2D DOA estimation with sparse uniform circular arrays in the presence of mutual coupling
- Julan Xie^{1}Email author,
- Zishu He^{1},
- Huiyong Li^{1} and
- Jun Li^{1}
https://doi.org/10.1186/1687-6180-2011-127
© Xie et al; licensee Springer. 2011
Received: 15 August 2011
Accepted: 8 December 2011
Published: 8 December 2011
Abstract
In this article, we consider the uniform circular arrays (UCAs) with the number of antenna elements insufficient to apply the traditional beamspace-based algorithms, which are labeled as sparse UCAs. For such UCAs, we propose a new hybrid approach for 2D direction-of-arrival (DOA) estimation in the presence of mutual coupling. Using the manifold decomposition technique, we present two new formulations of the steering vector in the presence of mutual coupling for sparse UCAs. Then, we introduce the adaptations to a modified uniform circular array rank reduction algorithm. This leads to an algorithm that is able to estimate the azimuth angle without the exact knowledge of mutual coupling. Next, we use a search-free rooting algorithm which expands the steering into a double Fourier series for each estimated azimuth to obtain the elevation angle estimates. The manifold decomposition technique introduces truncation errors. However, the accuracy of the DOA estimates is strongly affected by these errors when the array has a small number of elements. Therefore, expressions describing the truncation errors in the DOA estimates are derived. This allows us to choose an appropriate truncated degree in the manifold separation transformation to enhance the DOA estimate accuracy. Numerical examples are presented to demonstrate the effectiveness of the proposed method.
Keywords
1. Introduction
The problem of two-dimensional (2D) directions-of-arrival (DOAs) [1–7] estimation (i.e., azimuth and elevation angles) has received increasing attention in a variety of applications, such as radar, mobile communications, sonar, and seismology. In general, a planar array is needed when estimates of source azimuth and elevation are required. Such well-known planar arrays include the two-orthogonal uniform linear array (the L-shaped array) [1], the rectangular array [2], and the uniform circular array (UCA) [3–7]. The UCA is able to provide 360° of coverage in the azimuth plane. Moreover, the UCA has uniform performance regardless of angle of arrival. Thus, UCA attracts more attention than other planar arrays recently. Due to the circular symmetry, the beamspace transformation, based on the phase-mode excitation principle, is usually applied to obtain the desired Vandermode structure for the steering vector in the mode space. This transformation results in the development of several DOA estimation algorithms with low computational cost, such as UCA-RB-MUSIC [3], UCA-ESPRIT [3], and uniform circular array rank reduction (UCA-RARE) [4]. However, all these algorithms for UCAs ignore the mutual coupling effect, which ultimately destroys the underlying model assumptions needed for their efficient implementations. Moreover, all these algorithms, based on the traditional beamspace transformation, require a sufficiently large number of elements to avoid aliasing in the steering vector of the mode space.
In this article, we focus on UCAs with the number of antenna elements insufficient to apply the traditional beamspace-based algorithms. In [8], such UCAs are labeled as sparse UCAs and are allowed to adopt an efficient search-free and robust 1D DOA estimation algorithm. This algorithm is based on a modified beamspace transformation and is called as sparse UCA Root-MUSIC. In this algorithm, all relevant phased modes are able to be incorporated in a polynomial rooting procedure leading to biased free estimates when the number of elements of UCA is small. However, the algorithm in [8] only estimates the azimuth angle without considering the mutual coupling, given a fixed elevation angle. In this article, we estimate the 2D DOAs with such sparse UCAs in the presence of mutual coupling, acquiring both the azimuth and elevation angles estimates via a manifold decomposition technique. The straightforward extension of MUSIC to 2D DOA estimations brings on a 2D search over the MUSIC spectrum and has a high computational cost. In [5, 7], both the proposed algorithms take the mutual coupling into account and employ the UCA-RARE algorithm to estimate the azimuth angle first. With the open-circuit voltages of the antenna elements expanded in spherical mode, a Root-MUSIC algorithm is able to be performed in the elevation space to obtain the elevation estimates in [5]. In [7], a 1D parameter search replaces the implementation of Root-MUSIC algorithm in the elevation space. In the 1D parameter search for elevation estimates, the elevation-dependent mutual coupling effect can efficiently be compensated by the elevation-dependent receiving mutual impedances. However, this step results in higher computational load. Although these two algorithms are applied for the compact UCAs, they could inspire us for sparse UCAs.
In this article, we use the method proposed in [9] to calculate the mutual coupling. In [6, 7, 9], computer simulations have shown that this method can produce more accurate DOA estimation results than the open-circuit voltage method. The experiments in [6, 7] show that the mutual coupling matrix (MCM) depends on the elevation angle for UCAs. Moreover, the simulation results in [6, 7] have shown that the way compensating the mutual coupling with single-elevation-angle receiving-mutual-impedance, computed according to the method in [9], still produces better DOA estimation results than the open-circuit voltage method. In [7], it shows that the variation of the receiving mutual impedances with elevation angle is a process of gradual change. Hence, it is feasible to estimate the elevation angle using mutual coupling compensated with single-elevation-angle receiving-mutual-impedance. In [10], it is shown that any array steering vector can be expanded on a spherical surface to generate an expression containing spherical harmonics, which can be mapped to 2D Fourier basis [11]. In this article, we will extend this expansion to the steering vector in the presence of mutual coupling.
In this article, we propose a new hybrid algorithm for 2D DOA estimation in the presence of mutual coupling for sparse UCAs. Based on the manifold decomposition technique, we will present two new formulations of the steering vector in the presence of mutual coupling for sparse UCAs. One formulation, corresponding to Jacobi-Anger expansion [12], allows applying a modified UCA-RARE algorithm to estimate the azimuth angle without the exact knowledge of mutual coupling and elevation angle. The other formulation, corresponding to Bauer's formula [13], allows executing a Root-MUSIC algorithm in the elevation direction to estimate the elevation angle for each estimated azimuth angle. For sparse UCAs, compared with the original UCA-RARE, the modified UCA-RARE is able to avoid obtaining spurious estimates which only arise from the sparseness of the array elements. Note that the steering vector expansion for estimating the elevation angle in this article differs from that in [5] and has a more universal application [10, 14, 15]. In fact, these two kinds of decomposition techniques applied in this article can be considered as manifold decomposition transformations [11, 16–18]. It is shown that the DOA estimate accuracy usually depends on the truncation error introduced by the transformation [16–19]. Hence, we analyze the truncation errors for sparse UCAs and derive expressions describing the truncation errors in the DOA estimates. We find that the impact of the truncation error on the estimate accuracy of azimuth angle is weaker than it for the elevation angle estimate. Therefore, a method to choose an appropriate truncated degree for the elevation estimates is presented to enhance the estimate accuracy.
The rest of the article is organized as follows. First, the array signal model is presented in Section 2, followed by the description of the manifold decomposition technique in Section 3. Then, the proposed algorithm for sparse UCA is presented in Section 4. The impact of the truncation errors are analyzed in Section 5. Section 6 shows the simulation results. Finally, Section 7 concludes the article.
2. Array signal model
where $\stackrel{\u0303}{\mathbf{A}}\left(\theta ,\varphi \right)=\left[\stackrel{\u0303}{\mathbf{a}}\begin{array}{ccc}\hfill \left({\theta}_{1},{\varphi}_{1}\right)\hfill & \hfill \dots \hfill & \hfill \stackrel{\u0303}{\mathbf{a}}\left({\theta}_{D},{\varphi}_{D}\right)\hfill \end{array}\right]$ is the N × D matrix of the steering vectors, s(t) = [s_{1}(t)... s_{ D } (t)] ^{ T } is the D× 1 signal vector, n(t) = [n_{1}(t)... n_{ N } (t)] ^{ T } is the N× 1 noise vector. The signal vector s(t) and the vector n(t) of the additive and spatially white noise are assumed to be statistically independent and zero-mean.
where γ_{ n } = 2π(n-1)/N is the angular position of the n th element.
3. The manifold decomposition technique
Here, the concepts of the Wavefield modeling for scalar-fields are given. Two expressions for decomposing the steering vector (manifold) of array elements with sparse UCAs are presented. These form the theoretical basis of our algorithm.
where Γ^{ s } represents the so-called sampling matrix and b(θ, ϕ) is the basis functions of the decomposition. In general, the dimension, i.e., the number of basis functions is infinite in order to hold the equality exactly. Therefore, the sampling matrix can be considered as an operator defined as ${\mathbf{\Gamma}}^{s}:\phantom{\rule{2.77695pt}{0ex}}\mathscr{H}\to {\u2102}^{N\times 1}$. The coefficients of the expansion (sampling matrix) map functions defined on $\mathscr{H}$ into the N th-dimensional complex space $\left({\u2102}^{N\times 1}\right)$. Hence, the sampling matrix is a characteristic of the array only.
Since both J_{ m } (kr) and j_{ l } (kr) decay exponentially, we can assume that, for m ≫ kr and l ≫ kr, the higher-order Bessel functions and spherical Bessel functions are negligible. Therefore, the sampling matrix can be truncated by considering a finite number of modes or degrees. Ideally, the resulting truncation error can be made arbitrarily small just by increasing the number of modes or degrees. We assume that the truncated order is M and truncated degree is L for the first and second kinds of decomposition, respectively. The rule to select the truncated order or degree will be discussed in Section 5. In order to distinguish the sampling matrices and basis functions for two kinds of decomposition, let ${\mathbf{\Gamma}}_{1}^{s}$ and b_{1}(θ, ϕ) denote the sampling matrix and basis functions for the first kind of decomposition and ${\mathbf{\Gamma}}_{2}^{s}$ and b_{2}(θ, ϕ) be the sampling matrix and basis functions for the second kind of decomposition, respectively.
where the (L+1)^{2} × (L+1)^{2} diagonal matrix C_{ y } contains the diagonal elements ${\left[{\mathbf{C}}_{y}\right]}_{t,t}=\sqrt{\left(2l+1\right)\left(l-m\right)!/\left(4\pi \left(l+m\right)!\right)}$, t = l^{2}+l+m+1. ${\mathbf{D}}_{y}\left(\varphi \right)\in {\u2102}^{{\left(L+1\right)}^{2}\times {\left(L+1\right)}^{2}}$ is expressed as ${\mathbf{D}}_{y}\left(\varphi \right)=\text{diag}\left(\begin{array}{cccccc}\hfill 1\hfill & \hfill {e}^{-j\varphi}\hfill & \hfill 1\hfill & \hfill {e}^{j\varphi}\hfill & \hfill \dots \hfill & \hfill {e}^{jL\varphi}\hfill \end{array}\right)$. ${\mathbf{Z}}_{e}\in {\u2102}^{{\left(L+1\right)}^{2}\times \left(2L+1\right)}$ is the combination of the selection matrix and the coefficients vectors and is given by ${\mathbf{Z}}_{e}={\left[\begin{array}{ccccc}\hfill {\mathbf{Z}}_{e}^{0\phantom{\rule{2.77695pt}{0ex}}T}\hfill & \hfill {\mathbf{Z}}_{e}^{1\phantom{\rule{2.77695pt}{0ex}}T}\hfill & \hfill {\mathbf{Z}}_{e}^{2\phantom{\rule{2.77695pt}{0ex}}T}\hfill & \hfill \dots \hfill & \hfill {\mathbf{Z}}_{e}^{L\phantom{\rule{2.77695pt}{0ex}}T}\hfill \end{array}\right]}^{T}$, where ${\mathbf{Z}}_{e}^{l}={\left[\begin{array}{ccccc}\hfill {\mathbf{c}}_{l}^{-l\phantom{\rule{2.77695pt}{0ex}}T}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\mathbf{c}}_{l}^{0\phantom{\rule{2.77695pt}{0ex}}T}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\mathbf{c}}_{l}^{l\phantom{\rule{2.77695pt}{0ex}}T}\hfill \end{array}\right]}^{T}\in {\u2102}^{\left(2l+1\right)\times \left(2L+1\right)}$ and ${\mathbf{c}}_{l}^{m}=\left[\begin{array}{ccc}{\mathbf{0}}_{\left(2l+1\right)\times \left(L-l\right)}& {\tilde{\mathbf{c}}}_{l}^{m}{T}^{}& {\mathbf{0}}_{\left(2l+1\right)\times \left(L-l\right)}\end{array}\right]\in {\u2102}^{1\times \left(2L+1\right)}$. ${\stackrel{\u0303}{\mathbf{c}}}_{l}^{m}\in {\u2102}^{\left(2l+1\right)\times 1}$ can be obtained for an arbitrarily l and m using two recurrence expressions. The (2L+1) × 1 vector is described as d(θ) = [e^{-jLθ}··· 1 ··· e^{ jLθ } ] ^{ T } . More details about Equation 14 can be found in [11]. Apparently, Equation 14 is an expansion of 2D Fourier series.
4. The hybrid algorithm to DOA estimation
For the traditional beamspace transform, there is $N>2\u2308kr\u2309$ and the first term in (15) becomes dominant. However, for the beamspace transform applied to the UCA with N = 2K+1, where $K<\u2308kr\u2309$ and $N\le 2\u2308kr\u2309$, the value of the second term may be significant and cannot be neglected because of the contribution of J_{k±qN}(kr sin θ) with orders $K<\left|k\right|\le \u2308kr\u2309$. We also label such UCAs as sparse UCAs [8]. Obviously, the algorithms proposed in [3–5, 7], which are based on the traditional beamspace transform, cannot be employed directly for such UCAs. In this section, we will present a new hybrid algorithm applied to such UCAs. In order to avoid 2D search in MUSIC spectrum, we estimate the DOAs in two steps. Based on the beamspace transformation corresponding to the Jacobi-Anger expansion, first we estimate the azimuth angle using the modified UCA-RARE algorithm, which stems from the original UCA-RARE applied for compact UCA. This algorithm is attractive since it decouples azimuth estimation from elevation estimation and relaxes the assumption of omni-directional element patterns. Then, we perform a Root-MUSIC algorithm to estimate the elevation angle for every estimate azimuth angle using the expansion based on the Bauer's formula.
4.1. The azimuth angle estimation
The vector $\mathrm{\Delta}{\mathbf{a}}_{\text{b}}^{1}\left(\theta ,\varphi \right)$ represents the truncation errors term and its element is the summation of j^{ m }J_{ m } (kr sin θ)e^{ jmϕ } with M < |m| ≤ ∞. This term can be arbitrarily small just by increasing M. The matrix J_{ l } is a (2K+1) × (M-K) matrix consisting of the M-K last columns of the unity matrix I_{(2K+1) × (2K+1)}, whereas J_{ r } is a (2K+1) × (M-K) matrix consisting of the M-K first columns of the unity matrix I_{(2K+1) × (2K+1)}. Π is the M × M anti-diagonal matrix. Note that the expression of H in Equation 22 is restricted to M < 3K+1 [8]. The expression of Equation 21 is equivalent to the one for beamspace manifold in [8]. Now we will extend this transformation to the case considering the mutual coupling.
where $\stackrel{\u0303}{\mathbf{g}}\left(\theta \right)=\mathbf{m}\odot \mathbf{g}\left(\theta \right)$ and m is the first M+1 elements of the diagonal elements of M _{ s }. "⊙" denotes the Hadamard product of vectors. The beamspace steering vector in the presence of mutual coupling for compact UCAs [5, 7] is a special case of the one for sparse UCAs with H = I. Note that the components of $\stackrel{\u0303}{\mathbf{g}}\left(\theta \right)$ in Equation 29 have the same expression form with the ones in [7] and different expression form from the ones in [5].
Observing Equations 29 and 36, the beamspace manifolds for omni-directional elements and directional elements have the same expansion form with different components of g(θ).
However, there may be common roots for det{Ψ} and $\mathrm{det}\left\{\mathbf{I}-{\tilde{\mathbf{E}}}_{\text{s}}^{H}\mathbf{HT}\left(z\right){\mathbf{\Psi}}^{-1}{\mathbf{T}}^{T}\left(1/z\right){\mathbf{H}}^{H}{\tilde{\mathbf{E}}}_{\text{s}}\right\}$, which may be the true roots. In order to avoid eliminating such roots, it is better to use Equation 43 to acquire the azimuth estimate. Notice that the original UCA-RARE algorithm, based on the traditional beamspace transform, is a special case of the modified UCA-RARE algorithm in this study. For traditional beamspace transform, there is H = I and det{Ψ} = 2 ^{ M } . Hence, the roots of Equation 43 are equivalent to these of Equation 39 for original UCA-RARE algorithm.
Similar to the Root-MUSIC roots, RARE roots enjoy the so-called conjugate reciprocity property, i.e., if z_{0} is a root of P_{2}(z), then ${\stackrel{\u0303}{z}}_{0}=1/{z}_{0}^{*}$ is also a root of P_{2}(z). Therefore, there are spurious estimates ϕ_{ i } +π for ϕ_{ i } < π and ϕ_{ i } -π for ϕ_{ i > } π. Although there are still spurious estimates (ϕ_{i}+ ϕ_{ j } )/2 for the case of impinging sources with the same elevation angle (θ_{ i } = θ_{ j } ), we do not plan to eliminate them in order to avoid cancelling the real root at (ϕ_{i}+ ϕ_{ j } )/2 when there is a source exactly at (ϕ_{i}+ ϕ_{ j } )/2. Besides all these spurious estimates, there may be other spurious estimates introduced by the sparseness of the array elements. However, all spurious estimates can be eliminated in the final paired 2D DOA estimation by the elevation estimate in the next step.
4.2. The elevation angle estimation
A specifically designed closed-form algorithm similar to UCA-ESPRIT is proposed in the original UCA-RARE algorithm [4] to obtain the elevation estimates. Although it is a search-free implementation, there are some shortcomings that make it somewhat unsuitable for practical application, which are presented in detail in [7]. Hence, we apply for the Root-MUSIC algorithm via decomposing the steering vector into the double Fourier series to estimate the elevation angle. Note that the steering vector expansion in the presence of mutual coupling for estimating the elevation angle in this article differs from that in [5]. The method in [5], which estimates the DOAs for compact UCAs, is based on the open-circuit voltages of the antenna elements expanded in spherical mode, whereas our method, in which the mutual coupling is calculated by the proposed approach in [9], grounds on the manifold decomposition. In [5], the steering vector is expanded into a limited Fourier series of phase modes by considering a general multiport antenna, carrying a current distribution C(r, φ, z) on the surface S of a cylinder with radius r and height z_{max}. In our proposed method, the steering vector is expanded into a limited Fourier series of phase modes by considering an element on the surface of a unit sphere. Moreover, the truncation degree in [5] is determined by the radius r and height z_{max} together, while the one in our proposed algorithm is only relative to r.
In [7], it shows that the variation of the receiving mutual impedances with elevation angle is a process of gradual change. It means that the receiving mutual impedances do not vary with elevation angle significantly. The simulation results in [6, 7] have shown that estimating the elevation angle with single-elevation-angle receiving-mutual-impedance could achieve error accuracy around 1°. Therefore, it is feasible to estimate the elevation angle using mutual coupling compensated with single-elevation-angle receiving-mutual-impedance. We will estimate an initial elevation angle using the MCM obtained at θ = 45° first. Then we can get a more accurate result with the MCM obtained at the initial estimate.
That shows that the solutions to ϕ_{ i } + π or ϕ_{ i } - π are 2π-θ ∈ [3π/2, 2π] and 2π-(π-θ) = π+θ ∈ [π, 3π/2]. So, this algorithm allows eliminating the spurious estimate ϕ_{ i } +π or ϕ_{ i } -π automatically. Again the spurious estimates (ϕ_{i}+ϕ_{ j } )/2 and (ϕ_{i}+ϕ_{ j } )/2+π can only keep one result if there are sources with the same elevation angle (θ_{ i } = θ_{ j } ). Although all spurious azimuth estimates are considered, we only reserve the paired DOAs (θ_{ i } , ϕ_{ i } ) whose elevation estimates locate at [0, π/2]. The number of such paired estimate (θ_{ i } , ϕ_{ i } ) may be more than D. Hereby, it is necessary to calculate the MUSIC function for every paired estimate (θ_{ i } , ϕ_{ i } ). Only the D smallest values of the MUSIC function are considered as the final estimates for the DOAs (θ, ϕ).
Since the value of the azimuth angle has been estimated, g_{d}(θ, ϕ-γ_{ n } ) is only a function of elevation angle θ and can be labeled as ${g}_{\text{d}}^{n}\left(\theta \right)$. Usually the direction pattern g_{d}(θ) is able to be expressed as a function of cos θ and sin θ. We define w = e^{ jθ } . It is easy to get cos θ = (w+w^{-1})/2 and sin θ = -j(w-w^{-1})/2. In such case, the beamspace steering vector ${\stackrel{\u0303}{\mathbf{a}}}_{\text{b}}\left(\theta ,\varphi \right)$ has the same expansion form as it is in Equation 46 but with different components of the sampling matrix. It could sill be written as a polynomial in w. Hence, the Root-MUSIC algorithm is still able to be performed to estimate the elevation angle.
- (1)
Compute the sample covariance matrix $\hat{\mathbf{R}}=\left(1/P\right){\sum}_{p=1}^{P}\mathbf{x}\left(p\right){\mathbf{x}}^{H}\left(p\right)$ by averaging over P data snapshots. Compute the beamspace covariance matrix ${\hat{\mathbf{R}}}_{b}={\mathbf{W}}_{K}^{H}\hat{\mathbf{R}}{\mathbf{W}}_{K}$.
- (2)
Perform the eigenvalue decomposition of ${\hat{\mathbf{R}}}_{b}$. Form the matrix ${\hat{\mathbf{E}}}_{s}$ and ${\hat{\mathbf{E}}}_{n}$, which spans the estimated signal subspace and the noise subspace, respectively.
- (3)
Obtain the azimuth angle estimates with Equation 43. All spurious azimuth estimates are reserved.
- (4)
For every reserved azimuth estimate, perform the Root-MUSIC with the MCM obtained at θ = 45° to find an initial elevation estimates. Since the location of the real elevation angle is θ ∈ (0, π/2], except for the case that there is a source exactly at ϕ_{ i } + π or ϕ_{ i } -π, all spurious estimates ϕ_{ i } +π or ϕ_{ i } -π can be eliminated automatically (see Equation 49). Then, we perform the Root-MUSIC with the MCM obtained at initial estimate to get a more accurate estimate.
- (5)
Calculate the MUSIC function for every paired estimate (θ_{ i } , ϕ_{ i } ). Take the paired estimates (θ, ϕ) corresponding to the D smallest values of the MUSIC function as the final estimate.
5. The impact of the truncation errors on the estimation accuracy
As discussed in Sections 3 and 4, the manifold decomposition will introduce truncation errors. Here, a first-order approximation of the bias based on manifold decomposition is derived for sparse UCAs, and a thumb rule to choose the truncate degree is presented.
The receiving mutual impendence at different elevation angles of impinging sources
Elevation | Z _{12} | Z _{13} | Z _{14} | Z _{15} | Z _{16} |
---|---|---|---|---|---|
10° | i*11.7873+4.5314 | -2.4943-i*8.9404 | i*6.3951+1.4430 | -i*4.3664-1.1790 | i*3.3245+0.7546 |
20° | i*11.8023+4.4814 | -2.5015-i*8.9513 | i*6.3998+1.4395 | -i*4.3638-1.1815 | i*3.2703+0.7552 |
30° | i*11.8237+4.4157 | -2.5233-i*8.9649 | i*6.4139+1.4288 | -i*4.3570-1.1989 | i*3.2342+0.7563 |
40° | i*11.8401+4.3538 | -2.5602-i*8.9711 | i*6.4369+1.4101 | -i*4.3487-1.2203 | i*3.2053+0.7578 |
50° | i*11.8519+4.3054 | -2.6118-i*8.9714 | i*6.4659+1.3819 | -i*4.3375-1.2510 | i*3.1950+0.7586 |
60° | i*11.8583+4.2649 | -2.6728-i*8.9617 | i*6.4929+1.3353 | -i*4.3190-1.3035 | i*3.1834+0.7595 |
70° | i*11.8629+4.2227 | -2.7281-i*8.9483 | i*6.5087+1.3001 | -i*4.3032-1.3261 | i*3.1755+0.7609 |
80° | i*11.8663+4.1837 | -2.7606-i*8.9472 | i*6.5152+1.2800 | -i*4.2933-1.3392 | i*3.1676+0.7618 |
90° | i*11.8654+4.1721 | -2.7682-i*8.9458 | i*6.5202+1.2780 | -i*4.2900-1.3435 | i*3.1557+0.7621 |
5.1. Analysis of the bias in the azimuth angle estimation
5.2. Analysis of the bias in the elevation angle estimation
where $j\text{\_}max=max\left\{\left|{j}_{l}\left(kr\right)\right|,l\le 2\u2308kr\u2309\right\}$ and the predetermined ε is related to the truncated accuracy.
6. Simulations
7. Conclusions
Several algorithms for DOA estimation with UCAs are based on the traditional beamspace transform, which requires a sufficiently large number of elements to avoid aliasing in the steering vector of the mode space. Sometimes there may be a smaller number of antenna elements for application. We propose a new approach to estimate 2D DOAs for such UCAs. Two kinds of manifold decompositions are applied as the foundation of the proposed algorithm. In the first step, a modified sparse UCA-RARE is performed for the azimuth estimates. This step can be realized without the exact knowledge of elevation angle. It is proved by means of the Jacobi-Anger expansion (a decomposition of the element manifold into phase modes) that the sparse UCA-RARE is still applicable with a limited number of array elements. In the second step, the Root-MUSIC algorithm is used to obtain the elevation estimates via decomposing the manifold with Bauer's formula (an expansion of the array manifold into a double Fourier series). The influence of the truncation errors on the DOA estimate accuracy is analyzed and a method to choose the truncated degree for the elevation estimates is presented. Simulation results show that the proposed algorithm for sparse UCA can obtain good azimuth angle and elevation angle estimate results. The next challenge is to find a computational efficient method to handle the sparse UCAs with much wider inter-element spacing.
Appendix
Declarations
Acknowledgements
This work was supported by the University Basic Research Fund, P. R. China (nos. ZYGX2010J015 and ZYGX2009J015).
Authors’ Affiliations
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