Radar Signal Demixing via Convex Optimization

Transcription

1 Radar Signa Demixing via Convex Optimization Youye Xie Shuang Li Gongguo Tang and Michae B. Wain Department of Eectrica Engineering Coorado Schoo of Mines Goden CO USA Emai: {youyexie shuangi gtang Abstract The probem of decomposing a signa into components of different structures arises in many appications of signa processing and machine earning. In this paper we study such a signa demixing probem for airborne radar systems where the received signa consists of contributions from targets jammers and cutter. We promote the structures of the target and jammer components by designing two atomic norms whie we mode the cutter signa as ying in a nown subspace. This aows the formuation of a convex optimization that is abe to separate these components enabing the ocaization of targets and jammers as we as the supression of the cutter. Simuations show the superior performance of our approach compared to the cassica space-time adaptive processing (STAP) technique. A. Airborne radar systems I. INTRODUCTION Airborne radar systems are widey used in hazardous weather detection navigation aids ground mapping and sea searching [] [2]. Its motion fexibiity and capabiity of wide range imaging provide enormous benefits in these appications. This wor focuses on systems with a puse Dopper radar mounted on an airborne patform. The puse Dopper radar utiizes the Dopper effect to determine the targets veocities and the puse deay ranging to measure the targets distances [3]. The radar antenna is usuay a inear array with uniform spacing. During detection the transmitter antenna sends out puses at a constant puse repetition frequency (PRF) and the returned data is coected over a coherent processing interva (CPI). Within each puse repetition interva (PRI) the receiver sampes the received data and a digita processor performs signa processing to anayze the signa. A target contributes to the received signa a component that depends on the target s azimuth eevation and veocity reative to the airborne patform and can be ocated by its spatia and Dopper frequency. Besides the possibe target signa severa interferences particuary those from cutter jammers and noise can be captured by the antenna [4]. The cutter signas are mainy caused by the earth surface and may vary due to the specific surveiance environment. Since the ground cutter covers a searching azimuths and eevations its pattern may spread over a the spatia and Dopper frequencies [2]. For a stationary radar patform the cutter pattern is an edge occupying a spatia frequencies but zero Dopper frequency whie for a moving radar patform the aircraft motion induces a inear reationship between the Dopper frequency and the spatia frequency when the array aigns with the patform veocity. The barrage jamming is a tactica artificia signa which distributes over a Dopper frequencies with a fixed spatia frequency at a ong range [2]. Space time adaptive processing (STAP) first proposed by Brennan et a. [5] [6] is a we-nown adaptive fitering technique to demix the above signas. Based on the estimated interference covariance matrices STAP attempts to provide high gain on the target spatia and Dopper frequencies whie nuing the interferences. Fuy adaptive STAP weights a eements and puses adaptivey but suffers from high computationa compexity. Aternativey partiay adaptive STAP [7] is proposed to weight adaptivey in a smaer manageabe space. However besides the computationa compexity there are other imitations of STAP. First it is very chaenging for STAP to detect targets whose spatia and Dopper frequencies overap with those for jammers and the cutter. Second estimating the interference covariance matrices requires mutipesnapshot training data and the estimation error caused by the sampe support issue may ead to extremey sma signato-interference-pus-noise ratio (SINR) [2]. By everaging the nowedge of the signa structures we propose an atomic norm minimization based approach that is abe to demix the target jamming and cutter signas accuratey and staby with singe snapshot observations. B. Atomic norm minimization Simiar to norm minimization which promotes sparse modes in signa recovery probems atomic norm minimization promotes sparsity with respect to a dictionary whose eements (aso nown as atoms) are potentiay indexed by a continuous variabe [8]. Due to the continuous dictionary atomic norm minimization can avoid the basis mismatch conundrum in traditiona discretization methods especiay when the frequencies fa off the grid. The fexibiity in choosing dictionaries aows the atomic norm minimization framewor to mode and sove inverse probems invoving different signa structures usuay in a statisticay optima way. For exampe in ine spectra estimation atomic norm minimization achieves optima samping compexity up to ogarithmic factors for inear signa recovery [9] [] has near minimax denoising rate for noisy signa estimation [] [2] produces frequency estimators whose errors approach the Cramér-Rao ower bound [3] and is abe to correct the maxima number of outiers [4] [5]. Moreover it is aso appied to super-resoution probems [6] [7] [8] tensor inverse probems [9] and moda anaysis probems [2]. Since the received composite signa consists of signas with nown structures atomic norm minimization can aso be appied naturay to the airborne radar demixing probem with dictionaries consisting of target cutter and jamming atoms.

2 The paper is organized as foows. In Section II we present the signa mode formuate the atomic norm optimization probem and deveop its dua probem. In Section III we reformuate the atomic norm minimization probem approximatey as a semidefinite program (SDP) that can be soved efficienty. Preiminary numerica simuations to compare the performances of our approach and the STAP method are presented in Section IV. Finay we concude in Section V. II. PROBLEM FORMULATION The data coected by a puse Dopper radar with an antenna array in a CPI can be preprocessed and arranged into a radar datacube whose three axes are indexed respectivey by the range sampe the puse number and the antenna eement. Target bearing and veocity estimation within a range bin require appying signa processing techniques to a spacetime sice of the radar datacube with a fixed range gate. The space-time snapshot from a CPI datacube is a matrix whose (i j)th entry corresponds to the data received by the ith antenna eement in the jth PRI [2]. So the coumn and row contain spatia and tempora variations respectivey. Let N be the number of eements in the antenna array and M be the number of puses within each CPI. We vectorize the space-time snapshot into a MN vector and mode it as y = x T + x J + x C (II.) where x T x J x C represent respectivey contributions from targets jammers and cutter. By setting the first array eement as the reference we can define the (normaized) spatia and tempora steering vectors of the phase array respectivey as [2] [2]: a N (f) [ e j2πf e j2πf(n )] N a M (f) M [ e j2πf e j2πf(m )] where f is the spatia or normaized Dopper frequency of the detected object. Spatia frequency is determined by the object s eevation and azimuth anges and the Dopper frequency is determined by the object s veocity reative to the airborne radar. Throughout the paper we use the subscripts and H to denote transpose conjugate and conjugate transpose respectivey. For a moving target with a specific ange of arriva and veocity with respect to the airborne radar the received target signa is correated both spatiay and temporay [2]. More precisey the array response for a singe target is the Kronecer product of its spatia and Dopper steering vectors up to a magnitude scaing. A jamming signa is spatiay correated but temporay uncorreated from the radar point of view [2] contributing to the observation the Kronecer product of its tempora sampes and the spatia steering vector. In addition for a we aigned system the cutter from a anges ies on a inear ridge in the ange-dopper pane and can be modeed using a subspace whose dimension P can be determined using Brennan s rue [22]. We assume this subspace is nown since it is determined by the patform motion parameters ideay. Let B C MN P be a matrix whose coumns form an orthonorma basis for the cutter subspace and et z C P be the coefficient vector for the cutter signa component. In sum with K targets and L jammers the target jamming and cutter signa components are given respectivey as x T = x J = K = L = c a M (f TD ) a N (f TS ) d h a N (f JS ) and x C = Bz where c d are compex scaar coefficients. Note that we have used T J C D and S to denote Target Jamming Cutter Dopper and Spatia respectivey. For exampe under this notation f TD [ ) represents the normaized Dopper frequency for the th target. The vector h C M scaed to have unit 2 norm represents tempora sampes of the th jammer. Our goa in this wor is to separate the target component x T the jamming signa x J and the cutter contribution x C from the received signa y as shown in (II.) from which we can further extract the target Dopper frequencies {f TD and spatia frequencies {f TS the jamming spatia frequencies {f JS as we as the cutter coefficient vector z. To expoit the target and jamming signas structures we define two atomic sets as foows: A T { a M (f TD ) a N (f TS ) : f TD [ ) f TS [ ) A J { h a N (f JS ) : f JS [ ) h 2 =. Then the corresponding atomic norms are defined as [8] x T AT inf x J AJ inf { c : x T = { d : x J = c a M (f TD ) a N (f TS ) d h a N (f JS ) Note that both the target signa x T and jamming x J are inear combinations of a few atoms from the atomic sets A T and A J. Therefore we empoy the foowing atomic norm minimization program to separate different signa components from y (and recover the frequencies): x Tx Jz subject to x T AT + λ x J AJ y = x T + x J + Bz. (II.2) since it expoits the signa structures and the true targets and jamming coefficients woud give the minimum output. In addition this probem can be soved approximatey by reformuating itsef or its dua probem as the SDP. And we provide its dua probem as foows. The rea inner product

3 is defined as q y R Re( q y ) = Re(y H q). The dua atomic norms are defined as q A T sup q x R x AT = sup f T D f T S [) q A J sup x AJ q x R q am (f TD ) a N (f TS ) = sup (IM a H N(f JS ))q 2 f J S [) X H J where Toep(u) is a Toepitz matrix with u being the first coumn and row. We used tr( ) to denote the trace of a square matrix. For x T AT the authors of [24 Proposition ] derived the foowing SDP that produces a ower bound: T C (8M+) (8M+) t R { 2 tr(btoep(t)) + 2 t : [ ] btoep(t) xt x H T t where btoep(t) C MN MN is a boc Toepitz matrix generated by T. The SDP representation is exact if the optima btoep(t) has a Vandermonde decomposition. As a consequence we can approximatey sove the atomic norm minimization program in (II.2) via the foowing SDP x Tx Jz 2 tr(btoep(t)) + 2 t + λ 2 tr(toep(u)) + λ 2 tr(w) TtuW [ ] [ ] btoep(t) xt Toep(u) XJ subject to t W x H T y = x T + x J + Bz. IV. SIMULATIONS where I M is the identity matrix of size M M. Then the In this section we appy the atomic norm minimization dua probem of (II.2) is given as approach to target estimation for an airborne radar system maximize q y described in the previous sections and compare it with the R q (II.3) cassica STAP agorithm. subject to q A T q A J λ B H q =. A. Experimenta setup III. SDP FORMULATION The antenna array contains 8 uniform eements with hafwaveength spacing. There are 8 RPI within each CPI so the In this section we present an SDP approximation for the atomic norm minimization program in (II.2). Define received signa y for a fixed range gate can be represented by a 64 vector. The received signa contains contributions L X J = d a N (f JS )h from targets jammers the cutter as we as noise. = The spatia frequency of haf-waveength spacing array ranges from to. Without oss of generaity we shift as the reorganization of x J into a matrix of size N M the f TS range to [ ) to mae it consistent with our mode in such that x J = vec(x J ). Denote the atomic set for mutipemeasurement ine spectra signas as [23] section II and simiary for the normaized Dopper frequency. Ã J { a N (f JS )h H : f JS [ ) h 2 = We set the radar antenna array to be aigned with the airborne. veocity and the ratio of the patform speed times PRI over array eements spacing to 2. In this case ideay the cutter It foows that appears as a straight diagona ridge from bottom eft to top right on the ange-dopper pane [2] as shown in Fig.. In x J AJ = X J ÃJ order to demonstrate the infuence of cutter and jamming on target detection some of target ocations on the ange-dopper impying that x J AJ has an exact SDP representation [23] pane overap with the interferences. To be specific our { x J AJ = inf u C N 2 tr(toep(u)) + 2 tr(w) : targets are ocated at (.8) (.) ( ) ( ) (.7 ) (.7.7) (.9 ) (.7.) (.9.9) (.8) W C M M where the first coordinates are the spatia frequencies and the [ ] Toep(u) XJ second are the normaized Dopper frequencies as shown in x J = vec(x J ) Fig. 2. The jamming spatia frequencies f W JS = {.7. So a targets with and.7 spatia frequencies are interfered by the jammers and a the targets on the diagona ine from bottom eft to top right are interfered by the cutter as shown in Fig. 3. In addition compex white noise with zero mean and standard deviation. is aso added to simuate the therma noise. Since this is a synthetic experiment and the ground-truth frequency parameters of targets cutter and jamming are nown we can bypass the interference covariance estimation step and compute covariance matrices precisey for the optima fuy adaptive STAP [2]. However it shoud be noted that our atomic approach does not require nowedge of interference frequencies and no frequencies are given during simuation. The atomic norm minimization is impemented by soving its SDP approximation using CVX [25] [26] an off-the-shef toobox for soving convex optimizations. X H J

4 B. Resuts and anaysis The STAP resut is shown in Fig. 3 whie the atomic norm minimization demixing resuts for jamming and targets are shown in Fig. 4 and Fig. 5 respectivey. To better iustrate the demixing performance ground truth target parameters are highighted by red stars in a figures. In Fig. it can be seen that targets are buried in the interference. To better examine the target signa magnitudes we show the ground truth spectra density in Fig. 2 from which one can identify the ocations of targets. As a variant of matched fitering STAP suppresses both the cutter and jamming whie enhancing the gain on the targets ocations. In Fig. 3 it can be seen that STAP recovers some of the targets ceary. However if the targets are interfered with jamming or cutter STAP can no onger recover them which may ead to severey ow detection rate as demonstrated in our experiment. Atomic norm minimization resuts are much more promising. The jamming spatia frequencies are estimated amost perfecty as shown in Fig. 4. Moreover the targets interfered by jamming are recovered successfuy as shown in Fig. 5 but we can not recover targets buried in the cutter since they have the same structure. We summarize our major empirica findings from a numerica simuations we performed without presenting many of the detais of the experimenta resuts due to space imitation. We investigated four different experimenta scenarios: () targets have frequencies overapped with both jamming and cutter as shown in Fig. (2) targets have frequencies ony overapped with jamming (3) targets have frequencies ony overapped with the cutter and (4) a the frequencies in targets are we separated from those in jamming and cutter. In a of the above cases the jamming and cutter signas overap with each other on the ange-dopper pane. As a resut the energy of the jamming signas wi transfer to the cutter since this wi reduce the atomic norm of the jamming component (note the cutter signa is modeed using a subspace and is not penaized in any way). Therefore we shoud expect to identify the jamming frequencies accuratey but not their magnitude coefficients. In case (2) simiary we can recover the overapped targets and jamming frequencies as shown in Fig. 4 and Fig. 5 but not their magnitude coefficients via atomic norm minimization. In case (3) since the cutter signa can aso be written as a inear combination of target atoms with specia reations between the spatia and Dopper frequencies the targets overapping with the cutter can not be detected. The amount of energy transferred between target and jamming depends on the reguarization parameter λ and in our experiment the parameter in front of x T AT is eight whie it is two for x J AJ. Finay atomic norm minimization can achieve perfect target recovery in case (4). Fortunatey by recovering the target and jamming frequencies we can estimate the targets and suppress the jamming and cutter signas as shown in Fig. 5 which is the most important goa for an airborne radar system. Furthermore the targets overapping with jamming signas can aso be recovered via Normaized dopper frequency Spatia frequency Fig.. The spectrum of the composite received signa y which consists of target cutter jamming and noise contributions. Both the target and cutter signa coefficients are generated from compex standard norma distributions. The jamming signa coefficients are generated by a rea standard uniform distribution random variabe on the open interva ( 5). Normaized dopper frequency Spatia frequency Fig. 2. The spectrum of the targets-ony signa (the ground truth) seen by the airborne radar. Targets have varied signa ampitudes since the types of targets may infuence the echo signa strength in reaity. atomic norm minimization which eads to a higher detection rate and maes it more robust in strong jamming environments compared to STAP. V. CONCLUSION We propose an atomic norm minimization approach to sove the airborne radar system demixing probem. We define two atomic norms that are abe to expoit the target and jammer signa structures and mode the cutter as iving in a nown subspace. This aows the formuation of demixing as a convex atomic norm reguarization probem. We aso derive the SDP to approximatey sove the atomic norm minimization. Compared to the fuy adaptive STAP our approach can recover the targets overapped with jamming which eads to a higher detection rate without the need to estimate the inference covariance matrix. Numerica simuation shows that our approach outperforms STAP in severa aspects. We eave the deveopment of theoretica performance guarantees for the proposed approach to the future wor.

5 Normaized dopper frequency Spatia frequency Fig. 3. Space time adaptive processing (STAP) resut. Cutter and jamming are suppressed whie targets outside the interferences remain. Normaized dopper frequency Spatia frequency Fig. 5. Atomic norm minimization recovers targets outside the cutter ridge. In particuar the targets interfered by jamming are aso recovered Spatia frequency Fig. 4. Atomic norm minimization demixing resut for jamming. The groundtruth jamming spatia frequencies are indicated by the red stems whie according to the duaity theory the estimated jamming frequencies coincide with the paces where the bue dua poynomia curve achieves. ACKNOWLEDGMENT This wor was supported by NSF CAREER grant CCF NSF grant CCF and NSF grant CCF REFERENCES [] G. W. Stimson Introduction to airborne radar. SciTech Pub [2] J. Ward Space-time adaptive processing for airborne radar 998. [3] C. Aabaster Puse Dopper Radar. The Institution of Engineering and Technoogy 22. [4] D. K. Barton Modern radar system anaysis Norwood MA Artech House p. vo [5] L. E. Brennan and L. Reed Theory of adaptive radar IEEE Transactions on Aerospace and Eectronic Systems no. 2 pp [6] L. Brennan J. Maett and I. Reed Adaptive arrays in airborne mti radar IEEE Transactions on Antennas and Propagation vo. 24 no. 5 pp [7] H. Wang and L. Cai On adaptive spatia-tempora processing for airborne surveiance radar systems IEEE Transactions on Aerospace and Eectronic Systems vo. 3 no. 3 pp [8] V. Chandrasearan B. Recht P. A. Parrio and A. S. Wisy The convex geometry of inear inverse probems Foundations of Computationa Mathematics vo. 2 no. 6 pp [9] G. Tang B. N. Bhasar P. Shah and B. Recht Compressed sensing off the grid IEEE Transactions on Information Theory vo. 59 no. pp [] S. Li D. Yang G. Tang and M. B. Wain Atomic norm minimization for moda anaysis from random and compressed sampes arxiv preprint arxiv: [] G. Tang B. N. Bhasar and B. Recht Near minimax ine spectra estimation IEEE Transactions on Information Theory vo. 6 no. pp [2] B. N. Bhasar G. Tang and B. Recht Atomic norm denoising with appications to ine spectra estimation IEEE Transactions on Signa Processing vo. 6 no. 23 pp [3] Q. Li and G. Tang Approximate support recovery of atomic ine spectra estimation: A tae of resoution and precision arxiv preprint arxiv: [4] C. Fernandez-Granda G. Tang X. Wang and L. Zheng Demixing sines and spies: Robust spectra super-resoution in the presence of outiers arxiv preprint arxiv: [5] G. Tang P. Shah B. N. Bhasar and B. Recht Robust ine spectra estimation in 24 48th Asiomar Conference on Signas Systems and Computers pp IEEE 24. [6] E. J. Candès and C. Fernandez-Granda Towards a mathematica theory of super-resoution Communications on Pure and Appied Mathematics vo. 67 no. 6 pp [7] G. Tang Resoution imits for atomic decompositions via marovbernstein type inequaities in Samping Theory and Appications (SampTA) 25 Internationa Conference on pp IEEE 25. [8] D. Yang G. Tang and M. B. Wain Super-resoution of compex exponentias from moduations with unnown waveforms IEEE Transactions on Information Theory vo. 62 no. pp [9] Q. Li A. Prater L. Shen and G. Tang Overcompete tensor decomposition via convex optimization in Computationa Advances in Muti- Sensor Adaptive Processing (CAMSAP) 25 IEEE 6th Internationa Worshop on pp IEEE 25. [2] S. Li D. Yang G. Tang and M. B. Wain Atomic norm minimization for moda anaysis from random and compressed sampes arxiv preprint arxiv: [2] M. A. Richards Fundamentas of radar signa processing. Tata McGraw-Hi Education 25. [22] L. Brennan and F. Staudaher Subcutter visibiity demonstration tech. rep. Tech. Rep. RL-TR-92-2 Adaptive Sensors Incorporated 992. [23] Y. Li and Y. Chi Off-the-grid ine spectrum denoising and estimation with mutipe measurement vectors IEEE Transactions on Signa Processing vo. 64 no. 5 pp [24] Y. Chi and Y. Chen Compressive recovery of 2-d off-grid frequencies in 23 Asiomar Conference on Signas Systems and Computers pp IEEE 23. [25] M. Grant and S. Boyd CVX: Matab software for discipined convex programming version Mar. 24. [26] M. Grant and S. Boyd Graph impementations for nonsmooth convex programs in Recent Advances in Learning and Contro (V. Bonde S. Boyd and H. Kimura eds.) Lecture Notes in Contro and Information Sciences pp. 95 Springer-Verag Limited edu/ boyd/graph dcp.htm.