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fix: 行内分式用 \dfrac、行间用 \frac

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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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josh
2026-06-22 12:42:41 +08:00
parent 722a97f629
commit ae79e1d60b
4 changed files with 24 additions and 24 deletions
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parts/D-singleArrayCrb.tex
+10 -10
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@@ -25,15 +25,15 @@
\begin{equation}
\label{eq:wirtinger_derivative}
\begin{cases}
\dfrac{\partial}{\partial z} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right)
\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right)
\\
\dfrac{\partial}{\partial z^*} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right)
\frac{\partial}{\partial z^*} = \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right)
\end{cases}
\quad\Rightarrow\quad
\begin{cases}
\dfrac{\partial}{\partial x} = \dfrac{\partial}{\partial z} + \dfrac{\partial}{\partial z^*}
\frac{\partial}{\partial x} = \frac{\partial}{\partial z} + \frac{\partial}{\partial z^*}
\\
\dfrac{\partial}{\partial y} = j\left(\dfrac{\partial}{\partial z} - \dfrac{\partial}{\partial z^*} \right)
\frac{\partial}{\partial y} = j\left(\frac{\partial}{\partial z} - \frac{\partial}{\partial z^*} \right)
\end{cases},
\end{equation}
%
@@ -41,13 +41,13 @@
\begin{subequations}
\begin{equation}
\frac{\partial z^*}{\partial z}
= \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} - j\dfrac{\partial}{\partial y} \right) (x - jy)
= \frac{1}{2}\left(\frac{\partial}{\partial x} - j\frac{\partial}{\partial y} \right) (x - jy)
= \frac{1}{2}\left(\frac{\partial x}{\partial x} - j\frac{\partial (-jy)}{\partial y} \right) = 0,
\end{equation}
%
\begin{equation}
\frac{\partial z}{\partial z^*}
= \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + j\dfrac{\partial}{\partial y} \right) (x + jy)
= \frac{1}{2}\left(\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} \right) (x + jy)
= \frac{1}{2}\left(\frac{\partial x}{\partial x} + j\frac{\partial (jy)}{\partial y} \right) = 0,
\end{equation}
\end{subequations}
@@ -110,11 +110,11 @@
\begin{equation}
\frac{\partial\ln \mathcal{L}}{\partial \bm{\theta}}
= \begin{bmatrix}
\dfrac{\partial}{\partial \theta_1}
\frac{\partial}{\partial \theta_1}
\\
\vdots
\\
\dfrac{\partial}{\partial \theta_K}
\frac{\partial}{\partial \theta_K}
\end{bmatrix} \ln \mathcal{L}
= \frac{2}{\sigma} \sum_{t=1}^T \Re\left[ \mathbf{X}^\Herm _t \mathbf{D}^\Herm \mathbf{n}_t \right].
\end{equation}
@@ -394,9 +394,9 @@
%
\begin{align}
\mathrm{CRB}^{-1}(\bm{\theta}) &= \mathcal{I}_{\theta\theta} - \sum_{t=1}^T \Re\{\bm{\Delta}_t^\Herm \mathbf{G}\bm{\Delta}_t\} \nonumber
= \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
= \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
\\
&= \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\}.
&= \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\}.
\end{align}
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