josh
f665c5e259
新增 .gitignore 项目规则、CLAUDE.md（含 git 规范）、 agent_instruction.md 重构指导。作为 agent_instruction.md 阶段 1 之前的回滚锚点。
544 lines
18 KiB
TeX
544 lines
18 KiB
TeX
\section{多阵列确定信号 DoA--Doppler 估计的 CRB 推导}
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\label{sec:A6_multiArrayStatDetDoaDopplerCrb}
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本附录沿用第~\ref{sec:multiArrayStatDetDoaDopplerEst} 节，尤其是第~\ref{subsec:staticBlockModel} 小节中的静态多阵列确定信号 DoA--Doppler 观测模型与参数定义，只补充 \gls{crb} 推导所需的 Jacobian、\gls{fim} 分块与 Schur 补步骤。主参数仍记为
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\begin{equation}
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\boldsymbol\eta_{\mathrm{stat}}
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=
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\begin{bmatrix}
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\alpha \\
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\beta \\
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\nu_{\mathrm{ref}}
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\end{bmatrix}
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\in \mathbb R^{3}.
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\end{equation}
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与\ref{sec:det_doa_doppler_crb} 的单阵列情形相比，这里需要先联合堆叠 $L$ 颗协作卫星的观测，再将各星复干扰系数视为干扰参数并通过 Schur 补得到 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 \gls{fim}。本文统一采用 $\mathcal I$ 表示 Fisher 信息矩阵，采用 $\mathbf I$ 表示单位阵，采用 $\mathcal P$ 表示投影矩阵。
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正文中的静态干扰参数 $\boldsymbol\psi_{\mathrm{stat}}$ 同时包含复增益和噪声方差。本附录在构造主参数 $\boldsymbol\eta_{\mathrm{stat}}$ 的 EFIM 时，只显式保留影响均值向量的复增益干扰参数；噪声方差只进入协方差，并与均值参数 Fisher 块解耦，因此不改变下面关于复增益的 Schur 补形式。换言之，下文得到的是在给定真实噪声方差或将其作为解耦协方差参数时，关于 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 Fisher 信息。
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\subsection{联合观测模型与 Fisher 信息矩阵通式}
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沿用第~\ref{subsec:staticBlockModel} 小节中的向量化观测模型，第 $\ell$ 颗卫星的均值向量可写为 $\zeta_\ell \mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}})$。为便于后续统一构造 \gls{fim}，本小节先将所有卫星观测按块联合堆叠。
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对于第 $\ell$ 颗卫星，向量化后的观测模型为
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\begin{equation}
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\tilde{\mathbf y}_\ell
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=
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\zeta_\ell \mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}})
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+
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\tilde{\mathbf n}_\ell,
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\qquad
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\tilde{\mathbf n}_\ell \sim \mathcal{CN}(\mathbf 0,\sigma_{n,\ell}^2 \mathbf I_{M_\ell T}),
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\quad \ell=0,1,\ldots,L-1,
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\end{equation}
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其中
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\begin{equation}
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\mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}})
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\triangleq
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\mathbf q_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}}) \otimes \mathbf a_\ell(\boldsymbol\xi)
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\in \mathbb C^{M_\ell T \times 1},
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\end{equation}
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$\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sigma_{n,\ell}^2$ 为第 $\ell$ 颗卫星对应的噪声方差。
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将所有卫星的观测联合堆叠，定义
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\begin{equation}
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\tilde{\mathbf y}
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\triangleq
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\begin{bmatrix}
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\tilde{\mathbf y}_0 \\
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\vdots \\
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\tilde{\mathbf y}_{L-1}
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\end{bmatrix},
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\qquad
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\boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}})
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\triangleq
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\begin{bmatrix}
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\zeta_0 \mathbf g_0(\boldsymbol\eta_{\mathrm{stat}}) \\
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\vdots \\
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\zeta_{L-1} \mathbf g_{L-1}(\boldsymbol\eta_{\mathrm{stat}})
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\end{bmatrix},
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\qquad
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\tilde{\mathbf n}
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\triangleq
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\begin{bmatrix}
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\tilde{\mathbf n}_0 \\
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\vdots \\
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\tilde{\mathbf n}_{L-1}
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\end{bmatrix},
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\end{equation}
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则有
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\begin{equation}
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\tilde{\mathbf y} = \boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}}) + \tilde{\mathbf n},
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\qquad
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\tilde{\mathbf n} \sim \mathcal{CN}(\mathbf 0,\mathbf R_n),
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\end{equation}
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其中
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\begin{equation}
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\mathbf R_n
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=
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\operatorname{blkdiag}
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\big(
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\sigma_{n,0}^2 \mathbf I_{M_0 T},
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\ldots,
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\sigma_{n,L-1}^2 \mathbf I_{M_{L-1} T}
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\big).
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\end{equation}
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将各星复干扰系数按实部和虚部分解为
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\begin{equation}
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\zeta_\ell = \zeta_\ell^{(r)} + j\zeta_\ell^{(i)},
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\end{equation}
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并定义实参数化后的均值干扰参数向量
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\begin{equation}
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\boldsymbol\psi_{\zeta,\mathrm{stat}}
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\triangleq
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\begin{bmatrix}
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\zeta_0^{(r)} & \zeta_0^{(i)} & \cdots & \zeta_{L-1}^{(r)} & \zeta_{L-1}^{(i)}
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\end{bmatrix}^{\Tran}
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\in \mathbb R^{2L}.
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\end{equation}
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于是总实参数向量可写为
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\begin{equation}
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\boldsymbol\kappa_{\mathrm{stat}}
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\triangleq
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\begin{bmatrix}
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\boldsymbol\eta_{\mathrm{stat}}^{\Tran} & \boldsymbol\psi_{\zeta,\mathrm{stat}}^{\Tran}
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\end{bmatrix}^{\Tran}
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\in \mathbb R^{2L+3}.
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\end{equation}
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由于当前均值参数仅通过均值项 $\boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}})$ 进入，而协方差矩阵 $\mathbf R_n$ 与 $\boldsymbol\kappa_{\mathrm{stat}}$ 无关，因此对任意两个实参数 $\kappa_i,\kappa_j$，Fisher 信息矩阵元素满足
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\begin{equation}
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[\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})]_{ij}
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=
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2\Re\!\left\{
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\frac{\partial \boldsymbol\mu^{\Herm}(\boldsymbol\kappa_{\mathrm{stat}})}{\partial \kappa_i}
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\mathbf R_n^{-1}
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\frac{\partial \boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}})}{\partial \kappa_j}
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\right\}.
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\label{eq:A6_fim_general}
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\end{equation}
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\subsection{\texorpdfstring{关于 $\boldsymbol\eta_{\mathrm{stat}}$ 的导数}{关于 eta_stat 的导数}}
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由第~\ref{sec:multiArrayStatDetDoaDopplerEst} 节，单位方向向量为
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\begin{equation}
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\mathbf u(\boldsymbol\xi)
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=
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\begin{bmatrix}
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\cos\beta\cos\alpha \\
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\cos\beta\sin\alpha \\
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\sin\beta
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\end{bmatrix},
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\end{equation}
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其关于全局角度参数的一阶导数为
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\begin{equation}
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\mathbf u_{\alpha}
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\triangleq
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\frac{\partial \mathbf u}{\partial \alpha}
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=
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\begin{bmatrix}
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-\cos\beta\sin\alpha \\
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\cos\beta\cos\alpha \\
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0
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\end{bmatrix},
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\qquad
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\mathbf u_{\beta}
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\triangleq
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\frac{\partial \mathbf u}{\partial \beta}
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=
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\begin{bmatrix}
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-\sin\beta\cos\alpha \\
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-\sin\beta\sin\alpha \\
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\cos\beta
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\end{bmatrix}.
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\end{equation}
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根据第~\ref{subsec:refSatParameterization} 小节的参考星参数化，静态模型中第 $\ell$ 颗卫星对应的归一化 Doppler 统一写成
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\begin{equation}
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\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})
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=
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f_{\nu,\ell}(\boldsymbol\xi,\nu_{\mathrm{ref}})
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=
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\nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi),
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\end{equation}
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其中 $\Delta\nu_\ell(\boldsymbol\xi)$ 由已知几何、速度与姿态关系确定，且对参考星 $\ell_{\mathrm{ref}}$ 有
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\begin{equation}
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f_{\nu,\ell_{\mathrm{ref}}}(\boldsymbol\xi,\nu_{\mathrm{ref}})
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=
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\nu_{\mathrm{ref}}.
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\end{equation}
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因此其一阶导数统一写为
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\begin{equation}
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\frac{\partial \nu_\ell}{\partial \alpha}
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=
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\frac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \alpha},
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\qquad
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\frac{\partial \nu_\ell}{\partial \beta}
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=
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\frac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \beta},
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\qquad
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\frac{\partial \nu_\ell}{\partial \nu_{\mathrm{ref}}}=1.
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\label{eq:A6_domega}
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\end{equation}
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上式利用了正文中采用的
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$\nu_\ell=\nu_{\mathrm{ref}}+\Delta\nu_\ell(\boldsymbol\xi)$
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这一参考星链路参数化形式。特别地，对参考星有
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$\Delta\nu_{\ell_{\mathrm{ref}}}(\boldsymbol\xi)=0$，因而
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$\partial\nu_{\ell_{\mathrm{ref}}}/\partial\alpha=0$、
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$\partial\nu_{\ell_{\mathrm{ref}}}/\partial\beta=0$。若改用别的等价参数化，只需将上述三个 Doppler 链式导数替换为对应结果即可。
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再定义第 $\ell$ 颗卫星本地坐标系下的来向向量
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\begin{equation}
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\mathbf u_\ell^{\mathrm{loc}}
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=
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\boldsymbol{\Omega}_\ell^{\Tran}\mathbf u
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=
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\begin{bmatrix}
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u_{\ell,1}^{\mathrm{loc}} & u_{\ell,2}^{\mathrm{loc}} & u_{\ell,3}^{\mathrm{loc}}
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\end{bmatrix}^{\Tran}.
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\end{equation}
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相应的局部角度为
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\begin{equation}
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\alpha_\ell^{\mathrm{loc}}
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=
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\operatorname{atan2}\!\big(u_{\ell,2}^{\mathrm{loc}},u_{\ell,1}^{\mathrm{loc}}\big),
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\qquad
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\beta_\ell^{\mathrm{loc}}
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=
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\arcsin\!\big(u_{\ell,3}^{\mathrm{loc}}\big).
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\end{equation}
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对任意 $x\in\{\alpha,\beta\}$，有
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\begin{equation}
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\frac{\partial \mathbf u_\ell^{\mathrm{loc}}}{\partial x}
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=
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\boldsymbol{\Omega}_\ell^{\Tran}\mathbf u_x.
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\end{equation}
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进一步可得
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\begin{equation}
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\frac{\partial \alpha_\ell^{\mathrm{loc}}}{\partial x}
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=
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\frac{
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u_{\ell,1}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,2}^{\mathrm{loc}}}{\partial x}
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-
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u_{\ell,2}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,1}^{\mathrm{loc}}}{\partial x}
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}{
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\big(u_{\ell,1}^{\mathrm{loc}}\big)^2 + \big(u_{\ell,2}^{\mathrm{loc}}\big)^2
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},
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\qquad
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\frac{\partial \beta_\ell^{\mathrm{loc}}}{\partial x}
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=
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\frac{\dfrac{\partial u_{\ell,3}^{\mathrm{loc}}}{\partial x}}{\sqrt{1-\big(u_{\ell,3}^{\mathrm{loc}}\big)^2}}.
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\label{eq:A6_local_angle_chain}
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\end{equation}
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记局部阵列导向矢量关于局部方位角和局部俯仰角的一阶导数分别为
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\begin{equation}
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\dot{\mathbf a}_{\ell,\alpha}
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\triangleq
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\frac{\partial \mathbf a_\ell}{\partial \alpha_\ell^{\mathrm{loc}}},
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\qquad
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\dot{\mathbf a}_{\ell,\beta}
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\triangleq
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\frac{\partial \mathbf a_\ell}{\partial \beta_\ell^{\mathrm{loc}}}.
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\end{equation}
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则对任意 $x\in\{\alpha,\beta\}$，根据链式法则有
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\begin{equation}
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\mathbf a_{\ell,x}
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\triangleq
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\frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x}
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=
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\dot{\mathbf a}_{\ell,\alpha}\frac{\partial \alpha_\ell^{\mathrm{loc}}}{\partial x}
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+
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\dot{\mathbf a}_{\ell,\beta}\frac{\partial \beta_\ell^{\mathrm{loc}}}{\partial x}.
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\label{eq:A6_da_chain}
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\end{equation}
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由第~\ref{sec:multiArrayStatDetDoaDopplerEst} 节的离散时间模板定义，
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\begin{equation}
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[\mathbf q_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})]_t
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=
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s(t)e^{j\nu_\ell(\boldsymbol\xi,\nu_{\mathrm{ref}})t},
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\qquad t=0,1,\ldots,T-1.
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\end{equation}
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因此其关于 $\nu_\ell$ 的导数为
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\begin{equation}
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\dot{\mathbf q}_\ell
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\triangleq
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\frac{\partial \mathbf q_\ell}{\partial \nu_\ell}
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=
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j\,\operatorname{diag}(0,1,\ldots,T-1)\,\mathbf q_\ell.
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\label{eq:A6_qdot}
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\end{equation}
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从而对任意 $x\in\{\alpha,\beta\}$，有
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\begin{equation}
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\mathbf q_{\ell,x}
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\triangleq
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\frac{\partial \mathbf q_\ell}{\partial x}
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=
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\dot{\mathbf q}_\ell\frac{\partial \nu_\ell}{\partial x},
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\qquad
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\mathbf q_{\ell,\nu}
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\triangleq
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\frac{\partial \mathbf q_\ell}{\partial \nu_{\mathrm{ref}}}
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=
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\dot{\mathbf q}_\ell\frac{\partial \nu_\ell}{\partial \nu_{\mathrm{ref}}}
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=
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\dot{\mathbf q}_\ell.
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\label{eq:A6_dq}
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\end{equation}
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其中最后一步同样利用了
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$\partial \nu_\ell/\partial \nu_{\mathrm{ref}}=1$。
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对空时导向向量
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$\mathbf g_\ell=\mathbf q_\ell\otimes \mathbf a_\ell$
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应用 Kronecker 积求导法则，可得
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\begin{equation}
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\mathbf d_{\ell,\alpha}
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\triangleq
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\frac{\partial \mathbf g_\ell}{\partial \alpha}
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=
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\mathbf q_\ell \otimes \mathbf a_{\ell,\alpha}
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+
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\mathbf q_{\ell,\alpha} \otimes \mathbf a_\ell,
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\end{equation}
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\begin{equation}
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\mathbf d_{\ell,\beta}
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\triangleq
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\frac{\partial \mathbf g_\ell}{\partial \beta}
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=
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\mathbf q_\ell \otimes \mathbf a_{\ell,\beta}
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+
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\mathbf q_{\ell,\beta} \otimes \mathbf a_\ell,
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\end{equation}
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\begin{equation}
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\mathbf d_{\ell,\nu}
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\triangleq
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\frac{\partial \mathbf g_\ell}{\partial \nu_{\mathrm{ref}}}
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=
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\mathbf q_{\ell,\nu} \otimes \mathbf a_\ell
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=
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\dot{\mathbf q}_\ell \otimes \mathbf a_\ell.
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\end{equation}
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进一步定义
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\begin{equation}
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\mathbf D_\ell
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\triangleq
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\begin{bmatrix}
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\mathbf d_{\ell,\alpha} & \mathbf d_{\ell,\beta} & \mathbf d_{\ell,\nu}
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\end{bmatrix}
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\in \mathbb C^{M_\ell T \times 3}.
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\label{eq:A6_Dl}
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\end{equation}
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\subsection{Fisher 信息矩阵的分块构造}
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对第 $\ell$ 颗卫星，其均值向量为
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\begin{equation}
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\boldsymbol\mu_\ell(\boldsymbol\kappa_{\mathrm{stat}})
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=
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\zeta_\ell\mathbf g_\ell(\boldsymbol\eta_{\mathrm{stat}}).
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\end{equation}
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因此，其关于主参数向量 $\boldsymbol\eta_{\mathrm{stat}}$ 的 Jacobian 为
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\begin{equation}
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\mathbf J_{\eta,\ell}
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\triangleq
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\frac{\partial \boldsymbol\mu_\ell}{\partial \boldsymbol\eta_{\mathrm{stat}}}
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=
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\zeta_\ell \mathbf D_\ell
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\in \mathbb C^{M_\ell T \times 3},
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\label{eq:A6_Jeta}
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\end{equation}
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而其关于本星复干扰系数实部与虚部的 Jacobian 为
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\begin{equation}
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\mathbf J_{\psi,\ell}
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\triangleq
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\frac{\partial \boldsymbol\mu_\ell}{\partial
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\begin{bmatrix}
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\zeta_\ell^{(r)} & \zeta_\ell^{(i)}
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\end{bmatrix}^{\Tran}}
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=
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\begin{bmatrix}
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\mathbf g_\ell & j\mathbf g_\ell
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\end{bmatrix}
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\in \mathbb C^{M_\ell T \times 2}.
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\label{eq:A6_Jpsi}
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\end{equation}
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由于不同卫星噪声相互独立，总 Fisher 信息矩阵可按卫星求和。记总 Fisher 信息矩阵关于 $(\boldsymbol\eta_{\mathrm{stat}},\boldsymbol\psi_{\zeta,\mathrm{stat}})$ 的分块形式为
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\begin{equation}
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\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})
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=
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\begin{bmatrix}
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\mathcal I_{\eta\eta} & \mathcal I_{\eta\psi} \\
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\mathcal I_{\psi\eta} & \mathcal I_{\psi\psi}
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\end{bmatrix},
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\end{equation}
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则根据 \eqref{eq:A6_fim_general} 有
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\begin{equation}
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\mathcal I_{\eta\eta}
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=
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\sum_{\ell=0}^{L-1}
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\frac{2}{\sigma_{n,\ell}^2}
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\Re\!\left\{
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\mathbf J_{\eta,\ell}^{\Herm}\mathbf J_{\eta,\ell}
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\right\},
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\label{eq:A6_Itt}
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\end{equation}
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\begin{equation}
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\mathcal I_{\eta\psi}
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=
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\begin{bmatrix}
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\dfrac{2}{\sigma_{n,0}^2}\Re\!\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\}
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& \cdots &
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\dfrac{2}{\sigma_{n,L-1}^2}\Re\!\left\{\mathbf J_{\eta,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\}
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\end{bmatrix},
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\label{eq:A6_Itb}
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\end{equation}
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\begin{equation}
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\mathcal I_{\psi\psi}
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=
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\operatorname{blkdiag}
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\left(
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\frac{2}{\sigma_{n,0}^2}\Re\!\left\{\mathbf J_{\psi,0}^{\Herm}\mathbf J_{\psi,0}\right\},
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\ldots,
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\frac{2}{\sigma_{n,L-1}^2}\Re\!\left\{\mathbf J_{\psi,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\}
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\right).
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\label{eq:A6_Ibb}
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\end{equation}
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\subsection{消去复干扰系数后的等效 Fisher 信息矩阵}
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由于 $\boldsymbol\psi_{\zeta,\mathrm{stat}}$ 是均值干扰参数，主参数向量 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 Fisher 信息矩阵为
|
||
\begin{equation}
|
||
\mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}})
|
||
=
|
||
\mathcal I_{\eta\eta}
|
||
-
|
||
\mathcal I_{\eta\psi}
|
||
\mathcal I_{\psi\psi}^{-1}
|
||
\mathcal I_{\psi\eta}.
|
||
\label{eq:A6_schur}
|
||
\end{equation}
|
||
|
||
对每颗卫星，复增益实部和虚部对应的 Jacobian 为
|
||
$\begin{bmatrix}\mathbf g_\ell & j\mathbf g_\ell\end{bmatrix}$。虽然这两个列向量对应的是实参数化下的二维实线性干扰子空间，但其等价于由 $\mathbf g_\ell$ 张成的复一维子空间；相应的 Schur 补可以写成对该复一维子空间的正交投影形式。定义投影矩阵及其正交补为
|
||
\begin{equation}
|
||
\mathcal P_{\mathbf g_\ell}
|
||
\triangleq
|
||
\mathbf g_\ell(\mathbf g_\ell^{\Herm}\mathbf g_\ell)^{-1}\mathbf g_\ell^{\Herm},
|
||
\qquad
|
||
\mathcal P_{\mathbf g_\ell}^{\perp}
|
||
\triangleq
|
||
\mathbf I_{M_\ell T} - \mathcal P_{\mathbf g_\ell}.
|
||
\label{eq:A6_proj}
|
||
\end{equation}
|
||
将 \eqref{eq:A6_Jeta}--\eqref{eq:A6_proj} 代入 \eqref{eq:A6_schur} 并整理，可得
|
||
\begin{equation}
|
||
\mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}})
|
||
=
|
||
2\sum_{\ell=0}^{L-1}
|
||
\frac{1}{\sigma_{n,\ell}^2}
|
||
\Re\!\left\{
|
||
\mathbf J_{\eta,\ell}^{\Herm}
|
||
\mathcal P_{\mathbf g_\ell}^{\perp}
|
||
\mathbf J_{\eta,\ell}
|
||
\right\}.
|
||
\label{eq:A6_Ieff_proj}
|
||
\end{equation}
|
||
再利用 $\mathbf J_{\eta,\ell}=\zeta_\ell\mathbf D_\ell$，得到紧凑形式
|
||
\begin{equation}
|
||
\mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}})
|
||
=
|
||
2\sum_{\ell=0}^{L-1}
|
||
\frac{|\zeta_\ell|^2}{\sigma_{n,\ell}^2}
|
||
\Re\!\left\{
|
||
\mathbf D_\ell^{\Herm}
|
||
\mathcal P_{\mathbf g_\ell}^{\perp}
|
||
\mathbf D_\ell
|
||
\right\}.
|
||
\label{eq:A6_Ieff_final}
|
||
\end{equation}
|
||
式~\eqref{eq:A6_Ieff_final} 即为静态多阵列确定信号 DoA--Doppler 联合估计的等效 \gls{fim}。
|
||
|
||
\subsection{CRB 表达式}
|
||
|
||
因此，联合参数向量 $\boldsymbol\eta_{\mathrm{stat}}=[\alpha,\beta,\nu_{\mathrm{ref}}]^{\Tran}$ 的 \gls{crb} 为
|
||
\begin{equation}
|
||
\operatorname{CRB}(\boldsymbol\eta_{\mathrm{stat}})
|
||
=
|
||
\mathcal I_{\mathrm e}^{-1}(\boldsymbol\eta_{\mathrm{stat}}).
|
||
\label{eq:A6_crb_joint}
|
||
\end{equation}
|
||
|
||
若仅关心角度参数
|
||
\begin{equation}
|
||
\boldsymbol\xi
|
||
\triangleq
|
||
\begin{bmatrix}
|
||
\alpha \\
|
||
\beta
|
||
\end{bmatrix},
|
||
\end{equation}
|
||
则将 \eqref{eq:A6_Ieff_final} 按 $(\boldsymbol\xi,\nu_{\mathrm{ref}})$ 分块为
|
||
\begin{equation}
|
||
\mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}})
|
||
=
|
||
\begin{bmatrix}
|
||
\mathcal I_{\mathrm{ang},\mathrm{ang}} & \mathcal I_{\mathrm{ang},\nu} \\
|
||
\mathcal I_{\nu,\mathrm{ang}} & \mathcal I_{\nu\nu}
|
||
\end{bmatrix},
|
||
\end{equation}
|
||
其中 $\mathcal I_{\mathrm{ang},\mathrm{ang}}\in\mathbb R^{2\times 2}$，$\mathcal I_{\mathrm{ang},\nu}\in\mathbb R^{2\times 1}$，则 DoA 参数的等效 \gls{crb} 为
|
||
\begin{equation}
|
||
\operatorname{CRB}(\boldsymbol\xi)
|
||
=
|
||
\left(
|
||
\mathcal I_{\mathrm{ang},\mathrm{ang}}
|
||
-
|
||
\mathcal I_{\mathrm{ang},\nu}\,\mathcal I_{\nu\nu}^{-1}\,\mathcal I_{\nu,\mathrm{ang}}
|
||
\right)^{-1}.
|
||
\label{eq:A6_crb_ang}
|
||
\end{equation}
|
||
类似地，仅参考归一化 Doppler 参数 $\nu_{\mathrm{ref}}$ 的 \gls{crb} 为
|
||
\begin{equation}
|
||
\operatorname{CRB}(\nu_{\mathrm{ref}})
|
||
=
|
||
\left(
|
||
\mathcal I_{\nu\nu}
|
||
-
|
||
\mathcal I_{\nu,\mathrm{ang}}\,\mathcal I_{\mathrm{ang},\mathrm{ang}}^{-1}\,\mathcal I_{\mathrm{ang},\nu}
|
||
\right)^{-1}.
|
||
\label{eq:A6_crb_omega}
|
||
\end{equation}
|
||
|
||
\subsection{备注}
|
||
|
||
\begin{enumerate}
|
||
\item 若所有卫星噪声方差相同，即 $\sigma_{n,\ell}^2=\sigma^2$，则 \eqref{eq:A6_Ieff_final} 退化为
|
||
\begin{equation}
|
||
\mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{stat}})
|
||
=
|
||
\frac{2}{\sigma^2}
|
||
\sum_{\ell=0}^{L-1}
|
||
|\zeta_\ell|^2
|
||
\Re\!\left\{
|
||
\mathbf D_\ell^{\Herm}\mathcal P_{\mathbf g_\ell}^{\perp}\mathbf D_\ell
|
||
\right\}.
|
||
\end{equation}
|
||
|
||
\item 若进一步将噪声方差 $\{\sigma_{n,\ell}^2\}_{\ell=0}^{L-1}$ 也并入待估参数，则协方差项仅对这些方差参数自身有贡献，且与均值参数块解耦。具体而言，若把第 $\ell$ 条链路的方差参数取为 $\sigma_{n,\ell}^2$，则有
|
||
\begin{equation}
|
||
[\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})]_{\sigma_{n,\ell}^2,\sigma_{n,\ell}^2}
|
||
=
|
||
\frac{M_\ell T}{\sigma_{n,\ell}^4},
|
||
\qquad
|
||
[\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})]_{\sigma_{n,\ell}^2,i}=0,
|
||
\quad i\neq \sigma_{n,\ell}^2,
|
||
\end{equation}
|
||
因此不会改变 \eqref{eq:A6_Ieff_final} 给出的 $\boldsymbol\eta_{\mathrm{stat}}$ 的等效 \gls{fim}。
|
||
\end{enumerate}
|
||
|
||
因此，附录 A6 与正文静态多星模型一一对应：正文负责给出观测模型、参数分组和集中似然，本文附录负责给出 Jacobian、Fisher 信息分块与 Schur 补形式。动态模型的连续相位和独立相位 CRB 分别在 \ref{sec:A7_dynDoaDopplerCrb} 与 \ref{sec:A8_dynIndependentPhaseCrb} 中继续展开。
|