Coarray Polarization Smoothing for DOA Estimation with Coprime Vector Sensor Arrays

In this paper, the problem of direction-of-arrival (DOA) estimation of multiple sources using a coprime electromagnetic vector sensor (EMVS) array is addressed. Each EMVS consists of three mutually orthogonal electric dipoles and three mutually orthogonal magnetic loops, all collocated at a single point in space. By exploiting the polarization diversities embedded in the vector sensors, the coarray polarization smoothing (COPS) is presented to increase the degrees of freedom (DOFs) from the polarization coarray domain. In contrast to the widely used spatial smoothing technique, the COPS offers the following insights: (1) it is source polarization-dependent and is proved to offer 20 DOFs at most, (2) it does not reduce the efficient spatial coarray aperture of the coprime array so that it enables the use of all virtual coarray sensors without resorting to the nontrivial sparse recovery or array interpolation operations, and (3) it can be synthesized with the spatial smoothing to get the DOFs multiplied. Finally, the efficacy of the COPS scheme is verified by numerical examples.

In order to exploit the enhanced DOFs provided by the coprime array for identifying more sources than sensors, the spatial smoothing technique [15] is generally adapted to the virtual difference coarray output to form a full rank augmented sample data matrix. After that, the subspace-based approaches, e.g., MUSIC [31] and ESPRIT [30], can be used directly for DOA estimation. Because the virtual difference coarray of the coprime array is nonuniform in nature, the coarray smoothing technique based methods utilize only the consecutive virtual sensors, i.e., the central uniform segment, of the coarray. Of course, this would lead to induce performance loss since the entire coarray aperture is not completely exploited. Alternatively, by using the sparsity of the signal sources, the sparse recovery and the array interpolation techniques may be applied, see e.g., [14,18,34,45,[47][48][49]. However, such operations result in increasing the computational complexity due to the involvement of nonlinear optimization. Therefore, the problem of how to exploit the entire coarray aperture of the coprime array in a more effective way remains further exploration.
All the aforementioned references consider coprime scalar sensor arrays in that each sensor measures a single electromagnetic field component of the incident signals as a complex scalar. For space propagation electromagnetic signal, its propagation direction and polarization state are important characteristic parameters [27]. A complete electromagnetic signal is a six-dimensional complex vector signal with complete electric-field and magnetic-field information. Given the fact that the signal sources in practical applications like radar and wireless communications are usually diversely polarized, it is expected that the parameter estimation performance can be significantly improved with the use of coprime electromagnetic vector sensor (EMVS) arrays since an EMVS can simultaneously measure all six electromagnetic field components. Although DOA estimation using EMVS arrays has been investigated intensively [4,9,13,21,27,40,51,52], far fewer algorithms are specially designed for consideration of the coprime structure. For the DOA estimation of uniform linear EMVS arrays, subspace based methods can be directly used, such as MUSIC algorithm [21,52] and ESPRIT algorithm [13]. A spatial-invariance ESPRIT algorithm which regards an EMVS as a subarray was proposed to realize the estimation of the received signal angle and polarization parameters in [51]. It uses a sparse planar array structure, which is ULA in both the x and y directions. Different from the above structure, the coprime array discussed in this paper is a one-dimensional non-uniform array. The problem to be solved is that the spatial smoothing algorithm in the virtual domain reduces the effective spatial coarray aperture. Only recently, methods for DOA estimation using coprime EMVS arrays have been developed in [2,7]. The PARAFAC method is applied in [2], but this research is based on coprime subarray decomposition, does not use the extended array element of virtual domain, therefore, the DOFs is not improved. The corresponding low-rank covariance matrix is recovered by solving the nuclear norm minimization (NNM) problem, and a larger covariance matrix is constructed for DOA estimation in [7]. This algorithm does not make full use of the polarization information of electromagnetic signal. These two works exploit only the increased DOFs of the spatial coarray domain. In other words, the potential DOFs that can be gained from the polarization coarray domain are not explored in these two works.
By making full use of the additional polarization diversities embedded in the vector sensors, we propose a new scheme of what is called COarray Polarization Smoothing (COPS) for increasing the DOFs of a coprime EMVS array in DOA estimation. Specifically, the DOFs offered by the COPS are derived explicitly. Finally, compared with the commonly used spatial smoothing technique, the COPS would provide the following three new properties in more detail: -First, it will be proved subsequently that for diversely polarized sources, up to 20 DOFs can be obtained from the COPS. Interestingly, they are dependent on the source polarization states. When all signal sources exhibit exactly the same polarization, a minimum of 5 DOFs can still be gained from the COPS. -Second, the COPS is applicable to arbitrary array geometry and does not reduce the array spatial aperture. This implies that the COPS is able to exploit the entire spatial coarray aperture of the coprime EMVS array, without needing to perform the sparse reconstruction or array interpolation operations. -Third, the COPS can be combined with spatial smoothing to get the DOFs multiplied. Note that this combination can be employed directly to the coarray output associated with the consecutive virtual array segment. Moreover, by performing extra array interpolation to each of the polarization coarray components, the preceding combination can be applied to the entire coarray output as well.
Notation: Throughout the paper, italic bold uppercase and lowercase symbols, respectively, represent the matrices and vectors. Superscripts * , T and H denote the complex conjugate, transpose, and conjugate transpose operations, respectively. ⊗, and ⊕ denote the Kronecker product, the Khatri-Rao matrix product and Hadamard product, respectively. E{·} represents the mathematical expectation. D(·) represents the operator that forms a diagonal matrix with the entries in brackets. vec(·) represents the operator that vectorizes the entry in parentheses.
x and x represent the smallest integer not less than x and the largest integer not greater than x, respectively. I m , o m , and O m,n represent the m × m identity matrix, the m × 1 vector of all zeros, and the m × n zero matrix, respectively.

Coprime EMVS Array Signal Model
We now discuss the details of the linear coprime scalar sensor array model proposed in [23]. Consider a nonuniform linear array, which is composed of two uniform linear subarrays. The first one has 2M sensors, interspaced at a distance N d, while the second one has N sensors, interspaced at a distance Md. Here, M and N are mutual prime integers, and d is defined as half a wavelength. Further, let the two subarrays share the first sensor. Then, these two subarrays are interleaved to be a coprime array with L = 2 M + N − 1 sensors. The sensor locations of this coprime array can be expressed as a 2M × 1 vector T and an (N − 1) × 1 vector p 2 [Md, 2Md, . . . , (N − 1)Md] T , respectively. For illustration this, a coprime array configuration with M = 3 and N = 5 is depicted in Fig. 1.
The scenario considered herein includes K far-field, fully polarized narrowband signal sources, impinging upon the above-defined array from directions (θ k , φ k ), for k = 1, . . . , K , where θ k and φ k denote the elevation DOA (measured from positive zaxis) and the azimuth DOA (measured from positive x-axis) of the kth signal source, respectively. Furthermore, it is assumed in this paper that the sensor array and the incident signal sources are coplanar, i.e., θ k = π/2, ∀k. Taking the first sensor as the phase reference, the L × 1 spatial steering vector between the array and the source coming from DOA φ is thus given by q(φ) [1, . . . , e j2π/λ p sin φ , . . . , e j2π/λ p L sin φ ] T , where λ represents the wavelength and p is the th element of the L × 1 vector T . Now, suppose that full-component electromagnetic vector sensors are deployed instead of the above-considered scalar sensors. The EMVS considered is assumed to be spatially collocated, consisting of three mutually orthogonal electric dipoles and three mutually orthogonal magnetic loops, which collect three electric-field components and three magnetic-field components of electromagnetic waves respectively. The phase centers of each constituent unit of the structure coincide. This six-component electromagnetic vector sensor was first proposed by Professor Nehorai [22]. Theoretically, EMVS can be placed in any geometric structure. This paper adopts the coprime array structure. It follows from [22] that the EMVS thus produces the 6×1 polarization steering vector as follows: where γ ∈ (0, π/2) and η ∈ (−π, π] represent the incident signal's auxiliary polarization angle and polarization phase difference, respectively. Note that, the parameter φ determines the location of a source, while the parameter pair (γ , η) determines its polarization state. The 6 L × 1 steering vector of the coprime EMVS array can be expressed by (1) and (2) as For all K source signals, the data measured at time t can be written as the following 6L × 1 vector: where a k a(φ k , γ k , η k ) represents the 6L × 1 spatial-polarization array steering vector for the kth source signal, s k (t) represents the baseband waveform of the kth source signal, A [a 1 , . . . , a K ] = Q C represents the 6 L × K spatial-polarization array manifold matrix, in which Q . . , s K (t)] T represents the K × 1 baseband signal vector, and n(t) represents the 6 L × 1 additive noise vector.
The objective here is to estimate the DOAs To that end, the following assumptions are made: 1. the source DOAs are pairwise distinct, i.e., φ 1 = φ 2 = · · · = φ K ; 2. the number of sources K is correctly estimated; 3. the source signals are spatial-temporal uncorrelated with each other; 4. the noise is zero-mean white complex Gaussian, statistically independent to all source signals.

Derivation of the COPS
Under the assumptions made in Sect. 2, the 6 L × 6 L covariance matrix of x(t) is given by where σ 2 n denotes the noise variance, R s E{s(t)s H (t)} = diag[σ 2 1 , . . . , σ 2 K ] represents the signal covariance matrix, with σ 2 k being the power of the kth source signal. Vectorizing R yields the 36L 2 × 1 coarray output, which is expressed as where . Before proceeding to the COPS, we need to introduce the following theorem: Theorem 1 Let P be the matrix of size 36L 2 × 36L 2 as follows: where U P×Q is a P Q × P Q matrix, defined as in which E pq is a P × Q matrix with all zeros except a 1 at the ( p, q)th position and F qp is a Q × P matrix with all zeros except a 1 at the (q, p)th position. We have Proof See Appendix A.
Left multiplying P to r yields a row-exchanged version of r as whereC C * C andQ Q * Q represent the 36 × K polarization coarray manifold and the L 2 × K spatial coarray manifold, respectively, and e m is a vector of size m 2 × 1 with its ith entry being defined as By comparing (10) with (4), it is found that (10) can be viewed as a single sample received signal at an array with the manifoldC Q . Moreover, the manifoldC Q has two parts, in which the polarization coarray manifold C * C is determined by the response of the EMVS and the spatial coarray manifold Q * Q is determined by the geometry of the coprime array. The 36L 2 × 1 vectorr can be divided into 36 non-overlapping sub-vectors, with each has a size L 2 × 1. In this way, each sub-vector would correspond to the same spatial coarray but to a different polarization coarray component. For example, the ith sub-vector corresponds to the ith polarization coarray component, i.e., the ith row ofC, and constitutes the ((i − 1)L 2 + 1)th to (i L 2 )th elements ofr. Now, letr i denote the ith sub-vector ofr. Mathematically,r i can be extracted from r and expressed as wherec i is the ith row ofC,n i σ 2 n J i P i represents the corresponding noise component ofr i , and J i is the L 2 × 36 L 2 selective matrix, defined as the ((i − 1)L 2 + 1)th to (i L 2 )th rows of I 36L 2 , i.e., and According to [14], the columns ofQ have onlyL = 3M N + M − N non-repeated elements, which behave as the spatial steering vector of a virtual nonuniform array with sensors being located from − p max to p max . Here, p max is the location of the farthest sensor of the coprime array, i.e., p max = (2M − 1)N d. Moreover, the central segment of this virtual array is a filled uniform linear array ofL = 2M N + 2 M − 1 sensors. Selecting the elements inr i that are associated with the non-repeated entries ofQ yields aL × 1 short vector, denoted as whereQ [q 1 , . . . ,q K ] is defined as a subset ofQ that contains theL × K nonrepeated element ofQ andñ i represents the noise component ofr i . To make full use of the DOFs in the virtual array, a general solution is to use the array interpolation technique tor i for filling the missing elements in the derived nonuniform array. However, array interpolation involves nonlinear optimization operations. As a consequence, it leads to be computationally expensive, as aforementioned.
Referring to (15), it is observed that for i = 1, . . . , 36, each polarization coarray component provides a possibly different linear combination of the vectorsq 1 , . . . ,q K . This would help to obtain a full rank (rank K ) signal subspace. Using this idea, we define the following coarray polarization smoothing matrix R cops as As seen from (16), the COPS scheme averages the data correlation matrix along the polarization coarray components. Unlike the spatial smoothing technique, in which the averaging operation is performed across the array aperture, the COPS imposes no restriction on the array geometry and does not reduce the effective array aperture. Moreover, the COPS enables the direct application of the subspace-based techniques like MUSIC algorithm [31] on R cops to perform DOA estimation. For better understanding the core idea of the COPS, Fig. 2 shows a graphic illustration of the COPS for a coprime EMVS array with M = 2 and N = 3.

DOFs Analysis of the COPS Matrix
This subsection will derive explicitly the DOFs that can be obtained from the COPS. Observing (16), it is concluded that by applying the subspace-based techniques to R cops , the number of sources can be identified is limited by both the number of sensors in the virtual coarray and the number of independent smoothing matrices in constructing R cops . The former is determined by the DOFs provided by the spatial coarray, i.e., the number of the DOFs provided by the geometry of the coprime array, which isL. The latter is equivalent to the DOFs obtained from the polarization coarray, which is presented in the following theorem.
Proof See Appendix B.
Unlike the spatial smoothing technique, the DOFs obtained from the COPS are dependent on the polarization states of the source signals. Theorem 2 explicitly indicates that the COPS scheme combined with the MUSIC algorithm enables to identify at most 20 diversely polarized sources. This result is useful when considering some specific problems. For example, for the problem of DOA estimation of skywaves [29], it is reasonable to assume that the incoming signals have different polarizations. Indeed, when polarized signals are reflected from the various layers of the ionosphere, their polarization states are likely to vary with the electron densities of the ionosphere around the layers. Since the electron densities tend to vary across the different layers of the ionosphere, it can be expected that the polarization states of the incident signals, which are reflected from the various layers, are distinct from one another.
Even so, Theorem 2 does not reveal the important fact that how many DOFs can COPS provide if some of the signals have the same polarization. In fact, in case some source signals have the same polarization parameters, the 20 DOFs derived in Theorem 2 may not be guaranteed. It is unavailable to present an analytical derivation of DOFs if a portion of the sources have the identical auxiliary polarization angle and/or polarization phase difference. Nonetheless, the achievable DOFs can be derived for the cases when all the source signals have the same auxiliary polarization angle and/or polarization phase difference. For these cases, the achievable DOFs from the COPS are given as follows: Corollary 1 When source signals have the same auxiliary polarization angle but pairwise distinct polarization phase differences or have pairwise distinct auxiliary polarization angles but the same polarization phase difference, i.e., γ 1 = γ 2 = · · · = γ K and η 1 = η 2 = · · · = η K or γ 1 = γ 2 = · · · = γ K and η 1 = η 2 = · · · = η K , the maximum number of identifiable sources by applying the subspace-based techniques directly to R cops is K = min{L − 1, 15}.
Proof See Appendix C.
Corollary 1 is a complement of Theorem 2 for the special cases of γ 1 = γ 2 = · · · = γ K or η 1 = η 2 = · · · = η K . By Corollary 1, we know that if one of the polarization parameters of the source signals is identical, the COPS can offer 15 DOFs. This result may be considered in applications like passive DOA estimation of radar signals. In this case, the received signals may be assumed to be linearly polarized with different auxiliary polarization angles; that is η k = 0, ∀k and γ k = γ , k = . Interestingly, the COPS scheme may still be used even if all the source signals have exactly the same polarization state. The achievable DOFs in this case are presented in the following corollary: Corollary 2 When source signals have the same polarization state, i.e., γ 1 = γ 2 = · · · = γ K and η 1 = η 2 = · · · = η K , the maximum number of identifiable sources by applying the subspace-based techniques directly to R cops is K = min{L − 1, 5}.
Proof See Appendix D.
With the aid of Corollary 2, it is easy to see that under a very stringent condition, the COPS scheme can still offer 5 DOFs. In general, the COPS can provide 5 to 20 DOFs, depending on the polarization states of the source signals. Referring back to (16), it is inferred that at most 20 among all 36 correlation matrices are independent in constructing the R cops . Indeed, in order to obtain a full rank signal subspace from R cops , it is sufficient to compute 20 correlation matrices with i ∈ S, where S = {1, . . . , 6, 8, 9, 11, . . . , 20, 22, 23}.
Nonetheless, in noisy environments, it is suggested to compute all the 36 correlation matrices so as to facilitate coherent summation for better noise suppression performance.

Combination Between COPS and Spatial Smoothing
From the analysis presented in last subsection, there is a limitation on the DOFs obtained by the COPS. Nevertheless, this constraint can be readily eased by combining the COPS with the spatial smoothing technique. To apply the spatial smoothing technique, it is required that the spatial coarray is of a uniform structure. Therefore, this combination may be accomplished with the use of the data corresponding to the centralL ×36 parts out of the entire coarray output. Denote the vector that is associated with the centralL × 1 segment ofr i byȓ i . Then,ȓ i can be expressed as whereQ represents theL × K spatial coarray manifold corresponding to a uniform linear array with sensors locating from −Ld to −Ld andn i represents the corresponding noise vector. To proceed with the combination between the COPS and the spatial smoothing, it is assumed that the R cops as given in (16) is obtained by usingȓ i instead ofr i . Then, the synergy between the COPS and the spatial smoothing is proceeded as where P determines the subarray size for spatial smoothing, with F p being a L ×L selective matrix, defined as the pth to (L + p−1)th rows of IL , where L (L−P+1). The identifiability result for the synergy between the COPS and the spatial smoothing is given in the following corollary.

Corollary 3 By applying the subspace-based techniques toR cops , the maximum number of identifiable sources is K = min{L − 1, P J }, if the maximum number of identifiable sources obtained by applying the subspace-based techniques directly to R cops is J .
Proof See Appendix E.
By Corollary 3, with combination to the spatial smoothing technique, the DOFs offered by the COPS can be multiplied by a factor P ≥ 2, which is a significant improvement over the results obtained for the use of the COPS alone. Recently, it has been found that for a coprime scalar array, by performing the so-called array interpolation to the derived coarray, the "missing sensors" can be filled so that the spatial smoothing technique can be applied to the interpolated coarray for facilitating the use of the entire coarray aperture [47]. The idea of array interpolation can be applied to the coprime EMVS array as well. Specifically, for the coprime EMVS array considered, array interpolation can be implemented by applying the method given in [47] to each of the 36 polarization coarray components individually. Afterward, the entire spatial coarray can be applied to construct theR cops in (19), and accordingly, the identifiable result in Corollary 3 will be updated to K = min{L − 1, P J }.

DOA Estimation
For illustration purpose, the standard MUSIC technique developed in [31] is applied to R cops as follows: Firstly, obtain the estimate of R cops , denoted byR cops from the sample covariance matrix asR Then, calculate the eigenvalue decomposition ofR cops to estimate theL × (L − K ) noise subspace matrixÊ n , whose columns are the eigenvectors associated with the (L − K ) smallest eigenvalues ofR cops . Afterward, form the MUSIC spectrum scalar function V (φ), given by Finally, the source DOAs are estimated as the K values of φ that maximize V (φ).

Computational Complexity Analysis and CRB Comparison
The , where n is the number of steps in local search. The coprime EMVS array structure is also considered in [2,7]. The computational complexity of the algorithm proposed in [2] is O 3K 3 + (144 L + 144T + 24T L − 15)K 2 + (211T L − 36T − 36 L − 9)K + 98 L − 28 , and the computational complexity of the algorithm proposed in [7] is O 36 . In addition, the computational complexity also includes the complexity of using MATLAB CVX toolbox to solve NNM problems in the interpolation algorithm. The CRB for uniform linear EMVS can be expressed as This CRB expression is not applicable to the coprime array which can identify more sources than the number of sensors.

Numerical Results
Numerical simulation results for different scenarios are provided in this section. In what follows, a coprime EMVS array with M = 2 and N = 3 is utilized. The entire array thus has 6 physical sensors, locating at pd, with p = [0, 2, 3, 4, 6,9]. The spatial coarray derived from this coprime array geometry containsL = 17 virtual sensors, among which the centralL = 15 ones constitute a virtual uniform array. In addition, this spatial coarray would haveL = 19 virtual sensors after interpolation. The two recently proposed algorithms [2,7], which are designed for coprime EMVS arrays, are considered for comparison. In the following figures, the labels "COPS", "PAFAFAC", and "Interpolation" are used to represent the proposed algorithm, the algorithm in [2], and the algorithm in [7], respectively.
The identifiable performance of the COPS is firstly examined. The following four scenarios: (a) 16 diversely polarized sources with randomly generated polarization parameters, (b) 15 linearly polarized sources with randomly generated polarization angles, (c) 15 elliptically polarized sources (γ k = 45 • , ∀k) with randomly generated polarization phase differences, and (d) 10 identically polarized sources with (γ k , η k ) = (60 • , 90 • ), ∀k, are considered. The sources are assumed to be uniformly distributed in [−52 • , 50 • ] with an angular separation 6.8 • for scenario (a), in [−59 • , 60 • ] with an angular separation 8.5 • for scenarios (b) and (c), and in [−50 • , 49 • ] with an angular separation 11 • for scenario (d). Further, for all these four scenarios, the source signals are assumed to be of equal-power, with 25 dB signal-to-noise ratio (SNR). Altogether T = 1000 snapshots are used to generate the simulation data. The MUSIC algorithm is applied to R cops (for scenarios (a), (b), (c)) orR cops (for scenario (d)) to estimate the source DOAs. The corresponding results are shown in Fig. 3a-d. The results obtained from the array interpolation algorithm [7] are included in each sub-figure as well. It is seen from the figure that for all the four considered scenarios, the MUSIC algorithm with COPS successfully identifies the incident sources. The observation from Fig. 3 is in agreement with our theoretical derivation presented in Sects. 3.2 and 3.3. Also, it is readily observed from Fig. 3 that the array interpolation algorithm [7] fails to work properly for these four scenarios. The explanation for this phenomenon is that the array interpolation exploits merely the DOFs provided by spatial coarray but overlooks the potential DOFs offered by the polarization coarray. Incidentally, the PARAFAC algorithm [2] cannot handle the underdetermined cases like the presetting scenarios (a)-(d) in this example. Next, the estimation accuracy of the PAFARAC, Interpolation, and COPS algorithms is compared. Now, consider a two-source scenario, in which the first source is left-circularly polarized and the second one is right-circularly polarized, i.e., γ 1 = 45 • , η 1 = 90 • and γ 2 = 45 • , η 2 = −90 • . The DOAs of these two sources are assumed to be θ 1 = 10 • and θ 2 = 20 • , respectively. Figure 4 shows the rootmean-squared (RMS) errors of the DOA estimates, which are computed by means of 500 independent experiments, as a function of the SNR. The number of snapshots is fixed at T = 1000. The Cramér-Rao Bound (CRB) curves, which may be obtained from [35], are also plotted for the use of the performance benchmark. Upon inspection from Fig. 4, for all tested SNRs, the performance of the COPS is superior to that of  Figure 5 gives similar plots changing the number of snapshots at a fixed SNR of 10 dB. It is inferred from Fig. 5 that for a wide range of snapshot number, the COPS can provide a more accurate estimation performance than that of the PAFARAC and Interpolation. Additionally, Figs. 4 and 5 unanimously exhibit that the estimation errors of the COPS approach reasonably to the CRBs. The computational complexity comparison of the above three algorithms is shown in Fig. 6. It can be seen that the computational complexity of COPS and Interpolation is less than that of PARAFAC in the case of large snapshots. In addition, the Interpolation curve in Fig. 6 does not include the complexity required to solve the NNM problem, because it is completed by the CVX toolbox of MATLAB and is difficult to estimate. In conclusion, the computational complexity of COPS is smaller than the others in the case of large snapshots.

Conclusions
Using a coprime EMVS array, a new scheme has been presented in this paper to estimate DOAs of multiple completely polarized source signals. In this new scheme, the COarray Polarization Smoothing (COPS), which allows using the subspace-based techniques directly, is defined to increase the DOFs from the polarization coarray domain. In contrast to the widely used spatial smoothing technique, this COPS processing does not reduce the efficient coarray aperture and is proved to offer up to 20 DOFs. By combining with the spatial smoothing, the achievable DOFs can be further multiplied. Numerical simulations verify our theoretical results, thereby demonstrating the efficacy of the proposed COPS scheme.

Data availability
The data used to support the findings of this study are available from the corresponding author upon request.

Appendix A: Proof of Theorem 1
Using the following three matrix prosperities given in [11] A (B C) = ( A B) C, (22) we have and Let P = (I 6 ⊗ U 6×L 2 )(U 6×L ⊗ I 6 L ). The relationship (9) is established.  Therefore, the 7th, 10th, 21st, and 24-36th rows ofC are linearly dependent on the remaining 20 rows. For sources of pairwise distinct auxiliary polarization angles and pairwise distinct polarization phase differences, the rank ofC is up to 20 at most. That is, rank(C) ≤ min(20, K ). Since the spatial coarray manifoldQ and the diagonal matrix R s are always full rank, the rank of the signal part of R cops would equal to rank(Q R sC ) and satisfy rank(Q R sC ) ≤ min(20, K ,L) ≤ min(20, K ).
For the use of the subspace-based techniques, it is required that rank(R cops ) = rank(Q R sC ) = K .
The requirement in (29) together with the limitation in (28) imply that K ≤ 20. In addition, to guarantee that a signal subspace is available, it is required that K ≤L − 1.
Combining these two constraints yields Therefore, applying the subspace-based techniques directly to R cops , the maximum number of identifiable sources is K = min{L − 1, 20}. The proof is thus complete.

Appendix E: Proof of Corollary 3
It follows directly from (16) and (19) that the rank ofR cops satisfies rank(R cops ) ≤ rank Constraints (29) and (31) together yield K ≤ P J . To apply the subspace-based techniques, it is required that K ≤ L − 1. Combining these two constraints leads to K ≤ min(L − 1, P J ).
Therefore, applying the subspace-based techniques directly toR cops , the maximum number of identifiable sources is K = min{L − 1, P J }, completing the proof.
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