Atomic Norm Based Localization of Far-Field and Near-Field Signals with Generalized Symmetric Arrays
Most localization methods for mixed far-field (FF) and near-field (NF) sources are based on uniform linear array (ULA) rather than sparse linear array (SLA). In this paper, we propose a localization method for mixed FF and NF sources based on the gen…
Authors: Xiaohuan Wu, Wei-Ping Zhu, Jun Yan
A TOMIC NORM B ASED LOCALIZA TION OF F AR-FIELD AND NEAR-FIELD SIGNALS WITH GENERALIZED SYMMETRIC ARRA YS Xiaohuan W u 1 , W ei-Ping Zhu 1 , 2 and J un Y an 1 1 School of T elecommunication and Information Engineering, Nanjing Uni versity of Posts and T elecommunications, Nanjing, China 2 Department of Electrical and Computer Engineering, Concordia Uni versity , Montreal, Canada Email: { xiaohuanwu, zwp, yanj } @njupt.edu.cn ABSTRA CT Most localization methods for mixed far-field (FF) and near- field (NF) sources are based on uniform linear array (ULA) rather than sparse linear array (SLA). In this paper, we pro- pose a localization method for mixed FF and NF sources based on the generalized symmetric linear arrays, which include ULAs, Cantor array , Fractal array and many other SLAs. Our method consists of two steps. In the first step, the high-order statistics of the array output is exploited to increase the degree of freedom. Then the direction-of-arriv als (DO As) of the FF and NF sources are jointly estimated by us- ing the recently proposed atomic norm minimization ( ANM), which belongs to the gridless super-resolution method since the discretization of the parameter space is not required. In the second step, the ranges are giv en by MUSIC-like one- dimensional searching. Simulations results are provided to demonstrate the advantages of our method. Index T erms — Source localization, far-field, near-field, generalized symmetric arrays, atomic norm minimization. 1. INTRODUCTION Source localization is a fundamental problem in array signal processing and has receiv ed considerable attention. Based on the far-field (FF) source assumption where the impinged signals are assumed to be plane-wav e, numerous methods hav e been proposed for FF source localization, i.e., direction- of-arriv al (DOA) estimation [1 – 5] [6]. Howe ver , when the sources are close to the array and lie in the near -field (NF) region (i.e., the Fresnel re gion), the impinged signals are spherical wa ve rather than plane-wa ve. In this case, the steer- ing vectors are characterized by two independent parameters: DO A and range. Hence, the DOA estimation methods for FF source localization can not be applied to the NF source localization. In order to deal with this problem, the nonlinear time delay of the spherical wav efront model is approximated This work was supported by the National Natural Science Founda- tion of China (61801245, 61772287 and 61771256), the NSF of Jiangsu Province (BK20180748), the NSF of Jiangsu Higher Education Institutions (18KJB510032), NUPTSF (NY218101), open research fund of Ke y Lab of Broadband W ireless Communication and Sensor Network T echnology (JZNY201913) and the Key University Science Research Project of Jiangsu Province (18KJ A510004). into a quadratic wa vefront model by using the second-order T aylor expansion. Based on this approximation, a lar ge num- ber of NF source localization methods hav e been proposed. For instance, He et al. have proposed an oblique projection based MUSIC (OPMUSIC) algorithm for mix ed FF and NF sources localization [7]. An ef ficient subspace-based local- ization method without eigendecomposition has been pro- posed in [8]. Howe ver , the maximum number of resolv able sources of these methods are limited to half of the number of sensors. Liang et al. hav e proposed a cumulant based algorithm called two-stage MUSIC (TSMUSIC) by using the high-order statistics (HOS) of the array output [9]. By using the HOS and compressed sensing theory , a sparse method has been presented for mix ed sources localization [10]. Howe ver , sparse method requires to discretize the angle space hence its performance is limited by the so called ”basis mismatch” ef- fect. Note that, all the aforementioned methods are proposed based on the uniform linear array (ULA) and most of them can not be extended to the sparse linear array (SLA). As a result, the maximum number of resolvable sources of these methods can not exceed the number of sensors. Recently , the SLA has been studied in FF DO A estimation by exploiting the coarray of the physical array to increase the array aperture [11–14]. Due to the extended array aperture, the resolution and the maximum number of resolvable sources can be greatly improv ed. Nev ertheless, the classical SLAs, e.g., coprime array , nested array can not be directly emplo yed for NF source localization since most of the NF source local- ization methods require symmetric array structure. By using the special geometry of the nested array (i.e., two subarrays without ov erlapping), sev eral symmetric nested arrays hav e been proposed [15, 16]. Howe ver , these literatures focus on constructing specific SLA to e xtend the array aperture. A uni- fied algorithm for generalized symmetric SLA in NF source localization area is still missing. Furthermore, all these meth- ods choose MUSIC for DOA estimation which suffers from basis mismatch effect. In this paper , we propose a unified localization method for mixed FF and NF sources which can be applied to any sparse/uniform symmetric redundancy array . The high-order cumulants of the array output are exploited to increase the degrees-of-freedom (DoFs). The atomic norm is employed to eliminate the basis mismatch effect in DO A estimation and able to provide super -resolution. Moreov er, the proposed method does not require the prior kno wledge of the noise power as well as the number of sources or FF/NF sources. 2. SIGNAL MODEL W e first giv e the definition of the generalized symmetric lin- ear array that will be used throughout of this paper . Let [ N ] = {− N , · · · , − 1 , 0 , 1 , · · · , N } . The indices of the array sensors are denoted as Ω = { Ω − M , · · · , Ω − 1 , Ω 0 , Ω 1 , · · · , Ω M } ⊆ [ N ] where { Ω 0 , · · · , Ω M } are non-negati ve unequal integers and sorted ascendingly with Ω 0 = 0 and Ω M = N . Due to the symmetric property of the array , we hav e Ω − m = − Ω m , m ∈ [ M ] . Let | Ω | denote the cardinality of Ω , it is easy to see that M = | Ω |− 1 2 ≤ N . If N = M , Ω represents a symmetric ULA while if N > M , Ω represents a symmetric SLA. It should be noted that ULA can be reg arded as a spe- cial case of SLA. As a result, only the SLA case is considered in the rest of this paper . W e then provide the definition of coarray below . Definition 1 The coarray D of a physical array Ω is defined as, D = { Ω m − Ω n − N : m, n ∈ Ω , m ≥ n } ⊆ [ N ] . Ω is called a redundancy array if D = [ N ] (a.k.a. hole-fr ee coarray). Otherwise, Ω is called a non-r edundancy array . It is sho wn that the maximum number of detectable sources using the array Ω is determined by its coarray D rather than the number of physical sensors [17]. In fact, a redundancy array is able to detect up to 2 N sources, which is greater than its number of sensors 2 M + 1 if N > M . T o better understand our array , we here provide a simple example. Example 1 A symmetric SLA with Ω = {− 3 , − 2 , 0 , 2 , 3 } has the coarray D = [3] which is hole-fr ee. By using pr oper algorithms, the array can detect up to 2 N = 6 sour ces, whic h is gr eater than the number of sensors. It should be pointed out that the aforementioned symmetric SLAs with hole-free coarray not only include the array in Example 1, b ut also contain the recently proposed cantor ar - ray [18] and fractal array [19]. 1 . In fact, the proposed method can be applied to any symmetric redundancy array . T o avoid the angle ambiguity , the minimum sensor spacing d should satisfy d ≤ λ/ 4 where λ is the wa velength. W e then assume K narro wband uncorrelated mixed FF and NF sources impinge onto the array and the k -th source is characterized by a parameter pair ( θ k , r k ) . The array output is giv en by , x Ω ( t ) = A Ω s ( t ) + n Ω ( t ) , (1) where x Ω ( t ) = [ x Ω − M ( t ) , · · · , x Ω M ( t )] T is the array out- put, s ( t ) = [ s 1 ( t ) , · · · , s K ( t )] T is the signal w av eform, n Ω ( t ) denotes the additi ve white Gaussian noise with zero mean and A Ω = [ a Ω ( θ 1 , r 1 ) , · · · , a Ω ( θ K , r K )] denotes the manifold matrix of the array , in which a Ω ( θ k , r k ) = [ e j [ − Ω M ω k +( − Ω M ) 2 φ k ] , · · · , 1 , · · · , e j [Ω M ω k +Ω 2 M φ k ] ] T is the steering vector of the k -th signal with ω k = − 2 π d λ sin( θ k ) , (2) 1 Note that the factal array is able to extend any known symmetric redun- dancy array to a much lar ger one. φ k = π d 2 λr k cos 2 ( θ k ) . (3) Note that we unify the steering vectors of the FF and NF sources as a Ω ( θ k , r k ) since the steering vector of the FF source can be represented as a Ω ( θ k , + ∞ ) . 3. THE PROPOSED METHOD 3.1. DO A Estimation of FF and NF Sources Similar to TSMUSIC, the fourth-order cumulants are utilized to construct a Hermitian matrix without the range parameter for DO A estimation. Specifically , we define the fourth-order cumulant as C Ω ( ¯ m, ¯ n ) = − cum { x Ω m ( t ) , x ∗ − Ω m ( t ) , x − Ω n ( t ) , x ∗ Ω n ( t ) } = K X k =1 − c s k e j 2(Ω m − Ω n ) ω k , (4) where ¯ m = m + M + 1 , ¯ n = n + M + 1 , m, n ∈ [ M ] and c s k < 0 is the fourth-order cumulant of s k ( t ) . 2 C Ω can be compactly written as the following Hermitian matrix, C Ω = K X k =1 − c s k ¯ a Ω ( θ k ) ¯ a H Ω ( θ k ) = ¯ A Ω C s ¯ A H Ω , (5) where C s = diag ([ − c s 1 , · · · , − c s K ]) contains the fourth- order cumulants of the signals, ¯ A Ω = [ ¯ a Ω ( θ 1 ) , · · · , ¯ a Ω ( θ K )] and ¯ a Ω ( θ k ) = [ e j 2Ω − M ω k , · · · , 1 , · · · , e j 2Ω M ω k ] T . It can be seen from (5) that, the range parameter is remov ed from C Ω . Hence matrix C Ω can be regarded as the cov ariance matrix of a virtual array output with manifold matrix ¯ A Ω which is positiv e semidefinite (PSD). Although we can apply the subspace based methods such as MUSIC to C Ω to estimate the DO As, directly using MUSIC for DOA estimation can not utilize the coarray property to increase the DoFs. Belo w we use the atomic norm theory which can fully utilize the coarray to estimate the DO As. First, denote ¯ a ( θ k ) = [ e j 2( − N ) ω k , · · · , 1 , · · · , e j 2 N ω k ] T as the steering vector of the coarray D and Γ as a selecting matrix such that the j -th row of Γ contains all 0 s but a single 1 at the (Ω j − M − 1 + N + 1) -th position. It is clear that ¯ a Ω ( θ k ) = Γ ¯ a ( θ k ) and ¯ A Ω = Γ ¯ A . (6) Bring (6) into (5), we hav e, C Ω = Γ ¯ AC s ¯ A H Γ T , = Γ C Γ T , (7) where C = ¯ AC s ¯ A H denotes the cov ariance matrix of the coarray output. It can be seen that compared to C Ω , C has a 2 Since many practical signals such as sinusoidal signal, amplitude shift keying (ASK) signal and phase shift ke ying (PSK) signal are sub-Gaussian process, we assume s k ( t ) is zero-mean stationary random processes with negati ve kurtosis, i.e., c s k < 0 . much lar ger dimensionality which can be used to increase the DoF . T o do this, let’ s first rewrite C as, C = K X k =1 c s k ¯ a ( θ k ) ¯ a H ( θ k ) , K X k =1 c s k B ( θ k ) . (8) And then define an atom set A = { B ( ϑ k ) = ¯ a ( ϑ k ) ¯ a H ( ϑ k ) , ϑ k ∈ ( − 90 ◦ , 90 ◦ ] } . (9) Based on the atomic norm theory , the atomic norm of C can be defined as k C k A = inf ( X k c k | C = X k c k B ( θ k ) , B ( θ k ) ∈ A , c k > 0 ) . (10) Although (10) pro vides a proper decomposition of C by us- ing the atoms in A with respect to the DO As, it is still un- known how to compute the atomic norm from the definition. T o solv e this problem, the atomic norm can be transformed to a semidefinite programming (SDP). Formally , we hav e, Theorem 1 ( [20]) The atomic norm defined in (10) equals the optimal value of the following SDP , k C k A = min Z , u 1 2(2 N + 1) tr [ Z + T ( u )] s.t. Z C C T ( u ) ≥ 0 , (11) wher e T ( u ) ∈ C (2 N +1) × (2 N +1) denotes a T oeplitz matrix with u being its first r ow . It is sho wn that the DOAs are encoded in T ( u ) whose ( ¯ m, ¯ n ) - th element can be giv en as [21] ( T ( u )) ¯ m, ¯ n = K X k =1 c s k e j 2( m − n ) ω k m, n ∈ [ N ] . (12) Hence the DOAs can be retriev ed from the V andermonde de- composition of T ( u ) [22]. Consequently , we propose the fol- lowing atomic norm minimization (ANM) problem to retriev e T ( u ) , 3 min Z , u , C tr [ Z + T ( u )] s.t. Z C C T ( u ) ≥ 0 , Γ C Γ T = C Ω . (13) In practice, since we can only observe the sample cumulant matrix of the sparse array output b C Ω rather than the exact one C Ω , the DOAs can be retrieved based on the following ANM problem, min Z , u , C tr [ Z + T ( u )] s.t. Z C C T ( u ) ≥ 0 , k C Ω − b C Ω k F ≤ β , (14) 3 The term 1 2(2 N +1) is omitted for brevity . where β is an upper bound of the noise energy . Solving the problem (14) by using CVX gi ves the esti- mation of T ( u ) . Then the DOAs can be retrie ved by using V andermonde decomposition or the subspace methods such as root-MUSIC. Remark 1 By exploiting the coarray pr operty of the r edun- dancy array , our method is able to estimate more sour ces than sensors in terms of DO A estimation. In particular , the max- imum number of detectable sour ces is K max = 2 N , whic h is gr eater than the number of sensors 2 M + 1 if M < N . 3.2. Range Estimation of the NF Sources In the previous part, we constructed a fourth-order cumulant matrix for DO A estimation based on the special structure of the array . In this part, ho wever , it is unable to construct a fourth-order cumulant matrix for range estimation since the array is sparse. Hence, we utilize the co variance matrix of the array output and apply MUSIC-like algorithm to retrieve the ranges. In particular, we first construct the sample co variance matrix b R Ω and then apply the eigen-decomposition to b R Ω to obtain the matrix U n containing eigen vectors of b R Ω with respect to the 2 M + 1 − K minimum eigen values. W ith the estimated DO As ˆ θ k , the range r k of the k -th signal can be obtained by solving the following problem, r k = min r a Ω ( θ k , r ) U n U H n a H Ω ( θ k , r ) , (15) which can be solved by 1-D searching. Note that the FF source is a special NF source with r → + ∞ , we choose the range searching area as (0 . 62( D 2 /λ ) 0 . 5 , r max ) , where r max is selected to be larger than 2 D 2 /λ with D being the array aperture. If the estimated range is located in the Fresnel re- gion (0 . 62( D 2 /λ ) 0 . 5 , 2 D 2 /λ ) , the corresponding source is classified as the NF source. On the other hand, if the esti- mated range is larger than 2 D 2 /λ , the source belongs to the FF source and we let r = + ∞ . Note that the DOAs and ranges are automatically paired by using (15). 4. SIMULA TION RESUL TS In this section, we ev aluate the performance of our pro- posed method with comparison to OPMUSIC [7], TSMU- SIC [9], high-order statistics based sparse signal representa- tion method (HOS-SSR) [10] and the Crammer-Rao lower bound (CRLB) [23]. 4 The number of sources K is assumed to be known for OPMUSIC, TSMUSIC and HOS-SSR but unknown for our method. Meanwhile, OPMUSIC requires the prior knowledge of the number of FF sources. W e first consider a 7 -element ULA (i.e., Ω = [3] ) with the sensor spacing d = λ/ 4 and assume two narrowband equal-power source signals consisting one FF source from { 0 ◦ , + ∞} and one NF source from { 25 ◦ , 2 λ } impinge onto the ULA. W e set the number of snapshots L = 200 and ev aluate these methods by comparing their RMSEs of the estimates with the SNR v arying from − 10 dB to 25 dB and 4 T o reduce complexity , iterative grid refinement procedure is employed in HOS-SSR. -10 -5 0 5 10 15 20 25 SNR(dB) 10 -1 10 0 10 1 RMSE(Deg.) OPMUSIC TSMUSIC HOS-SSR Proposed CRLB (a) DO A performance of FF sources -10 -5 0 5 10 15 20 25 SNR(dB) 10 -1 10 0 10 1 RMSE(Deg.) OPMUSIC TSMUSIC HOS-SSR Proposed CRLB (b) DO A performance of NF sources -10 -5 0 5 10 15 20 25 SNR(dB) 10 -2 10 -1 10 0 10 1 RMSE(Deg.) OPMUSIC TSMUSIC HOS-SSR Proposed CRLB (c) Range performance of NF sources Fig. 1 : DO A and range estimation comparison for one FF source from { 0 ◦ , + ∞} and one NF source from { 25 ◦ , 2 λ } impinging onto a 7 -element ULA with L = 200 . -10 -5 0 5 10 15 20 25 SNR(dB) 10 -1 10 0 10 1 RMSE(Deg.) OPMUSIC TSMUSIC HOS-SSR Proposed CRLB (a) DO A performance of FF sources -10 -5 0 5 10 15 20 25 SNR(dB) 10 -1 10 0 10 1 RMSE(Deg.) OPMUSIC TSMUSIC HOS-SSR Proposed CRLB (b) DO A performance of NF sources -10 -5 0 5 10 15 20 25 SNR(dB) 10 -1 10 0 10 1 RMSE(Deg.) HOS-SSR Proposed CRLB (c) Range performance of NF sources Fig. 2 : DO A and range estimation comparison for one FF source from { 0 ◦ , + ∞} and one NF source from { 25 ◦ , 2 λ } impinging onto a 5 -element symmetric SLA with L = 200 . show the results in Fig. 1. It can be observed from Fig. 1(a) that our proposed method, OPMUSIC and TSMUSIC are able to approach and coincide with CRLB curve as the SNR grows. HOS-SSR deviates gradually from CRLB because of the model mismatch between the ` 0 -norm and ` 1 -norm mini- mization models. In Fig. 1(b), our method and TSMUSIC can also show satisfying performance whereas OPMUSIC shows worse accuracy since it only utilizes partial information of the cov ariance matrix of the array output to estimate the DO As of the NF sources. In Fig. 1(c), we can see that TSMU- SIC cannot provide good performance and there e xists a large gap between the curves of TSMUSIC and CRLB. In contrast, OPMUSIC and our method can coincide with the CRLB and OPMUSIC performs better in low SNR region. HOS-SSR deviates from the CRLB in range estimation as well. In the second experiment, we replace the ULA with a 5 -element symmetric SLA with Ω = {− 3 , − 2 , 0 , 2 , 3 } and other settings are the same as Fig. 1. From the RMSEs comparison displayed in Fig. 2 we can see that these four methods show similar performance as in the ULA case in DO A estimation of FF source. While for NF source local- ization, OPMUSIC requires spatial smoothing hence we can observe that it f ails in Fig. 2(b). For range estimation, TSMU- SIC requires the uniform array structure and thus fails in the SLA case. Therefore, we omit the curves of OPMUSIC and TSMUSIC in Fig. 2(c) from which it can be seen that our pro- posed method can still approach the CRLB while HOS-SSR deviates from the CRLB as the SNR grows. In summary , our method performs better than other compared methods in DO A and range estimation in both ULA and SLA cases. 5. CONCLUSIONS In this paper , we proposed a mixed source localization based on ANM with symmetric redundancy linear arrays, includ- ing ULA, Cantor array , Fractal array and any symmetric re- dundancy SLAs. 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