise ratio (SNR)=10 dB, by using the extended kalman filter of the 2-L shape arrays. We observe that the method gives a very close joint DOA estimation and the clear peaks appear around (60°, 20°). No failure can be observed in the estimation of the 2-D directions of arrival, but it is clear that the proposed algorithm improves the performance significantly and it also reduces the estimation error of both the azimuth and elevation angles.
estimted elevation real elevation
DOA(elevation)(deg.)
85 80 75 70 65 60
90
0
100
estimted elevation real elevation
200
300
Number of iteration
(a)
60
50
40 0
estimted zimuth real azimuth
35
70 DOA(azimuth)(deg.)
DOA(elevation)(deg.)
80
50
100
30 25 20 15
150
Number of iteration
10
(a)
0
100
200
300
Number of iteration
(b) 50
estimted zimuth real azimuth
Figure.3: Convergence results for the sources located at (70°,20º), (74°,24º) and (78°,28º), respectively. SNR =10dB. (a) Elevation angles. (b) Azimuth angles.
30 20
Histogramme angle elevation
250
10 200
0 -10
0
50
100
iteration
DOA(azimuth)(deg.)
40
150
150
100
Number of iteration
(b)
50
0 40
Figure.2: Convergence results for the sources located at (60°,20º), (70°,30º) and (80°,40º), respectively. SNR =10dB. (a) Elevation angles. (b) Azimuth angles
50
60 DOA(elevation)(deg.)
(a)
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70
80
SETIT2007 Histogramme angle azimuth
250
REFERENCES:
iteration
200
Greg Welch and Gary Bishop, “An Introduction to the Kalman Filter”, Department of Computer Science University of North Carolina at Chapel Hill , April 2004.
150
100
Hua Y., T.K. Sarkar, D.D.Weiner “an L-shape array for estimating 2-D directions of wave arrival”, IEEE Trans. Antennas Propagation 39 (2) (1991) 143-146.
50
Jinkuan W. , T.Takano and H.Kojiro, “estimation of arrival waves using an extended Kalman filter’’ IEICE Trans. Vol. E78-B-II, No.11, Nov.1995.
(b)
Kikuma N. , M.Anzai , M.Ogawa , K.Yamada, and N.Inagaki, ‘Estimation of direction of arrival land delay profile of multi-path waves by MUSIC algorithm for indoor radio communications’ IEICE Trans., vol.J73-B-II,no.11,pp.786-795,Nov.1990.
Figure.4: Histogram of elevation DOA estimations for a single source of DOA at (60°, 20°), SNR = 10 dB, (a) Elevation angles. (b) Azimuth angles.
Krekel P. , E. Deprettre, “A two dimensional version of matrix pencil method to solve the DOA problem”, European conference on circuit theory and design, pp.435-439, 1989.
0
0
10
20
30
DOA(azimuth)(deg.)
Li P. , B. Yu, J. Sun “a new method for two dimensional array”, signal processing 47(3), p. 325-332, December 1995. Liu T. , J. Mendel “azimuth and elevation direction finding using arbitrary array geometries”, IEEE Trans. Antenna and propagations, Vol. 53, No. 4, APRIL 2005
6. CONCLUSIONS: An antenna array configuration was proposed using the extended kalman filter method, for the 2-D azimuth and elevation angle estimation problem. The results obtained are summarized as follows.
Marcos S., “calibration of a distorted towed array using a propagator operator”, J. Acoust. Soc.Amer. 93(4) , p.1987-1994, march 1993.
(1) A space variable model, called spatial state equation is derived, using the elements array of the 2L shape antenna. (2) We show that the extended kalman filter algorithm, which can be applied to all array configurations, is simple and applicable to practical cases. (3) The proposed scheme reduces the estimation error of both the azimuth and elevation angles and it shows a good performance at low SNRs and at the proximity of the waves.
Shu C. , “General Two Stage Extended Kalman Filters”, IEEE Trans. Automatic control, Vol.48, No.2, p.289-293, February 2003. Tayem N. , H. Kwon, “L-shape-2-D arrival angle estimation with propagator method”, IEEE Trans. vol.53.No5, May 2005. Wu Y. , G. Liao, H.C.So “a fast algorithm for 2-D direction of arrival estimation” signal processing. 83 (8), p. 18271831, august 2003.
It should be noted that the idea of the proposed approach is different from the well-known MUSIC or the PM algorithms. The angles of the arrival waves are directly estimated from the signal received at each array element, though MUSIC and PM algorithms obtain the estimation based on the spectral analysis techniques. The performance of the proposed approach is not affected by the change of the incident waves during the computing time of the algorithm, because the signals at each array element are collected at the same sampling time.
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