TARGET ESTIMATION IN COLOCATED MIMO RADAR VIA MATRIX COMPLETION. Shunqiao Sun, Athina P. Petropulu and Waheed U. Bajwa

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1 TARGET ESTIMATION IN COLOCATED MIMO RADAR VIA MATRIX COMPLETION Shunqiao Sun, Athina P. Petopulu and Waheed U. Bajwa Depatment of Electical & Compute Engineeing, Rutges, The State Univesity of New Jesey ABSTRACT We conside a colocated MIMO ada scenaio, in which the eceive antennas fowad thei measuements to a fusion cente. Based on the eceived data, the fusion cente fomulates a matix which is then used fo taget paamete estimation. When the eceive antennas sample the taget etuns at Nyquist ate, and assuming that thee ae moe eceive antennas than tagets, the data matix at the fusion cente is lowank. When each eceive antenna sends to the fusion cente only a small numbe of samples, along with the sample index, the eceive data matix has missing elements, coesponding to the samples that wee not fowaded. Unde cetain conditions, matix completion techniques can be applied to ecove the full eceive data matix, which can then be used in conjunction with aay pocessing techniques, e.g., MUSIC, to obtain taget infomation. Numeical esults indicate that good taget ecovey can be achieved with occupancy of the eceive data matix as low as 5%. Index Tems Aay pocessing, compessed sensing, matix completion, MIMO ada, MUSIC 1. INTRODUCTION Multiple-input and multiple-output (MIMO ada systems have eceived consideable attention in ecent yeas due to thei supeio taget estimation pefomance. Colocated MIMO ada systems exploit wavefom divesity to fomulate a long vitual aay with numbe of elements equal to the poduct of the numbe of tansmit and eceive antennas. As a esult, they achieve highe esolution than taditional phased aay adas fo the same amount of data [1][2]. Compessed sensing (CS enables MIMO ada systems to maintain thei advantages while significantly educing the equied measuements pe eceive antenna [3][4]. In CS-based MIMO ada, taget paametes ae estimated by exploiting the spasity of tagets in the angle, Dopple and ange space, efeed to as the taget space. Fo CS-based spase taget estimation, the taget space needs to be discetized into a fine gid, based on which the CS sensing matix is constucted. Howeve, pefomance of CS-based MIMO ada degades when tagets This wok was suppoted by the Office of Naval Reseach unde Gant ONR-N fall between gid points, a case also known as basis mismatch [5] in the CS liteatue. Anothe appoach elated in spiit to CS is that of matix completion. Matix completion aims to ecove a low-ank data matix fom patial samples of its enties by solving a elaxed nuclea nom optimization poblem [6][7]. Aay signal pocessing with matix completion has been studied in [8][9]. To the best of ou knowledge, howeve, matix completion has not been exploited fo taget estimation in colocated MIMO ada. Ou pape is elated to the ideas in [9] in the sense that matix completion is applied to the eceived data matix fomed by an aay. Howeve, due to the unique stuctue of the eceived signal in MIMO ada, the poblem fomulation and teatment in hee is diffeent than that in [9]. The main idea of ou wok is as follows. We conside a colocated MIMO ada scenaio in which eceive antennas fowad thei measuements to a fusion cente. Based on the eceived data, the fusion cente fomulates a matix, which is then used fo estimating the taget paametes. When the eceive antennas sample the taget etuns at Nyquist ate, and assuming that thee ae moe eceive antennas than tagets, the data matix at the fusion cente is low-ank. When each eceive antenna sends to the fusion cente only a small numbe of samples, along with the sample index, the eceive data matix has missing elements, coesponding to the samples that wee not fowaded. Unde cetain conditions, matix completion techniques can be applied to ecove the full eceive data matix, which can then be used in conjunction with paametic methods such as MUSIC to obtain taget infomation. Compaed to CS MIMO ada, ou poposed method has the same advantage in tems of eduction of samples needed fo accuate estimation but it avoids the basis mismatch issue inheent in CS-based appoach. The est of the pape is oganized as follows. The colocated MIMO ada system model is descibed in Section 2. Some backgound of noisy matix completion is intoduced in Section 3. The applicability of matix completion to MIMO ada is discussed in Section 4, and numeical esults ae given in Section 5. Finally, Section 6 povides concluding emaks. 2. COLOCATED MIMO RADAR SYSTEM We conside a MIMO ada system that employs colocated tansmit and eceive antennas, shown in Fig. 1. We use

2 Taget Taget Hees(t = [s 1 (t,...,s Mt (t] T. Suppose that each eceive antenna samples the eceived signal at ate L/T s and fowads the samples to the fusion cente. Hee, T s is the sampling time, L is the numbe of nonzeo samples and we assume L K. At the fusion cente, eceived data duing theqth pulse can be witten as [1] X q = B(θΣD q A T (θs+w q = Z q +W q, (3 s1t 2dtsin Tansmittes d t d sin t d t s 1t s t M t M t M 1dsin Receives 2dsin d Fusion cente d sin d Matched filtebank Fig. 1. Colocated MIMO ada system unde the ULA model. whee Σ = diag([β 1,...,β K ]; D q = diag(d q, with d q = [ e j2π2ϑ1(q 1TPRI,...,e j2π2ϑk(q 1TPRI ] T ; S = [s(t s,...,s((l 1T s ]; and W q is a Gaussian noise matix. Note that both maticesσ andd q ae ank-k, while the ank of matix S is min{m t,l}. Thus, fo M t > K the ank of the noise fee data matixz q = B(θΣD q A T (θs isk. In othe wods, the data matixz q is low-ank based on the assumption thatm K. M t and M to denote the numbes of tansmit antennas and eceive antennas, espectively. Although ou esults can be extended to an abitay antenna configuation, the esults hee ae pesented fo the case in which the tansmit and eceive antennas fom a unifom linea aay (ULA with inte-element spacing between tansmit and eceive antennas d t and d, espectively. We futhe assume d t = d = λ/2, whee λ is the wavelength of the caie signal. Futhe, the wavefomss k (t,k = 1,...,M t, tansmitted fom the tansmit antennas ae assumed to be naow-band and othogonal. Now suppose thee ae K point tagets in the fa field at angles θ k,k = 1,...,K, each moving with speed ϑ k. The coespondingm t K dimensional tansmit steeing matix can be expessed asa(θ = [a(θ 1,...,a(θ K ], whee a(θ k = [ 1,e j 2π λ dt sin(θ k,...,e j 2π λ (Mt 1dt sin(θ k ] T, (1 and them K dimensional eceive steeing matixb(θ can be defined in a simila fashion based on the eceive steeing vectosb(θ k. In ode to estimate the speed of each taget, multiple pulses need to be tansmitted. Let Q be the numbe of tansmitted pulses, with T PRI being the pulse epetition inteval. Assume the taget eflection coefficients {β k },k = 1,...,K ae complex and emain constant duing the Q pulses. Fo slowly moving tagets, (2ϑT p /λ 1, whee T p is the pulse duation the Dopple shift within a pulse can be ignoed, while the Dopple changes fom pulse to pulse. Unde the naowband assumption the eceived signal at thelth eceive antenna can be appoximated as [4] x l (t K β k e j 2π λ 2ϑkt b l (θ k a T (θ k s(t+w l (t. (2 k=1 3. MATRIX COMPLETION WITH NOISE We now povide a bief oveview of the poblem of ecoveing a ank matix M C n1 n2 based on patial knowledge of its enties, possibly coupted by noise, i.e., [Y] ij = [M] ij + [E] ij,(i,j Ω, whee, [E] ij epesents noise and Ω is the set of obseved enties. This can also be expessed as P Ω (Y = P Ω (M+P Ω (E, wheep Ω epesents the sampling opeation. Accoding to [7], when M is low-ank and its singula vectos ae sufficiently spead, i.e., both the numbes of zeo and lage elements in the singula vectos ae not lage,m can be ecoveed by solving a elaxed nuclea nom optimization poblem, given by min X s.t. P Ω (X Y F δ (4 whee denotes the nuclea nom, i.e., the sum of singula values of X, while δ > is a constant. To test the sufficiently spead equiement, the stong incoheence popety of matix M with paamete µ has been intoduced in [6]. Conside the singula value decomposition (SVD of M, i.e., M = ρ k u k vk H, and k=1 define P U = u i u H i, P V = v i vi H, and T = 1 i 1 i u i vi H. When the following conditions ae satisfied, the 1 i matix M is said to satisfy the stong incoheence popety withµ = max(µ 1,µ 2. A1 Fo all pais(a,a [n 1 ] [n 1 ] and(b,b [n 2 ] [n 2 ], thee isµ 1 > such that e a,p U e a 1 a=a n 1 µ 1, (5 n 1 e b,p V e b 1 b=b n 2 µ 1, (6 n 2 whee e a is the vecto with the ath element equal to 1 and othes being zeo, while 1 a=a indicates that it is equal to 1 whena = a andothewise. A2 Fo all (a,b [n 1 ] [n 2 ], thee exists a constant µ 2 > such that T ab µ 2 n1n 2, whee T ab is the (a,b enty of the matixt.

3 Note that it has been shown in [6] that when the maximum (in magnitude values of the K left and ight singula vectos ae bounded, i.e., µb µb u k l, v k n l (7 1 n 2 with µ B = O(1, then the stong incoheence popety is guaanteed withµ µ B. Now define n = max(n 1,n 2 and suppose M satisfies the stong incoheence popety. Then [6] establishes in the noiseless case that, by obseving N andomly selected enties with N Cµ 2 nlog 6 n fo some constant C, the matix M can be ecoveed exactly with a pobability of at least1 n 3. Futhe, [7] establishes that, when obsevations ae coupted with white noise [E] ij that is zeo-mean Gaussian with vaiance σ 2, the ecovey eo is bounded as M ˆM F 4 1 p (2+pmin(n 1,n 2 δ+2δ, wheep = N n 1n 2 is the faction of obseved enties. 4. MATRIX COMPLETION FOR MIMO RADAR The left singula vectos ofz q defined in (3 ae the eigenvectos ofz q Z H q = HSSH H H, wheeh = B(θΣD q A T (θ. The ight singula vectos of Z q ae the eigenvectos of S H H H HS. Since S is othogonal, it holds that SS H = I. Thus, the left singula vectos ae only detemined by matix H, while the ight singula vectos ae affected by both tansmit wavefoms and matix H. Fo the poblem consideed in this pape, it is difficult to detemine analytically the behavio of the enties of the left and ight singula vectos of Z q. Instead, we get an idea of the behavio of the maximum values and the paametes that affect thei spead using simulations. We conside a MIMO ada setup in which the taget diection of aival (DOA angles ae unifomly distibuted in [ 9,9 ] and the coesponding taget speeds ae unifomly distibuted in the ange [15,45]m/s. In addition,β k ae following complex Gaussian distibution and kept unchanged foq = 1 pulses. The pulse epetition inteval is T PRI = 1/4 second, and the caie fequency is f = 1 9 Hz, esulting in λ = c/f =.3 mete. Two types of othogonal wavefoms ae consideed: Hadamad and Gaussian othogonal wavefoms. Seveal cases of paametes ae veified. Case I: M = 4, L = 128; Case II:M = 1,L = 128; Case III:M = 4, L = 124. Each case uns fo 3 iteations. Let m 1 andm 2 denote the maximum element value of K left and ight singula vectos of Z q. The complementay cumulative distibution function (CDF cuves of m 1 and m 2, i.e., P(m > m i,i = 1,2 ae plotted in Fig. 2 fo Cases I and II. It can be seen fom these plots that asm inceases, the bounds of m 1 fo both K = 2 and K = 1 decease, while the distibution of m 2 does not significantly change when L P(m m 1 Gaussian Othogonal K= m 1 P(m m 1 (a Gaussian Othogonal K= m 1 (c P(m m 2 Gaussian Othogonal K= m 2 P(m m 2 (b Gaussian Othogonal K= m 2 Fig. 2. Complementay CDF of m 1 and m 2. Case I: M = 4, L = 128; (a left singula vectos, (b ight singula vectos; Case II: M = 1,L = 128; (c left singula vectos, (d ight singula vectos. is fixed. Figue 2 (c shows that as M gets lage, m 1 gets bounded by a small numbe with high pobability. Space limitations pevent us fom displaying moe figues of Case III, which show thatm 2 is bounded by a small numbe with high pobability aslgets lage. Extensive simulations also show that the bounds on m 1 andm 2 scale as1 / M and1/ L, espectively, with some constant µ B. Fo Gaussian othogonal wavefom and K = 2, µ B 2.4 fo the bound of m 1 (see Fig. 3 (a and µ B 6.5 fo the bound of m 2 (see Fig. 3 (b. Fo K = 1 and Hadamad wavefom, the scaling laws also hold but with lage constants µ B. Theefoe, depending on the numbe of eceive antennas M and the numbe of samples L in one pulse, m 1 and m 2 can be assumed to be bounded by small numbes. Based on [6], theefoe, we conclude that the stong incoheence popety is likely satisfied in ou poblem. It is also woth noting that unde Gaussian othogonal wavefoms, m 2 is concentated in a smalle ange as compaed with the ange fo Hadamad wavefoms. This indicates that the wavefom indeed plays a ole fo the use of matix completion, and pehaps the wavefom can be optimally designed to esult in low pobability of lage values fom 2. Since the matix completion conditions appea to be satisfied in ou case, we popose that each antenna obtains and fowads to the fusion cente a small numbe of samples duing each pulse. Note that along with the samples, the antenna needs to infom the fusion cente on how the sample was obtained so that the fusion cente can detemine whee to position the eceived samples in the eceive data matix Z q. In a pactical setting, a pseudo-andom sampling ADC at each e- (d

4 Value Bound of m 1 ecipocal of sqt(m M Value Bound of m 2 ecipocal of sqt(l L Values.3.1 Recipocal of SNR Hadamad G Oth p Pobability of esolution Hadamad p=.3 Hadamad p=.5 G Oth p=.3 G Oth p= Minimum DOA sepaation (degee (a (b (a (b Fig. 3. Bounds of m 1 andm 2 scales with ecipocal of M and L, espectively. ceive antenna could be used in the place of the Nyquist sample, whee the pseudo-andom geneato seeds would be distibuted to the eceive antennas by the fusion cente. Finally, the fusion cente can ecove the data matix ˆX q by applying matix completion to the eceived data. Once the data matix ˆX q is ecoveed, it can go though a matched-filte bank to poduce Y q = 1 L ˆX q S H = B(θΣD q A T (θ+ W q, (8 whee W q is noise whose distibution is a function of the additive noise W q and the nuclea nom minimization poblem in (4. Next, stacking Y q C M Mt into a vecto y q, and based on the vectos coesponding to Q pulses, the following matix can be fomed: Y R = [y 1,...,y Q ] C MMt Q. ReshapingY R intoy C QMt M, we have Y = FΣ[b(θ 1,...,b(θ K ]+W, (9 whee F = [d(ϑ 1 a(θ 1,...,d(ϑ K a(θ K ], d(ϑ = [ ] 1,e j2π2ϑt T PRI,...,e j2π2ϑ(q 1TPRI, with denoting the Konecke poduct. The sampled covaiance matix of the eceive data signal can then be obtained as ˆR Y = 1 M YY H, based on which taget estimation can be implemented using any aay pocessing method such as MUSIC. 5. NUMERICAL RESULTS In this section we pesent some simulation esults on the pefomance of the poposed method. We use the simulation setting consideed in the pevious section, i.e.,m t = 2, M = 4, Q = 5, L = 128. The signal-to-noise atio (SNR is set to 25 db, while β k ae following complex Gaussian distibution and kept unchanged fo Q pulses. Fo matix completion, the TFOCS softwae package [11] is used. Fist, we plot elative eos (aveaged ove 5 Monte Calo uns of the eceived data matix ˆX q fo Hadamad and Gaussian othogonal (G-Oth wavefoms. The elative eo is defined as Zq ˆX q F Z q, whee Z q is the data matix F calculated without missing elements. Unde each wavefom, Fig. 4. Pefomance compaisons: (a Relative eo of the ecoveed data matix; (b Pobability of DOA estimation esolution. K = 2 point tagets in the fa field ae andomly geneated. The esult is shown in Fig. 4 (a fo q = 1. It can be seen fom this figue that, as p inceases, the elative ecovey eo of the data matix unde Gaussian othogonal wavefom educes to the ecipocal of the SNR faste than that unde Hadmad wavefom. A plausible eason fo this is that unde the Gaussian othogonal wavefom, the maximum value of elements in the singula vectos ofz q is bounded by a smalle numbe with high pobability, as compaed with that unde the Hadamad wavefom (see Fig. 2. Next, the pobabilities of DOA estimation esolution unde the two othogonal wavefoms ae plotted in Fig. 4 (b fo the following scenaio. Two tagets ae andomly geneated among DOA ange [ 15,15 ] with minimum DOA sepaationsd θ = [,.3,.4,.5,.7,1 ] and the coesponding speeds ae set to 15 and 4 m/s. The MUSIC algoithm is applied to obtain the taget DOA infomation. If the DOA estimates ˆθ i, i = 1,2 satisfy θ i ˆθ i εdθ,ε =.1, we declae this as a success. The pobability of DOA esolution is then defined as the faction of successful events in 5 iteations. It can be seen fom the figue that whenp =.3, the Gaussian othogonal wavefom has a much bette DOA estimation esolution compaed with Hadamad wavefom. As p inceases to.5, the pefomance diffeence becomes small since the elative ecovey eos unde both wavefoms ae simila (see Fig. 4 (a. Figue 4 confims that Gaussian othogonal wavefoms ae bette than Hadamad wavefoms fo matix completion-based DOA estimation. 6. CONCLUSIONS We have povided esults suggesting that matix completion can be used in MIMO ada to educe the numbe of data needed to be communicated to the fusion cente by each eceive antenna. Numeical esults show that matix completion in conjunction with MUSIC can achieve accuate taget estimation with sub-nyquist samples. Thus, the poposed method can esult in significant savings in tems of data that need to be obtained at the eceive antennas and subsequently tansmitted to the fusion cente.

5 7. REFERENCES [1] P. Stoica and J. Li, MIMO ada with colocated antennas, IEEE Signal Pocess. Magazine, vol. 24, no. 5, pp , 27. [2] C. Y. Chen and P. P. Vaidyanathan, MIMO ada spacetime adaptive pocessing using polate spheoidal wave functions, IEEE Tans. on Signal Pocessing, vol. 56, no. 2, pp , 28. [3] M. Heman and T. Stohme, High-esolution ada via compessed sensing, IEEE Tans. on Signal Pocessing, vol. 57, no. 6, pp , 29. [4] Y. Yu, A. P. Petopulu, and H. V. Poo, MIMO ada using compessive sampling, IEEE Jounal of Selected Topics in Signal Pocessing, vol. 4, no. 1, pp , 21. [5] Y. Chi, L. L. Schaf, A. Pezeshki, and A. R. Caldebank, Sensitivity to basis mismatch in compessed sensing, IEEE Tans. on Signal Pocessing, vol. 59, no. 5, pp , 211. [6] E. J. Candès and T. Tao, The powe of convex elaxation: Nea-optimal matix completion, IEEE Tans. on Infomation Theoy, vol. 56, no. 5, pp , 21. [7] E. J. Candès and Y. Plan, Matix completion with noise, Poceedings of the IEEE, vol. 98, no. 6, pp , 21. [8] A. Wates and V. Cevhe, Distibuted beaing estimation via matix completion, in Poc. of IEEE ICASSP, 21. [9] Z. Weng and X. Wang, Low-ank matix completion fo aay signal pocessing, in Poc. of IEEE ICASSP, 212. [1] D. Nion and N. D. Sidiopoulos, Tenso algeba and multi-dimensional hamonic etieval in signal pocessing fo MIMO ada, IEEE Tans. on Signal Pocessing, vol. 58, no. 11, pp , 21. [11] S. Becke, E. J. Candès, and M. Gant, Templates fo convex cone poblems with applications to spase signal ecovey, Math. Pog. Comp. vol. 3, no. 3, pp , 211.