Enhanced "vectorcrossproduct" directionfinding using a constrained sparse triangulararray
 Feng Luo^{1} and
 Xin Yuan^{2}Email author
https://doi.org/10.1186/168761802012115
© Luo and Yuan; licensee Springer. 2012
Received: 24 December 2011
Accepted: 24 May 2012
Published: 24 May 2012
Abstract
A new configuration of sparse array is proposed in this article to estimate the directionofarrivals (DOAs) and polarizations of multiple sources. This constrained sparse array is composed of a dipoletriad, a looptriad, and a single antenna, which can be a dipole, a loop, or a scalarsensor. These three units comprise a triangular geometry in the space. This geometry creatively synergizes the conventional interferometry method based on the spatial phasedelay across displaced antennas, and the "vectorcrossproduct" based on Poyntingvector estimator to enhance the DOA estimation accuracy. The investigated algorithm based on this configuration adopts the "vectorcrossproduct" DOA estimator to provide the coarse estimate and then derives the fine estimate by extracting the intersensor phase factors in the sparse array. Following this, the disambiguation approach is adapted to derive the unambiguous estimate, and this estimate is also fine in estimation resolution. The proposed configuration can extend the array aperture and also reduce the mutual coupling. The significant performance of the proposed sparse array composition is demonstrated by Monte Carlo simulations when the intersensor spacing far exceeds a halfwavelength.
Keywords
1. Introduction
The basic principle of the "vectorcrossproduct" directionfinding is to extract the relations between the electricfield e and the magneticfield h of an electromagnetic wave. The vectorcrossproduct between e and h, the Poyntingvector u, will provide the directioncosines of the incident source. It follows that the directionofarrival (DOA) of the source can be estimated.
where * denotes complex conjugation, × symbolizes the vector crossproduct operator,  ·  represents the Frobenius norm of the element inside  , and {u_{ x, k }, u_{ y, k }, u_{ z, k }} are the directioncosines of the k th source align to xaxis, yaxis, zaxis, respectively. From this u_{ k }, the DOA (θ_{1, k}, θ_{2, k}) of the k th source can be derived uniquely in the threedimensional space. Equation (2) also indicates that the azimuthangle and elevationangle of each source can be automatically paired without postprocessing [2].
Wong et al. advanced the "vectorcrossproduct" directionfinding algorithm by investigating some novel capabilities of the electromagnetic vectorsensor (array), for example, sparse array with sixcomponent electromagnetic vectorsensor [10], DOA estimation without the priori known sensors' locations [9], "selfinitiating MUSIC" [12] and blind geolocation, beamforming for frequencyhopping sources of unknown and arbitrary hopsequences [13].
 (1)
The collocated antennas are reduced from six to three, and thus the mutual coupling across the composed antennas are reduced greatly.
 (2)
Since the dipoletriad and the looptriad are spatially spread and there is no constraint of their relative locations, the spatially array aperture is extended and so the estimation accuracy for DOA can be improved distinctly.
Wong [44] proposed this configuration and presented an example with CRBs to show that this configuration can improve the direction finding accuracy. However, the algorithm used in [44] was only the "vectorcrossproduct" result, which is the same as the collocated electromagnetic vectorsensor. Therefore, the approach utilized in [44] can not investigate the advantage (2) described above. Based on this, the present article will propose a new configuration based on the "Displaced DipoleTriadPlusLoopTriad Pair" in [44], and will investigate an enhanced algorithm to extract the aperture extension property of the new configuration. This enhanced algorithm will then improve the directionfinding estimation accuracy by extracting the intersensor phase factors across the sensors, which were ignored in [44].
For the 2D elevationazimuth angle estimation, two phasefactors are necessary. Therefore, a constrained sparse triangulararray is proposed in this article. The triangulararray consists of a dipoletriad, a looptriad, and a single dipole/loop/scalarsensor. It is worth noting that an additional antenna is employed in this triangulararray, which is used to increase the arrayaperture. In the derived DOA estimation algorithm, it will provide another intersensor phase factor, which is used to derive the fine estimate of one directioncosine. The enhanced directioncosines' estimates will improve the direction finding estimation accuracy. In addition, this antenna can be a dipole, a loop, or a scalarsensor without polarization information. The single dipole or loop can be oriented along any one of the three Cartesian coordinate axes. The proposed array geometry is a sparse array with intersensor spacings far larger than a halfwavelength. Similar sparse vectorsensor arrays have been investigated in [10, 11, 36, 45].
Another remarkable innovation to adapt the "vectorcrossproduct" directionfinding algorithm is the "noncollocating electromagnetic vectorsensor" proposed in [45]. The six antennas composed of the electromagnetic vectorsensor are spatially spread in the space with some constrains. Then the "vectorcrossproduct" directionfinding algorithm can still be used. Furthermore, the mutual coupling is reduced and the angular resolution is enhanced. In addition, it is also shown in [45] that the "univectorsensor ESPRIT" algorithm proposed in [2] can still be utilized in the noncollocating electromagnetic vector for direction finding. This "univectorsensor ESPRIT" algorithm will thus be used in the present article in the following derivation and also in the simulation. However, the algorithm investigated in [45] can only offer one fine estimate of the directioncosine, and this is not sufficient for the 2D direction finding. Based on this reason, the present article proposes a new configuration with seven sensors to form a triangular array, in order to increase the array aperture and then to improve the two dimensional DOA estimation accuracy. It is worth noting that the dipoletriad and looptriad can be noncollocating but need to satisfy the conditions proposed in [45]. In order to simplify the exposition, the following derivation will be based on the collocated case.
The dipoletriad and looptriad have also been extensively investigated in other literature. The antijamming performance of the dipoletriad (a.k.a. tripole) has been investigated by Comption Jr. [46, 47]. The dipoletriad (array) was used for direction finding in [48–52]. The performance of a dipoletriad array for 1D direction finding and polarization estimation has been evaluated in [48] through the CRBs derivation, and it showed that the quality of the DOA estimate depends strongly on the polarization state. Zhang [49] investigated an ESPRIT based algorithm for direction finding and polarization estimation for uniform circular dipoletriad array, and ZainudDeen et al. [50] adopted the radial basis function neural network to the uniform circular dipoletriad array (and also crossdipole array) for direction finding and polarization estimation. Daldorff et al. [53] combined unitary matrix pencil method and a least squares solver to do the direction finding with a single dipoletriad. The linear dependence and uniqueness of dipoletriad (dipoletriad array) was developed in [54–57]. An H^{∞} approach was proposed in [58] to track polarized cochannel sources with dipoletriad array, and a new quasicrossproduct algorithm was proposed in [59] for tracking the direction of a moving and nonlinearly polarized electromagnetic source using a dipoletriad. Zhang and Xu [60] explored blind beamforming of the dipoletriad array and the parallel factor model was adopted. Ravinder and Pandharipande [61, 62] showed that a dipoletriad could minimize bit error rate better through polarization diversity when the desired user and other interfering users arrived from the same direction or were very close to the desired user direction but with different polarization states. Theoretical performance bounds for direction finding using the dipole/loop triad were derived in [63].
The remainder of this article is organized as follows. The geometry of the sparse triangulararray is provided in Section 2. The enhanced "vectorcrossproduct" directionfinding algorithm based on the proposed configuration is derived in Section 3. Section 4 presents the simulation results of the enhanced algorithm, and Section 5 concludes the whole article.
2. Spatial geometry of the sparse array used in this work
Wong [44] investigated the "vectorcross product" for direction finding with spatially spaced dipole and loop triad but ignored the effect of spatial phasefactor, which can improve the accuracy of direction finding. In order to get the finer estimate for the DOA, at least two finer directioncosines' estimates should be obtained. The dipoleloop triad pair can present three coarse directioncosines' estimates from the "vectorcross product" result and one finer directioncosine's estimate from the intertriad spacing phase. Thus, another antenna is employed to provide the other finer directioncosine's estimates from the intersensor spacing phase factor and at the same time to increase the array aperture. Figure 1 depicts the array geometry used in this work.
 (A)Cases (i)(iii), the dipoletriad is located at the origin of the Cartesian coordinate system, the looptriad is located at (Δ_{ x }, 0, 0) on the xaxis, and the single dipole is located on the yaxis at (0, Δ_{ y }, 0). The arraymanifold can be shown as:$\mathbf{a}=\left[\begin{array}{c}\hfill \mathbf{e}\hfill \\ \hfill {q}_{1}\mathbf{h}\hfill \\ \hfill {q}_{2}b\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \mathbf{e}\hfill \\ \hfill {e}^{j2\pi \frac{{\mathrm{\Delta}}_{x}{u}_{x}}{\lambda}}\mathbf{h}\hfill \\ \hfill {e}^{j2\pi \frac{{\mathrm{\Delta}}_{y}{u}_{y}}{\lambda}}b\hfill \end{array}\right]$(3)
where ${{q}_{1}}^{\underset{}{\underset{=}{\mathsf{\text{def}}}}}{e}^{j2\pi \frac{{\mathrm{\Delta}}_{x}{u}_{x}}{\lambda}},{{q}_{2}}^{\underset{}{\underset{=}{\mathsf{\text{def}}}}}{e}^{j2\pi \frac{{\mathrm{\Delta}}_{y}{u}_{y}}{\lambda}}$, and b is one of {e_{ x }, e_{ y }, e_{ z }} corresponding to the different cases.
 (B)Cases (iv)(vi), the looptriad is located at the origin of the Cartesian coordinate system, the dipoletriad is located at (Δ_{ x }, 0, 0) on the xaxis, and the single loop is located on the yaxis at (0, Δ_{ y }, 0). The arraymanifold can be shown as:$\mathbf{a}=\left[\begin{array}{c}\hfill {q}_{1}\mathbf{e}\hfill \\ \hfill \mathbf{h}\hfill \\ \hfill {q}_{2}b\hfill \end{array}\right]$(4)
where b is one of {h_{ x }, h_{ y }, h_{ z }} corresponding to the different cases.
Remarks:

The single dipole/loop can be replaced by one dipole/loop triad and thus the compositions will have three dipole/loop triads: (a) one dipoletriad and two looptriads, (b) one looptriad and two dipoletriads. Under those configurations, the "vectorcrossproduct" can be obtained three times and hence the average value can be used. Also the two fine estimates of directioncosines are both obtained from the intertriad phase factors of the "vectorcrossproduct" results.

The single dipole/loop in Figure 2 can be located at an arbitrary position (x_{2}, y_{2}), but not collinear with the dipoletriad and looptriad.

The relative locations of dipoletriad, looptriad and the single dipole/loop can be changed, Figure 2 just presents six examples.

The single dipole/loop in Figure 2 can be replaced by a scalarsensor, and this scalarsensor will not include the polarization information of the incident source. In this case, the b in (3)(4) will be replaced by 1.

The proposed sparsearray is different from the sparsearray in [10, 11], where the array is composed of sixcomponent electromagnetic vectorsensors. The proposed array pioneers the geometry with three different sensors and it has two advantages compared with the sparse array in [10, 11]: (a) The triad only has three collocated antennas, thus the new array configuration can reduce the mutual coupling; (b) The new array configuration can diminish the hardware cost as only the triad and single dipole/loop is used. Furthermore, the single dipole/loop can be replaced by the simple scalarsensor.
As the following analysis is similar for all the six cases, case (i) will be taken as an example to derive the enhanced "vectorcross product" algorithm for directionfinding. Please note that the classical far field and narrowband assumption is made throughout the article.
3. The enhanced "vectorcross product" algorithm for direction finding
All the following derivation for the enhanced "vectorcross product" algorithm will be based on (5). The derivation steps are similar to the algorithm proposed in [45]. First, the course but unambiguous estimates of directioncosines are derived from the "vectorcrossproduct" result. Then, the fine but cyclically ambiguous estimates of directioncosines are obtained from the intersensor phase factors. Finally, the coarse estimates of directioncosines are used to disambiguate the fine but cyclically ambiguous estimates to derive both fine and unambiguous estimates of directioncosines.
3.1. Get the coarse but unambiguous estimates of directioncosines from the "vectorcrossproduct" result
 (1)If θ_{2, k}∈ [0, π/2], which means u_{ z, k } ≥ 0, then:${\widehat{\mathbf{u}}}_{k}={\stackrel{\u0303}{\mathbf{u}}}_{k}{e}^{j\angle {\left[{\mathbf{\u0169}}_{k}\right]}_{3}}=\left[\begin{array}{c}\hfill {\mathit{\xfb}}_{x,k}^{\mathsf{\text{coarse}}}\hfill \\ \hfill {\mathit{\xfb}}_{y,k}^{\mathsf{\text{coarse}}}\hfill \\ \hfill {\mathit{\xfb}}_{z,k}\hfill \end{array}\right],$(8)
where [·]_{ i } is the i th element of the vector in [ ], and ∠ denotes the complex angle of the following complex number.
 (2)If θ_{2, k}∈ [π/2, 0), which is u_{ z, k } ≤ 0, then:${\widehat{\mathbf{u}}}_{k}={\stackrel{\u0303}{\mathbf{u}}}_{k}{e}^{j\angle {\left[{\stackrel{\u0303}{\mathbf{u}}}_{k}\right]}_{3}}=\left[\begin{array}{c}\hfill {\mathit{\xfb}}_{x,k}^{\mathsf{\text{coarse}}}\hfill \\ \hfill {\mathit{\xfb}}_{y,k}^{\mathsf{\text{coarse}}}\hfill \\ \hfill {\mathit{\xfb}}_{z,k}\hfill \end{array}\right].$(9)It follows that:${\mathit{\xfb}}_{x,k}^{\mathsf{\text{coarse}}}={\left[{\widehat{\mathbf{u}}}_{k}\right]}_{1},$(10)${\mathit{\xfb}}_{y,k}^{\mathsf{\text{coarse}}}={\left[{\widehat{\mathbf{u}}}_{k}\right]}_{2}.$(11)
3.2. Obtain the fine but cyclically ambiguous estimates of directioncosines
 (1)if θ_{2, k}∈ [0, π/2],${\mathit{\xfb}}_{x,k}^{\mathsf{\text{fine}}}=\frac{{\lambda}_{k}}{2\pi}\frac{1}{{\mathrm{\Delta}}_{x}}\angle {\left[{\stackrel{\u0303}{\mathbf{u}}}_{k}\right]}_{3};$(12)
 (2)if θ_{2, k}∈ [π/2, 0],${\mathit{\xfb}}_{x,k}^{\mathsf{\text{fine}}}=\frac{{\lambda}_{k}}{2\pi}\frac{1}{{\mathrm{\Delta}}_{x}}\left(\angle {\left[{\stackrel{\u0303}{\mathbf{u}}}_{k}\right]}_{3}+\pi \right).$(13)
Remarks:

The cases (1) and (2) in (8)(9) and (12)(13) are based on the condition whether u_{ z, k }is positive or negative, which leads to that the validity region of directionfinding is the upper hemisphere or lower hemisphere in the polar coordinate system. Therefore, the ∠[ũ_{ k }]_{3} is used in (12) and (13).
If the condition changes to be based on the positive or negative of u_{ x, k }, the validity region of directionfinding will be the front hemisphere or back hemisphere in the polar coordinate system. In this case, the ∠[ũ_{ k }]_{1} will be used in Section 2.
If the condition changes to be based on the positive or negative of u_{ y, k }, the validity region of directionfinding will be the left hemisphere or right hemisphere in the polar coordinate system. In this case, the ∠[ũ_{ k }]_{2} will be used in Section 2.

${\mathit{\xfb}}_{x,k}^{\mathsf{\text{coarse}}}$, ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{coarse}}}$, ${\mathit{\xfb}}_{x,k}^{\mathsf{\text{fine}}}$ are the same for all the six cases in Figure 2, but the ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{fine}}}$ varies for different cases based on the arraymanifold. For cases (i)(vi) in Figure 2, ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{fine}}}$ can be estimated from ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{fine}}}=\frac{\lambda}{2\pi}\frac{1}{{\mathrm{\Delta}}_{y}}\angle \left\{\frac{{\left[{\widehat{\mathbf{a}}}_{k}\right]}_{i}}{{\left[{\widehat{\mathbf{a}}}_{k}\right]}_{7}}\right\}$, ∀i = 1, 2, ..., 6, corresponding to each case.

If the scalarsensor is used to replace the single dipole/loop, since the response of h_{ z }is a positive real number, ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{fine}}}$ can be estimated from ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{fine}}}=\frac{{\lambda}_{k}}{2\pi}\frac{1}{{\mathrm{\Delta}}_{y}}\angle \left\{\frac{{\left[{\widehat{\mathbf{a}}}_{k}\right]}_{6}}{{\left[{\widehat{\mathbf{a}}}_{k}\right]}_{7}}\right\}$.

The single dipole/loop/scalarsensor in Figure 1 can be located at an arbitrary position (x_{2}, y_{2}), but not collinear with the dipoletriad and looptriad. Then the arraymanifold will be ${\mathbf{a}}_{k}={\left[{\mathsf{\text{e}}}_{k},{q}_{1,k}{\mathbf{h}}_{k},{e}^{j\frac{2\pi}{{\lambda}_{k}}\left({x}_{2}{u}_{x,k}+{y}_{2}{u}_{y,k}\right)}{b}_{k}\right]}^{T}$. In this case, after the û_{ x, k }is derived, ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{fine}}}$ can be obtained through $\left\{\frac{{\lambda}_{k}}{2\pi}\frac{1}{{y}_{2}}\angle \left\{\frac{{\left[{\widehat{\mathbf{a}}}_{k}\right]}_{i}}{{\left[{\widehat{\mathbf{a}}}_{k}\right]}_{7}}\right\}\frac{{x}_{2}}{{y}_{2}}{\mathit{\xfb}}_{x,k}\right\}$, where i = 1, 2, ..., 6 corresponds to the number of element in the arraymanifold. The disadvantage of this arbitrary location configuration is that it will increase the computation workload of the estimation algorithm.
3.3. Disambiguate the fine estimates by coarse estimates of directioncosines
In order to get the fine and unambiguous estimates of the directioncosines, the coarse estimates obtained in Section 1 will be used as the reference to disambiguate the fine estimates derived in Section 2. This disambiguation approach has been derived by Zoltowski and Wong [10], and has also been used in the other literature, i.e. [11, 45]. The main essence is summarized as follows [45].
where ⌈α⌉ denotes the smallest integer not less than α, and ⌊α⌋ refers to the largest integer not exceeding α.
However, in case that Δ_{ x } ≤ λ_{ k }, Δ_{ y } ≤ λ_{ k }, the spatial aperture is not much extended. We can set ${\mathit{\xfb}}_{x,k}={\mathit{\xfb}}_{x,k}^{\mathsf{\text{coarse}}}$, ${\mathit{\xfb}}_{y,k}={\mathit{\xfb}}_{y,k}^{\mathsf{\text{coarse}}}$, directly.^{a}
where $\phantom{\rule{0.3em}{0ex}}{\widehat{q}}_{1,k}={e}^{j\frac{2\pi}{{\lambda}_{k}}{\mathrm{\Delta}}_{x}{\mathit{\xfb}}_{x,k}}$.
4. Monte Carlo simulation for the algorithm obtained in Section 3
where $\left\{{\mathit{\xfb}}_{x,k}^{i},{\mathit{\xfb}}_{y,k}^{i}\right\}$ are the estimate of directioncosines at i th run.
4.1. Compare the proposed algorithm (EVC) with the conventional "vectorcross product" algorithm (CVC) and CRBs
4.2. The aperture extension of the proposed configuration
It is well known that the larger is the array's spatial aperture, the finer would be the resolution of the arrival angle estimates, so it is of interest to investigate the performance of the proposed sparse array when the spatial aperture becomes larger.
5. Conclusion
A constrained sparse array composed of one dipoletriad, one looptriad and one single dipole/loop/scalar sensor is investigated in this article. This new array configuration can support better estimation accuracy in directionfinding by synergizing the "vectorcross product" algorithm and intersenor spacing phase factors. Aperture extension is achieved by spacing the three different sensors much greater than a half wavelength. Monte Carlo simulation demonstrates the efficiency of the proposed array configuration and the algorithm. Unlike the sparse array investigated before, the aperture extension and fine estimates of DOA are implemented by only one dipoletriad, one looptriad and one single dipole/loop/scalarsensor. Therefore, the mutual coupling across the sensors is reduced and additionally the hardware cost is decreased.
Endnotes
^{a}The proposed arraygeometry has an improved identifiability compared with the electromagnetic vectorsensors in [2, 44, 45] since an additional antenna is employed. The basic principle of the subspacebased parameter estimation algorithms, such as ESPRIT, is to separate the signal and the noise, into the different subspaces (i.e. the signal subspace and the noise subspace), which are derived from the data covariance matrix [64]. It follows that the number of incident sources should be less than the maximal rank of the data covariance matrix. When the "univectorsensor" algorithm [2] is used in the present arraygeometry, the maximal rank of the data covariance matrix equals 7. On the other hand, For the collocated electromagnetic vectorsensor [2], noncollocating electromagnetic vectorsensor [45], or the "Displaced DipoleTriadPlusLoopTriad Pair" [44], it equals 6. Therefore, when the "univectorsensor" algorithm [2] is used with distinguishable DOAs and polarizations, the resolvable monochromatic sources number should be less than seven, which is one more that the electromagnetic vectorsensors in [2, 44, 45]. Thus, the additional antenna improves the identifiability compared with the electromagnetic vectorsensor. For more investigations of this identifiability issue with the electromagnetic vectorsensor, please refer to [65–67]. ^{b}The CRBs plotted in the figures in this section is computed by the same method as in [45], with the same signal model and noise model. Since the closedform results are too long to be listed here, we just plot the corresponding curves in the graphs.
Declarations
Authors’ Affiliations
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