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Enhanced "vectorcrossproduct" directionfinding using a constrained sparse triangulararray
EURASIP Journal on Advances in Signal Processingvolume 2012, Article number: 115 (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.
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.
This "vectorcrossproduct" directionfinding algorithm was proposed by Nehorai and Paldi based on the sixcomponent electromagnetic vectorsensor. A sixcomponent electromagnetic vectorsensor consists of three orthogonal dipoles and three orthogonal loops. These dipoles and loops are collocated at a point geometry in space, in order to measure the electricfield and magneticfield of the incident signal, respectively. In a multiple source scenario with K incident sources, the responses of the k th source a_{ k } can be represented by the 3 × 1 electricfield vector e_{ k } and the 3 × 1 magneticfield vector h_{ k } [1, 2]:
where {θ_{1, k}∈ [0, 2π), θ_{2, k}∈ [π/2, π/2]} are the azimuthangle and elevationangle of the source (please refer to Figure 1), respectively, and {θ_{3, k}∈ [0, π/2], θ_{4, k}∈ [π, π)} denote the auxiliary polarization angle and polarization phase difference of the incident signal, respectively (equating to {γ, η} in [3]). The unique arraymanifold in (1) has been exploited extensively by various eigenstructurebased directionfinding frameworks [2–43].
Based on (1), the Poyntingvector u_{ k } of the k th incident source can be obtained by [1]:
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].
One remarkable innovation to adapt the "vectorcrossproduct" directionfinding algorithm is the "Displaced DipoleTriadPlusLoopTriad Pair" proposed in [44]. The "vectorcrossproduct" directionfinding algorithm is found still applicable when the dipoletriad and looptriad are spatially spread in the space. Reference [44] adopted the "vectorcrossproduct" to do the directionfinding with this "Displaced DipoleTriadPlusLoopTriad Pair", and also compared the CramérRao bounds (CRBs) of this configuration with that of the collocated electromagnetic vectorsensor to show that the displaced pair can offer a lower CRBs. The advantages of the "Displaced DipoleTriadPlusLoopTriad Pair" compared with the collocated electromagnetic vectorsensor are significant:

(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.
Figure 2 illustrates the six different sparse array compositions, and each composition is made up of seven dipoles/loops. The seven dipoles/loops are categorized into three different units: (1) one collocated dipoletriad, (2) one collocated looptriad, and (3) one single dipole/loop of various orientations.
The arraymanifold of the compositions in Figure 2 can be classified into two groups:

(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
From various eigenstructurebased parameter estimation algorithms cited in Section 1, the steering vector of the k th incident source can be obtained, within an unknown complex number c [44, 45]. That is:
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
From (5), for case (i) in Figure 2, in the multiple source scenario with K incident sources,
and from the vectorcross product [44],
Note that ũ_{ k } is different from the Poyting vector u_{ k } (see Figure 1), but it can be seen as an estimate of u_{ k }, ${\stackrel{\u0303}{\mathbf{u}}}_{k}={q}_{1,k}^{*}{\mathbf{u}}_{k}$. It follows that u_{ k } can be estimated from this ũ_{ k }. Separately consider the following two cases:

(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
The intersensor phasefactors {q_{1}, q_{2}} can offer the fine estimates for the directioncosines. However, they will suffer the cyclically ambiguous because of the periodicity of the phase. From the vectorcross product result in (7),

(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)
From (6), ${\mathit{\xfb}}_{y,k}^{\mathsf{\text{fine}}}$ can be obtained by:
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].
Using {û_{ x, k }, û_{ y, k }} to denote the fine and unambiguous estimates of the directioncosines, there exist two integers $\left\{{m}_{x,k}^{\mathsf{\text{o}}},\phantom{\rule{2.77695pt}{0ex}}{m}_{y,k}^{\mathsf{\text{o}}}\right\}$ leading to [45]:
$\left\{{m}_{x,k}^{\mathsf{\text{o}}},\phantom{\rule{2.77695pt}{0ex}}{m}_{y,k}^{\mathsf{\text{o}}}\right\}$ can be derived by:
for
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}
Lastly, after the unique {û_{ x, k }, û_{ y, k }} has been obtained, the DOA of k th incident source {θ_{1, k}, θ_{2, k}} can be estimated by [2, 45]:
The polarization parameters can be estimated by â_{ k }:
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
The proposed algorithm's directionfinding efficacy and extendedaperture capability are demonstrated by Monte Carlo simulations and the accuracy is compared with the conventional "vectorcross product" (CVC) algorithm in [44]. In the plotted figures, the curves with the proposed algorithm are labeled with EVC and the curves with conventional "vectorcross product" algorithm are labeled with CVC. The "univectorsensor ESPRIT" algorithm in [2] was adopted to estimate the steering vectors of the incident sources (to derive Equation (5)) in the following simulations and thus the sources are modeled as uncorrelated pure tones with different frequencies. The estimates use 400 temporal snapshots and 500 independent runs. The root mean square error (RMSE) is utilized as the performance measure. The RMSE for the directioncosine of the k th source is defined as
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
Figure 3a shows a twosource scenario results, whereas Figure 3b shows a threesource scenario results. Both the estimation bias and RMSE of the directioncosine are plotted in Figure 3. Figure 3 clearly demonstrates that the performance of proposed algorithm (EVC) is better than that of the conventional "vectorcross product" algorithm (CVC), especially when SNR ≥ 5 dB. The RMSE for directioncosine with the proposed algorithm is ten times lower than the RMSE with the CVC, and they are very close to the CRBs.^{b} Figure 4 plots the standard deviations of estimates for the DOA (θ_{1, k}, θ_{2, k}) versus SNR for each source in a twosource scenario as in Figure 3a. It can be seen that when SNR ≥ 5 dB, the standard deviations of the (θ_{1, k}, θ_{2, k}) with the proposed algorithm are about 30 times lower than their counterparts with the CVC, and they are very close to the CRBs. Figures 3 and 4 clearly verify the performance of the proposed arraygeometry and also verify the efficacy of the proposed algorithm.
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.
Figure 5a shows a twosource scenario, whereas Figure 5b shows a threesource scenario with the same setting as in Figure 3 at SNR = 30 dB, by plotting the RMSE of the directioncosines estimates versus intersensor spacing $\frac{\mathrm{\Delta}}{\lambda}$, where λ is the minimum wavelength of the incident sources. Figure 5 clearly shows that the RMSE of the directioncosines estimates with the proposed algorithm decrease with the increase of the spatial aperture and they are very close to the CRBs. This proposed configuration and the enhanced algorithm lead to ordersofmagnitude improvement in estimation accuracy. However, the RMSE of the directioncosines estimates with the conventional "vectorcross product" algorithm remain the same with the increase of the spatial aperture. It can also be observed that when$\frac{\mathrm{\Delta}}{\lambda}\le 2$, the performance of the two algorithm is nearly the same and it is better to use the conventional "vectorcross product" algorithm since it needs less manipulation. It is worth noting that a breakdown phenomenon initiates in Figure 5 at an intersensor spacing of about Δ = 100λ (200 half wavelengths). This is because the coarse estimates of directioncosines begin to misidentify the estimation grid. For further investigation of this breakdown phenomenon, please refer to [10].
In order to investigate the increased aperture induced by the additional scalar sensor, Figure 6 plots the CRBs of the directioncosines in a three sources scenario. Both the CRBs with and without the scalar sensor are plotted. "With the scalar sensor" means that the array geometry in Figure 1 is used with the additional antenna as a scalar sensor. "Without the scalar sensor" means that only the dipoletriad and the looptriad are Figure 1 is used. It can be found from Figure 6a that at each point of SNR, the CRBs with the scalar sensor is about 20 times smaller than the CRBs without the scalar sensor. Thus, the additional scalar sensor increases the arrayaperture significantly and following this, the DOA estimation accuracy is improved. When SNR = 30 dB, Figure 6b plots the CRBs versus the intersensor spacing Δ/λ. It can also be found that when the intersensor spacings increase, the fallingrate of CRBs for the proposed array geometry with the scalar sensor is much faster than its counterpart for the array without the scalar sensor. Again, the additional scalar sensor increases the array aperture and so enhances the angular resolution.
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.
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Keywords
 antenna array mutual coupling
 antenna arrays
 aperture antennas
 array signal processing
 direction of arrival estimation
 polarization.