Direction of Arrival Estimation Using Co-prime Arrays: A Super Resolution Viewpoint

Transcription

1 Direction of Arrival Etimation Uing Co-prime Array: A Super Reolution Viewpoint Zhao Tan Student Member, IEEE, Yonina C Eldar Fellow, IEEE, and Arye Nehorai Fellow, IEEE arxiv:37793v [cit] 3 Dec 3 Abtract We conider the problem of direction of arrival (DOA etimation uing a newly propoed tructure of nonuniform linear array, referred to a co-prime array, in thi paper By exploiting the econd order tatitical information of the received ignal, co-prime array exhibit O(MN degree of freedom with only M + N enor A parity baed recovery method i propoed to fully utilize thee degree of freedom Unlike traditional pare recovery method, the propoed method i baed on the developing theory of uper reolution, which conider a continuou range of poible ource intead of dicretizing thi range into a dicrete grid With thi approach, off-grid effect inherited in traditional pare recovery can be neglected, thu improving the accuracy of DOA etimation In thi paper we how that in the noiele cae one can theoretically detect up to MN ource with only M + N enor The noie tatitic of co-prime array are alo analyzed to demontrate the robutne of the propoed optimization cheme A ource number detection method i preented baed on the pectrum recontructed from the pare method By extenive numerical example, we how the uperiority of the propoed method in term of DOA etimation accuracy, degree of freedom, and reolution ability compared with previou method, uch a MUSIC with patial moothing and the dicrete pare recovery method Index Term Direction of arrival etimation, co-prime array, uper reolution, pare recovery method, ource number detection I INTRODUCTION In the lat few decade, reearch on direction of arrival (DOA etimation uing array proceing ha focued primarily on uniform linear array (ULA [] It i well known that by implementing a ULA with N enor, the number of ource that can be reolved by MUSIC-like algorithm i N [] New geometrie [3], [] of non-uniform linear array have been recently propoed to increae the degree of freedom of the array by tudying the covariance matrix of the received ignal among different enor By vectorizing the covariance matrix, the ytem model can be viewed a a virtual array with a wider aperture In [3], a neted array tructure wa propoed to increae the degree of freedom from O(N to O(N, with only O(N enor However, ome of the enor in Z Tan and A Nehorai are with the Preton M Green Department of Electrical and Sytem Engineering Department, Wahington Univerity in St Loui, St Loui, MO, 633 USA {tanz, nehorai}@eewutledu Y C Eldar i with the Department of Electrical Engineering, Technion Irael Intitute of Technology, Haifa 3, Ireal yonina@eetechnionacil The work of Z Tan and A Nehorai wa upported by the AFOSR Grant FA955---, and ONR Grant N35 The work of Y C Eldar wa upported in part by the Irael Science Foundation under Grant no 7/, in part by the Ollendorf Foundation, and in part by a Magneton from the Irael Minitry of Indutry and Trade the neted array tructure are cloely located, which lead to mutual coupling among thee enor To overcome thi hortcoming, co-prime array were propoed in [], and it wa hown that by uing O(M + N enor, thi tructure can achieve O(MN degree of freedom In thi paper we will focu on co-prime array The increaed degree of freedom provided by the co-prime tructure can be utilized to improve DOA etimation Two main methodologie have been propoed One i ubpace method, uch a the MUSIC algorithm In [5], a patial moothing technique wa implemented prior to the application of MUSIC The author howed that an increaed number of ource can be detected by the co-prime array However, the application of patial moothing reduce of the obtained virtual array aperture [6] The econd methodology ue parity baed recovery to overcome thi diadvantage of ubpace method [6]-[9] Traditional parity baed recovery dicretize the range of interet into a grid The aumption made by parity method i that all ource are located exactly at the grid point However, off-grid target can lead to mimatche in the model and deteriorate the performance of pare recovery ignificantly [] In [], [] the grid mimatche were etimated imultaneouly with the original ignal, and they howed that by conidering grid mimatche one can achieve a better pare recovery performance than the traditional pare recovery method In [3], the joint parity between the original ignal and the mimatch wa exploited during the DOA etimation for co-prime array Due to the firt order approximation ued in [3], the etimation performance i till limited by the higher order modeling mimatch To overcome thi difficulty of traditional parity baed method, a recent developed mathematical theory of uper reolution [], [5] i utilized in thi paper to perform DOA etimation with co-prime array In [] it wa proved that the high frequency contant of a ignal pectrum can be perfectly recovered by ampling only the low end of it pectrum when the minimum ditance among different pike atifie certain requirement Robutne of thi theory to noie i analyzed in [5] One merit of thi theory i that it conider all the poible location within the intereted range, and thu doe not uffer from model mimatch Here we extend the mathematical theory of uper reolution to DOA etimation with co-prime array under Gauian noie The noie tructure reulting from the uage of co-prime array conit of a term with a known tructure and another term coniting of quadratic combination of Gauian noie Therefore, we modify the recontruction method to fit thi particular noie tructure and how the robutne of our approach by analyzing the

2 noie tatitic We alo demontrate theoretically that with M + N enor in co-prime array, one can detect up to MN ource Previou reearch [9] on identifiability uing coprime array wa baed on the idea of mutual coherence [6] Although uing mutual coherence can prove theoretically that by implementing co-prime array one can increae the number of ource being detected from O(M + N to O(MN, thi analyi baed on coherence allow to go from O(M + N to O(MN only for very mall value of the number of ource Source number detection i another main application of array proceing Variou method have been propoed over the year baed on the eigenvalue of the ignal pace, uch a the Akaike information criterion [7], the econd order tatitic of eigenvalue (SORTE [8], the predicted eigenthrehold approach [9], and an eigenvector-baed method that exploit the property of the variance of the rotational ubmatrix [] The author of [] howed that among thee method, SORTE often lead to a better detection performance We combine the SORTE method with pectrum recontructed from DOA etimation to detect the number of ource Through thi ource number detection, we referwhich recontructed pike are true detection and which are fale alarm The paper i organized a follow In Section II, we introduce the DOA etimation model and explain how co-prime array can increae the degree of freedom of the etimation ytem In Section III, we extend uper reolution theory to the application of co-prime array, and analyze the robutne of thi extenion by tudying the tatitic of the noie pattern in the model We propoe a numerical method to perform DOA etimation for co-prime array in Section IV We then extend thi approach to detect the number of ource in Section V Section VI preent extenive numerical imulation to how the advantage of our method in term of etimation accuracy, degree of freedom, and reolution ability Throughout the paper, we ue capital italic bold letter to repreent matrice and operator, and lowercae italic bold letter to repreent vector For a given matrix A, A denote the conjugate tranpoe matrix, A T denote the tranpoe, and A H repreent the conjugate matrix without tranpoe We ue A mn to denote the (m, nth element of A We ue to denote the Kronecker product of two matrice For a given operator F, F denote the conjugate operator of F Given vector x, we ue x and x to denote it l and l norm; x i and x[i] are both ued to repreent the ith element of x Given a function f, f L, f L, f L are it l, l, l norm x(t = a(θ k k (t + ε(t = A(t + ε(t, ( k= for t =,, T, in which ε(t R L i an iid white Gauian noie CN (, σ, A = [a(θ, a(θ, a(θ K ] R L K, and (t = [ (t, (t,, K (t] T preent the ource ignal vector with k (t ditributed a CN (, σk We aume that the ource are temporally uncorrelated The correlation matrix among the K ource can then be expreed a R xx =E[x(tx (t] =AR A + σ I = σka(θ k a (θ k + σ I, ( k= in which R i a R K K diagonal matrix with diagonal element σ, σ,, σk After vectorizing the correlation matrix R xx, we have where z = vec(r xx = Φ(θ, θ,, θ K + σ n, (3 Φ(θ,, θ K = A A = [a(θ H a(θ,, a(θ K H a(θ K ] The ignal of interet become = [σ, σ,, σk ], and n = [e T, e T,, e T L ]T, where e i denote a vector with all zero element, except for the ith element, which equal to one Comparing equation ( and (3, we ee that behave like a coherent ource and σ n become a determinitic noie term The ditinct row in Φ act a a larger virtual array with enor located at d i d j, with i, j L Traditional DOA etimation algorithm can be implemented to detect more ource when the tructure of the enor array i properly deigned Following thi idea, neted array [3] and co-prime array [] were introduced, and then hown to improve the degree of freedom from O(N to O(N, and from O(M + N to O(MN repectively In the following demontration, we focu only on co-prime array; the reult follow naturally for neted array II DIRECTION OF ARRIVAL ESTIMATION AND CO-PRIME ARRAYS Conider a linear enor with L enor which may be non-uniformly located Aume that there are K narrow band ource located at θ, θ,, θ K with ignal power σ, σ,, σk The teering vector for the kth ource located at θ k i a(θ k R L with l-th element e j(π/λd l in(θ k, in which d l i the location of the lth enor and λ i the wavelength The data collected by all the enor at time t can be expreed a Fig : Geometry of Co-prime Array Conider a co-prime array tructure conit of two array with N and M enor repectively The location of the N enor are in the et {Mnd, n N }, and the location of the M enor are in the et {Nmd, m

3 3 M } Pleae note that firt enor of thee two array are collocated The geometry of thee co-prime array i hown in Fig In thi cae the ening matrix A R (M+N K ha the ame tructure a that in ( Indeed, the location of the virtual enor in (3 are given by the cro difference et {±(Mn Nmd, n N, m M } and the two elf difference et In order to implement patial moothing of MUSIC, we are intereted in generating a conecutive range of virtual enor It wa hown in [5] that when M and N are coprime number, a conecutive range can be created from MNd to MNd, with { MNd, (MN d,, d, d, d, d,, (MN d, MNd} taken from the cro difference et and {d} taken from any one of elf difference et By removing repeated row of (3 and orting the remaining row from M N d to M N d, we have the linear model rearranged a z = Φ + σ w ( It i eay to verify that w R (MN+ i a vector whoe element all equal to zero, except that the (MN+-th element equal to one The matrix Φ R (MN+ K i expreed a e jmnd π λ in(θ e jmnd π λ in(θ K e Φ j(mn d π λ in(θ e j(mn d π λ in(θ K =, e jmnd π λ in(θ e jmnd π λ in(θ K which i the teering matrix for a uniform linear array (ULA with MN + enor Therefore, ( can be regarded a a ULA detecting a coherent ource with determinitic noie term σ w By applying MUSIC with patial moothing, the author in [5] howed that O(MN ource can be detected III DIRECTION OF ARRIVAL ESTIMATION WITH SUPER RESOLUTION THEORY In thi ection we firt aume that the ignal model (3 i accurate, which mean that the number of ample T i infinity, and alo that the noie power σ i known a priori The uper reolution theory developed in [] can be implemented for coprime array to demontrate that we can detect up to O(MN ource a long a the ditance between any two ource i on the order of MN Firt we briefly introduce uper reolution theory and extend the idea to the reearch of co-prime array We then conider the cae which the number of time ample T i limited and demontrate the robutne of uper reolution via tatitical analyi of the noie tructure A The Mathematical Theory of Super Reolution Super reolution eek to recover high frequency detail from the meaurement of low frequency component Mathematically, given a continuou ignal (τ with τ [, ], the Fourier erie coefficient are recorded a r(n = e jπnτ (dτ, n = f c, f c +,, f c Uing the operator F to denote the low frequency meauring operator, we can write r = F, in which r = [r( f c,, r(f c ] T and = (τ, τ Suppoe that the ignal (τ i pare, ie, (τ i a weighted um of everal pike: (τ = k δ τk, (5 k= in which k can be complex valued and τ k [, ] for all k Then r(n = k e jπnτ k, n = f c, f c +,, f c (6 k= Total variation minimization i introduced to encourage the parity in continuou ignal (τ, jut a l norm minimization produce pare ignal in the dicrete pace Total variation for the complex meaure i defined a TV = up (B j, j= the upremum being taken over all partition of the et [, ] into countable collection of dijoint meaurable et B j When take the form in (5, TV = K k= k, which i the l norm in the dicrete cae The following convex optimization formula wa propoed in [] to olve the uper reolution problem: min TV t F = r (7 When the ditance between any two τ i and τ j i larger than /f c, then the original pare ignal i the unique olution to the above convex optimization [] The continuou optimization (7 i olved via the following emidefinite programming []: max u,q Re[u r] [ ] Q u t u, (8 MN+ j { j =, Q i,i+j = j =,,, MN, i= where Q C (MN+ (MN+ i an Hermitian matrix B DOA etimation with Super Reolution DOA etimation with co-prime array can be related to (6 by a traightforward change of variable Letting τ k = d λ ( in(θ k for all k, the linear model of ( can be tranformed into r(n =e jπn d λ ( zn σ w n = e jπn d λ k e jπn d λ in(θ k = k= k e jπnτ k, (9 k= where n = MN, MN +,, MN, MN We ue T = {τ k, k K} to denote the upport et A theorem about the reolution and degree of freedom for co-prime array can be directly derived uing Theorem

4 in [] Before introducing the theorem, we firt define the minimum ditance between any two ource a (θ = min in(θ i in(θ j θ i,θ j,θ i θ j Theorem III Conider a co-prime array coniting of two linear array with N and M enor repectively The ditance between two conecutive enor are M d for the firt array and Nd for the econd array, where M and N are co-prime number, and d λ Suppoe we have K ource located at θ,, θ K If the minimum ditance follow the contraint that (θ λ MNd, then by olving the convex optimization (7 with the ignal model (9, one can recover the location θ k for k =,, K exactly The maximum number of ource that can be detected i given by K max = MNd λ Remark: With a traditional uniform linear array uing M + N enor, uper reolution theory can detect up to (M+Nd ource λ (M+Nd when (θ With the utilization of co-prime array, the ame number of enor can detect O(MN ource a indicated by traditional MUSIC theory [5] A we will how in the numerical example, implementing uper reolution framework provide with a larger degree of freedom and a finer reolution ability than thoe of MUSIC, ince the patial moothing in the MUSIC reduce the obtained virtual array aperture C Noiy Model for Super Reolution In a realitic enario, the covariance matrix R xx in ( cannot be obtained exactly except unle the number of ample T goe to infinity Normally the covariance matrix i approximated by the following equation: ˆR xx = x(tx (t T Subtracting the noie covariance matrix from both ide, we obtain ˆR xx σ I = AR A + E ( Here R i a diagonal matrix with k-th diagonal element ˆσ k = k (t T k(t The (m, n-th element in E i given a E mn = A mi A T nj i (t j (t + T + T i,j=,i j i= A mi i (tε n(t + T i= λ ε m (t i (ta ni ε m (tε n(t σ I mn, m, n L ( Similar to the operation in (3, vectorizing ( lead to, z = vec( ˆR xx = Φ(θ, θ,, θ K + σ n + e, ( where e i gained from vectorizing E For co-prime array, by removing repeated row in (, and orting them a conecutive lag from MNd to MNd, we get z = Φ + σ w + ẽ (3 Pleae note that only one element from ẽ correpond to the diagonal element from E Here = [ˆσ,, ˆσ K ]T By applying the tranformation technique in (9, we have r = F + e, ( where e(n = ẽ(ne jπn d λ Thu we can formulate the following uper reolution optimization problem, which conider the noie, a min TV t F r ɛ (5 The optimization can be olved via a emidefinite programming [5]: max u,q Re[u r] ɛ u [ ] Q u t u, (6 MN+ j { j =, Q i,i+j = j =,,, MN i= Here Q C (MN+ (MN+ i an Hermitian matrix To derive the tatitical behavior of each element in E we rely on two lemma about the concentration behavior of complex Gauian random variable Their proof are baed on the reult from [] and given in the Appendix Lemma III Let x(t and y(t, t =,, T be equence of iid, circularly-ymmetric complex normal ditribution with zero mean and variance equal to σx and σy repectively That i x(t CN (, σx and y(t CN (, σy Then ( Pr x(ty (t ɛ ( 8 exp ɛ 6σ x σ y (T σ x σ y + ɛ Lemma III Let x(t, t =,, T be a equence of iid, circularly-ymmetric complex normal ditribution with zero mean and variance equal to σx, ie, x(t CN (, σx When ɛ σxt, we obtain ( Pr x(tx (t T σx ɛ exp ( ɛ 6T σx With thee two concentration lemma, the probability of E mn being larger than a contant can be upper bounded For implicity of analyi, in the ret of thi paper we aume ε CN (, σ and i (t CN (, σ Lemma III3 Let E mn be given in ( Then for m n we have Pr( E mn ɛ 8 exp( C (ɛt + 6 exp( C (ɛt +8 exp( C 3 (ɛt

5 5 When m = n, we obtain Applying Lemma III, we have Pr( E mn ɛ 8 exp( C (ɛt + 6 exp( C (ɛt + exp( C (ɛt, when ɛ 6σ Here C (ɛ, C (ɛ, C 3 (ɛ and C (ɛ are increaing function of ɛ Proof: We ue T, T, and T 3 to denote the firt three term in ( The lat two term are denoted by T Firt we have Pr( E mn ɛ Pr( i= T i ɛ = Pr( i= T i ɛ Pr( T i ɛ, i= which lead to the inequality We alo have Pr( E mn ɛ T = T T T i,j=,i j i,j=,i j i,j=,i j i= Pr( T i ɛ (7 A mi A nj i (t j (t A mi A ni i (t j (t i (t j (t (8 The lat inequality follow from the fact that A mn for all m, n Thu Pr( T ɛ Pr i (t j (t ɛt i,j=,i j Then it i traightforward to find that ( Pr( T ɛ T Pr i (t j (t ɛt, K(K for ome i, j with i j Uing Lemma III Pr( T ɛ 8 exp( C (ɛt, (9 with C (ɛ = ɛ 6σ K(K (6σ K(K +ɛ For the econd term T, we have Pr( T ɛ 8 exp( C (ɛt, ( with C (ɛ = ɛ 6σ σk(6σ σk+ɛ For the third term, we have the ame reult a the econd one, given a Pr( T 3 ɛ 8 exp( C (ɛt ( When m n, the lat term T = T T ε m(tε n(t, and by Lemma III, Pr( T ɛ 8 exp( C 3(ɛT, (3 ɛ with C 3 (ɛ = 6σ (6σ +ɛ When m = n, the lat term i given a T = T T ε m(tε m(t σ, thu the probability i bounded a Pr( T ɛ exp( C (ɛt, ( where C (ɛ = ɛ 56σ and ɛ 6σ Applying the reult from ε (9, (, (, (3 and ( to inequality (7, we prove the remain In order to analyze the robutne of uper reolution, a high reolution kernel i introduced in [5] referred to a the Fejér kernel In our cae it ha a cut-off frequency f h > MN a i given by K h (t = f h fh T = K h ( opt L C A mi i (tε M N ɛ, T n(t with probability at leat αe γ(ɛt when ɛ i= A mi T i (tε n (t 6 MN + σ, where γ(ɛ i a increaing function of ɛ T i (tε n (t Here C i a poitive contant number i= i= Proof: With the fact that d ( λ, τ k [, ] for all k after tranformation (9 It wa hown in [5] that when the two Following imilar argument a for T, we obtain that condition opt TV TV and F F ( opt L ɛ ( Pr( T ɛ T Pr i (tε n (t ɛt hold, it uffice to obtain fh K K h ( opt L C M N ɛ f h = f h + (f h + k e jπkt k= f ( h in(π(fh + t in(πt (5 Uing the high reolution kernel K h (t introduced in (5, we can how that by olving the convex optimization problem in (5 the high reolution detail of the original ignal (τ can be recovered with high probability, even though the ample number T i finite for co-prime array Theorem III Conider a co-prime array coniting of two linear array with N and M enor repectively The ditance between two conecutive enor are Md for the firt array and Nd for the econd array, where M and N are coprime number, and d λ Let (τ = K k= kδ τk T time ample point are collected for each receiver, by taking the tranformation in (9 and olving the optimization (5 with opt a the optimal function, we can how that

6 6 In order to atify thee condition, the tatitical behavior of e in ( i analyzed firt Uing a imilar argument to (7, we have Pr( e ɛ = MN n= MN MN n= MN Pr( e(n Pr( ẽ(n ɛ MN + ɛ MN + (6 The inequality follow from the fact that e(n = ẽ(n MN element of ẽ are taken from E mn when m n, and one element of ẽ i taken from E mn when m = n Therefore, by applying the reult from Lemma III3, we can how that F r = e ɛ with a probability of at leat αe γ(ɛt, and γ(ɛ i a increaing function of ɛ The lemma require that ɛ 6 MN + σ The firt condition hold due to the optimization problem in (5, and i feaible with high probability Furthermore, F F ( opt L F F ( opt L = F ( opt L for all τ uch that opt (τ F opt r L + F r L ɛ The firt inequality follow from the Cauchy-Schwarz inequality Therefore the proof i complete Remark: K h defined in (5 i a low pa filter with cut-off frequency f h > MN By convolving it with the recontructed error opt we get the recontruction error detail up to the frequency f h By olving optimization (5, uing noiy meaurement one can recontruct the high frequency detail of with high probability Thi probability goe to one exponentially a the number of ample T goe to IV DOA ESTIMATION VIA SEMIDEFINITE PROGRAMMING AND ROOT FINDING We now derive an optimization framework to recontruct for co-prime array For DOA etimation the noie power σ i normally unknown Therefore, the optimization mut be modified to include thi effect A more realitic optimization i reformulated a min TV t F r σ w ɛ, (7,σ in which w n = w n e jπn d λ The dual problem take the form max u Re[u r] ɛ u t F u L, Re[u w] (8 The derivation of the dual problem i given in the Appendix Since u = i a feaible olution, trong duality hold according to the general Slater condition [3] Due to the firt contraint in (8, the problem itelf i till an infinite dimenional optimization It wa hown in [] that the firt contraint can be recat a a emidefinite matrix contraint Thu the infinite dimenional dual problem i equivalent to the following emidefinite programming (SDP: max u,q Re[u r] ɛ u [ ] Q u t u, Re[u w], (9 MN+ j { j =, Q i,i+j = j =,,, MN i= Here Q C (MN+ (MN+ i an Hermitian matrix The optimization problem can be eaily olved by uing the CVX package [3] Solving (9 yield the optimal olution only for the dual problem The following lemma i introduced to link the olution of the primal and dual problem Lemma IV Let opt and u opt be the optimal olution of the primal problem (7 and dual problem (9 repectively Then F u opt (τ = gn( opt (τ Proof: Let σ opt be the noie power etimated in the primal problem Since trong duality hold, we have opt TV = Re r, u opt ɛ u opt = Re r F opt σ optw, u opt ɛ u opt + Re F opt + σ optw, u opt Re F opt + σ optw, u opt Re F opt, u opt The firt inequality follow from the Cauchy-Schwarz inequality and the fact that r F opt σoptw ɛ The econd inequality reult from Re[u optw] Becaue F u opt L, we have opt TV Re opt, F u opt Therefore opt TV = Re opt, F u opt hold and we have the deired reult needed to atify thi equality The upport et T can be etimated by root-finding baed on the trigonometric polynomial F u(τ = Let T et denote the etimation of the upport et, and ue τ et [i] to denote element in T et with i K et A matrix F et C (MN+ Ket can be formulated, with meaurement r expreed a r = F et + σ w + e, (3 in which R Ket and e jmndπτet[] e jmndπτet[ket] e j(mn dπτet[] e j(mn dπτet[ket] F et = e jmndπτet[] e jmndπτet[ket] Due to the the numerical iue in the root finding proce, the cardinality of T et i normally larger than the cardinality of T, ie, K et K It i poible in ome cae that K et MN +, which lead to an ill conditional linear ytem (3 Sparity can then be exploited on thi ignal A convex optimization in the dicrete domain can be formulated a min t F et r σ w ɛ d (3,σ

7 7 The ɛ d in (3 i normally choen to be larger than ɛ in (7 ince the noie level i expected to be higher in (3 due to inevitable error introduced in the root finding proce Auming that the optimization olution of (3 i et R Ket, the etimation of in the continuou domain can be repreented a et opt = et [i]δ τet[i] i= V EXTENSION: SOURCE NUMBER DETECTION Traditional ource number detection for array proceing i typically performed by exploiting eigenvalue from the ample covariance matrix For coprime array, thi covariance matrix can be obtained by performing patial moothing on z The ame idea can alo be implemented on the pare ignal recovered from the previou ection Ideally, after orting it element in a decending order, the ignal et recontructed from (3 hould follow et [] et [] et [K] et [K + ] = = et [K et ] = The SORTE algorithm can be applied to thi erie The difference of the element from et i et [i] = et [i] et [i + ], for i =,, K et The gap meaure in SORTE i given a { var[i+] SORTE(i = var[i] var[i], i =,, K et, + var[i] =, (3 where ( K et K et var[i] = et [m] et [n] K et i K et i m=i n=i (33 The number of the ource can be determined by following the criteron ˆK = argmin i SORTE(i It only work when K et > due to the definition of SORTE(i in (3 When K et, ince T et i obtained from the rooting finding proce baed on the continuou pare recovery, we imply let ˆK = Ket We will refer to thi continuou pare recovery baed SORTE a CSORTE conidering grid mimatche [3] In [3], a LASSO formulation wa ued to perform the DOA etimation Here we implement an equivalent form of LASSO, ie, Bai puruit, to perform the comparion The MUSIC method in thi imulation follow the patial moothing technique in [5] For the dicrete pare recovery method, we take the grid from to, with tep ize 5 for in(θ The noie level ɛ in the optimization formula are choen by cro validation We conider 5 narrow band ignal located at in(θ = [ 8876, 76, 636, 596, 388, 55, 3, 6, 6,, 369, 97, 68, 75, 87] We how that the continuou pare recovery method yield better reult in term of detection ability, reolution, and etimation accuracy A Degree of Freedom In thi firt numerical example, we verify that continuou pare recovery increae the degree of freedom to O(MN by implementing the coprime array tructure The ɛ for CSR i taken a 5, and ɛ d i taken a while DSR ue ɛ = The number of time ample i 5 and the SNR i choen to be db In Fig, we ue a dahed line to repreent the true direction of arrival The CPU time for running CSR wa 73 econd DSR took 78 econd, while MUSIC algorithm only ued 8 econd For MUSIC we implement a root MUSIC algorithm to etimate the location of each ource, and the number of ource i aumed to be given The average etimation error for CSR, DSR, and root MUSIC are 3%, 6%, and % repectively We can ee that all the three method achieve O(M N In the following ubection, we tet the etimation accuracy of thee three method via Monte Carlo imulation Normalized Spectrum 8 6 Continuou SR MUSIC method Dicrete SR 8 VI NUMERICAL RESULTS In thi ection, we preent everal numerical example to how the merit of implementing uper reolution technique on co-prime array We conider a co-prime array with enor One et of enor i located at poition [, 3, 6, 9, ]d, and the econd et of enor i located at poition [, 5,, 5,, 5]d, where d i taken a half of the wavelength The firt enor from both et are collocated It i eay to how that the correlation matrix generate a virtual array with lag from 7d to 7d We compare the continuou pare recovery (CSR technique with MUISC and alo with the dicrete pare recovery method (DSR in(θ Fig : Normalized pectra for CSR, MUSIC, and DSR, with T = 5 and SNR= db B Etimation Accuracy In thi ection, we compare the continuou pare recovery method with the MUSIC algorithm and alo the dicrete

8 8 pare recovery method via Monte Carlo imulation Since traditional MUSIC doe not yield the DOA of each ource directly, we conider the Root MUSIC algorithm intead For implicity, we will till refer it a MUSIC in thi ection The number of ource i aumed to be known for the MUSIC algorithm in thi imulation, while pare method do not aume thi a priori ɛ and ɛ d are choen to be 5 and in thi imulation, while dicrete SR ue ɛ = DOA Etimation Error x SNR(dB MUSIC Dicrete SR Continuou SR Fig 3: DOA etimation error for CSR, MUSIC, and DSR, with T = 5 Figure 3 how the DOA etimation error with repect to changing SNR after 5 Monte Carlo imulation The etimation error i calculated baed on the ine function of the DOA The average CPU time for running CSR, DSR and MUSIC are 693, 93, and 6 repectively We can ee that CSR perform better than DSR uniformly with le computing time Both pare recovery method achieve better DOA etimation accuracy than MUSIC The accuracy of DSR can be further improved by taking finer grid with a maller tepize However, thi will low down the computing of DSR further In Fig we how that with a changing number of naphot the propoed CSR alo exhibit better etimation accuracy than either DSR or MUSIC The average CPU time for running CSR, DSR and MUSIC are 65, 79, and 3 repectively The performance of MUSIC and DSR approach the performance of CSR when the number of naphot approache 5 We can ee that implementing CSR can ave ampling time by taking a mall number of naphot to achieve the ame etimation accuracy a the MUSIC algorithm The parameter ɛ and ɛ d are equal to 5 and in thi imulation C Source Number Detection Performance Comparion In thi ection, we compare the ource number detection performance of the propoed CSORTE with that of traditional SORTE applied to the covariance matrix The SNR i et to db while the number of naphot i 3 We range the number of ource from to 7 Since thi co-prime array DOA Etimation Error x Number of Sample MUSIC Dicrete SR Continuou SR Fig : DOA etimation error for CSR, MUSIC, and DSR, with SNR= db tructure yield conecutive lag from 7d to 7d, 7 i the maximum number of ource that can be detected theoretically via technique baed on the covariance matrix Probability of Detection CSORTE SORTE Number of Target Fig 5: Source number detection uing CSORTE and SORTE, with SNR= db, T = 3 Figure 7 how the probability of detection with repect to the number of ource after 5 Monte Carlo imulation In the CSR, ɛ i choen to be 5σ, and ɛ d i et to be ɛ When the number of ource i le than 5, CSORTE and SORTE yield comparable reult However, SORTE fail after the number of ource i larger than 5, while CSORTE give table performance and alo exhibit perfect detection even when the number of ource reache the theoretical limit of 7 Dicrete pare recovery can alo be combined with SORTE to perform ource number detection However, the detection accuracy i jeopardized by the puriou ignal from the recontructed ignal uing DSR Therefore SORTE baed on DSR i not included here Thi imulation how that the parity baed method offer more degree of freedom than

9 9 the ubpace baed method D Reolution Ability Finally we compare the reolution abilitie of the MUSIC algorithm and the propoed continuou pare recovery method We how that CSR i capable of reolving very cloely located ignal In the firt imulation, two ource are cloely located at 3 and 3 The value of ɛ i choen to be 7σ and ɛ d i et to be ɛ in the CSR, where σ i the noie power thee two target becaue the traditional SORTE fail to etimate the number of ource correctly CSR reolve the two ource uccefully even though a priori information about the number of ource i not aumed to be given In Fig 7, we lower the SNR to 5 db, and we notice that even given the number of ource, the MUSIC algorithm fail to reolve the two cloely located ource while CSR reolve uccefully Next we conduct a imulation baed on Monte Carlo run to compare the reolution ability of the CSORTE and the traditional SORTE algorithm 5 Continuou SR 9 Normalized Spectrum MUSIC method A MUSIC method B in(θ Fig 6: Source number detection uing CSR and the MUSIC algorithm, with SNR= db, T = 5 Probability of Reolution CSORTE SORTE SNR(dB Fig 8: Comparion of reolution performance of CSORTE and SORTE, with T = Normalized Spectrum 5 Continuou SR MUSIC method A MUSIC method B in(θ Fig 7: Source number detection uing CSR and MUSIC algorithm, with SNR= 5 db, T = 5 Figure 6 how a numerical example when the SNR i db and the number of naphot i 5 Normalized pectra are plotted for three method MUSIC method A i the MUSIC algorithm with the aumption that the number of ource i known while MUSIC method B i the MUSIC method relying on traditional SORTE to provide the etimated number of ource We can ee that MUSIC method B fail to reolve Figure 8 how the reolution performance in detecting two ource located at 3 and 3, uing CSORTE and SORTE method after 5 Monte Carlo run The parameter ɛ i choen to be 7σ, and ɛ d i et to be ɛ in the CSR We can ee that CSORTE outperform the traditional SORTE when detecting the two cloely located ource VII CONCLUSIONS AND FUTURE WORK In thi work, we extended the mathematical theory of uper reolution to the topic of DOA etimation uing co-prime array A primal-dual approach wa utilized to tranform the original infinite dimenional optimization to a olvable emidefinite program After etimating the candidate upport et by olving the emidefinite program, a mall cale pare recovery problem can be olved efficiently The robutne of the propoed uper reolution approach wa verified by performing tatitical analyi of the noie inherit to co-prime array proceing A ource number detection algorithm wa then propoed by combining the exiting SORTE algorithm with the recontructed pectrum from convex optimization Via numerical example, we howed that the propoed method achieve a more accurate DOA etimation while providing more degree of freedom, and alo exhibit a more powerful reolution ability than the traditional MUSIC algorithm with patial moothing Although implementing the continuou pare recovery method ave ampling time in obtaining a certain etimation accuracy compared with MUSIC, one hortcoming of

10 the propoed pare method i that olving the emidefinite program i more time conuming than the MUSIC algorithm Fat algorithm development could be an intereting topic for future work It i alo of interet to develop a ytematic way to chooe ɛ and ɛ d in the optimization formula One major aumption made by current co-prime array reearch i that ource are uncorrelated Incoporating correlation among ource i alo an important topic for future work APPENDIX Proof of Lemma III: Firt we have x(ty (t = Re[x(t]Re[y(t] + Im[x(t]Im[y(t] j Re[x(t]Im[y(t] + j Im[x(t]Re[y(t] Firt we have the equation According to the ame procedure ued in the proof of lemma x(tx (t T σx = Re[x(t] + Im[x(t] T σx III3, we have ( Likewie, we obtain Pr x(ty (t ɛ ( Pr Re[x(t]Re[y(t] ɛ Pr( x(tx(t T σx ɛ ( +Pr Im[x(t]Im[y(t] ɛ =Pr( Re[x(t] T σ x ɛ ( +Pr Re[x(t]Im[y(t] ɛ +Pr( Im[x(t] T σ x ɛ ( With lemma A, we have the deired reult for lemma III +Pr Im[x(t]Re[y(t] ɛ Derivation of the dual problem in Section IV By introducing by variable z C MN+, the original primal problem i equivalent to the following optimization: Uing the lemma 6 from [], we finih the proof of lemma III min TV,σ,z Before the next proof, we need to how that the quare um t z of iid Gauian random variable concentrate around the um ɛ, z = F r σ w of the variance It utilize the reult in lemma 7 from [] Lemma A Let x(t, t =,, T be a equence of iid normal ditribution with zero mean and variance equal to σ, ie, x(t N (, σ Then ( Pr x(t T σ ɛ exp( ɛ 6σ T when ɛ σ T Proof: From the reult in [], we know that for any poitive c, we have two aymmetric bound a ( T Pr x(t T σ σ T c + σ c exp( c, ( T Pr x(t T σ σ T c exp( c When c T, we obtain ( T Pr x(t T σ σ T c exp( c, ( T Pr x(t T σ σ T c exp( c Combing the above two inequalitie, we get ( Pr x(t T σ σ T c exp( c, which yield the reult by replacing σ T c with ɛ while maintaining c T Proof of Lemma III: With the Lagrangian multiplier v R and u C MN+, the Lagrangian function i given a L(, z, σ, u, v = TV + v( z ɛ The dual function i given a g(u, v = Re[u r] vɛ +Re[u (r F σ w z] + inf,z,σ { TV Re[u F ] σ Re[u w] + v z u z} The Lagrangian multiplier u and v in the domain of the dual function have to atify the following three contraint: F u L, Re[u w], v z z = u From the third contraint, we have v = u Therefore, we obtain the dual problem tated in (9

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