Appendix 1: Proof of (25)
Substitution of (22) and (23) into (21) yields
$$\begin{aligned} {\varvec{\delta m}}&\approx ( {{\varvec{mi}}_L^{( 1)\mathrm{T}} -{\varvec{I}}_L })\cdot \left\{ {{\varvec{\varPhi }}\left[ {{\varvec{\theta }}_0 ,{\varvec{E}}_0 } \right] } \right\} ^\mathrm{\dag }\cdot \left( {\sum \limits _{n=1}^D {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot ( {{\varvec{\varPi }}_{\varvec{E}} -{\varvec{\varPi }}_{{\varvec{E}}_0 } })\cdot {\varvec{T}}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot {\varvec{m}}} }\right) \nonumber \\&= ( {{\varvec{mi}}_L^{( 1)\mathrm{T}} -{\varvec{I}}_L })\cdot \left\{ {{\varvec{\varPhi }}\left[ {{\varvec{\theta }}_0,{\varvec{E}}_0 } \right] } \right\} ^\mathrm{\dag }\cdot \sum \limits _{k=1}^D {\delta \theta _k \cdot \left( {\sum \limits _{n=1}^D {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot {\varvec{\dot{\varPi }}}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^{\bot ( k)} \cdot {\varvec{T}}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot {\varvec{m}}} }\right) }\nonumber \\ \end{aligned}$$
(48)
Putting (24) into (48) leads to
$$\begin{aligned} {\varvec{\delta m}}&\approx -( {{\varvec{mi}}_L^{( 1)\mathrm{T}} -{\varvec{I}}_L })\cdot \left\{ {{\varvec{\varPhi }}\left[ {{\varvec{\theta }}_0,{\varvec{E}}_0 } \right] } \right\} ^\mathrm{\dag } \nonumber \\&\times \sum \limits _{k=1}^D {\delta \theta _k \cdot \left( {\sum \limits _{n=1}^D {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot \left( {\begin{array}{l} {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot ( {{\varvec{\dot{b}}}\left[ {\theta _{0k} } \right] \otimes {\varvec{i}}_D^{( k)\mathrm{T}} })\cdot {\varvec{B}}^\mathrm{\dag }\left[ {{\varvec{\theta }}_0 } \right] \\ +{\varvec{B}}^{\dag \hbox {H}}\left[ {{\varvec{\theta }}_0 } \right] \cdot ( {{\varvec{\dot{b}}}^\mathrm{H}\left[ {\theta _{0k} } \right] \otimes {\varvec{i}}_D^{( k)} })\cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \\ \end{array}}\right) \cdot {\varvec{T}}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot {\varvec{m}}} }\right) } \nonumber \\&= -( {{\varvec{mi}}_L^{( 1)\mathrm{T}} -{\varvec{I}}_L })\cdot \left\{ {{\varvec{\varPhi }}\left[ {{\varvec{\theta }}_0 ,{\varvec{E}}_0 } \right] } \right\} ^\mathrm{\dag }\nonumber \\&\times \sum \limits _{k=1}^D {\delta \theta _k \cdot \left( {\sum \limits _{n=1}^D {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot ( {{\varvec{\dot{b}}}\left[ {\theta _{0k} } \right] \otimes {\varvec{i}}_D^{( k)\mathrm{T}} })\cdot {\varvec{B}}^\mathrm{\dag }\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{b}}\left[ {\theta _{0n} } \right] } }\right) } \nonumber \\&= -( {{\varvec{mi}}_L^{( 1)\mathrm{T}} -{\varvec{I}}_L })\cdot \left\{ {{\varvec{\varPhi }}\left[ {{\varvec{\theta }}_0 ,{\varvec{E}}_0 } \right] } \right\} ^\mathrm{\dag }\nonumber \\&\times \sum \limits _{k=1}^D {\delta \theta _k \cdot \left( {\sum \limits _{n=1}^D {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot ( {{\varvec{\dot{b}}}\left[ {\theta _{0k} } \right] \otimes {\varvec{i}}_D^{( k)\mathrm{T}} })\cdot {\varvec{i}}_D^{( n)} } }\right) } \end{aligned}$$
(49)
where the second equality follows from the orthogonal relationship as below:
$$\begin{aligned} {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{T}}\left[ {{\varvec{a}}\left[ {\theta _{0n} } \right] } \right] \cdot {\varvec{m}}={\varvec{O}}_{N\times 1} \quad ( {n=1, 2, \ldots , D}) \end{aligned}$$
(50)
and the third equality is a result of the following identity:
$$\begin{aligned} {\varvec{B}}^\mathrm{\dag }\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{b}}\left[ {\theta _{0n} } \right] ={\varvec{B}}^\mathrm{\dag }\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{i}}_D^{( n)} ={\varvec{i}}_D^{( n)} \quad ( {n=1, 2, \ldots , D}) \end{aligned}$$
(51)
In addition, it can be easily checked that
$$\begin{aligned} ( {{\varvec{\dot{b}}}\left[ {\theta _{0k} } \right] \otimes {\varvec{i}}_D^{( k)\mathrm{T}} })\cdot {\varvec{i}}_D^{( n)} =\delta _{nk} \cdot {\varvec{\dot{b}}}\left[ {\theta _{0k} } \right] \quad ( {n, k=1, 2, \ldots , D}) \end{aligned}$$
(52)
which combined with (49) yields
$$\begin{aligned} {\varvec{\delta m}}&\approx \sum \limits _{k=1}^D {\delta \theta _k \cdot ( {{\varvec{I}}_L -{\varvec{mi}}_L^{( 1)\mathrm{T}} })\cdot \left\{ {{\varvec{\varPhi }}\left[ {{\varvec{\theta }}_0 ,{\varvec{E}}_0 } \right] } \right\} ^\mathrm{\dag }\cdot {\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k} } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{\dot{b}}}\left[ {\theta _{0k} } \right] }\nonumber \\&= \sum \limits _{k=1}^D {\delta \theta _k \cdot {\varvec{\xi }}_k } \end{aligned}$$
(53)
where \({\varvec{\xi }}_k \) is defined in (26). At this point, the proof of (25) is ended.
Appendix 2: Proof of (30)
Differentiating (8) with respect to \(\theta _k \) gives
$$\begin{aligned} {\varvec{\dot{A}}}_{\varvec{W}}^{( k)} \left[ {{\varvec{\theta }}_0 } \right]&= \left. {\frac{\partial {\varvec{A}}_{\varvec{W}} \left[ {\varvec{\theta }} \right] }{\partial \theta _k }} \right| _{{\varvec{\theta }}={\varvec{\theta }}_0 } {=}\left. {\frac{\partial {\varvec{A}}\left[ {\varvec{\theta }} \right] }{\partial \theta _k }} \right| _{{\varvec{\theta }}={\varvec{\theta }}_0 } {\cdot } {\varvec{W}}{\cdot } {\varvec{A}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] {+}{\varvec{A}}\left[ {{\varvec{\theta }}_0 } \right] {\cdot } {\varvec{W}}{\cdot } \left. {\frac{\partial {\varvec{A}}^\mathrm{H}\left[ {\varvec{\theta }} \right] }{\partial \theta _k }} \right| _{{\varvec{\theta }}={\varvec{\theta }}_0 } \nonumber \\&=( {{\varvec{\dot{a}}}\left[ {\theta _k } \right] \otimes {\varvec{i}}_D^{( k)\mathrm{T}} })\cdot {\varvec{W}}\cdot {\varvec{A}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] +{\varvec{A}}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{W}}\cdot ( {{\varvec{\dot{a}}}^\mathrm{H}\left[ {\theta _k } \right] \otimes {\varvec{i}}_D^{( k)} }) \nonumber \\&={\varvec{\dot{P}}}^{( k)}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{P}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] +{\varvec{P}}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{\dot{P}}}^{( k)\text{ H }}\left[ {{\varvec{\theta }}_0 } \right] \end{aligned}$$
(54)
which together with the matrix identity
$$\begin{aligned} \text{ vec }\left[ {{\varvec{X}}_{1} {\varvec{X}}_{2} {\varvec{X}}_{3} } \right] =( {{\varvec{X}}_{3}^\mathrm{T} \otimes {\varvec{X}}_{1} })\cdot \text{ vec }\left[ {{\varvec{X}}_{2} } \right] \end{aligned}$$
(55)
Implies
$$\begin{aligned}&\text{ vec }\left[ {{\varvec{\dot{A}}}_{\varvec{W}}^{( k)} \left[ {{\varvec{\theta }}_0 } \right] } \right] \nonumber \\&\quad =\text{ vec }\left[ {\left( {{\varvec{\dot{a}}}\left[ {\theta _k } \right] \otimes {\varvec{i}}_D^{( k)\mathrm{T}} }\right) \cdot {\varvec{W}}\cdot {\varvec{A}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] +{\varvec{A}}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{W}}\cdot \left( {{\varvec{\dot{a}}}^\mathrm{H}\left[ {\theta _k } \right] \otimes {\varvec{i}}_D^{( k)} }\right) } \right] \nonumber \\&\quad =( {{\varvec{P}}^*\left[ {{\varvec{\theta }}_0 } \right] \otimes {\varvec{I}}_N })\cdot \text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)}\left[ {{\varvec{\theta }}_0 } \right] } \right] +( {{\varvec{I}}_N \otimes {\varvec{P}}\left[ {{\varvec{\theta }}_0 } \right] }){\varvec{\varPi }}_{N\bullet D} \cdot \text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)*}\left[ {{\varvec{\theta }}_0 } \right] } \right] \nonumber \\ \end{aligned}$$
(56)
where \({\varvec{\varPi }}_{N\bullet D} \) is the permutation matrix such that \(\text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)\mathrm{T}}\left[ {{\varvec{\theta }}_0 } \right] } \right] ={\varvec{\varPi }}_{N\bullet D} \cdot \text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)}\left[ {{\varvec{\theta }}_0 } \right] } \right] \).
Moreover, it can be readily shown from (56) that
$$\begin{aligned}&\overline{\text{ vec }\left[ {{\varvec{\dot{A}}}_{\varvec{W}}^{( k)} \left[ {{\varvec{\theta }}_0 } \right] } \right] }\nonumber \\&\quad =\overline{\overline{{\varvec{P}}^*\left[ {{\varvec{\theta }}_0 } \right] \otimes {\varvec{I}}_N }} \cdot \overline{\text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)}\left[ {{\varvec{\theta }}_0 } \right] } \right] } +\overline{\overline{( {{\varvec{I}}_N \otimes {\varvec{P}}\left[ {{\varvec{\theta }}_0 } \right] }){\varvec{\varPi }}_{N\bullet D} }} \cdot \overline{\text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)*}\left[ {{\varvec{\theta }}_0 } \right] } \right] } \nonumber \\&\quad =\left( {\overline{\overline{{\varvec{P}}^*\left[ {{\varvec{\theta }}_0 } \right] \otimes {\varvec{I}}_N }} +\overline{\overline{( {{\varvec{I}}_N \otimes {\varvec{P}}\left[ {{\varvec{\theta }}_0 } \right] }){\varvec{\varPi }}_{N\bullet D} }} \cdot {\varvec{J}}_{\text{2 }ND}^{( 1)} }\right) \cdot \overline{\text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)}\left[ {{\varvec{\theta }}_0 } \right] } \right] } \end{aligned}$$
(57)
which produces
$$\begin{aligned}&\text{ vec }\left[ {{\varvec{\dot{P}}}^{( k)}\left[ {{\varvec{\theta }}_0 } \right] } \right] \nonumber \\&\quad ={\varvec{J}}_{\text{2 }ND}^{( 2)} ( {\overline{\overline{{\varvec{P}}^*\left[ {{\varvec{\theta }}_0 } \right] \otimes {\varvec{I}}_N }} +\overline{\overline{( {{\varvec{I}}_N \otimes {\varvec{P}}\left[ {{\varvec{\theta }}_0 } \right] }){\varvec{\varPi }}_{N\bullet D} }} \cdot {\varvec{J}}_{\text{2 }ND}^{( 1)} })^\mathrm{\dag }\cdot \overline{\text{ vec }\left[ {{\varvec{\dot{A}}}_{\varvec{W}}^{( k)} \left[ {{\varvec{\theta }}_0 } \right] } \right] }\nonumber \\ \end{aligned}$$
(58)
It then follows directly from (58) that
$$\begin{aligned}&\!\!\!{\varvec{\dot{p}}}_n^{( k)} \left[ {{\varvec{\theta }}_0 } \right] \nonumber \\&~ =\text{ avec }\left[ {{\varvec{J}}_{\text{2 }ND}^{( 2)} ( {\overline{\overline{{\varvec{P}}^*\left[ {{\varvec{\theta }}_0 } \right] \otimes {\varvec{I}}_N }} \,\,{+}\,\,\overline{\overline{( {{\varvec{I}}_N \otimes {\varvec{P}}\left[ {{\varvec{\theta }}_0 } \right] }){\varvec{\varPi }}_{N\bullet D} }} \cdot {\varvec{J}}_{\text{2 }ND}^{( 1)} })^\mathrm{\dag }\cdot \overline{\text{ vec }\left[ {{\varvec{\dot{A}}}_{\varvec{W}}^{( k)} \left[ {{\varvec{\theta }}_0 } \right] } \right] } } \right] {\cdot } {\varvec{i}}_D^{( n)}\!\nonumber \\ \end{aligned}$$
(59)
Hence, Eq. (30) is proved.
Appendix 3: Proof of (44)
First, due to the structure of \(\overline{{\varvec{\tilde{I}}}_L {\varvec{m}}} =\left[ {{\begin{array}{ll} {( {{\varvec{\tilde{I}}}_L \cdot \text{ Re }\left\{ {\varvec{m}} \right\} })^\mathrm{T}} &{} {( {{\varvec{\tilde{I}}}_L \cdot \text{ Im }\left\{ {\varvec{m}} \right\} })^\mathrm{T}} \\ \end{array} }} \right] ^{\,\,\mathrm{T}}\), the \({\varvec{m}}\)-block of the Fisher information matrix \(\mathbf{FISH}_{{\varvec{mm}}} \) can be partitioned equally along the rows and columns, respectively, as below:
$$\begin{aligned} \mathbf{FISH}_{{\varvec{mm}}} =\left[ \begin{array}{l@{\quad }l} {{\varvec{\tilde{I}}}_L \cdot \mathbf{FISH}_{{\varvec{mm}}}^{( {11})} \cdot {\varvec{\tilde{I}}}_L^\mathrm{T} } &{} {{\varvec{\tilde{I}}}_L \cdot \mathbf{FISH}_{{\varvec{mm}}}^{( {12})} \cdot {\varvec{\tilde{I}}}_L^\mathrm{T} }\\ {{\varvec{\tilde{I}}}_L \cdot \mathbf{FISH}_{{\varvec{mm}}}^{( {21})} \cdot {\varvec{\tilde{I}}}_L^\mathrm{T} } &{} {{\varvec{\tilde{I}}}_L \cdot \mathbf{FISH}_{{\varvec{mm}}}^{( {22})} \cdot {\varvec{\tilde{I}}}_L^\mathrm{T} }\\ \end{array}\right] \end{aligned}$$
(60)
Applying (42), it follows that
$$\begin{aligned} \left\langle {\mathbf{FISH}_{{\varvec{mm}}}^{( {11})} } \right\rangle _{nl}&= \frac{\text{2 }K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\text{ trace }\left[ {\frac{\partial {\varvec{B}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] }{\partial \left\langle {\text{ Re }\left\{ {\varvec{m}} \right\} } \right\rangle _n }\cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot \frac{\partial {\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }{\partial \left\langle {\text{ Re }\left\{ {\varvec{m}} \right\} } \right\rangle _l }{\varvec{W}}_{\text{ NSF }} } \right] } \right\} \nonumber \\&= \frac{\text{2 }K}{\sigma ^2}\sum \limits _{k_1 =1}^D \sum \limits _{k_2 =1}^D \mathrm{Re}\left\{ \text{ trace }\left[ \left\langle {{\varvec{W}}_{\text{ NSF }} } \right\rangle _{k_2 k_1 } \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{T}}\left[ {{\varvec{a}}\left[ {\theta _{0k_2 } } \right] } \right] \cdot {\varvec{i}}_L^{( l)} {\varvec{i}}_L^{( n)\mathrm{T}} \right. \right. \nonumber \\&\left. \left. \cdot \,{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k_1 } } \right] } \right] \right] \right\} \nonumber \\&= \frac{\text{2 }K}{\sigma ^2}\sum \limits _{k_1 =1}^D {\sum \limits _{k_2 =1}^D {\mathrm{Re}\left\{ {\left\langle {{\varvec{W}}_{\text{ NSF }} } \right\rangle _{k_2 k_1 } {\cdot } \left\langle {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k_1 } } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot {\cdot } {\varvec{T}}\left[ {{\varvec{a}}\left[ {\theta _{0k_2 } } \right] } \right] } \right\rangle _{nl} } \right\} } }\nonumber \\ \end{aligned}$$
(61)
which implies
$$\begin{aligned} \mathbf{FISH}_{{\varvec{mm}}}^{( {11})}&= \frac{\text{2 }K}{\sigma ^2}\sum \limits _{k_1 =1}^D {\sum \limits _{k_2 =1}^D {\mathrm{Re}\left\{ {\left\langle {{\varvec{W}}_{\text{ NSF }} } \right\rangle _{k_2 k_1 } \cdot {\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k_1 } } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{T}}\left[ {{\varvec{a}}\left[ {\theta _{0k_{2} } } \right] } \right] } \right\} } } \nonumber \\&= \frac{\text{2 }K}{\sigma ^2}\sum \limits _{k_1 =1}^D {\sum \limits _{k_2 =1}^D {\mathrm{Re}\left\{ {{\varvec{\Sigma }}_{k_2 k_1 } } \right\} } } \end{aligned}$$
(62)
where \({\varvec{\Sigma }}_{k_2 k_1 } \) is defined in (45). Likewise, it can be readily seen that
$$\begin{aligned} \left\{ \begin{array}{l} \mathbf{FISH}_{{\varvec{mm}}}^{( {\text{12 }})} =-\mathbf{FISH}_{{\varvec{mm}}}^{( {21})} =-\frac{2K}{\sigma ^\mathrm{2}}\sum \limits _{k_1 =1}^D {\sum \limits _{k_2 =1}^D {\mathrm{Im}\left\{ {{\varvec{\Sigma }}_{k_2 k_1 } } \right\} } } \\ \mathbf{FISH}_{{\varvec{mm}}}^{( {\text{22 }})} =\mathbf{FISH}_{{\varvec{mm}}}^{( {\text{11 }})} =\frac{2K}{\sigma ^\mathrm{2}}\sum \limits _{k_1 =1}^D {\sum \limits _{k_2 =1}^D {\mathrm{Re}\left\{ {{\varvec{\Sigma }}_{k_2 k_1 } } \right\} } } \\ \end{array} \right. \end{aligned}$$
(63)
which proves the first equation in (44).
Also, the upper right \(\text{2 }( {L-1})\times D\) corner of the Fisher information matrix \(\mathbf{FISH}_{{\varvec{m\theta }}} \) can be uniformly partitioned along the rows as
$$\begin{aligned} \mathbf{FISH}_{{\varvec{m\theta }}} =\left[ \begin{array}{l} {{\varvec{\tilde{I}}}_L \cdot \mathbf{FISH}_{{\varvec{m\theta }}}^{( 1)} } \\ {{\varvec{\tilde{I}}}_L \cdot \mathbf{FISH}_{{\varvec{m\theta }}}^{( 2)} } \\ \end{array} \right] \end{aligned}$$
(64)
Recalling (42) leads to
$$\begin{aligned} \left\langle {\mathbf{FISH}_{{\varvec{m\theta }}}^{( 1)} } \right\rangle _{nl}&= \frac{\text{2 }K}{\sigma ^2}\cdot \mathrm{Re}\left\{ \text{ trace }\left[ \frac{\partial {\varvec{B}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] }{\partial \left\langle {\mathrm{Re}\left\{ {\varvec{m}} \right\} } \right\rangle _n }\cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot \frac{\partial {\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }{\partial \left\langle {{\varvec{\theta }}_0 } \right\rangle _l }\cdot {\varvec{W}}_{\text{ NSF }} \right] \right\} \nonumber \\&= \frac{\text{2 }K}{\sigma ^2}\sum \limits _{k=1}^D {\mathrm{Re}\left\{ {\text{ trace }\left[ {{\varvec{i}}_L^{( n)\mathrm{T}} \cdot {\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k} } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{\dot{B}}}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{i}}_D^{( l)} {\varvec{i}}_D^{( l)\mathrm{T}} {\varvec{W}}_{\text{ NSF }} {\varvec{i}}_D^{( k)} } \right] } \right\} } \nonumber \\&= \frac{\text{2 }K}{\sigma ^2}\sum \limits _{k=1}^D {\mathrm{Re}\left\{ {\left\langle {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k} } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{\dot{B}}}\left[ {{\varvec{\theta }}_0 } \right] } \right\rangle _{nl} \cdot \left\langle {{\varvec{W}}_{\text{ NSF }} } \right\rangle _{lk} } \right\} } \end{aligned}$$
(65)
which gives
$$\begin{aligned} \mathbf{FISH}_{{\varvec{m\theta }}}^{( 1)} =\frac{2K}{\sigma ^2}\sum \limits _{k=1}^D {\mathrm{Re}\left\{ {( {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k} } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{\dot{B}}}\left[ {{\varvec{\theta }}_0 } \right] })\bullet ( {{\varvec{1}}_{L\times 1} \otimes ( {{\varvec{W}}_{\text{ NSF }} {\varvec{i}}_D^{( k)} })^\mathrm{T}})} \right\} }\nonumber \\ \end{aligned}$$
(66)
Similarly, it can be easily deduced that
$$\begin{aligned} \mathbf{FISH}_{{\varvec{m\theta }}}^{( 2)} =\frac{2K}{\sigma ^2}\sum \limits _{k=1}^D {\mathrm{Im}\left\{ {( {{\varvec{T}}^\mathrm{H}\left[ {{\varvec{a}}\left[ {\theta _{0k} } \right] } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{\dot{B}}}\left[ {{\varvec{\theta }}_0 } \right] })\bullet ( {{\varvec{1}}_{L\times 1} \otimes ( {{\varvec{W}}_{\text{ NSF }} {\varvec{i}}_D^{( k)} })^\mathrm{T}})} \right\} }\nonumber \\ \end{aligned}$$
(67)
Equations (66) and (67) together prove the second equation in (44).
Finally, invoking (42) once again produces
$$\begin{aligned} \left\langle {\mathbf{FISH}_{{\varvec{\theta \theta }}} } \right\rangle _{nl}&= \frac{\text{2 }K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\text{ trace }\left[ {\frac{\partial {\varvec{B}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] }{\partial \left\langle {{\varvec{\theta }}_0 } \right\rangle _n }\cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot \frac{\partial {\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }{\partial \left\langle {{\varvec{\theta }}_0 } \right\rangle _l }\cdot {\varvec{W}}_{\text{ NSF }} } \right] } \right\} +\left\langle {{\varvec{R}}_{{\varvec{\theta \theta }}}^{-1} } \right\rangle _{nl} \!\!\!\nonumber \\&= \frac{\text{2 }K}{\sigma ^2}\mathrm{Re}\left\{ {\text{ trace }\left[ {{\varvec{i}}_D^{( n)} {\varvec{i}}_D^{( n)\mathrm{T}} \cdot {\varvec{\dot{B}}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{\dot{B}}}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{i}}_D^{( l)} {\varvec{i}}_D^{( l)\mathrm{T}} {\varvec{W}}_{\text{ NSF }} } \right] } \right\} \!\!\! \nonumber \\&= \frac{\text{2 }K}{\sigma ^2}\mathrm{Re}\left\{ {\left\langle {{\varvec{\dot{B}}}^\mathrm{H}\left[ {{\varvec{\theta }}_0 } \right] \cdot {\varvec{\varPi }}_{{\varvec{B}}\left[ {{\varvec{\theta }}_0 } \right] }^\bot \cdot {\varvec{\dot{B}}}\left[ {{\varvec{\theta }}_0 } \right] } \right\rangle _{nl} \cdot \left\langle {{\varvec{W}}_{\text{ NSF }}^\mathrm{T} } \right\rangle _{nl} } \right\} +\left\langle {{\varvec{R}}_{{\varvec{\theta \theta }}}^{-1} } \right\rangle _{nl} \end{aligned}$$
(68)
which proves the third equality in (44).
