Adaptive clutter rejection by angle of arrival tracking using unscented filters for atmospheric radars. Dr. Vijaykumar Chakka Arpit Gupta Dhirubhai Ambani Institute of information and communication technolgy Gujarat email: vijaykumar_chakka , [email protected] Abstract – Suppression of clutter caused by moving sources is an important and open issue for Atmospheric RADAR/SONAR. As atmospheric radars are highly sensitive on the main antenna beam pattern, it becomes all the more important to suppress the sidelobes caused by such sources. This paper presents the effective adaptive antenna techniques capable of suppressing moving clutter by classifying the incident sources based on DOA estimates given by Differential MUSIC as fixed or moving sources. Null constraints are added to suppress the fixed and moving clutter using the additional information in the DCMP-CN algorithm.

The Differential MUSIC algorithm proposed by Rajagopal and Rao [4] enjoys many advantages over conventional MUSIC and can be applied to estimate the DOA’s of multiple sources which are highly or fully coherent. To reduce the computation complexity of Eigen vectors QR decomposition of the difference matrix has been proposed [5]. Though this method reduces the computational load considerably it cannot be applied to the passive Radar environment. Vijaykumar et. al [6, 7] proposed model based DOA estimation using EKF and Unscented Kalman filter which is effectively applied to passive radar environment.

Index Terms - Atmospheric Radars, Unscented Kalman Filter (UKF), Moving Clutter Suppression, Differential MUSIC, Target Tracking

This paper presents a unified procedure, which combines Direction of Arrival (DOA) estimation and tracking by unscented filter to suppress the clutter (both fixed and moving) for atmospheric radars. Central idea of the algorithm is to better estimate the direction of clutter so that null constraints applied in these directions result in effective sidelobe canceling. The incident noise sources are classified based on their nature of direction of arrival i.e. whether it is coming from a stationary source or moving source. This is done by tracking all the incident sources on radar array based on the bearings-only estimate given by Differential MUSIC using Unscented Kalman Filter (UKF). The filter tracks all the measurements for each source using kinematic state space model.

1. Introduction Application of adaptive antenna techniques for clutter suppression in atmospheric radars has been limited primarily due to strict conditions on main antenna beam pattern. Atmospheric radars are used for meteorology and wind profiling. Guiding principle behind them is Doppler Beam Swinging (DBS) which depends heavily upon the sensitivity of the system i.e. beam pattern should not be altered by adaptive filtering. Use of adaptive antennas for atmospheric radar under the above constraint has been done effectively by Sato et.al. [1]. It is based on the Directionally Constrained Minimum Power (DCMP) algorithm [2] which uses a priori knowledge of direction of arrival in discriminating undesired echoes. Their algorithm modified DCMP by introducing an additional constraint on norm (CN) of weight of the receiving array to contain the sidelobe level. The (DCMP-CN) algorithm worked well for stationary clutter i.e. when the direction of undesired echo does not change with time. But when the direction of incident echo changes with time (like incident echoes from the aircraft), sidelobes are not effectively suppressed [1]. To estimate the direction of arrival of incident echoes from fixed and moving source, different DOA estimation methods are reported in the literature. Among the different high resolution DOA estimation methods using passive array, those based on the estimation of noise subspace vectors to estimate the DOA’s of the sources became very popular owing to their superior resolution capabilities and better bias performances. Most important among Eigen structure methods is MUSIC algorithm proposed by Schmidt [3].

If the incident signal at radar array is coming from fixed direction (like echoes coming from nearby mountains), then the resulting error in the actual DOA measurements and those estimated by the model would rise after every step. If this error increases successively the source is classified as a stationary source. All the estimated DOA values given by unscented kalman filter are discarded for stationary sources. DOA of stationary source is fixed with the original value given by the MUSIC algorithm. On the other hand if the incident signal is coming from a moving source (like aircraft) then DOA measurements would be consistent with the assumed model. For these moving sources kalman filter gives better state estimate as it takes into account the kinematics of moving object. So, DOA for moving source is the updated value of filter taken at every time step. Equipped with DOA estimates of noise sources, weight vector using DCMP-CN by adding nulls in the direction of noise sources adaptively is calculated. This paper is organized as follows. Section 2 describes DOA estimation method using Differential MUSIC. Section 3 describes adaptive DOA update by model based approach using unscented filter. Section 4 describes the incorporation of

DOA estimation to provide the nulls in direction of clutter. Section 5 deals with Simulation setup. Section 6 deals with the Conclusion. 2 DOA Estimation using Differential MUSIC Algorithm The following steps are involved in estimating the DOA’s using differential MUSIC [4] of incident sources: 1. 2. 3. 4.

X denotes the complex signal vector at the radar antenna array output. Calculate the covariance matrix RXX of the X. Compute the difference matrix D = R – E RXX E where E is the M x M exchange matrix. Compute the Eigen-decomposition of D. Estimate the DOA’s by using the pseudo-spectrum as in MUSIC.

Estimated measure of DOA for each incident source obtained after every step would be tracked using Unscented Kalman Filter based on the kinematic state space model. 3 Target tracking using Unscented Kalman Filter

3.2 Unscented Kalman Filter

The problem of estimating the position and velocity is referred to as passive tracking [8, 9]. The passive target-tracking problem is a problem of nonlinear filtering. In general Extended Kalman Filtering (EKF) is used to apply the estimation of the position and velocity of a target traveling from a series of passive measurements. However, the Extended Kalman Filter produces large errors in the mean and covariance of the transformed random variable. Unscented Kalman Filter based on Unscented Transform of non linear functions is more accurate [10]. Unscented Transform proposed by Julier [11] represents the statistics of a random variable that has been nonlinearly transformed to minimum of second order moments. All the DOA measurements of each source are tracked based on the state space model of moving object. Target tracking algorithm is based on tracking algorithm by Vijaykumar [6] using bearings only information for passive scenarios. Following steps are involved in unscented kalman filtering: 1.

Find (2n+1) sigma points [11] from the initial estimate of the state (where n is the size of state vector). Since our state vector is of size, n=4.

2.

Each of these points are then transformed by Unscented Transform through process model, xi(k +1 / k) = f [x (n), y (n), Vx (n), Vy (n)] (3.2.1)

3.

Based on the transformed points a new state estimate is predicted. Here (k+1/k) denotes that to predict the new state at (k+1)th instant all the measurements upto time k is taken.

3.1 Kinematic State Space Model

DOA measurements of each incident source are now tracked with a kinematic model of moving object. The moving object model is based on uniform velocity assumption. State variables that represent the motion of the moving source are x(n), y (n), Vx (n), Vy (n). Here x (n) and y (n) denotes position variables of the incident source. Vx (n) and Vy (n) denotes the velocity variable of the source. For constant velocity model the source can be represented as a Gauss Markov model with the noise sources representing uncertainties in the states and measurements. State space model is given by:

4.

(3.2.2) Predicted covariance based on this transformed values and new state is given by:

(3.1.1) Where X(n) is the state of the source at time instant n X(n) = [ x y Vx Vy ]T

(3.1.2)

Corresponding Measurement model is given by Θ (n) = arctan[ y(n) / x(n) ] + w(n)

(3.1.3)

v (n) is 4*1 is Gaussian noise vector for State space model and w(n) is 1*1 Gaussian noise vector for Measurement model

5.

(3.2.3) Each of the predicted point of the new state is transformed (reverse transform as that applied in part 2) and predicted observation is given

(3.2.4) Based on the these transformed value, expected value of new measurement at (k+1)th instant is calculated. Thereafter when actual measurement is taken, system state is updated to better estimate the DOA of target.

(3.2.5) 6.

Innovation covariance

Weight vector is calculated at every time step based on the above m constraints using DCMP-CN algorithm. DCMP-CN algorithm minimizes the output power and uses a priori knowledge of direction of arrival of desired signals in discriminating with undesired echoes and places an additional constraint on weight vector norm [2]. It can be expressed as

(3.2.6) 7.

Cross covariance between predicted state and predicted measurement is given as,

(4.2) Where C is the desired direction vector, and H is the constraint. So, the governing principle of our algorithm for weight vector calculation is thus expressed as

(3.2.7) 8.

Kalman Gain is

9.

Update on error covariance is

(3.2.8)

(3.2.9) 10. State update after new measurement is available

(3.2.10) 4 Clutter Rejection Method For each time step, equipped with additional direction of arrival information of stationary and moving clutter weight vector is calculated using DCMP-CN. For moving clutter updated DOA value of unscented kalman filter is taken for each step. DOA of the stationary clutter is the initial estimate given by Differential MUSIC algorithm. Based on optimum DOA estimates of noise sources, it suffices to place null constraints in the se direction. If m denotes the number of directions being tracked by unscented kalman filter then m constraints that efficiently results in suppression of clutter is given by s=m-1 ∑ NsT W* ≤ ∂ s=0 (4.1) where Ns is the direction of mth clutter being reported at each time step by unscented filter for moving clutter or DOA estimate given by MUSIC algorithm for stationary clutter W is the weight vector ∂ is the maximum side lobe level allowed in the these directions

(4.3) s=m-1 Subject to CTW* = H, WHW ≤ U and ∑ NsT W* ≤ ∂ s=0 This results in an effective cutter rejection ensuring the proper side lobe level for atmospheric radars. Solution to this minimization problem with an equality constraint and an inequality constraint is solved using the penalty function method [1]. 5 Simulation results In order to verify the effectiveness of our proposed algorithm in this paper, simulation experiments were carried out and the results are presented in this section. Three fully coherent sources form the clutter for our simulation. Two of these are moving and the third is a stationaryobject. These sources radiate electromagnetic energy and are received as narrowband planar wavefronts by a LES radar array of size n=500 with inter-element spacing is λ/2, where λ is the wavelength of the received signal. The signals were corrupted by spatially colored noise with an unknown covariance matrix. Initial DOA's are θ1(ni) = 60, θ2(ni) = 300 and θ3(ni) = 700 . Moving objects are moving with velocities v1= v2 = 300 Km/Hr. We applied the algorithm on the simulated data. We observed low tracking error for the two moving objects. The figure1 indicates the tracking of moving objects using UKF. Dark lines are the estimates of algorithm while dotted are the actual angle of object.

4.

R. Rajagopal & P.R. Rao: "A Generalized Algorithm for DOA Estimation in Passive Sonar", IEE Proc. Part F, Vol. 150. No. 1 Feb 1993. pp. 12-20.

5.

J.H.Wang & S. Wei: “Adaptive DOA Tracking by Rank Revealing QR-updating and Exponential Sliding Window Techniques”, Proc. ICASSP-944, Vol. III, pp. 225-228.

6.

Vijaykumar, C. and Rajagopal, R.: “Passive target tracking by unscented filters”, Proc. of IEEE International Conference on Industrial Technology 2000. Volume 2, 19-22 Jan. 2000 Page(s):129 - 134 vol.1.

7.

R. Rajagopal, Vijaykumar Chakka & Subhash Challa: “A Unified Approach for Passive DOA Estimation and Tracking Using Linear and Circular Arrays, Proc. SPCOM-1999. 5th Biennial Conference on Signal Processing and Communication July 2 1-24, IISC Bangalore.pp.23 -32.

8.

Allen G Lindgen and Kai F. Gong: “Position and Velocity Estimation Via Bearing Observations”, IEEE Trans. Aemsp. Elecrmn. Syst., Vol. AES-14, pp. 564-577, July. 1978.

9.

Vincent J. Aidala: “Kalman Filter Behavior in Bearings-only Tracking Applications”, IEEE Trans. Automat. Contc, Vol. AES15, pp. 29-39, January. 1979.

Measurement timeVs Angleof arrival usingUKF 100 90 80

A ngle of arrival in deg

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10. Yi Xu, & Li Liping: “Single observer bearings-only tracking with the unscented Kalman filter”, Communications, Circuits and Systems, ICCCAS 2004, Volume 2, 27-29 June 2004 Page(s):901 905 Vol.2

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11. Simon J. Julier and J.K. Uhlman: “Unscented filtering and nonlinear estimation”, Proc. of the IEEE. Volume 92, Issue 3, Mar 2004 Page(s):401 – 422.

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Fig1 6 Conclusion In this paper, we applied an adaptive DOA estimation technique and subsequent tracking by unscented kalman filter for better directional estimate of the clutter. The proposed algorithm would suppress the sidelobe effectively for moving and fixed clutter without disturbing the main antenna beam pattern as verified by simulations. References 1.

Kamio, K. , Nishimura, K., and Sato, T. : “Adaptive sidelobe control for clutter rejection of atmospheric radars”, Proc. of 10th International Workshop on Technical and Scientific Aspects of MST Radar (MST10). Page(s) 4005-4012. SRef-ID: 14320576/ag/2004-22-4005 ANNALES - Volume 22, Number 11, 2004.

2.

Takao, K., Fujita, M., and Nishi, T.: “An adaptive antenna array under directional constraint”, IEEE Trans. Antennas Propagation, 24, 662–669, 1976.

3.

R.O. Schmidt: "Multiple Emitter Location & Signal Parameter Estimation", IEEE Trans. On AP. Vol. Ap-31. No. 3 March 1986, pp. 276-288.

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