An efficient implementation of iterative adaptive approach for source localization
- Gang Li^{1}Email author,
- Hao Zhang^{1},
- Xiqin Wang^{1} and
- Xiang-Gen Xia^{2}
https://doi.org/10.1186/1687-6180-2012-7
© Li et al; licensee Springer. 2012
Received: 11 July 2011
Accepted: 12 January 2012
Published: 12 January 2012
Abstract
The iterative adaptive approach (IAA) can achieve accurate source localization with single snapshot, and therefore it has attracted significant interest in various applications. In the original IAA, the optimal filter is performed for every scanning angle grid in each iteration, which may cause the slow convergence and disturb the spatial estimates on the impinging angles of sources. In this article, we propose an efficient implementation of IAA (EIAA) by modifying the use of the optimal filtering, i.e., in each iteration of EIAA, the optimal filter is only utilized to estimate the spatial components likely corresponding to the impinging angles of sources, and other spatial components corresponding to the noise are updated by the simple correlation of the basis matrix with the residue. Simulation results show that, in comparison with IAA, EIAA has significant higher computational efficiency and comparable accuracy of source angle and power estimation.
Keywords
1. Introduction
Source localization is a fundamental problem in a wide range of applications including communications, radar, and acoustics, and many algorithms have been presented in the literature during recent decades. The Fourier-based algorithms suffer from the low resolution and the high sidelobes. Some methods based on subspace processing, e.g., Capon beamforming [1], MUSIC [2], ESPRIT [3], and other subspace-based algorithms [4, 5], provide super-resolution for uncorrelated sources with sufficient number of snapshots. However, in the case of few snapshots, the performances of these subspace-based methods will degrade sharply.
Recently, the source localization problem has been converted into a sparse recovery framework, because the number of actual sources of interest is generally much smaller than the number of potential source locations in the region to be observed. A kind of algorithms of sparse recovery is based on iterative weighted least squares, e.g., the FOCal Underdetermined System Solver (FOCUSS) [6], the Sparse Learning via Iterative Minimization (SLIM) [7], the iterative adaptive approach (IAA) [8], etc. Here, we are interested in IAA, which is able to provide accurate source localization with single snapshot and has attracted significant interest in various applications [9–11]. IAA is non-parametric and it achieves accurate estimates of angles and powers of the sources by iterative operations [8]. The spatial component on every potential angle is estimated by optimal filtering, which passes the signal from the current angle without distortion and fully suppresses the interferences from other angles. The iteration is terminated when the norm of the difference between two successive spatial estimates is smaller than a certain threshold. However, it is time consuming to perform optimal filtering on all potential angles, since in general we are only interested in several angles where the actual sources are located. Moreover, the excessive estimation of the spatial components on the angles that are outside the actual source position set may result in a slow convergence. In this article, we propose an efficient implementation of IAA (EIAA) by modifying the use of the optimal filtering, i.e., in each iteration, the optimal filter is only utilized to estimate the spatial components likely corresponding to the actual signal sources, and other spatial components corresponding to the noise are updated by the simple correlation of the basis matrix with the residue. It will be shown that EIAA has significant faster convergence speed and comparable accuracy of source angle and power estimation. In [12, 13], two fast implementations of IAA have been proposed by using the matrix computation technique such as Gohberg-Semencul decomposition, etc. It is noted that the way of the computational burden reduction in this article is different from [12, 13]: herein, we focus on reducing the number of running optimal filtering procedures, while [12, 13] focus on improving the computational efficiency of the optimal filtering procedure. In addition, similar to the algorithms mentioned above, we are only interested in the unambiguous angle solution, which depends on the ratio of interelement spacing of the array to the wavelength. In the case that the angle ambiguity occurs, we refer to [14–16] for resolving the ambiguity.
The remainder of this article is organized as follows. The signal model and the original IAA are introduced in Section 2. The EIAA algorithm is proposed in Section 3. The proposed EIAA is evaluated by some simulations in Section 4. Concluding remarks are presented in Section 5.
2. Signal model and IAA
IAA algorithm
Initialization: ${\widehat{p}}_{k}^{\left(0\right)}=\frac{{\left|{\mathbf{a}}^{H}\left({\theta}_{k}\right)\mathbf{y}\right|}^{2}}{{\left[{\mathbf{a}}^{H}\left({\theta}_{k}\right)\mathbf{a}\left({\theta}_{k}\right)\right]}^{2}}$ for k = 1,2,...,K. |
---|
Repeat: |
(a) Calculate the correlation matrix by ${\widehat{\mathbf{R}}}^{\left(i\right)}=\mathbf{A}{\widehat{\mathbf{P}}}^{\left(i\right)}{\mathbf{A}}^{H}$. |
(b) Estimate the spatial components by${\widehat{p}}_{k}^{\left(i\right)}={\left|\frac{{\mathbf{a}}^{H}\left({\theta}_{k}\right)\cdot {\left({\widehat{\mathbf{R}}}^{\left(i\right)}\right)}^{-1}\cdot \mathbf{y}}{{\mathbf{a}}^{H}\left({\theta}_{k}\right)\cdot {\left({\widehat{\mathbf{R}}}^{\left(i\right)}\right)}^{-1}\cdot \mathbf{a}\left({\theta}_{k}\right)}\right|}^{2}$, for k = 1, 2, ..., K. |
(c) If the norm of the difference between ${\widehat{\mathbf{P}}}^{\left(i-1\right)}$ and ${\widehat{\mathbf{P}}}^{\left(i\right)}$ is smaller than a threshold, i.e., ${\delta}^{\left(i\right)}\triangleq \sqrt{\sum _{k=1}^{K}{\left[{\widehat{p}}_{k}^{\left(i-1\right)}-{\widehat{p}}_{k}^{\left(i\right)}\right]}^{2}}<\epsilon $, the iteration is stopped; otherwise let i = i+1 and go to a). |
3. Efficient implementation of IAA
EIAA algorithm
Initialization: let ${\widehat{p}}_{k}^{\left(0\right)}=\frac{{\left|{\mathbf{a}}^{H}\left({\theta}_{k}\right)\mathbf{y}\right|}^{2}}{{\left[{\mathbf{a}}^{H}\left({\theta}_{k}\right)\mathbf{a}\left({\theta}_{k}\right)\right]}^{2}}$ for k = 1,2,...,K; let the residue r^{(0)} = y; |
---|
Repeat: |
(a) Calculate the correlation matrix by ${\widehat{\mathbf{R}}}^{\left(i\right)}=\mathbf{A}{\widehat{\mathbf{P}}}^{\left(i\right)}{\mathbf{A}}^{H}$; |
Let the index support set Λ^{(i)}= ∅ and the principal spatial component set Γ^{(i)}= ∅. |
(b) While the relative residue is larger than a threshold, i.e., $\frac{{\u2225{\mathbf{r}}^{\left(i\right)}\u2225}_{2}^{2}}{{\u2225\mathbf{y}\u2225}_{2}^{2}}>\xi $ |
Find the index n_{ l } corresponding to the largest entry in the vector $\left[{\widehat{p}}_{1}^{\left(i\right)},{\widehat{p}}_{2}^{\left(i\right)},\dots ,{\widehat{p}}_{K}^{\left(i\right)}\right]$; |
Expand the index support set by Λ^{(i)}= {Λ^{(i)}, n_{1}}; |
Expand the principal spatial component set by ${\mathrm{\Gamma}}^{\left(i\right)}=\left\{{\mathrm{\Gamma}}^{\left(i\right)},{\left|\frac{{\mathbf{a}}^{H}\left({\theta}_{{n}_{l}}\right)\cdot {\left({\widehat{\mathbf{R}}}^{\left(i\right)}\right)}^{-1}\cdot \mathbf{y}}{{\mathbf{a}}^{H}\left({\theta}_{{n}_{l}}\right)\cdot {\left({\widehat{\mathbf{R}}}^{\left(i\right)}\right)}^{-1}\cdot \mathbf{a}\left({\theta}_{{n}_{l}}\right)}\right|}^{2}\right\}$; |
Calculate the residue by ${\mathbf{r}}^{\left(i\right)}=\mathbf{y}-{\left({\mathbf{A}}_{{\mathrm{\Lambda}}^{\left(i\right)}}^{H}{\mathbf{A}}_{{\mathrm{\Lambda}}^{\left(i\right)}}\right)}^{-1}{\mathbf{A}}_{{\mathrm{\Lambda}}^{\left(i\right)}}^{H}\mathbf{y}$, where the matrix ${\mathbf{A}}_{{\mathrm{\Lambda}}^{\left(i\right)}}$ consists of the columns of A with indices k ∈ Λ^{(i)}; |
Update the spatial estimate by ${\widehat{p}}_{k}^{\left(i\right)}=\frac{{\left|{\mathbf{a}}^{H}\left({\theta}_{k}\right){\mathbf{r}}^{\left(i\right)}\right|}^{2}}{{\left[{\mathbf{a}}^{H}\left({\theta}_{k}\right)\mathbf{a}\left({\theta}_{k}\right)\right]}^{2}}$, for k = 1, 2, ..., K. |
end While |
(c) Restore the principal spatial components by ${\widehat{p}}_{k}^{\left(i\right)}={\mathrm{\Gamma}}^{\left(i\right)}\left(k\right)$, for k ∈ Λ^{(i)}. |
(d) If the norm of the difference between ${\widehat{\mathbf{P}}}^{\left(i-1\right)}$ and ${\widehat{\mathbf{P}}}^{\left(i\right)}$ is smaller than a threshold, i.e., ${\delta}^{\left(i\right)}\triangleq \sqrt{\sum _{k=1}^{K}{\left[{\widehat{p}}_{k}^{\left(i-1\right)}-{\widehat{p}}_{k}^{\left(i\right)}\right]}^{2}}<\epsilon $, the iteration is stopped; otherwise let i = i+1 and go to a). |
The main difference between the proposed EIAA and the original IAA lies in the estimation of spatial components that are outside the actual source location set. As seen from step (b) in Table 2 {θ_{ k } with index k ∈ Λ^{(i)}} are considered to be likely angle candidates where actual sources are located. Then, the spatial components corresponding to the actual source locations are updated by optimal filtering, and other spatial components corresponding to the noise are updated by simple correlation of the columns of basis matrix with the residue. This implies that the excessive estimation of noise components is avoided. Compared with the original IAA, EIAA can significantly reduce the computational burden thanks to the following facts: (1) In each iteration, the required times of optimal filtering procedure is equal to the number of the selected principle components in step (b) of Table 2. The step (b) of Table 2 is finished by the residual energy threshold, for example, in practice it is reasonable to let ξ = 0.05, which implies that the relative residue energy is smaller than 5%. In high SNR case, it is believable that the number of the selected principle components in step (b) of Table 2 is equal to the number of the actual sources; for lower SNR, the number of the selected principle components in step (b) of Table 2 may be slight larger than the number of the actual sources because the signal-subspace and the noise-subspace become undistinguishable. Anyway, the number of the selected principle components, i.e., the required times of optimal filtering procedure, is usually much smaller than K, which is guaranteed by the prior assumption of sparse signal property. (2) The relaxation of the estimation of spatial components corresponding to noise leads to stable and fast convergence.
4. Simulations
In this section, some examples are provided to evaluate the performance of the proposed EIAA in single snapshot case. Consider a uniform linear array of M = 14 sensors with the interelement spacing λ/2. The additional noise is assumed Gaussian with zero mean and variance σ^{2}, and the SNR is defined as $10{log}_{10}\left(\sum _{n=1}^{L}{p}_{n}\u2215\left(L{\sigma}^{2}\right)\right)$. The angle scanning grid is uniform in the range from 1° to 180° with 1° increment between adjacent angle candidates.
Performances of IAA and EIAA for various SNR
SNR | 0 dB | 5 dB | 10 dB | 15 dB | 20 dB |
---|---|---|---|---|---|
${\mathsf{\text{er}}}_{\mathsf{\text{EIAA}}}^{\mathsf{\text{angle}}}-{\mathsf{\text{er}}}_{\mathsf{\text{IAA}}}^{\mathsf{\text{angle}}}$ | -0.297° | -0.114° | -0.107° | -0.015° | -0.004° |
${\mathsf{\text{er}}}_{\mathsf{\text{EIAA}}}^{\mathsf{\text{power}}}-{\mathsf{\text{er}}}_{\mathsf{\text{IAA}}}^{\mathsf{\text{power}}}$ | -0.087 | -0.393 | -0.543 | -0.489 | -0.373 |
RTR of EIAA and IAA | 0.131 | 0.118 | 0.081 | 0.072 | 0.070 |
5. Conclusion
In this article, EIAA algorithm is proposed for source localization. By selecting the principal components of spatial estimate in each iteration, the optimal filter is only utilized to estimate the spatial components likely corresponding to the actual signal sources, and the other spatial components corresponding to noise are updated by the simple correlation of the basis matrix with the residue. Compared with the original IAA that performs optimal filtering on every scanning angle grid in each iteration, EIAA shows higher computational efficiency and slightly better accuracy of angle and power estimation.
Declarations
Acknowledgements
This study was supported in part by the National Natural Science Foundation of China under Grant 40901157, and in part by the National Basic Research Program of China (973 Program) under Grant 2010CB731901, in part by the Doctoral Fund of Ministry of Education of China under Grant 200800031050, and in part by Tsinghua National Laboratory for Information Science and Technology (TNList) Cross-discipline Foundation. Xia's work was supported by the National Science Foundation (NSF) under Grant CCF-0964500 and the World Class Univerrsity (WCU) Program, National Research Foundation, Korea.
Authors’ Affiliations
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