fix: 行内分式用 \dfrac、行间用 \frac
B/E/D 行间 equation/align 内 37 处 \dfrac 改 \frac；sec5 行内 1 处 \frac 改 \dfrac。行内已是 \dfrac 的保留；导数斜杠记号 ∂f/∂x 非分式、未动。
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@@ -117,7 +117,7 @@ $\mathbf{Z} \sim \mathcal{CN} \left( \bm{\mu}, \mathbf{R} \right)$ 的 \gls{pdf}
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\mathrm{var} ( \hat\theta )
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\geqslant
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\frac{1}{-\mathbb{E}_X \left[
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\dfrac{
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\frac{
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\partial^2 \ln f_X \left( x; \theta \right)
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}{
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\partial \theta^2
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+10
-10
@@ -25,15 +25,15 @@
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\begin{equation}
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\label{eq:wirtinger_derivative}
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\begin{cases}
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\dfrac{\partial}{\partial z} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right)
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\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right)
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\\
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\dfrac{\partial}{\partial z^*} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right)
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\frac{\partial}{\partial z^*} = \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right)
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\end{cases}
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\quad\Rightarrow\quad
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\begin{cases}
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\dfrac{\partial}{\partial x} = \dfrac{\partial}{\partial z} + \dfrac{\partial}{\partial z^*}
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\frac{\partial}{\partial x} = \frac{\partial}{\partial z} + \frac{\partial}{\partial z^*}
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\\
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\dfrac{\partial}{\partial y} = j\left(\dfrac{\partial}{\partial z} - \dfrac{\partial}{\partial z^*} \right)
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\frac{\partial}{\partial y} = j\left(\frac{\partial}{\partial z} - \frac{\partial}{\partial z^*} \right)
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\end{cases},
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\end{equation}
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%
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@@ -41,13 +41,13 @@
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\begin{subequations}
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\begin{equation}
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\frac{\partial z^*}{\partial z}
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= \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right) (x - jy)
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= \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right) (x - jy)
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= \frac{1}{2}\left(\frac{\partial x}{\partial x} - j\frac{\partial (-jy)}{\partial y} \right) = 0,
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\end{equation}
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%
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\begin{equation}
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\frac{\partial z}{\partial z^*}
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= \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right) (x + jy)
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= \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right) (x + jy)
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= \frac{1}{2}\left(\frac{\partial x}{\partial x} + j\frac{\partial (jy)}{\partial y} \right) = 0,
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\end{equation}
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\end{subequations}
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@@ -110,11 +110,11 @@
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\begin{equation}
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\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}}
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= \begin{bmatrix}
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\dfrac{\partial}{\partial \theta_1}
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\frac{\partial}{\partial \theta_1}
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\\
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\vdots
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\\
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\dfrac{\partial}{\partial \theta_K}
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\frac{\partial}{\partial \theta_K}
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\end{bmatrix} \ln \mathcal{L}
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= \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{X}^\Herm _t \mathbf{D}^\Herm \mathbf{n}_t \right].
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\end{equation}
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@@ -394,9 +394,9 @@
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%
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\begin{align}
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\mathrm{CRB}^{-1}(\bm{\theta}) &= \mathcal{I}_{\theta\theta} - \sum_{t=1}^T \Re\{\bm{\Delta}_t^\Herm \mathbf{G}\bm{\Delta}_t\} \nonumber
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= \dfrac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t - \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right\} \nonumber
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= \frac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{D} \mathbf{X}_t - \mathbf{X}_t^\Herm \mathbf{D}^\Herm \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \mathbf{D} \mathbf{X}_t \right\} \nonumber
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\\
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&= \dfrac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \left[ \mathbf{I} - \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \right] \mathbf{D} \mathbf{X}_t \right\}.
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&= \frac{2}{\sigma} \sum_{t=1}^T \Re\left\{ \mathbf{X}_t^\Herm \mathbf{D}^\Herm \left[ \mathbf{I} - \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm \right] \mathbf{D} \mathbf{X}_t \right\}.
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\end{align}
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参考 \cite{stoicaMusicMaximumLikelihood1989} 进一步定义阵列导向矩阵的列空间的投影矩阵 $\mathcal{P}_{\mathbf{A}} = \mathbf{A}(\mathbf{A}^\Herm \mathbf{A})^{-1}\mathbf{A}^\Herm$，及其正交补空间的投影矩阵 $\mathcal{P}_{\mathbf{A}}^\perp = \mathbf{I} -\mathcal{P}_{\mathbf{A}}$，投影矩阵下的等效信息矩阵 $\mathcal{H} = \mathbf{D}^\Herm \mathcal{P}_{\mathbf{A}}^\perp \mathbf{D}$，确定信号的经验协方差 $\mathbf R_{x,\mathrm{det}} = \dfrac{1}{T} \displaystyle\sum_{t=1}^T \mathbf{x}_t \mathbf{x}_t^\Herm$，则关于角度的 \gls{efim} 可以表示为
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@@ -193,7 +193,7 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig
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\frac{\partial \nu_\ell}{\partial \boldsymbol\eta^\Tran}
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=
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\begin{bmatrix}
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\dfrac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \boldsymbol\xi^\Tran}
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\frac{\partial \Delta\nu_\ell(\boldsymbol\xi)}{\partial \boldsymbol\xi^\Tran}
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& 1
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\end{bmatrix}
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\in\mathbb R^{1\times3},
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@@ -242,15 +242,15 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig
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\frac{\partial \boldsymbol\xi_\ell^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran}
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=
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\begin{bmatrix}
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\dfrac{
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u_{\ell,1}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,2}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran}
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\frac{
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u_{\ell,1}^{\mathrm{loc}}\,\frac{\partial u_{\ell,2}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran}
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-
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u_{\ell,2}^{\mathrm{loc}}\,\dfrac{\partial u_{\ell,1}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran}
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u_{\ell,2}^{\mathrm{loc}}\,\frac{\partial u_{\ell,1}^{\mathrm{loc}}}{\partial \boldsymbol\xi^\Tran}
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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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\\[1.2em]
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\dfrac{\partial u_{\ell,3}^{\mathrm{loc}}/\partial \boldsymbol\xi^\Tran}{\sqrt{1-\big(u_{\ell,3}^{\mathrm{loc}}\big)^2}}
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\frac{\partial u_{\ell,3}^{\mathrm{loc}}/\partial \boldsymbol\xi^\Tran}{\sqrt{1-\big(u_{\ell,3}^{\mathrm{loc}}\big)^2}}
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\end{bmatrix}
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\in\mathbb R^{2\times2},
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\label{eq:A6_local_angle_chain}
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@@ -264,9 +264,9 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig
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\frac{\partial \mathbf a_\ell}{\partial (\boldsymbol\xi_\ell^{\mathrm{loc}})^\Tran}
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=
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\begin{bmatrix}
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\dfrac{\partial \mathbf a_\ell}{\partial \alpha_\ell^{\mathrm{loc}}}
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\frac{\partial \mathbf a_\ell}{\partial \alpha_\ell^{\mathrm{loc}}}
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&
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\dfrac{\partial \mathbf a_\ell}{\partial \beta_\ell^{\mathrm{loc}}}
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\frac{\partial \mathbf a_\ell}{\partial \beta_\ell^{\mathrm{loc}}}
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\end{bmatrix}
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\in\mathbb C^{M_\ell\times2}.
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\label{eq:A6_a_local_jac}
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@@ -325,11 +325,11 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig
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\frac{\partial \mathbf g_\ell}{\partial \boldsymbol\eta^\Tran}
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=
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\begin{bmatrix}
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\mathbf q_\ell\otimes\dfrac{\partial \mathbf a_\ell}{\partial \boldsymbol\xi^\Tran}
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\mathbf q_\ell\otimes\frac{\partial \mathbf a_\ell}{\partial \boldsymbol\xi^\Tran}
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+
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\dfrac{\partial \mathbf q_\ell}{\partial \boldsymbol\xi^\Tran}\otimes\mathbf a_\ell
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\frac{\partial \mathbf q_\ell}{\partial \boldsymbol\xi^\Tran}\otimes\mathbf a_\ell
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&
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\dfrac{\partial \mathbf q_\ell}{\partial \nu_\ell}\otimes\mathbf a_\ell
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\frac{\partial \mathbf q_\ell}{\partial \nu_\ell}\otimes\mathbf a_\ell
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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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@@ -403,9 +403,9 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig
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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_0}\Re\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\}
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\frac{2}{\sigma_0}\Re\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\}
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& \cdots &
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\dfrac{2}{\sigma_{L-1}}\Re\left\{\mathbf J_{\eta,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\}
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\frac{2}{\sigma_{L-1}}\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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@@ -282,7 +282,7 @@ $\boldsymbol\xi$ 的增益主要来自空间视线与阵列流形几何多样性
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该 comb 是全局 profile likelihood 的周期窗口效应，与 \gls{efim} 局部曲率并不矛盾——\gls{efim} 只刻画某一个 tooth 内的二阶曲率，并不保证全局唯一 tooth。
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相邻 tooth 是否容易区分由两层因素共同决定：单帧 pilot block 的长度 $T$（控制块内 Doppler 相干带宽）与非周期 subset 的窗口形状（控制 comb 旁瓣压制）。
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当 $T\ll\Delta\tau_{\mathrm{f}}$ 时，相邻 tooth 的块内相干损失约为 $\frac{\pi^2}{3}(T/\Delta\tau_{\mathrm{f}})^2$，因此增大 $T$ 可增强相邻 tooth 的分离度。
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当 $T\ll\Delta\tau_{\mathrm{f}}$ 时，相邻 tooth 的块内相干损失约为 $\dfrac{\pi^2}{3}(T/\Delta\tau_{\mathrm{f}})^2$，因此增大 $T$ 可增强相邻 tooth 的分离度。
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非周期 subset 打破 comb 的严格周期性，使本来等高的假 tooth 退化为受压制的旁瓣、真峰凸显，因而更适合 tooth acquisition；完整周期数据在 tooth 选对之后由于保留更多相干能量，更适合同 tooth 内 refine。
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若定义 tooth 可分离性为不同 $k$ 下 profile optimum 之间的目标差
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%
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