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fix(equation): 删去公式中手动 \! 负间距

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\arcsin\!\big(、\Re\!\left\{、\mathbf a_\ell\!\big( 等"函数/算子+定界符"
处的手动负间距由 LaTeX 自动排版处理，无需手动加 \!。全文删 39 处。
主文件 \newcommand{\diff}{\mathop{}\!\mathrm{d}} 中的 \! 是微分号标准间距，保留。
This commit is contained in:
josh
2026-06-16 15:28:12 +08:00
parent 8885e3e7f6
commit 0932cc9f41
9 changed files with 39 additions and 39 deletions
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parts/C-classicalDoa.tex
+2 -2
View File
@@ -458,7 +458,7 @@ where $\angle(\cdot)$ denotes the phase angle of a complex number. Finally, the
\begin{equation}
\hat{\theta}_k
=
\arcsin\!\left(
\arcsin\left(
\frac{\lambda_{\mathrm{c}}}{2\pi d}\hat{\mu}_k
\right),
\qquad k = 1,\dots,K.
@@ -955,7 +955,7 @@ where $\angle(\cdot)$ denotes the phase angle of a complex number. Finally, the
\begin{equation}
\hat{\theta}_k
=
\arcsin\!\left(
\arcsin\left(
\frac{\lambda_{\mathrm{c}}}{2\pi d}\hat{\mu}_k
\right),
\qquad
parts/E-multiSatStaticCrb.tex
+14 -14
View File
@@ -134,7 +134,7 @@ $\zeta_\ell\in\mathbb C$ 为第 $\ell$ 条链路的未知复干扰系数，$\sig
\begin{equation}
[\mathcal I(\boldsymbol\kappa_{\mathrm{stat}})]_{ij}
=
2\Re\!\left\{
2\Re\left\{
\frac{\partial \boldsymbol\mu^{\Herm}(\boldsymbol\kappa_{\mathrm{stat}})}{\partial \kappa_i}
\mathbf R_n^{-1}
\frac{\partial \boldsymbol\mu(\boldsymbol\kappa_{\mathrm{stat}})}{\partial \kappa_j}
@@ -237,11 +237,11 @@ $\partial\nu_{\ell_{\mathrm{ref}}}/\partial\beta=0$。若改用别的等价参
\begin{equation}
\alpha_\ell^{\mathrm{loc}}
=
\operatorname{atan2}\!\big(u_{\ell,2}^{\mathrm{loc}},u_{\ell,1}^{\mathrm{loc}}\big),
\operatorname{atan2}\big(u_{\ell,2}^{\mathrm{loc}},u_{\ell,1}^{\mathrm{loc}}\big),
\qquad
\beta_\ell^{\mathrm{loc}}
=
\arcsin\!\big(u_{\ell,3}^{\mathrm{loc}}\big).
\arcsin\big(u_{\ell,3}^{\mathrm{loc}}\big).
\end{equation}
对任意 $x\in\{\alpha,\beta\}$，有
@@ -439,7 +439,7 @@ $\mathbf g_\ell=\mathbf q_\ell\otimes \mathbf a_\ell$
=
\sum_{\ell=0}^{L-1}
\frac{2}{\sigma_{n,\ell}^2}
\Re\!\left\{
\Re\left\{
\mathbf J_{\eta,\ell}^{\Herm}\mathbf J_{\eta,\ell}
\right\},
\label{eq:A6_Itt}
@@ -448,9 +448,9 @@ $\mathbf g_\ell=\mathbf q_\ell\otimes \mathbf a_\ell$
\mathcal I_{\eta\psi}
=
\begin{bmatrix}
\dfrac{2}{\sigma_{n,0}^2}\Re\!\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\}
\dfrac{2}{\sigma_{n,0}^2}\Re\left\{\mathbf J_{\eta,0}^{\Herm}\mathbf J_{\psi,0}\right\}
& \cdots &
\dfrac{2}{\sigma_{n,L-1}^2}\Re\!\left\{\mathbf J_{\eta,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\}
\dfrac{2}{\sigma_{n,L-1}^2}\Re\left\{\mathbf J_{\eta,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\}
\end{bmatrix},
\label{eq:A6_Itb}
\end{equation}
@@ -459,9 +459,9 @@ $\mathbf g_\ell=\mathbf q_\ell\otimes \mathbf a_\ell$
=
\operatorname{blkdiag}
\left(
\frac{2}{\sigma_{n,0}^2}\Re\!\left\{\mathbf J_{\psi,0}^{\Herm}\mathbf J_{\psi,0}\right\},
\frac{2}{\sigma_{n,0}^2}\Re\left\{\mathbf J_{\psi,0}^{\Herm}\mathbf J_{\psi,0}\right\},
\ldots,
\frac{2}{\sigma_{n,L-1}^2}\Re\!\left\{\mathbf J_{\psi,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\}
\frac{2}{\sigma_{n,L-1}^2}\Re\left\{\mathbf J_{\psi,L-1}^{\Herm}\mathbf J_{\psi,L-1}\right\}
\right).
\label{eq:A6_Ibb}
\end{equation}
@@ -502,7 +502,7 @@ $\begin{bmatrix}\mathbf g_\ell & j\mathbf g_\ell\end{bmatrix}$。虽然这两个
=
2\sum_{\ell=0}^{L-1}
\frac{1}{\sigma_{n,\ell}^2}
\Re\!\left\{
\Re\left\{
\mathbf J_{\eta,\ell}^{\Herm}
\mathcal P_{\mathbf g_\ell}^{\perp}
\mathbf J_{\eta,\ell}
@@ -517,7 +517,7 @@ $\begin{bmatrix}\mathbf g_\ell & j\mathbf g_\ell\end{bmatrix}$。虽然这两个
=
2\sum_{\ell=0}^{L-1}
\frac{|\zeta_\ell|^2}{\sigma_{n,\ell}^2}
\Re\!\left\{
\Re\left\{
\mathbf D_\ell^{\Herm}
\mathcal P_{\mathbf g_\ell}^{\perp}
\mathbf D_\ell
@@ -596,7 +596,7 @@ $\begin{bmatrix}\mathbf g_\ell & j\mathbf g_\ell\end{bmatrix}$。虽然这两个
\frac{2}{\sigma^2}
\sum_{\ell=0}^{L-1}
|\zeta_\ell|^2
\Re\!\left\{
\Re\left\{
\mathbf D_\ell^{\Herm}\mathcal P_{\mathbf g_\ell}^{\perp}\mathbf D_\ell
\right\}.
\end{equation}
@@ -707,7 +707,7 @@ $\mathbf J_{\psi,\ell}$ 的实参数化二维实子空间与复矩阵 $\mathbf G
=
\sum_{\ell=0}^{L-1}
\frac{2}{\sigma_{n,\ell}^2}
\Re\!\big\{\mathbf J_{\eta,\ell}^\Herm\mathbf J_{\eta,\ell}\big\}.
\Re\big\{\mathbf J_{\eta,\ell}^\Herm\mathbf J_{\eta,\ell}\big\}.
\end{equation}
%
由于 $\mathbf J_{\psi,\ell}$ 的实子空间等价于 $\mathbf G_\ell$ 的复列空间，关于复增益块的 Schur 补化为对 $\mathbf G_\ell$ 列空间正交补的投影。定义
@@ -730,7 +730,7 @@ $\mathbf J_{\psi,\ell}$ 的实参数化二维实子空间与复矩阵 $\mathbf G
=
2\sum_{\ell=0}^{L-1}
\frac{1}{\sigma_{n,\ell}^2}
\Re\!\big\{
\Re\big\{
\mathbf J_{\eta,\ell}^\Herm
\mathcal P_{\mathbf G_\ell}^\perp
\mathbf J_{\eta,\ell}
@@ -745,7 +745,7 @@ $\mathbf J_{\psi,\ell}$ 的实参数化二维实子空间与复矩阵 $\mathbf G
=
2\sum_{\ell=0}^{L-1}
\frac{1}{\sigma_{n,\ell}^2}
\Re\!\Big\{
\Re\Big\{
\zeta_{\ell,k}^*\zeta_{\ell,k'}\,
\mathbf D_{\ell,k}^\Herm
\mathcal P_{\mathbf G_\ell}^\perp
parts/F-timestampErrCrb.tex
+10 -10
View File
@@ -41,7 +41,7 @@
=
\sum_{\ell=0}^{L-1}
\frac{2}{\sigma_{n,\ell}^2}
\Re\!\left\{
\Re\left\{
\frac{\partial\boldsymbol\mu_\ell^\Herm}{\partial\kappa_i}
\frac{\partial\boldsymbol\mu_\ell}{\partial\kappa_j}
\right\}.
@@ -66,7 +66,7 @@
\big[\mathcal I_{\delta\delta}\big]_{\ell\ell'}
=
\frac{2}{\sigma_{n,\ell}^2}
\Re\!\left\{
\Re\left\{
\frac{\partial\boldsymbol\mu_\ell^\Herm}{\partial\delta_\ell}
\frac{\partial\boldsymbol\mu_\ell}{\partial\delta_\ell}
\right\}
@@ -108,16 +108,16 @@ $\mathcal I_{\delta\psi}$ 同样具有按 $\ell$ 的块对角结构：第 $\ell$
\mathcal I_{\eta\eta}'
&=
2\sum_{\ell}\frac{1}{\sigma_{n,\ell}^2}
\Re\!\left\{
\Re\left\{
\mathbf J_{\eta,\ell}^\Herm
\big(\mathbf I-\mathcal P_{\mathbf G_\ell}\big)
\mathbf J_{\eta,\ell}
\right\}\big|_{\delta_\ell=0},\\
\mathcal I_{\delta\delta}'
&=
\operatorname{diag}\!\Big(
\operatorname{diag}\Big(
\tfrac{2}{\sigma_{n,\ell}^2}
\Re\!\big\{\mathbf j_{\delta,\ell}^\Herm
\Re\big\{\mathbf j_{\delta,\ell}^\Herm
(\mathbf I-\mathcal P_{\mathbf G_\ell})
\mathbf j_{\delta,\ell}\big\}\big|_{\delta_\ell=0}
\Big)_{\ell\neq\ell_{\mathrm{ref}}},\\
@@ -126,7 +126,7 @@ $\mathcal I_{\delta\psi}$ 同样具有按 $\ell$ 的块对角结构：第 $\ell$
\begin{bmatrix}
\cdots\quad
\tfrac{2}{\sigma_{n,\ell}^2}
\Re\!\big\{\mathbf J_{\eta,\ell}^\Herm
\Re\big\{\mathbf J_{\eta,\ell}^\Herm
(\mathbf I-\mathcal P_{\mathbf G_\ell})\mathbf j_{\delta,\ell}\big\}\big|_{\delta_\ell=0}
\quad\cdots
\end{bmatrix},
@@ -196,10 +196,10 @@ $\boldsymbol\delta$ 在导频自相关半宽 $T_{\mathrm{ac}}$ 处的非线性
\mathcal I_{\mathrm e}^{(\mathrm{stat})}(\boldsymbol\eta_{\mathrm{stat}})
=
\frac{2|\zeta_0|^2}{\sigma_{n,0}^2}
\Re\!\big\{\mathbf D_0^\Herm\mathcal P_{\mathbf g_0}^\perp\mathbf D_0\big\}
\Re\big\{\mathbf D_0^\Herm\mathcal P_{\mathbf g_0}^\perp\mathbf D_0\big\}
+
\frac{2|\zeta_1|^2}{\sigma_{n,1}^2}
\Re\!\big\{\mathbf D_1^\Herm\mathcal P_{\mathbf g_1}^\perp\mathbf D_1\big\}\bigg|_{\delta_1=0},
\Re\big\{\mathbf D_1^\Herm\mathcal P_{\mathbf g_1}^\perp\mathbf D_1\big\}\bigg|_{\delta_1=0},
\end{equation}
%
其中 $\mathbf D_\ell$ 沿用 \eqref{eq:A6_Dl} 定义，$\delta_1=0$ 处求值即得到完全相干情形的 \gls{efim}。
@@ -225,7 +225,7 @@ $\boldsymbol\delta=[\delta_1]$ 退化为标量，\eqref{eq:F2_jdelta} 中
\frac{2|\zeta_1|^2}{\sigma_{n,1}^2}
\big\|\mathbf a_1(\boldsymbol\xi)\big\|_2^2
\cdot
\Re\!\big\{(\dot{\mathbf q}_1^{\,\mathrm{te}})^\Herm\mathcal P_{\mathbf q_1}^\perp\dot{\mathbf q}_1^{\,\mathrm{te}}\big\},
\Re\big\{(\dot{\mathbf q}_1^{\,\mathrm{te}})^\Herm\mathcal P_{\mathbf q_1}^\perp\dot{\mathbf q}_1^{\,\mathrm{te}}\big\},
\end{equation}
%
其中 $\mathcal P_{\mathbf q_1}^\perp=\mathbf I_T-\mathbf q_1\mathbf q_1^\Herm/\|\mathbf q_1\|^2$。化简利用了 Kronecker 内积关系 $(\mathbf u_1\otimes\mathbf v_1)^\Herm(\mathbf u_2\otimes\mathbf v_2)=(\mathbf u_1^\Herm\mathbf u_2)(\mathbf v_1^\Herm\mathbf v_2)$，并把空间部分因式 $\|\mathbf a_1\|^2$ 提出。
@@ -237,7 +237,7 @@ $\boldsymbol\delta=[\delta_1]$ 退化为标量，\eqref{eq:F2_jdelta} 中
\mathbf v'_{\eta\delta,1}
=
\frac{2}{\sigma_{n,1}^2}
\Re\!\Big\{
\Re\Big\{
\zeta_1^*\mathbf D_1^\Herm
\big(\mathbf I-\mathcal P_{\mathbf g_1}\big)
\big[\zeta_1\,\dot{\mathbf q}_1^{\,\mathrm{te}}\otimes\mathbf a_1\big]
parts/H-userPositionCrb.tex
+1 -1
View File
@@ -56,7 +56,7 @@ Doppler 部分，由 $\nu_\ell=\nu_{\mathrm{ref}}+\Delta\nu_\ell(\mathbf p)$ 与
\begin{equation}
\mathcal I_{\mathrm e}(\boldsymbol\eta_{\mathrm{pos}})
=2\sum_{\ell=0}^{L-1}\frac{|\zeta_\ell|^2}{\sigma_{n,\ell}^2}
\Re\!\big\{(\mathbf D_\ell^{\mathrm{pos}})^\Herm\mathcal P_{\mathbf g_\ell}^\perp\mathbf D_\ell^{\mathrm{pos}}\big\}\in\mathbb R^{4\times4}.
\Re\big\{(\mathbf D_\ell^{\mathrm{pos}})^\Herm\mathcal P_{\mathbf g_\ell}^\perp\mathbf D_\ell^{\mathrm{pos}}\big\}\in\mathbb R^{4\times4}.
\label{eq:H_efim}
\end{equation}
%
parts/sec1_singleArrayDoa.tex
+2 -2
View File
@@ -155,7 +155,7 @@ $M$ 元阵列在第 $t$ 个快拍的接收向量记为
\qquad
\widehat\sigma(\bm\Theta)
=
\frac{1}{M}\tr\!\left[\left(\mathbf I_M-\mathcal P_{\mathbf A}\right)\widehat{\mathbf R}\right],
\frac{1}{M}\tr\left[\left(\mathbf I_M-\mathcal P_{\mathbf A}\right)\widehat{\mathbf R}\right],
\end{equation}
%
其中
@@ -176,7 +176,7 @@ $\mathcal P_{\mathbf A}$ 是导向矩阵列空间的正交投影算子，将贯
\widehat{\bm\Theta}_{\mathrm{CDML}}
=
\argmin_{\bm\Theta}
\tr\!\left[\left(\mathbf I_M-\mathcal P_{\mathbf A(\bm\Theta)}\right)\widehat{\mathbf R}\right].
\tr\left[\left(\mathbf I_M-\mathcal P_{\mathbf A(\bm\Theta)}\right)\widehat{\mathbf R}\right].
\end{equation}
%
也即把 $\bm\Theta$ 的来向估计等价为最小化投影残差能量。集中似然的逐步矩阵代入与对应的 \gls{crb}/\gls{fim} 推导见附录~\ref{sec:det_doa_crb}。
parts/sec2_singleArrayDoaDoppler.tex
+1 -1
View File
@@ -112,7 +112,7 @@ $\mathbf b_{\mathrm{st},k}$ 的 Kronecker 结构把空间维度的 $\mathbf a(\b
\tilde{\mathbf y},
\end{equation}
%
其中 $\mathcal P_{\mathbf B_{\mathrm{st}}}\triangleq\mathbf B_{\mathrm{st}}\!\left(\mathbf B_{\mathrm{st}}^\Herm\mathbf B_{\mathrm{st}}\right)^{-1}\mathbf B_{\mathrm{st}}^\Herm$ 为空时字典列空间的投影算子。
其中 $\mathcal P_{\mathbf B_{\mathrm{st}}}\triangleq\mathbf B_{\mathrm{st}}\left(\mathbf B_{\mathrm{st}}^\Herm\mathbf B_{\mathrm{st}}\right)^{-1}\mathbf B_{\mathrm{st}}^\Herm$ 为空时字典列空间的投影算子。
对 $\boldsymbol\zeta$ 与 $\sigma$ 的闭式消元过程与第~\ref{sec:det_doa_est} 节的单阵列 \gls{doa} 完全平行，详细矩阵代入见附录~\ref{sec:det_doa_doppler_crb}。
若 Doppler 本身不感兴趣、仅作为干扰参数，可先做 profile 处理：
parts/sec3_multiSatSingleUser.tex
+3 -3
View File
@@ -70,11 +70,11 @@
\begin{equation}
\alpha_\ell^{\mathrm{loc}}
=
\operatorname{atan2}\!\big(u^{\mathrm{loc}}_{\ell,2},u^{\mathrm{loc}}_{\ell,1}\big),
\operatorname{atan2}\big(u^{\mathrm{loc}}_{\ell,2},u^{\mathrm{loc}}_{\ell,1}\big),
\qquad
\beta_\ell^{\mathrm{loc}}
=
\arcsin\!\big(u^{\mathrm{loc}}_{\ell,3}\big).
\arcsin\big(u^{\mathrm{loc}}_{\ell,3}\big).
\end{equation}
%
第 $\ell$ 颗卫星的阵列导向矢量统一定义为
@@ -82,7 +82,7 @@
\begin{equation}
\mathbf a_\ell(\boldsymbol\xi)
\triangleq
\mathbf a_\ell\!\big(\alpha_\ell^{\mathrm{loc}},\beta_\ell^{\mathrm{loc}}\big)
\mathbf a_\ell\big(\alpha_\ell^{\mathrm{loc}},\beta_\ell^{\mathrm{loc}}\big)
\in\mathbb C^{M_\ell\times 1},
\end{equation}
%
parts/sec6_multiFrameCp.tex
+3 -3
View File
@@ -126,8 +126,8 @@ $\chi_\ell$ 为第 $\ell$ 颗卫星在参考时刻的\emph{公共相位}，由
\zeta_{\ell,p}^{\mathrm{ip}}
\mathbf a_\ell(\boldsymbol\xi)
s_p(t)
\exp\!\left\{
j\!\left[
\exp\left\{
j\left[
\bar\nu_{\ell,p}t+\frac{1}{2}\dot\nu_\ell t^2
\right]
\right\}
@@ -228,7 +228,7 @@ CP 模型的干扰参数向量为
\begin{equation}
\mathcal I_{\mathrm{e}}
=
2\Re\!\left\{
2\Re\left\{
\bar{\mathbf J}_{\eta}^{\Herm}
\mathcal P^{\perp}_{\bar{\mathbf J}_{\psi}}
\bar{\mathbf J}_{\eta}
parts/sec7_userPositioning.tex
+3 -3
View File
@@ -23,15 +23,15 @@
\begin{equation}
\mathbf u_\ell^{\mathrm{loc}}(\mathbf p)=\boldsymbol\Omega_\ell^\Tran\mathbf u_\ell(\mathbf p),
\qquad
\alpha_\ell^{\mathrm{loc}}=\operatorname{atan2}\!\big(u^{\mathrm{loc}}_{\ell,2},u^{\mathrm{loc}}_{\ell,1}\big),
\alpha_\ell^{\mathrm{loc}}=\operatorname{atan2}\big(u^{\mathrm{loc}}_{\ell,2},u^{\mathrm{loc}}_{\ell,1}\big),
\qquad
\beta_\ell^{\mathrm{loc}}=\arcsin\!\big(u^{\mathrm{loc}}_{\ell,3}\big),
\beta_\ell^{\mathrm{loc}}=\arcsin\big(u^{\mathrm{loc}}_{\ell,3}\big),
\end{equation}
%
并据此代入与第~\ref{subsec:single_arr_array_model} 小节完全相同的 \gls{ula}/\gls{upa} 阵列导向模板，得到本地导向矢量
%
\begin{equation}
\mathbf a_\ell(\mathbf p)\triangleq\mathbf a_\ell\!\big(\alpha_\ell^{\mathrm{loc}}(\mathbf p),\beta_\ell^{\mathrm{loc}}(\mathbf p)\big)\in\mathbb C^{M_\ell\times1}.
\mathbf a_\ell(\mathbf p)\triangleq\mathbf a_\ell\big(\alpha_\ell^{\mathrm{loc}}(\mathbf p),\beta_\ell^{\mathrm{loc}}(\mathbf p)\big)\in\mathbb C^{M_\ell\times1}.
\end{equation}
%
也即\emph{每阵列内部仍是平面波模型，阵列导向 $\mathbf a_\ell$ 的形式一字不改}；变的只是``本地角由 $\mathbf p$ 而非 $\boldsymbol\xi$ 诱导''。