Localization and identification of unknown target signal using oblique projection

Localization and identification of unknown target signal using oblique projection The problem of source localization and waveform identification is the key of array signal processing. In this paper, an oblique projection-based localization and identification (OPLI) algorithm is proposed without known prior DOA or waveform information of the sources. The proposed OPLI is implemented iteratively. In each iteration, oblique projection is employed to separate the multiple incident signals into a series of single signal groups. After that, the procedure of waveform and DOA estimation for each single signal is implemented. Theoretical analysis and simulation result verify the performance and effectiveness of the proposed OPLI. Keywords: Beamforming, Direction of arrival estimation, Oblique projection, Interference suppression 1 Introduction the interference signals are suppressed or attenuated in Source localization and waveform identification are cen- advance, the performance of localization and waveform tral problems in antenna array processing, including in identification for SOI would be better. Haimovich and particular radar, sonar, or wireless communication [1, 2]. Bar-Ness [20], Haimovich [21], and Honig and Goldstein To this end, various sensor signal processing tools have [22] proposed eigenanalysis-based interference canceler. been developed over the last several decades, ranging Similarly, Gu and Leshem [23], Huang et al. [24], Chan and from direction-of-arrival (DOA) estimation algorithms to Chen [25], Boyer [26], Xu et al. [27], Mak and Manikas spatial beamforming algorithms [3–13]. It is noted that, in [28], Shi and Lin [29], Behrens and Scharf [30], Mao practical applications, the localization and identification et al. [31], and Mao et al. [32]employedbeamforming of target signal could be difficult, when any a priori knowl- techniques to suppress inference signals. Hassanien et al. edge of signal-of-interest (SOI) is unavailable or the SOI [33], Vaccaro and Harrison [34], and Han and Zhang [35] is sheltered by adjacent interference signals. One of the designed matrix filters to cancel out-of-sector interference general approaches to solve this problem is realized as fol- signals. It should be noted that these mentioned algo- lows: The DOA estimation is firstly implented for all the rithms are based on accurate prior DOA of the SOI, and array receiving signals, and then, the waveform of SOI is some of them may also require necessary DOA informa- obtained via beamforming techniques [14–19]. tion of interference signals. No matter in the process of DOA estimation or in The estimation accuracy of DOA has major influ- beamforming, the influence of the interference signals on ence on the beamforming performances. Generally, target localization and identification is generally not triv- the performance of a beamformer decreases severely ial, especially when the interference signals are spatially when the DOA error increases [26, 31, 32]. Different adjacent with the SOI in the presence of large power. If approaches, including the linearly constrained minimum variance (LCMV), diagonal loading, convex optimization, *Correspondence: mxp@hit.edu.cn and covariance matrix taper approaches, are developed School of Electronics and Information Engineering, Harbin Institute of [16, 36–46] to combat DOA errors. These algorithms Technology, Harbin 150001, China can combat DOA uncertainties, but only suitable for Collaborative Innovation Centre of Information Sensing and Understanding, Harbin Institute of Technology, Harbin 150001, China small DOA errors. Lam and Singer [47], Bell et al. [48], Full list of author information is available at the end of the article © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 2 of 11 K −1 and Han and Zhang [49] developed Bayesian beamform- x(t) = a (θ ) s (t) + n(t) k k ing, which is able to implement waveform identifica- k=0 tion when the DOA is uncertain or unknown, while it = AS(t) + n(t) (1) requires prior statistics that describes the level of DOA uncertainty. where Even if an accurate DOA of the SOI is given, the opti- A = [a (θ ) , a (θ ) , a (θ ) , ··· , a (θ )] (2) 0 1 2 K −1 mal performance of beamforming is hard to achieve since most of the beamformers rely on the sampled matrix and inversion to replace the theoretical one, especially when S(t) = [s (t), s (t), s (t), ··· , s (t)] (3) 0 1 2 K −1 in the presence of short data samples [3, 4, 44, 46]. With limited number of snapshots, in this paper, we discuss the where s (t) refers to the baseband signal waveform of the problem of localization and identification for unknown kth incident signal, and a (θ ) denotes the corresponding target signal, where there exist spatially adjacent inter- steering vector. n(t) represents the additive noise compo- H T ference signals and the DOAs of interference signals are nent with covariance R = E n(t)n (t) . E[ ·], (·) and also completely unknown. The well-known RELAX dis- (·) stand for statistical expectation, transposition and cuss the similar problem [50–52], where a simultaneous Hermitian transposition, respectively. realization of DOA estimation and spatial beamforming Both the received signals and the noise are assumed to for all receiving signals is achieved, but the performance be sampled from zero-mean and uncorrelated stationary of beamforming is not theoretically deduced. random processes. Furthermore, the noise is temporally With spatially adjacent interference signals, an oblique and spatially independent of the received signals, and 2 2 projection-based localization and identification (OPLI) R = σ I,where σ is referred to as the noise power and I algorithm is proposed for unknown target signals. Firstly, stands for the identity matrix. Thus, the covariance matrix the OPLI employs oblique projection to separate the of the received data is given by mixed array receiving signals into individual signal H H 2 R = E x(t)x (t) = AR A + σ I (4) x S groups, where each signal group contains only one sig- nal. Then, the OPLI sequentially estimates DOA and where R = E S(t)S (t) . waveform of the signal in each group. Finally, the OPLI recursively reduces DOA and waveform errors via min- 2.2 Problem description imizing the optimal maximum likelihood cost function. This subsection formulates the localization and waveform Theoretical analysis is provided to show the beamform- identification problem. Without loss of generality, the sig- ing performance of the proposed OPLI, and simulation nal s (t) is assumed to be the SOI and the remaining K −1 results indicate that the OPLI is computationally effec- received signals are treated as interference signals. For the tive for source localization and waveform identification; SOI and the interference signals, neither the source loca- besides, it is superior to the counterpart conventional tions (i.e., θ , θ , ··· , θ ) nor the signal waveforms (i.e., 1 2 K −1 algorithms at moderate to high input signal-to-noise ratio s (t), s (t), ··· , s (t))are priorknown. 0 2 K −1 (SNR) region in terms of output signal-to-interference- The localization and waveform identification problem noise ratio (SINR) and root mean square error (RMSE). addressed in this paper is to obtain the DOA θ and wave- The paper is organized as follows. Section 2 presents form s (t) from multiple snapshots {x t } ,where L ( ) 0 l l=1 the signal model for source localization and waveform represents thenumberofsnapshots. identification. The proposed OPLI algorithm as well In the next section, it will be shown that the waveform as its complexity analysis is developed in Section 3. of the SOI can be identified as follows, i.e., Section 4 discusses the proposed OPLI, and Section 5 s ˆ (t) = W x (t) (5) 0 0 presents the simulation results. The conclusion is drawn in Section 7. where W refers to the optimum weight, and x (t) = a (θ ) s (t) + n(t).(6) 0 0 0 2 Signal model and problem description 2.1 Signal model And, according to (6), the location of the SOI is given by This subsection presents the signal model which is the conventional method of single target localization. considered for source localization and waveform iden- tification. Assume that K far-field narrowband signals 3 Proposed OPLI algorithm impinge upon an antenna array from distinct directions Derivation and implementation of the proposed OPLI are θ , θ , θ , ··· , θ . The antenna array consists of N sen- presented in this section. The proposed OPLI attempts to 0 1 2 K −1 sors with arbitrary array geometry. The N × 1vectorof employ oblique projection to separate the multiple inci- received signals at time t can be expressed as [3] dent signals into a series of single signal groups. As a Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 3 of 11 result, the source localization and waveform identification can be estimated by solving the following minimization are implemented on each separated single signal. Note problem, i.e., that the procedure of oblique projection requires the DOA parameters for source separation. To this end, the OPLI 1 θ , s ˆ (t) = arg min x t − a(θ )s t  (15) ( ) ( ) 0 0 0 l l is implemented iteratively and the method of maximum θ,s(t) l=1 likelihood approximation [53]isemployedtoevaluatethe convergence. Solving (15), it is obtained that 3.1 Basic principle s ˆ (t) = a (θ ) x (t) (16) 0 0 0 To illustrate the principle of localization and waveform and identification of the SOI, the DOAs of all incident signals are temporarily assumed to be known in this subsec- tion. According to (1)and (6), it is natural that the SOI ˆ θ = arg min P x (t ) (17) 0 0 a(θ ) component in the received data can be given by l=1 K −1 = arg min tr P R . a(θ ) x (t) = x(t) − a θ s (t) ( ) 0 k k k=1 ⊥ † H where P = I − a(θ )a (θ ),and R = E x (t)x (t) . 0 0 a(θ ) 0 = x(t) − B S (t).(7) 0 B Comparing (5)and (16), it follows that the optimum weight W to obtain the SOI is given by where 0 −1 B = [a (θ ) , a (θ ) , ··· , a (θ )] 0 1 2 K −1 † H W = a (θ ) = a (θ ) a (θ ) a (θ ) (18) 0 0 0 0 0 = A \ a (θ ) (8) −1 where (·) denotes inverse. and Using (5), (7), (10), (17), and (18), both the location S (t) = s (t), s (t), ··· , s (t) [ ] θ and the waveform s (t) of the SOI can be com- B 1 2 K −1 0 0 puted. Besides, the localization and waveform identifica- = S(t) \ s (t) (9) tion processes for the SOI can also be extended, i.e., the where the symbol \ signifies set difference. localization and waveform identification process for the To separate the SOI from the multiple incident sig- interference signals are feasible if each of the interference nals by (7), we employ oblique projection to obtain the signalsisregardedasthe SOI. interference signals, i.e., 3.2 Practical considerations S (t) = B E x(t) (10) B B |a θ ( ) 0 0 0 0 Recall that the localization and waveform identification process in Section 3.2 requires known DOAs of all the where E denotes the oblique projection operator B|a(θ ) incident signals, which is infeasible in practical applica- whose range space is R {B} and whose null space contains tions. In this paper, it is considered that the DOA of the R {a (θ )},and SOI is prior unavailable, and the prior DOAs of the inter- −1 H ⊥ H ⊥ ference signals are also unknown. Toward the purpose, the E = B B P B B P (11) B |a(θ ) 0 0 0 0 0 a(θ ) 0 a(θ ) 0 0 presented localization and waveform identification pro- where cess is implemented iteratively. The proposed OPLI firstly estimate the parameters of the SOI, so that a rough infor- E B = B (12) B |a(θ ) 0 0 0 0 mation of the SOI is available. And then, the parameters of each interference signals are updated one after another. E a (θ ) = 0 (13) B |a(θ ) 0 0 0 Through iteration, not only the parameters of the SOI are estimated, but also the parameters of the interference and signals are gradually achieving high precision. ⊥ † (i) (i) ˆ ˆ P = I − a (θ ) a (θ ) (14) 0 0 At the ith iteration, the estimated DOAs θ , θ , ··· , a(θ ) 0 0 1 (i) (i−1) (i−1) (i−1) ˆ ˆ ˆ ˆ θ , θ , θ , ··· , θ are utilized as initial val- ⊥ † K −1 k−1 k k+1 where R {·}, (·) ,and (·) denote range space, orthogonal (i) (i) ues to compute θ and s ˆ (t),where i ≥ 1and k = complement, and Moore-Penrose pseudoinverse, respec- k k (i) (i) tively. 0, 1, 2, ··· , K − 1. θ and s ˆ (t) denote the estimation of k k According to (7), only the SOI is contained in x (t ), θ and s (t) at the ith iteration (similarly hereinafter), 0 l k k and the interferences signals are removed. Thus, the SOI respectively. Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 4 of 11 Table 1 Iteration procedure of OPLI If the kth incident signal is regarded as the SOI, using Initialization: (7) ∼ (10)and (17), it follows that (−1) (−1) (−1) (−1) ˆ ˆ ˆ ˆ a θ = 0, a θ = 0, a θ = 0, a θ = 0, (i) (i) ⊥ 0 1 2 K−1 ˆ ˆ θ = arg min tr P R (19) k a(θ ) k Main Loop: (0) calculate L by (27) where for k = 0, 1, 2, ··· , K − 1 do (i) (i) (i) R = E x ˆ (t) x ˆ (t) (20) k k k (0) calculate θ by (19) (0) calculate s ˆ (t) by (25) and (i) (i) repeat x ˆ (t) = x(t) − B S (i) (t) (21) k k for j = 0, 1, ··· , k do where (i) redetermine θ by (19) (i) redetermine s ˆ (t) by (25) S (t) = B E x(t) (22) (i) (i) (i−1) ˆ k ˆ B ˆ B |a θ k k end for (i) (i) (i) (i−1) redetermine L by (27) ˆ ˆ ˆ B = A \ a θ (23) k k k (i) (i−1) until (|L − L | is less than a specified tolerance) and (0) (i) (0) (i) (0) (i) ˆ ˆ ˆ ˆ ˆ ˆ a θ ← a θ , a θ ← a θ , ··· , a θ ← a θ 0 0 1 1 j j (i) (i) (i) ˆ ˆ ˆ A = a θ , ··· , a θ , end for k k−1 Output (i−1) (i−1) (i−1) ˆ ˆ ˆ a θ , a θ , ··· , a θ . (24) k k+1 K (i) (i) (i) ˆ ˆ ˆ ˆ ˆ ˆ θ ← θ , θ ← θ , ··· , θ ← θ 0 1 K−1 0 1 K−1 Besides, using (5)and (18), it follows that (i) (i−1) (i) s ˆ (t) = W x ˆ (t) (25) k k k (i) 2 2 where x ˆ (t ) /L, which takes about O LN + N flops. −1 Herein, a flop is defined as a complex floating-point (i) (i) (i) (i) ˆ ˆ ˆ ˆ W = a θ a θ a θ . (26) k k k k addition or multiplication operation. The number of flops roughly required to compute (21)is O (N (K − 1)L + NL) During the proposed iterative process, the relative flops. The calculation of (22) requires approximately change of the following cost function, which is derived 3 2 2 O (K − 1) + 2N (K − 1) + N (K − 1) + (K − 1)NL from maximum likelihood approximation [53], between flops, where the calculation of E additionally (i) (i−1) ˆ ˆ B |a θ two consecutive iterations is utilized to indicate the con- k k 2 2 3 vergence, i.e., takes about O 3N (K − 1) + 2N (K − 1) + (K − 1) + 2 2N + N flops. Thus, the computational complex- L K −1 1 2 (i) (i) (i) ity of (20) ∼ (22)isroughly O N (L + 4K − 1)+ L = x (t ) − a θ s ˆ (t ) . (27) l l k k 2 3 N (4(K − 1) + (2K − 1)L + 1) + 2(K − 1) flops in l=1 k=0 total. Referring to the well-known relaxed iterative approach in (i) The computation of θ in (19) requires roughly [50, 51], the localization and waveform identification pro- 3 2 ˜ ˜ O N + 2N + 2N N (i) flops, where N (i) denotes cedure of the proposed OPLI is summarized in Table 1.It ˆ ˆ θ θ k k is shown that the proposed OPLI starts with the initializa- the number of potential source locations in the region (−1) (−1) (−1) (i−1) (i−1) (i) ˆ ˆ ˆ ˆ ˆ tion a θ = 0, a θ = 0, ··· , a θ = 0.At scope θ −  , θ +  . The computation of s ˆ (t) 0 1 K −1 k k k k k (i) in (25) requires roughly O(NL) flops, where the cal- the ith iteration, θ is determined within the range scope (i) (i−1) (i−1) culation of W additionally takes about O (2N + 1) ˆ ˆ θ −  , θ +  ,where K − 1 ≥ j ≥ 0, and  is a j j j j j (i) flops. The computation of L in (27) requires roughly user-selected parameter which indicates the range of the O ((K + 2)NL) flops. jth signal in the spatial domain. Therefore, according to Table 1, the computational com- plexity of the proposed OPLI is roughly O ((K + 2)NL) + 3.3 Computational complexity K −1 3 2 2 O N + 2N + 2N N (0) + N (L + 4K − 1)+ In this subsection, the computational complexity of the k=0 ˆ 2 3 proposed OPLI is analyzed. N 4(K − 1) + (2K − 1)L + 1 +2(K −1) +NL+2N +1+ Using L number of snapshots, the computation k 3 2 2 (N +2N +2N )N (i) +N (L+4K − 1)+ (i) L (i) ˆ i=1 j=0 ˆ of matrix R in (20)can be givenas x ˆ t ( ) j l=1 k k Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 5 of 11 2 3 N (4(K − 1) + (2K − 1)L + 1) + 2(K − 1) + NL + 2N It follows from (31)to(33) that the proposed OPLI can effectively suppress interference signals and have distor- + 1 + (K + 2)NL flops in total, where N denotes tionless response for the SOI. Regarding to the result of thenumberofiterationsemployedinthe kth outer waveform identification for the SOI, the output SINR of loop. Particularly, the total complexity is approximately OPLI is given by K −1 k 3 k 3 ˜ ˜ O N N + N N flops, when (0) (i) ˆ i=1 j=0 ˆ k=0 θ θ k j E W a (θ ) s (t) 0 0 opt,0 L  N > K, N  L/N, which occurs often in practical (i) SINR = . (36) applications. E W [x(t) − a (θ ) s (t)] 0 0 opt,0 4 Discussion Substituting (1), (31) ∼ (33)into(36), it follows that TheproposedOPLIinvolvesthe procedureofDOA esti- mation, hence it is applicable for localization of unknown SINR = target signal. Additionally, the OPLI also can obtain accu- E W n(t) rate locations of the interference signals, which is valuable opt,0 in practical applications. 2 2 σ σ TheproposedOPLIalsoinvolvesthe procedureof 0 0 = = H 2 beamforming. Substituting (21), (22), (26)into(25), it W R W n opt,0 σ W opt,0 opt,0 follows that (i) (i−1) 2 † s ˆ (t) = a θ I − E x(t). (28) σ a (θ ) I − E (i) (i−1) 0 B |a(θ ) 0 0 k k ˆ B |a θ k k When OPLI achieves convergence, it follows from (28) 2 † σ a (θ ) E 0 a(θ )|B 0 0 that 0 H ⊥ = a (θ ) P a (θ ) (37) s (t) = W x(t) (29) 0 0 k B opt,k 2 0 where W is referred to as the optimum beamformer 2 2 opt,k where σ = E |s (t)| refers to the power of the SOI. employed by the proposed OPLI, k = 0, 1, 2, ··· K −1, and It is known that the optimal SINR, which follows from the maximum SINR beamformer principle, is given W = a (θ ) I − E . (30) opt,k k B |a(θ ) k k by [46] 2 H −1 Let k = 0, according to (2), (8), and (30), it is obtained SINR = σ a (θ ) R a (θ ) (38) opt 0 0 0 in that −1 −1 H 2 where R = B R B + σ I .Herein, R = 0 S S † B B in 0 0 0 W = a (θ ) I − E (31) opt,0 0 B |a(θ ) 0 0 H 2 2 2 2 E S (t)S (t) = diag σ , σ , ··· , σ ,where σ 0 B 1 2 K −1 k and denotesthe powerofthe kth signal. In particular, when 2 2 the incident signals have equipower, i.e., σ = σ = ··· = W a (θ ) = 1 (32) 0 0 1 opt,0 2 2 σ = σ ,itisobtainedthat K −1 −1 W B = 0. (33) opt,0 2 1 σ −1 i H R = B B + I . (39) in 2 2 Recall from [30]that σ σ † † a (θ ) E = a (θ ) P I − E 2 2 0 a(θ )|B 0 a(θ ) B |a(θ ) 0 0 0 0 0 Further, if σ  σ ,then[30] = a (θ ) I − E (34) 0 B |a(θ ) 0 0 −1 1 σ −1 i H ⊥ R = lim B B + I = P . (40) where in 0 B 2 2 0 2 2 σ σ σ /σ →∞ −1 H ⊥ H ⊥ E = a(θ ) a (θ ) P a (θ ) a (θ ) P . (35) a(θ )|B 0 0 0 0 0 0 B B 0 0 Substituting (40)into(38), it is obtained that −1 Substituting (34)into(31), it is obtained that W = opt,0 2 2 σ σ H 0 H i H SINR = lim a (θ ) B B + I a (θ ) a (θ ) E . So, the optimum beamformer of the opt 0 0 0 0 a(θ )|B 0 0 0 2 2 σ 2 σ σ /σ →∞ proposed OPLI is equivalence to the well-known oblique projection beamformer which has excellent performance 0 H ⊥ = a (θ ) P a (θ ) . (41) 0 0 2 0 and has been well researched in [26, 31, 32]. σ Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 6 of 11 Comparing (41)with(37), it follows that the output SINR of the proposed OPLI can coincide well with the optimal SINR, when the power of the incident signals are equal and much higher than that of the noise. 5 Simulation results Extensive simulation results are provided to verify the effectiveness of the proposed OPLI for source local- ization and waveform identification. The combinational algorithm, which firstly utilizes DOA estimator to obtain source locations and then utilizes beamformer for wave- form identification, is compared with OPLI. Besides, the well-known RELAX [51] is also employed for perfor- mance comparison. Specially, for source localization, the stochastic Cramér-Rao bound (CRB) [54]isalsousedfor Fig. 1 Output SINR versus input SNR performance evaluation. Recall that the procedure of OPLI not only contains beamforming but also involves DOA estimation. The out- of the proposed OPLI coincides well with the optimal out- put SINR is employed for measuring the beamforming put SINR when the input SNR within moderate to high performance, and the RMSE is employed for evaluating region. Herein, RELAX shows a similar performance to the DOA estimation performance, where the proposed OPLI. OPLI exhibits better performance than the counterpart MVDR, LSMI and GLC beamform- M K ˆ ers, and this is mainly due to the fact that OPLI employs RMSE = θ − θ (42) k,m k KM oblique projection, which has excellent performance for m=1 k=1 suppressing interference signals, to identify the waveform where M denotes the number of Monte Carlo trials, θ c k,m of the SOI. The proposed OPLI tends to have a perfor- refers to the DOA estimation of θ in the mth Monte mance drop when the input SNR is lower than 0 dB, this Carlo trial (similarly hereafter). In the following simula- is because the employed oblique projection increases the tions, M = 200, and a half-wavelength spaced uniform noise variance (cf. [26, 30]). The simulation result verifies linear array composed of N = 6 sensors, is considered. the theoretical analysis given in Section 4. Figure 2 presents the output SINR of the proposed 5.1 Waveform identification OPLI,where thenumberofsnapshots L ranges from 2 This subsection evaluates the waveform identification to 1000. The input SNR is fixed at 10 dB. Other simula- performance of the proposed OPLI. The minimum vari- tion conditions remain the same as previous experiment. ance distortionless response (MVDR) beamformer [36], It is seen that the output SINR gradually approximate to the diagonally loaded sample matrix inversion (LSMI) the optimal output SINR when the number of snapshots beamformer [38], the general linear combination (GLC)- based beamformer [39], and the optimal beamformer which is based on the maximum output SINR principle [46] are employed for performance comparison. The diag- onal loading factor of the LSMI beamformer is set to be equal to the noise power. Note that both OPLI and RELAX implement waveform identification without prior DOA information. Whereas the MVDR, LSMI, and GLC beamformers require prior DOA of the SOI to calculate beamforming weights. Unless otherwise stated, in the subsection, it is assumed that the prior DOA is considered without error. Figure 1 presents the output SINR of the proposed OPLI,where theinput SNRrangesfrom −20 to 30 dB. K = 3 signals are considered, and L = 200 snapshots are employed. The SOI impinges on the array from 5.3°, and the interference signals are impinging from − 5.2°and Fig. 2 Output SINR versus the number of snapshots 45.5°. The simulation results show that the output SINR Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 7 of 11 increases. Compared to RELAX, OPLI exhibits the same performance. While comparing to the MVDR, LSMI, and GLC beamformers, the proposed OPLI has a better per- formance for waveform identification even if the number of snapshots is much limited. Figure 3 presents the output SINR of the proposed OPLI, where the DOA error of the SOI ranges from − 5° to 5°. In this simulation, L = 200 snapshots are consid- ered, and the input SNR is fixed at 10 dB. Other simula- tion conditions remain the same as previous experiment, except that the prior DOA of the SOI is considered with different error for the MVDR, LSMI, and GLC beamform- ers. The prior DOA of the SOI is set to 5.3° + δ,where δ is varied from − 5° to 5°. The simulation results show that the MVDR, LSMI, and GLC beamformers are sensi- Fig. 4 RMSE versus input SNR tive to the prior DOA uncertainties of the SOI. The larger the DOA error goes, the quicker the output SINR drops. Comparing to the MVDR, LSMI, and GLC beamformers, a much better performance when the input SNR is high. the proposed OPLI as well as RELAX remains the same Compared to MUSIC and RELAX, the proposed PSBL output SINR. This is due to the fact that both RELAX and exhibits a better performance when the input SNR is low. OPLI implement waveform identification without prior Figure 5 verifies the influence of the number of snap- DOA information. shots on DOA estimation, where the number of snapshots L is varied from 1 to 400. K = 3 signals impinging from 5.2 Source localization − 5.2°, 5.3°, and 45.5° are considered, and the input SNR is This subsection evaluates the source localization perfor- fixedat15dB. It canbeseenthatthe proposed OPLI as mance of the proposed OPLI. The well-known RELAX well as RELAX is computationally efficient for DOA esti- [51], multiple signal classification (MUSIC) [55], and per- mation even if the number of snapshots is small, whereas turbed SBL (PSBL) [56] are included for performance the conventional MUSIC fails in this case. Compared to comparison. PSBL, the DOA estimation performance of the proposed Figure 4 presents the RMSE of the DOA estimates for OPLI is more attractive. When the number of snapshots theproposedOPLI, wherethe inputSNR ranges from −20 increases, the RMSE of the proposed OPLI decreases. The to 20 dB. K = 3 signals impinging from − 5.2°, 5.3°, RMSE of OPLI, MUSIC, and RELAX can coincide well and 45.5° are considered, and L = 200 snapshots are with the CRB when a large number of snapshots is given. employed. It is seen that the RMSE of the proposed OPLI Figure 6 examines the influence of angular separation coincides well with the CRB within a moderate to high on DOA estimation, where K = 2 signals impinging SNR region. Compared to PSBL, the proposed OPLI has from 5.3° −  and 5.3° are considered.  ranges from Fig. 3 Output SINR versus DOA uncertainty of the SOI Fig. 5 RMSE versus the number of snapshots Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 8 of 11 Fig. 6 RMSE versus angular separation Fig. 7 AveRMSE versus input SNR 2° to 16°, the input SNR is fixed at 10 dB, and L = 200 simulation results, it is seen that the proposed OPLI and snapshots are employed. The simulation results indicate the well-known RELAX exhibit better performance than that the RMSE of OPLI coincides well with the CRB the counterpart algorithms, when the input SNR within when the angular separation is large. Furthermore, the moderate to high region. With a low input SNR, the per- proposed OPLI exhibits a better DOA estimation perfor- formance of the proposed OPLI is not attractive, and this mance than RELAX, MUSIC and PSBL when the angular performance verifies the result of Fig. 1. separation is small. OPLI exhibits a better performance for Figure 8 illustrates the AveRMSE of the proposed OPLI, source localization of spatially adjacent sources than the where the number of snapshots ranges from 1 to 400. K = 3 counterpart algorithms. signals are considered, where the SOI is incident from 5.3° and the interference signals is incident from − 5.2° 5.3 Comprehensive performance evaluation and 45.5°. The input SNR is fixed at 15 dB. It follows This subsection compares the comprehensive localization from the simulation results that the proposed OPLI and and waveform identification performances of the pro- the well-known RELAX exhibit a better comprehensive posed OPLI with that of the combinational algorithms. performance than the other counterpart combinational Here, the combinational algorithms employ MUSIC and algorithms, when the number of snasphots is much lim- PSBL to estimate DOA of the SOI, and utilize MVDR, ited. With a few number of snasphots, the performance of LSMI, and GLC to estimate the waveform of the SOI. the proposed OPLI is much attractive. For comprehensive performance evaluation, the average Figure 9 illustrates the AveRMSE of the proposed OPLI, root mean square error (AveRMSE) is utilized, and it is where different angular separation is considered. K = 2 defined as M L L 1 1 AveRMSE = s ˆ (t ) − s (t ) |s (t )| 0,m l 0 l 0 l 2 M m=1 l=1 l=1 ˆ ⎠ + θ − θ  . 0,m 0 m=1 (43) Besides, the computational complexity of the proposed OPLI, which is evaluated by the running time of algo- rithm, is also compared with those of the combinational algorithms. Figure 7 illustrates the AveRMSE of the proposed OPLI, where the input SNR ranges from −20 to 20 dB. K = 3 signals are considered, and L = 200 snapshots are employed. The SOI is incident from 5.3°, and the interfer- Fig. 8 AveRMSE versus the number of snapshots ence signals is incident from − 5.2° and 45.5°. From the Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 9 of 11 Fig. 9 AveRMSE versus angular separation Fig. 11 Running time versus the number of snapshots signals are considered, the input SNR is fixed at 10 dB, and MATLAB 2013b running on a computer with a 2.3 GHz L = 200 snapshots are employed. The SOI impinges from Intel Quad-Core processor and 8GB RAM, under Win- 5.3°, and the interference signals impinge from 5.3° − , dows 8.1. It can be seen that the computational complexity where  ranges from 2° to 16°. It can be seen that the of the proposed OPLI is lower than that of the RELAX, proposed OPLI is superior to the well-known RELAX but is higher than that of the MUSIC-based combina- when the angular separation is small. With a small angular tional algorithms. This is mainly because the localization separation, comparing to the counterpart combinational and waveform identification processes of the proposed algorithms, the proposed OPLI also shows a better per- OPLI is implemented iteratively, whereas the MUSIC- formance, and this is mainly because the DOA estimation based combinational algorithms are not. Compared with accuracy of the proposed OPLI is much higher than those the PSBL-based combinational algorithms, the proposed of the counterpart algorithms (see also Fig. 5). OPLI illustrates a relatively stable running time when the Figure 10 shows the running time of the proposed OPLI input SNR and the number of snapshots within moderate at each input SNR. Figures 11 and 12 illustrate the run- to high SNR region. However, when the SNR, the number ning time of the proposed OPLI when different number of snapshots and the angular separation increase, the run- of snapshots and different angular separations are corre- ning time of the PSBL-based combinational algorithms spondingly considered. Parameters settings of Figs. 10, 11, fluctuates. This is due to the fact that the iteration num- and 12 remain the same as Figs. 7, 8,and 9,respec- ber of PSBL is not stable and keeps varying with these tively. All the simulation experiments are performed using parameters (cf. [56]). Fig. 10 Running time at each input SNR Fig. 12 Running time versus angular separation Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 10 of 11 6 Methods Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in The oblique projection-based localization and identi- published maps and institutional affiliations. fication (OPLI) algorithm is proposed without known prior DOA or waveform information of the sources. Author details School of Electronics and Information Engineering, Harbin Institute of The proposed OPLI employs oblique projection to Technology, Harbin 150001, China. The No.14 Institute of CETC, Nanjing separate the multiple incident signals into a series 3 210039, China. Qingdao Branch, Naval Aeronautical Engineering Institute, of single signal groups. Then, the source localiza- Qingdao 266041, China. Collaborative Innovation Centre of Information Sensing and Understanding, Harbin Institute of Technology, Harbin 150001, tion and waveform identification are implemented on China. Science and Technology on Electronic Information Control Laboratory, each separated single signal. To this end, the OPLI Chengdu 610036, China. is implemented iteratively. The method of maximum Received: 16 September 2017 Accepted: 10 May 2018 likelihood approximation is employed to evaluate the convergence. References 7Conclusions 1. S Sahnoun, P Comon, Joint source estimation and localization. IEEE Trans. Signal Process. 63(10), 2485–2495 (2015) A new OPLI algorithm, which is based on oblique projec- 2. A Khabbazibasmenj, SA Vorobyov, A Hassanien, Robust adaptive tion, is proposed for localization and waveform identifi- beamforming based on steering vector estimation with as little as possible cation of unknown target signal. The oblique projection prior information. IEEE Trans. Signal Process. 60(6), 2974–2987 (2012) is employed to separate the SOI from the received data of 3. H Krim, M Viberg, Two decades of array signal processing research: the parametric approach. IEEE Signal Process. Mag. 13(4), 67–94 (1996) the array, and can also be used to suppress the interference 4. H Cox, R Zeskind, M Owen, Robust adaptive beamforming. IEEE Trans. signals at the same time. Acoust. Speech Signal Process. ASSP-35(10), 1365–1375 (1987) The OPLI requires no prior information of the DOA or 5. BD Van Veen, KM Buckley, Beamforming: a versatile approach to spatial filtering. IEEE ASSP Mag. 5(2), 4–24 (1988) the signal waveforms, and it estimates the DOA and wave- 6. LC Godara, Application of antenna arrays to mobile communications, part form of the SOI iteratively. Comparing to the well known II: beam-forming and direction-of-arrival considerations. Proc. IEEE. 85(8), RELAX and the combinational algorithms, which esti- 1195–1245 (1997) 7. RG Lorenz, SP Boyd, Robust minimum variance beamforming. IEEE Trans. mates DOA via employing MUSIC or PSBL, and realizes Signal Process. 53(5), 1684–1696 (2005) beamforming via employing MVDR, LSMI or GLC, the 8. J Dmochowski, J Benesty, S Affes, Direction of arrival estimation using the simulation results show that the proposed OPLI exhibits a parameterized spatial correlation matrix. IEEE/ACM Trans. Audio Speech better performance when the angular separation is small. Lang. Process. 15(4), 1327–1339 (2007) 9. JC Chen, K Yao, RE Hudson, Source localization and beamforming. Especially, when the input SNR within moderate to high IEEE Signal Process. Mag. 19(2), 30–39 (2002) region, the proposed OPLI not only shows an attractive 10. Q Shen, W Liu, W Cui, SL Wu, Underdetermined DOA estimation under output SINR which can coincide well with the optimal the compressive sensing framework: A review. IEEE Access. 4, 8865–8878 (2016) one, but also can achieve a high estimation accuracy for 11. X Liu, F Li, N Zhenyu, Optimal resource allocation in simultaneous source localization since its RMSE could coincide well cooperative spectrum sensing and energy harvesting for multichannel with CRB. cognitive radio. IEEE Access. 5, 3801–3812 (2017) Both OPLI and RELAX are implemented iteratively, 12. M Zhou, Y Tang, Z Tian, L Xie, W Nie, Robust neighborhood graphing for semi-supervised indoor localization with light-loaded location and the running time of the former is lower. Whereas, fingerprinting. IEEE Internet Things J. 99, 1–1 (2017) when compared to MUSIC-based counterpart combina- 13. M Zhou, Y Wei, Z Tian, X Yang, L Li, Achieving Cost-efficient Indoor tional algorithms, the OPLI requires a higher computa- Fingerprint Localization on WLAN Platform: A Hypothetical Test Approach. IEEE Access. 5, 15865–15874 (2017) tional complexity to implement source localization and 14. SA Vorobyov, AB Gershman, ZQ Luo, Robust adaptive beamforming using waveform identification. Extensive experiments have been worst-case performance optimization: A solution to the signal mismatch undertaken to verify the effectiveness and superiority of problem. IEEE Trans. Signal Process. 51(2), 313–324 (2003) 15. B Liao, SC Chan, KM Tsui, Recursive steering vector estimation and the OPLI with uncorrelated sources. Future work includes adaptive beamforming under uncertainties. IEEE Trans. Aerosp. Electron. extension of the OPLI to correlated and coherent sources, Syst. 49(1), 489–501 (2013) and so on. 16. JW Xu, GS Liao, SQ Zhu, L Huang, Response vector constrained robust LCMV beamforming based on semidefinite programming. IEEE Trans. Signal Process. 63(21), 5720–5732 (2015) Funding 17. NY Wang, P Agathoklis, A Antoniou, A new DOA estimation technique This work was supported by the National Natural Science Foundation of China based on subarray beamforming. IEEE Trans. Signal Process. 54(9), (grant no. 61171180), by the Fundamental Research Funds for the Central 3279–3290 (2006) Universities (grants nos. HIT. MKSTISP. 2016 13 and HIT. MKSTISP. 2016 26), and by a fund from the Science and Technology on Electronic Information Control 18. AZ Hu, DOA-based beamforming for multi-cell massive MIMO systems. Laboratory. J. Commun. Netw. 18(5), 735–743 (2016) 19. X Liu, XZ Tan, Optimization algorithm of periodical cooperative spectrum Authors’ contributions sensing in cognitive radio. Int. J. Commun. Syst. 27(5), 705–720 (2012) All authors read and approved the final manuscript. 20. A Haimovich, Y Bar-Ness, An eigenanalysis interference canceler. IEEE Trans. Signal Process. 39(1), 76–84 (1991) Competing interests 21. A Haimovich, The eigencanceler: adaptive radar by eigenanalysis The authors declare that they have no competing interests. methods. IEEE Trans. Aerosp. Electron. Syst. 32(2), 532–542 (1996) Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 11 of 11 22. ML Honig, JS Goldstein, Adaptive reduced-rank interference suppression 49. YB Han, DQ Zhang, A recursive Bayesian beamforming for steering vector based on the multistage wiener filter. IEEE Trans. Comput. 50(6), 986–994 uncertainties. EURASIP J. Adv. Signal Process. 2013(108), 1–10 (2013) (2002) 50. J Li, P Stoica, Efficient mixed-spectrum estimation with applications to 23. YJ Gu, A Leshem, Robust adaptive beamforming based on interference target feature extraction. IEEE Trans. Signal Process. 44(2), 281–295 (1996) covariance matrix reconstruction and steering vector estimation. 51. J Li, DM Zheng, P Stoica, Angle and waveform estimation via relax. IEEE IEEE Trans. Signal Process. 60(7), 3881–3885 (2012) Trans. Aerosp. Electron. Syst. 33(3), 1077–1087 (1997) 52. T Yardibi, J Li, P Stoica, M Xue, AB Baggeroer, Source localization and 24. L Huang, J Zhang, X Xu, ZF Ye, Robust adaptive beamforming with a sensing: a nonparametric iterative adaptive approach based on weighted novel interference-plus-noise covariance matrix reconstruction method. least squares. IEEE Trans. Aerosp. Electron. Syst. 46(1), 425–443 (2010) IEEE Trans. Signal Process. 63(7), 1643–1650 (2015) 53. I Ziskind, M Wax, Maximum likelihood localization of multiple sources by 25. SC Chan, HH Chen, Uniform concentric circular arrays with frequency- alternating projection. IEEE Trans. Acoust. Speech Signal Process. 36(10), invariant characteristics—theory, design, adaptive beamforming and 1553–1560 (1988) DOA estimation. IEEE Trans. Signal Process. 55(1), 165–177 (2007) 54. P Stoica, A Nehorai, MUSIC, maximum likelihood, and Cramer-Rao bound. 26. R Boyer, Oblique projection for source estimation in a competitive IEEE Trans. Acoust. Speech Signal Process. 37(5), 720–741 (1989) environment: algorithm and statistical analysis. Signal Process. 89(12), 55. RO Schmidt, Multiple emitter location and signal parameter estimation. 2547–2554 (2009) IEEE Trans. Antennas Propag. 34(3), 276–280 (1986) 27. HB Xu, BR Zhu, J Liu, AZ Zhou, Robust beamforming design for secure 56. X Wu, WP Zhu, J Yan, Direction of arrival estimation for off-grid signals multiuser MISO interference channel. IEEE Commun. Lett. 21(4), 833–836 based on sparse bayesian learning. IEEE Sensors J. 16(7), 2004–2016 (2016) (2017) 28. K Mak, A Manikas, A superresolution wide null beamformer for undersampled signal reconstruction in SIMO SAR. IEEE J. Sel. Top. Signal Process. 9(8), 1548–1559 (2015) 29. LM Shi, Y Lin, Convex combination of adaptive filters under the maximum correntropy criterion in impulsive interference. IEEE Signal Process. Lett. 21(11), 1385–1388 (2014) 30. RT Behrens, LL Scharf, Signal processing applications of oblique projection operators. IEEE Trans. Signal Process. 42(6), 1413–1424 (1994) 31. XP Mao, AJ Liu, J HH, Oblique projection polarisation filtering for interference suppression in high-frequency surface wave radar. IET Radar Sonar Navig. 6(2), 71–80 (2012) 32. XP Mao, YL Yang, H Hong, WB Deng, Multi-domain collaborative filter for interference suppressing. IET Signal Process. 10(9), 1157–1168 (2016) 33. A Hassanien, SA Elkader, AB Gershman, Convex optimization based beam-space preprocessing with improved robustness against out-of-sector sources. IEEE Trans. Signal Process. 54(5), 1587–1595 (2006) 34. RJ Vaccaro, BF Harrison, Optimal matrix-filter design. IEEE Trans. Signal Process. 44(3), 705–709 (1996) 35. D Han, XH Zhang, Optimal matrix filter design with application to filtering short data records. IEEE Signal Process. Lett. 17(5), 521–524 (2010) 36. M Souden, J Benesty, S Affes, A study of the LCMV and MVDR noise reduction filters. IEEE Trans. Signal Process. 58(9), 4925–4935 (2010) 37. AI Koutrouvelis, RC Hendriks, R Heusdens, J Jensen, Relaxed binaural LCMV beamforming. IEEE/ACM Trans. Audio Speech Lang. Process. 25(1), 137–152 (2017) 38. J Li, P Stoica, ZS Wang, On robust capon beamforming and diagonal loading. IEEE Trans. Signal Process. 51(7), 1702–1715 (2003) 39. L Du, J Li, P Stoica, Fully automatic computation of diagonal loading levels for robust adaptive beamforming. IEEE Trans. Aerosp. Electron. Syst. 46(1), 449–458 (2010) 40. SA Vorobyov, AB Gershman, ZQ Luo, Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem. IEEE Trans. Signal Process. 51(2), 313–324 (2003) 41. ZL Yu, W Ser, MH Er, et al, Robust adaptive beamformers based on worst-case optimization and constraints on magnitude response. IEEE Trans. Signal Process. 57(7), 2615–2628 (2009) 42. JR Guerci, Theory and application of covariance matrix tapers for robust adaptive beamforming. IEEE Trans. Signal Process. 47(4), 977–985 (1999) 43. MH Er, BC Ng, A new approach to robust beamforming in the presence of steering vector errors. IEEE Trans. Signal Process. 42(7), 1826–1829 (1994) 44. F Shen, FF Chen, JY Song, Robust adaptive beamforming based on steering vector estimation and covariance matrix reconstruction. IEEE Commun. Lett. 19(9), 1636–1639 (2015) 45. H Li, YB Zhao, ZF Cheng, ZW Liu, PL Shui, Robust adaptive beamforming based on sparse representation technique. IET Radar Sonar Navig. 11(9), 1417–1424 (2017) 46. YX Ke, CS Zheng, RH Peng, XD Li, Robust adaptive beamforming using noise reduction preprocessing-based fully automatic diagonal loading and steering vector estimation. IEEE Access. 5, 12974–12987 (2017) 47. CJ Lam, AC Singer, Bayesian beamforming for DOA uncertainty: theory and implementation. IEEE Trans. Signal Process. 54(11), 4435–4445 (2006) 48. KL Bell, Y Ephraim, HL Van Trees, A Bayesian approach to robust adaptive beamforming. IEEE Trans. Signal Process. 48(2), 386–398 (2000) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png EURASIP Journal on Wireless Communications and Networking Springer Journals

Localization and identification of unknown target signal using oblique projection

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Engineering; Signal,Image and Speech Processing; Communications Engineering, Networks; Information Systems Applications (incl.Internet)
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Abstract

The problem of source localization and waveform identification is the key of array signal processing. In this paper, an oblique projection-based localization and identification (OPLI) algorithm is proposed without known prior DOA or waveform information of the sources. The proposed OPLI is implemented iteratively. In each iteration, oblique projection is employed to separate the multiple incident signals into a series of single signal groups. After that, the procedure of waveform and DOA estimation for each single signal is implemented. Theoretical analysis and simulation result verify the performance and effectiveness of the proposed OPLI. Keywords: Beamforming, Direction of arrival estimation, Oblique projection, Interference suppression 1 Introduction the interference signals are suppressed or attenuated in Source localization and waveform identification are cen- advance, the performance of localization and waveform tral problems in antenna array processing, including in identification for SOI would be better. Haimovich and particular radar, sonar, or wireless communication [1, 2]. Bar-Ness [20], Haimovich [21], and Honig and Goldstein To this end, various sensor signal processing tools have [22] proposed eigenanalysis-based interference canceler. been developed over the last several decades, ranging Similarly, Gu and Leshem [23], Huang et al. [24], Chan and from direction-of-arrival (DOA) estimation algorithms to Chen [25], Boyer [26], Xu et al. [27], Mak and Manikas spatial beamforming algorithms [3–13]. It is noted that, in [28], Shi and Lin [29], Behrens and Scharf [30], Mao practical applications, the localization and identification et al. [31], and Mao et al. [32]employedbeamforming of target signal could be difficult, when any a priori knowl- techniques to suppress inference signals. Hassanien et al. edge of signal-of-interest (SOI) is unavailable or the SOI [33], Vaccaro and Harrison [34], and Han and Zhang [35] is sheltered by adjacent interference signals. One of the designed matrix filters to cancel out-of-sector interference general approaches to solve this problem is realized as fol- signals. It should be noted that these mentioned algo- lows: The DOA estimation is firstly implented for all the rithms are based on accurate prior DOA of the SOI, and array receiving signals, and then, the waveform of SOI is some of them may also require necessary DOA informa- obtained via beamforming techniques [14–19]. tion of interference signals. No matter in the process of DOA estimation or in The estimation accuracy of DOA has major influ- beamforming, the influence of the interference signals on ence on the beamforming performances. Generally, target localization and identification is generally not triv- the performance of a beamformer decreases severely ial, especially when the interference signals are spatially when the DOA error increases [26, 31, 32]. Different adjacent with the SOI in the presence of large power. If approaches, including the linearly constrained minimum variance (LCMV), diagonal loading, convex optimization, *Correspondence: mxp@hit.edu.cn and covariance matrix taper approaches, are developed School of Electronics and Information Engineering, Harbin Institute of [16, 36–46] to combat DOA errors. These algorithms Technology, Harbin 150001, China can combat DOA uncertainties, but only suitable for Collaborative Innovation Centre of Information Sensing and Understanding, Harbin Institute of Technology, Harbin 150001, China small DOA errors. Lam and Singer [47], Bell et al. [48], Full list of author information is available at the end of the article © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 2 of 11 K −1 and Han and Zhang [49] developed Bayesian beamform- x(t) = a (θ ) s (t) + n(t) k k ing, which is able to implement waveform identifica- k=0 tion when the DOA is uncertain or unknown, while it = AS(t) + n(t) (1) requires prior statistics that describes the level of DOA uncertainty. where Even if an accurate DOA of the SOI is given, the opti- A = [a (θ ) , a (θ ) , a (θ ) , ··· , a (θ )] (2) 0 1 2 K −1 mal performance of beamforming is hard to achieve since most of the beamformers rely on the sampled matrix and inversion to replace the theoretical one, especially when S(t) = [s (t), s (t), s (t), ··· , s (t)] (3) 0 1 2 K −1 in the presence of short data samples [3, 4, 44, 46]. With limited number of snapshots, in this paper, we discuss the where s (t) refers to the baseband signal waveform of the problem of localization and identification for unknown kth incident signal, and a (θ ) denotes the corresponding target signal, where there exist spatially adjacent inter- steering vector. n(t) represents the additive noise compo- H T ference signals and the DOAs of interference signals are nent with covariance R = E n(t)n (t) . E[ ·], (·) and also completely unknown. The well-known RELAX dis- (·) stand for statistical expectation, transposition and cuss the similar problem [50–52], where a simultaneous Hermitian transposition, respectively. realization of DOA estimation and spatial beamforming Both the received signals and the noise are assumed to for all receiving signals is achieved, but the performance be sampled from zero-mean and uncorrelated stationary of beamforming is not theoretically deduced. random processes. Furthermore, the noise is temporally With spatially adjacent interference signals, an oblique and spatially independent of the received signals, and 2 2 projection-based localization and identification (OPLI) R = σ I,where σ is referred to as the noise power and I algorithm is proposed for unknown target signals. Firstly, stands for the identity matrix. Thus, the covariance matrix the OPLI employs oblique projection to separate the of the received data is given by mixed array receiving signals into individual signal H H 2 R = E x(t)x (t) = AR A + σ I (4) x S groups, where each signal group contains only one sig- nal. Then, the OPLI sequentially estimates DOA and where R = E S(t)S (t) . waveform of the signal in each group. Finally, the OPLI recursively reduces DOA and waveform errors via min- 2.2 Problem description imizing the optimal maximum likelihood cost function. This subsection formulates the localization and waveform Theoretical analysis is provided to show the beamform- identification problem. Without loss of generality, the sig- ing performance of the proposed OPLI, and simulation nal s (t) is assumed to be the SOI and the remaining K −1 results indicate that the OPLI is computationally effec- received signals are treated as interference signals. For the tive for source localization and waveform identification; SOI and the interference signals, neither the source loca- besides, it is superior to the counterpart conventional tions (i.e., θ , θ , ··· , θ ) nor the signal waveforms (i.e., 1 2 K −1 algorithms at moderate to high input signal-to-noise ratio s (t), s (t), ··· , s (t))are priorknown. 0 2 K −1 (SNR) region in terms of output signal-to-interference- The localization and waveform identification problem noise ratio (SINR) and root mean square error (RMSE). addressed in this paper is to obtain the DOA θ and wave- The paper is organized as follows. Section 2 presents form s (t) from multiple snapshots {x t } ,where L ( ) 0 l l=1 the signal model for source localization and waveform represents thenumberofsnapshots. identification. The proposed OPLI algorithm as well In the next section, it will be shown that the waveform as its complexity analysis is developed in Section 3. of the SOI can be identified as follows, i.e., Section 4 discusses the proposed OPLI, and Section 5 s ˆ (t) = W x (t) (5) 0 0 presents the simulation results. The conclusion is drawn in Section 7. where W refers to the optimum weight, and x (t) = a (θ ) s (t) + n(t).(6) 0 0 0 2 Signal model and problem description 2.1 Signal model And, according to (6), the location of the SOI is given by This subsection presents the signal model which is the conventional method of single target localization. considered for source localization and waveform iden- tification. Assume that K far-field narrowband signals 3 Proposed OPLI algorithm impinge upon an antenna array from distinct directions Derivation and implementation of the proposed OPLI are θ , θ , θ , ··· , θ . The antenna array consists of N sen- presented in this section. The proposed OPLI attempts to 0 1 2 K −1 sors with arbitrary array geometry. The N × 1vectorof employ oblique projection to separate the multiple inci- received signals at time t can be expressed as [3] dent signals into a series of single signal groups. As a Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 3 of 11 result, the source localization and waveform identification can be estimated by solving the following minimization are implemented on each separated single signal. Note problem, i.e., that the procedure of oblique projection requires the DOA parameters for source separation. To this end, the OPLI 1 θ , s ˆ (t) = arg min x t − a(θ )s t  (15) ( ) ( ) 0 0 0 l l is implemented iteratively and the method of maximum θ,s(t) l=1 likelihood approximation [53]isemployedtoevaluatethe convergence. Solving (15), it is obtained that 3.1 Basic principle s ˆ (t) = a (θ ) x (t) (16) 0 0 0 To illustrate the principle of localization and waveform and identification of the SOI, the DOAs of all incident signals are temporarily assumed to be known in this subsec- tion. According to (1)and (6), it is natural that the SOI ˆ θ = arg min P x (t ) (17) 0 0 a(θ ) component in the received data can be given by l=1 K −1 = arg min tr P R . a(θ ) x (t) = x(t) − a θ s (t) ( ) 0 k k k=1 ⊥ † H where P = I − a(θ )a (θ ),and R = E x (t)x (t) . 0 0 a(θ ) 0 = x(t) − B S (t).(7) 0 B Comparing (5)and (16), it follows that the optimum weight W to obtain the SOI is given by where 0 −1 B = [a (θ ) , a (θ ) , ··· , a (θ )] 0 1 2 K −1 † H W = a (θ ) = a (θ ) a (θ ) a (θ ) (18) 0 0 0 0 0 = A \ a (θ ) (8) −1 where (·) denotes inverse. and Using (5), (7), (10), (17), and (18), both the location S (t) = s (t), s (t), ··· , s (t) [ ] θ and the waveform s (t) of the SOI can be com- B 1 2 K −1 0 0 puted. Besides, the localization and waveform identifica- = S(t) \ s (t) (9) tion processes for the SOI can also be extended, i.e., the where the symbol \ signifies set difference. localization and waveform identification process for the To separate the SOI from the multiple incident sig- interference signals are feasible if each of the interference nals by (7), we employ oblique projection to obtain the signalsisregardedasthe SOI. interference signals, i.e., 3.2 Practical considerations S (t) = B E x(t) (10) B B |a θ ( ) 0 0 0 0 Recall that the localization and waveform identification process in Section 3.2 requires known DOAs of all the where E denotes the oblique projection operator B|a(θ ) incident signals, which is infeasible in practical applica- whose range space is R {B} and whose null space contains tions. In this paper, it is considered that the DOA of the R {a (θ )},and SOI is prior unavailable, and the prior DOAs of the inter- −1 H ⊥ H ⊥ ference signals are also unknown. Toward the purpose, the E = B B P B B P (11) B |a(θ ) 0 0 0 0 0 a(θ ) 0 a(θ ) 0 0 presented localization and waveform identification pro- where cess is implemented iteratively. The proposed OPLI firstly estimate the parameters of the SOI, so that a rough infor- E B = B (12) B |a(θ ) 0 0 0 0 mation of the SOI is available. And then, the parameters of each interference signals are updated one after another. E a (θ ) = 0 (13) B |a(θ ) 0 0 0 Through iteration, not only the parameters of the SOI are estimated, but also the parameters of the interference and signals are gradually achieving high precision. ⊥ † (i) (i) ˆ ˆ P = I − a (θ ) a (θ ) (14) 0 0 At the ith iteration, the estimated DOAs θ , θ , ··· , a(θ ) 0 0 1 (i) (i−1) (i−1) (i−1) ˆ ˆ ˆ ˆ θ , θ , θ , ··· , θ are utilized as initial val- ⊥ † K −1 k−1 k k+1 where R {·}, (·) ,and (·) denote range space, orthogonal (i) (i) ues to compute θ and s ˆ (t),where i ≥ 1and k = complement, and Moore-Penrose pseudoinverse, respec- k k (i) (i) tively. 0, 1, 2, ··· , K − 1. θ and s ˆ (t) denote the estimation of k k According to (7), only the SOI is contained in x (t ), θ and s (t) at the ith iteration (similarly hereinafter), 0 l k k and the interferences signals are removed. Thus, the SOI respectively. Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 4 of 11 Table 1 Iteration procedure of OPLI If the kth incident signal is regarded as the SOI, using Initialization: (7) ∼ (10)and (17), it follows that (−1) (−1) (−1) (−1) ˆ ˆ ˆ ˆ a θ = 0, a θ = 0, a θ = 0, a θ = 0, (i) (i) ⊥ 0 1 2 K−1 ˆ ˆ θ = arg min tr P R (19) k a(θ ) k Main Loop: (0) calculate L by (27) where for k = 0, 1, 2, ··· , K − 1 do (i) (i) (i) R = E x ˆ (t) x ˆ (t) (20) k k k (0) calculate θ by (19) (0) calculate s ˆ (t) by (25) and (i) (i) repeat x ˆ (t) = x(t) − B S (i) (t) (21) k k for j = 0, 1, ··· , k do where (i) redetermine θ by (19) (i) redetermine s ˆ (t) by (25) S (t) = B E x(t) (22) (i) (i) (i−1) ˆ k ˆ B ˆ B |a θ k k end for (i) (i) (i) (i−1) redetermine L by (27) ˆ ˆ ˆ B = A \ a θ (23) k k k (i) (i−1) until (|L − L | is less than a specified tolerance) and (0) (i) (0) (i) (0) (i) ˆ ˆ ˆ ˆ ˆ ˆ a θ ← a θ , a θ ← a θ , ··· , a θ ← a θ 0 0 1 1 j j (i) (i) (i) ˆ ˆ ˆ A = a θ , ··· , a θ , end for k k−1 Output (i−1) (i−1) (i−1) ˆ ˆ ˆ a θ , a θ , ··· , a θ . (24) k k+1 K (i) (i) (i) ˆ ˆ ˆ ˆ ˆ ˆ θ ← θ , θ ← θ , ··· , θ ← θ 0 1 K−1 0 1 K−1 Besides, using (5)and (18), it follows that (i) (i−1) (i) s ˆ (t) = W x ˆ (t) (25) k k k (i) 2 2 where x ˆ (t ) /L, which takes about O LN + N flops. −1 Herein, a flop is defined as a complex floating-point (i) (i) (i) (i) ˆ ˆ ˆ ˆ W = a θ a θ a θ . (26) k k k k addition or multiplication operation. The number of flops roughly required to compute (21)is O (N (K − 1)L + NL) During the proposed iterative process, the relative flops. The calculation of (22) requires approximately change of the following cost function, which is derived 3 2 2 O (K − 1) + 2N (K − 1) + N (K − 1) + (K − 1)NL from maximum likelihood approximation [53], between flops, where the calculation of E additionally (i) (i−1) ˆ ˆ B |a θ two consecutive iterations is utilized to indicate the con- k k 2 2 3 vergence, i.e., takes about O 3N (K − 1) + 2N (K − 1) + (K − 1) + 2 2N + N flops. Thus, the computational complex- L K −1 1 2 (i) (i) (i) ity of (20) ∼ (22)isroughly O N (L + 4K − 1)+ L = x (t ) − a θ s ˆ (t ) . (27) l l k k 2 3 N (4(K − 1) + (2K − 1)L + 1) + 2(K − 1) flops in l=1 k=0 total. Referring to the well-known relaxed iterative approach in (i) The computation of θ in (19) requires roughly [50, 51], the localization and waveform identification pro- 3 2 ˜ ˜ O N + 2N + 2N N (i) flops, where N (i) denotes cedure of the proposed OPLI is summarized in Table 1.It ˆ ˆ θ θ k k is shown that the proposed OPLI starts with the initializa- the number of potential source locations in the region (−1) (−1) (−1) (i−1) (i−1) (i) ˆ ˆ ˆ ˆ ˆ tion a θ = 0, a θ = 0, ··· , a θ = 0.At scope θ −  , θ +  . The computation of s ˆ (t) 0 1 K −1 k k k k k (i) in (25) requires roughly O(NL) flops, where the cal- the ith iteration, θ is determined within the range scope (i) (i−1) (i−1) culation of W additionally takes about O (2N + 1) ˆ ˆ θ −  , θ +  ,where K − 1 ≥ j ≥ 0, and  is a j j j j j (i) flops. The computation of L in (27) requires roughly user-selected parameter which indicates the range of the O ((K + 2)NL) flops. jth signal in the spatial domain. Therefore, according to Table 1, the computational com- plexity of the proposed OPLI is roughly O ((K + 2)NL) + 3.3 Computational complexity K −1 3 2 2 O N + 2N + 2N N (0) + N (L + 4K − 1)+ In this subsection, the computational complexity of the k=0 ˆ 2 3 proposed OPLI is analyzed. N 4(K − 1) + (2K − 1)L + 1 +2(K −1) +NL+2N +1+ Using L number of snapshots, the computation k 3 2 2 (N +2N +2N )N (i) +N (L+4K − 1)+ (i) L (i) ˆ i=1 j=0 ˆ of matrix R in (20)can be givenas x ˆ t ( ) j l=1 k k Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 5 of 11 2 3 N (4(K − 1) + (2K − 1)L + 1) + 2(K − 1) + NL + 2N It follows from (31)to(33) that the proposed OPLI can effectively suppress interference signals and have distor- + 1 + (K + 2)NL flops in total, where N denotes tionless response for the SOI. Regarding to the result of thenumberofiterationsemployedinthe kth outer waveform identification for the SOI, the output SINR of loop. Particularly, the total complexity is approximately OPLI is given by K −1 k 3 k 3 ˜ ˜ O N N + N N flops, when (0) (i) ˆ i=1 j=0 ˆ k=0 θ θ k j E W a (θ ) s (t) 0 0 opt,0 L  N > K, N  L/N, which occurs often in practical (i) SINR = . (36) applications. E W [x(t) − a (θ ) s (t)] 0 0 opt,0 4 Discussion Substituting (1), (31) ∼ (33)into(36), it follows that TheproposedOPLIinvolvesthe procedureofDOA esti- mation, hence it is applicable for localization of unknown SINR = target signal. Additionally, the OPLI also can obtain accu- E W n(t) rate locations of the interference signals, which is valuable opt,0 in practical applications. 2 2 σ σ TheproposedOPLIalsoinvolvesthe procedureof 0 0 = = H 2 beamforming. Substituting (21), (22), (26)into(25), it W R W n opt,0 σ W opt,0 opt,0 follows that (i) (i−1) 2 † s ˆ (t) = a θ I − E x(t). (28) σ a (θ ) I − E (i) (i−1) 0 B |a(θ ) 0 0 k k ˆ B |a θ k k When OPLI achieves convergence, it follows from (28) 2 † σ a (θ ) E 0 a(θ )|B 0 0 that 0 H ⊥ = a (θ ) P a (θ ) (37) s (t) = W x(t) (29) 0 0 k B opt,k 2 0 where W is referred to as the optimum beamformer 2 2 opt,k where σ = E |s (t)| refers to the power of the SOI. employed by the proposed OPLI, k = 0, 1, 2, ··· K −1, and It is known that the optimal SINR, which follows from the maximum SINR beamformer principle, is given W = a (θ ) I − E . (30) opt,k k B |a(θ ) k k by [46] 2 H −1 Let k = 0, according to (2), (8), and (30), it is obtained SINR = σ a (θ ) R a (θ ) (38) opt 0 0 0 in that −1 −1 H 2 where R = B R B + σ I .Herein, R = 0 S S † B B in 0 0 0 W = a (θ ) I − E (31) opt,0 0 B |a(θ ) 0 0 H 2 2 2 2 E S (t)S (t) = diag σ , σ , ··· , σ ,where σ 0 B 1 2 K −1 k and denotesthe powerofthe kth signal. In particular, when 2 2 the incident signals have equipower, i.e., σ = σ = ··· = W a (θ ) = 1 (32) 0 0 1 opt,0 2 2 σ = σ ,itisobtainedthat K −1 −1 W B = 0. (33) opt,0 2 1 σ −1 i H R = B B + I . (39) in 2 2 Recall from [30]that σ σ † † a (θ ) E = a (θ ) P I − E 2 2 0 a(θ )|B 0 a(θ ) B |a(θ ) 0 0 0 0 0 Further, if σ  σ ,then[30] = a (θ ) I − E (34) 0 B |a(θ ) 0 0 −1 1 σ −1 i H ⊥ R = lim B B + I = P . (40) where in 0 B 2 2 0 2 2 σ σ σ /σ →∞ −1 H ⊥ H ⊥ E = a(θ ) a (θ ) P a (θ ) a (θ ) P . (35) a(θ )|B 0 0 0 0 0 0 B B 0 0 Substituting (40)into(38), it is obtained that −1 Substituting (34)into(31), it is obtained that W = opt,0 2 2 σ σ H 0 H i H SINR = lim a (θ ) B B + I a (θ ) a (θ ) E . So, the optimum beamformer of the opt 0 0 0 0 a(θ )|B 0 0 0 2 2 σ 2 σ σ /σ →∞ proposed OPLI is equivalence to the well-known oblique projection beamformer which has excellent performance 0 H ⊥ = a (θ ) P a (θ ) . (41) 0 0 2 0 and has been well researched in [26, 31, 32]. σ Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 6 of 11 Comparing (41)with(37), it follows that the output SINR of the proposed OPLI can coincide well with the optimal SINR, when the power of the incident signals are equal and much higher than that of the noise. 5 Simulation results Extensive simulation results are provided to verify the effectiveness of the proposed OPLI for source local- ization and waveform identification. The combinational algorithm, which firstly utilizes DOA estimator to obtain source locations and then utilizes beamformer for wave- form identification, is compared with OPLI. Besides, the well-known RELAX [51] is also employed for perfor- mance comparison. Specially, for source localization, the stochastic Cramér-Rao bound (CRB) [54]isalsousedfor Fig. 1 Output SINR versus input SNR performance evaluation. Recall that the procedure of OPLI not only contains beamforming but also involves DOA estimation. The out- of the proposed OPLI coincides well with the optimal out- put SINR is employed for measuring the beamforming put SINR when the input SNR within moderate to high performance, and the RMSE is employed for evaluating region. Herein, RELAX shows a similar performance to the DOA estimation performance, where the proposed OPLI. OPLI exhibits better performance than the counterpart MVDR, LSMI and GLC beamform- M K ˆ ers, and this is mainly due to the fact that OPLI employs RMSE = θ − θ (42) k,m k KM oblique projection, which has excellent performance for m=1 k=1 suppressing interference signals, to identify the waveform where M denotes the number of Monte Carlo trials, θ c k,m of the SOI. The proposed OPLI tends to have a perfor- refers to the DOA estimation of θ in the mth Monte mance drop when the input SNR is lower than 0 dB, this Carlo trial (similarly hereafter). In the following simula- is because the employed oblique projection increases the tions, M = 200, and a half-wavelength spaced uniform noise variance (cf. [26, 30]). The simulation result verifies linear array composed of N = 6 sensors, is considered. the theoretical analysis given in Section 4. Figure 2 presents the output SINR of the proposed 5.1 Waveform identification OPLI,where thenumberofsnapshots L ranges from 2 This subsection evaluates the waveform identification to 1000. The input SNR is fixed at 10 dB. Other simula- performance of the proposed OPLI. The minimum vari- tion conditions remain the same as previous experiment. ance distortionless response (MVDR) beamformer [36], It is seen that the output SINR gradually approximate to the diagonally loaded sample matrix inversion (LSMI) the optimal output SINR when the number of snapshots beamformer [38], the general linear combination (GLC)- based beamformer [39], and the optimal beamformer which is based on the maximum output SINR principle [46] are employed for performance comparison. The diag- onal loading factor of the LSMI beamformer is set to be equal to the noise power. Note that both OPLI and RELAX implement waveform identification without prior DOA information. Whereas the MVDR, LSMI, and GLC beamformers require prior DOA of the SOI to calculate beamforming weights. Unless otherwise stated, in the subsection, it is assumed that the prior DOA is considered without error. Figure 1 presents the output SINR of the proposed OPLI,where theinput SNRrangesfrom −20 to 30 dB. K = 3 signals are considered, and L = 200 snapshots are employed. The SOI impinges on the array from 5.3°, and the interference signals are impinging from − 5.2°and Fig. 2 Output SINR versus the number of snapshots 45.5°. The simulation results show that the output SINR Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 7 of 11 increases. Compared to RELAX, OPLI exhibits the same performance. While comparing to the MVDR, LSMI, and GLC beamformers, the proposed OPLI has a better per- formance for waveform identification even if the number of snapshots is much limited. Figure 3 presents the output SINR of the proposed OPLI, where the DOA error of the SOI ranges from − 5° to 5°. In this simulation, L = 200 snapshots are consid- ered, and the input SNR is fixed at 10 dB. Other simula- tion conditions remain the same as previous experiment, except that the prior DOA of the SOI is considered with different error for the MVDR, LSMI, and GLC beamform- ers. The prior DOA of the SOI is set to 5.3° + δ,where δ is varied from − 5° to 5°. The simulation results show that the MVDR, LSMI, and GLC beamformers are sensi- Fig. 4 RMSE versus input SNR tive to the prior DOA uncertainties of the SOI. The larger the DOA error goes, the quicker the output SINR drops. Comparing to the MVDR, LSMI, and GLC beamformers, a much better performance when the input SNR is high. the proposed OPLI as well as RELAX remains the same Compared to MUSIC and RELAX, the proposed PSBL output SINR. This is due to the fact that both RELAX and exhibits a better performance when the input SNR is low. OPLI implement waveform identification without prior Figure 5 verifies the influence of the number of snap- DOA information. shots on DOA estimation, where the number of snapshots L is varied from 1 to 400. K = 3 signals impinging from 5.2 Source localization − 5.2°, 5.3°, and 45.5° are considered, and the input SNR is This subsection evaluates the source localization perfor- fixedat15dB. It canbeseenthatthe proposed OPLI as mance of the proposed OPLI. The well-known RELAX well as RELAX is computationally efficient for DOA esti- [51], multiple signal classification (MUSIC) [55], and per- mation even if the number of snapshots is small, whereas turbed SBL (PSBL) [56] are included for performance the conventional MUSIC fails in this case. Compared to comparison. PSBL, the DOA estimation performance of the proposed Figure 4 presents the RMSE of the DOA estimates for OPLI is more attractive. When the number of snapshots theproposedOPLI, wherethe inputSNR ranges from −20 increases, the RMSE of the proposed OPLI decreases. The to 20 dB. K = 3 signals impinging from − 5.2°, 5.3°, RMSE of OPLI, MUSIC, and RELAX can coincide well and 45.5° are considered, and L = 200 snapshots are with the CRB when a large number of snapshots is given. employed. It is seen that the RMSE of the proposed OPLI Figure 6 examines the influence of angular separation coincides well with the CRB within a moderate to high on DOA estimation, where K = 2 signals impinging SNR region. Compared to PSBL, the proposed OPLI has from 5.3° −  and 5.3° are considered.  ranges from Fig. 3 Output SINR versus DOA uncertainty of the SOI Fig. 5 RMSE versus the number of snapshots Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 8 of 11 Fig. 6 RMSE versus angular separation Fig. 7 AveRMSE versus input SNR 2° to 16°, the input SNR is fixed at 10 dB, and L = 200 simulation results, it is seen that the proposed OPLI and snapshots are employed. The simulation results indicate the well-known RELAX exhibit better performance than that the RMSE of OPLI coincides well with the CRB the counterpart algorithms, when the input SNR within when the angular separation is large. Furthermore, the moderate to high region. With a low input SNR, the per- proposed OPLI exhibits a better DOA estimation perfor- formance of the proposed OPLI is not attractive, and this mance than RELAX, MUSIC and PSBL when the angular performance verifies the result of Fig. 1. separation is small. OPLI exhibits a better performance for Figure 8 illustrates the AveRMSE of the proposed OPLI, source localization of spatially adjacent sources than the where the number of snapshots ranges from 1 to 400. K = 3 counterpart algorithms. signals are considered, where the SOI is incident from 5.3° and the interference signals is incident from − 5.2° 5.3 Comprehensive performance evaluation and 45.5°. The input SNR is fixed at 15 dB. It follows This subsection compares the comprehensive localization from the simulation results that the proposed OPLI and and waveform identification performances of the pro- the well-known RELAX exhibit a better comprehensive posed OPLI with that of the combinational algorithms. performance than the other counterpart combinational Here, the combinational algorithms employ MUSIC and algorithms, when the number of snasphots is much lim- PSBL to estimate DOA of the SOI, and utilize MVDR, ited. With a few number of snasphots, the performance of LSMI, and GLC to estimate the waveform of the SOI. the proposed OPLI is much attractive. For comprehensive performance evaluation, the average Figure 9 illustrates the AveRMSE of the proposed OPLI, root mean square error (AveRMSE) is utilized, and it is where different angular separation is considered. K = 2 defined as M L L 1 1 AveRMSE = s ˆ (t ) − s (t ) |s (t )| 0,m l 0 l 0 l 2 M m=1 l=1 l=1 ˆ ⎠ + θ − θ  . 0,m 0 m=1 (43) Besides, the computational complexity of the proposed OPLI, which is evaluated by the running time of algo- rithm, is also compared with those of the combinational algorithms. Figure 7 illustrates the AveRMSE of the proposed OPLI, where the input SNR ranges from −20 to 20 dB. K = 3 signals are considered, and L = 200 snapshots are employed. The SOI is incident from 5.3°, and the interfer- Fig. 8 AveRMSE versus the number of snapshots ence signals is incident from − 5.2° and 45.5°. From the Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 9 of 11 Fig. 9 AveRMSE versus angular separation Fig. 11 Running time versus the number of snapshots signals are considered, the input SNR is fixed at 10 dB, and MATLAB 2013b running on a computer with a 2.3 GHz L = 200 snapshots are employed. The SOI impinges from Intel Quad-Core processor and 8GB RAM, under Win- 5.3°, and the interference signals impinge from 5.3° − , dows 8.1. It can be seen that the computational complexity where  ranges from 2° to 16°. It can be seen that the of the proposed OPLI is lower than that of the RELAX, proposed OPLI is superior to the well-known RELAX but is higher than that of the MUSIC-based combina- when the angular separation is small. With a small angular tional algorithms. This is mainly because the localization separation, comparing to the counterpart combinational and waveform identification processes of the proposed algorithms, the proposed OPLI also shows a better per- OPLI is implemented iteratively, whereas the MUSIC- formance, and this is mainly because the DOA estimation based combinational algorithms are not. Compared with accuracy of the proposed OPLI is much higher than those the PSBL-based combinational algorithms, the proposed of the counterpart algorithms (see also Fig. 5). OPLI illustrates a relatively stable running time when the Figure 10 shows the running time of the proposed OPLI input SNR and the number of snapshots within moderate at each input SNR. Figures 11 and 12 illustrate the run- to high SNR region. However, when the SNR, the number ning time of the proposed OPLI when different number of snapshots and the angular separation increase, the run- of snapshots and different angular separations are corre- ning time of the PSBL-based combinational algorithms spondingly considered. Parameters settings of Figs. 10, 11, fluctuates. This is due to the fact that the iteration num- and 12 remain the same as Figs. 7, 8,and 9,respec- ber of PSBL is not stable and keeps varying with these tively. All the simulation experiments are performed using parameters (cf. [56]). Fig. 10 Running time at each input SNR Fig. 12 Running time versus angular separation Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 10 of 11 6 Methods Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in The oblique projection-based localization and identi- published maps and institutional affiliations. fication (OPLI) algorithm is proposed without known prior DOA or waveform information of the sources. Author details School of Electronics and Information Engineering, Harbin Institute of The proposed OPLI employs oblique projection to Technology, Harbin 150001, China. The No.14 Institute of CETC, Nanjing separate the multiple incident signals into a series 3 210039, China. Qingdao Branch, Naval Aeronautical Engineering Institute, of single signal groups. Then, the source localiza- Qingdao 266041, China. Collaborative Innovation Centre of Information Sensing and Understanding, Harbin Institute of Technology, Harbin 150001, tion and waveform identification are implemented on China. Science and Technology on Electronic Information Control Laboratory, each separated single signal. To this end, the OPLI Chengdu 610036, China. is implemented iteratively. The method of maximum Received: 16 September 2017 Accepted: 10 May 2018 likelihood approximation is employed to evaluate the convergence. References 7Conclusions 1. S Sahnoun, P Comon, Joint source estimation and localization. IEEE Trans. Signal Process. 63(10), 2485–2495 (2015) A new OPLI algorithm, which is based on oblique projec- 2. A Khabbazibasmenj, SA Vorobyov, A Hassanien, Robust adaptive tion, is proposed for localization and waveform identifi- beamforming based on steering vector estimation with as little as possible cation of unknown target signal. The oblique projection prior information. IEEE Trans. Signal Process. 60(6), 2974–2987 (2012) is employed to separate the SOI from the received data of 3. H Krim, M Viberg, Two decades of array signal processing research: the parametric approach. IEEE Signal Process. Mag. 13(4), 67–94 (1996) the array, and can also be used to suppress the interference 4. H Cox, R Zeskind, M Owen, Robust adaptive beamforming. IEEE Trans. signals at the same time. Acoust. Speech Signal Process. ASSP-35(10), 1365–1375 (1987) The OPLI requires no prior information of the DOA or 5. BD Van Veen, KM Buckley, Beamforming: a versatile approach to spatial filtering. IEEE ASSP Mag. 5(2), 4–24 (1988) the signal waveforms, and it estimates the DOA and wave- 6. LC Godara, Application of antenna arrays to mobile communications, part form of the SOI iteratively. Comparing to the well known II: beam-forming and direction-of-arrival considerations. Proc. IEEE. 85(8), RELAX and the combinational algorithms, which esti- 1195–1245 (1997) 7. RG Lorenz, SP Boyd, Robust minimum variance beamforming. IEEE Trans. mates DOA via employing MUSIC or PSBL, and realizes Signal Process. 53(5), 1684–1696 (2005) beamforming via employing MVDR, LSMI or GLC, the 8. J Dmochowski, J Benesty, S Affes, Direction of arrival estimation using the simulation results show that the proposed OPLI exhibits a parameterized spatial correlation matrix. IEEE/ACM Trans. Audio Speech better performance when the angular separation is small. Lang. Process. 15(4), 1327–1339 (2007) 9. JC Chen, K Yao, RE Hudson, Source localization and beamforming. Especially, when the input SNR within moderate to high IEEE Signal Process. Mag. 19(2), 30–39 (2002) region, the proposed OPLI not only shows an attractive 10. Q Shen, W Liu, W Cui, SL Wu, Underdetermined DOA estimation under output SINR which can coincide well with the optimal the compressive sensing framework: A review. IEEE Access. 4, 8865–8878 (2016) one, but also can achieve a high estimation accuracy for 11. X Liu, F Li, N Zhenyu, Optimal resource allocation in simultaneous source localization since its RMSE could coincide well cooperative spectrum sensing and energy harvesting for multichannel with CRB. cognitive radio. IEEE Access. 5, 3801–3812 (2017) Both OPLI and RELAX are implemented iteratively, 12. M Zhou, Y Tang, Z Tian, L Xie, W Nie, Robust neighborhood graphing for semi-supervised indoor localization with light-loaded location and the running time of the former is lower. Whereas, fingerprinting. IEEE Internet Things J. 99, 1–1 (2017) when compared to MUSIC-based counterpart combina- 13. M Zhou, Y Wei, Z Tian, X Yang, L Li, Achieving Cost-efficient Indoor tional algorithms, the OPLI requires a higher computa- Fingerprint Localization on WLAN Platform: A Hypothetical Test Approach. IEEE Access. 5, 15865–15874 (2017) tional complexity to implement source localization and 14. SA Vorobyov, AB Gershman, ZQ Luo, Robust adaptive beamforming using waveform identification. Extensive experiments have been worst-case performance optimization: A solution to the signal mismatch undertaken to verify the effectiveness and superiority of problem. IEEE Trans. Signal Process. 51(2), 313–324 (2003) 15. B Liao, SC Chan, KM Tsui, Recursive steering vector estimation and the OPLI with uncorrelated sources. Future work includes adaptive beamforming under uncertainties. IEEE Trans. Aerosp. Electron. extension of the OPLI to correlated and coherent sources, Syst. 49(1), 489–501 (2013) and so on. 16. JW Xu, GS Liao, SQ Zhu, L Huang, Response vector constrained robust LCMV beamforming based on semidefinite programming. IEEE Trans. Signal Process. 63(21), 5720–5732 (2015) Funding 17. NY Wang, P Agathoklis, A Antoniou, A new DOA estimation technique This work was supported by the National Natural Science Foundation of China based on subarray beamforming. IEEE Trans. Signal Process. 54(9), (grant no. 61171180), by the Fundamental Research Funds for the Central 3279–3290 (2006) Universities (grants nos. HIT. MKSTISP. 2016 13 and HIT. MKSTISP. 2016 26), and by a fund from the Science and Technology on Electronic Information Control 18. AZ Hu, DOA-based beamforming for multi-cell massive MIMO systems. Laboratory. J. Commun. Netw. 18(5), 735–743 (2016) 19. X Liu, XZ Tan, Optimization algorithm of periodical cooperative spectrum Authors’ contributions sensing in cognitive radio. Int. J. Commun. Syst. 27(5), 705–720 (2012) All authors read and approved the final manuscript. 20. A Haimovich, Y Bar-Ness, An eigenanalysis interference canceler. IEEE Trans. Signal Process. 39(1), 76–84 (1991) Competing interests 21. A Haimovich, The eigencanceler: adaptive radar by eigenanalysis The authors declare that they have no competing interests. methods. IEEE Trans. Aerosp. Electron. Syst. 32(2), 532–542 (1996) Huo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:138 Page 11 of 11 22. ML Honig, JS Goldstein, Adaptive reduced-rank interference suppression 49. YB Han, DQ Zhang, A recursive Bayesian beamforming for steering vector based on the multistage wiener filter. IEEE Trans. Comput. 50(6), 986–994 uncertainties. EURASIP J. Adv. Signal Process. 2013(108), 1–10 (2013) (2002) 50. J Li, P Stoica, Efficient mixed-spectrum estimation with applications to 23. YJ Gu, A Leshem, Robust adaptive beamforming based on interference target feature extraction. IEEE Trans. Signal Process. 44(2), 281–295 (1996) covariance matrix reconstruction and steering vector estimation. 51. J Li, DM Zheng, P Stoica, Angle and waveform estimation via relax. IEEE IEEE Trans. Signal Process. 60(7), 3881–3885 (2012) Trans. Aerosp. Electron. Syst. 33(3), 1077–1087 (1997) 52. T Yardibi, J Li, P Stoica, M Xue, AB Baggeroer, Source localization and 24. L Huang, J Zhang, X Xu, ZF Ye, Robust adaptive beamforming with a sensing: a nonparametric iterative adaptive approach based on weighted novel interference-plus-noise covariance matrix reconstruction method. least squares. IEEE Trans. Aerosp. Electron. Syst. 46(1), 425–443 (2010) IEEE Trans. Signal Process. 63(7), 1643–1650 (2015) 53. I Ziskind, M Wax, Maximum likelihood localization of multiple sources by 25. SC Chan, HH Chen, Uniform concentric circular arrays with frequency- alternating projection. IEEE Trans. Acoust. Speech Signal Process. 36(10), invariant characteristics—theory, design, adaptive beamforming and 1553–1560 (1988) DOA estimation. IEEE Trans. Signal Process. 55(1), 165–177 (2007) 54. P Stoica, A Nehorai, MUSIC, maximum likelihood, and Cramer-Rao bound. 26. R Boyer, Oblique projection for source estimation in a competitive IEEE Trans. Acoust. Speech Signal Process. 37(5), 720–741 (1989) environment: algorithm and statistical analysis. Signal Process. 89(12), 55. RO Schmidt, Multiple emitter location and signal parameter estimation. 2547–2554 (2009) IEEE Trans. Antennas Propag. 34(3), 276–280 (1986) 27. HB Xu, BR Zhu, J Liu, AZ Zhou, Robust beamforming design for secure 56. X Wu, WP Zhu, J Yan, Direction of arrival estimation for off-grid signals multiuser MISO interference channel. IEEE Commun. Lett. 21(4), 833–836 based on sparse bayesian learning. IEEE Sensors J. 16(7), 2004–2016 (2016) (2017) 28. K Mak, A Manikas, A superresolution wide null beamformer for undersampled signal reconstruction in SIMO SAR. IEEE J. Sel. Top. Signal Process. 9(8), 1548–1559 (2015) 29. LM Shi, Y Lin, Convex combination of adaptive filters under the maximum correntropy criterion in impulsive interference. IEEE Signal Process. Lett. 21(11), 1385–1388 (2014) 30. RT Behrens, LL Scharf, Signal processing applications of oblique projection operators. IEEE Trans. Signal Process. 42(6), 1413–1424 (1994) 31. XP Mao, AJ Liu, J HH, Oblique projection polarisation filtering for interference suppression in high-frequency surface wave radar. IET Radar Sonar Navig. 6(2), 71–80 (2012) 32. XP Mao, YL Yang, H Hong, WB Deng, Multi-domain collaborative filter for interference suppressing. IET Signal Process. 10(9), 1157–1168 (2016) 33. A Hassanien, SA Elkader, AB Gershman, Convex optimization based beam-space preprocessing with improved robustness against out-of-sector sources. IEEE Trans. Signal Process. 54(5), 1587–1595 (2006) 34. RJ Vaccaro, BF Harrison, Optimal matrix-filter design. IEEE Trans. Signal Process. 44(3), 705–709 (1996) 35. D Han, XH Zhang, Optimal matrix filter design with application to filtering short data records. IEEE Signal Process. Lett. 17(5), 521–524 (2010) 36. M Souden, J Benesty, S Affes, A study of the LCMV and MVDR noise reduction filters. IEEE Trans. Signal Process. 58(9), 4925–4935 (2010) 37. AI Koutrouvelis, RC Hendriks, R Heusdens, J Jensen, Relaxed binaural LCMV beamforming. IEEE/ACM Trans. Audio Speech Lang. Process. 25(1), 137–152 (2017) 38. J Li, P Stoica, ZS Wang, On robust capon beamforming and diagonal loading. IEEE Trans. Signal Process. 51(7), 1702–1715 (2003) 39. L Du, J Li, P Stoica, Fully automatic computation of diagonal loading levels for robust adaptive beamforming. IEEE Trans. Aerosp. Electron. Syst. 46(1), 449–458 (2010) 40. SA Vorobyov, AB Gershman, ZQ Luo, Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem. IEEE Trans. Signal Process. 51(2), 313–324 (2003) 41. ZL Yu, W Ser, MH Er, et al, Robust adaptive beamformers based on worst-case optimization and constraints on magnitude response. IEEE Trans. Signal Process. 57(7), 2615–2628 (2009) 42. JR Guerci, Theory and application of covariance matrix tapers for robust adaptive beamforming. IEEE Trans. Signal Process. 47(4), 977–985 (1999) 43. MH Er, BC Ng, A new approach to robust beamforming in the presence of steering vector errors. IEEE Trans. Signal Process. 42(7), 1826–1829 (1994) 44. F Shen, FF Chen, JY Song, Robust adaptive beamforming based on steering vector estimation and covariance matrix reconstruction. IEEE Commun. Lett. 19(9), 1636–1639 (2015) 45. H Li, YB Zhao, ZF Cheng, ZW Liu, PL Shui, Robust adaptive beamforming based on sparse representation technique. IET Radar Sonar Navig. 11(9), 1417–1424 (2017) 46. YX Ke, CS Zheng, RH Peng, XD Li, Robust adaptive beamforming using noise reduction preprocessing-based fully automatic diagonal loading and steering vector estimation. IEEE Access. 5, 12974–12987 (2017) 47. CJ Lam, AC Singer, Bayesian beamforming for DOA uncertainty: theory and implementation. IEEE Trans. Signal Process. 54(11), 4435–4445 (2006) 48. KL Bell, Y Ephraim, HL Van Trees, A Bayesian approach to robust adaptive beamforming. IEEE Trans. Signal Process. 48(2), 386–398 (2000)

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EURASIP Journal on Wireless Communications and NetworkingSpringer Journals

Published: Jun 4, 2018

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