Code Design to Optimize Radar Detection

Transcription

1 Code Design to Optimize Radar Detection 1 Performance Under Accuracy and Similarity Constraints A. De Maio, S. De Nicola, Y. Huang, S. Zhang, A. Farina Abstract This paper deals with the design of coded waveforms which optimize radar performances in the presence of colored Gaussian disturbance. We focus on the class of linearly coded pulse trains and determine the radar code which maximizes the detection performance under a control on the region of achievable Doppler estimation accuracies, and imposing a similarity constraint with a pre-fixed radar code. This last constraint is tantamount to requiring a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the pre-fixed sequence. The resulting optimization problem is non-convex and quadratic. In order to solve it, we propose a technique (with polynomial computational complexity) based on the relaxation of the original problem into a semidefinite program. Indeed the best code is determined through a rank-one decomposition of an optimal solution of the relaxed problem. At the analysis stage we assess the performance of the new encoding technique both in terms of detection performance, region of achievable Doppler estimation accuracies, and ambiguity function, under two different choices for the similarity code. Index Terms Radar Signal Processing, Non-Convex Quadratic Optimization, Semidefinite Programming Relaxation. A. De Maio (Corresponding Author) and S. De Nicola are with Università degli Studi di Napoli Federico II, Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, Via Claudio 21, I Napoli, Italy. ademaio@unina.it, silviodenicola@gmail.com. Y. Huang and S. Zhang are with Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. ywhuang@se.cuhk.edu.hk, zhang@se.cuhk.edu.hk. A. Farina is with Selex - Sistemi Integrati via Tiburtina Km.12.4, 00131, Roma, Italy; afarina@selex-si.com.

2 2 I. INTRODUCTION The huge advances in high speed signal processing hardware, digital array radar technology, and the requirement of better and better radar performances has promoted, during the last two decades, the development of very sophisticated algorithms for radar waveform design [1]. The synthesis of narrowband waveforms with a specified ambiguity function is considered in [2]. In [3] the theory of wavelets is exploited for the design of radar signals capable of operating in a dense target environment. The use of information theory to devise waveforms for the detection of extended radar targets, exhibiting resonance phenomena, is studied in [4]. Therein two design problems with constraints on signal energy and duration are solved. The concept of matched-illumination for optimized target detection and identification has also received a considerable attention [5], [6], [7, and references therein] due to the advent of adaptive radar transmitters which permit the use of advanced pulse shaping techniques. This theory derives the optimized transmission waveform and the corresponding receiver response which maximizes the Signal to Noise Ratio (SNR). The case of single polarimetric channel is considered in [6] whereas the most general situation of full polarization radar transmission is handled in [7]. The concept is also further investigated in [8] with reference to Gaussian point-target and stationary Gaussian clutter. The results show that the optimum procedure places the signal energy in the noise band having minimum power. Waveform optimization in the presence of colored disturbance with known covariance matrix has been addressed in [9]. Three techniques based on the maximization of the SNR are introduced and analyzed. Two of them also exploit the degrees of freedom provided by a rank deficient disturbance covariance matrix. In [10] a signal design algorithm relying on the maximization of the SNR under a similarity constraint with a given waveform is proposed and assessed. The solution potentially emphasizes the target contribution and de-emphasizes the disturbance. Moreover it also preserves some characteristics of the desired waveform. In [11] a signal subspace framework, which allows the derivation of the optimal radar waveform (in the sense of maximizing the SNR at the output of the detector) for a given scenario, is presented under the Gaussian assumption for the statistics of both the target and the clutter. A quite different signal design approach relies on the modulation of a pulse train parameters (amplitude, phase, and frequency) in order to synthesize waveforms with some specified prop-

3 3 erties. This technique is known as the radar coding and a substantial bulk of work is nowadays available in open literature about this topic. Here we mention Barker, Frank, and Costas codes which lead to waveforms whose ambiguity functions share good resolution properties both in range and Doppler. This list is not exhaustive and a comprehensive treatment can be found in [12], [13]. Nevertheless, it is worth pointing out that the ambiguity function is not the only relevant objective function for code design in operating situations where the disturbance is not white. This might be the case of optimum radar detection in the presence of colored disturbance, where the standard matched filter is no longer optimum and the most powerful radar receiver requires whitening operations. In this paper we attack the problem of code optimization in the presence of colored Gaussian disturbance. At the design stage we focus on the class of coded pulse trains and propose a code selection algorithm which maximizes the detection performance but, at the same time, is capable of controlling both the region of achievable values for the Doppler estimation accuracy and the degree of similarity with a pre-fixed radar code. Actually this last constraint is equivalent to force a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the pre-fixed sequence. The resulting optimization problem belongs to the family of non-convex quadratic programs [14], [15]. In order to solve it we first resort to a relaxation of the original problem into a convex one which belongs to the Semidefinite Program (SDP) class. Then an optimum code is derived exploiting a suitable rank-one decomposition [16] of an optimal solution of the relaxed problem. Remarkably the entire code search algorithm possesses a polynomial computational complexity. At the analysis stage we assess the performance of the new encoding algorithm both in terms of detection performance, region of estimation accuracies that an estimator of the Doppler frequency can theoretically achieve, and ambiguity function. The results show that it is possible to realize a trade off between the three aforementioned performance metrics. In other words detection capabilities can be swaped for desirable properties of the waveform ambiguity function and/or for an enlarged region of achievable Doppler estimation accuracies. The paper is organized as follows. In Section II we present the model for both the transmitted and the received coded signal. In Section III we discuss some relevant guidelines for code design. In Section IV we introduce the code design optimization problem and the algorithm which solves it exploiting SDP relaxation. In Section V we assess the performance of the proposed encoding

4 4 method also in comparison with standard radar codes. Finally, in Section VI, we draw conclusions and outline possible future research tracks. II. SYSTEM MODEL We consider a radar system which transmits a coherent burst of pulses s(t) = a t u(t) exp[j(2πf 0 t + φ)], where a t is the transmit signal amplitude, j = 1, u(t) = N 1 i=0 a(i)p(t it r ), is the signal s complex envelope (see Figure 1), p(t) is the signature of the transmitted pulse, T r is the Pulse Repetition Time (PRT), [a(0), a(1),..., a(n 1)] C N is the radar code (assumed without loss of generality with unit norm), f 0 is the carrier frequency, and φ is a random phase. Moreover the pulse waveform p(t) is of duration T p T r and has unit energy, i.e. Tp 0 p(t) 2 dt = 1, where denotes the modulus of a complex number. The signal backscattered by a target with a two-way time delay τ and received by the radar is r(t) = α r e j2π(f 0+f d )(t τ) u(t τ) + n(t), where α r is the complex echo amplitude (accounting for the transmit amplitude, phase, target reflectivity, and channels propagation effects), f d is the target Doppler frequency, and n(t) is additive disturbance due to clutter and thermal noise. This signal is down-converted to baseband and filtered through a linear system with impulse response h(t) = p ( t), where ( ) denotes complex conjugate. Let the filter output be v(t) = α r e j2πf 0τ N 1 i=0 a(i)e j2πif dt r χ p (t it r τ, f d ) + w(t), where χ p (λ, f) is the pulse waveform ambiguity function [12], i.e. χ p (λ, f) = + p(β)p (β λ)e j2πfβ dβ,

5 5 and w(t) is the down-converted and filtered disturbance component. The signal v(t) is sampled at t k = τ + kt r, k = 0,..., N 1, providing the observables 1 v(t k ) = αa(k)e j2πkf dt r χ p (0, f d ) + w(t k ), k = 0,...,N 1, where α = α r e j2πf0τ. Assuming that the pulse waveform time-bandwidth product and the expected range of target Doppler frequencies are such that the single pulse waveform is insensitive to target Doppler shift 2, namely χ p (0, f d ) χ p (0, 0) = 1, we can rewrite the samples v(t k ) as v(t k ) = αa(k)e j2πkf dt r + w(t k ), k = 0,...,N 1. Moreover, denoting by c = [a(0), a(1),..., a(n 1)] T the N-dimensional column vector containing the code elements 3, p = [1, e j2πf dt r,...,e j2π(n 1)f dt r ] T the temporal steering vector, v = [v(t 0 ), v(t 1 ),...,v(t N 1 )] T, and w = [w(t 0 ), w(t 1 ),..., w(t N 1 )] T, we get the following vectorial model for the backscattered signal v = αc p + w, (1) where denotes the Hadamard element-wise product [17]. III. RELEVANT PERFORMANCE MEASURES FOR CODE DESIGN In this section we introduce some key performance measures to be optimized or controlled during the selection of the radar code. As it will be shown, they permit to formulate the design of the code as a constrained optimization problem. The metrics considered in this paper are: A. Detection Probability. This is one of the most important performance measures which radar engineers attempt to optimize. We just remind that the problem of detecting a target in the presence of observables described by the model (1) can be formulated in terms of the following binary hypotheses test H 0 : v = w (2) H 1 : v = αc p + w 1 We neglect range straddling losses and also assume that there are no target range ambiguities. 2 Notice that this assumption might be restrictive for the cases of very fast moving targets such as fighters and ballistic missiles. 3 ( ) T is the transpose operator.

6 6 Assuming that the disturbance vector w is a zero-mean complex circular Gaussian vector with known positive definite covariance matrix E[ w w ] = M (E[ ] denotes statistical expectation and ( ) conjugate transpose), the Generalized Likelihood Ratio Test (GLRT) detector for (2), which coincides with the optimum test (according to the Neyman-Pearson criterion) if the phase of α is uniformly distributed in [0, 2π[ [18], is given by H 1 v M 1 (c p) 2 > < G, (3) H 0 where G is the detection threshold set according to a desired value of the false alarm probability (P fa ). An analytical expression of the Detection Probability (P d ), for a given value of P fa, is available both for the cases of non-fluctuating and fluctuating target. In the former case (NFT), P d = Q ( 2 α 2 (c p) M 1 (c p), ) 2 ln P fa, (4) while, for the case of Rayleigh fluctuating target (RFT) with E[ α 2 ] = σa 2 ( ), ln P fa P d = exp 1 + σa 2(c p) M 1, (5) (c p) where Q(, ) denotes the Marcum Q function of order 1. These last expressions show that, given P fa, P d depends on the radar code, the disturbance covariance matrix and the temporal steering vector only through the SNR, defined as α 2 (c p) M 1 (c p) NFT SNR = σa 2(c p) M 1 (c p) RFT Moreover P d is an increasing function of SNR and, as a consequence, the maximization of P d can be obtained maximizing the SNR over the radar code. (6) B. Doppler Frequency Estimation Accuracy. The Doppler accuracy is bounded below by Cramer-Rao Bound (CRB) and CRB-like techniques which provide lower bounds for the variances of unbiased estimates. Constraining the CRB is tantamount to controlling the region of achievable Doppler estimation accuracies, referred to in the following as A. We just highlight that a reliable measurement of the Doppler frequency

7 7 is very important in radar signal processing because it is directly related to the target radial velocity useful to speed the track initiation, to improve the track accuracy [19], and to classify the dangerousness of the target. In this subsection we introduce the CRB for the case of known α, the Modified CRB (MCRB) [20] and the Miller-Chang Bound (MCB) [21] for random α, and the Hybrid CRB (HCRB) [22, pp ] for the case of a random zero-mean α. Proposition I. The CRB for known α (Case 1), the MCRB and the MCB for random α (Case 2 and Case 3 respectively), and the HCRB for a random zero-mean α (Case 4) are given by where h = c p, CR (f d ) = Ψ 2 h 1 h M f d f d 1 α 2 Case 1, (7) Ψ = 1 E[ α 2 ] Cases 2 and 4 (8) E [ ] 1 α 2 Case 3 Proof. Cases 1, 2, and 3 follow respectively from [22, p. 928, Eq. 8.37], the definitions [20, Eq. 4], and [23, Eq. 20]. Case 4 is based on the division of the parameter vector θ into a non-random and a random component, i.e. θ = [θ 1, θ T 2 ]T, where θ 1 = f d, θ 2 = [α R, α I ] T, α R and α I are the real and the imaginary part of α. Then, according to the definition [22, pp , Section ], the Fisher Information Matrix (FIM) for the HCRB can be written as J H = J D + J P, (9) where J D and J P are given by [22, pp , Eq and 8.59]. Standard calculus implies that and ) J D = 2diag (E[ α 2 ] h 1 h M, h M 1 h, h M 1 h, (10) f d f d 0 0 T J P =, (11) 0 K

8 8 where diag( ) denotes a diagonal matrix, 0 = [0, 0] T, and K is a 2 2 matrix whose elements are related to the a-priori probability density function of α through [22, pp. 930, Eq. 8.51]. The HCRB for Doppler frequency estimation can be evaluated as [J 1 H ](1, 1) which provides (7). Notice that h f d = T r c p u, with u = [0, j2π,..., j2π(n 1)] T, and (7) can be rewritten as CR (f d ) = Ψ 2T 2 r (c p u) M 1 (c p u). (12) As already stated, forcing an upper bound to CR (f d ), for a specified Ψ value, results in a lower bound on the size of A. Hence, according to this guideline, we focus on the class of radar codes complying with the condition which can be equivalently written as CR (f d ) Ψ 2Tr 2δ, (13) a (c p u) M 1 (c p u) δ a, (14) where the parameter δ a rules the lower bound on the size of A (see Figure 2 for a pictorial description). Otherwise stated, suitably increasing δ a, we ensure that new points fall in the region A, namely new smaller values for the estimation variance can be theoretically reached by estimators of the target Doppler frequency. C. Similarity Constraint. Designing a code which optimizes the detection performance does not provide any kind of control to the shape of the resulting coded waveform. Precisely it can lead to signals with significant modulus variations, poor range resolution, high peak sidelobe levels, and more in general with an undesired ambiguity function behavior. These drawbacks can be partially circumvented imposing a further constraint to the sought radar code. Precisely it is required the solution to be similar to a known code c 0 ( c 0 2 = 1) 4, which shares constant modulus, reasonable range resolution and peak sidelobe level. This is tantamount to imposing that [10] c c 0 2 ǫ, (15) 4 denotes the Euclidean norm of a complex vector.

9 9 where the parameter ǫ 0 rules the size of the similarity region. In other words (15) permits to indirectly control the ambiguity function of the considered coded pulse train: the smaller ǫ the higher the degree of similarity between the ambiguity functions of the designed radar code and of c 0. IV. CODE DESIGN In this section we propose a technique for the selection of the radar code which attempts to maximize the detection performance but at the same time provides a control both on the target Doppler estimation accuracy and on the similarity with a given radar code. To this end we first observe that (c p) M 1 (c p) = c [ M 1 ( p p ) ] c = c Rc (16) and (c p u) M 1 (c p u) = c [ M 1 ( p p ) ( u u ) ] c = c R 1 c, (17) where R = M 1 ( p p ) and R 1 = M 1 ( p p ) ( u u ) are positive semidefinite [22, p. 1352, A.77]. It follows that P d and CR (f d ) are competing with respect to the radar code. In fact their values are ruled by two different quadratic forms, (16) and (17) respectively, with two different matrices R and R 1. Hence, only when they share the same eigenvector corresponding to the maximum eigenvalue (16) and (17) are maximized by the same code. In other words, maximizing P d results in a penalization of CR (f d ) and vice versa. Exploiting (16) and (17) the code design problem can be formulated as a non-convex optimization Quadratic Problem (QP) maxc c Rc s.t. c c = 1 c R 1 c δ a c c 0 2 ǫ which can be equivalently written as maxc c Rc s.t. c c = 1 (QP) c R 1 c δ a Re ( ) c c 0 1 ǫ/2 (18)

10 10 The feasibility of the problem, which not only depends on the parameters δ a and ǫ but also on the pre-fixed code c 0, is discussed in Appendix A. A. Solution to the Optimization Problem In this subsection we show that an optimal solution of (18) can be obtained from an optimal solution of the following Enlarged Quadratic Problem (EQP): maxc c Rc s.t. c c = 1 (EQP) (19) c R 1 c δ a Re ( ) 2 c c ( ) 0 + Im 2 c c 0 = c c 0 c 0 c δ ǫ where δ ǫ = (1 ǫ/2) 2. Since the feasibility region of EQP is larger than that of QP, every optimal solution of EQP, which is feasible for QP, is also an optimal solution for QP [14]. Indeed, assume that c is an optimal solution of EQP and let φ = arg ( c c 0 ). It is easily seen that ce jφ is still an optimal solution of EQP. Moreover, observing that ( ce jφ ) c 0 = c c 0, ce jφ is a feasible solution of QP. Thus ce j arg( c c 0 ) is optimal for both QP and EQP. Now we have to find an optimal solution of EQP and, to this end, we exploit the equivalent matrix formulation max C tr(c R) s.t. tr(c) = 1 (EQP) tr(c R 1 ) δ a (20) tr(cc 0 ) δ ǫ C = cc where C 0 = c 0 c 0 and tr( ) is the trace of a square matrix [17]. Problem (20) can be relaxed into a SemiDefinite Program (SDP) neglecting the rank-one constraint [24]. By doing so we obtain an Enlarged Quadratic Problem Relaxed (EQPR) max C tr(c R) s.t. tr(c) = 1 (EQPR) tr(c R 1 ) δ a (21) tr(cc 0 ) δ ǫ C 0

11 11 where the last constraint means that C is to be Hermitian positive semidefinite [14]. The dual problem of (21), EQPR Dual (EQPRD), is min y1,y 2,y 3 (EQPRD) s.t. y 1 y 2 δ a y 3 δ ǫ y 1 I y 2 R 1 y 3 C 0 R y 2 0, y 3 0 where I denotes the identity matrix. This problem is bounded below and is strictly feasible, so the optimal value is the same as the primal [15] and the complementary conditions are satisfied at the optimal point, due to the strict feasibility of the primal problem (see Appendix A). In the following we prove that a solution of EQP can be obtained from a solution of EQPR C, and from a solution of EQPRD (ȳ 1, ȳ 2, ȳ 3 ). Precisely we show how to obtain a rank-one feasible solution of EQPR that satisfies optimality conditions (complementary conditions) tr [ (ȳ 1 I ȳ 2 R 1 ȳ 3 C 0 R) C ] = 0 (22) [ tr( C R1 ) δ a ]ȳ2 = 0 (23) [ tr( CC0 ) δ ǫ ]ȳ3 = 0 (24) Such rank-one solution is also optimal for EQP. The proof we propose is based on the following Proposition II. Suppose that X is a N N complex Hermitian positive semidefinite matrix of rank R, and A, B are two N N given Hermitian matrices. There is a rank-one decomposition of X (synthetically denoted as D( X, A, B)), such that Proof. See [16]. x r A x r = X = tr( X A) R R r=1 x r x r (25) and x r B x r = Moreover we have to distinguish between four possible cases: 1) tr ( C R1 ) δa > 0 and tr ( CC0 ) δǫ > 0 2) tr ( C R1 ) δa = 0 and tr ( CC0 ) δǫ > 0 3) tr ( C R1 ) δa > 0 and tr ( CC0 ) δǫ = 0 4) tr ( C R1 ) δa = 0 and tr ( CC0 ) δǫ = 0 tr( X B) R (26)

12 12 Case 1: Using the decomposition D( C, I, R 1 ) of Proposition II, we can express C as C = R c r c r r=1 Now we show that there exists a k {1,...,R} such that Rc k is an optimal solution of EQP. Specifically we first prove that ( Rc k )( Rc k ) is a feasible solution of EQPR, and then that ( Rc k )( Rc k ) and (ȳ 1, ȳ 2, ȳ 3 ) comply with the optimality conditions, i.e. ( Rc k )( Rc k ) is a rank-one optimal solution of EQPR and hence Rc k is an optimal solution of EQP. The decomposition D( C, I, R 1 ) implies that every ( Rc r )( Rc r ), r = 1,..., R satisfies the first and the second constraints in EQPR. Moreover there must be a k {1,..., R} such that ( Rc k ) C 0 ( Rc k ) δ ǫ. In fact, if ( Rc r ) C 0 ( Rc r ) < δ ǫ for every r, then R ( Rc r ) C 0 ( Rc r ) < Rδ ǫ r=1 [( R ) ] tr Rcr c r R C 0 < Rδ ǫ r=1 tr( CC 0 ) < δ ǫ which is in contrast with the feasibility of C. This proves that there exists at least one k {1,..., R} for which ( Rc k )( Rc k ) is feasible for EQPR. As to fulfillment of the optimality conditions, tr ( C R1 ) δa > 0 and tr ( CC0 ) δǫ > 0 imply ȳ 2 = 0 and ȳ 3 = 0, namely (23) and (24) are verified for every ( Rc r )( Rc r ), with r = 1,...,R. Moreover (22) can be recast as tr [ (ȳ 1 I R) C ] [ ( R )] = tr (ȳ 1 I R) c r c r = 0 which, since c r c r 0, r = 1,...,R, and (ȳ 2 I R) 0 (from the first constraint of EQPRD), implies r=0 [ ( tr (ȳ 2 I R) Rcr c )] r R = 0 It follows that there exists one k {1,..., R} such that ( Rc k )( Rc k ) is an optimal solution of EQPR, and thus, Rc k is an optimal solution of EQP. Cases 2 and 3: The proof is very similar to Case 1, hence we omit it.

13 13 Case 4: In this case all the constraints of EQPR are active namely tr( C) = 1, tr( C R 1 ) = δ a, and tr( CC 0 ) = δ ǫ. It follows that and tr[ C ( R 1 /δ a I)] = 0 tr[ C (C 0 /δ ǫ I)] = 0 According to D( C, R 1 /δ a I, C 0 /δ ǫ I), we decompose C as and observe that with γ r > 1 such that R r=1 1/γ r = 1. C = R c r c r, r=1 tr ( c r c r ) = 1 γ r, r = 1,..., R, (27) We now prove that each ( γ r c r )( γ r c r ) is an optimal solution of EQPR. Precisely, we first show that ( γ r c r )( γ r c r ) is in the feasible region of EQPR and then we prove that ( γ r c r )( γ r c r ) satisfies the optimality conditions. Equation (27) implies that the first constraint in EQPR is satisfied. From the feasibility of C and from the used decomposition we can also prove that ( γ r c r )( γ r c r ) satisfies the second and the third constraints of EQPR. In fact, with reference to the second constraint we have tr[ C ( R 1 /δ a I)] = 0 tr[ C ( R 1 /δ a I)] R = 0 c r ( R 1 /δ a I)c r = 0 c r ( R 1 /δ a )c r = c r c r c r ( R 1 /δ a )c r = tr ( c r c ) r c r ( R 1 /δ a )c r = 1/γ r γr c r ( R 1 /δ a ) γ r c r = 1 γr c r R 1 γr c r = δ a

14 14 As to the third constraint we observe that tr[ C (C 0 /δ ǫ I)] = 0 tr[ C (C 0 /δ ǫ I)] R = 0 c r (C 0 /δ ǫ I) c r = 0 c r (C 0 /δ ǫ )c r = c r c r c r (C 0 /δ ǫ )c r = tr ( c r c ) r c r (C 0 /δ ǫ )c r = 1/γ r γr c r (C 0 /δ ǫ ) γ r c r = 1 γr c r C 0 γr c r = δ ǫ It remains to prove that ( γ r c r )( γ r c r ) complies with the three optimality conditions. As to the first we note that tr [ (ȳ 1 I ȳ 2 R 1 ȳ 3 C 0 R) C ] = [ ] R tr (ȳ 1 I ȳ 2 R 1 ȳ 3 C 0 R) c r c r = 0 which, since c r c r 0 and (ȳ 1 I ȳ 2 R 1 ȳ 3 C 0 R) 0, implies that r=1 tr [ (ȳ 1 I ȳ 2 R 1 ȳ 3 C 0 R)( γ r c r γr c r ) ] = 0, proving the first optimality condition. The compliance with the second optimality condition can be shown as follows c r ( R 1 /δ a I) c r = 0 ( γ r c r ) ( R 1 /δ a I) γ r c r = 0 [ γr c r ( R 1 /δ a I) γ r c r ]ȳ2 = 0 tr [ ( R 1 /δ a I) ( γr c r c r γr )ȳ2 ] = 0

15 15 As to the third optimality condition we have c r (C 0 /δ ǫ I)c r = 0 γr c r (C 0 /δ ǫ I) γ r c r = 0 [ γr c r (C 0 /δ ǫ I) γ r c r ]ȳ3 = 0 and the proof is completed. tr [ (C 0 /δ ǫ I) ( γr c r γr c r)ȳ3 ] = 0 In conclusion, using the decomposition of Proposition II, we have shown how to construct a rank-one optimal solution of EQPR, which is tantamount to finding an optimal solution of EQP. Summarizing the optimum code can be constructed according to the following procedure. 1) Enlarge the original QP into the EQP (i.e. substitute Re(c c 0 ) 1 ǫ/2 with c c 0 2 δ ǫ ); 2) Relax EQP into EQPR (i.e. relax C = cc into C 0); 3) Solve the SDP problem EQPR finding an optimal solution C; 4) Evaluate tr( C R 1 ) δ a and tr( CC 0 ) δ ǫ : if both are equal to 0 go to 5), else go to 7); 5) Using Proposition II evaluate D( C, R 1 /δ a I, C 0 /δ ǫ I), obtaining C = R r=1 c rc r ; 6) Compute c = γ 1 c 1, with γ 1 = 1/ c 1 2, and go to 9); 7) Using Proposition II evaluate D( C, R 1, I) obtaining C = R r=1 c rc r; 8) Find k such that c k C 0c k δ ǫ /R and compute c = Rc k ; 9) Evaluate the optimal solution of the original QP as ce jφ, with φ = arg( c c 0 ). The computational complexity connected with the implementation of the algorithm is polynomial as both the SDP problem 5 and the decomposition of Proposition II can be performed in polynomial time. V. PERFORMANCE ANALYSIS The present section is aimed at analyzing the performance of the proposed encoding scheme. To this end we assume that the disturbance covariance matrix is exponentially shaped with one-lag correlation coefficient ρ = 0.8, i.e. M(i, j) = ρ i j, 5 An SDP problem can be efficiently solved in polynomial time through interior point methods [14], namely iterative algorithms which terminate once a pre-specified accuracy is reached.

16 16 and fix P fa of the receiver (3) to The analysis is conducted in terms of P d, region of achievable Doppler estimation accuracies, and ambiguity function of the coded pulse train which results exploiting the proposed algorithm, i.e. χ(λ, f) = u(β)u (β λ)e j2πfβ dβ = N 1 N 1 l=0 m=0 ā(l)ā (m)χ p [λ (l m)t r, f], where [ā(0),...,ā(n 1)] is an optimum code. As to the temporal steering vector p, we set the normalized Doppler frequency f d T r = 0 6. The convex optimization MATLAB toolbox SElf-DUal-MInimization (SeDuMi) [25] is exploited for solving the SDP relaxation. The decomposition D(,, ) of the SeDuMi solution is performed using the technique described in [16]. Finally the MATLAB toolbox of [26] is used to plot the ambiguity functions of the coded pulse trains. In the following two distinct situations corresponding to different similarity codes are considered: generalized Barker sequence [12, pp ] and Px code [12, pp ]. A. Generalized Barker Code The generalized Barker codes are polyphase sequences whose autocorrelation function has minimal peak-to-sidelobe ratio excluding the outermost sidelobe. Examples of such sequences were found for all N 45 [27], [28] using numerical optimization techniques. In the simulations of this subsection, we assume N = 7 and set the similarity code equal to the generalized Barker sequence c 0 = [0.3780, , j, j, j, j, j] T. In Figure 3a, we plot P d of the optimum code (according to the proposed criterion) versus α 2 for several values of δ a, δ ǫ = 0.01, and for non-fluctuating target. In the same figure we also represent both the P d of the similarity code as well as the benchmark performance, namely the maximum achievable detection rate (over the radar code), given by ( P d = Q 2 α 2 λ max ( R), ) 2 ln P fa, (28) where λ max ( ) denotes the maximum eigenvalue of the argument. The curves show that increasing δ a we get lower and lower values of P d for a given α 2 value. This was expected since the higher δ a the smaller the feasibility region of the optimization 6 We have also considered other values for the target normalized Doppler frequency. The results, not reported here, confirm the performance behavior showed in the next two subsections.

17 17 problem to be solved for finding the code. Nevertheless the proposed encoding algorithm usually ensures a better detection performance than the original generalized Barker code. In Figure 3b the normalized CRB (CRB n = T 2 r CRB) is plotted versus α 2 for the same values of δ a as in Figure 3a. The best value of CRB n is plotted too, i.e. CRB n = 1 2 α 2 λ max ( R 1 ). (29) The curves highlight that increasing δ a better and better CRB values can be achieved. This is in accordance with the considered criterion, because the higher δ a the larger the size of the region A. Summarizing the joint analysis of Figures 3a-3b shows that a tradeoff can be realized between the detection performance and the estimation accuracy. Moreover there exist codes capable of outperforming the generalized Barker code both in terms of P d and size of A. The effects of the similarity constraint are analyzed in Figure 3c. Therein we set δ a = 10 6 and consider several values of δ ǫ. The plots show that increasing δ ǫ worse and worse P d values are obtained; this behavior can be explained observing that the smaller δ ǫ the larger the size of the similarity region. However this detection loss is compensated for an improvement of the coded pulse train ambiguity function. This is shown in Figures 4b-4e where such function is plotted assuming rectangular pulses, T r = 5T p and the same values of δ a and δ ǫ as in Figure 3c. Moreover, for comparison purposes, the ambiguity function of c 0 is plotted too (Figure 4a). The plots highlight that the closer δ ǫ to 1 the higher the degree of similarity between the ambiguity functions of the devised and the pre-fixed codes. This is due to the fact that increasing δ ǫ is tantamount to reducing the size of the similarity region. In other words we force the devised code to be similar and similar to the pre-fixed one and, as a consequence, we get similar and similar ambiguity functions. B. Px Code A Px code is a modified version of a Frank code [29]. It shares the same aperiodic peak sidelobe as the Frank code but a lower integrated sidelobe level. The mathematical definition of the code elements can be also found in [12]. In the numerical examples presented here, we assume N = 9 and set the similarity code equal to the Px sequence c 0 = [ j, j, , , , , j, j, ] T.

18 18 In Figure 5a, we plot P d of the optimum code versus α 2 for several values of δ a, δ ǫ = 0.01, and for non-fluctuating target. For comparison purposes we also represent both the P d of the similarity code as well as the benchmark performance. The curves confirm the behavior of Figure 3a. Specifically they show that increasing δ a, worse and worse values of P d are obtained for a given α 2. In Figure 5b the CRB n is plotted versus α 2 in correspondence of the same values of δ a and δ ǫ considered in Figure 5a. As expected, increasing δ a, lower and lower values of the CRB can be achieved. Otherwise stated, while an increase of δ a is detrimental for P d, it leads to improvements in the actual CRB. Moreover the new encoding technique leads to some codes which behave better that the chosen similarity sequence both in terms of detection capabilities and in terms of achievable region of Doppler estimation accuracies. In Figure 5c the impact of the similarity constraint on the detection performance is studied. Therein we plot P d versus α 2 for δ a = 10 6 and several values of δ ǫ. The performance behavior is similar to that of Figure 3c, namely P d is a decreasing function of δ ǫ. However the detection loss is compensated for a more regular behavior of the coded pulse train ambiguity function. This can be observed in Figures 6b-6e, where the aforementioned function is plotted for the considered values of δ ǫ along with the ambiguity of the pulse train encoded with the similarity sequence (Figure 6a). VI. CONCLUSION In this paper we have considered the design of coded waveforms in the presence of colored Gaussian disturbance. We have devised and assessed an algorithm which attempts to maximize the detection performance under a control both on the region of achievable values for the Doppler estimation accuracy, and on the similarity with a given radar code. The proposed technique, whose implementation requires a polynomial computational complexity, is based on the SDP relaxation of non-convex quadratic problems and on a suitable rank-one decomposition of a positive semidefinite Hermitian matrix. The analysis of the algorithm has been conducted in terms of the following performance metrics detection performance; region of achievable Doppler estimation accuracies; ambiguity function of the coded pulse waveform;

19 19 under two setups for the similarity code. Hence the trade off between the three considered performance measures has been thoroughly studied and commented. Possible future research tracks might concern the possibility to make the algorithm adaptive with respect to the disturbance covariance matrix, namely to devise techniques which jointly estimate the code and the covariance. Moreover it should be investigated the introduction in the code design optimization problem of knowledge-based constraints, ruled by the a-priori information that the radar has about the surrounding environment. It might also be of interest to consider the cases of non-gaussian clutter and of a system equipped with multiple transmitters and/or receivers. Finally the extension of the framework to the target classification problem might represent another area for future researches. VII. APPENDIX A: FEASIBILITY OF QP, EQPR, AND EQPRD A. Feasibility of QP The feasibility of QP depends on the three parameters δ a, δ ǫ and c 0. Accuracy parameter δ a. This parameter rules the constraint c R 1 c δ a. Since c has unitary norm, c R 1 c ranges between the minimum (λ min ( R 1 )) (which is zero in this case) and the maximum (λ max ( R 1 )) eigenvalue of R 1. As a consequence a necessary condition on δ a in order to ensure feasibility is δ a [0, λ max ]. Similarity parameter δ ǫ. This parameter rules the similarity between the sought code and the pre-fixed code, through the constraint c c 0 2 ǫ. If ǫ < 0 then QP is infeasible. Moreover 0 ǫ 2 2Re(c c 0 ) 2, where the last inequality stems from the observation that we choose the phase of c such that Re(c c 0 ) = c c 0 0. It follows that a necessary condition on δ ǫ = (1 ǫ/2) 2 in order to ensure feasibility is δ ǫ [0, 1]. Pre-fixed code c 0. Choosing 0 δ a λ max and 0 δ ǫ 1 is not sufficient in order to ensure the feasibility of QP. A possible way to construct a strict feasible QP problem is to reduce the range of δ a according to value of c 0. In fact, denoting by δ max = c 0 R 1c 0, the problem QP is strictly feasible, assuming that 0 δ a < δ max and 0 δ ǫ < 1. Moreover a strict feasible solution of QP is c 0 itself.

20 20 B. Feasibility of EQPR The primal problem is strictly feasible due to the strict feasibility of QP. Indeed, assume that c s is a strictly feasible solution of QP, i.e. c sc s = 1, c s R 1 c s > δ a and c sc 0 2 Re 2 (c sc 0 ) > δ ǫ. Then there are u 1, u 2,..., u N 1 such that U = [c s, u 1, u 2,..., u N 1 ] is a unitary matrix [17], and for a sufficient small η > 0 the matrix C s (1 η) Ue 1 e 1 U + η ( ) N 1 U I e 1 e 1 U, (30) with e 1 = [1, 0,..., 0] T, is a strictly feasible solution of the SDP problem EQPR. In fact tr[c s ] = 1 (31) tr[c s R 1 ] = (1 η)c s R 1c s + tr[c s R 1 ] = (1 η)c s C 0c s + η N 1 N 1 n=1 η N 1 N 1 n=1 u n R 1 u n (32) u n C 0 u n (33) which highlight that, if η is suitable choosen, then C s is a strictly feasible solution, i.e. tr[c s ] = 1, tr[c s R 1 ] > δ a and tr[c s C 0 ] > δ ǫ. C. Feasibility of EQPRD The dual problem is strictly feasible because, for every finite value of y 2 and y 3, say ŷ 2, ŷ 3, we can choose a ŷ 1, such that ŷ 1 I ŷ 2 R 1 ŷ 3 C 0 R. It follows that (ŷ 1, ŷ 2, ŷ 3 ) is a strictly feasible solution of EQPRD [15].

21 21 REFERENCES [1] A. Farina, Waveform Diversity: Past, Present, and Future, Third International Waveform Diversity & Design Conference, Plenary Talk, Pisa, June [2] C. H. Wilcox, The Synthesis Problem for Radar Ambiguity Functions, Math Res. Center, U.S. Army, Univ. Wisconsin, Madison, WI, MRC Tech. Summary Rep. 157, April 1960; reprinted in R. E. Blahut, W. M. Miller, and C. H. Wilcox, Radar and Sonar, Part 1, New York, Springer-Verlang, [3] H. Naparst, Dense Target Signal Processing, IEEE Transactions on Information Theory, Vol. 37, n. 2, pp , March [4] M. R. Bell, Information Theory and Radar Waveform Design, IEEE Transactions on Information Theory, Vol. 39, n. 5, pp , September [5] A. Farina and F. A. Studer, Detection with High Resolution Radar: Great Promise, Big Challenge, Microwave Journal, pp , May [6] S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, Optimum Transmit-Receiver Design in the Presence of Signal- Dependent Interference and Channel Noise, IEEE Transactions on Information Theory, Vol. 46, n. 2, pp , March [7] D. A. Garren, A. C. Odom, M. K. Osborn, J. S. Goldstein, S. U. Pillai, and J. R. Guerci, Full-Polarization Matched- Illumination for Target Detection and Identification, IEEE Transactions on Aerospace and Electronic Systems, Vol. 38, n. 3, pp , July [8] S. Kay, Optimal Signal Design for Detection of Point Targets in Stationary Gaussian Clutter/Reverberation, IEEE Journal on Selected Topics in Signal Processing, Vol. 1, n. 1, pp , June [9] J. S. Bergin, P. M. Techau, J. E. Don Carlos, and J. R. Guerci, Radar Waveform Optimization for Colored Noise Mitigation, 2005 IEEE International Radar Conference, pp , Alexandria, VA, 9-12 May [10] J. Li, J. R. Guerci, and L. Xu, Signal Waveform s Optimal-under-Restriction Design for Active Sensing, IEEE Signal Processing Letters, Vol. 13, n. 9, pp , September [11] B. Friedlander, A Subspace Framework for Adaptive Radar Waveform Design, Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers 2005, Pacific Grove, CA, pp , October 28 - November 1, [12] N. Levanon and E. Mozeson, Radar Signals, John Wiley & Sons, [13] M. Bell, Information Theory of Radar and Sonar Waveforms, Wiley Encyclopedia of Electrical and Electronic Engineering, J. G. Webster, Ed., Vol. 10, pp , New York, NY: Wiley-Interscience, [14] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, [15] A. Nemirovski, Lectures on Modern Convex Optimization, Class Notes, Georgia Institute of Technology, Fall [16] Y. Huang and S. Zhang, Complex Matrix Decomposition and Quadratic Programming, to appear in Mathematics of Operations Research, [17] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, [18] J. S. Goldstein, I. S. Reed, and P. A. Zulch, Multistage Partially Adaptive STAP CFAR Detection Algorithm, IEEE Transactions on Aerospace and Electronic Systems, Vol. 35, n. 2, pp , April [19] A. Farina and S. Pardini, A Track-While-Scan Algorithm Using Radial Velocity in a Clutter Environment, IEEE Transactions on Aerospace and Electronic Systems, Vol. 14, n. 5, pp , September [20] A. N. D Andrea, U. Mengali, and R. Reggiannini, The Modified Cramer-Rao Bound and Its Application to Synchronization Problems, IEEE Transactions on Communications, Vol. 42, n. 3, pp , February-April 1994.

22 22 [21] R. W. Miller, and C. B. Chang, A Modified Cramér-Rao Bound and its Applications, IEEE Transactions on Information Theory, Vol. 24, n. 3, pp , May [22] H. L. Van Trees, Optimum Array Processing. Part IV of Detection, Estimation and Modulation Theory, John Wiley & Sons, [23] F. Gini and R. Reggiannini, On the Use of Cramer-Rao-like Bounds in the Presence of Random Nuisance Parameters, IEEE Transactions on Communications, Vol. 48, n. 12, pp , December [24] A. d Aspermont and S. Boyd, Relaxations and Randomized Methods for Nonconvex QCQPs, EE392o Class Notes, Stanford University, Autumn [25] J. F. Sturm, Using SeDuMi 1.02, a MATLAB Toolbox for Optimization over Symmetric Cones, Optimization Methods and Software, Vol , pp , August [26] E. Mozeson and N. Levanon, MATLAB Code for Plotting Ambiguity Functions, IEEE Transactions on Aerospace and Electronic Systems, Vol. 38, n. 3, pp , July [27] L. Bomer and M. Antweiler, Polyphase Barker Sequences, Electronics Letters, Vol. 25, n. 23, pp , November [28] M. Friese, Polyphase Barker Sequences up to Length 36, IEEE Transactions on Information Theory, Vol. 42, n. 4, pp , July [29] P. B. Rapajic and R. A. Kennedy, Merit Factor Based Comparison of New Polyphase Sequences, IEEE Communication Letters, Vol. 2, n. 10, pp , October 1998.

23 23 Figure 1: Coded pulse train u(t) for N = 5 and p(t) with rectangular shape. Figure 2: Lower bound to the size of the region A for two different values of δ a (δ a < δ a).

24 increasing δ a 0.6 P d α 2 [db] Figure 3a: P d versus α 2 for P fa = 10 6, N = 7, δ ǫ = 0.01, non-fluctuating target, and several values of δ a {10 6, , , }. Generalized Barker code (dashed curve). Code which maximizes the SNR for a given δ a (solid curve). Benchmark code (dotted-marked curve). Notice that the curve for δ a = 10 6 perfectly overlaps with the benchmark P d increasing δ a 40 CRB n [db] α 2 [db] Figure 3b: CRB n versus α 2 for N = 7, δ ǫ = 0.01 and several values of δ a {10 6, , , }. Generalized Barker code (dashed curve). Code which maximizes the SNR for a given δ a (solid curve). Benchmark code (dotted-marked curve). Notice that the curve for δ a = perfectly overlaps with the benchmark CRB n.

25 P d 0.5 increasing δ ε α 2 [db] Figure 3c: P d versus α 2 for P fa = 10 6, N = 7, δ a = 10 6, non-fluctuating target, and several values of δ ǫ {0.01, , , }. Generalized Barker code (dashed curve). Code which maximizes the SNR for a given δ ǫ (solid curve). Benchmark code (dotted-marked curve). Notice that the curve for δ ǫ = 0.01 perfectly overlaps with the benchmark P d. Figure 4a: Ambiguity function modulus of the generalized Barker code c 0 = [0.3780, , j, j, j, j, j] T.

26 26 Figures 4b-4e: Ambiguity function modulus of code which maximizes the SNR for N = 7, δ a = 10 6, c 0 generalized Barker code, and several values of δ ǫ : (b) δ ǫ = , (c) δ ǫ = , (d) δ ǫ = , (e) δ ǫ = 0.01.

27 increasing δ a 0.6 P d α 2 [db] Figure 5a: P d versus α 2 for P fa = 10 6, N = 9, δ ǫ = 0.01, non-fluctuating target, and several values of δ a {10 6, 10940, 13245, 14000}. Px code (dashed curve). Code which maximizes the SNR for a given δ a (solid curve). Benchmark code (dotted-marked curve). Notice that the curve for δ a = 10 6 perfectly overlaps with the benchmark P d increasing δ a 40 CRB n [db] α 2 [db] Figure 5b: CRB n versus α 2 for N = 9, δ ǫ = 0.01 and several values of δ a {10 6, 10940, 13245, 14000}. Px code (dashed curve). Code which maximizes the SNR for a given δ a (solid curve). Benchmark code (dotted-marked curve). Notice that the curve for δ a = perfectly overlaps with the benchmark CRB n.

28 increasing δ ε P d α 2 [db] Figure 5c: P d versus α 2 for P fa = 10 6, N = 9, δ a = 10 6, non-fluctuating target, and several values of δ ǫ {0.01, , , }. Px code (dashed curve). Code which maximizes the SNR for a given δ ǫ (solid curve). Benchmark code (dotted-marked curve). Notice that the curve for δ ǫ = 0.01 perfectly overlaps with the benchmark P d. Figure 6a: Ambiguity function modulus of the Px code c 0 = [ j, j, , ,0.3333, , j, j, ] T.

29 29 Figures 6b-6e: Ambiguity function modulus of code which maximizes the SNR for N = 9, δ a = 10 6, c 0 Px code, and several values of δ ǫ : (b) δ ǫ = , (c) δ ǫ = , (d) δ ǫ = , (e) δ ǫ = 0.01.