$\displaystyle F(\bar{\bm{\eta}}_{q};\\{{\hat{{\bf
Y}}}_{q}^{(k)}[n]\\}_{n=1,\cdots,N})$ $\displaystyle=$ $\displaystyle
2{\mathfrak{Re}}\left\\{\frac{{\bf a}_{{\rm B}}^{H}(\vartheta_{{\rm
r},0})\left\\{\sum\limits_{n=1}^{N}(e^{j2\pi\tau_{q}\frac{(n-1)B}{N}}{\hat{{\bf
Y}}}_{q}^{(k)}[n]\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf a}_{{\rm
M}}(\vartheta_{{\rm t},q})}{N_{{\rm b}}{\bf a}_{{\rm M}}^{H}(\vartheta_{{\rm
t},q})\left\\{\sum\limits_{n=1}^{N}({\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf
a}_{{\rm M}}(\vartheta_{{\rm t},q})}\right.$ $\displaystyle\cdot\left.{\bf
a}_{{\rm M}}^{H}(\vartheta_{{\rm
t},q})\left(\sum\limits_{n=1}^{N}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}{\bf
X}[n]\bm{{\Sigma}}_{q}(\hat{\bf{Y}}_{q}^{(k)}[n])^{H}\right){\bf a}_{{\rm
B}}(\vartheta_{{\rm r},0})\right\\}$ $\displaystyle-\left\\{\left|\frac{{\bf
a}_{{\rm B}}^{H}(\vartheta_{{\rm
r},0})\left\\{\sum\limits_{n=1}^{N}(e^{j2\pi\tau_{q}\frac{(n-1)B}{N}}{\hat{{\bf
Y}}}_{q}^{(k)}[n]\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf a}_{{\rm
M}}(\vartheta_{{\rm t},q})}{N_{{\rm b}}{\bf a}_{{\rm M}}^{H}(\vartheta_{{\rm
t},q})\left\\{\sum\limits_{n=1}^{N}({\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf
a}_{{\rm M}}(\vartheta_{{\rm t},q})}\right|^{2}\right.$
$\displaystyle\cdot\left.N_{{\rm b}}{\bf a}_{{\rm M}}^{H}(\vartheta_{{\rm
t},q})\left(\sum\limits_{n=1}^{N}{\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n]\right){\bf a}_{{\rm
M}}(\vartheta_{{\rm t},q}))\right\\}$ $\displaystyle=$ $\displaystyle
2{\mathfrak{Re}}\left\\{\frac{\left|{\bf a}_{{\rm M}}^{H}(\vartheta_{{\rm
t},q})\left\\{\sum\limits_{n=1}^{N}(e^{-j2\pi\tau_{l}\frac{(n-1)B}{N}}{\bf
X}[n]\bm{{\Sigma}}_{q}({\hat{{\bf Y}}}_{q}^{(k)}[n])^{H})\right\\}{\bf
a}_{{\rm B}}(\vartheta_{{\rm r},0})\right|^{2}}{N_{{\rm b}}{\bf a}_{{\rm
M}}^{H}(\vartheta_{{\rm t},q})\left\\{\sum\limits_{n=1}^{N}({\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf
a}_{{\rm M}}(\vartheta_{{\rm t},q})}\right\\}$ $\displaystyle-\frac{\left|{\bf
a}_{{\rm B}}^{H}(\vartheta_{{\rm
r},0})\left\\{\sum\limits_{n=1}^{N}(e^{j2\pi\tau_{q}\frac{(n-1)B}{N}}{\hat{{\bf
Y}}}_{q}^{(k)}[n]\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf a}_{{\rm
M}}(\vartheta_{{\rm t},q})\right|^{2}}{N_{{\rm b}}{\bf a}_{{\rm
M}}^{H}(\vartheta_{{\rm t},q})\left\\{\sum\limits_{n=1}^{N}({\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf
a}_{{\rm M}}(\vartheta_{{\rm t},q})}$ $\displaystyle=$
$\displaystyle\frac{2\left|{\bf a}_{{\rm M}}^{H}(\vartheta_{{\rm
t},q})\left\\{\sum\limits_{n=1}^{N}(e^{-j2\pi\tau_{l}\frac{(n-1)B}{N}}{\bf
X}[n]\bm{{\Sigma}}_{q}({\hat{{\bf Y}}}_{q}^{(k)}[n])^{H})\right\\}{\bf
a}_{{\rm B}}(\vartheta_{{\rm r},0})\right|^{2}}{N_{{\rm b}}{\bf a}_{{\rm
M}}^{H}(\vartheta_{{\rm t},q})\left\\{\sum\limits_{n=1}^{N}({\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf
a}_{{\rm M}}(\vartheta_{{\rm t},q})}$ $\displaystyle-\frac{\left|{\bf a}_{{\rm
B}}^{H}(\vartheta_{{\rm
r},0})\left\\{\sum\limits_{n=1}^{N}(e^{j2\pi\tau_{q}\frac{(n-1)B}{N}}{\hat{{\bf
Y}}}_{q}^{(k)}[n]\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf a}_{{\rm
M}}(\vartheta_{{\rm t},q})\right|^{2}}{N_{{\rm b}}{\bf a}_{{\rm
M}}^{H}(\vartheta_{{\rm t},q})\left\\{\sum\limits_{n=1}^{N}({\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf
a}_{{\rm M}}(\vartheta_{{\rm t},q})}$ $\displaystyle=$
$\displaystyle\frac{\left|{\bf a}_{{\rm B}}^{H}(\vartheta_{{\rm
r},0})\left\\{\sum\limits_{n=1}^{N}(e^{j2\pi\tau_{q}\frac{(n-1)B}{N}}{\hat{{\bf
Y}}}_{q}^{(k)}[n]\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf a}_{{\rm
M}}(\vartheta_{{\rm t},q})\right|^{2}}{N_{{\rm b}}{\bf a}_{{\rm
M}}^{H}(\vartheta_{{\rm t},q})\left\\{\sum\limits_{n=1}^{N}({\bf
X}[n]\bm{{\Sigma}}_{q}\bm{{\Sigma}}_{q}^{H}{\bf X}^{H}[n])\right\\}{\bf
a}_{{\rm M}}(\vartheta_{{\rm t},q})}.$ (119)

##  Appendix C Derivation of the FIM and Transformation Matrix

###  C-A Derivation of the FIM

We find that the key to obtain the FIM ${\bf J}_{\bm{\eta}}$ in (74) is to
calculate $\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial[\bm{\eta}]_{u}},u=1,2,\cdots,6(Q+1)$, which are given as
follows.

$\displaystyle\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial[\bm{\eta}]_{6q+1}}=\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial\tau_{q}}$
$\displaystyle=\frac{\partial\sum\limits_{q=0}^{Q}\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})}{\partial\tau_{q}}$
$\displaystyle=-j2\pi\frac{(n-1)B}{N}\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q}).$ (120) $\displaystyle\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial[\bm{\eta}]_{6q+2}}=\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial\delta_{q,\mathrm{R}}}$
$\displaystyle=\frac{\partial\sum\limits_{q=0}^{Q}\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})}{\partial\delta_{q,\mathrm{R}}}$
$\displaystyle=[\frac{\partial(\delta_{q,\mathrm{R}}+j\delta_{q,\mathrm{I}})}{\partial\delta_{q,\mathrm{R}}}]e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})$
$\displaystyle=e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q}).$ (121) $\displaystyle\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial[\bm{\eta}]_{6q+3}}=\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial\delta_{q,\mathrm{I}}}$
$\displaystyle=\frac{\partial\sum\limits_{q=0}^{Q}\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})}{\partial\delta_{q,\mathrm{I}}}$
$\displaystyle=[\frac{\partial(\delta_{q,\mathrm{R}}+j\delta_{q,\mathrm{I}})}{\partial\delta_{q,\mathrm{I}}}]e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})$
$\displaystyle=je^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q}).$ (122) $\displaystyle\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial[\bm{\eta}]_{6q+4}}=\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial\theta_{{\rm t},q}}$
$\displaystyle=\frac{\partial\sum\limits_{q=0}^{Q}\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})}{\partial\theta_{{\rm t},q}}$
$\displaystyle=\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}\bm{{\Sigma}}_{q}^{T}\otimes\frac{\partial{\tilde{{\bf
H}}}_{q}}{\partial\theta_{{\rm t},q}}$
$\displaystyle=\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}\left\\{\bm{{\Sigma}}_{q}^{T}\otimes[{\bf
a}_{{\rm B}}(\vartheta_{{\rm r},0})\frac{\partial{\bf
a}_{\mathrm{M}}^{H}(\vartheta_{{\rm t},q})}{\partial\theta_{{\rm
t},q}}]\right\\}$ $\displaystyle=j2\pi\frac{d}{\lambda}\cos(\theta_{{\rm
t},q})\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}\left[\bm{{\Sigma}}_{q}^{T}\otimes({\tilde{{\bf
H}}}_{q}{\bf D}_{N_{{\rm m}}})\right],$ (123)

where $q=0,1,\cdots,Q$, $\frac{\partial{\bf a}_{{\rm M}}^{H}(\vartheta_{{\rm
t},q})}{\partial\theta_{{\rm t},q}}=j2\pi\frac{d}{\lambda}\cos(\theta_{{\rm
t},q}){\bf a}_{{\rm M}}^{H}(\vartheta_{{\rm t},q}){\bf D}_{N_{{\rm m}}}$, and
${\bf D}_{N_{{\rm m}}}\triangleq{\rm diag}(0,1,\cdots,N_{{\rm m}}-1)$.

$\displaystyle\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial[\bm{\eta}]_{6q+5}}=\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial\phi_{{\rm in},q}}$
$\displaystyle=\frac{\partial\sum\limits_{q=0}^{Q}\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})}{\partial\phi_{{\rm in},q}}$
$\displaystyle=\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\frac{\partial\bm{{\Sigma}}_{q}^{T}}{\partial\phi_{{\rm
in},q}}\otimes{\tilde{{\bf H}}}_{q}),$ (124)

where $q=0,1,\cdots,Q$. The
$\frac{\partial\bm{{\Sigma}}_{q}^{T}}{\partial\phi_{{\rm in},q}}$ can be
represented as follows:

$\displaystyle\frac{\partial\bm{{\Sigma}}_{q}^{T}}{\partial\phi_{{\rm in},q}}$
$\displaystyle=\left\\{{\rm diag}[\frac{\partial{\bf a}_{{\rm
R}}^{H}(\triangle\omega_{q}^{{\rm a}},\triangle\omega_{q}^{{\rm
e}})}{\partial\phi_{{\rm in},q}}{\bf g}_{1},\cdots,\frac{\partial{\bf a}_{{\rm
R}}^{H}(\triangle\omega_{q}^{{\rm a}},\triangle\omega_{q}^{{\rm
e}})}{\partial\phi_{{\rm in},q}}{\bf g}_{T}]\right\\}^{T}.$ (125)

The $\frac{\partial{\bf a}_{{\rm R}}^{H}(\triangle\omega_{q}^{{\rm
a}},\triangle\omega_{q}^{{\rm e}})}{\partial\phi_{{\rm in},q}}$ in (125) can
be calculated by

$\displaystyle\frac{\partial{\bf a}_{{\rm R}}^{H}(\triangle\omega_{q}^{{\rm
a}},\triangle\omega_{q}^{{\rm e}})}{\partial\phi_{{\rm in},q}}=$
$\displaystyle\frac{\partial[{\bf f}(\triangle\omega_{q}^{{\rm e}},N_{{\rm
e}})\otimes{\bf f}(\triangle\omega_{q}^{{\rm a}},N_{{\rm
a}})]^{H}}{\partial\phi_{{\rm in},q}}$ $\displaystyle=$
$\displaystyle\frac{\partial{\bf f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm
e}})}{\partial\phi_{{\rm in},q}}\otimes{\bf f}^{H}(\triangle\omega_{q}^{{\rm
a}},N_{{\rm a}})+{\bf f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm
e}})\otimes\frac{\partial{\bf f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm
a}})}{\partial\phi_{{\rm in},q}}$ $\displaystyle=$ $\displaystyle
j2\pi\frac{d_{{\rm e}}}{\lambda}\sin\phi_{{\rm in},q}[{\bf
f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm e}}){\bf D}_{N_{{\rm
e}}}]\otimes{\bf f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm a}})$
$\displaystyle-j2\pi\frac{d_{{\rm a}}}{\lambda}\sin\psi_{{\rm
in},q}\cos\phi_{{\rm in},q}{\bf f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm
e}})\otimes[{\bf f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm a}}){\bf
D}_{N_{{\rm a}}}]$ $\displaystyle\overset{(k)}{=}$ $\displaystyle
j2\pi\frac{d_{{\rm e}}}{\lambda}\sin\phi_{{\rm in},q}[{\bf
f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm e}})\otimes{\bf
f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm a}})]({\bf D}_{N_{{\rm
e}}}\otimes{\bf I}_{N_{{\rm a}}})$ $\displaystyle-j2\pi\frac{d_{{\rm
a}}}{\lambda}\sin\psi_{{\rm in},q}\cos\phi_{{\rm in},q}[{\bf
f}{}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm e}})\otimes{\bf
f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm a}})]({\bf I}_{N_{{\rm
e}}}\otimes{\bf D}_{N_{{\rm a}}})$ $\displaystyle=$ $\displaystyle j2\pi{\bf
a}_{{\rm R}}^{H}(\triangle\omega_{q}^{{\rm a}},\triangle\omega_{q}^{{\rm
e}})[\frac{d_{{\rm e}}}{\lambda}\sin\phi_{{\rm in},q}({\bf D}_{N_{{\rm
e}}}\otimes{\bf I}_{N_{{\rm a}}}){\\!}-{\\!}\frac{d_{{\rm
a}}}{\lambda}\sin\psi_{{\rm in},q}\cos\phi_{{\rm in},q}({\bf I}_{N_{{\rm
e}}}\otimes{\bf D}_{N_{{\rm a}}})]$ $\displaystyle\overset{(l)}{=}$
$\displaystyle{\bf a}_{{\rm R}}^{H}(\triangle\omega_{q}^{{\rm
a}},\triangle\omega_{q}^{{\rm e}}){\bf D}_{\phi_{{\rm in},q}},$ (126)

where the step $(k)$ is obtained due to
$(\mathbf{A}\mathbf{B})\otimes(\mathbf{CD})=(\mathbf{A\otimes}\mathbf{C})(\mathbf{B}\mathbf{\otimes
D})$. In the step $(l)$ of (126), we define ${\bf D}_{\phi_{{\rm in},q}}$ as

$\displaystyle{\bf D}_{\phi_{{\rm in},q}}$ $\displaystyle\triangleq
j2\pi[\frac{d_{{\rm e}}}{\lambda}\sin\phi_{{\rm in},q}({\bf D}_{N_{{\rm
e}}}\otimes{\bf I}_{N_{{\rm a}}})-\frac{d_{{\rm a}}}{\lambda}\sin\psi_{{\rm
in},q}\cos\phi_{{\rm in},q}({\bf I}_{N_{{\rm e}}}\otimes{\bf D}_{N_{{\rm
a}}})],$ (127)

where ${\bf D}_{N_{{\rm e}}}\triangleq{\rm diag}(0,1,\cdots,N_{{\rm e}}-1)$
and ${\bf D}_{N_{{\rm a}}}\triangleq{\rm diag}(0,1,\cdots,N_{{\rm a}}-1)$.

$\displaystyle\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial[\bm{\eta}]_{6q+6}}=\frac{\partial{\tilde{{\bf
H}}}[n]}{\partial\psi_{{\rm in},q}}$
$\displaystyle=\frac{\partial\sum\limits_{q=0}^{Q}\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\bm{{\Sigma}}_{q}^{T}\otimes{\tilde{{\bf
H}}}_{q})}{\partial\psi_{{\rm in},q}}$
$\displaystyle=\delta_{q}e^{-j2\pi\tau_{q}\frac{(n-1)B}{N}}(\frac{\partial\bm{{\Sigma}}_{q}^{T}}{\partial\psi_{{\rm
in},q}}\otimes{\tilde{{\bf H}}}_{q}),$ (128)

where $q=0,1,\cdots,Q$. The
$\frac{\partial\bm{{\Sigma}}_{q}^{T}}{\partial\psi_{{\rm in},q}}$ can be
represented as follows:

$\displaystyle\frac{\partial\bm{{\Sigma}}_{q}^{T}}{\partial\psi_{{\rm in},q}}$
$\displaystyle=\left\\{{\rm diag}[\frac{\partial{\bf a}_{{\rm
R}}^{H}(\triangle\omega_{q}^{{\rm a}},\triangle\omega_{q}^{{\rm
e}})}{\partial\psi_{{\rm in},q}}{\bf g}_{1},\cdots,\frac{\partial{\bf a}_{{\rm
R}}^{H}(\triangle\omega_{q}^{{\rm a}},\triangle\omega_{q}^{{\rm
e}})}{\partial\psi_{{\rm in},q}}{\bf g}_{T}]\right\\}^{T}.$ (129)

The $\frac{\partial{\bf a}_{{\rm R}}^{H}(\triangle\omega_{q}^{{\rm
a}},\triangle\omega_{q}^{{\rm e}})}{\partial\psi_{{\rm in},q}}$ in (129) can
be calculated by

$\displaystyle\frac{\partial{\bf a}_{{\rm R}}^{H}(\triangle\omega_{q}^{{\rm
a}},\triangle\omega_{q}^{{\rm e}})}{\partial\psi_{{\rm in},q}}=$
$\displaystyle\frac{\partial[{\bf f}(\triangle\omega_{q}^{{\rm e}},N_{{\rm
e}})\otimes{\bf f}(\triangle\omega_{q}^{{\rm a}},N_{{\rm
a}})]^{H}}{\partial\psi_{{\rm in},q}}$ $\displaystyle=$ $\displaystyle{\bf
f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm e}})\otimes\frac{\partial{\bf
f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm a}})}{\partial\psi_{{\rm in},q}}$
$\displaystyle=$ $\displaystyle-j2\pi\frac{d_{{\rm a}}}{\lambda}\cos\psi_{{\rm
in},q}\sin\phi_{{\rm in},q}{\bf f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm
e}})\otimes[{\bf f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm a}}){\bf
D}_{N_{{\rm a}}}]$ $\displaystyle\overset{(m)}{=}$
$\displaystyle-j2\pi\frac{d_{{\rm a}}}{\lambda}\cos\psi_{{\rm
in},q}\sin\phi_{{\rm in},q}[{\bf f}^{H}(\triangle\omega_{q}^{{\rm e}},N_{{\rm
e}})\otimes{\bf f}^{H}(\triangle\omega_{q}^{{\rm a}},N_{{\rm a}})]({\bf
I}_{N_{{\rm e}}}\otimes{\bf D}_{N_{{\rm a}}})$ $\displaystyle=$
$\displaystyle-j2\pi\frac{d_{{\rm a}}}{\lambda}\cos\psi_{{\rm
in},q}\sin\phi_{{\rm in},q}{\bf a}_{{\rm R}}^{H}(\triangle\omega_{q}^{{\rm
a}},\triangle\omega_{q}^{{\rm e}})({\bf I}_{N_{{\rm e}}}\otimes{\bf
D}_{N_{{\rm a}}})$ $\displaystyle\overset{(n)}{=}$ $\displaystyle{\bf a}_{{\rm
R}}^{H}(\triangle\omega_{q}^{{\rm a}},\triangle\omega_{q}^{{\rm e}}){\bf
D}_{\psi_{{\rm in},q}},$ (130)

where the step $(m)$ is obtained by using
$(\mathbf{A}\mathbf{B})\otimes(\mathbf{CD})=(\mathbf{A\otimes}\mathbf{C})(\mathbf{B}\mathbf{\otimes
D})$. In the step $(n)$ of (130), we define ${\bf D}_{\psi_{{\rm in},q}}$ as

$\displaystyle{\bf D}_{\psi_{{\rm in},q}}$
$\displaystyle\triangleq-j2\pi\frac{d_{{\rm a}}}{\lambda}\cos\psi_{{\rm
in},q}\sin\phi_{{\rm in},q}({\bf I}_{N_{{\rm e}}}\otimes{\bf D}_{N_{{\rm
a}}}),$ (131)

where ${\bf D}_{N_{{\rm a}}}\triangleq{\rm diag}(0,1,\cdots,N_{{\rm a}}-1)$.

###  C-B Derivation of the Transformation Matrix

The transformation matrix $\mathbf{T}$ can be calculated according to the
definition of
$\mathbf{T}\triangleq\frac{\partial\bm{\eta}^{T}}{\partial\tilde{\bm{{\eta}}}}$,
$\bm{{\eta}}$, and $\tilde{\bm{{\eta}}}$. We give the derivatives of each
channel parameters with respect to different final parameters as follows.

(1) The derivatives of TOA $\tau_{q}$ with respect to different final
parameters are given as follows:

$\displaystyle\frac{\partial\tau_{q_{1}}}{\partial\delta_{q_{2},{\rm R}}}$
$\displaystyle=\frac{\partial\tau_{q_{1}}}{\partial\delta_{q_{2},{\rm
I}}}=\frac{\partial\tau_{q_{1}}}{\partial\alpha}=0,{\quad}\forall
q_{1},q_{2}=0,1,\cdots,Q,$ (132)
$\displaystyle\frac{\partial\tau_{q_{1}}}{\partial{\bf m}}$
$\displaystyle=\begin{cases}\frac{{\bf m}-{\bf r}}{c\cdot\left\|{\bf m}-{\bf
r}\right\|_{2}},&q_{1}{\rm=0}\\\ \frac{{\bf m}-{\bf
s}^{q_{1}}}{c\cdot\left\|{\bf m}-{\bf
s}^{q_{1}}\right\|_{2}},&q_{1}{\rm=1,}\cdots,Q\end{cases},$ (133)
$\displaystyle\frac{\partial\tau_{q_{1}}}{\partial{\bf s}^{q_{2}}}$
$\displaystyle=\begin{cases}{\bf 0},&q_{1}{\rm=0}\,\textrm{or}\,q_{1}\neq
q_{2}\\\ \frac{{\bf s}^{q_{1}}-{\bf r}}{c\cdot\left\|{\bf s}^{q_{1}}-{\bf
r}\right\|_{2}}-\frac{{\bf m}-{\bf s}^{q_{1}}}{c\cdot\left\|{\bf m}-{\bf
s}^{q_{1}}\right\|_{2}},&q_{1}=q_{2}={\rm 1,}\cdots,Q\end{cases}.$ (134)

(2) The derivatives of the real part of channel gain $\delta_{q,\mathrm{R}}$
with respect to different final parameters are given as follows:

$\displaystyle\frac{\partial\delta_{q_{1},{\rm R}}}{\partial\tilde{\eta}}$
$\displaystyle=\begin{cases}1,&\tilde{\eta}=\delta_{q_{1},{\rm R}}\\\
0,&\tilde{\eta}\neq\delta_{q_{1},{\rm
R}},\forall\tilde{\eta}\in\tilde{\bm{{\eta}}}\end{cases}.$ (135)

(3) The derivatives of the imaginary part of channel gain
$\delta_{q,\mathrm{I}}$ with respect to different final parameters are given
as follows:

$\displaystyle\frac{\partial\delta_{q_{1},{\rm I}}}{\partial\tilde{\eta}}$
$\displaystyle=\begin{cases}1,&\tilde{\eta}=\delta_{q_{1},{\rm I}}\\\
0,&\tilde{\eta}\neq\delta_{q_{1},{\rm
I}},\forall\tilde{\eta}\in\tilde{\bm{{\eta}}}\end{cases}.$ (136)

(4) The derivatives of the DOA $\theta_{\mathrm{t},q}$ with respect to
different final parameters are given as follows:

$\displaystyle\frac{\partial\theta_{\mathrm{t},q_{1}}}{\partial\delta_{q_{2},{\rm
R}}}$
$\displaystyle=\frac{\partial\theta_{\mathrm{t},q_{1}}}{\partial\delta_{q_{2},{\rm
I}}}=0,{\quad}\forall q_{1},q_{2}=0,1,\cdots,Q.$ (137)

By defining $\mathbf{a}=[\cos\alpha,-\sin\alpha,0]^{T}$, we have

$\displaystyle\frac{\partial\theta_{\mathrm{t},q_{1}}}{\partial{\bf m}}$
$\displaystyle=\begin{cases}\frac{\left[\mathbf{a}^{T}({\bf r}-{\bf
m})\right]({\bf r}-{\bf m})-\mathbf{a}\left\|{\bf r}-{\bf
m}\right\|_{2}^{2}}{\left\|{\bf r}-{\bf m}\right\|_{2}^{2}\sqrt{\left\|{\bf
r}-{\bf m}\right\|_{2}^{2}-\left[\mathbf{a}^{T}({\bf r}-{\bf
m})\right]^{2}}},&q_{1}{\rm=0}\\\ \frac{\left[\mathbf{a}^{T}({\bf
s}^{q_{1}}-{\bf m})\right]({\bf s}^{q_{1}}-{\bf m})-\mathbf{a}\left\|{\bf
s}^{q_{1}}-{\bf m}\right\|_{2}^{2}}{\left\|{\bf s}^{q_{1}}-{\bf
m}\right\|_{2}^{2}\sqrt{\left\|{\bf s}^{q_{1}}-{\bf
m}\right\|_{2}^{2}-\left[\mathbf{a}^{T}({\bf s}^{q_{1}}-{\bf
m})\right]^{2}}},&q_{1}{\rm=1,}\cdots,Q\end{cases},$ (138)
$\displaystyle\frac{\partial\theta_{\mathrm{t},q_{1}}}{\partial\alpha}$
$\displaystyle=\begin{cases}\frac{-(r_{x}-m_{x})\sin\alpha-(r_{y}-m_{y})\cos\alpha}{\sqrt{\left\|{\bf
r}-{\bf
m}\right\|_{2}^{2}-\left[(r_{x}-m_{x})\cos\alpha-(r_{y}-m_{y})\sin\alpha\right]^{2}}},&q_{1}{\rm=0}\\\
\frac{-(s_{x}^{q_{1}}-m_{x})\sin\alpha-(s_{y}^{q_{1}}-m_{y})\cos\alpha}{\sqrt{\left\|{\bf
s}^{q}-{\bf
m}\right\|_{2}^{2}-\left[(s_{x}^{q_{1}}-m_{x})\cos\alpha-(s_{y}^{q_{1}}-m_{y})\sin\alpha\right]^{2}}},&q_{1}{\rm=1,}\cdots,Q\end{cases},$
(139) $\displaystyle\frac{\partial\theta_{\mathrm{t},q_{1}}}{\partial{\bf
s}^{q_{2}}}$ $\displaystyle=\begin{cases}\mathbf{0},&q_{1}=0{\ }\textrm{or}{\
}q_{1}\neq q_{2}\\\ \frac{\mathbf{a}\left\|{\bf s}^{q_{1}}-{\bf
m}\right\|_{2}^{2}-\left[\mathbf{a}^{T}({\bf s}^{q_{1}}-{\bf m})\right]({\bf
s}^{q_{1}}-{\bf m})}{\left\|{\bf s}^{q_{1}}-{\bf
m}\right\|_{2}^{2}\sqrt{\left\|{\bf s}^{q_{1}}-{\bf
m}\right\|_{2}^{2}-\left[\mathbf{a}^{T}({\bf s}^{q_{1}}-{\bf
m})\right]^{2}}}.&q_{1}=q_{2}\neq 0\end{cases}.$ (140)

(5) The derivatives of the elevation angle $\phi_{\mathrm{in},q}$ at the RIS
with respect to different final parameters are given as follows:

$\displaystyle\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial\delta_{q_{2},{\rm
R}}}$
$\displaystyle=\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial\delta_{q_{2},{\rm
I}}}=\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial\alpha}=0,{\quad}\forall
q_{1},q_{2}=0,1,\cdots,Q,$ (141)
$\displaystyle\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial m_{x}}$
$\displaystyle=\begin{cases}\frac{-(r_{x}-m_{x})(r_{z}-m_{z})}{\left\|{\bf
r}-{\bf m}\right\|_{2}^{2}\sqrt{\left\|{\bf r}-{\bf
m}\right\|_{2}^{2}-(r_{z}-m_{z})^{2}}},&q_{1}{\rm=0}\\\
0,&q_{1}{\rm=1,}\cdots,Q\end{cases},$ (142)
$\displaystyle\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial m_{y}}$
$\displaystyle=\begin{cases}\frac{-(r_{y}-m_{y})(r_{z}-m_{z})}{\left\|{\bf
r}-{\bf m}\right\|_{2}^{2}\sqrt{\left\|{\bf r}-{\bf
m}\right\|_{2}^{2}-(r_{z}-m_{z})^{2}}},&q_{1}{\rm=0}\\\
0,&q_{1}{\rm=1,}\cdots,Q\end{cases},$ (143)
$\displaystyle\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial m_{z}}$
$\displaystyle=\begin{cases}\frac{\sqrt{\left\|{\bf r}-{\bf
m}\right\|_{2}^{2}-(r_{z}-m_{z})^{2}}}{\left\|{\bf r}-{\bf
m}\right\|_{2}^{2}},&q_{1}{\rm=0}\\\ 0,&q_{1}{\rm=1,}\cdots,Q\end{cases},$
(144) $\displaystyle\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial
s_{x}^{q_{2}}}$ $\displaystyle=\begin{cases}0,&q_{1}=0{\ }\textrm{or}{\
}q_{1}\neq q_{2}\\\
\frac{-(r_{x}-s_{x}^{q_{2}})(r_{z}-s_{z}^{q_{2}})}{\left\|{\bf r}-{\bf
s}^{q_{2}}\right\|_{2}^{2}\sqrt{\left\|{\bf r}-{\bf
s}^{q_{2}}\right\|_{2}^{2}-(r_{z}-s_{z}^{q_{2}})^{2}}},&q_{1}=q_{2}\neq
0\end{cases},$ (145)
$\displaystyle\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial s_{y}^{q_{2}}}$
$\displaystyle=\begin{cases}0,&q_{1}=0{\ }\textrm{or}{\ }q_{1}\neq q_{2}\\\
\frac{-(r_{y}-s_{y}^{q_{2}})(r_{z}-s_{z}^{q_{2}})}{\left\|{\bf r}-{\bf
s}^{q_{2}}\right\|_{2}^{2}\sqrt{\left\|{\bf r}-{\bf
s}^{q_{2}}\right\|_{2}^{2}-(r_{z}-s_{z}^{q_{2}})^{2}}}&q_{1}=q_{2}\neq
0\end{cases},$ (146)
$\displaystyle\frac{\partial\phi_{\mathrm{in},q_{1}}}{\partial s_{z}^{q_{2}}}$
$\displaystyle=\begin{cases}0,&q_{1}=0{\ }\textrm{or}{\ }q_{1}\neq q_{2}\\\
\frac{\sqrt{\left\|{\bf r}-{\bf
s}^{q_{2}}\right\|_{2}^{2}-(r_{z}-s_{z}^{q_{2}})^{2}}}{\left\|{\bf r}-{\bf
s}^{q_{2}}\right\|_{2}^{2}}&q_{1}=q_{2}\neq 0\end{cases}.$ (147)

(6) The derivatives of the azimuth angle $\psi_{\mathrm{in},q}$ at the RIS
with respect to different final parameters are given as follows:

$\displaystyle\frac{\partial\psi_{\mathrm{in},q_{1}}}{\partial\tilde{\eta}}=0,$
$\displaystyle\forall\tilde{\eta}\in\\{\delta_{q_{2},{\rm
R}},\delta_{q_{2},{\rm I}},\alpha,m_{z},s_{z}^{q_{2}}\\},\forall
q_{1},q_{2}=0,1,\cdots,Q,$ (148)
$\displaystyle\frac{\partial\psi_{\mathrm{in},q_{1}}}{\partial m_{x}}$
$\displaystyle=\begin{cases}\frac{-(r_{x}-m_{x})(r_{y}-m_{y})}{\left[(r_{x}-m_{x})^{2}+(r_{y}-m_{y})^{2}\right]\sqrt{(r_{x}-m_{x})^{2}}},&q_{1}{\rm=0}\\\
0,&q_{1}{\rm=1,}\cdots,Q\end{cases},$ (149)
$\displaystyle\frac{\partial\psi_{\mathrm{in},q_{1}}}{\partial m_{y}}$
$\displaystyle=\begin{cases}\frac{(r_{x}-m_{x})^{2}}{\left[(r_{x}-m_{x})^{2}+(r_{y}-m_{y})^{2}\right]\sqrt{(r_{x}-m_{x})^{2}}},&q_{1}{\rm=0}\\\
0,&q_{1}{\rm=1,}\cdots,Q\end{cases},$ (150)
$\displaystyle\frac{\partial\psi_{\mathrm{in},q_{1}}}{\partial s_{x}^{q_{2}}}$
$\displaystyle=\begin{cases}0,&q_{1}=0{\ }\textrm{or}{\ }q_{1}\neq q_{2}\\\
\frac{-(r_{x}-s_{x}^{q_{2}})(r_{y}-s_{y}^{q_{2}})}{\left[(r_{x}-s_{x}^{q_{2}})^{2}+(r_{y}-s_{y}^{q_{2}})^{2}\right]\sqrt{(r_{x}-s_{x}^{q_{2}})^{2}}},&q_{1}=q_{2}\neq
0\end{cases},$ (151)
$\displaystyle\frac{\partial\psi_{\mathrm{in},q_{1}}}{\partial s_{y}^{q_{2}}}$
$\displaystyle=\begin{cases}0,&q_{1}=0{\ }\textrm{or}{\ }q_{1}\neq q_{2}\\\
\frac{(r_{x}-s_{x}^{q_{2}})^{2}}{\left[(r_{x}-s_{x}^{q_{2}})^{2}+(r_{y}-s_{y}^{q_{2}})^{2}\right]\sqrt{(r_{x}-s_{x}^{q_{2}})^{2}}},&q_{1}=q_{2}\neq
0\end{cases}.$ (152)

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