Information Theoretic Radar Waveform Design for Multiple Targets

Transcription

1 1 Information Theoretic Radar Waveform Design for Mutipe Targets Amir Leshem and Arye Nehorai Abstract In this paper we use information theoretic approach to design radar waveforms suitabe for simutaneousy estimating and tracking parameters of mutipe targets. Our approach generaizes the information theoretic water-fiing approach of Be. The paper has three main contributions: A new information theoretic design criteria for singe transmit waveform with a receiving array using a weighted inear sum of the mutua informations between targets radar signatures and the corresponding received beams (given the transmitted waveforms, we proivde a famiy of design criteria that weight the various targets according to priorities. Then we generaize the information theoretic design criteria for designing mutipe waveforms under joint power constraint when beamforming is used both at transmitter and receiver. Finay we provide a highy efficient optimization agorithm for optimizing the transmitted waveforms both for singe target and mutipe targets. We show that the optimization probem in both cases can be decouped into a parae set of ow dimensiona search probems at each frequency, with dimension defined by the number of targets, instead of the number of frequency bands used. The power constraint is forced through the optimization of a singe Lagrange mutipier for the dua probem. We end with comments on the generaization of the proposed technique for other design criteria, e.g., for the ineary weighted MMSE design criterion. I. INTRODUCTION The probem of radar waveform design is of fundamenta importance in designing state of the art radar systems. The possibiity to vary the transmitted signa on a puse by puse basis opens the door to great enhancement in estimation and detection capabiity as we as improved robustness to jamming. Furthermore modern radars have the possibiity to detect and track mutipe targets simutaneousy. Therefore designing the transmitted puses for estimating mutipe targets becomes a critica issue in radar waveform design. Most of existing waveform design iterature deas with design for a singe target. One of the important toos in such design is the use of information theoretic techniques. The pioneering work of Woodward and Davies [1], [2], [3], [4] were the first to suggest that information theoretic toos are important for deveopment of radar receivers. Grettenberg [5] proposed using information theoretic criteria for optimizing radar sensitivity to estimated parameters by formuating the estimation probem as a mutipe hypothesis testing at a given parameter space resoution. He proposed to maximize the minimum diveregence between any two hypotheses tested. Amir Leshem is with Schoo of Engineering, Bar-Ian university, Ramat- Gan, 52900, Israe. Arye Nehorai is with Department of EE Washington University, St. Louis, MO. This work was supported by the department of defense through the Air Force Office of Scientific Research MURI grant FA and AFOSAR Grant FS This reduced to minimizing a argest vaue of the ambiguity function at a given distance from the origin. Schweppe and Gray [6] proposed design criteria for radar signas under both average and peak power constrains. In [7] that radar signa is designed to optimize the probabiity of detection of a point target in a cutter. The authors of [7] are optimizing the radar signa over a famiy of uniformy spaced puse trains with compex gains to each puse. Be [8] has been the first to propose using the mutua information between a random extended target and the received signa. His optimization ed to a water-fiing type strategy. In his paper he assumes that the radar signature is a reaization of random Gaussian process with a known power spectra density (PSD. However when considering rea-time signa design his approach can be used to enhance the next transmitted waveform based on the a-priori known signature. It is interesting to note that Be s formuation is equivaent to the design of the best communication channe intended to deiver a specific Gaussian random signa (under power constraint on channe response. Whereas waveform design iterature concentrated on estimation of a singe target, this competey contrasts the capabiities of modern radars to treat mutipe targets. Therefore deveopment of design techniques for mutipe targets are of critica importance to modern radar waveform design. Recent advances in convex optimization, see [9] and the references therein, open the way to design techniques specificay taiored for designing of radar waveforms suitabe for estimating the parameters of mutipe targets. In this paper we study the probem of radar waveform design for mutipe target estimation and tracking based on information theoretic concepts. The paper has three main contributions: First we extend Be s resuts to the design of a singe wavform for simutaneous estimation and tracking of mutipe targets using phased array techniques at the receiver. This approach is then generaized to the case of mutipe transmit waveforms, when the transmitter empoys beamforming as we. Finay an optimization agorithm is proposed for both cases. For both singe and mutipe waveform design we show that using duaity theory the probem can be reduced to a search over a singe parameter and mutipe ow-dimensiona optimization probems at each frequency. Interestingy even though the proposed design criteria for mutipe waveforms is non-convex strong duaity [9] sti hods, which aows us to sove the simper dua probem. Finay we comment that the same observation enabes optimization of a weighted inear sum of the non-causa mean square error.

2 2 II. INFORMATION THEORETIC APPROACH TO WAVEFORM DESIGN In this section we extend the waveform design paradigm of Be [8] to the case of mutipe radar transmitters and receivers. The section is divided into three parts: After a brief review of the resut of [8] we anayze the case of singe waveform design for spatiay resoved targets. This is interesting when transmitter is simpe e.g., in bi-static radar situations. We end up with generaization of our approach to the case of mutipe transmit wavefoms, each optimized for a specific target. In order to study the trade off between various radar receivers we use a inear convex combination of the mutua information between the targets and the received signa at each receiver beam oriented at that specific target. A. The design of a singe waveform for a singe target We begin with a brief overview of Be s information theoretic approach to the waveform design probem. In this paper we imit ourseves to the case of estimation waveforms for extended targets as described in [8]. We assume that the targets are acting on the transmitted waveform as a random inear time invariant system with discrete time frequency response taken from a Gaussian ensembe with known PSD. Denote by h(f =[h(f 1,..., h(f K ] T the target s radar signature and by σh 2(f k its PSD at frequency f k. As noted in [8] extension for the deay-dopper case is possibe, but compicates the formuation. A reaization of the received signa is given by x(f k =h(f k s(f k +w(f k k =1,...,K (1 s(f k, w(f k are respectivey the discrete time waveform and cutter at frequency f k and K is the number of frequency sub-bands. Under our assumptions and assuming compex enveope signaing over sufficienty narrowband division of the transmit bandwidth the mutua information between the target frequency response and the received signa at frequency f k is given by I (h(f k ; x(f k s(f k = f og (1+ σ2 h (f k s(f k 2 σw(f 2 (2 k σw 2 (f k is the cutter PSD at frequency f k, and f is the bandwidth used. The tota mutua information between target frequency response and received signa is now given by I (h; x s = f og (1+ σ2 h (f k s(f k 2 σw 2 (f. (3 k To maximize the mutua information Be proves [8] that a water-fiing strategy is required the transmit PSD is given by { } s(f k 2 = max 0,A σ2 w(f k σh 2(f (4 k and A is a constant chosen so that the tota power constraint is met. It is interesting to note that unike the usua communication probem the waveform design is simiar to optimization of a communication channe for a given signa famiy rather than optimization of the signa to achieve capacity. Note that since the target signature is the desired Gaussian signa we have no imitation on the distribution of s(f k, and the phase can be chosen arbitrariy. This means that we can use amost constant ampitude, by proper frequency scanning using a inear sum of propery deayed and windowed compex exponentia with durations proportiona to the ampitude s(f k 2. B. Designing a singe transmit waveform for mutipe spatiay resoved targets We now turn to the case of mutipe targets. We use simiar approach to [8] we use the mutua information as the basis for the waveform design. Moreover simiary to the notion of rate region of the broadcast channe that has been soved recenty [10], [11], [12], [13] we ook at the waveform design probem as a broadcast channe design probem, the signaing is given and we are free to choose our optima channe under tota power constraint. We assume a singe transmit waveform and mutipe receive eements that are used for reception of the mutipe targets. Foowing [8] we assume that L many targets are taken from a Gaussian ensembe with an a-priori known power spectra densities. In this paper we assume that L is known. However in an extended version of this work we wi provide detais on the adaptive estimation of L. The PSD of the th target at frequency f k is given by σ h (f k. We aso assume a tota power constraint on the transmitted signa, i.e., P s = s(f k 2 f (5 The received signa for the th beam at frequency f k in compex enveop form can be described as ( L z (f k =w (f k a(θ i,f k h i (f k s(f k +ν(f k (6 i=1 a(θ i,f k and h i (f k are the array response towards the direction and frequency response of the i th target at frequency f k respectivey, and w (f k is the beamformer vector of the th beam at frequency f k. The anaysis described can be appied to any beamforming techniques underying the radar operation, e.g., zero forcing,mvdr, SMI, LCMV, GSC or derivative constrained beamforming [14]. However we assume that a received beams are known to the radar processing unit. Since the transmitted waveforms are deterministic and the target response is assumed Gaussian, we obtain that the mutua information between the received signa and the th target radar signature is given by: we define and I (h( (f k ; z (f k s(f k = og 1+ σ2 h (f k g, (f k 2 s(f k 2 f g,j (f k 2 s(f k 2 f+σ 2 ν (f k f (7 g,i (f k =w (f k a(θ i,f k (8 σ 2 ν (f k = w ν 2 f (9

3 3 to be the compex beam gain of the th beam towards the i th target. Since we assume that the targets are spatiay resoved by a inear receive beamformer, a the information for a singe target is captured by z =[z (f 1,..., z (f K ] T Integrating over a frequencies we obtain that for each target the mutua information of the target and the received beam is given by: I (h ; z p = f ( K og 1+ σ 2 h (f k p k g, (f k 2 j σ2 (f h k p k g,j (f k 2 +σ 2 j ν (f k (10 p = [p 1,..., p K ] and p k = p(f k = s(f k 2 f is the power aocation at frequency f k. It is important to understand that this type of design does not constrain the phase of the signa, therefore it opens the way to incorporating other constraints on the transmitted signa, such as ow peak to average. We define the array gain by min j g, (f k 2 g,j (f k 2 (11 There are two imiting cases. The first is when the array gain for each target is sufficienty arge so that the received beam contains ony the desired signa and the Gaussian noise of the cutter. The second is when the main interference is caused by other targets inside the fied of view. In the atter case the gain in designing the signa is ess substantia since the expression in the denominator is dominated by a term which is inear in the waveform and therefore the waveform is canceed as ong as the signa to noise ratio of a targets is positive. Therefore we sha assume that the array gain is sufficient for suppressing interfering targets. In this case we woud ike to maximize for each I (h ; z p = f ( og 1+ σ2 h (f k p k g, (f k 2 σ 2 ν (f k (12 Note however that for each target we have a different cost function, and a waveform that is good for one beam is not necessary good for another. This situation is equivaent to the concept of rate region in mutiuser communication a singe node transmits simutaneousy to independent nodes. To overcome this we can try to find a L-tupes of mutua informations between targets and their respective beams. To that end we define the ineary weighted sum of mutua informations by I(p α = α I (h ; z p (13 α =[α 1,..., α L ] T is a vector of positive weights and α =1. Our waveform design probem with weight vector α can now be formuated as maxp I(p α subject to K p k P (14 or more expicity maxp f ( L K α og 1+ σ2 h (f k p k g, (f k 2 σ 2 ν (f k subject to 1 T p P (15 1 =[1,..., 1] T is a K-dimensiona vector of a ones. In the next section we wi describe an agorithm to perform the optimizaion probem (15. The choice of α is an interesting probem reated to dynamic management of radar resources and targets priroritization. We wi not pursue this issue here. C. Mutipe waveforms for spatiay unresoved targets We now extend our work to the case of mutipe unresoved targets, and the design of mutipe transmitted signas. We begin with revising the received signa mode. Assume that an array with p eements transmits simutaneousy L many waveforms. The transmitted signa at frequency f k is given by t(f k = u (f k s (f k,,k =1,..., K (16 u (f k are the beamformer coefficients for the th waveform designed for the th target at frequency f k, and s (k is the corresponding waveform at frequency f k. We assume channe reciprocity, i.e., that if the receive steering vector is a(θ,f k then the transmitted signa arrives at the target with channes a (θ,f k. The signa refected from the th target having signature h = h (f k,k =1,..., K is therefore given by y (f k = (a (θ,f k u m (f k h (f k s (f k (17 m=1 for k =1,..., K. Hence the received signa at the array is given by x(f k = R(f k u m (f k s m (f k +ν(f k (18 m=1 R(f k = R (f k (19 R is the rank one matrix given by R (f k =h (f k a(θ,f k a (θ,f k (20 As before assume that a beamformer w (f k is used to receive the th target resuting in z (f k = w (f kx(f k = w (f k L m=1 R(f ku m (f k s m (f k +ν (f k (21 ν (f k=w (f kν(f k is the received noise and cutter component of the th beam. Let σ 2 ν (f k =E ν (f k 2 f be the th beam noise power at frequency f k. To obtain the

4 4 mutua information between the th received beam and the th target we rewrite (21 as z (f k = w (f kr u (f k + m w (f kr m (f k u m (f k +ν (f k. (22 In this paper we sha not discuss the adaptive design of the beamforming vectors w (f k, u m (f k but we wi assume that they are given and optimize the transmitted waveforms. However we assume that the radar aocates a beam towards each target and does not perform non-inear processing jointy on the received beams z (f k for different s. Hece the mutua information of z = z (f k :k =1,..., K and the th target signature h = h (f k :k =1,..., K is given by I (h ; z P = f ( K og p 1+,k g, 2 (23 gn,m(f kp m,k +σ 2 m ν (f k g n,m (f k =w n(f k R m (f k u m (f k p n,k = s n (f k 2 f, p n =[p n,1,..., p n,k ] T (24 is the power aocation for the n th target and P =[p 1,..., p L ] is the tota power aocation matrix. Finay we shoud notice that the PSD of the target signature is impicity incuded in g m,n. Simiary to the singe waveform case we woud ike to optimize a inear combination of the mutua informations. The optimization proem can now be posed as L max P I (h ; z P subject to L K p (25,k P max III. WAVEFORM OPTIMIZATION FOR SINGLE AND MULTIPLE TARGETS We now turn to the soution of (15. To that end we note that since each of the terms in the sum is a concave function of the signa power at the reevant frequency. Therefore since a coefficients are positive the cost function I(p α is aso concave. Furthermore the constraint is inear and hence this can be posed as a convex optimiazation by transating the probem to minp f L α I (h ; z p subject to 1 T (26 p P The convex nature of the probem enabe us to use Lagrange duaity [9]. Writing the Lagrangian of the probem we obtain L(p,λ= f α I (h ; z p+λ ( 1 T p P (27 The Lagrangian dua function is now given by ( L d (λ =infp f α I (h ; z p+λ ( 1 T p P (28 Since the probem is convex we have a zero duaity gap which means that the soution to the dua probem max L d (λ (29 subject toλ 0 or more expicity max λ minp L(p,λ (30 subject toλ 0 achieves the same optima vaue as the prima probem. Furthermore foowing the KKT conditions the soution to the prima probem is given by the vector p which minimizes the Lagrangian for the optima λ soving the dua probem. Furthermore note that the Lagrangian can be writen as L(p,λ= f α L k (p k,λ λp (31 L k (p k,λ= f α I (h (f k ; z (f k p k +λp k (32 Therefore given λ, the optima vaue of p minimizing the dua Lagrangian function is computed coordinatewise across frequencies, transforming (30 into p k =argminl k (p, λ (33 p Hence we have divided the high dimensiona probem into an unconstrained search over the Lagrange mutipier and mutipe one-dimensiona unconstrained optimization probems for each frequency in order to evauate the dua Lagrange function. Furthermore since λ is determined by the tota power constraint it can be evauated very efficienty using a bisection method that has an exponentia convergence. This is done by noting that increasing λ reduces a p k since arge vaues of p k increase the Lagrangian. We begin with λ =0and if the tota power constraint is not met we increase λ unti we find a feasibe soution. This is computationay very attractive. We now discuss the mutipe waveform design probem. Unike the probem (15 we cannot reduce (25 to a convex optimization probem. However due to the specia structure as a sum of functions each depending on different variabes, strong duaity sti hods (this foows from [15]. Therefore duaity theory can sti be appied resuting in a simpe optimization, the reative powers at each frequency are optimized independenty and ony one dimensiona search for the singe Lagrange mutipier is performed. The detais of this wi be given in [16] IV. CONCLUSIONS AND EXTENSIONS In this paper we have shown that radar waveform design for mutipe target estimation can be done using a inear combination of mutua informatons between each target signa and the reated received beam. We have then devised a computationay efficient agorithm for soving the probem in the case of a singe waveform. In future work [16] wi wi show how to appy duaity theory to the probem of design of mutipe transmit waveforms for mutipe receive

5 5 beams under joint power constraint. We wi aso show that minimizing the MSE criterion can be efficienty soved using simiar optimization techniques and the design and tracking of the beamformers w (f k and u (f k. REFERENCES [1] P.M. Woodward and I.L. Davies, A theory of radar information, Phi. Mag, vo. 41, pp , oct [2] P.M. Woodward, Information theory and the design of radar receivers, Proc. IRE, vo. 39, pp , dec [3] P.M. Woodward and I.L. Davies, Information theory and inverseprobabiity in teecommunications, Proc. IEE, Part III, vo. 99, pp , mar [4] P.M. Woodward and I.L. Davies, Probabiity and information theory with appications to radar. Pergamon, London, [5] T.I. Grettenberg, Signa seection in communications and radar systems, IEEE trans. on IT, pp , [6] F.C. Schweppe and D.L. Gray, Radar signa design subject to simutaneous peak and average power constraints, IEEE trans. on IT, vo. 12, pp , jan [7] D.F. De Long and E.M. Hofstetter, On the design of optimum radar waveforms for cutter rejection, IEEE trans. on IT, vo. 13, pp , juy [8] M.R. Be, Information theory and radar waveform design, IEEE trans. on IT, vo. 39, pp , sep [9] S. Boyd and L. Vandenberge, Convex optimization. Cambridge university press, [10] Wei Yu and J.M. Cioffi, Sum capacity of vector broadcast channes, IEEE trans. on IT, vo. 50, pp , sep [11] G. Caire and S. Shamai, On the achievabe throughput of a mutiantenna Gaussian broadcast channe, IEEE trans. on IT, vo. 49, pp , juy [12] P. Viswanath and D.N.C. Tse, Sum capacity of the vector Gaussian broadcast channe and upink-downink duaity, IEEE trans. on IT, vo. 49, pp , aug [13] N. Jindah, P. Vishwanath, and A. Godsmith, Duaity, achievabe rates and sum rate capacity of Gaussian MIMO broadcast channes, IEEE trans. on IT, vo. 49, pp , oct [14] H. Van Trees, Optimum array processing. J. Wiey, [15] Wei Yu, R. Lui, and R. Cendrion, Dua optimization methods for mutiuser OFDM systems, in Proceedings of Gobecom 2004, pp , [16] A. Leshem and A. Nehorai, Information theoretic design of mutipe waveforms for mutipe target estimation. To be submitted to IEEE trans. on SP, 2006.