Incompletely Polarized MIMO Radar for Target Direction Estimation

MIMO radars with electromagnetic vector sensor (EMVS) antennas (also known as EMVS-MIMO radars) have been investigated intensively for target direction estimation in recent years. Many existing works on this topic are based on an unrealistic assumption that each EMVS antenna transmits six identically polarized and mutually orthogonal waveforms. Furthermore, target-fluctuation induced non-coherent backscattering in polarization has been entirely overlooked. This paper aims to develop a general transmit-receive signal model for EMVS-MIMO radar direction estimation. Specifically, by taking transmit polarization agility and/or polarization non-coherent backscattering into account, the collected pulse signals are no longer of complete polarization, but of incomplete polarization. By regarding an incomplete polarized (IP) signal as two incoherent completely polarized (CP) signals with the same azimuth-elevation directions and drawing upon the idea of ESPRIT, a new azimuth-elevation direction estimation algorithm is thus derived. The advantage of the new algorithm is that it can provide closed-form, automatically paired azimuth-elevation direction estimates and require no information on the location/displacement of the transmit-receive antennas. Finally, simulations are conducted to demonstrate the effectiveness of the proposed algorithm.

In many passive sensing systems, polarization information is often utilized in conjunction with spatial information for improving the direction estimation performance [26], [27], [28], [29], [30], [31], [32].In fact, a full electromagnetic field at a given location in space is composed of three co-orthogonal electric fields and three co-orthogonal magnetic fields, manifesting as a six-dimensional vector field.It is found that this six-dimensional electromagnetic field can be simultaneously measured with a single zero-aperture electromagnetic vector sensor (EMVS) [26].Indeed, MIMO radar with EMVS antennas, which is also referred to as EMVS-MIMO radar, has been shown to be versatile for radar target direction estimation.The concept of EMVS-MIMO radar was introduced in [33], where a simple ESPRIT-based algorithm is also developed using arbitrarily placed EMVS arrays.Since then, many efficient EMVS-radar direction estimation algorithms have appeared in the open literature, with core ideas drawing upon the modified ESPRIT technique [34], [35] the propagator technique [36], the parallel factor (PARAFAC) analysis technique [37], and the MUSIC technique [38].However, from the consideration of practical applications, the algorithms presented in [34], [35], [36], [37], and [38] have the following two problems: 1) The algorithms in [34], [35], [36], [37], and [38] implicitly assume that the received signals are generated from the coherent scattering of targets so that their polarization states are time-invariant.That is, the target reflected signals are of complete polarization.However, this complete polarization assumption would be practically inappropriate due to the fluctuant nature of radar targets.In fact, such fluctuation would lead to non-coherent backscattering of targets and incomplete polarization of the target reflected signals [39].Unlike the completely polarized (CP) signals, the polarization states of incompletely polarized (IP) signals vary with time.Consequently, the algorithms mentioned before, which are tailored for CP signals, would be unsuitable for handling IP signals.
2) The algorithms in [34], [35], [36], [37], and [38] are based on the premise that each EMVS antenna transmits six identically polarized and mutually orthogonal waveforms (one waveform by one EMVS antenna component).Unfortunately, this transmit system architecture is practically infeasible because, in the wave plane, the signals transmitted by an EMVS antenna have only two degrees-of-freedom, according to Maxwell's equations [26].Furthermore, in these algorithms, the polarizations of transmitting signals and the scattered signals are taken as identical parameters, which is also unrealistic.Consequently, these algorithms would be questionable for engineering practice.The main contribution of the present work is aimed at proposing a new azimuth-elevation direction estimation algorithm that avoids the aforementioned problems.Specifically, a general transmit-receive system model for EMVS-MIMO radar direction estimation is first developed.It will be shown that by describing the transmit signals in electric field vector form, the received signals after matched-filtering are of incomplete polarization.The name of incompletely polarized MIMO radar is hence coined as opposite to the previous EMVS-MIMO radars, where completely polarized states are implicitly assumed.By regarding an IP signal as two incoherent CP signals with the same azimuth-elevation directions and using the idea of ESPRIT, a new azimuth-elevation direction estimation is thus derived.The presented algorithm provides closed-form and automatically paired direction estimates.Moreover, it requires neither the location/displacement information of the transmit antennas nor that of the transmit antennas.Simulation results show that the proposed algorithm offers performance better than that of the conventional MIMO radar systems with spatially separated single-polarized antennas of comparable receive data size.
The notation used in the paper is given in Table I.

II. SYSTEM MODEL
The signal (electromagnetic wave) radiated and/or received by an antenna consists of an electric field and a magnetic field that are orthogonal to each other and to the direction of propagation.Moveover, the magnitude of the electric field and magnetic field can be found from each other.Hence, only the electric field vector is considered in establishing the system model.Now, let unit vectors r, r H , and r V form a right-handed coordinate system.Then, for a plane electromagnetic wave propagating along r, its electric field is orthogonal to r and lies in the plane spanned by (r H , r V ).The description of the coordinate system is shown in Fig. 1.

A. Transmit Signals
The transmit antenna array comprises M six-component EMVS antennas, numbered by m = 1, • • • , M. Each EMVS antenna consists of three electrically short dipoles and three magnetically small loops.The dipoles are electrically isolated from but spatially orthogonally integrated with the loops, wherein the dipoles and the loops are physically co-located with respect to each other.In the coordinate system (r H , r V ), the 6 × 2 response of an EMVS antenna is given by [26] where with θ ∈ [−π/2, π/2] and φ ∈ [0, 2π ) are the elevation direction and the azimuth direction, respectively.
represent the responses of an EMVS antenna to a signal at (θ, φ) with linear polarizations along the φ-direction and the θ -direction, respectively.u(θ, φ), v(θ, φ), and w(θ ) in ( 2) are direction-cosines along the three Cartesian coordinate axes.The unit vector r in ( 2) is simply the normalized Poynting vector (PV) of the signal at the direction (θ, φ).Based on the above definitions, the baseband signal emitted by the mth EMVS antenna can be expressed in 2 × 1 electric field vector form as where ω m denotes the 6 × 1 weighting vector used to control the polarization state of the transmit signal [40], s m (t) is the waveform of the mth transmit signal, and ] T is a 2 × 1 vector, describing the polarization state of the transmit signal.For any g m,1 ̸ = 0 and g m,2 ̸ = 0, g m can be uniquely represented as [26] where ψ m ∈ (−π, π], α m ∈ (−π/2, π/2] is the orientation angle, and β m ∈ [−π/4, π/4] is the ellipticity angle.In this way, the polarization state can be visually interpreted using the concept of the electric polarization ellipse.Interested readers are referred to [26] for more details.
It is reasonable to assume that the signals emitted by the M EMVS antennas have the same polarization, i.e., Then, for the nth pulse, the signal transmitted by the entire EMVS antenna array can be expressed as where g(n) = C T ω(n) represents the polarization state of the transmit signals, ] T is the M × 1 spatial response vector of the transmit antenna array, 2π λ p T m r represents the spatial phase delay arisen from the propagation time difference between the mth EMVS antenna and the reference point (Cartesian origin), and p m ≜ [x m , y m , z m ] T is a 3 × 1 vector, denoting the spatial location of the mth transmit EMVS antenna.
Remark 1: The transmit signal model established in ( 7) is fundamentally different from the model that is widely used in previous EMVS-MIMO radar studies [34], [35], [36], [37], [38].In the latter case, the signals radiated from a single EMVS antenna are assumed to have six mutually orthogonal but identically polarized waveforms (see, e.g., Eq. ( 4) in [34]).This assumption is clearly impractical because, according to Maxwell s equations, only two spatial degrees-of-freedom are available for signals transmitting from an EMVS antenna [26].In contrast, no unrealistic assumption, as such, is made in establishing (7).Indeed, the system model ( 7) can be considered as a corrected model for developing EMVS-MIMO radar signal processing algorithms.
Remark 2: In pulse radar systems, although the waveforms of the transmitted signals usually maintain unchanged in all pulses, their polarization states may change in different pulses.That is, g(n) may vary with n.Such agility in waveform polarization has many advantages, e.g., anti-jamming and low probability of intercept, in electronic warfare applications.However, this would lead to incomplete polarization of the received signals, as will be discussed in the subsequent sections.

B. Receive Signals
It is assumed that the receive antenna array is composed of L EMVS antennas, numbered by ℓ = 1, • • • , L. There are K targets in the far-field but within the same range cell of the radar.The kth target is located at azimuth direction φ k and elevation direction θ k .Then, for the nth transmitted pulse, the signal collected by the ℓth receive EMVS antennas can be modeled as the following 6 × 1 vector: where ρ k (n) is the complex envelope of the kth target at the nth pulse, e j 2π λ pT ℓ r k is the spatial phase factor related to the kth target and the ℓth receive EMVS antenna at the location is the 2×2 polarization scatter matrix, describing the polarization transform property of the kth target at the nth pulse [41], [42], q T,k ≜ q T (θ k , φ k ), and w ℓ (n, t) is the 6 × 1 noise vector measured by the ℓth EMVS receive antenna.
Further, it is assumed that the targets are fluctuant during the observation period and follow the Swerling model of type II [17].Under this assumption, the complex envelopes and the polarization scatter matrices remain invariant during the period of one pulse, but vary independently from pulse to pulse.The objective herein is to estimate azimuth-elevation directions Remark 3: The receive signal model established in ( 8) is also different from the model used in [34], [35], [36], [37], and [38].In the latter case, only the variation in complex envelopes is considered.The resulting signals after matched-filtering are thus of complete polarization (see, e.g., Eq. ( 5) in [34]).The variation in polarization, however, is completely overlooked.This complete polarization model may be appropriate for stationary (non-fluctuating) targets.However, non-fluctuating targets are rarely encountered in practice.As a consequence, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
this model would provide a poor representation of the received signals of a real radar-target scenario.

A. Matched Filtering of Received Signals
It is supposed that the waveforms emitted by the M transmit EMVS antennas are mutually orthogonal.That is, where τ denotes the transmit pulse duration.Based on ( 9), applying matched-filtering of the mth transmit waveform to the received signals, the echo due to this transmit waveform can be split out of the received signals.Mathematically, after matched-filtering with the mth transmit waveform s m (t), the signals received by the ℓth receive EMVS antenna become where denotes the corresponding noise vector.Eq. ( 10) can be expressed compactly as where A m,ℓ ≜ F(H m,ℓ ⊗ I 2 ) denotes the 6 × 2K response matrix relating the mth transmit and the ℓth receive EMVS antenna, For all m ∈ [1, M] and ℓ ∈ [1, L], altogether M L 6 × 1 vectors can be obtained.Stacking these vectors yields the 6M L × 1 long vector as follows: where have the same meaning as c φ (θ k , φ k ) and c θ (θ k , φ k ), respectively, except that they correspond to the response of the virtual antenna array of the MIMO radar, (iii) q(θ k , φ k ) ≜ q T (θ k , φ k ) ⊗ q R (θ k , φ k ) denotes the M L × 1 virtual spatial response vector derived from the spatial geometry of the MIMO radar, in which q T denotes the 6M L × 1 additive noise vector.
There are two essential observations about the signal vector b k (n).First, when the polarization state of the transmit signals maintains unchanged and for the non-fluctuating targets, the receive signals would have constant polarization, i.e., the signals are CP.The polarization state of a CP signal is often parameterized by ḡk = [sin γ k , cos γ k e jη k ], in which γ k ∈ [0, π/2] and η k ∈ (−π, π] are referred as the auxiliary polarization angle and polarization phase difference, respectively.In fact, the polarization state of a CP signal is related statistically to the rank of its covariance matrix.Clearly, the rank of ] is always one.In other words, the covariance matrix of a CP signal is of rank deficiency.Second, when the polarization states of the transmit signals vary across different pulses and/or the targets are fluctuating, the polarization states of receive signals would vary from pulse to pulse, i.e., the signals are IP.Specifically, for an IP signal, the rank of covariance matrix R b k is necessarily two; that is, the covariance matrix of an IP signal is of full rank.Given the fact that the problem of direction estimation of CP signals has been studied intensively in EMVS-MIMO radar signal processing, it would be of significance to investigate a similar problem for IP signals, which will be addressed in what follows.

B. Eigen-Decomposition of Matched-Filtering Data
The covariance matrix of the matched-filtering data can be expressed as where is the signal covariance matrix, which is nonsingular, and σ 2 is the noise variance.Since the covariance matrix of an IP signal has a rank two, the signal subspace of R would have a dimension 2K , rather than K as the common case in the subspace-based algorithms.
Let the eigenvalues and the corresponding eigenvectors of R be λ 1 , • • • , λ 6M L and u 1 , • • • , u 6M L , respectively.Further, assume that the eigenvalues are arranged in descending order.Then, the eigen-decomposition of R can be expressed as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where denotes a 2K × 2K diagonal matrix whose diagonal elements are the 2K largest eigenvalues of R, and diagonal matrix whose diagonal elements are the (6M L −2K ) smallest eigenvalues of R. Remark 4: Generally, an accurate signal subspace estimation requires a sufficiently high signal-to-noise ratio (SNR).In radar applications, target direction estimation is usually processed after a declaration of successful detection, which is commonly achieved by the pre-processing of the measured raw data with matched-filtering, pulse-compression, and Doppler processing.Although the SNR of the raw data is very low, e.g., below 0 dB, the SNR level after pre-processing would usually be relatively high, e.g., above 10 dB.Once a successful detection is made, the corresponding SNR condition would be sufficient for an accurate estimation of signal subspace.

C. Direction Estimation Using ESPRIT
The core idea of ESPRIT lies in the establishment of the relationship between the response matrix A and the signal subspace matrix U s as [43] where T is a 2K × 2K nonsingular but unique matrix.Basically, the ESPRIT algorithm forms two identical subarrays that are translated by a known separation and exploits the so-called translational invariance between them.Referring to (13), z(n) is indeed a connection of M L 6 × 1 subvectors, where each is associated with a different pair of transmit-receive EMVS antennas.Since each EMVS antenna itself can be regarded as a subarray of six components, any two transmit-receive EMVS antennas may be used to form an ESPRIT pair of subarrays.Thus, altogether M L(M L − 1)/2 different ESPRIT pairs may be formed from the EMVS-MIMO radar with M transmit and L receive antennas.Given the availability of the signal-subspace U s , the 6 × 2K signal-subspace, which is associated with the mth transmit antenna and the ℓth receive antenna, can be extracted from U s .That is, for p = (m − 1)L + ℓ.Then, for any 1 ≤ p 1 < p 2 ≤ M L, a pair of signal subspace matrices may be formed as and Since T and H p 1 are invertible, ( 18) and ( 19) together yield Therefore, the diagonal elements of H p 1 p 2 ≜ (H −1 p 1 H p 2 ) ⊗ I 2 and the columns of T can be estimated as the eigenvalues and the corresponding eigenvectors of U † p 1 U p 2 .
Fig. 2. Scatter plot of the azimuth-elevation direction estimates of the proposed algorithm.M = 2 and L = 2. 500 independent realization.Estimates are marked in "dots", true values are marked in "X".
Note that, from the eigenvalue decomposition of U † p 1 U p 2 , the estimation of T denoted by T p 1 p 2 is equal to T only within some column permutation.This is because (20) still holds if permuting the columns of T and reordering the elements of H p 1 p 2 accordingly.As a consequence, the columns of T p 1 p 2 and T p1 p2 , which are estimated from the eigenvalue decomposition of U † p 1 U p 2 and U † p1 U p2 , respectively, may not be pair-matched.Obviously, in the absence of such permutation ambiguity, we have T p 1 p 2 = T p1 p2 = T .In this case, T −1 p 1 p 2 T p1 p2 = I 2K .In contrast, when there exists such permutation ambiguity, T −1 p 1 p 2 T p1 p2 would be column-exchanged versions of the identity matrix I 2K .To solve the permutation ambiguity problem using this property, let h p 1 p 2 ≜ diag −1 (H p 1 p 2 ) and h p1 p2 ≜ diag −1 (H p1 p2 ).Then, the kth entries of h p 1 p 2 and (T −1 p 1 p 2 T p1 p2 )h p1 p2 are correctly paired.Having removed the permutation ambiguities among all estimates obtained from the M L(M L − 1)/2 ESPRIT pairs, the matrix F can be estimated as follows.
First, let F(p 1 , p 2 ) be the estimate of F obtained from the eigenvalue decomposition of U † p 1 U p 2 .The kth column of F(p 1 , p 2 ) , which represents the estimate of the kth column of F, is expressed as It should be noted that ( 21) is called the self-normalization estimator that is widely used for removing the scalar ambiguities in the EMVS manifold estimation [26], [27], [29].Then, for all 1 ≤ p 1 < p 2 ≤ M L, obtain the estimate of F as The estimation in (22) performs coherent summation among all M L(M L − 1)/2 candidates, thereby ensuring the preservation of the signal power and maximization of the noise cancellation.These M L(M L − 1)/2 candidates may be obtained concurrently by parallel computation so that only extra computational resources but no extra computational time are required.Referring to (1), every two columns of F correspond to one IP signal.Moreover, according to (13), an IP signal can be regarded as two incoherent CP signals with the same direction but different polarizations.Then, using the fact that the cross product between r H and r V is the PV pointing towards the target direction, i.e., it is concluded that the PVs derived from the 2K columns of F would exhibit as K pairs, with each corresponding to one IP signal.However, the PV estimate, obtained from F would be quite distinct.Hence, an additional association operation is required.Let rκ be the PV estimate obtained from the κth column of F, with where f κ,H and f κ,V are the first three and the last three rows of the κth column of F.Then, the PV estimate rκ ′ associated with rκ can be obtained by Afterwards, the PV estimate of the kth target may be obtained by Finally, the azimuth-elevation direction of the kth target may be estimated as and

IV. ANALYSIS AND DISCUSSIONS
The new algorithm accomplishes the azimuth-elevation direction estimation of multiple targets with EMVS-MIMO radars of arbitrarily and irregularly spatial configurations.Specifically, an EMVS-MIMO radar with two transmit and one receive antennas suffices for the implementation of the algorithm.In addition, since in the proposed algorithm, the target directions are indeed estimated from the vector cross-product computation (24), the exact locations of both transmit and receive antennas need not be known in advance.This flexibility in the layout of antenna arrays is not offered in previous studies, where certain regular array configurations, e.g., linearly [37], L-shaped [35], with precisely known antenna locations, are required in deriving the algorithms therein.
The transmit and receive antenna displacements p m and pℓ are not constrained to be within half a wavelength as required by the spatial Nyquist sampling theory.In fact, many of the M L(M L − 1)/2 ESPRIT pairs may have a translational separation beyond half a wavelength.This would result in a phase ambiguity with some integer multiple of 2π in the eigenvalues of U † p 1 U p 2 .Although such phase ambiguity would violate the one-to-one relationship between the phase angles of ESPRIT's eigenvalues and the direction-cosines of the targets, it does not affect the implementation of the new algorithm.This is because it is the ESPRIT's eigenvectors T , not the eigenvalues, which are required in the algorithm.In other words, this phase ambiguity issue is irrelevant to the objective at hand.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.RMS errors of azimuth-elevation direction estimates versus the number of pulses over 500 independent trials.Two targets are located As for the identifiability issue, the algorithm introduced in previous sections can identify K ≤ 3 targets.This three-target constraint arises from the 6×2 size of the response of an individual EMVS to an IP signal.Nevertheless, this three-target limit may be easily raised by dividing the M L virtual antennas into several translational invariant groups, with each group containing L > 1 EMVS antennas.In this way, the algorithm proposed herein can be modified to accommodate K ≤ 3 L targets.
As for the computational complexity issue, the major computations of the proposed algorithm lie in calculating the covariance matrix R and performing its eigenvalue decomposition.The resulting flops required are approximately (6M L) 2 N +(6M L) 3 .Here, a flop is defined as a floating-point addition or multiplication computation.Since K ≪ M L and K ≪ N , the computational costs for calculating the eigenvalue decomposition of U † p 1 U p 2 are negligible, compared to those for calculating the eigenvalue decomposition of R.

V. SIMULATION RESULTS
Simulation results provided in this section demonstrate the performance of the new algorithm.In all simulations below, the following settings are used in generating the simulation data.The first transmit antenna is set to serve as the phase reference.In each pulse, both the transmit polarizations and the polarization scatter matrices are randomly generated.In addition, 1024 data samples are used for matched-filtering.The SNR is defined as the ratio between the power of the reflected signals and the noise variance before matched-filtering, and the SNR is defined relative to each target signal.Scatter plot of the azimuth-elevation direction estimates of the proposed algorithm.M = 4 and L = 8. 500 independent realization.Estimates are marked in "dots", true values are marked in "X".
antennas and receive antennas are placed at the coordinates (0, 0), (9λ, 9λ) and (3λ, 5λ), (1λ, 4λ) in the x-y plane, respectively.Consider that there are K = 3 targets located at the azimuth-elevation directions (θ 1 , φ 1 ) = (60 • , 50 • ), (θ 2 , φ 2 ) = (40 • , 60 • ), and (θ 3 , φ 3 ) = (20 • , 70 • ), respectively.Altogether N = 32 pulse samples are collected.The SNR is set at 0 dB.Fig. 2 gives a scatter plot, in which the azimuth-elevation direction estimates of the proposed algorithm over 500 independent realizations are displayed.It is seen from Fig. 2 that the proposed algorithm successfully resolves and correctly identifies these three targets in all 500 trials.The result of Fig. 2 is in agreement with our analysis presented in Section IV that the proposed algorithm can handle K ≤ 3 targets.It is shown in [33] that with the same system setting, at most six CP signals can be identified.This result does not contradict our finding of handling K ≤ 3 targets.There is no contradiction because an IP signal can be viewed as two CP signals with the same directions.In this way, the estimated six azimuth-elevation directions would appear in three pairs, just as shown in Fig. 2.
The performance of the proposed algorithm is further evaluated by analyzing the statistical root mean squared (RMS) errors of the direction estimates.The ESPRIT algorithm for MIMO radars with single polarized spatially displaced receive antenna arrays is selected for comparison.Horizontally polarized (labeled as H-pol) and Vertically polarized (labeled as V-pol) antennas are both considered for data collection.For either case, a 12-element receive array, which is composed of a 3-element half-wavelength spaced triangular subarray and another 9-element arbitrarily placed subarray, is deployed.Therefore, the total data dimensions processed by these two algorithms are identical.In addition, in order to assess the impact of model mismatch on estimation performance, the ESPRIT algorithm designed by presuming CP signals (labeled as ESPRIT algorithm with CP modeling) is compared as well.Note that, the existing EMVS-MIMO radar direction estimation algorithms [34], [35], [36], [37], [38] are not considered for comparison as these algorithms are derived from an unrealistic transmit-receive model.Now, consider that there are two K = 2 targets located at the azimuth-elevation directions (θ 1 , φ 1 ) = (60 • , 40 • ) and (θ 2 , φ 2 ) = (40 • , 50 • ), respectively.The RMS errors of direction estimates versus SNR are plotted in Fig. 3, in which the curves corresponding to θ 1 , θ 2 , φ 1 , and φ 2 are displayed individually in each subfigure.The curves corresponding to H-pol and V-pol are nearly overlapped, indicating that H-pol and V-pol exhibit almost the same performance.This is because there is no difference for H-pol or V-pol antennas since both the transmit polarizations and the polarization scatter matrices are randomly generated.For elevation direction estimation, the performance of the proposed algorithm is superior to those of the ESPRIT algorithm using single polarized antennas.However, for azimuth direction estimation, the performance of both algorithms is approximately identical.The RMSEs of the ESPRIT algorithm with CP modeling remain relatively high and maintain nearly unchanged as the SNR increases.This phenomenon implies that the ESPRIT algorithm with CP modeling cannot work reliably as the estimation errors caused by the model mismatch between CP and IP assumptions become dominant.The simulation results for varying the number of pulses at 0 dB SNR are shown in Fig. 4. Clearly, the RMS error curves obtained exhibit similar behaviors as those observed from Fig. 3.
The resolution performance is finally examined.Consider a scenario of two targets, in which the first and second targets are located at (θ 1 , φ 1 ) = (10 • , 20 • ) and (θ 2 , φ 2 ) = (13 • , 20 • + ), respectively.The parameter = φ 2 − φ 1 represents the azimuthal separation between the two targets.The RMS errors of direction estimates versus the parameter are shown in Fig. 6, in which the curves corresponding to θ 1 , θ 2 , φ 1 , and φ 2 are displayed in each subfigure.For H-pol and V-pol algorithms, a 48-element receive array with a 3-element halfwavelength spaced triangular subarray and another 45-element arbitrarily placed subarray is used for data collection.It is seen from the figure that the performance of the proposed algorithm is superior to those of the other three algorithms for all the tested scenarios.The ESPRIT algorithm with CP modeling does not work properly as the estimation errors maintain very high for all cases.For the remaining three algorithms, the estimation accuracy improves as the increase of the angular separation | |.Incidentally, the proposed algorithm can provide successful resolving and correct identification even for two closely-spaced targets at = 0.

VI. CONCLUSION
In this paper, a new ESPRIT-based algorithm to estimate the azimuth-elevation directions of multiple targets for EMVS-MIMO radar is presented.As different from most of the existing algorithms, the proposed algorithm is based on the realistic transmit-receive signal propagation model by taking transmit polarization agility and/or polarization non-coherent backscattering into account.The nontrivial incompletely polarization issue of the receive signals is addressed by regarding one incompletely polarized signal as two incoherent completely polarized signals with the same azimuth-elevation directions.The presented algorithm can offer automatically paired azimuth-elevation direction estimates in closed-form.Additionally, no antenna location/displacement information is involved in the derivation of the algorithm.
representing the 2K ×1 signal vector of all targets, and b k (n) ≜ ḡk (n)ρ k (n) denotes the 2×1 signal vector of the kth target at the nth pulse.
A. Results for M = 2 and L = 2 An EMVS-MIMO radar containing M = 2 transmit antennas and L = 2 receive antennas is deployed.The transmit

Fig. 5 .
Fig. 5.Scatter plot of the azimuth-elevation direction estimates of the proposed algorithm.M = 4 and L = 8. 500 independent realization.Estimates are marked in "dots", true values are marked in "X".

TABLE I GLOSSARY
NOTATIONS Fig. 1.Description of the coordinate system.