Nested array receiver with time‐delayers for joint target range and angle estimation
2016; Institution of Engineering and Technology; Volume: 10; Issue: 8 Linguagem: Inglês
10.1049/iet-rsn.2015.0450
ISSN1751-8792
Autores Tópico(s)Antenna Design and Optimization
ResumoIET Radar, Sonar & NavigationVolume 10, Issue 8 p. 1384-1393 Research ArticleFree Access Nested array receiver with time-delayers for joint target range and angle estimation Wen-Qin Wang, Corresponding Author Wen-Qin Wang wqwang@uestc.edu.cn School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu, People's Republic of China Prof. Wang also is a Marie Curie International Incoming Fellow of Imperial College London, UKSearch for more papers by this authorChenglong Zhu, Chenglong Zhu School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu, People's Republic of ChinaSearch for more papers by this author Wen-Qin Wang, Corresponding Author Wen-Qin Wang wqwang@uestc.edu.cn School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu, People's Republic of China Prof. Wang also is a Marie Curie International Incoming Fellow of Imperial College London, UKSearch for more papers by this authorChenglong Zhu, Chenglong Zhu School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu, People's Republic of ChinaSearch for more papers by this author First published: 01 October 2016 https://doi.org/10.1049/iet-rsn.2015.0450Citations: 15 AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Phased-array provides range-independent beampattern and consequently it cannot detect/suppress range-dependent targets/interferences. Moreover, an M-element phased-array can resolve at most M − 1 targets simultaneously. This study proposes a nested array receiver using diverse time-delayers (DTDs) for jointly estimating the range and angle of targets. The essence is to construct a new array structure by systematically nesting two uniform linear arrays through DTDs instead of phase controlling in traditional nested array technique. Using the second-order statistics of the received data, an M-element array can resolve O(M2) targets. Moreover, the nested DTD receiver design is exploited for minimum variance distortionless response beamforming and direction finding. The improvements offered by the proposed method, as compared with basic DTD receivers, are demonstrated by examining the output signal-to-interference-plus-noise ratio, the probability of detection and the Cramér–Rao lower bounds for estimating the range and angle of targets. 1 Introduction Phased-array antenna is known for its capability to steer the beam electronically with high effectiveness [1, 2]. The directional gain offered by a phased-array antenna is useful for detecting weak targets and nulling interferences from other directions. However, the phased-array has a limitation in that the beam steering is range independent, but the ability to control range-dependent transmit energy distribution becomes an increasingly desirable trait in some applications such as range-dependent interference suppression [3] and range ambiguity suppression. Since range-dependent beampattern is beneficial in many applications [4, 5], frequency diverse array (FDA) using a small frequency increment across the array elements has received much attention in recent years [6-8]. The frequency increment results in a range-dependent beampattern for which the beam focusing direction will change as a function of the range, angle and time [9-12]. Therefore, instead of angle-dependent only beampattern such as the phased-array, FDA beampattern depends on frequency, range and time, which could be exploited for some specific applications [13-16]. Inspired by the FDA range-dependent beampattern characteristics, this paper proposes a nested array receiver using diverse time-delayers (DTDs) to equivalently implement FDA functionality for target range–angle estimation and increased degrees-of-freedom (DOFs) in target localisation. Note that the idea of using DTD in array design has been proposed in [17]. This paper is motivated also by the requirement of increasing the DOFs of antenna array systems. Moffet [18] increase the DOFs with the minimum redundancy array. The disadvantage is that the constructed augmented covariance matrix is not positive semi-definite for finite number of snapshots and the method is restricted to non-Gaussian sources. Moreover, sparsensing array elements have the potential drawbacks of generating grating lobes [19]. To overcome these disadvantages, an interesting linear nested phased-array based on the concept of difference co-array [20] has been designed [21]. It is a systematic way to increase the array DOFs. However, the nested phased-array cannot jointly estimate the range and angle of targets because it provides phase-dependent only beampattern. Therefore, we further apply the difference co-array processing to the proposed nested DTD receiver. Our contributions can be summarised as follows: (i) different from existing FDA literatures considering only on transmitter design, a nested array receiver architecture using DTDs for range-dependent receive beamforming is proposed and the corresponding signal models are derived in detail. (ii) An accurate range and angle estimation algorithm is developed by applying the difference co-array processing algorithm. Its detection performance and Cramér–Rao lower bounds (CRLBs) are derived and verified by numerical results. The rest of this paper is organised as follows. In Section 2, a brief introduction to difference co-array processing-based nested array, which motivates the work, is given. In Section 3, we propose the nested FDA receiver design scheme, along with the signal model. Then, the spatial smoothing and range–angle estimation are presented in Section 4. Next, the performance analysis including the probability of detection and CRLB for estimating range and angle are derived in Section 5. Finally, extensive simulation results are provided in Section 6 and concluding summaries are drawn in Section 7. 2 Background and motivation Consider an array with N elements which are divided into two levels, as shown in Fig. 1a. Let b(θ) be the N × 1 receive steering vector. Assuming there are D sources with directions {θi, i = 1, …, D} and powers , the array observations can be written as (1) where B = [b(θ1), b(θ2), …, b(θD)] stands for the array manifold matrix and s(t) = [s1(t), s2(t), …, sD(t)]T with T being the transpose operator accounts for the source signal vector. The noise n(t) is assumed to be temporally and spatially white and uncorrelated from the sources. Fig. 1Open in figure viewerPowerPoint Illustration of the proposed nested array receiver using DTDs When the sources are also assumed to be temporally uncorrelated, the received signal autocorrelation matrix can be represented by (2) where H is the conjugate transpose operator, is the noise power, Rss is the signal covariance matrix and I is the unit matrix. Rxx can be vectorised as [22] (3) where * is the conjugate operator, ⊗ is the Khatri–Rao product according to [22], and whose element ei stands for a column vector of all zeros except a 1 at the ith position. Comparing (3) with (1), we can regard z as the received data at an array whose manifold is given by B* ⊗ B. Accordingly, the equivalent source signal is represented by p and noise becomes a deterministic vector given by . The distinct rows of B* ⊗ B are the manifold associated with a virtual array whose sensor positions are given by the distinct values in the following set: (4) where di (dj) is the position vector of the ith (jth) element. The difference co-array is defined as the array whose elements are located at positions corresponding to the set De which consists of all the distinct elements in D. The cardinality of De gives the DOFs that can be obtained from the difference co-array. Then, the maximum attainable number of DOFs from a difference co-array with N elements, denoted by DOFmax, is (5) Certainly, if a difference occurs more than once, it implies a decrease in the overall cardinality of De. Consider a linear array with d being the minimum spacing of the underlying grid and define the function c[n] which takes a value 1 if there is an element located at nd and 0 otherwise. The number of same occurrences in each position, denoted by γ, is expressed as [20] (6) where is the convolution operator. That is to say, the difference co-array of an N-element nested array is another uniform linear array (ULA) with (2N − 1) elements. 3 Nested array receiver based on DTDs 3.1 Array arrangement To achieve more DOFs and offer range-dependent beampattern, we propose the nested DTD receiver. Note that phased-array was used in [21], but here we use time-delayer-based array instead of phased-array. It is also basically a concatenation of two ULA DTDs: namely, inner and outer DTDs where they consist of N1 and N2 elements with inter-element spacings d1 and d2 = (N1 + 1)d1, respectively. More precisely, it is a linear array with element locations given by the union of the sets Sinner = {n1d1, n1 = 1, 2, …, N1} and Souter = {n2(N1 + 1)d1, n2 = 1, 2, …, N2}. Fig. 1 illustrates the proposed nested receiver scheme. It is similar to the union of the transmit and receive arrays of a multiple-input–multiple-output (MIMO) radar; However, it uses a single such array for receiving only instead of active sensing as done in MIMO radar. Since the nested DTD receiver provides not only phase-dependent, but also range-dependent beampattern, it is rather different from the conventional nested phased-array [21] in beampattern, signal model and direction-of-arrival (DOA) estimation. In the following we derive first the nested DTD receiver signal model. 3.2 Array signal model formulation We derive the proposed nested DTD receiver pattern in a rather general way by assuming the signal coming to the array is a linearly frequency modulated (LFM) waveform (7) where f0 is the starting frequency (equal to the carrier frequency) and kr is the chirp rate. Specially, when kr = 0, s(t) simplifies into a monochromatic signal. Let ω0 = 2πf0, the far-field array pattern associated with the receiving nested DTD receiver can be defined as (8) where τn denotes the delay to the observation point for the nth array element, σ is the target radar cross-section and c0 is the speed of light. Note that, here, the element patterns are taken as unity. The range of observation point to the nth array element can be approximated as (9) where r0 is the range of observation point to the first element, d is the inter-element spacing in the first level of the nested DTD receiver and θ is the observation azimuth angle. The sensors are assumed to be placed on a linear grid with xn denoting the position of the nth sensor, which is an integer in times of the smallest inter-element spacing. Suppose the inter-element time-delay is Te, we have (10) In amplitude sense, rn can be replaced by r0. Then (8) can be rewritten as [17] (11) Since r0 ≫ (xnc0Te/2), the phase term can be regarded as a constant independent of the variables r, θ and t in (11) and thus can be ignored in subsequent target localisation discussion. As xnTe is a constant depending on the physical parameters of the array, ignoring (krxnd sin (θ)Te/c0) will cause deformation in the pulse shapes for large observation point angles and a shift in the timings. However, the deformation can be ignored in analytical derivations and thus it is also ignored in the derivations. In this case, (11) can simplify to (12) Letting (13) We rewrite (12) as a more compact form (14) The summation represents the amplitude modulation effect due to the nested array receiver. It creates a range–angle-dependent beampattern. This provides a possibility to detect/suppress range-dependent targets/interferences: namely, beamforming range-dependent nulls. The exponential term outside the summation is associated with the incoming LFM signal, which has useful information associated with the range of the target and the time-of-arrival. Thus, this term has useful information about the observed target position. Substituting N = 1 to (14) yields (15) This is just the signal received by the first element. Various signal estimation algorithms can be employed to estimate the signal parameters [23-25]. In this paper, we use the fractional Fourier transform (FrFT) technique because it is a linear transform and thus has no cross-term interferences [26]. Applying the FrFT to (15), we get [27] (16) where A0 denotes the amplitude term. The maximum arrives when (17) If ω0 is a prior-known parameter, the range of targets can then be estimated by (17) (18) At the same time, using (15) as the reference signal for matched filtering (14), we then have (19) For description convenience, amplitude terms are ignored in (19). Note that LFM waveform is assumed in above derivations. Specially, if a monochromatic signal: namely, kr = 0, is employed for the array system, (19) simplifies to (20) It creates a range-independent beampattern similar to the basic nested phased-array. 4 Difference co-array processing-based target range and angle estimation 4.1 Difference co-array processing on nested DTD receiver data For notation convenience, the receive steering vector corresponding to the range r and direction θ is represented by a(r, θ, t) whose nth element is . Different from the phased-array has direction-dependent only steering vector, the proposed array steering vector depends also on the range r and time t. Here, we take the first element as the reference. Suppose D sources impinge on this nested DTD receiver from distinct positions (ri, θi) (i = 1, 2, …, D) with powers . According to (14), the received signal is expressed as (21) where is the array manifold matrix, with is the source signal vector and n(t) is the noise vector which is assumed to be temporally and spatially white and uncorrelated with the sources. The covariance matrix is (22) When the sources are temporally uncorrelated, Ryy can be vectorised as [22] (23) The N2 × 1 vector z can be seen as the signal received at the difference co-array with the amplitudes of the source signal vector replaced by their corresponding powers. Correspondingly, the equivalent difference co-array steering vector, denoted by b(r, θ, t), is (24) The distinct elements of b(r, θ, t) behave such as the manifold of a (longer) array whose element positions are given by the distinct values in the set {xi − xj}. Therefore, similar to (5) for nested phased-array, the maximum DOFs that can be obtainable for an N-element nested DTD receiver is N(N − 1) + 1. This implies that, by exploiting the DOF of the difference co-array, we can get O(N2) DOFs with only O(N) physical DTD elements. The equivalent virtual array is a filled ULA with 2N2(N1 + 1) − 1 elements whose positions are given by the set Svirtual(25) It can attain 2N2(N1 + 1) − 1 DOFs in the difference co-array using only N = N1 + N2 physical elements. However, we should be aware of that the nested DTD receiver always produces less DOFs than the minimum redundancy array. However, the structure of the minimum redundancy array for a given N cannot be exactly predicted and it can only be found through numerical simulations individually for each N. In contrast, the nested DTD receiver can be easily constructed and the element positions and DOFs are analytically tractable and is also O(N2). For given N elements, the total DOFs can be maximised by using N1 = N2 = N/2 when N is an even number or N1 = (N − 1)/2, N1 = (N + 1)/2 for odd N. 4.2 Beamforming-based DOA estimation Since the virtual data z of (23) can be seen as the signal received at the difference co-array of nested DTD receiver with the amplitudes of the sources replaced by their corresponding powers, beamforming can be further applied at the receiver. When the non-adaptive beamformer is employed, the corresponding non-windowed weight vector is w = a* (r, θ) ⊗ a(r, θ). Correspondingly, the difference co-array beampattern, denoted by Bpower(r, θ), is (26) Though the incident jammers are originally assumed uncorrelated, after the difference co-array processing, they are represented by their powers and consequently the resulting covariance matrix will be of rank 1. In [21], this problem is tackled by the use of the forward spatial smoothing method at an expense of aperture length loss because it essentially halves the total DOFs offered by the difference co-array processing, meaning that less sources can be identified. To reduce the aperture length loss, we use a two-way smoothing method based on the forward–backward spatial smoothing technique [28]. We divide the virtual L-element co-array into K overlapping sub-arrays, each with P elements: namely, L = P + K − 1. That is, the number of smoothing times is K. Define two P × L matrices (27a) (27b) where k represents the kth smoothing sub-array, and are zero matrices and JP is the P × P exchange matrix whose anti-diagonal elements are 1 and zero otherwise. Multiplying Fk and Gk by (27) and its conjugate, respectively, yields (28a) (28b) Their covariance matrices are (29a) (29b) The forward and backward spatial smoothing data covariance matrices are expressed, respectively, as (30a) (30b) Correspondingly, the forward–backward spatial smoothing data covariance is (31) It has already been proved that when P ≥ D and 2K ≥ D, the forward–backward smoothing data covariance will be of full-rank [28]. We then have (32) On the other hand, for the forward spatial smoothing solution used in [21], the data covariance is full-rank only when P ≥ D and K ≥ D [28]. In this case, the maximum number of sources that can be identified by a virtual L-element difference co-array is determined as (33) Obviously, our forward–backward smoothing method can identify much more sources than the forward smoothing method [21]. Equivalently, when the beamformers can identify the same maximum number of sources, the forward–backward smoothing method has less aperture length loss than the forward smoothing method. Therefore, after forward–backward spatial smoothing [29, 30], we can use the full-rank covariance matrix for minimum variance distortionless response (MVDR) beamforming. The corresponding MVDR weight vector, denoted by wMVDR, is (34) where is the forward–backward spatial smoothing data covariance and as(θ) is the sub-array steering vector of the form (35) The eigendecomposition of is (36) where the diagonal matrix Λs contains the P largest eigenvalues and the columns of Es are the corresponding eigenvectors, and the diagonal matrix Λn contains the remaining (L − P) eigenvalues and the columns of En are the corresponding eigenvectors. According to the principle of the element-space multiple signal classification (MUSIC) estimator [31-33], we obtain the transmit beamspace-based MUSIC cost function [34, 35] (37) where Pn is the projection matrix onto the noise subspace (38) The range and angle of targets can be estimated from the L magnitude peaks of (37) (39) Now, the targets can then be localised in range–angle dimension by combining (18) with (39). 5 Performance analysis 5.1 Detection performance The detection problem can be mathematically formulated as (40) According to the Neyman–Pearson criterion, optimal detector is the likelihood ratio test (LRT) given by [36] (41) where and are the probability density functions (pdfs) of the observation vector with and without targets, respectively, and the threshold γ is determined by the desired probability of false alarm. The and are given, respectively, by (42) (43) where the μ and C are the complex mean vector and the complex covariance matrix, respectively, denotes the determinant of the matrix and T stands for the number of snapshots. Then the LRT detector (41) can be rewritten as (44) Denoting (45) with and M = N2/4 + N/2, we then have (46) For a false alarm (Pfa), the probability of detection is (47) where . The new threshold γ′ is determined by (48) where Q(·), Q−1(·) denote the complementary cumulative distribution function (CCDF) and inverse CCDF, respectively. 5.2 CRLB analysis The CRLB of target range estimation and angle estimation can be derived with the following data model [21]: (49) where A1i is a ((N2/4) + (N/2)) × D matrix consists of the ((N2/4) + (N/2) + 1 − i)th to ((N2 − 2/2) + N + 1 − i)th rows of A1 which is obtained from A* ⊗ A by removing the repeated rows and ei′ is a vector of all zeros expect a 1 at the ith position. The CRLBs can be obtained as the inverse of the Fisher information matrix (FIM). i. Angle is known and range is unknown: In this case, the range FIM can be derived as (50) where SNR denotes the signal-to-noise ratio: namely, , Cn is the spatial noise covariance matrix and −1 is the inverse of a matrix. In doing so, the range CRLB can then be calculated as . ii. Angle is unknown and range is Known: As the range is known, the angle FIM can be derived as (51) Accordingly, the angle CRLB is . iii. Both angle and range are unknown: In this case, the range and angle should be jointly estimated. The corresponding FIM with respect to ψ = [r, θ] is (52) where (53) (54) (55) Note that there is Iθr = Irθ. We then have . 6 Simulation results In the following simulations, we assume that the array with N1 = 3 and N2 = 3 operates at a carrier frequency 10 GHz. The additive noise is modelled as a complex Gaussian zero-mean spatially and temporally white random sequence that has identical variance in the array. 6.1 Example 1: Matched filtering and target range estimation Suppose there are two signal sources coming from the ranges of 10 and 12 km. They have the same parameters in baseband: namely, f0 = 10 GHz, pulse duration 10 μs and chirp rate kr = 1 × 1013. The sampling frequency is 200 MHz. Applying the FrFT algorithm to the signal received by the first element, we get the FrFT spectra, as shown in Fig. 2. Since FrFT is a linear transform technique, there are no cross-term interferences. The returned chirp parameters of the two targets can be estimated from the FrFT peaks as and . This implies that the ranges of targets can be estimated in this way. Fig. 2Open in figure viewerPowerPoint FrFT of signal received by the first element Then, we use the signal received by the first element or the estimated signal parameters as the reference signal for matched filtering all the received array signals. In doing so, we get the baseband signal model of (21) and further formulate the nested DTD received data model of (23) for following beamforming and DOA estimation. 6.2 Example 2: Difference co-array processing-based beamforming Consider a nested DTD receiver with N1 = 3 and N2 elements arranged as shown in Fig. 3. Suppose the array direction angle is 10° and the inter-element delay is Te = 5 ns. We compare the proposed nested array denoted by 'nested' with the same array structure FDA (also with six sensors) represented by 'SAS' and the equal aperture length DTD receiver (with 12 sensors) denoted as 'EAL'. As all the elements receive the signals, when the difference co-array processing algorithm is employed, we can calculate the difference co-array beampattern according to (24). Fig. 3Open in figure viewerPowerPoint Illustration of nested FDA positions with six physical elements First, we use conventional non-adaptive beamformers. For a monochromatic signal: namely, kr = 0, the corresponding difference co-array beampatterns are given in Fig. 4. It is noted that the proposed array has the narrowest beamwidth and lowest sidelobe level in the angle dimension as compared with the SAS FDA and EAL FDA methods due to the difference co-array processing. All the three arrays have the same (0 dB) beampattern. This is because the monochromatic signal misses the frequency increment effect and thus the beampattern will be range-independent. In contrast, when an LFM signal is received, Fig. 5 compares their difference co-array beampatterns. Our method has the narrowest mainlobe and lowest sidelobe levels in both range and angle dimensions. More importantly, all the three DTD receivers yield range-dependent beampattern. This provides a potential to detect/suppress range-dependent targets/interferences. Fig. 4Open in figure viewerPowerPoint Comparative beampattern for receiving a monochromatic signal at t = 1 ns when non-adaptive beamformer is applied a In angle dimension b In range dimension Fig. 5Open in figure viewerPowerPoint Comparative beampattern for receiving LFM signal t = 1 ns when non-adaptive beamformer is applied a In angle dimension b In range dimension Next, we use the MVDR beamformer to test the three DTD receivers. Suppose one interesting target is located at the direction of θs = 10° and the range of rs = 100 km. Moreover also, we assume four interferences are located at directions −50°, −20°, 25° and 40° and the range of ri = 105 km (i = 1, …, 4), respectively. The target power is fixed to 0 dB while the interference power is fixed to 50 dB. Figs. 6 and 7 shows the MVDR beampatterns for receiving a monochromatic signal and an LFM signal, respectively. It is observed that both our proposed array and the EAL exhibit nulls at the locations of the powerful interferences while the SAS has a poor beampattern. Moreover, our method has a relatively narrower mainlobe than the other two arrays. When the interference-to-noise ratio (INR) is fixed to −30 dB: namely, the interference power can be neglected as compared with the noise power, Fig. 8a compares the output signal-to-interference-plus-noise ratio (SINR) versus SNR performance. It is noted that our method yields the best output SINR performance. Fig. 8b compares also the output SINR versus SNR when the INR is fixed to 30 dB: namely, under strong interference-dominant environment. In this case, our method still has better output SINR performance than the SAS, and similar output SINR to the EAL due to that they have the same virtual array structure. Fig. 6Open in figure viewerPowerPoint Comparative beampattern for receiving a monochromatic signal at t = 1 ns when MVDR beamformer is applied a In angle dimension b In range dimension Fig. 7Open in figure viewerPowerPoint Comparative beampattern for receiving LFM signal t = 1 ns when MVDR beamformer is applied a In angle dimension b In range dimension Fig. 8Open in figure viewerPowerPoint Non-adaptive receive output SINR versus SNR a at fixed INR = −30 dB b at fixed INR = 30 dB 6.3 Example 3: MUSIC-based DOA estimation Suppose there are five sources located at the directions of −50°, −20°, −5°, 10° and 45°. Fig. 9a shows the MUSIC spectra for the three tested arrays. We can see that they all perform well in resolving five sources because six physical array sensors provide five DOFs, and thus five sources can be well resolved. However, when there are 11 sources, Fig. 10b shows that the nested array outperforms the EAL. The latter misses the sources located at − 40° and 45°, and also has an obvious deviation at 50°. This superior performance is expected since the former enjoys the advantages of nested array in increasing DOFs. Note that, due to its limited DOFs, when there are 11 sources, the MUSIC-based algorithm is not suitable for the SAS anymore and thus no corresponding comparisons are given in Fig. 9b. Fig. 9Open in figure viewerPowerPoint MUSIC spectrum as a function of the DOA a CRLB for estimating range versus SNR when the angle is unknown b CRLB for estimating angle versus SNR when the range is unknown Fig. 10Open in figure viewerPowerPoint Comparisons of probability of detection performance a Pd versus SNR b Pd versus Pfa We analysed also the probability of detection performance. Fig. 10a compares the probability of detection Pd versus SNR while the Pfa is fixed to 10−6. When SNR < −1 dB, the EAL has a little higher Pd than the proposed nested receiver. However, the proposed receiver has the best Pd performance when SNR ≥ −1 dB. Additionally, Fig. 10b compares also the Pd versus Pfa while the SNR is fixed to 0 dB. It can be seen that the proposed method enjoys higher Pd curve than the other two receivers. Finally, Fig. 11 depicts the range and angle CRLBs versus SNR. As expected, our method exhibits better CRLBs than both the EAL and SAS. This is attributed to the reason that the proposed method fully exploits the statistical information of the received signal. Fig. 11Open in figure viewerPowerPoint Comparisons of CRLB results 7 Conclusion We proposed a nested array receiver using time-delayers and difference co-array processing for joint target range and angle estimation. This technique is based on the advantages of range-dependent beampattern and increased DOFs. By combining two ULA DTDs with increasing inter-sensor spacing, the nested array receiver can be constructed. Then applying the second-order statistic of the received signal, it is capable of providing dramatically increased DOFs. More importantly, it enables a range–angle-dependent beamformer for joint target range and angle estimation. Superior performance is verified by extensive simulation results. Furthermore, the probability of detection and the CRLB performance were also investigated. 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