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Communicatio n http://dx.doi.or g Copyright © 2 0 Sp a ABSTRA C This paper pr o in the case o f b asis and the p ervised offli n an increment a need the num b location esti m Keywords: D P r 1. Introdu c Wireless loc a fields like co m radio astrono m p ast decade. T zation proble m signal param e time of arriva l dinates of un k p loiting the p though most l centrate on t h eral as expla i novel DPD m indeed intere s have been p r nique that is s step methods criterion gath e single emitte r p roach to tre a measurement- DPD method n s and Networ k g /10.4236/cn.2 0 0 13 SciRes. a rsity- B 1 The N a 2 The Jiangs u 3 Key L C T o poses an ada p f time-varying sparse soluti o n e DL by usi n a l fashion to a d b er of emitter s m ation accurac y D ictionary Lea r r ogramming c tion a lization as a m munications, m y has draw n T he traditiona l m consists of t e ters such as l (TOA) are e s k nown-locatio n p arameters es t ocalization al g h e two-step m e i ned in refere n m ethods that d i s t to the en d - u r oposed, as a s hown to out p [2]. Weiss e t e ring all sign a r [3] and the n a t the multipl e to-association can provide s u k , 2013, 5, 421- 4 0 13.53B2077 P u B ased D Two-s t Tin g Jiangsu Engin e a njing Universi t u Key Laborato ry N L aboratory of D i ap tive sparsity - channels. Th e o n based on a n g the quadrat i d apt to the ti m s a prior. Si m y . r ning; Compr e fundamental radar, sonar, n increasing a l approach to s t wo-step proc e angle of arri v s timated, and s n targets are c a t imated in th e g orithms prese n e thod, it is su b n ces [1]. Rec e i rectly estima t u ser, i.e., posit i promising po p erform the co n t al p roposed a ls of all stati o n proposed a e emitters cas e step is avoide d u perior locali z 4 25 u blished Online D irect L t ep Di c g ting Wang 1, 2 e ering Center o f t y of Informatio y on Optoelectr N anjing Norm U i saster Reducti o Ministry of Ci v Email: w Rece i - based direct p e novel featur e two-step dicti i c programmi n m e-varying ch a m ulation result s e ssive Sensing ; task in vario u seismology a n a ttention in t h s olve the loca l e dure. First, t h v al (AOA) a n s econd the co o a lculated b y e x e first step. A n ted so fa r co n b optimal in ge n e ntly, a kind o t e the results o i on coordinat e o sitioning tec h n ventional tw o a unique DP o ns firstly for decoupled a p e [4]. Since t h d , the decou p l e z ation capabili t September 201 3 ocatio n c tionar y 2 ,3 , Wei Ke 2, 3 f Meteorologica l n Scie nce and T onic Technolog U niversity, Nanj o n and Emergen v il Affairs, Beiji n w kykw @sina.c o i ve d May 2013 p osition deter m e of this meth o onary learnin g n g approach, a a nnel during t h s demonstrate ; Direct Locat i u s n d h e l i- h e n d o r- x - A l- n - n - o f o f e s, h - o - D a p - h e e d t y in mul t initial p to b e k n metho d [5,6]. B of sup p functio n higher exploit Co m of atte n in the t mensi o are fe w rate D P Picard a sity re p fitting m mise t h tionary In pra c channe misma t mation 3 (http://www.s c n Estim a y Lear n 3 , Gang Liu 2 l Sensor Netwo r T echnology, Na n y, School of Ph y ing, China cy Response E n n g, China om m ination (DP D o d is to dyna m g (DL) frame w a nd then the d h e online stag e the perfor ma n i on; Time -Var y t i-emitter cont e p osition estim a n own a priori. d to handle th B asically, the a p ort points in w n is evaluated. complexity th a the explicit g e m pressed sensi n n tion in recent t wo-step loca l o ns of measur e w works discu s P D estimation. a nd Weiss tre a p resentation b m ethod in [9]. h at the predef i ) is ideal and i c tice, due to t ls and rando m t ch the actual s performance c irp.org/journal / a tion B n ing r k Technology, n jing, China y sics and Tech n n gineering of th e D ) appoach to m ically adjust b w ork. The me t d ictionary is c o e . Furthermor e n ce of the pro p y ing Channel; e xt. But this m a tes, and the n u Other studies h e global navi g a bove DPD al g w hich the ma x Therefore, th e a n the two-ste p e ometric relati n g (CS), whic h years, has be e l ization meth o e ment vectors s sing the CS p To the best o a t the DPD pr o b y exploiting However, thi i ned overcom p i nvariable in t h t he dynamica l m noise, the s ignals stocha s is degraded. I / cn) B ased o n n ology, e locate multipl b oth the over c t hod first perf o o ntinuously u p e , the method p osed algorith m Quadratic m ethod depen d u mber of sour c h ave extend ed g ation satellit e g orithms gene r x imum likelih e se DPD meth o p approach, w onship. h receives a g r e n successfull y o d by reducin g [7,8]. Howe v p attern for m o o f our knowle d o ble m as a spa t the covarian c s work makes p lete basis (a. h e localizatio n l change of m predefined b a s tically so that I n this paper, CN n e targets c omplete o rms su- p dated in does not m on the d s on the c es ne eds the DPD e system r ate a set ood cost o ds have w hich can r eat deal y applie d g the di- v er, there o re accu- d ge, only t ial sp ar - c e-matrix the pre- k.a. dic- n process. m ultipath a sis may the esti- we p r o- T. T. WANG ET AL. Copyright © 2013 SciRes. CN 422 pose an adaptive sparsity-based DPD (ASDPD) algori thm to dynamically adjust both the overcomplete basis and th e sparse solution so that the solution can better match the actual scenario. The method first performs supervised of- fline dictionary training by using the quadratic program- ming approach. Dur ing the online stage, the dictionary is continuously updated in an incremental fashion to adapt to time-varying factors. The notation used in this paper is according to the convention. Symbols for matrices (upper case) and vec- tors (lower case) are in boldface. () H ⋅, 0 θ, 1 θ, 2 θ, N I, ⊗ and CN denote conjugate transpose (Hermitian), 0 l norm, 1 l norm, 2 l norm, identity matrix with the dimension N, the Kronecker product and complex Gaus- sian distribution, respectively. For any matrix Y, vec( )Y is denoted as the vertical concatenation of the columns of Y. Finally, ˆ x denotes the estimate of the parameter of interest x. The remainder of the paper is organized as follows. Section 2 briefly describes the system model assumed throughout th is p aper and formulates as a sparse recovery problem. In Section 3, we introduce a scheme calibrating the overcomplete basis dynamically and estimating the sparse solution adaptively. Simulation results are given in Section 4. Finally, Section 5 concludes the paper. 2. System Model and Problem Formulation Consider N base stations (BS) intercepting the narrow- band signals transmitted by L possible sources. Each BS which knows its coordinates is equipped with an antenna array consisting of M elements. Denote the lth unknown target position by the vector of coordinates l P. We use the far-field point-target model, which is commo nly used for source localization du e to its simplicity [3,4,9]. Based on this model, the received signal observed by the nth BS is given by 1 ()() (())(),0 L nnllnln l tst ttT τ = =−+≤≤ rappv (1) where () l s t is the signal waveform considered known. () nl ap is the array response at the nth BS from a signal transmitted position, and the propagation delay from the lth transmitter to the nth BS is given by () nl τ p. The vector ,{1,,} nn n nN∈=+rHθv represents noise terms, which is assumed as the independent and identi- cally distributed (i.i.d.) complex Gaussian process, un- correlated with the signals. We divide the area of interest into K grids. In general, KML>. Then, we formulate the location problem as a following CS problem ,{1,,} nn n nN∈=+rHθv (2) where () () 1 [,,] nn nK =Hh h is an overcomplete basis ma- trix at the nth BS, and ()n i h corresponds to the noiseless signal vector between the ith grid and the nth BS. 1 [, ,] K θθ =θis a sparse vector that having in total L nonzero entries, where the indices of nonzero entries in θ which represents the actual locations. It should be emphasized that the above matrix n H is constructed by ideal signals, where the parameters such as AOA and TOA can be calculated according to the geometric rela- tionship directly. Denote H the matrix obtained by concatenation of all the matricesn H, i.e., 1 [,, ] TTT N =HH H. Similarly, by denoting 1 [,,] TTT N =Rr r and 1 [,, ] TTT N =Vv v, we can obtain =+RHθV (3) Note that H is known under the id eal channel condi- tion, which means that we can estimate the actual coor- dinates of targets as long as we find the positions of nonzero values in θ. That is, the problem of localization is converted into one of sparse signal recovery from (3). Moreover, the number of these dominant nonzero values gives L. However, the non-ideal factors are inevitable in a prac- tical localization system. These factors include the chan- nel attenuation, phase error, time-varying fluctuations of the radio channel and so forth. When these happen, the predefined dictionary cannot effectively express the ac- tual signal, which will cause performance degradation in sparse recovery process. For avoiding the difficulty of estimate all kinds of the time-varying factors, we assume the error dictionar y ma- trix Γ which describe the difference between the prede- fined dictionary and the practical received signals. Note that the error matrix Γ is time-varying and cannot be known in advance. In this scenario, the sparse position- ing model is correspondingly modified as: =+ +RΓHθVDθV (4) where =DΓH denotes the actual overcomplete basis with the time-varying interference. To prevent D from having arbitrarily large values (which would lead to arbi- trarily small values of θ), it is common to constrain its columns 1,, K dd to have a 2 l norm less than or equal to one. Obviously, the mismatch exists between the columns of D and the corresponding columns of the pre- defined basis H, and thus the performance degradation is inevitable in the sparse recovery process. Focused on this problem, an adaptive sparse recovery algorithm is proposed in this paper, which dynamically calibrate the overcomplete basis so that the sparse solution can better fit the actual scenario. 3. Sparse Representation Based on the Two-stage Dictionary Learning The key feature of adaptive sparse recovery is the adap- T. T. WANG ET AL. Copyright © 2013 SciRes. CN 423 tive adjustment of the overcomplete basis. This process generally learns the uncertainty of the dictionary, which is not available from the prior knowledge, but rather has to be estimated using a given set of training samples. Several different DL algorithms have been presented re- cently [10]. However, these methods generally cannot effectively handle very large training sets or dynamic training data changing over time. To overcome these shortcomings, we propose a two-stage DL approach that can adapt to the varied upcoming samples. So far, the most DL methods are generally based on alternating minimization. In one step, a sparse recovery algorithm finds sparse representations of the training sam- ples with a fixed dictionary. In the other step, the dictio- nary is updated to decrease the average approximation error while the sparse coefficients remain fixed. The pro- posed method in this paper also uses this formulation of alternating minimization. 3.1. Sparse Recovery Phase The above problem of noisy sparse signal recovery can then be converted into a following optimization problem 2 21 min/ 2 λ −+RDθθ (5) where λ is the regularization parameter. However, it should be emphasized that larger coefficients in θ are penalized more heavily in the 1 l norm than smaller coef- ficients, unlike the more democratic penalization of the 0 l norm [11]. In practice, large coefficients are usually the entries corresponding to the actual positions of tar- gets, while small coefficients commonly represent the noise entries. The imbalance of the 1 l norm penalty will seriously influence the recovery accuracy, which may result in many false targets. Therefore, in this paper we choose the reweighted 1 l norm minimization algorithm in [11] as our sparse recovery method, which can over- come the mismatch between 0 l norm minimization and 1 l norm minimization while keeping the problem solva- ble with convex estimation tools. 3.2. Dictionary Learning Phase In this paper, we propose a two-stage DL framework in which the offline DL method allows to train the dictio- nary in a supervised manner to integrate the large train- ing sets, and the incremental DL method based on the results in the offline stage handles the unseen online var- iation to enhance its adaptability. 1) Offline dictionary learning In this stage, the ideal overcomplete basis H is op- timized to better represent the data of the training sets. Since the sparse coefficients θ are fixed in the DL stage, the resulting optimization problem becomes: 2 2 min/ 2,. .1,1,, H ii s tiK−≤=RDθdd (6) in which 2 2 −RDθcan be written as 2 2tr[( )( )] tr() 2tr()tr() vec() ()vec() 2vec() vec()tr() H HHHH H HHH H HH HH −=− − =−+ =⊗ −+ RDθRDθRDθ Dθθ DRθDRR DIθθD θRDRR (7) Let’s introduce several new expressions for clarity of notation vec( ) vec( ) H H H ⊗ αD GIθθ γθR Omitting the terms that do not depend on D, the objec- tive function in (6) can be equivalent to 1 min,..1,1,, 2 HH H ii s tiK−≤=αGαγα dd (8) Note that (8) is a standard form of constrained qua- dratic programming problem which can be solved by any standard optimization method, such as the gradient pro- jection algorithm in [12]. Moreover, the matrix G is ob- viously a positively definite matrix, and thus (8) is con- vex function and can be guaranteed to find a global op- timum [13] in this DL phase. 2) Online dictionary learning Although the of fline DL stage has adjust the overcom- plete basis according the training data, it is impossible to be fit for all kinds of time-varying interference patterns. Moreover, its computation load is quite large for real- time localization. On the contrary, the online incremental learning is especially applicable when one seeks to find the variation in the sense that the time-varying channels pattern might not be specifically learned offline but can be distinguished from the past online observations. Based on the incremental learning pattern, the online learning algorithm in [14] can use the result of the offline DL stage as a warm restart for computing the next dictionary where the new samples will be fed into the online dictio- nary learning procedure, and thus a single iteration has empirically been found to be enough [14]. For completeness, a full description of the algorithm is given in Algorithm 1. Algorithm 1 Two-stage DL algorithm Initialization: set the training sample set; generate the ideal dictionary H Offline DL stage: Input: the training sample set; (0) off ˆ=DH; the number of itera- tions T; for j=1 to T 1) use the reweighted 1-norm algorithm to compute the vecto r T. T. WANG ET AL. Copyright © 2013 SciRes. CN 424 (j) off ˆ θ with (1) off ˆj− D fixed for each sample; 2) use the gradient projection algorithm to minimize the objec- tive function in (8) with respect to () off ˆj D keeping (j) off ˆ θ fixed; end for Output: (T) off off ˆˆ =DD; (T) off off ˆˆ =θθ; Online DL stage: Input: (0) on off ˆˆ =DD; (0) on off ˆˆ =θθ; the latest observation samples; 1) update the pre-trained dictionary according to the lates t observations using the online DL algorithm in [14] with off ˆ D as warm restart, and return the learned dictionary (1) on ˆ D; 2) use the reweighted 1-norm algorithm to minimize the objec- tive function in (6) with re sp ec t to (1 ) on ˆ θ keeping (1) on ˆ D fixed; 3) (0)(1) on on ˆˆ =DD; (0) (1) on on ˆˆ =θθ; Output: (1) on on ˆˆ =DD; (1) on on ˆˆ =θθ; 4. Simulation Results In order to examine the performance of the proposed ASDPD method, we compare it with the decoupled DPD approach in [4] and covariance-based sparse DPD (CDP D) method [9]. Consider four BSs placed at the corners of a 1 km × 1 km square. Assume the number of grids in the location area is 26 26K=×, which means yielding a 40m resolution along both x and y axes. The carrier fre- quency of the simulated signal is assumed to be 900 MHz. Each BS is equipped with a uniform linear array of ten antenna elements with the adjacent elements spacing of half a wavelength. The locations of targets are selected at random, uniformly, within the square. All the simula- tion results are obtained based on 200 Monte Carlo rea- lizations. In each simulation, we consider the following multipath channel model 1 ()( ) Q ii i c τ β δτ τ = =− (9) to obtain a set of channel data. {} i β is a et of indepen- dent and identically distributed (i.i.d.) random variables which satisfy (0, ) i b iCNe τ β − . b = 1/16 is the expo- nential power delay profile and i τ is the delay spread for the ith path [15]. As shown in Figure 1, the improvement in location accuracy for the proposed method can be seen in terms of the root mean square error (RMSE), when the number of emitters is two and the SNR is set to 5 dB. We can ob- serve that the location performance of DPD and CDPD algorithms decreases evidently as the number of multi- path increases. On the contrary, the variation of RMSE in the ASDPD algorithm is very small due to its adaptive ability through DL technique. This result reveals that our method is very robust to multipath channels and effec- tively enhances location accuracy. Figure 2 illustrates the location error with respect to the number of emitters when the SNR is set to 5 dB. Here, real lines describe the case of single-path channel for three algorithms, while dashed lines represent the case of three paths. With the increase in the number of emitters, the RMSE of DPD algorithm increases quickly due to the high sensitivity to the estimated number of targets. Note that the CDPD method does not rely on a good estimate of the number of emitters in the single-case, but its per- formance decreases evidently as the number of multipath increases. On the contrary, the ASDPD algorithm is very robust to two scenarios. The importance of the low sensi- tivity of our algorithm to the number of targets is twofold: first, the number of sources is usually unknown, and second low sensitivity provides robustness against mistakes in estimating the number of targ ets. Figure 1. The localization error with respect to the number of multipaths. Figure 2. The localization error with respect to the number of emitters. 12345 20 40 60 80 100 120 140 The number of mult i pahs RMSE (m) ASDPD CDPD DPD 23 4 5 6 7 20 40 60 80 100 120 140 The number of emi t t ers R M SE (m ) ASDPD CDPD DPD ASDPD CDPD DPD T. T. WANG ET AL. Copyright © 2013 SciRes. CN 425 5. Conclusion In this paper, we exploit the inherent spatial sparsity to present a novel direct location method by combining the offline training and online learning into a unified DL framework, thereby better matching time-varying scena- rios. The effectiveness of the proposed scheme has been demonstrated by simulation results where substantial im- provement for localization performance is achieved. Fur- ther research will emphasize on the off-grid error analy- sis and the theoretic bound on the location estimation precision. 6. Acknowledgements This work is supported in part by the priority academic program development of Jiangsu higher education insti- tutions, and in part by the Open Research Fund of Key Laboratory of Disaster Reduction and Emergency Response Engineering of the Ministry of Civil Affairs under grant No. LDRERE20120303. REFERENCES [1] J. Bosse, A. Ferréol, C. Germond and P. 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