ITERATIVE ALGORITHMSFOR RADAR SIGNAL PROCESSING

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1 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) ITERATIVE ALGORITHMSFOR RADAR SIGNAL PROCESSING Dib Samira, Grimes Morad, Ghemit Amal and Hamel Sara Dept. of Electronics Engineering, University of Jijel, Algeria Barkat Mourad, IEEE Fellow Dept. of Computer Engineering, King Saud University, Riyadh, KSA ABSTRACT In this paper, we are interested in the application and the evaluation of the performances of adaptive recursive subspace-based algorithms of linear complexity for the suppression of interferences in Space Time Adaptive Processing (STAP), namely PAST, OPAST, API and FAPI. To highlight their application in the STAP, we present the reduction of the rank by the principal components (PC) method. The simulation results will be presented and the performances of the STAP for a reduced rank will be discussed with a comparative study made between the used methods.performance curves show that these algorithms allow good indeed detection of slow moving targets even with a low rank covariance matrix and in a Doppler ambiguous environment. Keywords:PAST, OPAST, API, FAPI, STAP,airborne radar. 1 INTRODUCTION A target in a scenario of airborne surveillance is darkened by the clutter of ground and the jammer in multiple dimensions. It was shown in [1,2] that the target is separated from the clutter and jammer in two-dimensional angle/doppler. Space-time processing can provide a rejection of such clutter and thus, be able to detect the slow targets. Brennan and Reed [3] first introduced STAP to the radar community in With the recent advancement of high speed, high performance digital signal processors, STAP is becoming an integral part of airborne or space-borne radars for MTI functions. However, it is well-known that optimum STAP detection implies a large computational cost, since it utilizes complex matrix operations and often in an iterative fashion. In fact, the methods of the STAP with full rank use all the available degrees of freedom to eliminate the interferences, so requiring a cost of high calculation. For this reason, some reduced-rank STAP algorithms have been developed. In [2-1] and references therein, it was shown that STAP has the unique property of compensating for the Doppler spread induced by the platform motion and thus, making the detection of slows targets possible. In this paper, we analyze, at first, the STAP with reduced rank by using the method of the principal components (PC). Then, we apply four iterative and adaptive algorithms of subspace tracking to reduce the rank. Finally a comparison is made to justify the use of these algorithms in the radar processing. The paper is organized as follows. In Section 2, the mathematical model is given. In Section 3, we give a brief description of STAP with reduced rank and define the PC method. The iterative algorithms proposed are treated in Section 4. Results and discussions are presented in Section. Finally, the conclusion is presented in Section 6 highlighting the main results obtained in the paper. 2 MATHEMATICAL MODEL OF DATA We consider a space time network with N antennas uniformly spaced and M delay elements for any antenna at a constant pulse-repetition frequency (PRF). The data are then processed on one range of interest which corresponds to one slice of the data cube as shown in Fig. 1. A space time snapshot at range k in the presence of a target is given by [1] X = αs + X i (1) where, X i is the vector of interferences (noise, jamming and clutter), α is the target amplitude and S is the space-time steering vector given by UbiCC Journal Page 18

2 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) and thermal noise. More details for the computation of these matrices are given in [4]. The performance of the processor can be discussed in terms of the Improvement Factor (IF). IF is defined as the ratio of the SINR of the output to that of the input of the Direct Form Processor (DFP) and is given by [1] IF = W H S.S H W.tr( R) opt W H R.W.S H S () (a) Where W is the optimum weights of the interference plus noise rejection filter. Note that a notch, which is a reversed peak of the clutter, appears at the frequency in the direction of sight of the radar, while the width of this notch gives a measurement of the detection of slow moving targets. (b) Figure 1: (a) Conventional chain of STAP; (b) Data cube of STAP S = S t S s, S t and S s are the temporal and space vectors given respectively by S t = [1;e and j 2πF t j 2πF s ;e j 2π 2 F t ;...;e j 2π ( M 1) F t ] j 2π 2 F s j 2π ( N 1) F s (2) 3 STAP WITH REDUCED RANK The fully adaptive techniques of signal processing cannot be applied for a real-time processing. The methods with reduced rank realize the adaptation on a space of reduced dimensions obtained after an adaptive transformation on the data. They also exploit the nature of the low rank matrix of interferences. The idea is the separation of the overall space into an interference subspace and a noise subspace [4,]. A common method to obtain these subspaces is via Singular Value Decomposition (SVD) of the interference-plus-noise covariance matrix. Such method can reduce the sample support requirement to O(2r) (number of snapshots), where r is the rank of the covariance matrix to achieve an SINR of 3 db below the optimum, but at the expense of a S s = [1;e ;e ;...;e ] (3) considerable computational complexity due to the SVD O((MN) 3 ). This complexity prevents the realtime applications. where F t = F d / PRF and F s = d.sinθ / λ are, respectively, the normalized Doppler and spatial frequency. d is the distance between the antennas and θ is the azimuth angle. The optimum weight of the STAP, which maximizes the signal to interference noise ratio SINR, is obtained to be [1] 1 W opt = αr S (4) Where R is the covariance matrix of the interferences. It is supposed to be known and is the sum of covariance matrices of the clutter, jammers The partially adaptive algorithms of the STAP consists in transforming the data with a matrix V C MN r where r<<mn. There are several methods for the covariance matrix rank reduction [3-1], which may differ in the shape of the processor as well as in the selection of the columns of the matrix. The principal component is based on the eigenvectors conservation of the matrix of covariance of interferences corresponding to the dominant eigenvalues [4]. If we assume that the r columns of V are a subset of the eigenvectors of R, the improvement factor of the reduced rank can then be written as UbiCC Journal Page 19

3 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) IF = S H V (V H RV ) 1 V H S tr(v H RV ) RR S H V.V H S 4 ITERATIVE / ADAPTIVE ALGORITHMS (6) The iterative/adaptive algorithms of subspace tracking allow to follow the temporal variations of the subspace and to update it in every new observation. One very large number of algorithms were proposed in the literature [11-17] and references therein. They showed their efficiency in several domains of signal processing in particular in antennas processing and in spectrum analysis. They offer interesting perspectives in the STAP. They can be classified according to their computational complexity. Schemes requiring O(MN 2 ) or O((MN) 2 r), operations will be classified as high complexity; algorithms with complexity O(MNr 2 ) as medium complexity, and schemes requiring only O(MNr) operations are said of low complexity [17]. In this context, we consider the class of the fastest, most robust and effective algorithms said of linear or low complexity because they are the most Table 1: PAST and OPAST Algorithms. Initialization : W () = I MXN ; Z () = I r For t = 1 to Number Snapshots do PAST Section: y(t ) = W (t 1) H x(t ) h(t ) = Z (t 1) y(t ) h(t ) g(t ) = µ + y(t ) H h(t ) Z (t ) = 1 ( Z (t 1) g(t ) y(t ) H Z (t 1)) µ e(t ) = x(t ) W (t 1) y(t ) W (t ) = W (t 1) + e(t) g(t ) H OPAST Section: (t ) = γ (t )(x (t ) W (t 1) y (t )) important ones due to their suitability for real-time µ 2 1 τ (t) = 2 applications. We propose the application and the 2 2 h(t ) ( ) ( ) evaluation of the performances of the algorithm t h t µ 2 PAST and its orthogonal version (OPAST) and the algorithm API and its Fast version (FAPI). A brief e(t ) = τ (t) W (t 1)h(t ) + (1+ τ (t) h(t ) 2 (t) description of key idea of these algorithms is as µ µ 2 follows. Z (t) = 1 ( Z (t 1) 1 g(t ) y(t ) H Z (t 1)) 4.1 PAST and OPAST algorithms µ µ The Projection Approximation Subspace Tracking (PAST) [12] is based on the optimization of the following criterion: W (t ) =W (t 1) + e (t )g H (t ) End t J (W (t )) = µ t i i =1 x(i) W (t ). y(i) 2 (7) where x is the observed data vector and W is the estimated interference subspace basis and µ is a forgetting factor, µ < 1and y(i) = W H (i 1).x(t ) this cost function has a global minimum which yields a non orthonormal basis of the interference subspace and which may be attained by a RLS adaptive algorithm given in table 1. In order to resolve the problem of convergence and orthogonality of the weighty matrix, the Orthonormal PAST (OPAST) was presented [13]. 4.2 API and FAPI algorithms The Approximated Power Iteration (API) and Fast Approximated Power Iteration (FAPI) algorithms derive from the power method [14]. A less restrictive approximation than for PAST is used. Indeed it concerns the projection on the estimated subspace instead of the estimated subspace itself: W (i).w H (i) W (i 1).W H (i 1) (8) The obtained subspace is found to be orthonormalized. The FAPI algorithm is a fast implementation of API. API and FAPI codes are presented in table 2. UbiCC Journal Page 2

4 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) Table 2: API and FAPI Algorithms. Initialization : W () = I MXN ; Z () = I r For t = 1 to Number Snapshots do PAST Section(see Table 1) API Section: Θ(t ) = 1 I + e(t ) 2 g(t ) g(t ) H (CNR) is set equal to 8dB. This clutter covers the band [3, 3 ]. For the iterative algorithms, we use the improvement factor given by Eq. () where H W = (I ww )S and w is the estimated subspace of interferences. The forgetting factor of both algorithms is fixed at µ =.99. Fig. 2 shows the eigenspectra for the covariance matrix. We note a clear distinction between the interferences subspace and noise subspace. An extension to Brenann s rule, for a number of an effective rank for the covariance matrix of a sidelooking radar, has been derived recently in [8] and is Z (t) = 1 (I g(t ) y(t ) H Z (t 1)Θ(t ) H given by r=n+(β+j)(m 1), where J denotes the ) µ number of jammers. The high-input eigenvalue is W (t) = (W (t 1) + e(t) g(t ) H )Θ(t ) about λmax=cnr+ 1 log(nm)(db)). This causes a slow convergence which is inherent to adaptive algorithms [17]. These values can actually be read FAPI Section: directly from Fig. 2. ε 2 (t) = x(t) 2 y(t) 2 We recall that the aim of this paper is to ε 2 demonstrate the robustness of the proposed iterative (t) 1 τ (t) = algorithms PAST, OPAST, API and FAPI by 1 + ε 2 (t) g(t ) ε 2 (t) g(t ) 2 comparing their performances to PC method for low η(t) = 1 τ (t) g(t) 2 y' (t ) = η (t ) y(t ) + τ (t ) g(t ) h' (t ) = Z (t 1) H y' (t ) (t) = τ (t) (Z(t 1)g(t) (h'(t)g(t))g(t) η(t) Z (t ) = 1 (Z (t 1) g(t )h' H (t )+ (t ) g H (t )) µ e' (t ) = η(t ) x(t ) W (t 1) y' (t ) W (t ) = W (t 1) + e' (t ) g H (t ) End RESULTS AND DISCUSSIONS In this Section, we discuss the influence of the PC method, based on the reduction of the rank of the covariance matrix, and the iterative algorithms (PAST, OPAST, API and FAPI) on the detection of a target with a low power (SNR=dB) and with a slow speed. The simulated environment is a linear side 4 looking network of N=8 antennas spaced out by half of the emitted wavelength d = λ / 2 and M=1 3 impulses in the coherent processing cube. The dimension of the adaptive process is thus MN = 8. 2 The elevation angle is fixed to 2. The speed of the 1 airborne radar is V R =1m/s, and the frequency of transmission is.3ghz. The environment of interferences consists of five jammers and ground clutter. The jammers are at azimuth angles of, 18, 6, 9, and 72, respectively, and with respective ratios of jammer to noise (JNRs) of 13dB, 12dB, 11dB, 1dB and 9dB. The clutter to noise ratio rank covariance matrices and in an ambiguous scenario. We start by studying the influence of the dimension of the STAP on the number of eigenvalues of the covariance matrix that has given us guidance on environment statistics in which the target is to be tested. It can be seen clearly from Fig.2that an increase in MN leads to an increase in powers and in the number of degrees of freedom of the system filter. We also note that the interference-noise space can be separated into two subspaces: the interference subspace and the noise subspace. As the number of eigenvalues is a measure of the degree of freedom of clutter elimination filter, we can notice that the number of small eigenvalues is high, while the number of those with a low value is significantly large, which increases the degree of freedom. Eigenvalues (db) 6 MN=2 MN=4 MN=8 MN= Numbers of eigenvalues Figure 2: Eigenvalues of the space-time covariance matrix R in the presence of two jammers: with JNR=3 db, CNR = 2 db UbiCC Journal Page 21

5 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) If we work with all size of the covariance matrix (MNxMN), there will be difficulties in implementing real-time especially since it also requires very cumbersome equipment. So the use of rank reduction methods reduce the computational complexity and therefore the cost, and this by taking a dimension (rxr) with r the rank which must be very inferior to MN. This choice is motivated by the fact that the observation space is divided into two areas (noise and interference) as it was commented on in Fig. 2. On the other hand, it is important to consider the performances of the optimal processor according to the values of the PRF. Using the spectral analysis and varying the PRF, Fig. 3shows the minimum variance spectra of the received signal in the presence of two jammers for different values of PRF. There is the presence of aliasing of the clutter for high values of PRF. We can therefore say that the choice of the PRF is essential to ensure good detection. We can conclude that high values of PRF lead to Doppler ambiguities and the clutter spectrum becomes ambiguous so that blind velocities occur. This analysis can be addressed by using the Improved Factor and varying the PRF (Fig. 4 and Fig. ). In the case where PRF is equal to (2V R /λ), we see the appearance of ambiguous notches with speeds associated are called blind velocities and the detection becomes relatively difficult. However, for the case of PRF equal to (4V R /λ), we notice that the ambiguous notches are eliminated and thus the detection is improved. Fig. 4 illustrates the effect of the rank on the detection of slow targets for an environment without ambiguities. We notice that there is a strong degradation in the performances for a small rank. The same detection results are obtained when r = 2 and r = 4, and thus to save calculation time, we choose r=2 as a good reduction of the rank. We observe that if the rank is very small, the slow moving targets will be removed with the clutter, and thus will not be detected. Similar findings were made for an ambiguous scenario, Fig.. We notice the appearance of new weak undulations in the bandwidth due to the estimate used for the rank reduction. Also, the appearance of ambiguous notches and the width of the notch do not change. Thus the rank reduction does not eliminate the ambiguities of the clutter during the suppression of the noise. We can then say that we should not reduce up to the rank to very low values. Normalised Doppler Frequency, Normalised Doppler Frequency, Normalised Doppler Frequency, Azimut ( ) (a) Azimut ( ) (b) Azimut ( ) (c) Figure 3: Angle/Doppler spectrum for known R in the presence of two jammers at -6 and 6 with JNR=4 db, N=8, M=1, CNR = 2 db: (a) PRF = 8.VR / λ ;(b) PRF = 4.VR / λ and (c) PRF = 2.VR / λ UbiCC Journal Page 22

6 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) IF(dB) r=4 r=2 r= Figure 4: Improvement factor for the PC-DFP with different values of r and with PRF = 4.VR / λ For the iterative algorithms, we use the improvement factor given by the Eq. () where W = (I ww H )S, and w is the estimated subspace of interferences. The factor of forgetting of both algorithms is fixed at µ =.99. To compare the different algorithms presented in this paper, we consider the two cases of scenarios and two limited values of the rank. In fact, we first consider the case without ambiguities, PRF = 4.V R / λ, and we draw the variations of the improvement factor, IF, according to the normalized Doppler frequency (). For medium value of rank r = 2, we can see from Fig. 6(a) that all the tested algorithms are globally equivalent and present good performances: the notch is relatively thin comparing to the optimal processor which leads to the detection of slow targets. We observe from Fig.6(b) that recursive algorithms perform better in the case of low rank value. In the presence of ambiguities ( PRF = 2.V R / λ ), we see from Fig. 7, the appearance of ambiguous notches. This is due to the aliasing phenomenon well known, which becomes more important with the decline in PRF and the effect of Doppler ambiguities will be more important and makes target detection more difficult. Fig. 7(a) shows that the iterative algorithms present acceptable detection performances than those of PC method. We notice from Fig. 7(b) that there is a strong degradation in the performances for a small rank and the detection becomes impossible for the PC method. Contrarily, the use of recursive algorithms overcomes this problem and improves the detection. In this case, PAST and API performances are comparable to that obtained by the optimal processor. In conclusion, we can note that PAST, OPAST, API and FAPI algorithms give a much better performance than the PC method, in terms of detection when the radar operates in an unambiguous environment and with low rank of reduction. IF(dB) IF(dB) IF(db) r=4 r=2 r= Figure : Improvement factor for the PC-DFP with different values of r and with PRF = 2.VR / λ PC -3 PAST OPAST -4 API FAPI (a) -3 PC -3 PAST OPAST -4 API FAPI (b) Figure 6: Improvement factor for the iterative algorithms and PC-DFP with PRF = 4.V R / λ : (a) r = 2; (b) r = 8 UbiCC Journal Page 23

7 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) IF(dB) IF(dB) PC -3 PAST OPAST -4 API FAPI (a) -3 PC -3 PAST OPAST -4 API FAPI (b) Figure 7: Improvement factor for the iterative algorithms and PC-DFP with PRF = 2.V R / λ : (a) r = 2; (b) r = 8 6 CONCLUSION In this paper, we considered the application and the evaluation of the performances of four adaptive recursive subspace-based algorithms of linear complexity and a comparison with reduced rank methods and the optimal filter. We observed that the presence of ambiguities and the reduction of rank to low values degraded the performance of STAP in suppressing interferences and detecting slow targets. We showed that we can mitigate these problems by using recursive algorithms which can estimate recursively the weights of the clutter rejection filter. With a very low covariance matrix rank and with Doppler ambiguities, the simulation results confirmed the superior performance of the proposed algorithms compared to the PC method even with the optimal filter. This comparative study proved that iterative algorithms could be applied for the reduction of the rank for the STAP because they give similar performances as those given by the methods of rank reduction, but gave a much better performance for low rank values. In addition, they present a very low computational complexity. In fact, it can be viewed From Tables 1 and 2, that the complexity burden is O(MNr) instead of O((MN) 3 ) for the PC processor. That s why these algorithms can be considered as an economical approach in comparison with the other techniques. 7 REFERENCES [1] J. Ward: Space-Time Adaptive Processing for airborne radar, Technical Report 11, Lincoln Laboratory MIT (1994). [2] R. Klemm: Space Time Adaptive Processing Principles and applications, The Institution of Electrical Engineers, London (1998). [3] L.E. Brennan and I.S. Reed: Theory of radar, IEEE transactions on Aerospace and Electronics, Vol. AES-9, n 2, pp (1973). [4] H. Nguyen: Robust Steering Vector Mismatch Techniques for reduced Rank Adaptive Array Signal Processing, PhD Thesis in Electrical and Computer Engineering, Virginia, USA (22). [] J.S. Goldstein and I.S. Reed: Theory Theory of partially adaptive radar, IEEE Transactions on Aerospace and Electronics Systems In Proceedings of the IEEE National Radar Conference, Vol. 33, n 4, pp (1997). [6] S.D. Berger, and B.M. Welsh: Selecting a reduced-rank transformation for STAP, a direct form perspective, IEEE Transactions on Aerospace and Electronics Systems, Vol. 3, n 2, pp (1999). [7] J.R. Guerci et al.,: and adaptive reduced-rank STAP, IEEE Transactions on Aerospace and Electronics Systems, Vol. 36, n 2, pp (2). [8] P. G. Richardson: STAP covariance matrix structure and its impact on clutter plus jamming suppression solutions, Electronics Letters, Vol. 37, n 2, pp (21). [9] S. Dib et al.,: A Reduced Rank STAP with Change of PRF, Eusipco27, Poznan, Poland, pp (27). UbiCC Journal Page 24

8 Special Issue on The International Conference on Information and Communication Systems (ICICS 211) [1] S. Dibet al.,: Two-dimensional signal adaptive processing for airborne radar, ICDIPC 211, Part II, CCIS 189, Springer, pp (211). [11] P. Comon and G.H. Golub:Tracking a few extreme singular values and vectors in signal processing, Proc. Of the IEEE, Vol. 78, n 8, pp (199). [12] B. Yang: Projection Approximation SubspaceTracking, IEEE-T-SP,Vol. 44, n 1, pp (1996). [13] K. Abed-Meraim A. Chkeif and Y. Hua: Fast Orthonormal PAST Algorithm, IEEE Signal Processing Letters, Vol. 7, pp (2). [14] R. Badeau et al.,: Fast approximated power iteration subspace tracking, IEEE Transactions on Signal Processing, Vol. 3, pp (2). [1] A.Valizadeh and M. Karimi: Fast Subspace tracking algorithm based on the constrained projection approximation, EURASIP Journal on Advances in Signal Processing (29). [16] M. Kamenetsky and B. Widrow: A variable leaky LMS adaptive algorithm, in Proceedings of the IEEE conference of the 38thAsilomar on Signals, Systems and Computers, Vol. 1, pp , Pacific Grove, Calif, USA (24). [17] L. Yang: Design and analysis of adaptive noise subspace estimation algorithms, PhD Thesis, Department of Electrical and Computer Engineering National University of Singapore (29). UbiCC Journal Page 2