From 70c931b1fd7c2df916afc38640a08e1685175ca3 Mon Sep 17 00:00:00 2001
From: Josh Gao <josh_gao@foxmail.com>
Date: Wed, 1 Jul 2026 12:25:35 +0800
Subject: [PATCH] =?UTF-8?q?fix(symbol):=20F=20=E5=BA=9F=E5=BC=83=20a=5F{l,?=
 =?UTF-8?q?x}=20=E4=B8=8B=E6=A0=87=E5=AF=BC=E6=95=B0=E7=AE=80=E8=AE=B0?=
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a_{l,x} ≜ ∂a_l(ξ)/∂x 的下标逗号当求导，违背 A 规则14
(一阶导数用分子布局 Jacobian ∂f/∂x^T、点号只表时间速率)：
下标逗号既像"第 x 个分量"又像求导、且是逐标量而非 Jacobian，
正是规则14 要淘汰的写法。E/H 早已按规则14 重排，F 的 CP/IP
两处遗留此简记，现补齐一致。

删去 CP/IP 两处 a_{l,x} 定义式(无 label、下游无 eqref)，保留
"对角度参数 x∈{α,β}"引子(后文 ∂ν_l/∂x 仍需通用 x)；均值
Kronecker 导数 d_{l,p,α/β} 内的 a_{l,α}/a_{l,β} 直接写成显示式
∂a_l/∂α、∂a_l/∂β。
---
 parts/F-multiFrameCrb.tex | 28 ++++++----------------------
 1 file changed, 6 insertions(+), 22 deletions(-)

diff --git a/parts/F-multiFrameCrb.tex b/parts/F-multiFrameCrb.tex
index a9ae40b..040350b 100644
--- a/parts/F-multiFrameCrb.tex
+++ b/parts/F-multiFrameCrb.tex
@@ -78,15 +78,7 @@
   \in\mathbb C^{M_\ell T\times 1}.
 \end{equation}
 
-对角度参数 $x\in\{\alpha,\beta\}$，记导向矢量导数为
-%
-\begin{equation}
-  \mathbf a_{\ell,x}
-  \triangleq
-  \frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x}.
-\end{equation}
-%
-由链式法则可得时间模板导数
+对角度参数 $x\in\{\alpha,\beta\}$，由链式法则可得时间模板导数
 %
 \begin{equation}
   \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial x}
@@ -127,7 +119,7 @@
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \alpha}
   =
-  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \mathbf a_{\ell,\alpha}
+  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\alpha}
   +
   \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \alpha}\otimes \mathbf a_\ell,
 \end{equation}
@@ -136,7 +128,7 @@
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{cp}}_{\ell,p}}{\partial \beta}
   =
-  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \mathbf a_{\ell,\beta}
+  \mathbf q^{\mathrm{cp}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\beta}
   +
   \frac{\partial \mathbf q^{\mathrm{cp}}_{\ell,p}}{\partial \beta}\otimes \mathbf a_\ell,
 \end{equation}
@@ -1170,15 +1162,7 @@ $\mathcal T_{\mathrm{sub}}=\{\tau_{p_1},\ldots,\tau_{p_Q}\}$，则
 %
 这里的 $\vartheta_{\ell,p}$ 是第 $(\ell,p)$ 个观测块的独立未知初始相位；与\gls{cp} 模型相比，跨帧绝对相位连续性已经被这些块级相位参数吸收。
 
-对角度参数 $x\in\{\alpha,\beta\}$，记
-%
-\begin{equation}
-  \mathbf a_{\ell,x}
-  \triangleq
-  \frac{\partial \mathbf a_\ell(\boldsymbol\xi)}{\partial x}.
-\end{equation}
-%
-由链式法则可得
+对角度参数 $x\in\{\alpha,\beta\}$，由链式法则可得
 %
 \begin{equation}
   \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial x}
@@ -1235,7 +1219,7 @@ $\mathcal T_{\mathrm{sub}}=\{\tau_{p_1},\ldots,\tau_{p_Q}\}$，则
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \alpha}
   =
-  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \mathbf a_{\ell,\alpha}
+  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\alpha}
   +
   \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \alpha}\otimes \mathbf a_\ell,
 \end{equation}
@@ -1244,7 +1228,7 @@ $\mathcal T_{\mathrm{sub}}=\{\tau_{p_1},\ldots,\tau_{p_Q}\}$，则
   \triangleq
   \frac{\partial \mathbf g^{\mathrm{ip}}_{\ell,p}}{\partial \beta}
   =
-  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \mathbf a_{\ell,\beta}
+  \mathbf q^{\mathrm{ip}}_{\ell,p}\otimes \frac{\partial\mathbf a_\ell}{\partial\beta}
   +
   \frac{\partial \mathbf q^{\mathrm{ip}}_{\ell,p}}{\partial \beta}\otimes \mathbf a_\ell,
 \end{equation}