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.

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 (i, j) of the left-hand side of (52) is equal to entry (i, j) 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 (i, j, k, l) 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 (I, J)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)