Statistical performance analysis of direct position determination method based on doppler shifts in presence of model errors

Abstract

The direct position determination (DPD) method based on Doppler shifts for narrowband emitter is first presented by Amar and Weiss. Compared to the conventional differential Doppler localization method, this DPD method has higher localization accuracy. In this paper, the statistical performance of this DPD method in presence of model errors (i.e., the observer position and velocity uncertainty) is analyzed mathematically. First, the DPD method for narrowband emitter using Doppler shifts is introduced. Then, applying the matrix eigen-perturbation theory that relates the eigenvalue perturbations to the additive noise on the Hermitian matrix, the first-order asymptotic expression of the direct localization errors is derived. As a consequence, the theoretical variance of position estimation can be approximately determined. Subsequently, two kinds of exact formulas for calculating the localization success probability are deduced using the analytical results obtained in the previous section of this paper. Furthermore, the Cramér-Rao bound (CRB) expression for the DPD method in absence of model errors is presented, which takes on more explicit form than the formulation given by Amar and Weiss. The obtained CRB can be viewed as a reasonable benchmark to assess the performance degradation due to model mismatch. Simulation results support the validation of the theoretical analysis in this paper.

Appendices

Appendix 1: Proof of Proposition 1

Assume the normalized eigenvectors of matrix ${\hat{\varvec{X}}}$ are defined as ${\hat{\varvec{\beta }}}_{1} , {\hat{\varvec{\beta }}}_{2}, \ldots , {\hat{\varvec{\beta }}}_n$. According to the Hermitian matrix eigen-perturbation theory (Kaveh and Barabell 1986), it follows that

$$\begin{aligned} \left\{ {\begin{array}{l} \hat{{\lambda }}_k =\lambda _k^{(0)} +\tilde{\lambda }_k^{(1)} +\tilde{\lambda }_k^{(2)} +\cdots , \\ {\hat{\varvec{\beta }}}_k =\left( {1-\frac{1}{2}{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\left| {\tilde{\varepsilon }_{ki}^{(1)}} \right| ^{2}}} \right) {\varvec{\beta }}_k^{(0)} +{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} {\varvec{\beta }}_i^{(0)}} + {\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(2)} {\varvec{\beta }}_i^{(0)}} +\cdots , \\ \end{array}} \right. \quad \left( {k=1 , 2 , \ldots , n} \right) \nonumber \\ \end{aligned}$$

(80)

where $\tilde{\lambda }_k^{(1)}$ and $\tilde{\varepsilon }_{ki}^{(1)}$ are the first-order perturbation terms, i.e., $\tilde{\lambda }_k^{(1)} =O\left( { \left\| { {\tilde{\varvec{X}}}} \right\| _2} \right)$ and $\tilde{\varepsilon }_{ki}^{(1)} =O\left( { \left\| { {\tilde{\varvec{X}}}} \right\| _2} \right)$, and $\tilde{\lambda }_k^{(2)}$ and $\tilde{\varepsilon }_{ki}^{(2)}$ are the second-order perturbation terms, i.e., $\tilde{\lambda }_k^{(2)} =O\left( { \left\| {\tilde{\varvec{X}}} \right\| _{2}^{2}} \right)$ and $\tilde{\varepsilon }_{ki}^{(2)} =O\left( { \left\| {\tilde{\varvec{X}}} \right\| _{2}^{2}} \right)$. It can be obtained from the matrix eigen-equation that

$$\begin{aligned} {\hat{\varvec{X}}\hat{\varvec{\beta }}}_k= & {} \hat{{\lambda }}_k {\hat{\varvec{\beta }}}_k \Leftrightarrow \left( {{\varvec{X}}+{\tilde{\varvec{X}}}} \right) \left( {\left( {1-\frac{1}{2}{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\left| {\tilde{\varepsilon }_{ki}^{(1)}} \right| ^{2}}} \right) {\varvec{\beta }}_k^{(0)} +{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} {\varvec{\beta }}_i^{(0)}} +{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(2)} {\varvec{\beta }}_i^{(0)}} +\cdots } \right) \nonumber \\= & {} \left( {\lambda _k^{(0)} +\tilde{\lambda }_k^{(1)} +\tilde{\lambda }_k^{(2)} +\cdots } \right) \left( {\left( {1-\frac{1}{2}{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\left| {\tilde{\varepsilon }_{ki}^{(1)}} \right| ^{2}}} \right) {\varvec{\beta }}_k^{(0)} +{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} {\varvec{\beta }}_i^{(0)}} +{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(2)} {\varvec{\beta }}_i^{(0)}} +\cdots } \right) \nonumber \\ \end{aligned}$$

(81)

By comparing the first-order perturbation terms between both sides of (81) leads to

$$\begin{aligned} {\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} \lambda _i^{(0)} {\varvec{\beta }}_i^{(0)}} +{\tilde{{\varvec{X}}}{\varvec{\beta }}}_k^{(0)} ={\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} \lambda _k^{(0)} {\varvec{\beta }}_i^{(0)}} +{\varvec{\beta }}_k^{(0)} \tilde{\lambda }_k^{(1)} \end{aligned}$$

(82)

Premultiplying both sides of (82) by ${\varvec{\beta }}_k^{(0)\mathrm{H}}$ and ${\varvec{\beta }}_i^{(0)\mathrm{H}} \left( {i\ne k} \right)$, respectively, produces

$$\begin{aligned} \left\{ {\begin{array}{l} \tilde{\lambda }_k^{(1)} ={\varvec{\beta }}_k^{(0)\mathrm{H}} {\tilde{\varvec{X}}\varvec{\beta }}_k^{(0)} \\ \tilde{\varepsilon }_{ki}^{(1)} =\left( {\lambda _k^{(0)} -\lambda _i^{(0)}} \right) ^{-1}{\varvec{\beta }}_i^{(0)\mathrm{H}} {\tilde{\varvec{X}}\varvec{\beta }}_k^{(0)} \\ \end{array}} \right. \end{aligned}$$

(83)

By comparing the second-order perturbation terms between both sides of (81), it follows that

$$\begin{aligned} {\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(2)} \lambda _i^{(0)} {\varvec{\beta }}_i^{(0)}} +{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} {\tilde{\varvec{X}}\varvec{\beta }}_i^{(0)}} ={\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(2)} \lambda _k^{(0)} {\varvec{\beta }}_i^{(0)}} +{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} \tilde{\lambda }_k^{(1)} {\varvec{\beta }}_i^{(0)}} +\tilde{\lambda }_k^{(2)} {\varvec{\beta }}_k^{(0)} \end{aligned}$$

(84)

Premultiplying both sides of (84) by ${\varvec{\beta }}_k^{(0)\mathrm{H}}$ yields

$$\begin{aligned} \tilde{\lambda }_k^{(2)}= & {} {\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\tilde{\varepsilon }_{ki}^{(1)} {\varvec{\beta }}_k^{(0)\mathrm{H}} {\tilde{\varvec{X}}\varvec{\beta }}_i^{(0)}} ={\varvec{\beta }}_k^{(0)\mathrm{H}} {\tilde{\varvec{X}}}\left( {{\mathop {\mathop {\sum }\limits _{i=1}}\limits _{i\ne k}^{n}} {\left( {\lambda _k^{(0)} -\lambda _i^{(0)}} \right) ^{-1}{\varvec{\beta }}_i^{(0)} {\varvec{\beta }}_i^{(0)\mathrm{H}}}} \right) {\tilde{\varvec{X}}\varvec{\beta }}_k^{(0)}\nonumber \\= & {} {\varvec{\beta }}_k^{(0)\mathrm{H}} {\tilde{\varvec{X}}\varvec{E}}_k {\tilde{\varvec{X}}\varvec{\beta }}_k^{(0)} \end{aligned}$$

(85)

Combining (80), (83) and (85), Eqs. (20)–(21) hold true and hence the proof is completed.

Appendix 2: Proof of (22) to (24)

Expanding the matrix function ${\bar{\varvec{A}}}\left( {{\hat{\varvec{q}}} , {\hat{{\bar{\varvec{p}}}}}_k} \right)$ in a second-order Taylor series around point $\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right)$ gives

$$\begin{aligned} {\bar{\varvec{A}}}\left( {{\hat{\varvec{q}}} , {\hat{{\bar{\varvec{p}}}}}_k} \right)= & {} {\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) +\sum _{d=1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) } +\sum _{d=1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }\nonumber \\&+\frac{1}{2}\sum _{d_1 =1}^D {\sum _{d_2 =1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_2} \cdot {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }} \nonumber \\&+\frac{1}{2}\sum _{d_1 =1}^{2DL} {\sum _{d_2 =1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_2} \cdot {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }}\nonumber \\&+\sum _{d_1 =1}^D {\sum _{d_2 =1}^{2DL} {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_2} \cdot {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }} +o\left( {\varepsilon ^{2}} \right) \end{aligned}$$

(86)

where ${\ddot{\bar{\varvec{A}}}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right)$, ${\ddot{\bar{\varvec{A}}}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right)$, ${\ddot{\bar{\varvec{A}}}}_{d_1 d_2 }^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right)$, ${\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right)$ and ${\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right)$ are given in (24), $o\left( {\varepsilon ^{2}} \right)$ denotes all the three- and higher-order terms.

Inserting (86) into (13) leads to

$$\begin{aligned} {\varvec{V}}_k \left( {{\hat{\varvec{q}}} , {\hat{{\bar{\varvec{p}}}}}_k , {\varvec{n}}_k} \right)= & {} {\bar{\varvec{A}}}\left( {{\hat{\varvec{q}}} , {\hat{{\bar{\varvec{p}}}}}_k} \right) \left( {{\varvec{X}}_k +{\varvec{N}}_k} \right) \nonumber \\\approx & {} {\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k +\sum _{d=1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k}\nonumber \\&+\sum _{d=1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k} +{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{N}}_k \nonumber \\&+\frac{1}{2}\sum _{d_1 =1}^D {\sum _{d_2 =1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_2} \cdot {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k}}\nonumber \\&+\frac{1}{2}\sum _{d_1 =1}^{2DL} {\sum _{d_2 =1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_2} \cdot {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k}} \nonumber \\&+\sum _{d_1 =1}^D {\sum _{d_2 =1}^{2DL} {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_2} \cdot {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k}}\nonumber \\&+\sum _{d=1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{N}}_k} +\sum _{d=1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{N}}_k} \nonumber \\= & {} {\varvec{V}}_k^{(0)} +{\tilde{\varvec{V}}}_k^{(1)} +{\tilde{\varvec{V}}}_k^{(2)} \end{aligned}$$

(87)

where the three- and higher-order perturbation terms are ignored. At this point, the proof of (22) to (24) is ended.

Appendix 3: Specified expressions for the matrices included in (24)

We begin by deriving the first- and second-order partial derivative of $\phi \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right)$ as follows:

$$\begin{aligned} {\dot{\varvec{\varphi }}}^\mathrm{(a)}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) =\frac{\partial \phi \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) }{\partial {\varvec{q}}}=\frac{1}{c}\left( {\frac{{\dot{\varvec{p}}}_{l, k}}{\left\| {{\varvec{q}}-{\varvec{p}}_{l, k}} \right\| _2}-\frac{\left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) \left( {{\dot{\varvec{p}}}_{l, k}^\mathrm{T} \left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) } \right) }{\left\| {{\varvec{q}}-{\varvec{p}}_{l, k}} \right\| _2^3}} \right) \end{aligned}$$

(88)

(89)

$$\begin{aligned} {\ddot{\varvec{{\varPsi }}}}^\mathrm{(aa)}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right)= & {} \frac{\partial \phi ^{{2}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) }{\partial {\varvec{q}}\partial {\varvec{q}}^{\mathrm{T}}}\nonumber \\= & {} \frac{1}{c}\left( {\begin{array}{l} {3}\frac{\left( {{\dot{\varvec{p}}}_{l, k}^\mathrm{T} \left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) } \right) }{\left\| {{\varvec{q}}-{\varvec{p}}_{l, k}} \right\| _2^5}\left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) \left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) ^{\mathrm{T}}-\frac{{\dot{\varvec{p}}}_{l, k} \left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) ^{\mathrm{T}}}{\left\| {{\varvec{q}}-{\varvec{p}}_{l, k}} \right\| _2^3} \\ -\frac{{\dot{\varvec{p}}}_{l, k}^\mathrm{T} \left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) }{\left\| {{\varvec{q}}-{\varvec{p}}_{l, k}} \right\| _2^3}{\varvec{I}}_D -\frac{\left( {{\varvec{q}}-{\varvec{p}}_{l, k}} \right) {\dot{\varvec{p}}}_{l, k}^\mathrm{T}}{\left\| {{\varvec{q}}-{\varvec{p}}_{l, k}} \right\| _2^3} \\ \end{array}} \right) \end{aligned}$$

(90)

(91)

(92)

According to the fifth equation in (14), it follows

$$\begin{aligned}&{\dot{\bar{\varvec{A}}}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) =\frac{\partial {\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _d}=\left[ {{\begin{array}{llll} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _d}}&{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _d}}&{} \cdots &{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _d}} \\ \end{array}}} \right] \nonumber \\&\quad =\left[ {{\begin{array}{llll} {\left( {{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\dot{\varvec{\varDelta }}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) } \right) ^{\mathrm{H}}}&{} {\left( {{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\dot{\varvec{\varDelta }}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) } \right) ^{\mathrm{H}}}&{} \cdots &{} {\left( {{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\dot{\varvec{\varDelta }}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) } \right) ^{\mathrm{H}}} \\ \end{array}}} \right] \end{aligned}$$

(93)

$$\begin{aligned}&{\dot{\bar{\varvec{A}}}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) =\frac{\partial {\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _d}=\left[ {{\begin{array}{llll} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _d}}&{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _d}}&{} \cdots &{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _d}} \\ \end{array}}} \right] \nonumber \\&\quad =\left\{ {\begin{array}{l} \left[ {{\begin{array}{llll} {\left( {{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\dot{\varvec{\varDelta }}}_{\left[ d \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) } \right) ^{\mathrm{H}}}&{} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N} \\ \end{array}}} \right] \quad 1\le d\le 2D \\ \left[ {{\begin{array}{lllll} {{\varvec{O}}_N}&{} {\left( {{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\dot{\varvec{\varDelta }}}_{\left[ d \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) } \right) ^{\mathrm{H}}}&{} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N} \\ \end{array}}} \right] \quad 2D+1\le d\le 4D \\ \cdots \cdots \\ \left[ {{\begin{array}{llll} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N}&{} {\left( {{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\dot{\varvec{\varDelta }}}_{\left[ d \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) } \right) ^{\mathrm{H}}} \\ \end{array}}} \right] \quad \quad 2D\left( {L-1} \right) +1\le d\le 2DL \\ \end{array}} \right. \end{aligned}$$

(94)

$$\begin{aligned}&{\ddot{\bar{\varvec{A}}}}_{d_{1} d_{2} }^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) =\frac{\partial ^{{2}}{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {\varvec{q}} \right\rangle _{d_{2}}}=\left[ {{\begin{array}{llll} {\frac{\partial ^{{2}}{\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {\varvec{q}} \right\rangle _{d_{2}}}}&{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {\varvec{q}} \right\rangle _{d_{2}}}}&{} \cdots &{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {\varvec{q}} \right\rangle _{d_{2}}}} \\ \end{array}}} \right] \nonumber \\&\quad =\left[ {{\begin{array}{llll} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\dot{\varvec{\varDelta }}}_{d_{2}}^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\dot{\varvec{\varDelta }}}_{d_{1}}^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\ddot{\varvec{\varDelta }}}_{d_{1} d_{2}}^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}}&{} \cdots &{} \cdots &{} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\dot{\varvec{\varDelta }}}_{d_{2}}^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\dot{\varvec{\varDelta }}}_{d_{1}}^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\ddot{\varvec{\varDelta }}}_{d_{1} d_{2}}^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}} \\ \end{array}}} \right] \nonumber \\ \end{aligned}$$

(95)

$$\begin{aligned}&{\ddot{\bar{\varvec{A}}}}_{d_{1} d_{2} }^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) =\frac{\partial ^{{2}}{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}=\left[ {{\begin{array}{llll} {\frac{\partial ^{{2}}{\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}}&{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}}&{} \cdots &{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) }{\partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}} \\ \end{array}}} \right] \nonumber \\&\quad =\left\{ {\begin{array}{l} \left[ {{\begin{array}{llll} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_{2}} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_{1}} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_1} \right] _{2D} \left[ {d_2} \right] _{2D}}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}}&{} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N} \\ \end{array}}} \right] \;\;\left( {\begin{array}{l} 1\le d_1 \le 2D \\ 1\le d_{2} \le 2D \\ \end{array}} \right) \\ \left[ {{\begin{array}{lllll} {{\varvec{O}}_N}&{} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_{2}} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_{1}} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_1} \right] _{2D} \left[ {d_2} \right] _{2D}}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}}&{} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N} \\ \end{array}}} \right] \;\;\left( {\begin{array}{l} 2D+1\le d_1 \le 4D \\ 2D+1\le d_2 \le 4D \\ \end{array}} \right) \\ \cdots \cdots \\ \left[ {{\begin{array}{llll} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N}&{} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_{2}} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_{1}} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\ddot{\varvec{\varDelta }}}_{\left[ {d_1} \right] _{2D} \left[ {d_2} \right] _{2D}}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}} \\ \end{array}}} \right] \;\;\left( {\begin{array}{l} 2D\left( {L-1} \right) +1\le d_1 \le 2DL \\ 2D\left( {L-1} \right) +1\le d_2 \le 2DL \\ \end{array}} \right) \\ \end{array}} \right. \end{aligned}$$

(96)

$$\begin{aligned}&{\ddot{\bar{\varvec{A}}}}_{d_1 d_2 }^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) =\frac{\partial ^{{2}}{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}=\left[ {{\begin{array}{llll} {\frac{\partial ^{{2}}{\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}}&{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}}&{} \cdots &{} {\frac{\partial {\varvec{A}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) }{\partial \left\langle {\varvec{q}} \right\rangle _{d_{1}} \partial \left\langle {{\bar{\varvec{p}}}_k} \right\rangle _{d_{2}}}}\\ \end{array}}} \right] \nonumber \\&\quad =\left\{ {\begin{array}{l} \left[ {{\begin{array}{llll} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\dot{\varvec{\varDelta }}}_{\left[ {d_2} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\dot{\varvec{\varDelta }}}_{d_1}^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) {\dot{\varvec{\varDelta }}}_{d_1 \left[ {d_2} \right] _{2D}}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{1, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}}&{} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N}\\ \end{array}}} \right] \quad 1\le d_2 \le 2D \\ \left[ {{\begin{array}{lllll} {{\varvec{O}}_N}&{} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\dot{\varvec{\varDelta }}}_{\left[ {d_2} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\dot{\varvec{\varDelta }}}_{d_1}^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) {\dot{\varvec{\varDelta }}}_{d_1 \left[ {d_2} \right] _{2D}}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{2, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}}&{} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N} \\ \end{array}}} \right] \quad 2D+1\le d_2 \le 4D \\ \cdots \cdots \\ \left[ {{\begin{array}{llll} {{\varvec{O}}_N}&{} \cdots &{} {{\varvec{O}}_N}&{} {\left( {\begin{array}{l} {\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\dot{\varvec{\varDelta }}}_{\left[ {d_2} \right] _{2D}}^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\dot{\varvec{\varDelta }}}_{d_1}^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) \\ +{\varvec{A}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) {\dot{\varvec{\varDelta }}}_{d_1 \left[ {d_2} \right] _{2D}}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{L, k}} \right) \\ \end{array}} \right) ^{\mathrm{H}}} \\ \end{array}}} \right] \quad 2D\left( {L-1} \right) +1\le d_2 \le 2DL \\ \end{array}} \right. \end{aligned}$$

(97)

where

$$\begin{aligned} \left\{ {\begin{array}{l} {\dot{\varvec{\varDelta }}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) =\left\langle {{\dot{\varvec{\varphi }}}^\mathrm{(a)}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) } \right\rangle _d \cdot \hbox {diag}\left[ {{\begin{array}{llll} 0&{} {\hbox {i2}{\uppi }f_c T_s}&{} \cdots &{} {\hbox {i2}{\uppi }f_c \left( {N-1} \right) T_s} \\ \end{array}}} \right] \\ {\dot{\varvec{\varDelta }}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) =\left\langle {{\dot{\varvec{\varphi }}}^\mathrm{(b)}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) } \right\rangle _d \cdot \hbox {diag}\left[ {{\begin{array}{llll} 0&{} {\hbox {j2}{\uppi }f_c T_s}&{} \cdots &{} {\hbox {j2}{\uppi }f_c \left( {N-1} \right) T_s} \\ \end{array}}} \right] \\ {\ddot{\varvec{\varDelta }}}_{d_1 d_2}^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) =\left\langle {{\ddot{\varvec{{\varPsi }}}}^\mathrm{(aa)}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) } \right\rangle _{d_1 d_2} \cdot \hbox {diag}\left[ {{\begin{array}{llll} 0&{} {\hbox {j2}{\uppi }f_c T_s}&{} \cdots &{} {\hbox {j2}{\uppi }f_c \left( {N-1} \right) T_s} \\ \end{array}}} \right] \\ {\ddot{\varvec{\varDelta }}}_{d_1 d_2}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) =\left\langle {{\ddot{\varvec{{\varPsi }}}}^\mathrm{(ab)}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) } \right\rangle _{d_1 d_2} \cdot \hbox {diag}\left[ {{\begin{array}{llll} 0&{} {\hbox {j2}{\uppi }f_c T_s}&{} \cdots &{} {\hbox {j2}{\uppi }f_c \left( {N-1} \right) T_s} \\ \end{array}}} \right] \\ {\ddot{\varvec{\varDelta }}}_{d_1 d_2}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) =\left\langle {{\ddot{\varvec{{\varPsi }}}}^\mathrm{(bb)}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_{l, k}} \right) } \right\rangle _{d_1 d_2} \cdot \hbox {diag}\left[ {{\begin{array}{llll} 0&{} {\hbox {j2}{\uppi }f_c T_s}&{} \cdots &{} {\hbox {j2}{\uppi }f_c \left( {N-1} \right) T_s} \\ \end{array}}} \right] \\ \end{array}} \right. \end{aligned}$$

(98)

At this point, the derivation is completed.

Appendix 4: Proof of (33) to (37)

With the first equation in (23), it follows for any vector ${\varvec{z}}\in \mathbf{C}^{L\times {1}}$ that

$$\begin{aligned} {\tilde{\varvec{V}}}_k^{(1)} {\varvec{z}}= & {} \sum _{d=1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k {\varvec{z}}} +\sum _{d=1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _d \cdot {\dot{\bar{\varvec{A}}}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k {\varvec{z}}} +{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{N}}_k {\varvec{z}} \nonumber \\= & {} \frac{\partial \left( {{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k {\varvec{z}}} \right) }{\partial {\varvec{q}}^{\mathrm{T}}}{\tilde{\varvec{q}}}+\frac{\partial \left( {{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k {\varvec{z}}} \right) }{\partial {\bar{\varvec{p}}}_k^\mathrm{T}}{\tilde{\bar{\varvec{p}}}}_k +{\bar{\varvec{A}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) \left( {\hbox {diag}\left[ {\varvec{z}} \right] \otimes {\varvec{I}}_N} \right) {\varvec{{\varPi }}}_1 {\bar{\varvec{n}}}_k \nonumber \\= & {} {\varvec{T}}_{k1}^\mathrm{(a)} \left( {\varvec{z}} \right) {\tilde{\varvec{q}}}+{\varvec{T}}_{k2}^\mathrm{(a)} \left( {\varvec{z}} \right) {\tilde{\bar{\varvec{p}}}}_k +{\varvec{T}}_{k3}^\mathrm{(a)} \left( {\varvec{z}} \right) {\bar{\varvec{n}}}_k \end{aligned}$$

(99)

$$\begin{aligned} {\tilde{\varvec{V}}}_k^{(1)\mathrm{H}} {\varvec{z}}= & {} \sum _{d=1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _d \cdot {\varvec{X}}_k^\mathrm{H} {\dot{\bar{\varvec{A}}}}_d^\mathrm{(a)H} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{z}}} +\sum _{d=1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _d \cdot {\varvec{X}}_k^\mathrm{H} {\dot{\bar{\varvec{A}}}}_d^\mathrm{(b)H} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{z}}} +{\varvec{N}}_k^\mathrm{H} {\bar{\varvec{A}}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{z}}\nonumber \\= & {} \frac{\partial \left( {{\varvec{X}}_k^\mathrm{H} {\bar{\varvec{A}}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{z}}} \right) }{\partial {\varvec{q}}^{\mathrm{T}}}{\tilde{\varvec{q}}}+\frac{\partial \left( {{\varvec{X}}_k^\mathrm{H} {\bar{\varvec{A}}}^{\mathrm{H}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{z}}} \right) }{\partial {\bar{\varvec{p}}}_k^\mathrm{T}}{\tilde{\bar{\varvec{p}}}}_k +\left( {{\varvec{I}}_L \otimes {\varvec{z}}^{\mathrm{T}}} \right) {\bar{{\bar{\varvec{A}}}}}\left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{{\varPi }}}_2 {\bar{\varvec{n}}}_k \nonumber \\= & {} {\varvec{T}}_{k1}^\mathrm{(b)} \left( {\varvec{z}} \right) {\tilde{\varvec{q}}}+{\varvec{T}}_{k2}^\mathrm{(b)} \left( {\varvec{z}} \right) {\tilde{\bar{\varvec{p}}}}_k +{\varvec{T}}_{k3}^\mathrm{(b)} \left( {\varvec{z}} \right) {\bar{\varvec{n}}}_k \end{aligned}$$

(100)

where $\left\{ {{\varvec{T}}_{kn}^\mathrm{(a)} \left( \cdot \right) } \right\} _{n=1}^3$ and $\left\{ {{\varvec{T}}_{kn}^\mathrm{(b)} \left( \cdot \right) } \right\} _{n=1}^3$ are given in (35) and (36), respectively.

It can be seen from (99) and (100) that

$$\begin{aligned} \tilde{J}_{\mathrm{cost}}^{(1)}= & {} \sum _{k=1}^K {\left( {{\varvec{\beta }}_{k, L}^{(0)\mathrm{H}} {\varvec{V}}_k^{(0)\mathrm{H}} {\varvec{T}}_{k1}^\mathrm{(a)} \left( {{\varvec{\beta }}_{k, L}^{(0)}} \right) {\tilde{\varvec{q}}}+{\varvec{\beta }}_{k, L}^{(0)\mathrm{H}} {\varvec{V}}_k^{(0)\mathrm{H}} {\varvec{T}}_{k2}^\mathrm{(a)} \left( {{\varvec{\beta }}_{k, L}^{(0)}} \right) {\tilde{\bar{\varvec{p}}}}_k +{\varvec{\beta }}_{k, L}^{(0)\mathrm{H}} {\varvec{V}}_k^{(0)\mathrm{H}} {\varvec{T}}_{k3}^\mathrm{(a)} \left( {{\varvec{\beta }}_{k, L}^{(0)}} \right) {\bar{\varvec{n}}}_k} \right) } \nonumber \\&+\,\sum _{k=1}^K {\left( {{\varvec{\beta }}_{k, L}^{(0)\mathrm{H}} {\varvec{T}}_{k1}^\mathrm{(b)} \left( {{\varvec{V}}_k^{(0)} {\varvec{\beta }}_{k, L}^{(0)}} \right) {\tilde{\varvec{q}}}+{\varvec{\beta }}_{k, L}^{(0)\mathrm{H}} {\varvec{T}}_{k2}^\mathrm{(b)} \left( {{\varvec{V}}_k^{(0)} {\varvec{\beta }}_{k, L}^{(0)}} \right) {\tilde{\bar{\varvec{p}}}}_k +{\varvec{\beta }}_{k, L}^{(0)\mathrm{H}} {\varvec{T}}_{k3}^\mathrm{(b)} \left( {{\varvec{V}}_k^{(0)} {\varvec{\beta }}_{k, L}^{(0)}} \right) {\bar{\varvec{n}}}_k} \right) } \nonumber \\= & {} \sum _{k=1}^K {{\varvec{t}}_{k1}^\mathrm{H} {\tilde{\varvec{q}}}} +\sum _{k=1}^K {{\varvec{t}}_{k2}^\mathrm{H} {\tilde{\bar{\varvec{p}}}}_k} +\sum _{k=1}^K {{\varvec{t}}_{k3}^\mathrm{H} {\bar{\varvec{n}}}_k} \end{aligned}$$

(101)

where $\left\{ {{\varvec{t}}_{kn}} \right\} _{n=1}^3$ are given in (34). At this point, the proof of (33) to (37) is ended.

Appendix 5: Proof of (38) to (45)

Using (99), it follows for any vectors ${\varvec{z}}_{1}$ and ${\varvec{z}}_{2}$ and matrix ${\varvec{Z}}$ of compatible dimensions

$$\begin{aligned} {\varvec{z}}_1^\mathrm{H} {\tilde{\varvec{V}}}_k^{(1)\mathrm{H}} {\varvec{Z}\tilde{\varvec{V}}}_k^{(1)} {\varvec{z}}_2= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{T}}_{k1}^\mathrm{(a)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k1}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{T}}_{k2}^\mathrm{(a)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k2}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}\left( {{\varvec{T}}_{k1}^\mathrm{(a)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k2}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k1}^\mathrm{(a)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k2}^\mathrm{(a){*}} \left( {{\varvec{z}}_1} \right) } \right) {\tilde{\bar{\varvec{p}}}}_k\nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}\left( {{\varvec{T}}_{k1}^\mathrm{(a)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k1}^\mathrm{(a)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k3}^\mathrm{(a){*}} \left( {{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} \left( {{\varvec{T}}_{k2}^\mathrm{(a)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k2}^\mathrm{(a)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k3}^\mathrm{(a){*}} \left( {{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{T}}_{k3}^\mathrm{(a)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \nonumber \\= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k1}^\mathrm{(a)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k2}^\mathrm{(a)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k3}^\mathrm{(a)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k4}^\mathrm{(a)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k +{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k5}^\mathrm{(a)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{{\varPhi }}}_{k6}^\mathrm{(a)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \end{aligned}$$

(102)

where $\left\{ {{\varvec{{\varPhi }}}_{kn}^\mathrm{(a)} \left( {{\varvec{z}}_1 , {\varvec{Z}} , {\varvec{z}}_2} \right) } \right\} _{n=1}^6$ are given in (42).

Analogously, it follows from (100) for any vectors ${\varvec{z}}_{1}$ and ${\varvec{z}}_{2}$ and matrix ${\varvec{Z}}$ of compatible dimensions

$$\begin{aligned} {\varvec{z}}_1^\mathrm{H} {\tilde{\varvec{V}}}_k^{(1)} {\varvec{Z}\tilde{\varvec{V}}}_k^{(1)\mathrm{H}} {\varvec{z}}_2= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{T}}_{k1}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k1}^\mathrm{(b)} \left( {{\varvec{z}}_2} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{T}}_{k2}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k2}^\mathrm{(b)} \left( {{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k\nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}\left( {{\varvec{T}}_{k1}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k2}^\mathrm{(b)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k1}^\mathrm{(b)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k2}^\mathrm{(b){*}} \left( {{\varvec{z}}_1} \right) } \right) {\tilde{\bar{\varvec{p}}}}_k\nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}\left( {{\varvec{T}}_{k1}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(b)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k1}^\mathrm{(b)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k3}^\mathrm{(b){*}} \left( {{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} \left( {{\varvec{T}}_{k2}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(b)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k2}^\mathrm{(b)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k3}^\mathrm{(b){*}} \left( {{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{T}}_{k3}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(b)} \left( {{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k\nonumber \\= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k1}^\mathrm{(b)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k2}^\mathrm{(b)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k3}^\mathrm{(b)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k4}^\mathrm{(b)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k +{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k5}^\mathrm{(b)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{{\varPhi }}}_{k6}^\mathrm{(b)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \end{aligned}$$

(103)

where $\left\{ {{\varvec{{\varPhi }}}_{kn}^\mathrm{(b)} \left( {{\varvec{z}}_1 , {\varvec{Z}} , {\varvec{z}}_2} \right) } \right\} _{n=1}^6$ are given in (43).

In addition, combining (99) and (101) it can be obtained for any vectors ${\varvec{z}}_{1}$ and ${\varvec{z}}_{2}$ and matrix ${\varvec{Z}}$ of compatible dimensions that

$$\begin{aligned} {\varvec{z}}_1^\mathrm{H} {\tilde{\varvec{V}}}_k^{(1)} {\varvec{Z}\tilde{\varvec{V}}}_k^{(1)} {\varvec{z}}_2= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{T}}_{k1}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k1}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{T}}_{k2}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k2}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k\nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}\left( {{\varvec{T}}_{k1}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k2}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k1}^\mathrm{(a)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k2}^\mathrm{(b){*}} \left( {{\varvec{z}}_1} \right) } \right) {\tilde{\bar{\varvec{p}}}}_k\nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}\left( {{\varvec{T}}_{k1}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k1}^\mathrm{(a)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k3}^\mathrm{(b){*}} \left( {{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} \left( {{\varvec{T}}_{k2}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) +{\varvec{T}}_{k2}^\mathrm{(a)T} \left( {{\varvec{z}}_2} \right) {\varvec{Z}}^{\mathrm{T}}{\varvec{T}}_{k3}^\mathrm{(b){*}} \left( {{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{T}}_{k3}^\mathrm{(b)H} \left( {{\varvec{z}}_1} \right) {\varvec{ZT}}_{k3}^\mathrm{(a)} \left( {{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \nonumber \\= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k1}^\mathrm{(c)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k2}^\mathrm{(c)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k3}^\mathrm{(c)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k4}^\mathrm{(c)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k +{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k5}^\mathrm{(c)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \nonumber \\&+\,{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{{\varPhi }}}_{k6}^\mathrm{(c)} \left( {{\varvec{z}}_1 ,{\varvec{Z}},{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k \end{aligned}$$

(104)

$$\begin{aligned} {\varvec{z}}_1^\mathrm{H} {\tilde{\varvec{V}}}_k^{(1)\mathrm{H}} {\varvec{Z}\tilde{\varvec{V}}}_k^{(1)\mathrm{H}} {\varvec{z}}_2= & {} \left( {{\varvec{z}}_2^\mathrm{H} {\tilde{\varvec{V}}}_k^{(1)} {\varvec{Z}}^{\mathrm{H}}{\tilde{\varvec{V}}}_k^{(1)} {\varvec{z}}_1} \right) ^{\mathrm{H}} \nonumber \\= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k1}^\mathrm{(c){*}} \left( {{\varvec{z}}_2 ,{\varvec{Z}}^{\mathrm{H}},{\varvec{z}}_1} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k2}^\mathrm{(c){*}} \left( {{\varvec{z}}_2 ,{\varvec{Z}}^{\mathrm{H}},{\varvec{z}}_1} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k3}^\mathrm{(c){*}} \left( {{\varvec{z}}_2 ,{\varvec{Z}}^{\mathrm{H}},{\varvec{z}}_1} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k4}^\mathrm{(c){*}} \left( {{\varvec{z}}_2 ,{\varvec{Z}}^{\mathrm{H}},{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3 {\bar{\varvec{n}}}_k +{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k5}^\mathrm{(c){*}} \left( {{\varvec{z}}_2 ,{\varvec{Z}}^{\mathrm{H}},{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3 {\bar{\varvec{n}}}_k\nonumber \\&+\,{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{{\varPhi }}}_{k6}^\mathrm{(c)H} \left( {{\varvec{z}}_2 ,{\varvec{Z}}^{\mathrm{H}},{\varvec{z}}_1} \right) {\bar{\varvec{n}}}_k \end{aligned}$$

(105)

where $\left\{ {{\varvec{{\varPhi }}}_{kn}^\mathrm{(c)} \left( {{\varvec{z}}_1 , {\varvec{Z}} , {\varvec{z}}_2} \right) } \right\} _{n=1}^6$ are given in (44).

Furthermore, applying the second equation in (23), it follows for any vectors ${\varvec{z}}_{1}$ and ${\varvec{z}}_{2}$ of compatible dimensions that

$$\begin{aligned} {\varvec{z}}_1^\mathrm{H} {\tilde{\varvec{V}}}_k^{(2)} {\varvec{z}}_2= & {} \frac{1}{2}\sum _{d_1 =1}^D {\sum _{d_2 =1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_2} \cdot {\varvec{z}}_1^\mathrm{H} {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(aa)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k {\varvec{z}}_2}}\nonumber \\&+\,\frac{1}{2}\sum _{d_1 =1}^{2DL} {\sum _{d_2 =1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_2} \cdot {\varvec{z}}_1^\mathrm{H} {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(bb)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k {\varvec{z}}_2}} \nonumber \\&+\,\sum _{d_1 =1}^D {\sum _{d_2 =1}^{2DL} {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _{d_1} \cdot \left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _{d_2} \cdot {\varvec{z}}_1^\mathrm{H} {\ddot{\bar{\varvec{A}}}}_{d_1 d_2}^\mathrm{(ab)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{X}}_k {\varvec{z}}_2}}\nonumber \\&+\,\sum _{d=1}^D {\left\langle {{\tilde{\varvec{q}}}} \right\rangle _d \cdot {\varvec{z}}_1^\mathrm{H} {\dot{\bar{\varvec{A}}}}_d^\mathrm{(a)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{N}}_k {\varvec{z}}_2} +\sum _{d=1}^{2DL} {\left\langle {{\tilde{\bar{\varvec{p}}}}_k} \right\rangle _d \cdot {\varvec{z}}_1^\mathrm{H} {\dot{\bar{\varvec{A}}}}_d^\mathrm{(b)} \left( {{\varvec{q}} , {\bar{\varvec{p}}}_k} \right) {\varvec{N}}_k {\varvec{z}}_2} \nonumber \\= & {} {\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k1}^\mathrm{(d)} \left( {{\varvec{z}}_1 ,{\varvec{z}}_2} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k2}^\mathrm{(d)} \left( {{\varvec{z}}_1 ,{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k3}^\mathrm{(d)} \left( {{\varvec{z}}_1 ,{\varvec{z}}_2} \right) {\tilde{\bar{\varvec{p}}}}_k +{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k4}^\mathrm{(d)} \left( {{\varvec{z}}_1 ,{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k +{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k5}^\mathrm{(d)} \left( {{\varvec{z}}_1 ,{\varvec{z}}_2} \right) {\bar{\varvec{n}}}_k\end{aligned}$$

(106)

$$\begin{aligned} {\varvec{z}}_1^\mathrm{H} {\tilde{\varvec{V}}}_k^{(2)\mathrm{H}} {\varvec{z}}_2= & {} \left( {{\varvec{z}}_2^\mathrm{H} {\tilde{\varvec{V}}}_k^{(2)} {\varvec{z}}_1} \right) ^{\mathrm{H}}={\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k1}^\mathrm{(d){*}} \left( {{\varvec{z}}_2 ,{\varvec{z}}_1} \right) {\tilde{\varvec{q}}}+{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k2}^\mathrm{(d){*}} \left( {{\varvec{z}}_2 ,{\varvec{z}}_1} \right) {\tilde{\bar{\varvec{p}}}}_k \nonumber \\&+\,{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k3}^\mathrm{(d){*}} \left( {{\varvec{z}}_2 ,{\varvec{z}}_1} \right) {\tilde{\bar{\varvec{p}}}}_k +{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varPhi }}}_{k4}^\mathrm{(d){*}} \left( {{\varvec{z}}_2 ,{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3 {\bar{\varvec{n}}}_k \nonumber \\&+\,{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varPhi }}}_{k5}^\mathrm{(d){*}} \left( {{\varvec{z}}_2 ,{\varvec{z}}_1} \right) {\varvec{{\varPi }}}_3 {\bar{\varvec{n}}}_k \end{aligned}$$

(107)

where $\left\{ {{\varvec{{\varPhi }}}_{kn}^\mathrm{(d)} \left( {{\varvec{z}}_1 ,{\varvec{z}}_2} \right) } \right\} _{n=1}^6$ are given in (45).

It can be obtained from (102) to (107) that

$$\begin{aligned} \tilde{J}_{\mathrm{cost}}^{(2)}= & {} \sum _{k=1}^K {{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varOmega }}}_{k1} {\tilde{\varvec{q}}}} +\sum _{k=1}^K {{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varOmega }}}_{k2} {\tilde{\bar{\varvec{p}}}}_k} +\sum _{k=1}^K {{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varOmega }}}_{k3} {\tilde{\bar{\varvec{p}}}}_k} +\sum _{k=1}^K {{\tilde{\varvec{q}}}^{\mathrm{T}}{\varvec{{\varOmega }}}_{k4} {\bar{\varvec{n}}}_k} \nonumber \\&+\sum _{k=1}^K {{\tilde{\bar{\varvec{p}}}}_k^\mathrm{T} {\varvec{{\varOmega }}}_{k5} {\bar{\varvec{n}}}_k} +\sum _{k=1}^K {{\bar{\varvec{n}}}_k^\mathrm{H} {\varvec{{\varOmega }}}_{k6} {\bar{\varvec{n}}}_k} \end{aligned}$$

(108)

where $\left\{ {{\varvec{{\varOmega }}}_{kn}} \right\} _{n=1}^3$ are given in (39), $\left\{ {{\varvec{{\varOmega }}}_{kn}} \right\} _{n=4}^5$ are given in (40), ${\varvec{{\varOmega }}}_{k6}$ is given in (41). At this point, the proof of (38) to (45) is completed.

Appendix 6: Proof of (58)

Assume that the unit eigenvectors of matrix ${\varvec{Q}}$ associated with eigenvalues $\kappa _1 , \kappa _2 , \ldots , \kappa _D$ are denoted as ${\varvec{\eta }}_1 , {\varvec{\eta }}_2 , \ldots , {\varvec{\eta }}_D$. Then it follows

$$\begin{aligned} \left\| {{\tilde{\varvec{q}}}} \right\| _{2}^{2} \mathop =\limits ^\mathrm{d} {\tilde{\varvec{q}}}_0^\mathrm{T} {\varvec{Q}\tilde{\varvec{q}}}_0 \mathop =\limits ^\mathrm{d} \sum _{d=1}^D {\gamma _d \left( {{\tilde{\varvec{q}}}_0^\mathrm{T} {\varvec{\eta }}_d} \right) ^{2}} =\sum _{d=1}^D {\gamma _d \kappa _d} \end{aligned}$$

(109)

where $\kappa _d =\left( {{\tilde{\varvec{q}}}_0^\mathrm{T} {\varvec{\eta }}_d} \right) ^{2}$. In can be straightforwardly proved that $\kappa _d$ is chi-square distributed with one degree of freedom. Consequently, the characteristic function of $\kappa _d$ can be formulated as $\varphi _{\kappa _d} \left( t \right) =\left( {1-2\hbox {j}t} \right) ^{-1/2}$, and furthermore, the characteristic function of $\gamma _d \kappa _d$ can be expressed as $\varphi _{\gamma _d \kappa _d} \left( t \right) =\left( {1-2\hbox {j}\gamma _d t} \right) ^{-1/2}$. In addition, it can be easily verified that the random variables $\kappa _{d_{1}}$ and $\kappa _{d_{2}}$ are statistically independent for all $d_1 \ne d_2$ due to the orthogonality between ${\varvec{\eta }}_{d_{1}}$ and ${\varvec{\eta }}_{d_{2}}$. As a result, Eq. (58) holds due to the fact that the characteristic function of the sum of several independent random variables is equal to the product of all the single characteristic function corresponding to each random variable. Then, the proof of (58) is ended.

Appendix 7: Detailed derivation of matrices ${G}_{1}$, ${G}_{2}$ and ${G}_{3}$

According to the first equality in (78) we obtain

$$\begin{aligned} {\varvec{G}}_1= & {} {\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varPi }}}_{{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }}^\bot {\varvec{{\varSigma }}}_{\varvec{q}} ={\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varSigma }}}_{\varvec{q}} -{\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} } \left( {{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }} \right) ^{-1}{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }^\mathrm{H} {\varvec{{\varSigma }}}_{\varvec{q}} \nonumber \\= & {} \sum _{k=1}^K {\sum _{l=1}^L {\left| {\alpha _{l, k}} \right| ^{{2}}\left( {\frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l, k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}} \right) ^{\mathrm{H}}\cdot \hbox {diag}\left[ {\left| {{\varvec{s}}_k} \right| ^{2}} \right] \cdot \frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l, k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}}} \nonumber \\&-\sum _{k=1}^K \frac{{1}}{\left\| {{\varvec{\alpha }}_k} \right\| _2^2}\sum _{l_{1} =1}^L \sum _{l_{2} =1}^L \left| {\alpha _{l_{1} , k}} \right| ^{{2}}\left| {\alpha _{l_{2} , k}} \right| ^{{2}}\left( {\frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l_1 , k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}} \right) ^{\mathrm{H}}\nonumber \\&\cdot \; \hbox {diag}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l_1 , k}} \right) {\varvec{s}}_k^{*}} \right] \cdot {\bar{\bar{\varvec{I}}}}_N \cdot \hbox {diag}\left[ {{\varvec{A}}^{{*}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l_{2} , k}} \right) {\varvec{s}}_k} \right] \cdot \frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l_2 , k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}\qquad \end{aligned}$$

(110)

where ${\bar{\bar{\varvec{I}}}}_N =\hbox {blkdiag}\left[ {{\begin{array}{ll} {0}&{} {{\varvec{I}}_{N-1}} \\ \end{array}}} \right]$. Using the second equality in (78) leads to

$$\begin{aligned} {\varvec{G}}_2 ={\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varPi }}}_{{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }}^\bot \cdot \left[ {{\begin{array}{lll} {{\varvec{{\varSigma }}}_{{\varvec{\varDelta }\varvec{f}}}}&{} {{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{\alpha }} \right\} }}&{} {{\varvec{{\varSigma }}}_{\mathrm{Im}\left\{ {\varvec{\alpha }} \right\} }} \\ \end{array}}} \right] =\left[ {{\begin{array}{lll} {{\varvec{G}}_{2,1}}&{} {{\varvec{G}}_{2,2}}&{} {{\varvec{G}}_{2,3}} \\ \end{array}}} \right] \end{aligned}$$

(111)

where

$$\begin{aligned} \left\{ {\begin{array}{l} {\varvec{G}}_{2,1} ={\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varPi }}}_{{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }}^\bot {\varvec{{\varSigma }}}_{{\varvec{\varDelta }\varvec{f}}} ={\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varSigma }}}_{{\varvec{\varDelta }\varvec{f}}} -{\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} } \left( {{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }} \right) ^{-1}{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }^\mathrm{H} {\varvec{{\varSigma }}}_{{\varvec{\varDelta }\varvec{f}}} \\ \quad =\left[ {{\begin{array}{llll} {{\varvec{G}}_{2,1}^{(1)}}&{} {{\varvec{G}}_{2,1}^{(2)}}&{} \cdots &{} {{\varvec{G}}_{2,1}^{(K)}} \\ \end{array}}} \right] \\ {\varvec{G}}_{2,2} ={\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varPi }}}_{{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }}^\bot {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{\alpha }} \right\} } ={\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{\alpha }} \right\} } -{\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} } \left( {{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }} \right) ^{-1}{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }^\mathrm{H} {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{\alpha }} \right\} } \\ \quad =\left[ {{\begin{array}{llll} {{\varvec{G}}_{2,2}^{(1)}}&{} {{\varvec{G}}_{2,2}^{(2)}}&{} \cdots &{} {{\varvec{G}}_{2,2}^{(K)}} \\ \end{array}}} \right] \\ {\varvec{G}}_{2,3} ={\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varPi }}}_{{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }}^\bot {\varvec{{\varSigma }}}_{\mathrm{Im}\left\{ {\varvec{\alpha }} \right\} } =\hbox {j}{\varvec{{\varSigma }}}_{\varvec{q}}^\mathrm{H} {\varvec{{\varPi }}}_{{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }}^\bot {\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{\alpha }} \right\} } =\hbox {j}{\varvec{G}}_{2,2} \\ \end{array}} \right. \nonumber \\ \end{aligned}$$

(112)

in which

$$\begin{aligned} \left\{ {\begin{array}{l} {\varvec{G}}_{2,1}^{(k)} =\sum _{l=1}^L \left| {\alpha _{l, k}} \right| ^{{2}}\left( {\frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l, k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}} \right) ^{\mathrm{H}}\\ \quad \left( {\hbox {diag}\left[ {{\varvec{B}}_k^{*} {\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l, k}} \right) \left| {{\varvec{s}}_k} \right| ^{2}} \right] -\hbox {diag}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l, k}} \right) {\varvec{s}}_k^{*}} \right] \cdot {\bar{\bar{\varvec{I}}}}_N \cdot \hbox {diag}\left[ {{\varvec{B}}_k^{*} {\varvec{s}}_k} \right] } \right) {\dot{\varvec{b}}}_k \\ {\varvec{G}}_{2,2}^{(k)} =\left[ {{\begin{array}{lll} {\alpha _{1, k}^{*} \left( {\frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{1, k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}} \right) ^{\mathrm{H}}{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{1, k}} \right) \left| {{\varvec{s}}_k} \right| ^{{2}}}&{} {\cdots \;\;\;\cdots }&{} {\alpha _{L, k}^{*} \left( {\frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{L, k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}} \right) ^{\mathrm{H}}{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{L, k}} \right) \left| {{\varvec{s}}_k} \right| ^{{2}}} \\ \end{array}}} \right] \\ -\sum _{l=1}^L {\frac{\left| {\alpha _{l, k}} \right| ^{2}}{\left\| {{\varvec{\alpha }}_k} \right\| _2^2}\left( {\frac{\partial \hbox {vecd}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l, k}} \right) } \right] }{\partial {\varvec{q}}^{\mathrm{T}}}} \right) ^{\mathrm{H}}\cdot \hbox {diag}\left[ {{\varvec{A}}\left( {{\varvec{q}}\;,\;{\bar{\varvec{p}}}_{l, k}} \right) {\varvec{s}}_k^{*}} \right] \cdot {\bar{\bar{\varvec{I}}}}_N \left( {{\varvec{\alpha }}_k^\mathrm{H} \otimes {\varvec{s}}_k} \right) } \\ \end{array}} \right. \nonumber \\ \end{aligned}$$

(113)

Applying the third equality in (78) produces

$$\begin{aligned} {\varvec{G}}_3= & {} \left[ {{\begin{array}{lll} {{\varvec{{\varSigma }}}_{{\varvec{\varDelta }\varvec{f}}}}&{} {{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{\alpha }} \right\} }}&{} {{\varvec{{\varSigma }}}_{\mathrm{Im}\left\{ {\varvec{\alpha }} \right\} }} \\ \end{array}}} \right] ^{\mathrm{H}}\cdot {\varvec{{\varPi }}}_{{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{s}} \right\} }}^\bot \cdot \left[ {{\begin{array}{lll} {{\varvec{{\varSigma }}}_{{\varvec{\varDelta }\varvec{f}}}}&{} {{\varvec{{\varSigma }}}_{\mathrm{Re}\left\{ {\varvec{\alpha }} \right\} }}&{} {{\varvec{{\varSigma }}}_{\mathrm{Im}\left\{ {\varvec{\alpha }} \right\} }} \\ \end{array}}} \right] \nonumber \\= & {} \left[ {{\begin{array}{l@{\quad }l@{\quad }l} {{\varvec{G}}_{3,11}}&{} {{\varvec{G}}_{3,12}}&{} {{\varvec{G}}_{3,13}} \\ {{\varvec{G}}_{3,12}^\mathrm{H}}&{} {{\varvec{G}}_{3,22}}&{} {{\varvec{G}}_{3,23}} \\ {{\varvec{G}}_{3,13}^\mathrm{H}}&{} {{\varvec{G}}_{3,23}^\mathrm{H}}&{} {{\varvec{G}}_{3,33}} \\ \end{array}}} \right] \end{aligned}$$

(114)

where

(115)

in which

$$\begin{aligned} \left\{ {\begin{array}{l} {\varvec{G}}_{3,11}^{(k)} =\left\| {{\varvec{\alpha }}_k} \right\| _2^2 {\dot{\varvec{b}}}_k^\mathrm{H} \left( {\hbox {diag}\left[ {\left| {{\varvec{s}}_k} \right| ^{2}} \right] -\hbox {diag}\left[ {{\varvec{B}}_k {\varvec{s}}_k^{*}} \right] \cdot {\bar{\bar{\varvec{I}}}}_N \cdot \hbox {diag}\left[ {{\varvec{B}}_k^{*} {\varvec{s}}_k} \right] } \right) {\dot{\varvec{b}}}_k \\ {\varvec{G}}_{3,12}^{(k)} =\left( {{\dot{\varvec{b}}}_k^\mathrm{H} \cdot \hbox {diag}\left[ {{\varvec{B}}_k {\varvec{s}}_k^{*}} \right] \cdot {\varvec{s}}_k} \right) {\varvec{\alpha }}_k^\mathrm{H} -{\dot{\varvec{b}}}_k^\mathrm{H} \cdot \hbox {diag}\left[ {{\varvec{B}}_k {\varvec{s}}_k^{*}} \right] \cdot {\bar{\bar{\varvec{I}}}}_N \left( {{\varvec{\alpha }}_k^\mathrm{H} \otimes {\varvec{s}}_k}\right) \\ {\varvec{G}}_{3,22}^{(k)} =\left\| {{\varvec{s}}_k} \right\| _2^2 {\varvec{I}}_L -\frac{1}{\left\| {{\varvec{\alpha }}_k} \right\| _2^2}\left( {{\varvec{\alpha }}_k \otimes {\varvec{s}}_k^\mathrm{H}} \right) {\bar{\bar{\varvec{I}}}}_N \left( {{\varvec{\alpha }}_k^\mathrm{H} \otimes {\varvec{s}}_k} \right) \\ \end{array}} \right. \quad \left( {k=1 , 2 , \ldots , K} \right) \nonumber \\ \end{aligned}$$

(116)

At this point, the derivation is completed.