Abstract
In this paper, we proposed a new direction-of-arrival estimation approach using Volterra signal model in spatial domain. The new technique basically uses additionally the second-order terms of Volterra series to produce augmented Volterra snapshots, the extension for higher-order case is straightforward. The resolution of the proposed method is high comparing with the standard multiple signal classification (MUSIC) algorithm and the 2q-MUSIC algorithm. Simulation results are presented to show the performance of the proposed technique.
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Birot, G., Albera, L., & Chevalier, P. (2010). Sequential high-resolution direction finding from higher order statistics. IEEE Transactions on Signal Processing, 58(8), 4144–4155.
Chevalier, P., & Ferreol, A. (1999). On the virtual array concept for the fourth-order direction finding problem. IEEE Transactions on Signal Processing, 47(9), 2592–2595.
Chevalier, P., Albera, L., Ferreol, A., & Comon, P. (2005). On the virtual array concept for higher order array processing. IEEE Transactions on Signal Processing, 53(4), 1254–1271.
Chevalier, P., Ferreol, A., & Albera, L. (2006). High-resolution direction finding from higher order statistics: The 2q -MUSIC algorithm. IEEE Transactions on Signal Processing, 54(8), 2986–2997.
Chevalier, P., Ferreol, A., Albera, L., & Birot, G. (2007). Higher order direction finding from arrays with diversely polarized antennas: The pd-2q-music algorithms. IEEE Transactions on Signal Processing, 55(11), 5337–5350.
Diniz, P. S. R. (2008). Adaptive filtering : Algorithms and practical implementation. Berlin: Springer.
Dogan, M. C., & Mendel, J. M. (1995a). Applications of cumulants to array processing. I. Aperture extension and array calibration. IEEE Transactions on Signal Processing, 43(5), 1200–1216.
Dogan, M. C., & Mendel, J. M. (1995b). Applications of cumulants to array processing. II. Non-gaussian noise suppression. IEEE Transactions on Signal Processing, 43(7), 1663–1676.
Gonen, E., Mendel, J. M., & Dogan, M. C. (1997). Applications of cumulants to array processing. IV. Direction finding in coherent signals case. IEEE Transactions on Signal Processing, 45(9), 2265–2276.
Kaveh, M., & Barabell, A. (1986). The statistical performance of the MUSIC and the minimum-norm algorithms in resolving plane waves in noise. IEEE Transactions on Acoustics, Speech, and Signal Processing, 34(2), 331–341.
Lee, H. B., & Wengrovitz, M. S. (1990). Resolution threshold of beamspace MUSIC for two closely spaced emitters. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(9), 1545–1559.
Liu, Z., Ruan, X., & He, J. (2013). Efficient 2-d doa estimation for coherent sources with a sparse acoustic vector-sensor array. Multidimensional Systems and Signal Processing, 24(1), 105–120.
Mathews, V. J. (1991). Adaptive polynomial filters. IEEE Signal Processing Magazine, 8(3), 10–26.
Pal, P., & Vaidyanathan, P. P. (2012). Multiple level nested array: An efficient geometry for th order cumulant based array processing. IEEE Transactions on Signal Processing, 60(3), 1253–1269.
Porat, B., & Friedlander, B. (1991). Direction finding algorithms based on high-order statistics. IEEE Transactions on Signal Processing, 39(9), 2016–2024.
Rao, B. D., & Hari, K. V. S. (1989). Performance analysis of root-MUSIC. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37(12), 1939–1949.
Schmidt, R. (1986). Multiple emitter location and signal parameter estimation. IEEE Transactions on Antennas and Propagation, 34(3), 276–280.
Stoica, P., & Nehorai, A. (1989). MUSIC, maximum likelihood, and Cramer–Rao bound. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37(5), 720–741.
Stoica, P., & Nehorai, A. (1990). MUSIC, maximum likelihood, and Cramer–Rao bound: further results and comparisons. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(12), 2140–2150.
Van Trees, H. L. (2002). Detection, estimation, and modulation theory, part IV, optimum array processing. London: Wiley.
Xu, X. L., & Buckley, K. M. (1992). Bias analysis of the MUSIC location estimator. IEEE Transactions on Signal Processing, 40(10), 2559–2569.
Zhang, X., Chen, C., Li, J., & Xu, D. (2012). Blind DOA and polarization estimation for polarization-sensitive array using dimension reduction MUSIC. Multidimensional Systems and Signal Processing, 1–16. doi:10.1007/s11045-012-0186-3.
Zhang, Q. T. (1995). Probability of resolution of the MUSIC algorithm. IEEE Transactions on Signal Processing, 43(4), 978–987.
Zheng, Z., & Li, G. (2012). Fast DOA estimation of incoherently distributed sources by novel propagator. Multidimensional Systems and Signal Processing, 1–9. doi:10.1007/s11045-012-0185-4.
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Appendix: The dimension of the signal subspace \(N_s\)
Appendix: The dimension of the signal subspace \(N_s\)
The dimension of the signal subspace \(N_s\) is equivalent to the Rank of \(\mathbf{R}_v\) for the noiseless case. In order to analysis the Rank of \(\mathbf{R}_v\), let us look at some of the properties of Rank:
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(1)
For a matrix \(\mathbf{{B}}\in \mathbb{C }^{m\times n}\), \(\mathrm{Rank}(\mathbf{{B}})\le \min \left(m,n\right).\)
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(2)
For two matrices \(\mathbf{{B}}\in \mathbb{C }^{m\times n}\) and \(\mathbf{{C}}\in \mathbb{C }^{n\times t}\), \(\mathrm{Rank}(\mathbf{{BC}})\le \min \big (\mathrm{Rank}(\mathbf{{B}}),\mathrm{Rank}(\mathbf{{C}})\big ).\)
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(3)
For two matrics \(\mathbf{{B}}\in \mathbb{C }^{m\times n}\) and \(\mathbf{{D}}\in \mathbb{C }^{t\times n}\), \(\mathrm{Rank}\left(\left[\begin{array}{c} \mathbf{{B}}\\ \mathbf{{D}}\end{array}\right]\right)\le \min \left(m+t,n\right).\)
For the noiseless case, the Volterra snapshots \(\mathbf{{X}}_{v}\) is given by
where \(\mathbf{S} = \left[\mathbf{s}(1),\, \mathbf{s}(2), \ldots , \mathbf{s}(k), \ldots , \mathbf{s}(K)\right]\) and \(\odot \) denotes Khatri-Rao product, which is a column-wise Kronecker product.
The estimated noiseless covariance matrix is obtained by
Based on the properties (1) and (2) of Rank, \(\mathrm{Rank}\big (\mathbf{{A}}(\theta )\big )\le L\) and \(\mathrm{Rank}(\mathbf{{S}})\le L\), then \(\mathrm{Rank}\big (\mathbf{{A}}(\theta )\mathbf{{S}}\big )\le L\).
The Kronecker product can be rewritten as
where \(\mathrm{vec}\left(\mathbf{X}\right)\) denotes the vectorization of the matrix \(\mathbf{X}\) formed by stacking the columns of \(\mathbf{X}\) into a single column vector.
Since
then the rank of \(\mathbf{{S}~\odot }~\mathbf{{S}}^*\) is
and
and
Based on the above property \((3)\), we obtain
The dimension of the signal subspace
The number of noise eigenvectors \(N_{e}\) used to perform DOA estimation is given by
Table 1 shows the dimension of the signal subspace \(N_s\) and the number of noise eigenvectors \(N_{e}\) used to perform DOA estimation, for different number of sensors \(M\) and different number of sources \(L\).
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Wen, F., Ng, B.P. A new DOA estimation approach using Volterra signal model. Multidim Syst Sign Process 25, 741–758 (2014). https://doi.org/10.1007/s11045-013-0228-5
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DOI: https://doi.org/10.1007/s11045-013-0228-5