Optimal estimation of sensor biases for asynchronous multi-sensor data fusion

Abstract

An important step in a multi-sensor surveillance system is to estimate sensor biases from their noisy asynchronous measurements. This estimation problem is computationally challenging due to the highly nonlinear transformation between the global and local coordinate systems as well as the measurement asynchrony from different sensors. In this paper, we propose a novel nonlinear least squares formulation for the problem by assuming the existence of a reference target moving with an (unknown) constant velocity. We also propose an efficient block coordinate decent (BCD) optimization algorithm, with a judicious initialization, to solve the problem. The proposed BCD algorithm alternately updates the range and azimuth bias estimates by solving linear least squares problems and semidefinite programs. In the absence of measurement noise, the proposed algorithm is guaranteed to find the global solution of the problem and the true biases. Simulation results show that the proposed algorithm significantly outperforms the existing approaches in terms of the root mean square error.

Appendices

Appendix A: Proof of Theorem 1

We first reformulate problem (4.2) as

$$\begin{aligned} \min _{\varDelta \rho }\min _{{\mathbf {v}}}~~\left\| {\mathbf {H}}_{0}\varDelta \rho +{\mathbf {H}}_{1}{\mathbf {v}}-{\mathbf {y}} \right\| ^2. \end{aligned}$$

(7.1)

The closed-form solution of the above problem with respect to ${\mathbf {v}}$ is

$$\begin{aligned} {\mathbf {v}}^*=-\left( {\mathbf {H}}_1^T{\mathbf {H}}_1\right) ^{-1}{\mathbf {H}}_1^T\left( {\mathbf {H}}_0\varDelta \rho -{\mathbf {y}}\right) . \end{aligned}$$

(7.2)

Substituting (7.2) into (7.1), we obtain the following linear LS problem with respect to $\varDelta \rho$:

$$\begin{aligned} \min _{\varDelta \rho }~~\left\| \left( {\mathbf {I}}_{2(K-1)}-\bar{\mathbf {H}}_1\right) \left( {\mathbf {H}}_0\varDelta \rho -{\mathbf {y}}\right) \right\| ^2, \end{aligned}$$

(7.3)

where $\bar{\mathbf {H}}_1={\mathbf {H}}_1\left( {\mathbf {H}}_1^T{\mathbf {H}}_1\right) ^{-1}{\mathbf {H}}_1^T$. By the definitions of ${\mathbf {H}}_0,~{\mathbf {H}}_1,$ and ${\mathbf {y}}$ in (4.3), problem (7.3) can be further rewritten as

$$\begin{aligned} \min _{\varDelta \rho }~~\sum _{k=1}^{K-1}\displaystyle \left\| {\mathbf {a}}_k\varDelta \rho - {\mathbf {b}}_k \right\| ^2, \end{aligned}$$

where

$$\begin{aligned} {\mathbf {a}}_k=\left[ \displaystyle \sum _{\ell =1}^{K-1}\bar{T}_\ell ^kc_\ell ,~\displaystyle \sum _{\ell =1}^{K-1}\bar{T}_\ell ^ks_\ell \right] ^T,\quad {\mathbf {b}}_k=\left[ \displaystyle \sum _{\ell =1}^{K-1}\bar{T}_\ell ^ky_\ell ^c,~\displaystyle \sum _{\ell =1}^{K-1}\bar{T}_\ell ^ky_\ell ^s\right] ^T, \end{aligned}$$

and

$$\begin{aligned} \bar{T}_\ell ^k=\left\{ \begin{aligned}&1-\frac{T_k^2}{\sum _{i=1}^{K-1}T_i^2},&\ell =k,\\&-\frac{T_kT_\ell }{\sum _{i=1}^{K-1}T_i^2},&\ell \ne k.\end{aligned}\right. \end{aligned}$$

To prove Theorem 1, it suffices to show that ${\mathbf {a}}_k^T{\mathbf {a}}_k$ and ${\mathbf {a}}_k^T{\mathbf {b}}_k$ for all $k=1,2,\ldots ,K-1$ are independent of $\varDelta \phi .$ Next, we show that this is indeed true.

It is simple to compute, for $k=1,2,\ldots ,K,$

$$\begin{aligned} {\mathbf {a}}_k^T{\mathbf {a}}_k =\sum _{\ell =1}^{K-1}\sum _{j=1}^{K-1}\bar{T}_\ell ^k\bar{T}_j^k\left( c_\ell c_j+s_\ell s_j\right) ,\quad {\mathbf {a}}_k^T{\mathbf {b}}_k =\sum _{\ell =1}^{K-1}\sum _{j=1}^{K-1}\bar{T}_\ell ^k\bar{T}_j^k\left( c_\ell y^c_j+s_\ell y^s_j\right) . \end{aligned}$$

Moreover, by the definitions of $c_k, s_k, y_k^c,$ and $y_k^s$ in (4.4), (4.5), (4.6), and (4.7) and the fact

$$\begin{aligned} \cos (\alpha _1+\varDelta \phi )\cos (\alpha _2+\varDelta \phi )+\sin (\alpha _1+\varDelta \phi )\sin (\alpha _2+\varDelta \phi ) =\cos (\alpha _1-\alpha _2), \end{aligned}$$

we can obtain, for $\ell , j=1,2,\ldots ,K-1,$

$$\begin{aligned} c_\ell c_j+s_\ell s_j=&\lambda ^{-2}\left[ \cos \left( \phi _{\ell +1}-\phi _{j+1}\right) +\cos \left( \phi _{\ell }-\phi _{j}\right) \right. \\&\left. -\,\cos \left( \phi _{\ell +1}-\phi _{j}\right) -\cos \left( \phi _{j+1}-\phi _{\ell }\right) \right] ,\\ c_\ell y_j^c+s_\ell y_j^s=&\lambda ^{-2}\left[ \rho _{j+1}\cos \left( \phi _{\ell +1}-\phi _{j+1}\right) +\rho _{j}\cos \left( \phi _{\ell }-\phi _{j}\right) \right. \\&\left. -\,\rho _{j}\cos \left( \phi _{\ell +1}-\phi _{j}\right) -{{\rho _{j+1}}}\cos \left( \phi _{j+1}-\phi _{\ell }\right) \right] . \end{aligned}$$

This immediately shows that ${\mathbf {a}}_k^T{\mathbf {a}}_k$ and ${\mathbf {a}}_k^T{\mathbf {b}}_k$ for all $k=1,2,\ldots ,K-1$ are independent of $\varDelta \phi .$ This completes the proof of Theorem 1.

Appendix B: Proof of Theorem 2

To show that SDP (5.10) admits a unique solution of rank one, it is sufficient to show that, if $\mathbf {\varDelta H}$ is sufficiently small, there always exists ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$ such that ${\mathbf {1}}{\mathbf {1}}^T\in {\mathbb {H}}^{M+1}$ is the unique solution of the following SDP

$$\begin{aligned} \begin{aligned} \min _{{\mathbf {X}}\in {\mathbb {H}}^{M+1}}\quad&\text {Tr}(\hat{\mathbf {C}}X)\\ \text {s.t.}\quad \quad&\text {diag}({\mathbf {X}})={\mathbf {1}}, \\&{\mathbf {X}}\succeq {\mathbf {0}}, \end{aligned} \end{aligned}$$

(7.4)

where

$$\begin{aligned} \hat{\mathbf {C}}=\begin{bmatrix}{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}&\quad {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}} \\ {\mathbf {c}}^\dagger \mathbf {PHW}&\quad 0\end{bmatrix}\in {\mathbb {H}}^{M+1}. \end{aligned}$$

By Lemma 1, we only need to show that, if $\mathbf {\varDelta H}$ is sufficiently small, there always exists ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$ such that ${\mathbf {1}}{\mathbf {1}}^T\in {\mathbb {H}}^{M+1}$ and some ${\mathbf {y}}\in {\mathbb {R}}^{M+1}$ jointly satisfy

$$\begin{aligned}&\begin{aligned}&\hat{\mathbf {C}}+\text {diag}({\mathbf {y}})\\&\quad =\begin{bmatrix}{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {Diag}({\mathbf {y}}_{1:M})&\quad {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}} \\ {\mathbf {c}}^\dagger \mathbf {PHW}&\quad {\mathbf {y}}_{M+1}\end{bmatrix}\succeq {\mathbf {0}}, \end{aligned} \end{aligned}$$

(7.5)

$$\begin{aligned}&\begin{bmatrix}{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {Diag}({\mathbf {y}}_{1:M})&\quad {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}} \\ {\mathbf {c}}^\dagger \mathbf {PHW}&\quad {\mathbf {y}}_{M+1}\end{bmatrix}{\mathbf {1}}={\mathbf {0}}\in {\mathbb {R}}^{M+1}, \end{aligned}$$

(7.6)

and

$$\begin{aligned} {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {Diag}({\mathbf {y}}_{1:M}) \succ {\mathbf {0}}. \end{aligned}$$

(7.7)

Notice that the above (7.5), (7.6), and (7.7) correspond to conditions 2, 3, and 4 in Lemma 1, respectively, and ${\mathbf {1}}{\mathbf {1}}^T\in {\mathbb {S}}^{M+1}$ obviously satisfies condition 1 in Lemma 1. Moreover, we rewrite (7.6) as follows:

$$\begin{aligned}&\left[ {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {Diag}({\mathbf {y}}_{1:M}) \right] {\mathbf {1}}+{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}}={\mathbf {0}}\in {\mathbb {R}}^{M}, \end{aligned}$$

(7.8a)

$$\begin{aligned}&\left[ {\mathbf {c}}^\dagger \mathbf {PHW}\right] {\mathbf {1}}+{\mathbf {y}}_{M+1}=0. \end{aligned}$$

(7.8b)

Next, we prove that, if $\mathbf {\varDelta H}$ is sufficiently small, there always exist ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$ and ${\mathbf {y}}\in {\mathbb {R}}^{M+1}$that satisfy (7.5), (7.8a), (7.8b), and (7.7). We divide the proof into two parts. More specifically, we first show that, if there exist ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$ and ${\mathbf {y}}\in {\mathbb {R}}^{M+1}$ satisfying (7.8a) and (7.7), then we can construct ${\mathbf {y}}_{M+1}$ that simultaneously satisfies (7.5) and (7.8b). Then, we show that, if $\mathbf {\varDelta H}$ is sufficiently small, there indeed exist ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$ and ${\mathbf {y}}_{1:M}$ that satisfy (7.8a) and (7.7).

Let us first assume that ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ (with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$) and ${\mathbf {y}}_{1:M}$ satisfy (7.7) and (7.8a). We show that there exists ${\mathbf {y}}_{M+1}$ that simultaneously satisfies (7.5) and (7.8b). Let

$$\begin{aligned} {\mathbf {y}}_{M+1}={\mathbf {c}}^\dagger \mathbf {PHW}\left[ {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {Diag}({\mathbf {y}}_{1:M}) \right] ^{-1}{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}}. \end{aligned}$$

(7.9)

This, together with (7.7), immediately implies (7.5). Moreover, from (7.8a), we have

$$\begin{aligned} {\mathbf {1}}=-\left[ {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {Diag}({\mathbf {y}}_{1:M}) \right] ^{-1}{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}}\in {\mathbb {R}}^{M}. \end{aligned}$$

Combining the above and (7.9) immediately yields (7.8b).

Now, let us argue that, if $\mathbf {\varDelta H}$ is sufficiently small, then there always exist ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ (with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$) and ${\mathbf {y}}_{1:M}$ that satisfy (7.8a) and (7.7). In particular, the following Claim 1 and Claim 2 guarantee the existence of ${\mathbf {W}}=\text {Diag}({\mathbf {w}})$ (with $|{\mathbf {w}}_m|=1$ for all $m=1,2,\ldots ,M$) and ${\mathbf {y}}_{1:M}$ that satisfy (7.8a) and (7.7), respectively.

Claim 1. There always exist a neighborhood ${{\mathcal {H}}}\subseteq {\mathbb {C}}^{(K-1)\times M}$ containing $\tilde{\mathbf {H}}$ and two unique continuously differentiable functions $d_{\varvec{\psi }}:{\mathbb {C}}^{(K-1)\times M}\mapsto {\mathbb {R}}^{M}$ and $d_{{\mathbf {y}}}:{\mathbb {C}}^{(K-1)\times M}\mapsto {\mathbb {R}}^{M}$ such that (7.8a) holds true for all ${\mathbf {H}}\in {\mathcal {H}}$ with ${\mathbf {W}}=\text {Diag}\left( \left[ e^{j\psi _1}, e^{j\psi _2},\ldots , e^{j\psi _M}\right] ^T\right)$, $\varvec{\psi }=d_{\varvec{\psi }}({\mathbf {H}}),$ and ${\mathbf {y}}=d_{{\mathbf {y}}}({\mathbf {H}})$.

Proof

To prove Claim 1, we first reformulate complex equation (7.8a) as an equivalent real form; then we apply the implicit function theorem [3] based on the equivalent form to show the existence of the neighborhood ${\mathcal {H}}$ and two unique continuously differentiable functions $d_{\varvec{\psi }}$ and $d_{{\mathbf {y}}}.$

The equivalent form of (7.8a) is

$$\begin{aligned} f\left( \varvec{\psi },{\mathbf {y}},{\mathbf {H}}\right) =\varvec{0}, \end{aligned}$$

(7.10)

where $f\left( \varvec{\psi },{\mathbf {y}},{\mathbf {H}} \right) :{\mathbb {R}}^M\times {\mathbb {R}}^M\times {\mathbb {C}}^{(K-1)\times M}\mapsto {\mathbb {R}}^{2M}$ is

$$\begin{aligned} \begin{aligned} f\left( \varvec{\psi },{\mathbf {y}},{\mathbf {H}}\right) =&\begin{bmatrix}\text {Re}\left\{ \left[ {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {diag}({\mathbf {y}}_{1:M}) \right] {\mathbf {1}}+{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}}\right\} \\ \text {Im}\left\{ \left[ {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {diag}({\mathbf {y}}_{1:M}) \right] {\mathbf {1}}+{\mathbf {W}}^\dagger {\mathbf {H}}^\dagger {\mathbf {P}}{\mathbf {c}}\right\} \end{bmatrix}\\ =&\begin{bmatrix} \mathbf {C_{\varvec{\psi }}\text {Re}\{ A_H \}}\mathbf {C_{\varvec{\psi }}}+\mathbf {S_{\varvec{\psi }}\text {Im}\{ A_H \}}\mathbf {C_{\varvec{\psi }}}\\ \mathbf {C_{\varvec{\psi }}\text {Re}\{ A_H \}}\mathbf {S_{\varvec{\psi }}}+\mathbf {S_{\varvec{\psi }}\text {Im}\{ A_H \}}\mathbf {S_{\varvec{\psi }}} \end{bmatrix}{\mathbf {1}}\\&+ \begin{bmatrix} -\mathbf {C_{\varvec{\psi }}\text {Im}\{ A_H \}}\mathbf {S_{\varvec{\psi }}}+\mathbf {S_{\varvec{\psi }}\text {Re}\{ A_H \}}\mathbf {S_{\varvec{\psi }}}\\ \mathbf {C_{\varvec{\psi }}\text {Im}\{ A_H \}}\mathbf {C_{\varvec{\psi }}}-\mathbf {S_{\varvec{\psi }}\text {Re}\{ A_H \}}\mathbf {C_{\varvec{\psi }}} \end{bmatrix}{\mathbf {1}}\\&+ \begin{bmatrix} \mathbf {C_{\varvec{\psi }}}\mathbf {\text {Re}\{ b_H \}}+\mathbf {S_{\varvec{\psi }}}\mathbf {\text {Im}\{ b_H \}}\\ \mathbf {C_{\varvec{\psi }}}\mathbf {\text {Im}\{ b_H \}}-\mathbf {S_{\varvec{\psi }}}\mathbf {\text {Re}\{ b_H \}} \end{bmatrix} +\begin{bmatrix}{\mathbf {y}}\\\varvec{0}\end{bmatrix}. \end{aligned} \end{aligned}$$

(7.11)

In the above, $\mathbf {C_{\varvec{\psi }}}$ and $\mathbf {S_{\varvec{\psi }}}$ are diagonal matrices related to $\varvec{\psi }$ as follows:

$$\begin{aligned} \begin{aligned} \mathbf {C_{\varvec{\psi }}}=\text {Diag}\left( \left[ \cos \varvec{\psi }_1,\ldots ,\cos \varvec{\psi }_M\right] ^T\right) ,\\ \mathbf {S_{\varvec{\psi }}}=\text {Diag}\left( \left[ \sin \varvec{\psi }_1,\ldots ,\sin \varvec{\psi }_M\right] ^T\right) ; \end{aligned} \end{aligned}$$

(7.12)

and $\mathbf {A_H}\in {\mathbb {H}}^{M}$ and $\mathbf { b_H}\in {\mathbb {C}}^M$ are determined by ${\mathbf {H}}$ as follows:

$$\begin{aligned} \mathbf {A_H} = \mathbf {H^\dagger PH},~\mathbf { b_H} = \mathbf {H^\dagger Pc}. \end{aligned}$$

Now, we apply the implicit function theorem [3, Theorem 5] to Eq. (7.10). Notice that it is the same to define $f\left( \varvec{\psi },{\mathbf {y}},{\mathbf {H}}\right)$ in (7.11) over $\left( \varvec{\psi },{\mathbf {y}},{\mathbf {H}}\right)$ or define $f(\varvec{\psi },{\mathbf {y}},\text {Re}\{\mathbf {{H}}\},\text {Im}\{\mathbf {{H}}\}): {\mathbb {R}}^M\times {\mathbb {R}}^M\times {\mathbb {R}}^{(K-1)\times M}\times {\mathbb {R}}^{(K-1)\times M}\mapsto {\mathbb {R}}^{2M}$ over $(\varvec{\psi },{\mathbf {y}},\text {Re}\{\mathbf {{H}}\},\text {Im}\{\mathbf {{H}}\}).$ For notational simplicity, we will keep using (7.11) (but we can think it as $f(\varvec{\psi },{\mathbf {y}},\text {Re}\{\mathbf {{H}}\},\text {Im}\{\mathbf {{H}}\})$) in our proof. First of all, $f(\varvec{\psi },{\mathbf {y}},\mathbf {{H}})$ is continuously differentiable (i.e., $f(\varvec{\psi },{\mathbf {y}},\text {Re}\{\mathbf {{H}}\},\text {Im}\{\mathbf {{H}}\})$ is continuously differentiable).

Moreover, it follows from (5.17) that

$$\begin{aligned} f\left( \varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}}\right) ={\mathbf {0}}, \end{aligned}$$

(7.13)

where $\varvec{\tilde{\psi }}=\varvec{\varDelta \bar{\phi }},~\tilde{\mathbf {y}}={\mathbf {0}},$ and $\tilde{\mathbf {H}}$ is defined in (5.14). If the Jacobian matrix $D_{(\varvec{\psi },{\mathbf {y}})}f$ is invertible at point $(\varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}})$, then it follows from the implicit function theorem [3, Theorem 5] that the desired $\mathcal {{H}}$, ${d}_{\varvec{\psi }},$ and ${d}_{{\mathbf {y}}}$ in Claim 1 must exist.

It remains to prove that the Jacobian matrix $D_{(\varvec{\psi },{\mathbf {y}})}f$ at point $(\varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}} )$ is invertible.

By some calculations, we obtain

$$\begin{aligned} D_{(\varvec{\psi },{\mathbf {y}})}f=\begin{bmatrix} \frac{\partial f}{\partial \varvec{\psi }}&\frac{\partial f}{\partial {\mathbf {y}}} \end{bmatrix}=\begin{bmatrix} *&{\mathbf {I}}_M\\{\mathbf {G}}&{\mathbf {0}} \end{bmatrix}\in {\mathbb {C}}^{2M\times 2M}, \end{aligned}$$

(7.14)

where

$$\begin{aligned} \begin{aligned} {\mathbf {G}}=&~{\mathbf {C}}_{\varvec{\psi }}\mathbf {\text {Re}\{ A_H \}}{\mathbf {C}}_{\varvec{\psi }}- {\mathbf {S}}_{\varvec{\psi }}\text {Diag}(\mathbf {\text {Re}\{ A_H \}}{\mathbf {s}}_{\varvec{\psi }})\\ {}&+ {\mathbf {S}}_{\varvec{\psi }}\mathbf {\text {Im}\{ A_H \}}{\mathbf {C}}_{\varvec{\psi }}+ {\mathbf {C}}_{\varvec{\psi }}\text {Diag}(\mathbf {\text {Im}\{ A_H \}}{\mathbf {s}}_{\varvec{\psi }})\\ {}&- {\mathbf {C}}_{\varvec{\psi }}\mathbf {\text {Im}\{ A_H \}}{\mathbf {S}}_{\varvec{\psi }}- {\mathbf {S}}_{\varvec{\psi }}\text {Diag}(\mathbf {\text {Im}\{ A_H \}}{\mathbf {c}}_{\varvec{\psi }})\\ {}&+ {\mathbf {S}}_{\varvec{\psi }}\mathbf {\text {Re}\{ A_H \}}{\mathbf {S}}_{\varvec{\psi }}- {\mathbf {C}}_{\varvec{\psi }}\text {Diag}(\mathbf {\text {Re}\{ A_H \}}{\mathbf {c}}_{\varvec{\psi }})\\ {}&+ {\mathbf {S}}_{\varvec{\psi }}\text {Diag}(\mathbf {\text {Im}\{ b_H \}})+{\mathbf {C}}_{\varvec{\psi }}\text {Diag}(\mathbf {\text {Re}\{ b_H \}}). \end{aligned} \end{aligned}$$

(7.15)

By (7.13), we have

$$\begin{aligned} \begin{aligned} {\text {Re}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}}\mathbf {{c}}_{\tilde{\varvec{\psi }}}-{\text {Im}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}}\mathbf {{s}}_{\varvec{\tilde{\psi }}}={\text {Re}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}},\\ {\text {Im}\{\mathbf {A}_{\tilde{\mathbf{H}}}\}}\mathbf {{c}}_{\varvec{\tilde{\psi }}}+{\text {Re}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}}\mathbf {{s}}_{\varvec{\tilde{\psi }}}={\text {Im}\{ \mathbf {b}_{\tilde{\mathbf{H}}}\}}. \end{aligned} \end{aligned}$$

(7.16)

Substituting (7.16) into (7.15), we obtain

$$\begin{aligned} {\mathbf {G}}|_{(\varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}})}=\begin{bmatrix}{\mathbf {C}}_{\varvec{\tilde{\psi }}}&{\mathbf {S}}_{\varvec{\tilde{\psi }}}\end{bmatrix} \begin{bmatrix}{\text {Re}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}}&-{\text {Im}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}}\\{\text {Im}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}}&{\text {Re}\{ \mathbf {A}_{\tilde{\mathbf{H}}}\}}\end{bmatrix} \begin{bmatrix}{\mathbf {C}}_{\varvec{\tilde{\psi }}}\\{\mathbf {S}}_{\varvec{\tilde{\psi }}}\end{bmatrix}. \end{aligned}$$

Since ${\mathbf {A}_{\tilde{\mathbf{H}}}}={{\mathbf {H}}^\dagger \mathbf P \tilde{\mathbf{H}}}$ is positive definite and the matrix $\left[ {\mathbf {C}}_{\varvec{\tilde{\psi }}}, {\mathbf {S}}_{\varvec{\tilde{\psi }}}\right]$ is of full row rank, we immediately have

$$\begin{aligned} {\mathbf {x}}^T{\mathbf {G}}|_{(\varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}})}{\mathbf {x}}>0,~\forall ~{\mathbf {x}}\in {\mathbb {R}}^{M}, \end{aligned}$$

which further implies that ${\mathbf {G}}|_{(\varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}})}$ is invertible. Consequently, the Jacobian matrix $D_{(\varvec{\psi },{\mathbf {y}})}f$ in (7.14) at point $(\varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}})$ is invertible. The proof of Claim 1 is completed. $\square$

Let ${\mathbf {F}}$ and ${\mathbf {D}}$ be the Jacobian matrices of function f (defined in (7.11)) with respect to $\text {vec}\left( {\mathbf {H}}\right)$ and $(\varvec{\psi },{\mathbf {y}})$ at point $(\varvec{\tilde{\psi }},\tilde{\mathbf {y}},\tilde{\mathbf {H}})$, respectively. From the proof of Claim 1, we know that both ${\mathbf {F}}$ and ${\mathbf {D}}$ are well defined.

Claim 2. Consider ${\mathbf {H}}\in {{\mathcal {H}}}\subseteq {\mathbb {C}}^{(K-1)\times M},$ where ${{\mathcal {H}}}$ is the one in Claim 1. Suppose

$$\begin{aligned} \lambda _{\text {min}}({\mathbf {H}}^\dagger {{\mathbf {P}}}{{\mathbf {H}}})>\min _{1\le m\le M}\left\{ -\text {Re}\left\{ \left[ {\mathbf {D}}^{-1}{\mathbf {F}}\text {vec}\left( {\mathbf {H}} - {{\tilde{\mathbf{H }}}}\right) \right] _{M+m} \right\} \right\} .\end{aligned}$$

(7.17)

Then, ${\mathbf {W}}$ and ${\mathbf {y}}$ defined in Claim 1 satisfy (7.7).

Proof

It follows from Claim 1 that

$$\begin{aligned} {\mathbf {y}}_{m}{{({\mathbf {H}})}}:=\left[ d_{{\mathbf {y}}}({\mathbf {H}})\right] _m,~m=1,2,\ldots ,M \end{aligned}$$

are continuously differentiable functions with respect to ${\mathbf {H}}\in {{\mathcal {H}}}\subseteq {\mathbb {C}}^{(K-1)\times M}.$

Therefore, for $m=1,2,\ldots ,M,$ we get

$$\begin{aligned} {\mathbf {y}}_m{{({\mathbf {H}})}}&= {\mathbf {y}}_m({{\tilde{\mathbf{H }}}}) + \text {Re}\left\{ \text {Tr}\left( \nabla {\mathbf {y}}_m({{\tilde{\mathbf{H }}}}) {{ \left( {\mathbf {H}}-{{\tilde{\mathbf{H }}}}\right) }}\right) \right\} + {{o}}(\Vert { {{\mathbf {H}}-{{\tilde{\mathbf{H }}}}}}\Vert )\nonumber \\&= -\text {Re}\left\{ \left[ {\mathbf {D}}^{-1}{\mathbf {F}}\text {vec} {{\left( {\mathbf {H}}-{{\tilde{\mathbf{H }}}}\right) }} \right] _{M+m} \right\} + {{o}}(\Vert {{{\mathbf {H}}-{{\tilde{\mathbf{H }}}}}}\Vert ), \end{aligned}$$

(7.18)

where (7.18) is due to the differentiation rule of the implicit function and ${\mathbf {y}}_m({{\tilde{\mathbf{H }}}})=0.$ Combining (7.18) and (7.17), we immediately have

$$\begin{aligned} {\mathbf {W}}^\dagger {\mathbf {H}}^\dagger \mathbf {PHW}+\text {Diag}({\mathbf {y}}_{1:M}) ={\mathbf {W}}^\dagger \left[ {\mathbf {H}}^\dagger {{\mathbf {P}}}{{\mathbf {H}}}+\text {Diag}({\mathbf {y}}_{1:M})\right] {\mathbf {W}}\succ {\mathbf {0}}, \end{aligned}$$

(7.19)

which shows that ${\mathbf {W}}$ and ${\mathbf {y}}$ satisfy (7.7). The proof of Claim 2 is completed. $\square$