Prior Knowledge Based Transmit Nulling Method for MIMO Radar
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1 Prior Knowledge Based Transmit Nulling ethod for IO Radar ao Zheng, Bo Jiu, ongwei iu, Yuan iu National aboratory of Radar Signal Processing Xidian University Xi an, China Xu Wang Xi an Eleconic Engineering Research Institute Xi an, China Absact In this paper, a multiple-input-multiple-output (IO) radar ansmit nulling method is proposed in the presence of fast moving interference. The IO radar ansmit waveform design is divided into two steps: ansmit waveform covariance mai optimization and ansmit waveform synthesis to approimate the optimized covariance mai. In the first step, the prior knowledge of interference is utilized to optimize covariance mai and the derivative consaint is inoduced to widen nulls. In the second step, most eisting methods, e.g. cyclic algorithm (CA) and sequential iterative algorithm (SIA), synthesize the waveform with constant modulus under leastsquares criterion. As the value at the null even can be neglected in the calculation of least-squares, these methods cannot form deep enough nulls at the interference direction. Aiming at this problem, a block coordinate descent (BCD) method is proposed to optimize the initial waveform synthesized by CA. Besides, the optimized waveform is still constant modulus. Numerical results show the efficiency of the proposed method. Keywords IO radar; ansmit nulling; interference; cyclic algorithm; block coordinate descent; constant modulus waveform I. INTRODUCTION Each ansmit antenna of multiple-input-multiple-output (IO) radar can freely choose the ansmit waveform, which provides IO radar with the advantage of waveform diversity. In general, IO radar system can be classified into two categories according to the configuration of antennas: IO radar with widely separated antennas and IO radar with colocated antennas. In this paper, we focus on the colocated IO radar, in which ansmitters and receivers are close so that all ansmitters see the same target radar cross section and this provides more fleibility in the beampattern design []. The waveform design for IO radar beampattern formulation has recently received the considerable attention in the literature [ ]. In summary, there are two ways to obtain the waveform for desired ansmit beampattern. Since IO radar ansmit beampattern is characterized by the covariance mai, the first one considers the two-step process by focusing on the optimization of the waveform covariance mai and waveform synthesis to approimate the optimized covariance mai under the practical consaints. For the optimization of covariance mai, a semi-definite quadratic programming algorithm was presented in [], where both the design of the beampattern matching and the minimum sidelobe beampattern were taken into account to optimize the waveform covariance mai. In [3], the consained optimization problem about waveform covariance mai was converted into an unconsained optimization problem. For waveform synthesis problem, the waveform is usually required to satisfy the constant modulus consaint (CC) so that the radio frequency amplifiers of the antennas have maimum power efficiency. In [4], the cyclic algorithm (CA) was inoduced to synthesize the waveform with CC from an infinite alphabet. In [5], an algorithm that generates binary-phase shift keying waveform to approimate the given beampattern was proposed. owever, this algorithm ust can apply to the symmeic beampattern or the shift of symmeic beampattern. In [6], a sequential iterative algorithm (SIA) to synthesize the ansmit waveform enforcing the CC was proposed. The second way is to design the waveform directly without covariance mai optimization. In [7], Benamin Friedlander obtained ansmit waveform by designing rank- beamformers. In [8], two methods of designing weight maice for a set of finite-alphabet waveform were proposed. In [9], discrete Fourier ansform coefficients were chosen to represent the region of interest and then ansmit waveform was obtained by a combination of a set of orthogonal waveform. In [0], in the presence of signal-dependent interference, both the ansmit waveform and receive filter are designed to maimize the signal to interference plus noise ratio. In [], the oint design problem of the space-time ansmit waveform and the spacetime receive filter for a moving target is considered. Because of the advantage of waveform diversity, compared with the single-input-multiple-output (SIO) radar, IO radar can realize simultaneous multibeam and ansmit nulling in one pulse by designing waveform. Therefore, in this paper, we investigate the IO radar ansmit nulling with simultaneous multibeam in the presence of fast moving interference. A conve problem is obtained to design the ansmit waveform covariance mai with low sidelobe and wide nulls. For waveform synthesis, because eisting methods are unable to form deep enough nulls at the interference direction, we propose a block coordinate descent (BCD) method to optimize the initial ansmit waveform synthesized by CA. After the BCD process, the optimized waveform still This work is partially supported by the National Natural Science Foundation of China (66735), the National Science Fund for Distinguished Young Scholars (65505), the Fund for Foreign Scholars in University Research and Teaching Programs (the Proect) (No. B8039).
2 satisfies the CC. Besides, because we regard every row of waveform mai as a block of variables in BCD, the block number of the optimization problem is small and the optimization process is fast. The rest sections of this paper is organized as follows. In section II, the signal model and the null consaint are inoduced and then a conve model is proposed to obtain the optimal waveform covariance mai. In section III, a BCD method is proposed to optimize the initial ansmit waveform from CA. In section IV, some simulation results are provided to demonsate the superiorities of the designed waveform. Finally, in section V, we conclude the paper. Notation: Boldface upper case letters denote maices, boldface lower case letters denote vectors and italics denote scalars. Re denotes the real part. angle denotes the phase of a comple number. The superscripts T, and denote anspose, comple conugate and conugate anspose, respectively. denotes the adamard product. denotes the ace of a mai. diag ( ) is a diagonal mai formed with as its principal diagonal. denotes the Euclidean norm of a vector. i is the imaginary unit. II. WAVEFOR COVARIANCE ATRIX OPTIIZATION Consider a colocated IO radar consisting of ansmit antennas. Each antenna radiates a continuous phase encoding signal with signal samples. The ansmit waveform mai can be written as X = m = m m m is a row vector, which represents the waveform ansmitted by the m-th antenna. Then the waveform covariance mai can be written as where, (, ), R = XX The received signal power at azimuth angle θ can be represented as = R P θ a θ a θ (3) where the ansmit steering vector is given by i π z sin θ λ i π z sin θ λ i π z sin θ λ T a θ = e, e,, e z m represents the location of antenna and λ is the ansmission wavelength. The interference s azimuth angle θ k and the number of interference K are the prior knowledge. For the passive interference e.g. aluminum chaff, these prior knowledge can be obtained by the parameter estimation method in [5] and [6]. To form nulls at interference s direction, the waveform covariance mai should satisfy the following consaint where a( θ ) a( θ ) a( θ ) V RV ρ (4) V =,,, K represents interference subspace. ρ is a parameter set in advance, which decides the orthogonality between interference subspace and ansmit waveform. ρ can be roughly regarded as the tolerable maimal power that radiates the interference. Consaint (4) only can form sharp nulls on the beampattern, so the fast moving interference may escape the null in a short time. For this reason, the derivative consaint in [] is used to etend the interference subspace and widen the nulls where U RU ρ (5) U B V B V 0 p =,, r r m r r r m= ( ) B = z diag z, z,, z, r =,, p 0 p represents the order of the derivative consaint. B is defined as an identity mai, so when the zero-order derivative consaint is used, the derivative consaint is same as (4). Finally, to obtain the beampattern with the desired nulls and the low sidelobe, the following problem model is used to optimize the waveform covariance mai min ma R θs Ωside a a ( θs) Ra( θs) ( θ ) Ra( θ ), s t i i Pi i = I ( U RU) ρ Rmm = c, m =,, R 0,, where, Ω side represents the sidelobe region, P i is the desired power at the interesting azimuth angle θ i. R mm is the m-th diagonal elements of the mai R. c is the total ansmit power. Above problem is conve [3], the optimal R can be obtained easily by using the conve optimization toolbo CVX in ATAB. III. OPTIIZE WAVEFOR VIA BCD According to the optimal R, we first use CA to obtain the initial waveform mai X. The details of CA are referred to [4]. Because CA approimates the optimal R based on the (6)
3 least-squares criterion, the small residuals are put on very small weight [3], that leads the designed waveform mai by CA can t form deep enough nulls. In order to form the desired nulls at the interference s direction, a BCD method is proposed to optimize the initial waveform mai X row by row to fulfill consaint (5). A proof of the convergence for the BCD method can be found in [4]. For the waveform mai, consaint (5) can be rewritten as X is divided into blocks as : U XX U ρ (7) X = Suppose that the -th block variable is optimized at present, the following problem model can be obtained where min ( U XX ˆˆ U) s t l =, l =,, Xˆ = is a vector which represents the phase change of the -th block variable so that ˆX still satisfies the CC. et u mq represent the (m-th, q-th) element of U. The obective function of (8) can be written as ( U XX ˆ ˆ U) = ) q= m= u mq m + u + ( )( ) q uqunq ( ) n n= n uqumq m ( ) m= + + umqunq m n m= n= n n m (8) (9) where K is the number of interference and p is the order of the derivative consaint. It is obvious that the first item and the last item in (9) are constant and because only changes the phase of, the second item in (9) is also constant. Ignoring the constant items in (9), problem (8) can be written as and because min ) q= m= Re st l =, l =,, (0) can be rewritten as ( uqumq m ( ) ) ( ) = ( ) m m ) min Re u u q= m= st l =, l =,, q mq m (0) et u represent the -th row of U, the following formula can be obtained η is defined as So, is equivalent to ) uqumqm = uu X u q= m= U X η= u u ( η ) min Re s t l =, l =,, The closed-form solution of is ( angle ) = ep i η (3) where i is the imaginary unit. We replace by and conduct the similar procedure to optimize the remaining blocks in X. The above process is repeated until the waveform mai fulfills (7). Finally, the total iteration procedure of the BCD method is summarized in Algorithm. Each iteration of the proposed algorithm need solve problems, each of which corresponds to the computational compleity of O( ).
4 Algorithm : CA+BCD Algorithm Input: Optimal covariance mai R, the number of antenna,the length of signal sample and the parameter ρ ; Output: The waveform mai X satisfies (7); : According to R, use CA to obtain the initial waveform X ; : Set = ; 3: Select the -th row of U and X, obtain through (3); 4: Use to update the -th row of X ; 5: Set = + ; 6: If, return to step 3; Otherwise turn to net step; 7: If ( U XX U ) ρ, output the waveform mai X ; Otherwise return to step. IV. NUERICA RESUTS In this section, we evaluate the performance of the proposed algorithm considering a uniform linear array (UA) of 6 ansmit antennas (i.e., = 6 ) with a halfwavelength interelement spacing. We focus on a scenario where 35, 0 and 45 are regarded as interesting azimuth angle (i.e., I = 3) and the interference s azimuth angles are 60 and 0 (i.e., K = ). And we assume that the sidelobe region Ω side is as follows: 90, 45 5, 0 0,35 55,90 In all simulation eperiments, P i ( i =,, 3 ), c and ρ are set as 0.9 3, and respectively and the number of signal simple = 00. Besides, the stop criterion of CA and SIA refer to [4] and [6] respectively and the stop threshold of CA and SIA are both set to 0-5. In the first simulation eperiment, only the zero order derivative consaint is considered, that is U = V. The optimal R is obtained by solving model (6). We compare the proposed algorithm with CA and SIA. The waveform synthesized by CA is used as the initial waveform mai for BCD. Fig. shows the ansmit beampattern of the optimal R and the ansmit beampattern of the waveforms synthesized by CA, SIA and CA+BCD respectively. The ansmit beampattern is obtained by and (3). In the second simulation eperiment, the first order 0 derivative consaint is considered, that is U = B V, B V. Fig. shows the ansmit beampattern of the optimal R and the ansmit beampattern of the waveforms synthesized by CA, SIA and CA+BCD in the case of considering first order derivative consaint. As can be seen in Fig. and Fig., because both CA and SIA approimate the optimal R based on the least-squares criterion, they can t form deep enough nulls at interference s azimuth angles. From Fig. and Fig., we also can find that after optimizing by BCD, the ansmit beampattern has desired nulls at interference s azimuth angles and in Fig., because of the first order derivative consaint, the nulls become wider compared with Fig.. In the third simulation eperiment, we show the behavior of the value of ( U XX U ) (hereinafter referred to as null consaint value) versus iteration number in Fig. 3(a) and versus computational time in Fig. 3(b) in the case of considering the first order derivative consaint. To compare these three algorithms more eactly, we still run CA and SIA when they reach the stop threshold 0-5 and the maimum iteration number is 700 for both CA and SIA. We choose the waveform mai which CA output when it reaches the stop threshold 0-5 as the initial waveform for BCD. In Fig. 3(a), for both CA and SIA, the null consaint value decreases fast in early and then is stuck at a local minimum. Additionally, as we can see, the null consaint value is pushed to reach the value of optimal R because of using BCD. From Fig. 3(b), we can find that because the waveform mai is divided into blocks, the BCD process is fast and only takes up little computational time of the whole process. Fig.. Transmit beampattern of CA, SIA, CA+BCD and optimal R considering the zero order derivative consaint. Fig.. Transmit beampattern of CA, SIA, CA+BCD and optimal R considering the first order derivative consaint.
5 REFERENCES null consaint value (db) null consaint value (db) Fig. 3(a). The value of the null consaint versus iteration number. Fig. 3(b). The value of the null consaint versus computational time. V. CONCUSION In this paper, we propose a ansmit nulling method for IO radar. To realize ansmit nulling, we first set up a conve model to obtain the desired covariance mai. Then, aiming at the shortcoming of eisting waveform synthesis methods, we propose a BCD method to optimize ansmit waveform to form desired nulls at interference direction and the waveform optimized by BCD fulfills the constant modulus consaint. The simulation eperiments show that the nulls formed by the BCD are closer to the optimal nulls than the nulls formed by CA and SIA. In addition, due to dividing the waveform mai into blocks, the process of producing nulls is very fast. It should be noted that while forming the nulls, the BCD process inevitably affects the mainlobe and the sidelobe of the initial waveform synthesized by CA. In our eperiments, it is shown that this method has very little effect on the mainlobe and the sidelobe. The theoretical analysis about the effect of BCD on the mainlobe and the sidelobe needs to be further studied in the future. [] J. i and P. Stoica, IO radar with colocated antennas, IEEE Signal Process. ag., vol. 4, no. 5, pp. 06 4, Sep [] P. Stoica, J. i, and Y. Xie, On probing signal design for IO radar, IEEE Trans. Signal Process., vol. 55, no. 8, pp , Aug [3] S. Ahmed, J. S. Thompson, Y. R. Petillot, and B. ulgrew, Unconsained synthesis of covariance mai for IO radar ansmit beampattern, IEEE Trans. Signal Process., vol. 59, no. 8, pp , Aug. 0. [4] P. Stoica, J. i, and X. Zhu, Waveform synthesis for diversity-based ansmit beampattern design, IEEE Trans. Signal Process., vol. 56, no. 6, pp , Jun [5] S. Ahmed, J. S. Thompson, Y. R. Petillot, and B. ulgrew, Finite alphabet constant-envelope waveform design for IO radar, IEEE Trans. Signal Process., vol. 59, no., pp , Aug. 0. [6] X Yu, G Cui, T Zhang and Kong, Consained ansmit beampattern design for colocated IO radar, Signal Process., in press. [7] B. Friedlander, On ansmit beamforming for IO radar, IEEE Trans. Aerosp. Elecon. Syst., vol. 48, no. 4, pp , Oct. 0. [8] S. Ahmed and.-s. Alouini, IO radar ansmit beampattern design without synthesising the covariance mai, IEEE Trans. Signal Process., vol. 6, no. 9, pp , ay 04. [9] J. ipor, S. Ahmed, and.-s. Alouini, Fourier-based ansmit beampattern design using IO radar, IEEE Trans. Signal Process., vol. 6, no. 9, pp. 6 35, ay 04. [0] G. Cui,. i, and. Rangaswamy, IO radar waveform design with constant modulus and similarity consaints, IEEE Trans. Signal Process., vol. 6, no., pp , Oct. 04. [] X. Yu, G. Cui,. Kong, and V. Carotenuto, Space-time ansmit code and receive filter design for colocated IO radar, in Proc. IEEE Radar Conf., ay 06, pp. 6. [] A. B. Gershman, U. Nickel, and J. F. Böhme, Adaptive beamforming algorithms with robustness against ammer motion, IEEE Trans. Signal Processing, vol. 45, pp , July 997. [3] Boyd S, and Vandenberghe, Conve Optimization, Cambridge Univ. Press, Cambridge, UK, 004. [4] P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization, Journal of Optimization Theory and Applications, vol. 09, pp , 00. [5]. Xu, J. i, and P. Stoica, Target detection and parameter estimation for IO radar systems, IEEE Trans. Aeros., Eleco. Syst., vol. 44, no. 3, pp , July 008. [6]. Xu and J. i, Iterative generalized-likelihood ratio test for IO radar, IEEE Trans. Signal Process., vol. 55, no. 6, pp , Jun. 007.
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