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Anglerangepolarizationdependent beamforming for polarization sensitive frequency diverse array
EURASIP Journal on Advances in Signal Processing volume 2019, Article number: 23 (2019)
Abstract
In this paper, we present a polarization sensitive frequency diverse array (PSFDA), which benefits both from frequency diverse array (FDA) and polarization sensitive array (PSA), producing anglerangepolarizationdependent beamforming capability. To simultaneously improve the angle, range and polarization resolution, the ℓ_{1}norm minimizationbased sparse constraint and ℓ_{∞}norm constraint for sidelobe region are further applied. Besides, the performance of the optimal output signaltointerferenceplusnoise ratio (SINR) is analyzed for such a highdimensional arrays. Numerical results demonstrate that the proposed PSFDA provides both range and polarization resolution as well as target angle estimation. Moreover, the PSFDA virtually increases the array size by exploiting multiple electromagnetic fields at each physical point. In addition, the proposed beamforming method can effectively null interferences, without degrading the array gain for the signals of interest.
Introduction
Multidomain signal processing can improve the array performance by exploiting space, time, polarization, frequency resources [1–3]. Nevertheless, the range domain information has not been sufficiently explored in literature. Although existing interference suppression approaches can be employed to exploit the angle as well as polarization domain information for polarization sensitive array (PSA), the joint information between range domain and other domains are rarely explored [4].
Recently, a new kind of array named frequency diverse array (FDA), which applies a small frequency increment between contiguous two elements, has gained substantial interest [5–8]. This small frequency increment, as compared to the carrier frequency, results in a anglerangepolarizationdependent beampattern [9–12] that is different from conventional phasedarray providing only angledependent beampattern and frequency scanning array whose beam pointing angle changes with working frequency. These characteristics make the FDA attractive for a wide range of applications in target imaging [13], localization [14], and bistatic radar [15–18]. Instead of using a fixed frequency offset, W. Khan et al. presented a variational frequency offset scheme to achieve timeindependent but anglerangedependent beampattern [19, 20]. In [21], the authors derived the FDA CramérRao lower bounds for estimating target direction, range and velocity. The work about the joint applications of FDA and multiinput and multioutput (MIMO) radar can be found in [16], where the FDA elements radiate coherent waveforms, rather than orthogonal waveforms like an MIMO radar. Additionally, an analytical investigation of FDA multipath characteristics was carried out in [22]. The authors of [23] extended the linear FDA to planar array.
In this paper, to further enhance the interference mitigation capability, we propose a PSFDA by combining the PSA and the FDA. A anglerange and polarizationdependent beamforming algorithm is also proposed by jointly using the normfunction constraint theory [24] and convex optimization [25]. The proposed method can effectively suppress both clutter and interference that is not accessible for the conventional phasedarray systems. The new contributions of this work is different from our previous work [26] in at least two aspects: One is that, a more general signal model is presented for the PSFDA: the PSFDA considered in [26] is just arranged with uniformly spaced orthogonal dipole, while the PSFDA considered in this work is constituted by distributed collocated electromagnetic vector sensor (EMVS), and the antenna elements in each EMVS with 2≤P≤6 dipole elements, which contains the special case in [26]. Therefore, the signal model of the PSFDA in this work is more general when compared to the one in [26]; The second is that the output SINR performance of the highdimensional PSFDAs are analyzed and simulated in a much more extensive way.
Note that, [·]^{T} and [·]^{H} denote the transpose and conjugate transpose operators, respectively. ∥·∥_{1}, ∥·∥_{2}, and ∥·∥_{∞} indicate the ℓ_{1}norm, ℓ_{2}norm, and ℓ_{∞}norm of a vector or matrix.
The remainder of this paper is organized as follows. Section 2 introduces the PSFDA, along with its measurement model. Section 3 proposes the anglerange and polarization dependent beamforming algorithm. Next, numerical simulation results and discussions are given in Section 4. Finally, conclusions are drawn in Section 5.
PSFDA and its signal model
Frequency diverse array
Consider a uniform Melement linear FDA with the element spacing d. The radiated frequency of the mth element is given by [5]
where the frequency offset between the mth element and the reference element is Δf_{m}=(m−1)·Δf, f_{0}, and Δf represent the carrier frequency and the frequency increment, respectively. Taking the first element as the reference for the array, then the phase difference between the mth element and the reference element can be expressed as
where the approximation r_{m}≈r−(m−1)d sinθ, where θ, r, and c are the direction and range of the given farfield point and the speed of light, respectively. The steering vector of a standard FDA can be expressed as (3), if the third term of (2) is ignored due to the fact that f_{0}≫Δf and r≫dsinθ, we can obtain the steering vector of FDA as
It is seen that the FDA generates a rangedependent beampattern that is different from a phasedarray. Therefore, FDA provides a more flexible beam scan option in either transmission or receive mode. The FDA beam direction will vary as a function of range r and angle θ, even for a fixed Δf. Specifically, the FDA is just a conventional phasedarray when Δf=0Hz. If the range r is fixed, the beam direction will vary as a function of Δf. Therefore, the anglerangedependent beampattern of FDA provides a potential to focus transmit energy in the desired rangeangle section. More importantly, additional degrees of freedom can be provided if polarized sensors are adopted.
Polarization sensitive array
Considering a righthand spherical coordinate system with the orthogonal basis defined by orthogonal components v_{1} and v_{2} (see Figs. 1 and 2), the measurement model of the vector sensor is given by [27]
where I_{3} is the third order identity matrix, \((\mathbf {u}\times)=\left [ \begin {array}{ccc} \ \ 0 \ \ \ \ u_{z} \ \ \ \ u_{y} \\ \ \ u_{z} \ \ \ \ \ 0 \ \ \ u_{x} \\ u_{y} \ \ \ \ u_{x} \ \ \ \ 0 \end {array} \right ]\), with u_{x},u_{y},u_{z} being the x,y,z components of the unit direction for the propagation vector u=[cosθcosϕ,sinθcosϕ,sinϕ]^{T}. The realvalued matrices \(\mathbf {V}\in \mathcal {R}^{{3\times 2}}\), \(\mathbf {Q}\in \mathcal {R}^{{2\times 2}}\) and complex vector \(\mathbf {h} \in {\mathcal {C}^{2\times 1}}\) are given, respectively, by
where θ∈[0,2π),ϕ∈[−π/2,π/2],α∈(−π/2,π/2] and β∈[−π/4,π/4] are the azimuth, elevation, polarized ellipse’s orientation and eccentricity angles, respectively. s(t) denotes the complex envelope of the transmitted signal. e_{E}(t) and e_{H}(t) are the noise components of electric and magnetic fields. \(\mathbf {V}^{(E)}_{x}\), \(\mathbf {V}^{(E)}_{y}\), and \(\mathbf {V}^{(E)}_{z}\) denote the response of the three electric dipoles along the x, y, and zaxis, respectively. Similarly, the definitions for the three magnetic dipoles along the x, y, and zaxis are given by
Consider a general model of (4), the measurement formulation of a distributed electromagnetic vector sensor (DEMVS), which is also named as distributed electromagnetic component sensor array (see Fig. 3), can be expressed as [28]
where Λ_{k}=[θ_{k},ϕ_{k},α_{k},β_{k}] denotes the direction and polarization parameters of the kth source signal. Γ(θ_{k},ϕ_{k}) is a diagonal matrix with the nth diagonal entry being \(\left [\mathbf {\Gamma }(\theta _{k}, \phi _{k}) \right ]_{n}=e^{j2\pi \mathbf {q}^{T}_{n}\mathbf {u}_{k}/\lambda } \ \ (n=1,2,\ldots,N,~(N\leq 6)\), where λ being the wavelength and N is the number of array elements constituting the DEMVS), which provides the phase shift between the DEMVS center and the position q_{n} of the nth element of the DEMVS, Ω is an N×6 selection matrix with elements of “1” and “0,” indicating the component of the electromagnetic field measured by the nth sensor.
Extending (7) to an array with MDEMVS, we can define the array directional response as b(θ_{k},ϕ_{k})⊗d(θ_{k},ϕ_{k}), where \({\mathbf {d}(\theta _{k}, \phi _{k})}=\mathbf {\Gamma }(\theta _{k}, \phi _{k}) \mathbf {\Omega } \left [ \begin {array}{c} \mathbf {I}_{3} \\ (\mathbf {u}_{k} \times) \end {array} \right ]\mathbf {V}_{k}\) embeds all the directional information of the electromagnetic sources, \(\mathbf {b}(\theta _{k}, \phi _{k})=\left [ e^{j2 \pi \mathbf {p}^{T}_{1} \mathbf {u}_{k} / \lambda }, e^{j2 \pi \mathbf {p}^{T}_{2} \mathbf {u}_{k} / \lambda }, \ldots, e^{j2 \pi \mathbf {p}^{T}_{M} \mathbf {u}_{k}/ \lambda } \right ]^{T}\) represents the phase of the planewave at the position p_{i} of the ith DEMVS center (i=1,…,M), and the ⊗ is the Kronecker Product. Then, the measurements can be expressed as
According to Kronecker Product relationship (a⊗b)c=a⊗(bc), where a is a column vector, the number of columns in b is the same as the number of rows in c. (10) can be rewritten as
where a(Λ_{k})=d(θ_{k},ϕ_{k})Q_{k}h_{k}.
Different from the DEMVS, for the EMVS, the dipole elements of the EMVS are collocated, so both Γ and Ω are an 6×6 identity matrix I_{6}, the d(θ_{k},ϕ_{k}) can be simplified to \({\mathbf {d}(\theta _{k}, \phi _{k})}=\left [ \begin {array}{c} \mathbf {I}_{3} \\ (\mathbf {u}_{k} \times) \end {array} \right ]\mathbf {V}_{k}\).
Signal model of PSFDA
For the array constituted by M collocated EMVS, where each vector located at yaxis transmits a different frequency. That is, the radiated frequency from the mth EMVS is taken as the same of Eq. 1. Similarly, the phase difference between the mth EMVS and the first EMVS is
Accordingly, the PSFDA signal model can be reformulated as
where c(Λ_{k})=
\({\mathbf {b}_{a,k}(\theta _{k}, \phi _{k})}=\left [1~~e^{j\varphi _{a,k}}~~\cdots ~~e^{j(M1)\varphi _{a,k}} \right ]^{T}\) with φ_{a,k}=2πf_{0}dsinθ_{k}cosϕ_{k}/c, \({\mathbf {b}_{r,k}(r_{k})}=\left [1~~e^{j\psi _{r,k}}~~\cdots ~~e^{j(M1)\psi _{r,k}} \right ]^{T}\) with ψ_{r,k}=2πr_{k}Δf/c, and the ⊙is the KhatriRao product. In doing so, we get a spatiopolarizedrange manifold g(𝜗_{k}), denoting the spatiopolarizedrange information 𝜗_{k}=(θ_{k},ϕ_{k},α_{k},β_{k},r_{k}) of the kth signal. Assume an xelectriccomponent sensor and a yelectriccomponent sensor are used for each vector sensor, namely, the PSFDA array arrangement with uniformly spaced orthogonal dipoles as shown in Fig. 4. In this case, then we have
Similarly, the signal models for the FDA and a conventional uniform linear phasedarray (ULA) can be described, respectively, as
The steering vector of PSFDA depends on the angle, range, frequency and polarization, which means the PSFDA has potential advantage in terms of discriminating the targets from interferences.
Proposed adaptive angle, range, and polarization beamformer
The PSFDA measurement model for the M collocated EMVS can be expressed as
where g(𝜗_{s}) and g(𝜗_{j}) (j=1,…,J=K−1) are the steering vectors for signal and interference, respectively. In the following, we assume that the signal of interest is s(t), interferences is i_{j}(t) (j=1,…,J) and n(t) is a zero mean white Gaussian process.
According to the minimumnoisevariance beamformer for PSA [29], we can get the classical minimum variance for the anglepolarizationrange sensitive (MVAPRS) beamformer as
where R_{z}=E{Z(t)Z^{H}(t)} denotes the 2M×2M covariance matrix of the array snapshot vectors, and (·)^{−1} indicates the matrix inversion operator. To further enhance the interference mitigation capability, we propose an adaptive anglerangepolarizationdependent beamforming method. For a small number of interferences in the sidelobe domain, and nonsparse array gains in the mainlobe, to make the sidelobe of beampattern sparse, we sample the whole observation domain and form a overcomplete basis \({\mathbf {G}}_{s}= \left [\mathbf {g}\left (\bar {\mathbf {\vartheta }}_{1}\right), \mathbf {g}\left (\bar {\mathbf {\vartheta }}_{2}\right), \dots, \mathbf {g}\left (\bar {\mathbf {\vartheta }}_{sq}\right), \mathbf {g}\left (\bar {\mathbf {\vartheta }}_{s+q}\right)\right.\), \(\left.\dots \,\mathbf {g}(\bar {\mathbf {\vartheta }}_{L_{0}})\right ] ({L_{0}} \gg K)\) corresponding to the sidelobe region of 𝜗_{s}, where q is an integer associate with the bounds between the mainlobe and sidelobes of the beampattern, and we add the ℓ_{1}norm penalization on the array gains G_{s}. To further suppress the sidelobe level and interference, the ℓ_{∞}norm penalization is further imposed on the G_{s}. In doing so, the proposed beamformer w can be recast as,
where the weighting factor ξ makes a tradeoff between the minimum variance constraint on total output energy and the sparse constraint on the sidelobe of beampattern, the parameters ε and η are designed according to the desired array performance. So the array gain including both sidelobes and interferences can be suppressed by ∥w^{H}G_{s}∥_{∞}≤η. The ∥w^{H}g(φ_{s})−1∥_{∞}≤ε in (Eq. 18) aims to maintain the signal of interest. \(\left \ \mathbf {c} \right \_{\hbar }^{\hbar } = {\sum \nolimits }_{i} {{{\left  {{c_{i}}} \right }^{\hbar }}}\) is \({\ell _{\hbar }}\)norm of the vector c, \(\hbar \le 1\) leads to sparse solutions, while \(\hbar =2\) is the ℓ_{2}norm criterion. Due to the sparsity of the beam pattern,and to make the optimization problem (Eq. 18) be an easytohandle convex problem, we choose \(\hbar =1\), that is ∥w^{H}G_{s}∥_{1}. Compared with the ℓ_{2}norm criterion that favors solutions with many nonzero entries, this sparse constraint has advantages for the cases with few nonzero elements for interference suppression and sidelobe reduction. The validity of our proposed method will be further demonstrated via numerical analysis in Section 4.
Simulation Results and Discussions
We first analyze the PSFDA performance by examing its spatial domain beampattern, spatialpolarization beampattern and spatialrange beampattern. Statistical simulations are carried out in an ideal scenario without steering vector mismatches.
In the following simulations, we consider a PSFDA with M=8 orthogonal dipole pairs and d=0.5m as shown in the Fig. 4. Suppose there are one signal of interest and one interference with fixed signaltointerference ratio (SIR) equal to 20dB, 100 independent runs in spatially white gaussian noise. We assume that the powers of the polarized signal, interference and noise are \({\sigma _{s}^{2}}\), \({\sigma _{j}^{2}}\) and \({\sigma _{n}^{2}}\), respectively. The average SINR is determined by:
where \(\mathbf {I} \in \mathcal {C}^{2M \times 2M}\).
Beampattern
Anglerangedependent beampattern
Suppose the signal and interference have the anglerangepolarization characteristics (θ_{s},ϕ_{s},α_{s},β_{s},r_{s})=(20^{∘},60^{∘},−20^{∘},30^{∘},10km) and (θ_{j},ϕ_{j},α_{j},β_{j},r_{j})=(20^{∘},60^{∘},−20^{∘},30^{∘},8km), respectively. For an ideal scenario with the exact knowledge of the steering vector, Fig. 5 compares the PSA and PSFDA beampatterns. The anglerangedependent Sshaped PSFDA beampattern has additional degrees of freedom and potential applications in suppressing the range interference and ambiguous clutter with same angle but different range from the target, while the rangeindependent PSA beampattern has no such advantages. Besides, compared with PSFDA, the FDA does not provide the polarization information.
We also compare the blockshaped anglerangedependent beampattern obtained by the MVAPRS beamformer and our proposed beamformer for the PSFDA with the frequency offsets as Δf_{m}=(m−1)^{2}·Δf,m=1,2,…,M. The results given in Fig. 6 demonstrate that our proposed algorithm can successfully suppress the interference as well as the MVAPRS.
Spatiopolarized beampattern
We compare the PSFDA anglepolarizationdependent beampattern and the angleelevationdependent beampattern like that of a PSA beampattern, as shown in the Figs. 7 and 8, respectively. From Fig. 7, It is seen that the anglepolarizationdependent beampattern can keep the maximum energy at the target positions (20^{∘},60^{∘},20^{∘},30^{∘},10km) and suppress simultaneously the interference from (40^{∘},60^{∘},−40^{∘},30^{∘},10km). Similarly, the angleelevationdependent beampattern effectively suppresses the interference from (40^{∘},60^{∘},−20^{∘},30^{∘},10km), without reducing energy at the target (20^{∘},30^{∘},−20^{∘},30^{∘},10km). Moreover, the PSFDA not only has the same performance as that of PSA, but the former also outperforms the latter in rangedimension, as shown in Fig. 5.
In addition, we consider the statistical results for the anglerangedependent blockshaped beampattern. Fig. 9 displays the output SINRs versus signaltointerference ratio (SNR) for T=500. For a fixed SNR=0dB, the SINRs versus the number of snapshots T are shown in Fig. 10. The results clearly demonstrate that our proposed beamformer substantially outperforms the MVAPRS for SNR >0dB. Since the proposed algorithm is a convex optimization problem, it can be efficiently resolved by the openly CVX software [30].
Highdimensional arrays
Finally, we analyze the performance of the optimal output SINR for highdimensional arrays. The anglerangepolarization characteristics are same as that of Fig. 5. In each antenna, we consider the antenna elements with 2≤p≤6. For each p, the dipole elements are chosen according to Table 1. Note that at each row, the symbol “ √” denotes that a certain dipole element is selected in the antenna. For example, when p=2, two electric dipoles at the xaxis and yaxis are selected, respectively. Accordingly, the antenna response is \(\left [\begin {array} {cc} \mathbf {V}^{(E)}_{x} \\ \mathbf {V}^{(E)}_{y} \end {array} \right ]\).
The simulated optimal output SINRs versus SNR for p=2,3,4,5,6, under the anglerangedependent blockshaped beampattern, are given in Fig. 11. For fixed SNR=0dB and SIR=20dB, the optimal SINRs versus T for p=2,3,4,5,6 are shown in Fig. 12. It is noticed along with the increase of the sensor dimensionality p, the output SINRs will increase even under the same SNR (or number of snapshots). Figures 13 and 14 show that the optimal output SINR is approximately linearly proportional to p. This verify that the electromagnetic vector array has the advantage of enabling the control of beampattern polarization, which achieves more degrees of freedom.
Conclusions
In this paper, we formulated the PSFDA signal model and proposed a corresponding beamformer based on the MVAPRS, ℓ_{1}norm minimization and ℓ_{∞}norm constraint. The proposed adaptive beamformer is superior to the conventional MVAPRS on maintaining array gain for the signal of interest, and PSA beamformers in terms of both nulling interferences and maintaining signal of interest. In addition, the performance of the optimal output SINR is also analyzed for the highdimensional arrays. Simulation results demonstrate that the electromagnetic vector array has the advantage of enabling the control of beampattern polarization and virtually increasing the array size.
Abbreviations
 DEMVS:

Distributed electromagnetic vector sensor
 DOA:

Directionofarrival
 EMVS:

Electromagnetic vector sensor
 FDA:

Frequency diverse array
 MIMO:

Multiinput and multioutput
 MVAPRS:

Minimum variance beamformer for the anglepolarizationrange sensitive
 PSA:

Polarization sensitive array
 PSFDA:

Polarization sensitive frequency diverse array
 SINR:

signaltointerferenceplusnoise ratio
 SIR:

signaltointerference ratio
 SNR:

signaltonoise ratio
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (61501089 & 61471103), Sichuan Science and Technology Program (18ZDYF2551), and Fundamental Research Funds for the Central Universities (ZYGX2018J005).
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HC formulated the idea and wrote the paper, all authors discussed the idea together. Additionally, all authors read and approved the final manuscript.
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Correspondence to Hui Chen.
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Chen, H., Shao, H. & Chen, H. Anglerangepolarizationdependent beamforming for polarization sensitive frequency diverse array. EURASIP J. Adv. Signal Process. 2019, 23 (2019). https://doi.org/10.1186/s136340190620x
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Keywords
 Array signal processing
 FDA
 PSA
 Beamforming
 Sparse constraint
 Convex optimization