- Research
- Open Access
Decoupled 2D direction-of-arrival estimation based on sparse signal reconstruction
- Feng Wang^{1}Email author,
- Xiaowei Cui^{1},
- Mingquan Lu^{1} and
- Zhenming Feng^{1}
https://doi.org/10.1186/s13634-015-0198-x
© Wang et al.; licensee Springer. 2015
- Received: 26 August 2014
- Accepted: 17 January 2015
- Published: 29 January 2015
Abstract
A new two-dimensional direction-of-arrival estimation algorithm called 2D- l _{1}-singular value decomposition (SVD) and its improved version called enhanced-2D- l _{1}-SVD are proposed in this paper. They are designed for rectangular arrays and can also be extended to rectangular arrays with faulty or missing elements. The key idea is to represent direction-of-arrival with two decoupled angles and then successively estimate them. Therefore, two-dimensional direction finding can be achieved by applying several times of one-dimensional sparse reconstruction-based direction finding methods instead of directly extending them to two-dimensional situation. Performance analysis and simulation results reveal that the proposed method has a much lower computational complexity and a similar statistical performance compared with the well-known l _{1}-SVD algorithm, which has several advantages over conventional direction finding techniques due to the application of sparse signal reconstruction. Moreover, 2D- l _{1}-SVD has better robustness to the assumed number of sources over l _{1}-SVD.
Keywords
- Antenna arrays
- Direction-of-arrival estimation
- Sparse signal reconstruction
1 Introduction
Direction-of-arrival (DOA) estimation has been playing a significant role in many applications such as radar and wireless communication, and it has been well studied in literature [1,2]. Conventional DOA estimation algorithms can be classified into three broad categories: beamforming [3], subspace-based methods [4,5], and maximum likelihood methods [6]. In the last decade, DOA estimation algorithms based on sparse signal reconstruction (SSR) were proposed. They enforce sparsity on the spatial spectrum and pose source localization as an over-complete basis representation problem. Many of them such as l _{1}-singular value decomposition (SVD) [7] and JLZA-DOA [8] work directly on the input data while others such as SPICE [9] and SpSF [10] work on the covariance matrix. DOA estimation algorithms based on SSR have several advantages over conventional methods, including increased resolution, better robustness to noise, limitations in data quantity, and correlated sources, as well as not requiring an accurate initialization [7]. Improvements like [11] which are based on weighted l _{1} minimization can get a better performance.
Although one-dimensional (1D) DOA estimation has been widely investigated, two-dimensional (2D) DOA estimation is of greater practical importance. However, most existing 2D DOA estimation methods have met problems of high arithmetic complexity and pair-matching due to a more complicated array manifold [12]. What is more, 2D methods extended from 1D conventional algorithms may still have shortcomings such as requirement of sufficient samples and performance degradation in the presence of correlated sources. In [13], the authors proposed a 2D algorithm whose key idea is to successively apply several times of 1D multiple signal classification (MUSIC) [4] in tree structure. It has a much lower complexity than 2D MUSIC but is limited to uniform rectangular arrays (URAs) and needs extra processes to deal with coherent sources. Sparse methods can also be directly extended to 2D situation, but they will significantly increase the dimension of over-complete basis or the dictionary [14]. A fast orthogonal matching pursuit (OMP) method for 2D angle estimation in multiple-input multiple-output (MIMO) radar has been proposed in [15], which decomposes the 2D dictionary into two sub-dictionaries. However, the bistatic MIMO radar considered in [15] consists of a transmit array and a receive array, and the signal at the receiver is analyzed to estimate the transmit angle and the receive angle. So its model differs from 2D DOA estimation. Some 2D DOA estimation algorithms using L-shaped array have also been proposed [16,17]. They may have a low computational complexity and need less antenna elements than those algorithms using rectangular arrays. However, these methods are based on the second-order statistics or the cross-correlation matrix of the received data, so their performances may deteriorate in the presence of correlated sources or insufficient snapshots.
In this paper, a 2D DOA estimation method based on SSR for rectangular arrays called 2D- l _{1}-SVD is proposed. The key idea is that 2D DOA estimation, which is usually implemented by estimating azimuth and elevation angles jointly, can be accomplished by successively solving several 1D direction finding problems. It is illustrated in this paper that 2D- l _{1}-SVD reduces the computational load due to the successive parameter estimation and has a similar performance with the direct extension of l _{1}-SVD [7] to 2D situation. What is more, the other SSR based DOA estimation methods that work directly on the input data, including the weighted version of l _{1}-SVD [11] or JLZA-DOA [8], can also be extended to 2D situations similar with a much lower complexity than direct extensions. Moreover, it is possible that the array is conformal to follow some prescribed shape or a few elements fail to work in practical applications, and 2D- l _{1}-SVD also works properly for these nonrectangular applications. Therefore, 2D- l _{1}-SVD keeps robustness to non-uniform array manifolds. It is illustrated in both theoretical analysis and simulation results that the proposed algorithm performs properly and effectively.
The rest of this paper is organized as follows. Section 1 addresses the problem formulation. Section 1 presents the proposed algorithm. Section 1 investigates the performance of the proposed method and Section 1 demonstrates the simulation results. Section 1 concludes the paper.
In this paper, we use a bold small letter to represent a vector and a bold capital letter to represent a matrix.
2 Problem formulation
2.1 Array configuration
so they are denoted as (x _{ m },y _{ n }) and the element at the mth row and nth column is indexed as (m,n). The rectangular array may be non-uniform.
In order to deal with irregular arrays mentioned above, consider the rectangular array that consists of M×N antenna elements in Equation 1. And it is assumed that there are D invalid (missing or faulty) sensors indexed as (m _{ d },n _{ d }), d=1,⋯,D, where m _{ d } and n _{ d } are the row and column number of the dth invalid sensor, correspondingly. The location of all invalid sensors is supposed to be known a priori in this paper.
2.2 Representation of 2D DOA
where the exponential term is written as the sum of factors related to α and β, respectively, hence the two parameters to be estimated can be decoupled from each other.
where \(\boldsymbol {n}\left (k \right) = [\!{n_{1,1}}\left (k \right), \cdots,{n_{M,1}}\left (k \right), \cdots,{n_{1,N}}\left (k \right), \cdots,{n_{M,N}}\left (k \right)]^{\mathrm {T}} \in {\mathbb {C}^{MN \times 1}}\), and \(\boldsymbol {a}\left ({{\alpha _{p}},{\beta _{p}}} \right) = \left [{a_{1,1}}\left ({{\alpha _{p}},{\beta _{p}}} \right),\! \cdots \!,{a{M,1}}\left ({{\alpha _{p}},{\beta _{p}}} \right), \cdots,{a_{1,N}}\left ({{\alpha _{p}},{\beta _{p}}} \right), \cdots, {a_{M,N}}\left ({{\alpha _{p}},{\beta _{p}}} \right)\right ]^{\mathrm {T}} \in {\mathbb {C}^{MN \times 1}}\) is the manifold vector.
where \(\boldsymbol {Y}\left (k \right) \buildrel \Delta \over = \left [ {\begin {array}{*{20}{c}} {{y_{1,1}}\left (k \right)}& \cdots &{{y_{1,N}}\left (k \right)}\\ \vdots & \ddots & \vdots \\ {{y_{M,1}}\left (k \right)}& \cdots &{{y_{M,N}}\left (k \right)} \end {array}} \right ] \in {\mathbb {C}^{M \times N}}\) is the matrix form of y(k), \({\boldsymbol {A}^{L}} \buildrel \Delta \over = \left [ {{\boldsymbol {a}^{L}}\left ({{\alpha _{1}}} \right), \cdots,{\boldsymbol {a}^{L}}\left ({{\alpha _{P}}} \right)} \right ] \in {\mathbb {C}^{M \times P}}\) denotes manifold matrix whose columns are comprised of P manifold vectors of the linear sub-array on the x-axis, \(\boldsymbol {X}\left (k \right) \buildrel \Delta \over = \left [ {\begin {array}{*{20}{c}} {{s_{1}}\left (k \right){{\left ({{\boldsymbol {b}^{L}}\left ({{\beta _{1}}} \right)} \right)}^{\mathrm {T}}}}\\ \vdots \\ {{s_{P}}\left (k \right){{\left ({{\boldsymbol {b}^{L}}\left ({{\beta _{P}}} \right)} \right)}^{\mathrm {T}}}} \end {array}} \right ] \in {\mathbb {C}^{P \times N}}\)denotes the signal matrix in the decoupled signal model, and \(\boldsymbol {N}\left (k \right) \buildrel \Delta \over = \left [ {\begin {array}{*{20}{c}} {{n_{1,1}}\left (k \right)}& \cdots &{{n_{1,N}}\left (k \right)}\\ \vdots & \ddots & \vdots \\ {{n_{M,1}}\left (k \right)}& \cdots &{{n_{M,N}}\left (k \right)} \end {array}} \right ] \in {\mathbb {C}^{M \times N}}\) is the matrix form of n(k).
From Equation 7, it can be observed that the information of α and β is in the column and row spaces of Y(k), respectively. The manifold matrix A ^{ L } is decided by α and is not related to β, while the signal matrix X(k) is determined by β. Equation 7 can also be explained in terms of the rectangular manifold. Alpha can be estimated by analyzing the samples of the sub-array that lies on the x-axis and consists of sensors at {(x _{ m },y _{1})},m=1,⋯,M. The linear sub-arrays that are parallel to x-axis in the rectangular array share the same manifold when only the estimation of α is taken into account. Therefore, in order to estimate α, each snapshot received by the rectangular array can be regarded as N correlated snapshots of the linear sub-array that lies on the x-axis. Then, 2D DOA can be estimated by solving two 1D DOA estimation problems successively instead of directly estimating 2D DOA. Since α is decoupled from β, we can also estimate β firstly and then estimate α. What is more, we will show in Section 1 that integrating the results of these two problems helps to improve the performance.
where \(\boldsymbol {Y}^{f}=\left [ {\begin {array}{*{20}{c}} {{y_{1,1}}\left (k \right)}& \cdots & \cdots & \cdots &{{y_{1,N}}\left (k \right)}\\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots &0& \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ {{y_{M,1}}\left (k \right)}& \cdots & \cdots & \cdots &{{y_{M,N}}\left (k \right)} \end {array}} \right ]\).
In Equation 9, the received signals at the elements indexed (m _{ d },n _{ d }),d=1,⋯,D are disregarded and set to zeros while the noise signals at these sensors change into \(n_{{m_{d}},{n_{d}}}^{f}\left (k \right) = {n_{{m_{d}},{n_{d}}}}\left (k \right) - {y_{{m_{d}},{n_{d}}}}\left (k \right)\). The superscript f denotes ‘faulty.’ The case of faulty sensors can be processed in the above form. If there are D invalid elements, then no restriction is placed on the range of \({y_{{m_{d}},{n_{d}}}}\left (k \right),d = 1, \cdots,D\) since there are no valid samples at these invalid sensors. As a result, \(n_{{m_{d}},{n_{d}}}^{f}\left (k \right),d = 1, \cdots,D\) in Equation 9 is unconstrained. In this way, the complex amplitudes of the sensors at a faulty rectangular array can still be written in the form of the decoupled signal model (Equation 9). However, it should be taken into account that the noise signals at the faulty sensors are unconstrained and no longer distribute as noise signals at valid sensors. As a result, when the decoupled signal model is exploited, it should be noticed that the received signals at the faulty elements are set to be zero and the noise signals at these faulty elements are totally unknown and do not provide any additional information.
3 Proposed algorithm
Based on the above decoupled model, 2D DOA estimation can be achieved by two steps. Firstly, solve a 1D direction finding problem to estimate the first parameter α and get the information about the second parameter β. This process can be accomplished by 1D DOA estimator directly working on the input data instead of using the covariance estimator, and information about the second parameter (signal matrix in Equation 7) can be obtained as well. Secondly, solve another several 1D direction finding problems to estimate β based on the rows extracted from the signal matrix and the corresponding α can be obtained from the row number.
DOA estimation methods based on SSR have been found to have several advantages over conventional direction finding methods and many of them work directly on the input data. Here, we pick l _{1}-SVD [7] as our 1D DOA estimation algorithm and propose a new 2D DOA estimation method called 2D- l _{1}-SVD. Exploiting the decoupled signal model, 2D- l _{1}-SVD has a much lower computational load than l _{1}-SVD due to the successive estimation of 2D DOA parameters. Moreover, an improved version called enhanced-2D- l _{1}-SVD is also proposed in this section to deal with multiple sources that are close to each other in α or β domain but well separated in the other domain.
Obviously, l _{1}-SVD does not need specific array shape. However, rectangular arrays are more suitable to exploit the decoupled signal model. As a result, although techniques such as manifold separation [18] can be used for modeling the steering vector of antenna arrays with practical interest with arbitrary geometry, we only consider rectangular arrays and faulty rectangular arrays in this paper. On the other hand, it is obvious that JLZA-DOA [8] and the other sparse reconstruction based methods that work directly on the input data can also be extended to 2D situation using the aforementioned decoupled signal model similarly.
3.1 2D- l _{1}-SVD algorithm
Notations used in section 1
Notation | Size | Definition |
---|---|---|
MN | 1×1 | The number of all the elements in the rectangular array |
D | 1×1 | The number of faulty elements in the rectangular array |
P _{0} | 1×1 | The number of incoming sources |
P | 1×1 | The number of different α |
T | 1×1 | The number of snapshots |
K | 1×1 | The assumed number of incoming sources |
Y ^{ant} | M N×T | The data matrix received at the array |
A | M N×P _{0} | The manifold matrix of the rectangular array |
S | P _{0}×T | The signal matrix of incoming sources |
N ^{ant} | M N×T | The noise-signal matrix at the array |
Y ^{sg} | M N×K | The reduced matrix containing most of the signal power |
S ^{sg} | P _{0}×K | The reduced signal matrix |
N ^{sg} | M N×K | The reduced noise signal matrix |
Y | M×N K | The rearrangement of Y ^{ s g } |
A ^{ L } | M×P | The manifold matrix of the linear sub-array on x-axis |
X | P×N K | The signal matrix in the decoupled model |
N | M×N K | The rearrangement of N ^{ s g } |
\(\boldsymbol {\tilde {A}}^{L}\) | M×K _{ α } | The dictionary of manifold vector of the linear sub-array on x-axis |
\(\boldsymbol {\tilde {X}}\) | K _{ α }×N K | The sparse signal matrix in the SSR in the α domain |
\(\boldsymbol {\tilde {B}}^{L}\) | N×K _{ β } | The dictionary of manifold vector of the linear sub-array on y-axis |
\(\boldsymbol {\tilde {S}}_{i}\) | K _{ β }×K | The sparse signal matrix in the SSR in the β domain |
where \(\boldsymbol {A} = \left [ \boldsymbol {a}\left ({{\alpha _{1}},{\beta _{1,1}}} \right), \cdots, \boldsymbol {a}\left ({{\alpha _{1}},{\beta _{1,{f_{1}}}}} \right), \cdots,\boldsymbol {a}\left ({{\alpha _{P}},{\beta _{P,1}}} \right)\!, \cdots,\boldsymbol {a}\left ({{\alpha _{P}},{\beta _{P,{f_{P}}}}} \right) \right ] \in {\mathbb {C}^{MN \times {P_{0}}}}\) is the manifold matrix of P _{0} incoming signals, \(\boldsymbol {S} = \left [ {\boldsymbol {s}\left (1 \right), \cdots,\boldsymbol {s}\left (T \right)} \right ] \in {\mathbb {C}^{P_{0} \times T}}\) is the signal matrix of the incoming signals and \({\boldsymbol {N}^{\text {ant}}} = \left [ {{\boldsymbol {n}^{\mathrm {\,ant}}}\left (1 \right), \cdots,{\boldsymbol {n}^{\mathrm {\,ant}}}\left (T \right)} \right ] \in {\mathbb {C}^{MN \times T}}\) is the noisesignal matrix at the sensor array. Here, \(\boldsymbol {s}\left (k \right) = \;[\!{s_{1,1}}\left (k \right),\! \cdots \!,{s_{1,{f_{1}}}}\left (k \right), \cdots,{s_{P,1}}\left (k \right), \cdots,{s_{P,{f_{P}}}}\left (k \right) ]^{\mathrm {T}} \in {\mathbb {C}^{P_{0} \times 1}}\) and \({\boldsymbol {n}^{\text {ant}}}\left (k \right) = [ n_{1,1}^{\text {ant}}\left (k \right), \cdots,n_{1,N}^{\text {ant}}(k), \cdots, n_{M,1}^{\text {ant}}(k), \cdots,n_{M,N}^{\text {ant}}(k) ]^{\mathrm {T}} \in {\mathbb {C}^{MN \times 1}}\) denote the incoming signals and the noise signals sampled at time instant k (1≤k≤T), respectively. As mentioned before, faulty arrays can be treated in a similar way as a complete rectangular array. For rectangular arrays with faulty or missing elements, the manifold vector a(α _{ p },β _{ p,i }) (1≤i≤f _{ p }) refers to the manifold vector of the complete rectangular array and the data matrix Y ^{ant} satisfies \({y_{{m_{d}},{n_{d}}}}\left (k \right) = 0\), d=1,⋯,D, k=1,⋯,T. Also, it should be noticed that noise signals at the faulty sensors \({n_{{m_{d}},{n_{d}}}}\left (k \right)\) are unconstrained.
It is easy to see that \(y_{{m_{d}},{n_{d}}}^{\text {sg}}\left (k \right) = 0\) and \(n_{{m_{d}},{n_{d}}}^{\text {sg}}\left (k \right)\) is unconstrained for d=1,⋯,D, k=1,⋯,T.Step 2: Estimate α from multiple measurement vectors (MMV)
where \({{\tilde {f}}_{{k_{\alpha }}}},{k_{\alpha }} = 1, \cdots,{K_{\alpha }}\) are positive integers and \({{\tilde {f}}_{{k_{\alpha }}}} = f_{p}\) if \({{\tilde {\alpha }}_{{k_{\alpha }}}} = \alpha _{p}\). The source signal \(\tilde {s}_{{k_{\alpha }},i}^{\text {sg}}\left (k \right)\) is nonzero and equal to \(s_{p,i}^{\text {sg}}(k)\) if \({{\tilde {\alpha }}_{{k_{\alpha } }}} = \alpha _{p}\) and zero otherwise. So each column in \(\boldsymbol {\tilde {X}}\) is sparse.
where the l _{1}-term enforces sparsity of the representation. \({{\boldsymbol {\tilde {x}}}^{{l_{2}}}} \buildrel \Delta \over = \left ({{{\left \| {{{\boldsymbol {\tilde {x}}}_{1}}} \right \|}_{2}}, \cdots,{{\left \| {{{\boldsymbol {\tilde {x}}}_{{K_{\alpha } }}}} \right \|}_{2}}} \right) \in {\mathbb {C}^{1 \times {K_{\alpha } }}}\) and ∥·∥_{ F } denotes the Frobenius norm which is defined as \(\left \| {\boldsymbol {Y} - {{\boldsymbol {\tilde {A}}}^{L}}\boldsymbol {\tilde {X}}} \right \|_{F}^{2} = \left \| {{\text {vec}}\left ({\boldsymbol {Y} - {{\boldsymbol {\tilde {A}}}^{L}}\boldsymbol {\tilde {X}}} \right)} \right \|_{2}^{2}\). Let \(\boldsymbol {\hat {X}} \in {\mathbb {C}^{{K_{\alpha } } \times NK}}\) denote the result of the above MMV problem. Then, the pseudo spectrum of α is obtained by calculating the l _{2}-norm of each row in the signal matrix \(\boldsymbol {\hat {X}}\), i.e., the amplitude of the pseudo spectrum at \({{\tilde {\alpha }}_{{k_{\alpha }}}}\) is \({\left \| {{{\boldsymbol {\hat {x}}}_{{k_{\alpha } }}}} \right \|_{2}}\).
In the MMV problems based on l _{1}-norm minimization mentioned above, l _{1}-term enforces sparsity while l _{2}-term forces the residual to be small. The residual specifies how much noise we wish to allow. For rectangular arrays with invalid elements, the noise signals or residuals of invalid elements are unknown while the residuals of valid elements are forced to be small. Hence, the constrained condition of the l _{2}-term should be revised, taking into consideration of invalid elements. On the other hand, the spatial sparsity still exists, so l _{1}-term remains unchanged.
where the l _{1}-term enforces sparsity of the representation. We will specify the sufficient condition of correct recovery of the signal matrix \(\boldsymbol {\hat X}\) in Section 1. Then, the pseudo spectrum of α is obtained by calculating the l _{2} norm of each row in the signal matrix \(\boldsymbol {\hat X}\) similarly.
Just like [7], with the knowledge of the distribution of noise, we can find a confidence interval for \(\left \| {{\boldsymbol {N}}} \right \|_{F}^{2}\), then use its upper value for \({\sigma _{1}^{2}}\).Step 3: Estimate β based on the signal matrix
where \({{\boldsymbol {\tilde {S}}}_{{p}}} = \left [ {\begin {array}{*{20}{c}} {\boldsymbol {\tilde {s}}_{p,1}^{\mathrm {T}}}\\ \vdots \\ {\boldsymbol {\tilde {s}}_{p,{K_{\beta }}}^{\mathrm {T}}} \end {array}} \right ] \in {\mathbb {C}^{{K_{\beta }} \times K}}\) and \({\boldsymbol {\tilde {s}}_{p}^{{l_{2}}}}\) denotes the vector whose k _{ β }th element is the l _{2}-norm of the k _{ β }th row in \({{\boldsymbol {\tilde {S}}}_{p}}\). Then, the pseudo spectrum of β is \(\|{\boldsymbol {\tilde {s}}_{p}^{{l_{2}}}}\|\) and the estimation of β is accomplished.
Although the distribution of the elements in \({{\boldsymbol {\hat {X}}}_{i}}\) is difficult to estimate, \({\sigma _{2}^{2}}\) can be determined by the upper value of \(K\left \| {\left ({n_{m1}^{\text {sg}}\left (k \right), \cdots,n_{\textit {mN}}^{\text {sg}}\left (k \right)} \right)} \right \|_{2}^{2}\) empirically.
Step 4: Calculate (θ,φ) based on (α,β) if needed. The effects of invalid elements on the degree of freedom will be studied in Section 1.
3.2 Enhanced 2D- l _{1}-SVD algorithm
where \(\boldsymbol {A} \buildrel \Delta \over = \left [ {\boldsymbol {a}({{{\tilde \theta }_{1}}}), \cdots, \boldsymbol {a}({{{\tilde \theta }_{{K_{\theta } }}}})} \right ]\) and \(\left \{ {{{\tilde \theta }_{1}}, \cdots, {{\tilde \theta }_{{K_{\theta } }}}} \right \}\) are all potential θ. a(θ) is the steering vector of a linear array. It is assumed that there are only two sources, which are from θ _{1} and θ _{2}, impinging on the linear array. Obviously, the sparsity condition is satisfied with proper \(\left \{ {{{\tilde \theta }_{1}}, \cdots, {{\tilde \theta }_{{K_{\theta }}}}} \right \}\) and the sources can be well resolved if they are not too close to each other. However, there is notable bias when two sources are too close to each other although the sparsity condition is still satisfied [14].
The problem still exists when l _{1}-SVD is extended to 2D cases. If two sources, which are from (α _{1},β _{1}) and (α _{2},β _{2}), are close to each other in both α and β domains so a(α _{1},β _{1}) is quite similar to a(α _{2},β _{2}), then l _{1}-SVD will get biased results. However, l _{1}-SVD can work properly and get nearly unbiased results when distinguished signals are close to each other in α domain but well separated to each other in β domain while 2D- l _{1}-SVD gets biased results in the α domain. An enhanced 2D- l _{1}-SVD is proposed to solve this problem, and it still has a much lower complexity than l _{1}-SVD.
Although 2D- l _{1}-SVD- α cannot accurately estimate α when there exist multiple sources close to each other in α domain, it can provide precise β-DOA if the sources are not too close to each other in β domain since the information of β is in the column of the input data matrix. Similar results can be obtained if β is firstly estimated using 2D- l _{1}-SVD- β. So the enhanced algorithm is designed to combine the estimation results in conjunction with a selection strategy.
The selection strategy is based on the condition that any two sources are not too close to each other in both α and β domains. The main idea is to make pairs of the estimation results of 2D- l _{1}-SVD- α and 2D- l _{1}-SVD- β so that in each pair, the DOA estimated by 2D- l _{1}-SVD- α and 2D- l _{1}-SVD- β corresponds to the same source. In Figure 4a, sources 1 and 2 are close to each other in α domain, so \({{\hat \alpha }_{1}}\) is inaccurate while \({{\hat \beta }_{1,2}}\) and \({{\hat \beta }_{1,1}}\) are precise. In Figure 4b, we get three different β, in which \({{\hat \beta }_{3}}\) is closest to \({{\hat \beta }_{1,2}}\) and \({{\hat \beta }_{1}}\) is closest to \({{\hat \beta }_{1,1}}\). Consider the estimation of α corresponding to \({{\hat \beta }_{3}}\) and we find that \({{\hat \alpha }_{3,1}}\) is the closest to \({{\hat \alpha }_{1}}\). Therefore, \(({{{\hat \alpha }_{1}},{{\hat \beta }_{1,2}}})\) and \(({{{\hat \alpha }_{3,1}},{{\hat \beta }_{3}}})\) are considered to be a pair. Similarly, we have the other four pairs. For pairs corresponding to sources 1, 2, 4, and 5, 2D- l _{1}-SVD- α has got more precise estimation of β and 2D- l _{1}-SVD- β has got more precise estimation of α. Therefore, primary results of 2D- l _{1}-SVD- α and 2D- l _{1}-SVD- β are corrected, as illustrated in Figure 4c. It should be noticed that the pairing process in enhanced-2D- l _{1}-SVD is quite different from the pair-matching in [16]. Given the set of elevation angles and azimuth angles, which is {θ _{1},⋯,θ _{ P }} and {φ _{1},⋯,φ _{ P }}, respectively, the conventional pair-matching process in [16] choose P final estimates from P ^{2} possible pairs by making use of the statistical analysis of the received data. In enhanced-2D- l _{1}-SVD, final DOA estimates are extracted from two existed set of 2D DOA without using information of the received data.
- Step 1:
Complete the 2D- l _{1}-SVD- α.
Estimate α firstly and then estimate β. There are P different α and the estimation results are denoted as \({\Omega _{\alpha } } = \{ ({{{\hat \alpha }_{1}},{{\hat \beta }_{1,1}}}), \cdots,({{{\hat \alpha }_{1}},{{\hat \beta }_{1,{f_{1}}}}}), \cdots,\) \(({{{\hat \alpha }_{P}},{{\hat \beta }_{P,1}}}), \cdots,({{{\hat \alpha }_{P}},{{\hat \beta }_{P,{f_{P}}}}}) \}\). For \({{\hat \alpha }_{p}}\), there are f _{ p } different β.
- Step 2:
Complete the 2D- l _{1}-SVD- β.
Estimate β firstly and then estimate α. There are Q different β and the estimation results are denoted as \({\Omega _{\beta } } = \{ ({{{\hat \alpha }_{1,1}},{{\hat \beta }_{1}}}), \cdots,({{{\hat \alpha }_{1,{g_{1}}}},{{\hat \beta }_{1}}}), \cdots,({{{\hat \alpha }_{Q,1}},{{\hat \beta }_{Q}}}), \cdots,({{{\hat \alpha }_{Q,{g_{Q}}}},{{\hat \beta }_{Q}}}) \}\). For \({{\hat \beta }_{q}}\), there are g _{ q } different α.
- Step 3:
Selection strategy.
For each \(({{{\hat \alpha }_{p}},{{\hat \beta }_{\textit {pq}}}}) \in {\Omega _{\alpha } }\), 1≤q≤g _{ p }, find \({{\hat \beta }_{q}}\) in \(\{ {{{\hat \beta }_{1}}, \cdots,{{\hat \beta }_{Q}}} \}\) that is closest to \({{\hat \beta }_{p,q}}\). The estimation of α corresponding to \({{\hat \beta }_{q}}\) is \(\{ {{{\hat \alpha }_{q,1}}, \cdots,{{\hat \alpha }_{q,{f_{q}}}}} \}\). Find \({{\hat \alpha }_{\textit {qk}}}\) in \(\{ {{{\hat \alpha }_{q,1}}, \cdots,{{\hat \alpha }_{q,{f_{q}}}}} \}\) that is closest to \({{\hat \alpha }_{p}}\). Then, \(({{{\hat \alpha }_{p}},{{\hat \beta }_{p,q}}})\) and \(({{{\hat \alpha }_{q,k}},{{\hat \beta }_{q}}})\) are considered to be a pair.
If \(\sqrt {{{| {{{\hat \alpha }_{p}} - {{\hat \alpha }_{q,k}}} |}^{2}} + {{| {{{\hat \beta }_{p,q}} - {{\hat \beta }_{q}}} |}^{2}}} > \delta \), then, \(({{{\hat \alpha }_{q,k}},{{\hat \beta }_{p,q}}})\) was considered to be a false peak. If \(\sqrt {{{| {{{\hat \alpha }_{p}} - {{\hat \alpha }_{q,k}}} |}^{2}} + {{| {{{\hat \beta }_{p,q}} - {{\hat \beta }_{q}}} |}^{2}}} \le \delta \), then \(({{{\hat \alpha }_{q,k}},{{\hat \beta }_{p,q}}})\) is the correct DOA.
4 Performance
Using l _{1}-SVD as 1D DOA estimators during successive estimation, 2D- l _{1}-SVD has several advantages over conventional 2D methods, including high resolution, robustness to the number of snapshots, low SNR, and coherent sources due to the use of sparse reconstruction [7]. In this section, we compare the performance of 2D- l _{1}-SVD and l _{1}-SVD and study about the effect of invalid elements. Firstly, we demonstrate that the computational complexity of 2D- l _{1}-SVD is much lower than that of l _{1}-SVD. Secondly, we study about the degree of freedom of 2D- l _{1}-SVD. Then, we will show that 2D- l _{1}-SVD keeps better robustness over the assumed number of sources in the presence of multiple sources. And finally, we investigate the number of sources that 2D- l _{1}-SVD can process with both rectangular arrays and faulty rectangular arrays.
4.1 Computational complexity
Arithmetic complexity of 2D- l _{ 1 } -SVD and l _{ 1 } -SVD
Methods | Arithmetic complexity | ||
---|---|---|---|
l _{1}-SVD | \(O\left ({{K^{3}}K_{\alpha }^{3}K_{\beta }^{3}} \right)\) | ||
2D- l _{1}-SVD | α | \(O\left ({{K^{3}}{N^{3}}K_{\alpha }^{3}} \right)\) | \(O\left ({{K^{3}}{N^{3}}K_{\alpha }^{3}} \right) + {P}O\left ({{K^{3}}K_{\beta }^{3}} \right)\) |
β | \(O\left ({{K^{3}}K_{\beta }^{3}} \right)\) | ||
Enhanced-2D- l _{1}-SVD | \(O\left ({{K^{3}}{N^{3}}K_{\alpha }^{3}} \right) + {P_{\alpha } }O\left ({{K^{3}}K_{\beta }^{3}} \right) + O\left ({{K^{3}}{M^{3}}K_{\beta }^{3}} \right) + {P_{\beta } }O\left ({{K^{3}}K_{\alpha }^{3}} \right)\) |
4.2 Degrees of freedom
where the spark of A is defined as the smallest number of columns in A that are linearly dependent. For DOA estimation problems, the measurement matrix X and the signal matrix S stand for the sampled snapshots at the array and the incoming source signals, respectively. The dictionary A is a matrix composed of the steering vectors corresponding to each potential DOA as its columns. And the number of columns in A is much bigger than the number of rows in A. Let L _{0} denote the number of sensors in the array. For linear or rectangular arrays whose manifold matrix is similar to a Vandermonde matrix, spark(A) is equal to L _{0}+1 if the potential DOAs in the dictionary are not too close to each other so that the mutual coherence [20] is small. On the hand, rank(X) = 1 if the number of snapshots is only one and rank(X) = K if the number of singular vectors used is sufficient, e.g., equal to the number of sources. So empirically, the l _{1}-SVD technique can resolve M−1 sources with an M-sensors array if they are not located too close to each other [7]. This holds under the assumption that the number of singular vectors used in l _{1}-SVD is sufficient, e.g., equal to the number of sources. When fewer singular vectors are taken than the number of sources, the number of resolvable sources may decrease.
Degrees of freedom of 2D- l _{ 1 } -SVD and l _{ 1 } -SVD
Methods | Degrees of freedom | |||
---|---|---|---|---|
Single snapshot | Multiple snapshots | |||
l _{1}-SVD | M N/2 | M N−1 | ||
2D- l _{1}-SVD- α | α | M/2 | α | M−1 |
β for each α | N/2 | β for each α | N−1 | |
Maximum | M N/4 | Maximum | (M−1)(N−1) | |
2D- l _{1}-SVD- β | β | N/2 | β | N−1 |
α for each β | M/2 | α for each β | M−1 | |
Maximum | M N/4 | Maximum | (M−1)(N−1) |
4.3 The assumed number of sources
The l _{1}-SVD technique works on the K singular vectors where K is the assumed number of sources. It has been illustrated in [7] that l _{1}-SVD has a low sensitivity to K.
In [7], it is illustrated that l _{1}-SVD can resolve M−1 sources using an M-sensor array. However, when fewer singular vectors are taken than the number of sources, the condition in (22) may be not satisfied and the number of resolvable sources may decrease. This limitation still exists in 2D situation. Assume that the 2D DOA of incoming sources are \(\phantom {\dot {i}\!}\{(\alpha _{1},\beta _{1,1}), \cdots, (\alpha _{1},\beta _{1,f_{1}}), \cdots, (\alpha _{P},\beta _{P,1}), \cdots, (\alpha _{P},\beta _{P,f_{P}})\}\), and \({P_{0}} \buildrel \Delta \over = \sum \limits _{p = 1}^{P} {f_{p}}\). To resolve P _{0} sources that have P different α, the assumed number of sources for l _{1}-SVD should be not less than P _{0} if P _{0} is sufficiently large (for example, P _{0} is close to the number of sensors). The number of singular vectors should not be less than P or f _{ p } to resolve P different α and f _{ p } different β, correspondingly. As a result, if the assumed number of sources for 2D- l _{1}-SVD is not less than max{P,f _{1},⋯,f _{ P }}, the 2D- l _{1}-SVD can resolve P _{0} sources. So 2D- l _{1}-SVD may resolve more sources with fewer singular vectors. Moreover, 2D- l _{1}-SVD may have an even smaller computational complexity by taking a smaller number of singular vectors.
4.4 Faulty or nonrectangular arrays
When there are only a small number of elements with failure in the array, the faulty elements have little effect on the performance of 2D- l _{1}-SVD. The faulty array is no longer rectangular while it can be regarded as a rectangular array with missing elements. Under this circumstance, the degree of freedom may be affected while 2D- l _{1}-SVD is still able to produce correct DOA estimates. Empirically, 2D- l _{1}-SVD- α can resolve M _{0}−1 different α and N _{0}−1 different β for each α when the sources are not too close to each other. Here, M _{0} and N _{0} denote the minimum number of valid elements in each row or column of the rectangular array. On the other hand, there should not be too many invalid elements, or the distances between valid elements become so large that there will be grating lobes.
5 Simulation results
In this section, several simulations are conducted to validate the advantages of 2D- l _{1}-SVD. All simulations are performed using MATLAB 2012a running on an Intel Core i7 3770 CPU @ 3.4 GHz with 16 GB RAM, under Windows 7.
5.1 Time cost
Time cost of different methods
Time (s) | K _{ αβ } | |||
---|---|---|---|---|
46 | 91 | 181 | 361 | |
2D-MUSIC | 0.3315 | 1.2961 | 5.1043 | 20.2029 |
l _{1}-SVD | 2.7922 | 12.9736 | 59.9812 | 397.105 |
2D- l _{1}-SVD | 0.4381 | 0.4929 | 0.5964 | 0.8356 |
Enhanced-2D- l _{1}-SVD | 1.0704 | 1.4574 | 1.6928 | 2.5747 |
5.2 Pseudo spectrum with multiple sources
5.3 Bias
5.4 Resolution capability
5.5 RMSE
where \({{\text {RMSE}}\left ({{\alpha _{p}}} \right)\! =\!\! {\left [ {\sum \limits _{l = 1}^{L} {{\left ({{{\hat \alpha }_{p}}\left (l \right)\! -\! {\alpha _{p}}} \right)}^{2}}/L} \right ]^{1/2}}, {\text {RMSE}}\left ({{\beta _{p}}} \right)\! =} {\left [ {\sum \limits _{l = 1}^{L} {{\left ({{{\hat \beta }_{p}}\left (l \right) - {\beta _{p}}} \right)}^{2}}/L} \right ]^{1/2}}\) and \(\left ({{{\hat \alpha }_{p}}\left (l \right),{{\hat \beta }_{p}}\left (l \right)} \right)\) is the lth estimates of the DOA of pth signal (α _{ p },β _{ p }). L is the number of realizations.
6 Conclusions
In this paper, a new 2D estimation method called 2D- l _{1}-SVD and its improved version enhanced-2D- l _{1}-SVD are proposed for rectangular arrays. They are proved to be able to work for rectangular arrays even with missing or faulty elements. Theoretical analysis and simulation results indicate that 2D- l _{1}-SVD has a much lower arithmetic complexity due to successive parameter estimation while performing close to the popular l _{1}-SVD. What is more, 2D- l _{1}-SVD has better robustness to the assumed number of sources and enhanced-2D- l _{1}-SVD is able to detect spurious peaks which are caused by inappropriate regularization parameter.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments that improved the manuscript.
Authors’ Affiliations
References
- H Krim, M Viberg, Two decades of array signal processing research: the parametric approach. IEEE Signal Process. Mag. 13(4), 67–94 (1996).View ArticleGoogle Scholar
- E Tuncer, B Friedlander, Classical and Modern Direction-of-Arrival Estimation (Elsevier Academic Press, Burlington, USA, 2009).Google Scholar
- J Capon, High-resolution frequency-wavenumber spectrum analysis. Proc IEEE. 57(8), 1408–1418 (1969).View ArticleGoogle Scholar
- R Schmidt, Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34(3), 276–280 (1986).View ArticleGoogle Scholar
- R Roy, T Kailath, ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoustics Speech Signal. Process. 37(7), 984–995 (1989).View ArticleGoogle Scholar
- P Stoica, K Sharman, Maximum likelihood methods for direction-of-arrival estimation. IEEE Trans. Acoustics Speech Signal. Process. 38(7), 1132–1143 (1990).View ArticleMATHGoogle Scholar
- D Malioutov, M Cetin, A Willsky, A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process. 53(8), 3010–3022 (2005).View ArticleMathSciNetGoogle Scholar
- M Hyder, K Mahata, Direction-of-arrival estimation using a mixed l _{2,0} norm approximation. IEEE Trans. Signal Process. 58(9), 4646–4655 (2010).View ArticleMathSciNetGoogle Scholar
- P Stoica, P Babu, J Li, A sparse covariance-based estimation method for array processing. IEEE Trans. Signal Process. 59(2), 629–638 (2011).View ArticleMathSciNetGoogle Scholar
- J Zheng, M Kaveh, Sparse spatial spectral estimation: a covariance fitting algorithm, performance and regularization. IEEE Trans. Signal Process. 61(11), 2767–2777 (2013).View ArticleMathSciNetGoogle Scholar
- X Xu, X Wei, Z Ye, DOA estimation based on sparse signal recovery utilizing weighted l _{1}-norm penalty. IEEE Signal Process. Lett. 19(3), 155–158 (2012).View ArticleGoogle Scholar
- A Gershman, M Rubsamen, M Pesavento, One-and two-dimensional direction-of-arrival estimation: an overview of search-free techniques. Signal Process. 90, 1338–1349 (2010).View ArticleMATHGoogle Scholar
- Y Wang, L Lee, A tree structure one-dimensional based algorithm for estimating the two-dimensional direction of arrivals and its performance analysis. IEEE Trans. Antennas Propag. 56(1), 178–188 (2008).View ArticleMathSciNetGoogle Scholar
- D Malioutov, A sparse signal reconstruction perspective for source localization with sensor arrays. PhD thesis, Massachusetts Institute of Technology (2003).Google Scholar
- Y Liu, M Wu, S Wu, Fast OMP algorithm for 2D angle estimation in MIMO radar. Electronics Lett. 46(6), 444–445 (2010).View ArticleGoogle Scholar
- S Kikuchi, H Tsuji, A Sano, Pair-matching method for estimating 2-D angle of arrival with a cross-correlation matrix. IEEE Antennas Wireless Propag. Lett. 5(1), 35–40 (2006).View ArticleGoogle Scholar
- Q Liu, S OuYang, L Jin, Two-dimensional DOA estimation with L-shaped array based on a jointly sparse representation. Inf. Technol. J. 12, 2037–2042 (2013).View ArticleGoogle Scholar
- F Belloni, A Richter, V Koivunen, DOA estimation via manifold separation for arbitrary array structures. IEEE Trans. Signal Process. 55(10), 4800–4810 (2007).View ArticleMathSciNetGoogle Scholar
- M Lobo, L Vandenberghe, S Boyd, H Lebret, Applications of second-order cone programming. Linear Algebra Its Applicat. Special Issue Linear Algebra Control Signals Image Process. 284, 193–228 (1998).View ArticleMATHMathSciNetGoogle Scholar
- M Davies, Y Eldar, Rank awareness in joint sparse recovery. IEEE Trans. Inf. Theory. 58(2), 1135–1146 (2012).View ArticleMathSciNetGoogle Scholar
- Q Zhang, Probability of resolution of the MUSIC algorithm. IEEE Trans. Signal Process. 43(4), 978–987 (1995).View ArticleGoogle Scholar
- Y Hua, T Sarkar, A note on the Cramer-Rao bound for 2-D direction finding based on 2-D array. IEEE Trans. Signal Process. 39(5), 1215–1218 (1991).View ArticleGoogle Scholar
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