Direction of arrival estimation in vector-sensor arrays using higher-order statistics

Abstract

MUSIC algorithm is an effective method in solving the direction-finding problems. Due to the good performance of this algorithm, many variations of it including tesnor-MUSIC for verctor-sensor arrays, have been developed. However, these MUSIC-based methods have some limitations with respect to the number of sources, modeling errors and the noise power. It has been shown that using 2qth-order \((q>1)\) statistics in MUSIC algorithm is very effective to overcome these drawbacks. However, the existing 2q-order MUSIC-like methods are appropriate for scalar-sensor arrays, which only measure one parameter, and have a matrix of measurements. In vector-sensor arrays, each sensor measures multiple parameters, and to keep this multidimensional structure, we should use a tensor of measurements. The contribution of this paper is to develop a new tensor-based 2q-order MUSIC-like method for vector-sensor arrays. In this regard, we define a tensor of the cumulants which will be used in the proposed algorithm. The new method is called tensor-2q-MUSIC. Computer simulations have been used to compare the performance of the proposed method with a higher-order extension of the conventional MUSIC method for the vector-sensor arrays which is called matrix-2q-MUSIC. Moreover, we compare the performance of tensor-2q-MUSIC method with the existing second-order methods for the vector-sensor arrays. The simulation results show the better performance of the proposed method.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  1. Abeida, H., & Delmas, J. P. (2006). MUSIC-like estimation of direction of arrival for noncircular sources. IEEE Transactions on Signal Processing, 54(7), 2678–2690. https://doi.org/10.1109/TSP.2006.873505.

    Article  MATH  Google Scholar 

  2. Becker, H., Chevalier, P., & Haardt, M. (2017). Higher order direction finding from rectangular cumulant matrices: The rectangular 2q-MUSIC algorithms. Signal Processing, 133, 240–249. https://doi.org/10.1016/j.sigpro.2016.10.020.

    Article  Google Scholar 

  3. Boizard, M., Ginolhac, G., Pascal, F., Miron, S., & Forster, P. (2013). Numerical performance of a tensor MUSIC algorithm based on HOSVD for a mixture of polarized sources. In 21st European signal processing conference (EUSIPCO 2013).

  4. Cardoso, J. F., & Moulines, E. (1995). Asymptotic performance analysis of direction-finding algorithms based on fourth-order cumulants. IEEE Transactions on Signal Processing, 43(1), 214–224. https://doi.org/10.1109/78.365301.

    Article  Google Scholar 

  5. Carroll, J. D., & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart–Young” decomposition. Psychometrika, 35(3), 283–319.

    Article  Google Scholar 

  6. Charge, P., Wang, Y., & Saillard, J. (2003). An extended cyclic MUSIC algorithm. IEEE Transactions on Signal Processing, 51(7), 1695–1701. https://doi.org/10.1109/TSP.2003.812834.

    MathSciNet  Article  MATH  Google Scholar 

  7. Chevalier, P., Ferréol, A., & Albera, L. (2006). High-resolution direction finding from higher order statistics: The \(2rmq \)-MUSIC algorithm. IEEE Transactions on Signal Processing, 54(8), 2986–2997. https://doi.org/10.1109/TSP.2006.877661.

    Article  MATH  Google Scholar 

  8. Choi, J. H., & Yoo, C. D. (2015). Underdetermined high-resolution doa estimation: A \(2\rho \)th-order source-signal/noise subspace constrained optimization. IEEE Transactions on Signal Processing, 63(7), 1858–1873. https://doi.org/10.1109/TSP.2015.2401531.

    MathSciNet  Article  MATH  Google Scholar 

  9. De Lathauwer, L. (1997). Signal processing based on multilinear algebra. Leuven: Katholieke Universiteit Leuven.

    Google Scholar 

  10. De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000). A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications, 21(4), 1253–1278.

    MathSciNet  Article  Google Scholar 

  11. Forster, P., Ginolhac, G., & Boizard, M. (2016). Derivation of the theoretical performance of a tensor MUSIC algorithm. Signal Processing, 129, 97–105. https://doi.org/10.1016/j.sigpro.2016.05.033.

    Article  Google Scholar 

  12. Giannakis, G. B. (1998). Cyclostationary signal analysis. In Digital signal processing handbook (Vol. 31, pp. 17–1) Citeseer.

  13. Gong, X., Xu, Y., & Liu, Z. (2006). On the equivalence of tensor-MUSIC and matrix-MUSIC. In 7th International symposium on antennas, propagation & EM theory, 2006. ISAPE’06 (pp. 1–4). IEEE. https://doi.org/10.1109/ISAPE.2006.353448.

  14. Haardt, M., Roemer, F., & Del Galdo, G. (2008). Higher-order SVD-based subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems. IEEE Transactions on Signal Processing, 56(7), 3198–3213. https://doi.org/10.1109/TSP.2008.917929.

    MathSciNet  Article  MATH  Google Scholar 

  15. Han, K., & Nehorai, A. (2014). Nested vector-sensor array processing via tensor modeling. IEEE Transactions on Signal Processing, 62(10), 2542–2553. https://doi.org/10.1109/TSP.2014.2314437.

    MathSciNet  Article  MATH  Google Scholar 

  16. Harshman, R. A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-mode factor analysis. UCLA Working Papers in Phonetics, Vol. 16, pp. 1–84.

  17. Hu, C., Wu, Y., Huang, L., & Yan, G. (2019). Unitary root-MUSIC based on tensor mode-R algorithm for multidimensional sinusoidal frequency estimation without pairing parameters. Multidimensional Systems and Signal Processing,. https://doi.org/10.1007/s11045-019-00672-5.

    Article  MATH  Google Scholar 

  18. McCullagh, P. (2018). Tensor methods in statistics: Monographs on statistics and applied probability. Boca Raton: Chapman and Hall/CRC.

    Google Scholar 

  19. Mendel, J. M. (1991). Tutorial on higher-order statistics (spectra) in signal processing and system theory: Theoretical results and some applications. Proceedings of the IEEE, 79(3), 278–305. https://doi.org/10.1109/5.75086.

    Article  Google Scholar 

  20. Miron, S., Le Bihan, N., & Mars, J. I. (2005). Vector-sensor MUSIC for polarized seismic sources localization. EURASIP Journal on Applied Signal Processing, 2005, 74–84. https://doi.org/10.1155/ASP.2005.74.

    Article  MATH  Google Scholar 

  21. Nehorai, A., & Paldi, E. (1994). Acoustic vector-sensor array processing. IEEE Transactions on Signal Processing, 42(9), 2481–2491. https://doi.org/10.1109/78.317869.

    Article  Google Scholar 

  22. Nehorai, A., & Paldi, E. (1994). Vector-sensor array processing for electromagnetic source localization. IEEE Transactions on Signal Processing, 42(2), 376–398. https://doi.org/10.1109/78.275610.

    Article  Google Scholar 

  23. Picinbono, B. (1994). On circularity. IEEE Transactions on Signal Processing, 42(12), 3473–3482. https://doi.org/10.1109/78.340781.

    Article  Google Scholar 

  24. Roy, R., & Kailath, T. (1989). Esprit-estimation of signal parameters via rotational invariance techniques. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37(7), 984–995. https://doi.org/10.1109/29.32276.

    Article  MATH  Google Scholar 

  25. Schmidt, R. (1986). Multiple emitter location and signal parameter estimation. IEEE Transactions on Antennas and Propagation, 34(3), 276–280. https://doi.org/10.1109/TAP.1986.1143830.

    Article  Google Scholar 

  26. Stoica, P., & Gershman, A. B. (1999). Maximum-likelihood doa estimation by data-supported grid search. IEEE Signal Processing Letters, 6(10), 273–275. https://doi.org/10.1109/97.789608.

    Article  Google Scholar 

  27. Stoica, P., & Sharman, K. C. (1990). Maximum likelihood methods for direction-of-arrival estimation. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(7), 1132–1143. https://doi.org/10.1109/29.57542.

    Article  MATH  Google Scholar 

  28. Swindlehurst, A. L., & Kailath, T. (1992). A performance analysis of subspace-based methods in the presence of model errors, part i: The MUSIC algorithm. IEEE Transactions on Signal Processing, 40(7), 1758–1774. https://doi.org/10.1109/78.143447.

    Article  MATH  Google Scholar 

  29. Tuncer, T., & Friedlander, B. (2009). Classical and modern direction-of-arrival estimation. Amsterdam: Elsevier Science.

    Google Scholar 

  30. Xiaofei, Z., Ming, Z., Han, C., & Jianfeng, L. (2014). Two-dimensional DOA estimation for acoustic vector-sensor array using a successive MUSIC. Multidimensional Systems and Signal Processing, 25(3), 583–600. https://doi.org/10.1007/s11045-012-0219-y.

    MathSciNet  Article  MATH  Google Scholar 

  31. Zhang, X., Chen, C., Li, J., & Xu, D. (2014). Blind DOA and polarization estimation for polarization-sensitive array using dimension reduction MUSIC. Multidimensional Systems and Signal Processing, 25(1), 67–82. https://doi.org/10.1007/s11045-012-0186-3.

    Article  MATH  Google Scholar 

  32. Zhang, Y., & Ng, B. P. (2010). MUSIC-like DOA estimation without estimating the number of sources. IEEE Transactions on Signal Processing, 58(3), 1668–1676. https://doi.org/10.1109/TSP.2009.2037074.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mahmood Karimi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Derivation of equality (1) in (43)

In the general case, if there would be four matrices \(\varvec{A}\in \mathbb {C}^{M\times N}\), \(\varvec{B}\in \mathbb {C}^{M\times N}\), \(\varvec{C}\in \mathbb {C}^{P\times Q} \), and \(\varvec{D}\in \mathbb {C}^{P\times Q}\), we want to show that

$$\begin{aligned} \left( \varvec{A}+\varvec{B}\right) \overline{\otimes }\left( \varvec{C} +\varvec{D}\right) = \left( \varvec{A}\overline{\otimes }\varvec{C}\right) +\left( \varvec{B} \overline{\otimes }\varvec{D}\right) \end{aligned}$$
(52)

It suffices to prove that, entry (ij) of the left-hand side of (52) is equal to entry (ij) of the right-hand side of it.

$$\begin{aligned} \begin{aligned} \left( \left( \varvec{A}+\varvec{B}\right) \overline{\otimes } \left( \varvec{C}+\varvec{D}\right) \right) _{ij}=&\,\left( \left( \varvec{A} +\varvec{B}\right) _{mn},\left( \varvec{C}+\varvec{D}\right) _{pq}\right) =\left( a_{mn}+b_{mn},c_{pq}+d_{pq}\right) \\ =&\,\left( a_{mn},c_{pq}\right) +\left( b_{mn},d_{pq}\right) \\ =&\,\left( \varvec{A}\overline{\otimes }\varvec{C}\right) _{ij} +\left( \varvec{B}\overline{\otimes }\varvec{D}\right) _{ij} = \left( \left( \varvec{A}\overline{\otimes }\varvec{C}\right) +\left( \varvec{B}\overline{\otimes }\varvec{D}\right) \right) _{ij} \end{aligned} \end{aligned}$$
(53)

where m, n, p, and q are defined in (35). Replacing \(\varvec{C} \) and \(\varvec{D} \) by \(\varvec{A} \) and \(\varvec{B}\), respectively, and using mathematical induction, it is straightforward to show that

$$\begin{aligned} \left( \varvec{A}+\varvec{B}\right) ^{\overline{\otimes }q}=\varvec{A}^{\overline{\otimes }q}+\varvec{B}^{\overline{\otimes }q} \end{aligned}$$
(54)

Derivation of equality (2) in (43)

Assume four matrices \(\varvec{A}\in \mathbb {C}^{M\times N}\), \(\varvec{B}\in \mathbb {C}^{M\times N}\), \(\varvec{C}\in \mathbb {C}^{P\times Q}\), and \(\varvec{D}\in \mathbb {C}^{P\times Q}\). Also, assume that every entry of \(\varvec{A}\), and \(\varvec{B} \) is an ordered q-tuple, and every entry of \(\varvec{C}\), and \(\varvec{D} \) is an ordered p-tuple. In the general case, we want to show that

$$\begin{aligned} \left( \varvec{A}+\varvec{B}\right) \overline{\circ } \left( \varvec{C}+\varvec{D}\right) =\left( \varvec{A}\overline{\circ } \varvec{C}\right) +\left( \varvec{B}\overline{\circ }\varvec{D}\right) \end{aligned}$$
(55)

Consider an arbitrary entry with indices (ijkl) of the left-hand side of (55)

$$\begin{aligned} \begin{aligned}&\left( \left( \varvec{A}+\varvec{B}\right) \overline{\circ }\left( \varvec{C}+\varvec{D}\right) \right) _{ijkl}\\&\quad =\left( \left( \varvec{A}+\varvec{B}\right) _{ij},\left( \varvec{C}+\varvec{D}\right) _{kl}\right) =\left( \left( a_{ij}^1+b_{ij}^1,\ldots ,a_{ij}^q+b_{ij}^q\right) ,\left( c_{kl}^1+d_{kl}^1,\ldots ,c_{kl}^p+d_{kl}^p\right) \right) \\&\quad =\left( \left( a_{ij}^1,\ldots ,a_{ij}^q,c_{kl}^1,\ldots ,c_{kl}^p\right) +\left( b_{ij}^1,\ldots ,b_{ij}^q,d_{kl}^1,\ldots ,d_{kl}^p\right) \right) \\&\quad =\left( a_{ij},c_{kl}\right) +\left( b_{ij},d_{kl}\right) \\&\quad =\left( \varvec{A}\overline{\circ }\varvec{C}\right) _{ijkl}+\left( \varvec{B}\overline{\circ }\varvec{D}\right) _{ijkl} =\left( \left( \varvec{A}\overline{\circ }\varvec{C}\right) +\left( \varvec{B}\overline{\circ }\varvec{D}\right) \right) _{ijkl} \end{aligned} \end{aligned}$$
(56)

where

$$\begin{aligned} (\varvec{A})_{ij}&=\left( a^1_{ij},\ldots ,a^q_{ij}\right) ,\quad (\varvec{B})_{ij}=\left( b^1_{ij},\ldots ,b^q_{ij}\right) ,\quad (\varvec{C})_{kl}=\left( c^1_{kl},\ldots ,c^p_{kl}\right) ,\nonumber \\ (\varvec{D})_{kl}&=\left( d^1_{kl},\ldots ,d^p_{kl}\right) \end{aligned}$$
(57)

Therefore, (55) holds.

Derivation of equality (4) in (43)

First, Consider the following

$$\begin{aligned} \begin{aligned} \left( \left( \varvec{\mathcal {A}}\times _3\varvec{m}\right) \overline{\otimes }\left( \varvec{\mathcal {A}}\times _3\varvec{m}\right) \right) _{ij}=&\, \left( \left( \varvec{\mathcal {A}}\times _3\varvec{m}\right) _{mn},\left( \varvec{\mathcal {A}}\times _3\varvec{m}\right) _{pq}\right) =\left( \sum _{k=1}^{M}a_{mnk}m_k,\sum _{k=1}^{M}a_{pqk}m_k\right) \\ =&\,\sum _{k=1}^{M}\left( a_{mnk}m_k,a_{pqk}m_k\right) =\sum _{k=1}^{M}\left( a_{mnk},a_{pqk}\right) m_k \end{aligned} \end{aligned}$$
(58)

where m, n, p, and q are defined in (35). From (58), it is straightforward to show that

$$\begin{aligned} \left( \left( \varvec{\mathcal {A}}\times _3\varvec{m}\right) ^{\overline{\otimes }q}\right) _{ij} =\sum _{k=1}^{M}\left( a_{i_1 i_2 k},\ldots ,a_{i_{2q-1}i_{2q} k}\right) m_k \end{aligned}$$
(59)

where

$$\begin{aligned} i_{2t-1}=\left\lceil \frac{i-1}{N^{q-t}}\right\rceil + 1,\quad \; \;\; i_{2t}=\left\lceil \frac{j-1}{N_c^{q-t}}\right\rceil + 1,\quad \; \;\; t=1,\ldots ,q. \end{aligned}$$
(60)

To simplify the following calculations, we denote by \(\varvec{\alpha }_{ijk}\), and \(\varvec{\beta }_{ijk} \) the q-tuple \( \left( a_{i_1 i_2 k},\ldots ,a_{i_{2q-1}i_{2q} k}\right) \), and the product of its entries \( \left( a_{i_1 i_2 k}\times \cdots \times a_{i_{2q-1}i_{2q} k}\right) \), respectively. Now, we define two tensors \(\varvec{\mathcal {Y}} \) and \(\varvec{\mathcal {Z}} \) as

$$\begin{aligned} \varvec{\mathcal {Y}}=\text {Cum}\left[ (\varvec{\mathcal {A}}\times _3\varvec{m})^{\overline{\otimes } q}\overline{\circ } (\varvec{\mathcal {A}}\times _3\varvec{m})^{*\overline{\otimes } q}\right] ,\quad \varvec{\mathcal {Z}}=\varvec{\mathcal {A}}^{\otimes q} \times _3\text {Cum}\left[ \varvec{m}^{\overline{\otimes } q}\overline{\circ } \varvec{m}^{*\overline{\otimes } q}\right] \times _3 \varvec{\mathcal {A}}^{*\otimes q} \end{aligned}$$
(61)

Using (59), we have

$$\begin{aligned} \begin{aligned} \varvec{\mathcal {Y}}_{ijrs}=&\,\text {Cum}\left[ \left( \left( \varvec{\mathcal {A}}\times _3\varvec{m}\right) ^{\overline{\otimes } q}\right) _{ij}, \left( \left( \varvec{\mathcal {A}}\times _3\varvec{m}\right) ^{*\overline{\otimes } q}\right) _{rs}\right] =\text {Cum}\left[ \sum _{k=1}^{M}\varvec{\alpha }_{ijk}m_k,\sum _{k=1}^{M}\varvec{\alpha }^*_{rsk}m^*_k\right] \\ =&\,\text {Cum}\left[ \sum _{k=1}^{M}\left( \varvec{\alpha }_{ijk}m_k,\varvec{\alpha }^*_{rsk}m_k^*\right) \right] =^1\sum _{k=1}^{M}\left( \text {Cum}\left[ \varvec{\alpha }_{ijk}m_k,\varvec{\alpha }^*_{rsk}m_k^*\right] \right) \\ =&\,^2\sum _{k=1}^{M}\left( \varvec{\beta }_{ijk}\text {Cum}\left[ m_k^{\overline{\otimes }q},m_k^{*\overline{\otimes }q}\right] \varvec{\beta }^*_{rsk}\right) \end{aligned} \end{aligned}$$
(62)

where the second and first equalities are, respectively, due to the 3rd and 4th properties of the cumulants, and the fact that sources are independent from each other. Now, from (61) it can be written

$$\begin{aligned} \begin{aligned} \varvec{\mathcal {Z}}_{ijrs}=&\,\left( \varvec{\mathcal {A}}^{\otimes q} \times _3\text {Cum}\left[ \varvec{m}^{\overline{\otimes } q}\overline{\circ } \varvec{m}^{*\overline{\otimes } q}\right] \times _3 \varvec{\mathcal {A}}^{*\otimes q}\right) _{ijrs}\\ =&\,\sum _{k_1=1}^{M^q}\left( \varvec{\mathcal {A}}^{\otimes q} \times _3\text {Cum}\left[ \varvec{m}^{\overline{\otimes } q}\overline{\circ } \varvec{m}^{*\overline{\otimes } q}\right] \right) _{ijk_1}\varvec{\beta }^*_{rsk_1}\\ =&\,\sum _{k_1=1}^{M^q}\left( \sum _{k_2=1}^{M}\varvec{\beta }_{ijk_2}\left( \text {Cum}\left[ \varvec{m}^{\overline{\otimes } q}\overline{\circ } \varvec{m}^{*\overline{\otimes } q}\right] \right) _{k_1k_2}\right) \varvec{\beta }^*_{rsk_1}\\ =&\,^3\sum _{k=1}^{M}\left( \varvec{\beta }_{ijk}\text {Cum}\left[ m_{k}^{\overline{\otimes } q}, m_{k}^{*\overline{\otimes } q}\right] \varvec{\beta }^*_{rsk}\right) \end{aligned} \end{aligned}$$
(63)

where the third equality holds because the sources are independent, and thus, \( \left( \text {Cum}\left[ \varvec{m}^{\overline{\otimes } q}\overline{\circ } \varvec{m}^{*\overline{\otimes } q}\right] \right) _{k_1k_2}=0\), for \( k_1\ne k_2\). So, as the right-hand side of (62) and (63) are equal, it can be deduced that

$$\begin{aligned} \varvec{\mathcal {Y}}_{ijrs}=\varvec{\mathcal {Z}}_{ijrs}\Rightarrow \varvec{\mathcal {Y}}=\varvec{\mathcal {Z}} \end{aligned}$$
(64)

Sixth-order cumulant

For a zero-mean random vector \(\varvec{x} \) the sixth-order cumulants can be calculated as

$$\begin{aligned}&\text {Cum}\left[ x_i,x_j,x_k,x_l^*,x_m^*,x_n^*\right] \nonumber \\&\quad =\text {E}\left[ x_i x_j x_k x_l^* x_m^* x_n^* \right] -\text {E}\left[ x_ix_j\right] \text {E}\left[ x_kx_l^*x_m^*x_n^*\right] -\text {E}\left[ x_ix_k\right] \text {E}\left[ x_jx_l^*x_m^*x_n^*\right] \nonumber \\&\qquad -\text {E}\left[ x_ix_l^*\right] \text {E}\left[ x_jx_kx_m^*x_n^*\right] -\text {E}\left[ x_ix_m^*\right] \text {E}\left[ x_jx_kx_l^*x_n^*\right] -\text {E}\left[ x_ix_n^*\right] \text {E}\left[ x_jx_kx_l^*x_m^*\right] \nonumber \\&\qquad -\text {E}\left[ x_jx_k\right] \text {E}\left[ x_ix_l^*x_m^*x_n^*\right] -\text {E}\left[ x_jx_l^*\right] \text {E}\left[ x_ix_kx_m^*x_n^*\right] -\text {E}\left[ x_jx_m^*\right] \text {E}\left[ x_ix_kx_l^*x_n^*\right] \nonumber \\&\qquad -\text {E}\left[ x_jx_n^*\right] \text {E}\left[ x_ix_kx_l^*x_m^*\right] -\text {E}\left[ x_kx_l^*\right] \text {E}\left[ x_ix_jx_m^*x_n^*\right] -\text {E}\left[ x_kx_m^*\right] \text {E}\left[ x_ix_jx_l^*x_n^*\right] \nonumber \\&\qquad -\text {E}\left[ x_kx_n^*\right] \text {E}\left[ x_ix_jx_l^*x_m^*\right] -\text {E}\left[ x_l^*x_m^*\right] \text {E}\left[ x_ix_jx_kx_n^*\right] -\text {E}\left[ x_l^*x_n^*\right] \text {E}\left[ x_ix_jx_kx_m^*\right] \nonumber \\&\qquad -\text {E}\left[ x_m^*x_n^*\right] \text {E}\left[ x_ix_jx_kx_l^*\right] -\text {E}\left[ x_ix_jx_k\right] \text {E}\left[ x_l^*x_m^*x_n^*\right] -\text {E}\left[ x_ix_jx_l^*\right] \text {E}\left[ x_kx_m^*x_n^*\right] \nonumber \\&\qquad -\text {E}\left[ x_ix_jx_m^*\right] \text {E}\left[ x_kx_l^*x_n^*\right] -\text {E}\left[ x_ix_jx_n^*\right] \text {E}\left[ x_kx_l^*x_m^*\right] -\text {E}\left[ x_ix_kx_l^*\right] \text {E}\left[ x_jx_m^*x_n^*\right] \nonumber \\&\qquad -\text {E}\left[ x_ix_kx_m^*\right] \text {E}\left[ x_jx_l^*x_n^*\right] -\text {E}\left[ x_ix_kx_n^*\right] \text {E}\left[ x_jx_l^*x_m^*\right] -\text {E}\left[ x_ix_l^*x_m^*\right] \text {E}\left[ x_jx_kx_n^*\right] \nonumber \\&\qquad -\text {E}\left[ x_ix_l^*x_n^*\right] \text {E}\left[ x_jx_kx_m^*\right] -\text {E}\left[ x_ix_m^*x_n^*\right] \text {E}\left[ x_jx_kx_l^*\right] +2\left( \text {E}\left[ x_ix_j\right] \text {E}\left[ x_kx_l^*\right] \text {E}\left[ x_m^*x_n^*\right] \right. \nonumber \\&\qquad +\text {E}\left[ x_ix_j\right] \text {E}\left[ x_kx_m^*\right] \text {E}\left[ x_l^*x_n^*\right] +\text {E}\left[ x_ix_j\right] \text {E}\left[ x_kx_n^*\right] \text {E}\left[ x_l^*x_m^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_k\right] \text {E}\left[ x_jx_l^*\right] \text {E}\left[ x_m^*x_n^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_k\right] \text {E}\left[ x_jx_m^*\right] \text {E}\left[ x_l^*x_n^*\right] +\text {E}\left[ x_ix_k\right] \text {E}\left[ x_jx_n^*\right] \text {E}\left[ x_l^*x_m^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_l^*\right] \text {E}\left[ x_jx_k\right] \text {E}\left[ x_m^*x_n^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_l^*\right] \text {E}\left[ x_jx_m^*\right] \text {E}\left[ x_kx_n^*\right] +\text {E}\left[ x_ix_l^*\right] \text {E}\left[ x_jx_n^*\right] \text {E}\left[ x_kx_m^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_m^*\right] \text {E}\left[ x_jx_k\right] \text {E}\left[ x_l^*x_n^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_m^*\right] \text {E}\left[ x_jx_l^*\right] \text {E}\left[ x_kx_n^*\right] +\text {E}\left[ x_ix_m^*\right] \text {E}\left[ x_jx_n^*\right] \text {E}\left[ x_kx_l^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_n^*\right] \text {E}\left[ x_jx_k\right] \text {E}\left[ x_l^*x_m^*\right] \nonumber \\&\qquad +\text {E}\left[ x_ix_n^*\right] \text {E}\left[ x_jx_l^*\right] \text {E}\left[ x_kx_m^*\right] \nonumber \\&\qquad \left. +\text {E}\left[ x_ix_n^*\right] \text {E}\left[ x_jx_m^*\right] \text {E}\left[ x_kx_l^*\right] \right) \end{aligned}$$
(65)

The (IJ)th element of the sixth-order cumulant matrix \(\varvec{C}_{6,\varvec{x}} \) for K snapshots, can be estimated as

$$\begin{aligned} c_{IJ} =&\,\frac{1}{K}\left( \sum _{g=1}^{G}x_i x_j x_k x_l^* x_m^* x_n^* -\left( \sum _{g=1}^{G}x_ix_j\right) \left( \sum _{g=1}^{G}x_kx_l^*x_m^*x_n^*\right) \right. \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_k\right) \left( \sum _{g=1}^{G}x_jx_l^*x_m^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_l^*\right) \left( \sum _{g=1}^{G}x_jx_kx_m^*x_n^*\right) -\left( \sum _{g=1}^{G}x_ix_m^*\right) \left( \sum _{g=1}^{G}x_jx_kx_l^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_n^*\right) \left( \sum _{g=1}^{G}x_jx_kx_l^*x_m^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_jx_k\right) \left( \sum _{g=1}^{G}x_ix_l^*x_m^*x_n^*\right) -\left( \sum _{g=1}^{G}x_jx_l^*\right) \left( \sum _{g=1}^{G}x_ix_kx_m^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_jx_m^*\right) \left( \sum _{g=1}^{G}x_ix_kx_l^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_jx_n^*\right) \left( \sum _{g=1}^{G}x_ix_kx_l^*x_m^*\right) -\left( \sum _{g=1}^{G}x_kx_l^*\right) \left( \sum _{g=1}^{G}x_ix_jx_m^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_kx_m^*\right) \left( \sum _{g=1}^{G}x_ix_jx_l^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_kx_n^*\right) \left( \sum _{g=1}^{G}x_ix_jx_l^*x_m^*\right) -\left( \sum _{g=1}^{G}x_l^*x_m^*\right) \left( \sum _{g=1}^{G}x_ix_jx_kx_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_l^*x_n^*\right) \left( \sum _{g=1}^{G}x_ix_jx_kx_m^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_m^*x_n^*\right) \left( \sum _{g=1}^{G}x_ix_jx_kx_l^*\right) -\left( \sum _{g=1}^{G}x_ix_jx_k\right) \left( \sum _{g=1}^{G}x_l^*x_m^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_jx_l^*\right) \left( \sum _{g=1}^{G}x_kx_m^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_jx_m^*\right) \left( \sum _{g=1}^{G}x_kx_l^*x_n^*\right) -\left( \sum _{g=1}^{G}x_ix_jx_n^*\right) \left( \sum _{g=1}^{G}x_kx_l^*x_m^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_kx_l^*\right) \left( \sum _{g=1}^{G}x_jx_m^*x_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_kx_m^*\right) \left( \sum _{g=1}^{G}x_jx_l^*x_n^*\right) -\left( \sum _{g=1}^{G}x_ix_kx_n^*\right) \left( \sum _{g=1}^{G}x_jx_l^*x_m^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_l^*x_m^*\right) \left( \sum _{g=1}^{G}x_jx_kx_n^*\right) \nonumber \\&-\left( \sum _{g=1}^{G}x_ix_l^*x_n^*\right) \left( \sum _{g=1}^{G}x_jx_kx_m^*\right) -\left( \sum _{g=1}^{G}x_ix_m^*x_n^*\right) \left( \sum _{g=1}^{G}x_jx_kx_l^*\right) \nonumber \\&+2\left( \left( \sum _{g=1}^{G}x_ix_j\right) \left( \sum _{g=1}^{G}x_kx_l^*\right) \left( \sum _{g=1}^{G}x_m^*x_n^*\right) \right. \nonumber \\&+\left( \sum _{g=1}^{G}x_ix_j\right) \left( \sum _{g=1}^{G}x_kx_m^*\right) \left( \sum _{g=1}^{G}x_l^*x_n^*\right) +\left( \sum _{g=1}^{G}x_ix_j\right) \left( \sum _{g=1}^{G}x_kx_n^*\right) \left( \sum _{g=1}^{G}x_l^*x_m^*\right) \nonumber \\&+\left( \sum _{g=1}^{G}x_ix_k\right) \left( \sum _{g=1}^{G}x_jx_l^*\right) \left( \sum _{g=1}^{G}x_m^*x_n^*\right) +\left( \sum _{g=1}^{G}x_ix_k\right) \left( \sum _{g=1}^{G}x_jx_m^*\right) \left( \sum _{g=1}^{G}x_l^*x_n^*\right) \nonumber \\&+\left( \sum _{g=1}^{G}x_ix_k\right) \left( \sum _{g=1}^{G}x_jx_n^*\right) \left( \sum _{g=1}^{G}x_l^*x_m^*\right) +\left( \sum _{g=1}^{G}x_ix_l^*\right) \left( \sum _{g=1}^{G}x_jx_k\right) \left( \sum _{g=1}^{G}x_m^*x_n^*\right) \nonumber \\&+\left( \sum _{g=1}^{G}x_ix_l^*\right) \left( \sum _{g=1}^{G}x_jx_m^*\right) \left( \sum _{g=1}^{G}x_kx_n^*\right) +\left( \sum _{g=1}^{G}x_ix_l^*\right) \left( \sum _{g=1}^{G}x_jx_n^*\right) \left( \sum _{g=1}^{G}x_kx_m^*\right) \nonumber \\&+\left( \sum _{g=1}^{G}x_ix_m^*\right) \left( \sum _{g=1}^{G}x_jx_k\right) \left( \sum _{g=1}^{G}x_l^*x_n^*\right) +\left( \sum _{g=1}^{G}x_ix_m^*\right) \left( \sum _{g=1}^{G}x_jx_l^*\right) \left( \sum _{g=1}^{G}x_kx_n^*\right) \nonumber \\&+\left( \sum _{g=1}^{G}x_ix_m^*\right) \left( \sum _{g=1}^{G}x_jx_n^*\right) \left( \sum _{g=1}^{G}x_kx_l^*\right) +\left( \sum _{g=1}^{G}x_ix_n^*\right) \left( \sum _{g=1}^{G}x_jx_k\right) \left( \sum _{g=1}^{G}x_l^*x_m^*\right) \nonumber \\&+\left. \left. \left( \sum _{g=1}^{G}x_ix_n^*\right) \left( \sum _{g=1}^{G}x_jx_l^*\right) \left( \sum _{g=1}^{G}x_kx_m^*\right) +\left( \sum _{g=1}^{G}x_ix_n^*\right) \left( \sum _{g=1}^{G}x_jx_m^*\right) \left( \sum _{g=1}^{G}x_kx_l^*\right) \right) \right) \end{aligned}$$
(66)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Barat, M., Karimi, M. & Masnadi-Shirazi, M.A. Direction of arrival estimation in vector-sensor arrays using higher-order statistics. Multidim Syst Sign Process 32, 161–187 (2021). https://doi.org/10.1007/s11045-020-00734-z

Download citation

Keywords

  • Direction finding
  • Vector sensor
  • Multilinear algebra
  • Higher-order statistics
  • 2q-MUSIC