Appendix
The Cramer–Rao lower bound (CRB) of 1D-DOA estimation in the presence of mutual coupling is given in [12, 37]. In this Appendix, the CRB of joint 2D-DOA and mutual coupling estimation is derived. Consider the array output vector \(\mathbf {x}\left( t \right) \) as a complex Gaussian vector with zero mean. Define \(\mathbf {A} \buildrel \varDelta \over = \left[ {{\mathbf {A}_c},{\mathbf {A}_u}} \right] \), \(\mathbf {E} \buildrel \varDelta \over = blkdiag\left[ {\varvec{\Gamma } ,{\mathbf {I}_{{K_u}}}} \right] \) , the covariance matrix of \(\mathbf {x}\left( t \right) \) is expressed as
$$\begin{aligned} {\mathbf {R}_x} = E\left\{ {\mathbf {x}\left( t \right) \mathbf {x}{{\left( t \right) }^H}} \right\} = \mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} \end{aligned}$$
(64)
For L statistically independent observations of \(\mathbf {x}\left( t \right) \), the logarithm of the likelihood function (the joint probability density function, PDF) can be written as [2, 38]
$$\begin{aligned} \varvec{\Theta }&= \ln \left\{ {f\left( {\mathbf {x}\left( 1 \right) ,\mathbf {x}\left( 2 \right) , \ldots ,\mathbf {x}\left( L \right) } \right) } \right\} \nonumber \\&= const - L \cdot \ln \left\{ {\det \left\{ {{\mathbf {R}_x}} \right\} } \right\} \nonumber \\&-\sum \limits _{t = 1}^L {\mathbf {x}{{\left( t \right) }^H}\mathbf {R}_x^{ - 1}\mathbf {x}\left( t \right) }\nonumber \\&= const - L \cdot \ln \left\{ {\det \left\{ {{\mathbf {R}_x}} \right\} } \right\} \nonumber \\&- L \cdot tr\left\{ {\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x}} \right\} \end{aligned}$$
(65)
where the estimate of auto-correlation covariance \({\widehat{\mathbf {R}}_x}\) is given by
$$\begin{aligned} {\widehat{\mathbf {R}}_x} = \frac{1}{L}\sum \limits _{l = 1}^L {\mathbf {x}\left( l \right) \mathbf {x}{{\left( l \right) }^H}}. \end{aligned}$$
(66)
The unknown parameter vector of \({\mathbf {R}_x}\) is defined as
$$\begin{aligned} \varvec{\eta }&= {\left[ {{\varvec{\alpha } ^T},{\varvec{\beta } ^T},{\varvec{\mu } ^T},{\varvec{\nu } ^T},{\varvec{\kappa } ^T},{\varvec{\varsigma } ^T}} \right] ^T} \nonumber \\ \varvec{\alpha }&= \left[ {{\alpha _{11}}, \ldots ,{\alpha _{1{L_1}}}, \ldots ,{\alpha _{P_1}}, \ldots ,{\alpha _{P{L_P}}},} \right. \nonumber \\&\,{\left. {{\alpha _{{K_c} + 1}}, \ldots ,{\alpha _K}} \right] ^T} \nonumber \\ \varvec{\beta }&= \left[ {{\beta _{11}}, \ldots ,{\beta _{1{L_1}}}, \ldots ,{\beta _{P_1}}, \ldots ,{\beta _{P{L_P}}},} \right. \nonumber \\&\,{\left. {{\beta _{{K_c} + 1}}, \ldots ,{\beta _K}} \right] ^T} \end{aligned}$$
(67)
In order to obtain the unique CRB of \({\varvec{\rho } _k}\), the fading coefficient of the signal in the pth group is normalized to unity. For the sake of simplify, \({\alpha _{p1}}\) is assumed to be the smallest DOA in the pth group in the following derivation. \(\varvec{\mu } = {\left[ {{\mu _{12}}, \ldots ,{\mu _{1{L_1}}}, \ldots ,{\mu _{P_2}}, \ldots ,{\mu _{P{L_P}}}} \right] ^T}\) and \(\varvec{\nu } = {\left[ {{\nu _{12}}, \ldots ,{\nu _{1{L_1}}}, \ldots ,{\nu _{P_2}}, \ldots ,{\nu _{P{L_P}}}} \right] ^T}\) are defined as the real part and \(\varvec{\varpi } = {\left[ {{\varvec{\rho } _1}{{\left( {2:{P_1}} \right) }^T}, \ldots ,{\varvec{\rho } _D}{{\left( {2:{P_D}} \right) }^T}} \right] ^T}\) is defined as the imaginary part, respectively. \(\varvec{\kappa } \) and \(\varvec{\varsigma } \) are defined as the real part and imaginary part of \({\mathbf {c}_1}\), respectively. The kth element in a vector, for example, \(\varvec{\eta } \) is defined as \({\eta _k}\). Then the general expression of the \(\left( {m,n} \right) \)th element in Fisher information matrix (FIM) can be expressed as
$$\begin{aligned} {\mathbf {F}_{{\eta _m}{\eta _n}}} = - E\left\{ {\frac{{{\partial ^2}\varTheta }}{{\partial {\eta _m}\partial {\eta _n}}}} \right\} \end{aligned}$$
(68)
Based on the relationship
$$\begin{aligned}&\frac{{\partial \mathbf {R}_x^{ - 1}}}{{\partial {\eta _m}}} = - \mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}\mathbf {R}_x^{ - 1}\end{aligned}$$
(69)
$$\begin{aligned}&\frac{{\partial \ln \left\{ {\det \left\{ {{\mathbf {R}_x}} \right\} } \right\} }}{{\partial {\eta _m}}} = tr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}} \right\} \end{aligned}$$
(70)
the first derivative of \(\varTheta \) is obtained
$$\begin{aligned} \frac{{\partial \varTheta }}{{\partial {\eta _m}}}&= - Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}} \right\} \nonumber \\&+\,Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x}} \right\} \nonumber \\&= Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}\left( {\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x} - \mathbf {I}} \right) } \right\} . \end{aligned}$$
(71)
The second derivative of \(\varTheta \) is given by
$$\begin{aligned} \frac{{\partial \varTheta }}{{\partial {\eta _m}{\eta _n}}}&= Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}\left( {\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x} - \mathbf {I}} \right) } \right\} \nonumber \\&= Ltr\left\{ {\left( {{{\partial \left( {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}} \right) } \Big / {\partial {\eta _n}}}} \right) \left( {\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x} - \mathbf {I}} \right) } \right\} \nonumber \\&+\,Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}\frac{{\partial \left( {\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x} - \mathbf {I}} \right) }}{{\partial {\eta _n}}}} \right\} \nonumber \\&= Ltr\left\{ {\left( {{{\partial \left( {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}} \right) } \Big / {\partial {\eta _n}}}} \right) \left( {\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x} - \mathbf {I}} \right) } \right\} \nonumber \\&-\,Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}\left( {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _n}}}\mathbf {R}_x^{ - 1}{{\widehat{\mathbf {R}}}_x}} \right) } \right\} . \end{aligned}$$
(72)
Due to \(E\left\{ {{{\widehat{\mathbf {R}}}_x}} \right\} = {\mathbf {R}_x}\), the expectation of both sides of (70) is taken, we have
$$\begin{aligned} {\mathbf {F}_{{\eta _m}{\eta _n}}} = Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _m}}}\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\eta _n}}}} \right\} . \end{aligned}$$
(73)
In the subsequent derivation process, the FIM expression of mixed signals is derived. The notation \({\widetilde{\mathbf {R}}_{{\eta _m}}}\) defines \({{\partial \mathbf {R}} / {\partial {\eta _m}}}\).
1.1 Derivatives with respect to DOA
Based on the expression of covariance matrix (64), the partial derivative of \({\mathbf {R}_x}\) with respect to the mth element \({\alpha _m}\) of \(\varvec{\alpha } \) can be written as
$$\begin{aligned} \frac{{\partial {\mathbf {R}_x}}}{{\partial {\alpha _m}}}&= \mathbf {C}{\widetilde{\mathbf {A}}_{{\alpha _m}}}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} \nonumber \\&+\,\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}\widetilde{\mathbf {A}}_{{\alpha _m}}^H{\mathbf {C}^H}. \end{aligned}$$
(74)
Based on \(tr\left\{ {\mathbf {R} + {\mathbf {R}^H}} \right\} = 2\mathrm{Re} \left\{ {tr\left\{ \mathbf {R} \right\} } \right\} \), we have
$$\begin{aligned} {\mathbf {F}_{{\alpha _m}{\alpha _n}}}&= Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\alpha _m}}}\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\alpha _n}}}} \right\} \nonumber \\&= Ltr\left\{ {\mathbf {R}_x^{ - 1}\left( {\mathbf {C}{{\widetilde{\mathbf {A}}}_{{\alpha _m}}}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} } \right. } \right. \nonumber \\&\left. +\,{\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}\widetilde{\mathbf {A}}_{{\alpha _m}}^H{\mathbf {C}^H}} \right) \times \mathbf {R}_x^{ - 1} \nonumber \\&\left( {\mathbf {C}{{\widetilde{\mathbf {A}}}_{{\alpha _n}}}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} } \right. \nonumber \\&\left. {\left. +\, {\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}\widetilde{\mathbf {A}}_{{\alpha _n}}^H{\mathbf {C}^H}} \right) } \right\} \nonumber \\&= 2L\mathrm{Re} \left\{ {tr\left\{ {\mathbf {R}_x^{ - 1}C{{\widetilde{\mathbf {A}}}_{{\alpha _m}}}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H} } \right. } \right. \nonumber \\&\times \, {\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\widetilde{\mathbf {A}}_{{\alpha _n}}}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} \nonumber \\&+\,\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}\widetilde{\mathbf {A}}_{{\alpha _m}}^H \nonumber \\&\left. {\left. \times \,{ {\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\widetilde{\mathbf {A}}}_{{\alpha _n}}}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}} \right\} } \right\} \end{aligned}$$
(75)
Since only the mth column of \({\widetilde{\mathbf {A}}_{{\alpha _m}}}\) is nonzero, then \({\widetilde{\mathbf {A}}_{{\alpha _m}}}\) can be represented as \({\widetilde{\mathbf {A}}_{{\alpha _m}}} = {\mathbf {A}_\alpha }\varvec{\gamma } _K^m{\left( {\varvec{\gamma } _K^m} \right) ^T}\), where the mth column of the identity matrix is defined as \(\varvec{\gamma } _K^m\). \({\mathbf {A}_ {\varvec{\alpha }} }\) is the derivative matrix of the array manifold matrix, which is expressed as
$$\begin{aligned} {\mathbf {A}_ {\varvec{\alpha }} }&= \left[ {\frac{{d\mathbf {a}\left( {{\alpha _{11}}} \right) }}{{d{\alpha _{11}}}}, \ldots ,\frac{{d\mathbf {a}\left( {{\alpha _{1{P_1}}}} \right) }}{{d{\alpha _{1{P_1}}}}}, \ldots ,\frac{{d\mathbf {a}\left( {{\alpha _{D1}}} \right) }}{{d{\alpha _{D1}}}},} \right. \nonumber \\&{\left. { \ldots ,\frac{{d\mathbf {a}\left( {{\alpha _{D{P_D}}}} \right) }}{{d{\alpha _{D{P_D}}}}},\frac{{d\mathbf {a}\left( {{\alpha _{{K_c} + 1}}} \right) }}{{d{\alpha _{{K_c} + 1}}}}, \ldots ,\frac{{d\mathbf {a}\left( {{\alpha _K}} \right) }}{{d{\alpha _K}}}} \right] ^T} \end{aligned}$$
(76)
Then (73) can be written as
$$\begin{aligned} {\mathbf {F}_{{\alpha _m}{\alpha _n}}}&= 2L\mathrm{Re} \left\{ {tr} \right. \left\{ {\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }\varvec{\gamma } _K^m{{\left( {\varvec{\gamma } _K^m} \right) }^T}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}} \right. \nonumber \\&\times \,{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }\varvec{\gamma } _K^n{\left( {\varvec{\gamma } _K^n} \right) ^T}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} \nonumber \\&+\, \mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}\varvec{\gamma } _K^m{\left( {\varvec{\gamma } _K^m} \right) ^T}\mathbf {A}_ {\varvec{\alpha }} ^H \times \nonumber \\&\left. {\left. {{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }\varvec{\gamma } _K^n{{\left( {\varvec{\gamma } _K^n} \right) }^T}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}} \right\} } \right\} \nonumber \\&= 2L\mathrm{Re} \left\{ {\left( {{{\left( {\varvec{\gamma } _K^m} \right) }^T}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }\varvec{\gamma } _K^n} \right) } \right. \nonumber \\&\times \,\left( {{{\left( {\varvec{\gamma } _K^n} \right) }^T}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }\varvec{\gamma } _K^m} \right) \nonumber \\&+ \left( {{{\left( {\varvec{\gamma } _K^m} \right) }^T}\mathbf {A}_ {\varvec{\alpha }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }\varvec{\gamma } _K^n} \right) \times \nonumber \\&\left. {\left( {{{\left( {\varvec{\gamma } _K^n} \right) }^T}\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}\varvec{\gamma } _K^m} \right) } \right\} \nonumber \\ \end{aligned}$$
(77)
Then the FIM that corresponds to \(\varvec{\alpha } \) can be expressed as
$$\begin{aligned} {\mathbf {F}_{\varvec{\alpha \alpha } }}&= 2L\mathrm{Re} \left\{ {\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }} \right) } \right. \nonumber \\&\odot \,{\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }} \right) ^T} \nonumber \\&+ \left( {\mathbf {A}_ {\varvec{\alpha }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_ {\varvec{\alpha }} }} \right) \nonumber \\&\odot \,\left. {{{\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}} \right) }^T}} \right\} \end{aligned}$$
(78)
where \( \odot \) denotes the Hadamard product. Similarly, the FIM that corresponds to \(\varvec{\beta } \) can be expressed as
$$\begin{aligned} {\mathbf {F}_{\varvec{\beta \beta } }}&= 2L\mathrm{Re} \left\{ {\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) } \right. \nonumber \\&\odot \,{\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) ^T} \nonumber \\&+\,\left( {\mathbf {A}_{\varvec{\beta }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) \nonumber \\&\odot \,\left. {{{\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}} \right) }^T}} \right\} \end{aligned}$$
(79)
where
$$\begin{aligned} {\mathbf {A}_{\varvec{\alpha }} }&= \left[ {\frac{{d\mathbf {a}\left( {{\beta _{11}}} \right) }}{{d{\beta _{11}}}}, \ldots ,\frac{{d\mathbf {a}\left( {{\beta _{1{P_1}}}} \right) }}{{d{\beta _{1{P_1}}}}}, \ldots ,\frac{{d\mathbf {a}\left( {{\beta _{D1}}} \right) }}{{d{\beta _{D1}}}},} \right. \nonumber \\&{\left. { \ldots ,\frac{{d\mathbf {a}\left( {{\beta _{D{P_D}}}} \right) }}{{d{\beta _{D{P_D}}}}},\frac{{d\mathbf {a}\left( {{\beta _{{K_c} + 1}}} \right) }}{{d{\beta _{{K_c} + 1}}}}, \ldots ,\frac{{d\mathbf {a}\left( {{\beta _K}} \right) }}{{d{\beta _K}}}} \right] ^T}\!.\nonumber \\ \end{aligned}$$
(80)
The FIM that corresponds to the cross terms between \(\varvec{\alpha } \) and \(\varvec{\beta } \) is
$$\begin{aligned} {\mathbf {F}_{\varvec{\alpha \beta } }}&= 2L\mathrm{Re} \left\{ {\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) } \right. \nonumber \\&\odot \,{\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\alpha }} }} \right) ^T} \nonumber \\&+ \left( {\mathbf {A}_{\varvec{\alpha }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) \nonumber \\&\odot \,\left. {{{\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}} \right) }^T}} \right\} \end{aligned}$$
(81)
1.2 Derivatives with respect to fading coefficients
\({\overline{\mathbf {A}} _c} = \left[ {{\mathbf {A}_{c1}}\left( {1:N,2:{L_1}} \right) , \ldots ,\left. {{\mathbf {A}_{cD}}\left( {1:N,2:{L_P}} \right) } \right] } \right. \), and \(\overline{\varvec{\Gamma }} = blkdiag\left\{ {{\varvec{\rho } _1}\left( {2:{L_1}} \right) , \ldots ,{\varvec{\rho } _D}\left( {2:{L_P}} \right) } \right\} \). The matrix \({\varvec{\Psi } _\mathbf {r}} = blkdiag\left\{ {{\mathbf {1}_{\left( {{L_1} - 1} \right) \times 1}}, \ldots ,{\mathbf {1}_{\left( {{L_P} - 1} \right) \times 1}}} \right\} \) and \({\varvec{\Psi } _\mathbf {i}} = blkdiag\left\{ {{\mathbf {j}_{\left( {{L_1} - 1} \right) \times 1}}, \ldots ,{\mathbf {j}_{\left( {{L_P} - 1} \right) \times 1}}} \right\} \), where all the elements of the vector \(\mathbf {1}\) are equal to 1 and all the elements of the vector \(\mathbf {j}\) are equal to the imaginary unit \(j\). According to (73), the \(\left( {{r_m},{r_n}} \right) \)th element of the FIM with respect to fading coefficients can be expressed as
$$\begin{aligned} {\mathbf {F}_{{\mu _m}{\mu _n}}}&= Ltr\left\{ {\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\mu _m}}}\mathbf {R}_x^{ - 1}\frac{{\partial {\mathbf {R}_x}}}{{\partial {\mu _n}}}} \right\} \nonumber \\&= Ltr\left\{ {\mathbf {R}_x^{ - 1}\left( {\mathbf {CA}{{\widetilde{\mathbf {E}}}_{{\mu _m}}}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} } \right. } \right. \nonumber \\&+\,\left. {\mathbf {CAE}{\mathbf {R}_s}\mathbf {E}_{{\mu _m}}^H{\mathbf {A}^H}{\mathbf {C}^H}} \right) \nonumber \\&\times \, \mathbf {R}_x^{ - 1}\left( {\mathbf {CA}{{\widetilde{\mathbf {E}}}_{{\mu _n}}}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} } \right. \nonumber \\&+\left. {\left. {\mathbf {CAE}{\mathbf {R}_s}\mathbf {E}_{{\mu _n}}^H{\mathbf {A}^H}{\mathbf {C}^H}} \right) } \right\} \nonumber \\&= Ltr\left\{ {\mathbf {R}_x^{ - 1}\left( {\mathbf {C}{{\overline{\mathbf {A}}_c}{{\widetilde{\overline{\varvec{\Gamma }}} }}_{{\mu _m}}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}} \right. } \right. \nonumber \\&+ \left. {\mathbf {C}{\mathbf {A}_c}\varvec{\Gamma } {\mathbf {R}_{cs}}{\widetilde{\overline{\varvec{\Gamma }}}}_{ {\mu _m}}^H\overline{\mathbf {A}} _c^H{\mathbf {C}^H}} \right) \nonumber \\&\times \, \mathbf {R}_x^{- 1}\left( {\mathbf {C}{{\overline{\mathbf {A}}}}_c} {{\widetilde{\overline{\varvec{\Gamma }}} }}_{{\mu _n}} {\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H} \right. \nonumber \\&+ \,\left. {\left. {\mathbf {C}{\mathbf {A}_c}\varvec{\Gamma } {\mathbf {R}_{cs}}\widetilde{\overline{\varvec{\Gamma }}}_{{\mu _n}}^H\overline{\mathbf {A}} _c^H{\mathbf {C}^H}} \right) } \right\} \nonumber \\&= 2L\mathrm{Re} \left\{ {tr\left\{ {\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}{{\widetilde{\overline{\varvec{\Gamma }}}}_{{\mu _m}}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H } \right. } \right. \nonumber \\&\times \,{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\overline{\mathbf {A}} _c}{\widetilde{\overline{\varvec{\Gamma }}}_{{\mu _n}}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H} \nonumber \\&+\,\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_c}\varvec{\Gamma } {\mathbf {R}_{cs}}\widetilde{\overline{\varvec{\Gamma }}}_{{\mu _m}}^H\overline{\mathbf {A}} _c^H \nonumber \\&\left. {\left. {{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}{{\widetilde{\overline{\varvec{\Gamma }}}}_{{\mu _n}}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}} \right\} } \right\} \end{aligned}$$
(82)
Based on \({\overline{\varvec{\Gamma }}}_{{\mu _m}} = {\varvec{\upgamma }} _{{K_c} - P}^m{\left( {\varvec{\upgamma } _{{K_c} - P}^m} \right) ^T}{\varvec{\Psi } _\mathbf {r}}\), the real part of fading coefficients of the FIM can be represented as
$$\begin{aligned} {F_{\mu \mu }}&= 2L\mathrm{Re} \left\{ {\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}{\widetilde{\overline{\varvec{\Gamma }}}}_{\mu _m}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H \right. \nonumber \\&\times \,{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\overline{\mathbf {A}} _c}{\widetilde{\overline{\varvec{\Gamma }}}_{{\mu _n}}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H} \nonumber \\&+\,\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_c}\varvec{\Gamma } {\mathbf {R}_{cs}}\widetilde{\overline{\varvec{\Gamma }}}_{{\mu _m}}^H\overline{\mathbf {A}} _c^H \nonumber \\&\times \, \left. {{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}{{\widetilde{\overline{\varvec{\Gamma }}}}_{{\mu _n}}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}} \right\} \nonumber \\&= 2L\mathrm{Re} \left\{ {\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right. \nonumber \\&\odot \,{\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) ^T} \nonumber \\&+\,\left( {\overline{\mathbf {A}} _c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&\odot \left. {{{\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_c}\varvec{\Gamma } {\mathbf {R}_{cs}}\varvec{\Psi } _\mathbf {r}^H} \right) }^T}} \right\} \end{aligned}$$
(83)
Based on \({\overline{\varvec{\Gamma }} _{{\nu _m}}} = \varvec{\gamma } _{{K_c} - P}^m{\left( {\varvec{\gamma } _{{K_c} - P}^m} \right) ^T}{\varvec{\Psi } _\mathbf {i}}\), the imaginary part of fading coefficients of the FIM can be represented as
$$\begin{aligned} {\mathbf {F}_{\varvec{\nu \nu } }}&= 2L\mathrm{Re} \left\{ {\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right. \nonumber \\&\odot \,{\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) ^T} \nonumber \\&+\left( {\overline{\mathbf {A}} _c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&\odot \,\left. {{{\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_c}\varvec{\Gamma } {\mathbf {R}_{cs}}\varvec{\Psi } _\mathbf {i}^H} \right) }^T}} \right\} \end{aligned}$$
(84)
The FIM that corresponds to the cross terms between \(\varvec{\mu } \) and \(\varvec{\nu }\) is
$$\begin{aligned} {\mathbf {F}_{\varvec{\mu \nu } }}&= 2L\mathrm{Re} \left\{ {\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right. \nonumber \\&\odot \,{\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) ^T} \nonumber \\&+\left( {\overline{\mathbf {A}} _c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&\odot \,\left. {{{\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_c}\varvec{\Gamma } {\mathbf {R}_{cs}}\varvec{\Psi } _\mathbf {r}^H} \right) }^T}} \right\} \end{aligned}$$
(85)
1.3 Derivatives with respect to mutual coupling coefficients
Based on (73), the mth and nth element of \({\mathbf {F}_{\varvec{\kappa \kappa } }} \), \({\mathbf {F}_{\varvec{\varsigma } \varvec{\varsigma } }} \) and \({\mathbf {F}_{\varvec{\kappa \varsigma } }} \) can be given respectively as follows
$$\begin{aligned} {\mathbf {F}_{{\varvec{\kappa } _m}{\varvec{\kappa } _n}}}&= 2L\mathrm{Re} \left\{ {tr\left\{ {\mathbf {R}_x^{ - 1}{{\widetilde{\mathbf {C}}}_{{\varvec{\kappa } _m}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}} \right. } \right. \nonumber \\&\times \, {\mathbf {C}^H}\mathbf {R}_x^{ - 1}{\widetilde{\mathbf {C}}_{{\varvec{\kappa } _n}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} \nonumber \\&+\,\mathbf {R}_x^{ - 1}{\widetilde{\mathbf {C}}_{{\varvec{\kappa } _m}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H} \nonumber \\&\times \, \left. {\left. {{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\mathbf {C}_{{\varvec{\kappa } _n}}^H} \right\} } \right\} \end{aligned}$$
(86)
$$\begin{aligned} {\mathbf {F}_{{\varvec{\varsigma } _m}{\varvec{\varsigma } _n}}}&= 2L\mathrm{Re} \left\{ {tr\left\{ {\mathbf {R}_x^{ - 1}{{\widetilde{\mathbf {C}}}_{{\varvec{\varsigma } _m}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}} \right. } \right. \nonumber \\&\times \, {\mathbf {C}^H}\mathbf {R}_x^{ - 1}{\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _n}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} \nonumber \\&+\,\mathbf {R}_x^{ - 1}{\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _m}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H} \nonumber \\&\times \, \left. {\left. {{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\mathbf {C}_{{\varvec{\varsigma } _n}}^H} \right\} } \right\} \end{aligned}$$
(87)
$$\begin{aligned} {\mathbf {F}_{{\varvec{\kappa } _m}{\varvec{\varsigma } _n}}}&= 2L\mathrm{Re} \left\{ {tr\left\{ {\mathbf {R}_x^{ - 1}{{\widetilde{\mathbf {C}}}_{{\varvec{\kappa } _m}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}} \right. } \right. \nonumber \\&\times \,{\mathbf {C}^H}\mathbf {R}_x^{ - 1}{\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _n}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H} \nonumber \\&+\,\mathbf {R}_x^{ - 1}{\widetilde{\mathbf {C}}_{{\varvec{\kappa } _m}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H} \nonumber \\&\times \,\left. {\left. {{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\mathbf {C}_{{\varvec{\varsigma } _n}}^H} \right\} } \right\} \end{aligned}$$
(88)
where
$$\begin{aligned} {\widetilde{\mathbf {C}}_{{\varvec{\kappa } _m}}}&= toeplitz\left\{ {\left[ {0,{{\left( {\varvec{\gamma } _{{M_0}}^m} \right) }^T},{\mathbf {0}_{1,\left( {M - {M_0} - 1} \right) }}} \right] } \right\} , \nonumber \\ m&= 1, \ldots ,{M_0}. \end{aligned}$$
(89)
\(toeplitz\left\{ \mathbf {z} \right\} \) stands for the symmetric Toeplitz matrix constructed by the vector \(\mathbf {z}\).
1.4 DOA-fading coefficients cross terms
$$\begin{aligned} {\mathbf {F}_{\varvec{\alpha \mu } }}&= 2L\mathrm{Re} \left\{ {\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right. \nonumber \\&\odot \,{\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\alpha }} }} \right) ^T} \nonumber \\&+\,\left( {\mathbf {A}_{\varvec{\alpha }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&\odot \left. {{{\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}} \right) }^T}} \right\} \end{aligned}$$
(90)
$$\begin{aligned} {\mathbf {F}_{\varvec{\alpha \nu } }}&= 2L\mathrm{Re} \left\{ {\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right. \nonumber \\&\odot {\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\alpha }} }} \right) ^T} \nonumber \\&+\,\left( {\mathbf {A}_{\varvec{\alpha }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&\odot \left. {{{\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}} \right) }^T}} \right\} \end{aligned}$$
(91)
$$\begin{aligned} {\mathbf {F}_{\varvec{\beta \mu } }}&= 2L\mathrm{Re} \left\{ {\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right. \nonumber \\&\odot \,{\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta } }}} \right) ^T} \nonumber \\&+\,\left( {\mathbf {A}_{\varvec{\beta }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&\odot \,\left. {{{\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}} \right) }^T}} \right\} \end{aligned}$$
(92)
$$\begin{aligned} {\mathbf {F}_{\varvec{\beta \nu }}}&= 2L\mathrm{Re} \left\{ {\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right. \nonumber \\&\odot \,{\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) ^T} \nonumber \\&+\left( {\mathbf {A}_{\varvec{\beta }} ^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&\odot \,\left. {{{\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}} \right) }^T}} \right\} \end{aligned}$$
(93)
1.5 DOA-mutual coupling coefficients cross terms
$$\begin{aligned} {\mathbf {F}_{{\varvec{\alpha }} {\kappa _n}}}&= 2L\mathrm{Re} \left\{ {diag\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}{{\widetilde{\mathbf {C}}}_{{\varvec{\kappa } _n}}}} \right. } \right. \nonumber \\&\times \,\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\alpha }} }} \right) \nonumber \\&+\,diag\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}} \right. \nonumber \\&\times \, \left. {\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\kappa } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\alpha }} }} \right) } \right\} \end{aligned}$$
(94)
$$\begin{aligned} {\mathbf {F}_{\varvec{\alpha } {\varvec{\varsigma } _n}}}&= 2L\mathrm{Re} \left\{ {diag\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}{{\widetilde{\mathbf {C}}}_{{\varvec{\varsigma } _n}}}} \right. } \right. \nonumber \\&\times \, \left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\alpha }} }} \right) \nonumber \\&+\,\,diag\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}} \right. \nonumber \\&\times \, \left. {\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\alpha }} }} \right) } \right\} \end{aligned}$$
(95)
$$\begin{aligned} {\mathbf {F}_{\varvec{\beta } {\varvec{\kappa } _n}}}&= 2L\mathrm{Re}\left\{ {diag} \right. \left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}{{\widetilde{\mathbf {C}}}_{{\varvec{\kappa } _n}}}} \right. \nonumber \\&\times \, \left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) \nonumber \\&+\,\,diag \left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C} \times } \right. \nonumber \\&\left. {\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\kappa } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) } \right\} \end{aligned}$$
(96)
$$\begin{aligned} {\mathbf {F}_{\varvec{\beta } {\varvec{\varsigma } _n}}}&= 2L\mathrm{Re} \left\{ {diag} \right. \left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}{{\widetilde{\mathbf {C}}}_{{\varvec{\varsigma } _n}}}} \right. \nonumber \\&\times \, \left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) \nonumber \\&+\,\,diag\left( {\mathbf {E}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}} \right. \nonumber \\&\times \, \left. {\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{\mathbf {A}_{\varvec{\beta }} }} \right) } \right\} \end{aligned}$$
(97)
where \(diag\left( \mathbf {A} \right) \) is a column vector constructed by the main diagonal elements of matrix \(\mathbf {A}\).
1.6 Fading coefficients-mutual coupling coefficients cross terms
$$\begin{aligned} {\mathbf {F}_{\varvec{\mu } {\varvec{\kappa } _n}}}&= 2L\mathrm{Re} \left\{ {diag} \right. \left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}} \right. \nonumber \\&\times \left. {{\widetilde{\mathbf {C}}}_{{\varvec{\kappa } _n}}}{\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&+diag\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}} \right. \nonumber \\&\times \left. {\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\kappa } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right\} \end{aligned}$$
(98)
$$\begin{aligned} {\mathbf {F}_{\varvec{\mu } {\varvec{\varsigma } _n}}}&= 2L\mathrm{Re} \left\{ {diag} \right. \left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1} } \right. \nonumber \\&\times \left. {{{\widetilde{\mathbf {C}}}_{{\varvec{\varsigma } _n}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&+\,diag\left( {{\varvec{\Psi } _\mathbf {r}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}} \right. \nonumber \\&\times \left. {\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right\} \end{aligned}$$
(99)
$$\begin{aligned} {F_{\nu {\kappa _n}}}&= 2L\mathrm{Re} \left\{ {diag} \right. \left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}} \right. \nonumber \\&\times \left. {{{\widetilde{\mathbf {C}}}_{{\varvec{\kappa } _n}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) \nonumber \\&+\,diag\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1} } \right. \nonumber \\&\times \left. {\left. {\mathbf {CAE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\kappa } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right\} \end{aligned}$$
(100)
$$\begin{aligned} {\mathbf {F}_{\varvec{\nu } {\varvec{\varsigma } _n}}}&= 2L\mathrm{Re}\left\{ {diag} \right. \left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}} \right. \nonumber \\&\times {\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _n}}}\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}{\overline{\mathbf {A}} _c} \nonumber \\&+\,diag\left( {{\varvec{\Psi } _\mathbf {i}}{\mathbf {R}_{cs}}{\varvec{\Gamma } ^H}\mathbf {A}_c^H{\mathbf {C}^H}\mathbf {R}_x^{ - 1}\mathbf {C}} \right. \nonumber \\&\times \left. {\left. {\mathbf {AE}{\mathbf {R}_s}{\mathbf {E}^H}{\mathbf {A}^H}\widetilde{\mathbf {C}}_{{\varvec{\varsigma } _n}}^H\mathbf {R}_x^{ - 1}\mathbf {C}{{\overline{\mathbf {A}} }_c}} \right) } \right\} \end{aligned}$$
(101)
Based on the above formulations, the whole FIM can be expressed as
$$\begin{aligned} {\mathbf {F}_{\varvec{\eta \eta }}}&= \left[ {\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {{\mathbf {F}_{\varvec{\alpha \alpha } }}} &{} {{\mathbf {F}_{\varvec{\alpha \beta } }}} &{} {{\mathbf {F}_{\varvec{\alpha \mu } }}} &{} {{\mathbf {F}_{\varvec{\alpha \nu } }}} &{} {{\mathbf {F}_{\varvec{\alpha \kappa } }}} &{} {{\mathbf {F}_{\varvec{\alpha \varsigma } }}} \\ {\mathbf {F}_{\varvec{\alpha \beta } }^T} &{} {{\mathbf {F}_{\varvec{\beta \beta } }}} &{} {{\mathbf {F}_{\varvec{\beta \mu } }}} &{} {{\mathbf {F}_{\varvec{\beta \nu } }}} &{} {{\mathbf {F}_{\varvec{\beta \kappa } }}} &{} {{\mathbf {F}_{\varvec{\beta \varsigma } }}} \\ {\mathbf {F}_{\varvec{\alpha \mu } }^T} &{} {\mathbf {F}_{\varvec{\beta \mu } }^T} &{} {{\mathbf {F}_{\varvec{\mu \mu } }}} &{} {{\mathbf {F}_{\varvec{\mu \nu } }}} &{} {{\mathbf {F}_{\varvec{\mu \kappa } }}} &{} {{\mathbf {F}_{\varvec{\mu \varsigma } }}} \\ {\mathbf {F}_{\varvec{\alpha \nu } }^T} &{} {\mathbf {F}_{\varvec{\beta \nu } }^T} &{} {\mathbf {F}_{\varvec{\mu \nu } }^T} &{} {{\mathbf {F}_{\varvec{\nu \nu } }}} &{} {{\mathbf {F}_{\varvec{\nu \kappa } }}} &{} {{\mathbf {F}_{\varvec{\nu \varsigma } }}} \\ {\mathbf {F}_{\varvec{\alpha \kappa } }^T} &{} {\mathbf {F}_{\varvec{\beta \kappa } }^T} &{} {\mathbf {F}_{\varvec{\mu \kappa } }^T} &{} {\mathbf {F}_{\varvec{\nu \kappa } }^T} &{} {{\mathbf {F}_{\varvec{\kappa \kappa } }}} &{} {{\mathbf {F}_{\varvec{\kappa \varsigma } }}} \\ {\mathbf {F}_{\varvec{\alpha \varsigma } }^T} &{} {\mathbf {F}_{\varvec{\beta \varsigma } }^T} &{} {\mathbf {F}_{\varvec{\mu \varsigma } }^T} &{} {\mathbf {F}_{\varvec{\nu \varsigma } }^T} &{} {\mathbf {F}_{\varvec{\kappa \varsigma } }^T} &{} {{\mathbf {F}_{\varvec{\varsigma \varsigma } }}} \\ \end{array}} \right] \end{aligned}$$
(102)
Define \(\mathbf {H} = \mathbf {F}_{\varvec{\eta \eta } }^{ - 1}\), the CRBs of coherent signals, uncorrelated signals and mutual coupling coefficients can be given, respectively, as
$$\begin{aligned} CR{B_{{\Omega _c}}}&= \sqrt{\frac{1}{{2{K_c}}}\left( {\sum \limits _{k = 1}^{{K_c}} {{\mathbf {H}_{kk}}} + \sum \limits _{k = K + 1}^{K + {K_c}} {{\mathbf {H}_{kk}}} } \right) }\end{aligned}$$
(103)
$$\begin{aligned} CR{B_{{\Omega _u}}}&= \nonumber \\&\sqrt{\frac{1}{{2{K_c}}}\left( {\sum \limits _{k = {K_c} + 1}^K {{\mathbf {H}_{kk}}} + \sum \limits _{k = K + {K_c} + 1}^{2K} {{\mathbf {H}_{kk}}} } \right) }\end{aligned}$$
(104)
$$\begin{aligned} CR{B_c}&= \sqrt{\frac{1}{{\left\| {{\mathbf {c}_1}} \right\| }}\left( {\sum \limits _{k = 2\left( {K + {K_c} - P} \right) }^{2\left( {K + {K_c} - P + {M_0}} \right) } {{\mathbf {H}_{kk}}} } \right) } \end{aligned}$$
(105)
